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University
LECTURE
Series
Volume 10
Vertex Algebras for Beginners
Second Edition VictorL--Kac
American Mathematical Society Providence. Rhode Island

EDITORIAL COMMITTEE Jerry L. Bona
Nicolai Reshetikhin Leonard L. Scott (Chair)
1991 Mathematics Subject Classification. Primary l 7B69; Secondary 17B65, 81T05, 81T40.
ABSTRACT. This book is an introduction to vertex algebras, a new mathematical structure that has appeared recently in quantum physics. It can be used by researchers and graduate students working on representation theory and mathematical physics.

Library of Congress Cataloging-in-Publication Data

Kac, Victor G., 1943-

Vertex algebras for beginners / Victor Kac. - 2nd ed.

p. cm. - (University lecture series, ISSN 1047-3998 ; v. 10)

Includes bibliographical references and index.

ISBN 0-8218-1396-X (alk. paper)

1. Vertex operator algebras. 2. Quantum field theory. 3. Mathematical physics. I. Title.

II. Series: University lecture series (Providence, R.I.) ; 10.

QCl74.52.O6K33 1998

512'.55-dc21

98-41276

CIP

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© 1997, 1998 by the American Mathematical Society. All rights reserved.
First edition 1997 The American Mathematical Society retains all rights except those granted to the United States Government.
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Contents

Preface

1

Preface to the second edition

3

Chapter 1. Wightman axioms and vertex algebras

5

1.1. Wightman axioms of a QFT

5

1.2. d = 2 QFT and chiral algebras

8

1.3. Definition of a vertex algebra

13

1.4. Holomorphic vertex algebras

15

Chapter 2. Calculus of formal distributions

17

2.1. Formal delta-function

17

2.2. An expansion of a formal distribution a(z, w) and formal Fourier

transform

19

2.3. Locality of two formal distributions

24

2.4. Taylor's formula

29

2.5. Current algebras

31

2.6. Conformal weight and the Virasoro algebra

34

2.7. Formal distribution Lie superalgebras and conformal superalgebras 39

2.8. Conformal modules and modules over conformal superalgebras

50

2.9. Representation theory of finite conformal algebras

56

2.10. Associative conformal algebras and the general conformal algebra 61

2.11. Cohomology of conformal algebras

67

Chapter 3. Local fields

81

3.1. Normally ordered product

81

3.2. Dong's lemma

84

3.3. Wick's theorem and a "non-commutative" generalization

87

V

vi

CONTE~TS

3.4. Bounded and field representations of formal distribution Lie

superalgebras

91

3.5. Free (super)bosons

93

3.6. Free (super)fermions

98

Chapter 4. Structure theory of vertex algebras

103

4.1. Consequences of translation covariance and vacuum axioms

103

4.2. Skewsymmetry

105

4.3. Subalgebras, ideals, and tensor products

106

4.4. Uniqueness theorem

108

4.5. Existence theorem

110

4.6. Borcherds OPE formula

111

4.7. Vertex algebras associated to formal distribution Lie superalgebras 113

4.8. Borcherds identity

116

4.9. Graded and Mobius conformal vertex algebras

119

4.10. Conformal vertex algebras

125

4.11. Field algebras

129

Chapter 5. Examples of vertex algebras and their applications

133

5.1. Charged free fermions and triple product identity

133

5.2. Boson-fermion correspondence and KP hierarchy

137

5.3. gl 00 and Wl+oo

143

5.4. Lattice vertex algebras

148

5.5. Simple lattice vertex algebras

152

5.6. Root lattice vertex algebras and affine vertex algebras

158

5.7. Conformal structure for affine vertex algebras

161

5.8. Super boson-fermion correspondence and sums of squares

168

5.9. Superconformal vertex algebras

178

5.10. On classification of conformal superalgebras

185

Bibliography

193

Index

199

Preface
The notion of a vertex algebra was introduced ten years ago by Richard Bor-
cherds [Bl]. This is a rigorous mathematical definition of the chiral part of a
2-dimensional quantum field theory studied intensively by physicists since the landmark paper of Belavin, Polyakov and Zamolodchikov [BPZ]. However, implicitly this notion was known to physicists much earlier. Some of the most important
precursors are Wightman axioms [W] and Wilson's notion of the operator product expansion [Wi]. In fact, as I show in Sections 1.1 and 1.2, the axioms of a vertex
algebra can be deduced from Wightman axioms. The exposition of these two sections is somewhat terse. The rest of the book, written at a more relaxed pace, is motivated by these sections but can be read independently of them.
Axioms of a vertex algebra used in this book are essentially those of [FKRW]
and were inspired by Goddard's lectures [G]. These axioms are much simpler than
the original Borcherds' axioms and are very easy to check. One of the objectives of this book is to show that these systems of axioms are equivalent (see Section 4.8).
Another objective of the book is to lay rigorous grounds for the notion of the operator product expansion (OPE) and demonstrate how to use it to perform calculations that are otherwise very painful. The classical Wick theorem allows one to compute OPE in free field theories. A "non-commutative" generalization of Wick's formula allows one to compute OPE of arbitrary fields (see Section 3.3).
The main objective of the book is to show how to construct a variety of examples of vertex algebras, and how to perform calculations using the formalism of vertex algebras to get applications in many different directions (Chapter 5).
In Sections 2.7 and 5.10, I present some new material on a topic closely related to vertex algebras - the theory of conformal superalgebras.
These notes represent a part of the course given at MIT in 1994 and 1995. Unfortunately, I didn't have time to write down the chapters on representation

2

PREFACE

theory of vertex algebras and some other applications. (~lost quoted literature is related to these unwritten chapters, and I hope that the present book will facilitate the reading of these papers.) In fact, another important application of vertex algebra theory is that it picks out the most interesting representations of infinitedimensional Lie (super)algebras and provides means for their detailed study.
There is nothing in this book on the application to the ~Ionster simple group (there is a book [FLM] on this, after all), nothing on Borcherds' solution of the Conway-Norton problem [B2], and nothing on Borcherds· marvelous applications to generalized Kac-Moody algebras and automorphic forms [B3).
A technical remark is in order. What I call a ""Yertex algebra" should probably
be called a "N = 0 vertex superalgebra" (see Section 5.9 for the definition of a
N = n vertex superalgebra), but I decided on this simpler name. (Also, I call a
"conformal vertex algebra" what is called in [FLM]. with some additional restrictions, a "vertex operator algebra.") The reader who detests ··supermathematics" may assume that the Z 2-gradation is trivial, that -Lie superalgebra" means "Lie algebra", etc. But then he skips fermions and beautiful applications to identities and to soliton equations, the rich variety of superconformal theories, etc.
The bibliography is by no means complete. It is already quite a task to compile a complete list that would include all the relevant work done by physicists. However, it includes all items that influenced my thinking on the subject. One may also find there further references.
In addition to the sources mentioned above. the most important for the present book were the work of Todorov on the Wightman axioms point of view on CFT, the paper by Li from which I learned the unified formula for n-th products and Dong's lemma, the paper by Getzler from which I learned the "non-commutative" Wick formula, and the work of Lian and Zuckerman on "quantum operator algebras."
A preliminary version of these notes has been published in the proceedings of the summer school in Bulgaria in 1995 where I lectured on this subject. I am grateful to Ivan Todorov and Kiyokazu Nagatomo for reading the manuscript and correcting errors, and to Maria Golenishcheva-Kutuzova, Mike Hopkins, Andrey Radul, and Ivan Todorov for numerous illuminating discussions.

Vienna, June 1996

Preface to the second edition
This improved and enlarged edition is based on a course given at M.I.T. in the spring of 1997 and in Rome University in May and June of 1997. Below is a list of the most important improvements and additions.
Chapter 2. The notion of formal Fourier transform is introduced in Section 2.2.
This reduces significantly the calculations and leads to the important notion of >.-
bracket in the theory of conformal algebras. Four new Sections 2.8-2.11 on the theory of conformal algebras are added and Section 2.7 is reworked. Thus, Sections 2.7-2.11 present the foundations of this rapidly developing area of algebraic conformal field theory.
Conformal algebra is an axiomatic description of the singular part of the operator product expansion of chiral fields in conformal field theory. It is, to some extent, related to a vertex algebra in the same way Lie algebra is related to its universal enveloping algebra. A structure theory of vertex algebras, similar, for example, to the structure theory of finite-dimensional Lie algebras, seems to be far away. Conformal algebras turned out to be a much more tractable object; as shown in Sections 2.7-2.11, for finite conformal algebras such a theory can be developed.
In Section 2.7 an explicit correspondence between an important class of infinitedimensional Lie algebras, called formal distribution Lie algebras, and certain new structures, called conformal algebras, is established and a classification of finite conformal algebras is outlined. In Sections 2.8 and 2.9 representation theory of conformal algebras is developed, and in Section 2.11 the corresponding cohomology theory is explained. In Section 2.10 elements of conformal linear algebra are presented.
Chapter 3. The "non-commutative" Wick formula is expressed via >.-bracket (formula (3.3.12)), which greatly facilitates the use of this formula.
3

4

PREFACE

Chapter 4. The exposition of Sections 4.4-4.6 is simplified by making a more systematic use of the Uniqueness Theorem (a similar simplification was independently found in [MN]). Section 4.11 on field algebras is corrected.
Chapter 5. A new Section 5.8 on super boson-fermion correspondence is added. Comparing characters leads to a beautiful identity, whose specializations give classical results on sums of squares which go back to Gauss and Jacobi. In Section 5.10 a complete list of finite simple conformal superalgebras is given.
I wish to thank the participants of the course at M.I.T. for many discussions and suggestions, especially Bojko Bakalov, Alessandro D'Andrea, Eddie Karat, and Alexandre Soloviev. In particular, Bakalov gave a proof of Proposition 3.2 and suggested Example 4.11, and Karat and Soloviev gave proofs of Lemma 2.7. I am grateful to D. Fattori, A. Rudakov and J. van de Leur for sending corrections, and to Jan Wetzel for technical help in preparation of the manuscript. I am enormously indebted to Bojko Bakalov, Shun-Jen Cheng, Alessandro D'Andrea, Alexander Voronov, and Minoru Wakimoto for collaboration. It is due to their efforts that the theory of conformal algebras reached this level of maturity in such a short period of time.

Brookline, Massachusetts, December 1997

CHAPTER 1
Wightman axioms and vertex algebras
1. 1. Wightman axioms of a QFT Let M be the d-dimensional Minkowski space (space-time), i.e., the d-dimensional real vector space with metric
(As usual, x0 = ct where c is the speed of light and tis time, and x1, ... , Xd-l are
space coordinates.) Two subsets A and B of Mare called space-like separated if for any a EA and
b EB one has la-bl 2 < 0. The forward cone is the set {x EM I lxl 2 ~ 0, xo ~ O}.
Define causal order on M by x ~ y iff x - y lies in the forward cone. The Poincare group is the unity component of the group of all transformations
of M preserving the metric. It is the semidirect product of the group of translations (= M) and the Lorentz group L, the group of all unimodular linear transformations of M preserving the forward cone. Hence the Poincare group preserves the causal order and therefore the space-like separateness.
A quantum field theory (QFT) is the following data: the space of states-a complex Hilbert space H; the vacuum vector-a vector IO) E H; a unitary representation (q, A) i-+ U(q, A) of the Poincare
group in H; a collection of fields <I>a (a an index)-operator-valued distri-
butions on M (that is continuous linear functionals f i-+ cI>a (J) on the space of rapidly decreasing C 00 tensor valued test functions on M with values in the space of linear operators densely defined on H).
5

6

1. WIGHTMAN AXIOMS AND VERTEX ALGEBRAS

One requires that these data satisfy the following Wightman axioms:
Wl (Poincare covariance): U(q,A)<I>a(f)U(q,A)- 1 = <I>a((q,A)f)), q EM,
A EL.
Note that U(q,l) = expi'5:,f:;,,~qkPk, where Pk are self-adjoint commuting opera-
tors on 1{.
W2 (stable vacuum): The vacuum vector ID) is fixed by all the operators U(q, A). The joint spectrum of all the operators P0, ... , Pd-I lies in the forward cone.
W3 (completeness): The vacuum vector IO) is in the domain of any polynomial in the <I>a(f)'s and the linear subspace D of 1l spanned by all of them applied to IO) is dense in N.
W4 (locality): <I>a(f)<I>b(h) = <I>b(h)<I>a(f) on D if the supports off and h
are spacelike separated.
The physical meaning of axoim W2 is that vacuum has zero energy and it is the minimal energy state. The last axiom means that the measurements in spacelike separated points are independent. (According to the main postulate of special relativity the speed of a signal does not exceed the speed of light.)
Actually, these are axioms of a purely "bosonic" QFT. In order to include
"fermions" one considers even and odd fields by introducing parity p(a) = 0 or
I E Z /2.Z. Then only the axiom W4 is modified:
W4super (locality): <I>a(f)<I>b(h) = (-l)P(a)p(b)<J>b(h)<I>a(f) on D if the sup-
ports of f and h are spacelike separated.
Axiom Wl gives, in particular, translation covariance (q E M):

(1.1.1)

Here and further, by abuse of notation, we often write if>a ( x) in place of if>a (f (x)) .
Note that, by definition, D lies in the domain of definition and is invariant with respect to all the operators <I>a(f). It follows from Wl and W2 that D is U(q, 1)-invariant. Since the translation covariance means

(1.1.2)

and Pk IO) = 0 by W2, we see that Dis invariant with respect to all the operators Pk.

1.1. WIGHTMAN AXIOMS OF A QFT

7

Note that applying both sides of (1.1.1) to the vacuum vector and using its U(q, 1)-invariance, we obtain (q EM):

(1.1.3)

<I>a(x+q)I0) = (expi~qkPk) <I>a(x)I0).

Now, the Poincare group preserves distances on M. One considers also a larger group - the group of conformal transformations of M (preserving only angles). The simplest conformal transformation is the inversion

x H- -x/lxl2 •
Conjugating a translation x H- x - b by the inversion, we get a special conformal transformation (b E M):

(1.1.4)

The group generated by the translations and the special conformal transformations is called the conformal group. It includes the Poincare group and also the group of dilations:

X H- AX, A -:j:. 0.

Conformal transformations of the Minkowski space are important for QFT since they preserve causality (hence space-like separatenees).
A quantum field theory is called conformal if the unitary representation of the Poincare group in 1i extends to a unitary representation of the conformal group: (q, A, b) H- U(q, A, b) such that the vacuum vector I0) is still fixed and also the special conformal covariance holds for the given collection of fields; in the case of a scalar field it means

( 1.1.5)

where ~a is a real number called the conformal weight of the field <I>a and

( 1.1.6)

cp(b,x) = 1 + 2x •b + lxl2 1bl2-

Note that cp(b, x)-d is the Jacobian of the transformation (1.1.4). It follows that axiom Wl and (1.1.5) together give conformal covariance:

8

1. WIGHTMAN AXIOMS AND VERTEX ALGEBRAS

In particular, we have dilation covariance:

(1.1. 7)

where >. r-+ U(>.) denotes the representation of the dilation subgroup. Formula (1.1.5) implies that the infinitesimal special conformal generators are
represented by selfadjoint operators Qk (k = 0, ... , d - 1) on 1l such that

(1.1.8)
= where E I::~~o XmOx.,,. is the Euler operator and 'T/k are the coefficients of the metric ('T/o = 1, 'T/k = -1 fork 2:: 1).

= 1.2. d 2 QFT and chiral algebras
Consider now the case d = 2. Introduce the light cone coordinates t = x0 - x1 , t = x0 + x1 , so that lxl2 = tt. (In this section the overbar does not mean the
complex conjugate.) Let P =-1(Po-Pi), 2
Then formula (1.1.3) becomes:

(1.2.1)
By the vacuum axiom the joint spectrum of the operators P and P lies in the
domain t 2:: 0, t 2:: 0, hence the operator exp i(tP+tP) is defined on 'D for all values Imt 2:: 0, Imt 2:: 0. Moreover, by formula (1.2.1) the 'D-valued distribution <I>alO)
extends analytically to a function in the domain
{t I Imt > 0} X {ti Imt> 0} C C2 .

Indeed, by the spectral decomposition, ei(qP+qP) is the Fourier transform of a (op-
erator valued) function whose support is in the domain p 2:: 0, p 2:: 0, by the second part of axiom W2. Hence we may take the value <I>a (t, t) IO) when Im t > 0,
Im f > 0. It follows from (1.2.1) that this value is non-zero unless <I>a = 0.
The locality axiom means

1.2. d = 2 QFT AND CHIRAL ALGEBRAS

9

In the light cone coordinates the special conformal transformations decouple:

(1.2.3)

where b± = b0 ± b1 . Hence the conformal group consists of transformations of the
form:

"'
I

(t,

t- )

=

(a-ctt-++db,

-ca--ft=++--l-d=.i)

,

fl where (~ and (~ !) are from SL2(1R). Then the Poincare covariance (axiom

Wl) and special conformal covariance (formula (1.1.5)) give together the following
conformal covariance (with -6.a = ~a):

(1.2.4)
= Because of the decoupling (1.2.3) one usually does not assume that -6.a ~a and
considers more general conformal covariance of the form (1.2.4). Introduce further the operators

Then formulas (1.1.2) and (1.1.8) become:

(1.2.5a) (1.2.5b) (1.2.5c) (1.2.5d)

i [P, <I>a (t,f)] = 8t<I>a (t, [),

i[P, <I>a (t, [)]

8r<I>a (t, f) ,

i [Q, 4>a (t, f)] = (t28t + 2.6.at) 4>a (t, f),

i[Q, 4>a (t,f)]

(f28r + 2~af) <I>a (t, f).

In order to make conformal transformations defined everywhere, consider the

compactification of the Minkowski space given by:

1 + it

z

=-
1-

-it'

z =11-+-- -iift-

This maps the domain Im t > 0, Im f > 0 to the domain izl < 1, izl < 1. Consider

the new fields defined in izl < 1, lzl < 1:

= Y(a,z,z)

1 (l+z)Ma(l+z)Ma <I>a(t,l),

where t=i1---z, l+z

t_ =

1- z
i--.

l+z

Note that Y (a, z, z) IO)lz=O,z=O is a well defined vector in 'D which we denote by a,

and (due to the above remark) Y (a, z, z) r-+ a is a linear injective map.

10

1. WIGHTMAN AXIOMS A~D VERTEX ALGEBRAS

We let

T

21 (P + [P, Q] - Q),

H

1 2(P+

Q),

1

T*

2(P- [P, Q] - Q),

and similarly we define T, fI, T*. It is straightforward to check that formulas
(1.2.5a-d) imply:

(1.2.6a) (1.2.6b) (1.2.6c)

[T, Y(a, z, z)]
[H, Y(a, z, z)] [T*, Y(a, z, z)]

and similarly for f, fl, T*. Also, of course, all the operators T, t, ... annihilate
the vacuum vector 10). Note that (1.2.6b) means:

Note also that the operators T, H, and T* satisfy the following commutation relations:

(1.2. 7)

[H,T] = T, [H,T*] = -T*, [T*,T] = 2H.

Applying both sides of (l.2.6b and c) to the vacuum vector and letting z = z = 0, we get:

Ha= ~aa, T*a = 0.

Recall that P and P are positive semidefinite self-adjoint operators on 1i (due to axiom W2). The same is true for Q and Q since they are operators similar to P and P respectively. Hence H is a positive semidefinite self-adjoint operator as well. Thus, conformal weights are non-negative numbers.
If in our QFT, Ta= 0 = Ta is possible only for the multiples of the vacuum
vector, then ~a= -6.a = 0 imply that a= IO).

= 1.2. d 2 QFT AND CHIRAL ALGEBRAS

11

Now consider the right chiral fields, namely those fields for which q <I>a = 0.
Then (1.2.2) becomes

This implies that the (super) commutator (i.e., the difference between the left- and the right-hand sides) has the following form:
L [<I>a(t), <I>b(t')] = 15(j) (t - t')\Jlj (t')
j?_O
for some fields \JfJ (t'). For these fields the Wightman axioms still hold (but the conformal covariance does not necessarily hold), hence we may add them to our QFT to obtain:
L [Y(a, z), Y(b, w)] = 15Ul(z - w)Y(cj, w).
j?_O
Commuting H with both sides of this equality and using (l.2.6b) we see that the
field Y(cj,w) has conformal weight ~a+ ~b - j -1 (in the sense of (l.2.6b)). Due
to the positivity of conformal weights we conclude that the sum on the right is finite. It follows that
(z - w)N[Y(a, z), Y(b, w)] = 0 for N » 0.
(A detailed explanation of this will be given in Section 2.3.) We expand a chiral field Y(a, z) in a Fourier series:
L Y(a, z) = a(n)Z-n-l,
n
where a(n) E EndD and denote by V the subspace of D spanned by all polynomials in the a(n) applied to the vacuum vector /0). It is clear that V is invariant with respect to all a(n) and, by (1.2.6a), with respect to T. By the argument proving
Corollary 4.6(f), V is spanned by all polynomials in the a(n) with n < 0 applied
to /0). We thus arrived at the following data called the right chiral algebra:
the space of states-a vector space V; the vacuum vector-a non-zero vector /0) E V; the infinitesimal translation operator T E EndV;

12

1. WIGHTMAN AXIOMS AND VERTEX ALGEBRAS

fields Y(a, z) for each a E A, some subset of V endowed with

the parity p(a), where
L Y(a, z) = a(n)Z-n-l
nEZ
is a series with a(n) E EndV.

These data satisfy the following properties for a E A (we ignore the remaining

properties for a while):

(translation covariance) [T, Y(a, z)] = 8Y(z, a);

(vacuum) (completeness)

TIO)= 0, Y(a, z)IO)/z=O = a;
polynomials in the a(n) 's with n < 0 applied to /0)

span V;

(locality)

(z -w)NY(a,z)Y(b, w) = (-l)P(a)p(b) (z - w)NY(b, w)Y(a, z) for some

NEZ+ (depending on a, b EA). By the vacuum property we have (a E A):

(1.2.8)

Applying both sides of the translation covariance property to /0) and letting z = 0, we obtain (using TIO)= 0 and (1.2.8)):

(1.2.9)

Ta= a(-2)10), a EA.

Thus, the infinitesimal translation operator on A is built in the collection of fields. The positivity of conformal weights imply, due to (l.2.6b):

(1.2.10)

a(n)V = 0 for n » O(depending on a EA and v E V).

Later (in Section 4.5) we shall prove the existence theorem that asserts that, using (1.2.10), one can construct fields Y(a, z) for all a EV (using the so-called normally ordered product) such that (1.2.10), translation covariance, vacuum and locality properties still hold (completeness then automatically holds). We thus arrive at the definition of a chiral algebra. This name is used by physicists. Mathematicians, following Borcherds, use the name vertex algebras, or vertex operator algebras, since (for historical reasons) the fields Y (a, z) are called vertex operators.
Similarly, one may consider the left chiral fields, that is those fields for which 8t<I>a = 0. In the same way as above, we construct the left chiral algebra V with the

1.3. DEFINITION OF A VERTEX ALGEBRA

13

same vacuum vector /0), the infinitesimal translation operator T and fields Y(a, z),
a EV. Due to locality (1.2.2) we see that <I>a(t)cI>a;(f) = (-l)P(a)p(a)cI>a(f)<I>a(t) for
all t and t, hence

[Y(a, z), Y(a, z)] = 0 for all a E V, a E V.

The left and right chiral algebras are the most important invariants of a conformally covariant 2-dimensional QFT. Under certain assumptions and with certain additional data one may reconstruct the whole QFT from these chiral algebras, but we shall not discuss this problem here.

REMARK 1. 2. One may also consider the case of d = 1 conformal QFT. Then the only coordinate is time t = xo and the forward cone is the set of non-negative
numbers. Then conformal covariance reads:

u(at+b) cI> t u(at+b)-i = 1 cI> (at+b)

ct+d a() ct+d

(at+d)~a a ct+d •

It follows that there exist self-adjoint operators P and Q in 1i such that

Compactifying by z = ~~!~, letting 1
Y(a,z) = (l + z) 2~" <I>a(z) and defining T, H, and T* as in d = 2 case, we find that Y(a, z) satisfies formulas
(l.2.6a-c). As ind = 2 case, we see that Y(a, z)/0) lz=O is a well-defined vector.
The only property that is completely missing is locality since there are no spacelike separated points.
1.3. Definition of a vertex algebra
Let V be a superspace, i.e., a vector space decomposed in a direct sum of two subspaces:

Here and further Oand I stand for the cosets in Z /2Z of 0 and l. We shall say that an element a of V has parity p(a) E Z/2Z if a E Vp(a)· If dim V (= dim Va +dim Vr) < oo, we let
sdim V = dim Va - dim Vr

14

1. WIGHTMAN AXIOMS AND VERTEX ALGEBRAS

to be the superdimension of V. In what follows, whenever p(a) is written, it is to be understood that a E Vp(a) ·
A field is a series of the form a(z) = EnEZ a(n)Z-n-l where a(n) E EndV and
for each v E V one has

( 1.3.1)

a(n)(v) = 0 for n » 0.

We say that a field a(z) has parity p(a) E Z/2Z if

(1.3.2)

a(n) Va C Va+p(a) for all a E Z/2Z, n E Z.

A vertex algebra is the following data:
the space of states-a superspace V,
the vacuum vector-a vector 10) E Vo,
the state-field correspondence-a parity preserving linear map
of V to the space of fields, at--+ Y(a,z) = EnEZa(n)z-n-l,
satisfying the following axioms:
(translation covariance): [T, Y(a, z)] = 8Y(a, z),
where TE EndV is defined by

(1.3.3)

(vacuum): Y(IO),z) = Iv, Y(a,z)IO)lz=O = a, (locality): (z - w)NY(a, z)Y(b, w)
» = (-l)P(a)p(b)(z - w)NY(b, w)Y(a, z) for N 0.
Note that the infinitesimal translation operator T is an even operator, i.e., TVa C Va, and the bracket in the translation covariance axiom is the usual bracket:
= [T, Y] TY - YT, so that this axiom says

(1.3.4)

The first of the vacuum axioms says that

(1.3.5a)

IO)(n) = bn,-li in particular TIO)= 0.

The second of the vacuum axioms says that

(1.3.5b)

a(n) IO) = 0 for n ~ 0, a(-l) IO) = a.

1.4. HOLOMORPHIC VERTEX ALGEBRAS

15

The locality axiom is to be understood as a coefficient-wise equality of two series in z and w of the form I::m,nEZ am,nZmwn.

REMARK 1.3. Applying T to both sides of (1.3.3) n-1 times, and using (1.3.4)
~7 = and T[0) 0, we obtain (a)= a(-n-i)[0), for n E Z+, which is equivalent, by
(1.3.5b), to

(1.3.6)

1.4. Holomorphic vertex algebras

A vertex algebra Vis called holomorphic if a(n) = 0 for n ~ 0, i.e., Y(a, z) = I::nEZ+ a(-n-l)zn are fo~mal power series in z.
Let V be a holomorphic vertex algebra. Since the algebra of formal power series in z and w has no zero divisors, it follows that locality for V turns into a usual supercommutativity:

(1.4.1)

Y(a, z)Y(b, w) = (-l)p(a)p(b)y(b, w)Y(a, z).

Define a bilinear product ab on the space V by the formula

(1.4.2)

and let [0) = 1. Then applying both sides of (1.4.1) to c and letting z = w = 0
gives:

(1.4.3)

a(bc) = (-l)P(a)p(b)b(ac).

The vacuum axioms give

(1.4.4)

l ·a= a· l = a.

It is easy to see that properties (1.4.3) and (1.4.4) are equivalent to the axioms of a (super)commutative associative unital super algebra. Indeed, letting
c = 1 in (1.4.3), we see by (1.4.4) that V is (super)commutative. But using (super)commutativity, we can rearrange (1.4.3) to get a(cb) = (ac)b, which is associa-
tivity. The converse is clear. Furthermore, apply Y(b, w) to both sides of (1.3.6):
= Y(b,w)Y(a,z)l Y(b,w)e 2 T(a).

16

l. WIGHTMAN AXIOMS AND VERTEX ALGEBRAS

Applying commutativity to the left-hand side and then (1.3.6), we obtain
(-l)P(a)p(b)Y(a, z)ewT(b) = Y(b, w)ezT (a).

Letting w = 0 and using the comrrru-tativity of our product on V we get

(1.4.5)

Y(a,z)(b) = ezT(a)b.

Thus, the fields Y(a, z) are defined entirely in terms of the product on V and the operator T.
Finally, by (1.4.5), translation covariance axiom becomes:

(1.4.6)

Letting z = 0 we see that T is an even derivation of the associative commutative unital superalgebra V and that (1.4.6) is equivalent to this.
Thus, we canonically associated to a holomorphic vertex algebra V a pair consisting of an associative commutative unital superalgebra structure on V and an even derivation T. Conversely, to such a pair we canonically associate a holomorphic vertex algebra with fields defined by (1.4.5).
If T = 0, then Y(a,z)(b) = ab. Therefore we may view vertex algebras as a generalization of unital commutative associative superalgebras where the multiplication depends on the parameter z via

However, as we shall see, a general vertex algebra is very far from being a "commutative" object.

CHAPTER 2

Calculus of formal distributions

2.1. Formal delta-function

In the previous chapter we considered formal expressions

(2.1.1)

L am,n,... zmwn ... ,
m,n, ... EZ

where am,n, ... are elements of a vector space U over <C. Series of the form (2.1.1) are called formal distributions in the indeterminates z, w, . . . with values in U. They form a vector space over C denoted by U [[z, z-1, w, w- 1, ...]] .

We can always multiply a formal distribution and a Laurent polynomial (pro-

vided that product of coefficients is defined), but cannot in general multiply two

formal distributions. Each time when a product of two formal distribution occurs,
we need to check that it converges in the algebraic sense, i.e. the coefficient of each monomial zmwn ... is a finite (or convergent) sum.
I
Given a formal distribution a(z) = LnEZ anzn, we define the residue by the
usual formula

Since Resz oa(z) = 0, we have the usual integration by parts formula (provided
that ab is defined):

(2.1.2)

Resz 8a(z)b(z) = - Resz a(z)ob(z).

Here and further oa(z) = Ln nanzn-l is the derivative of a(z).
Let C [z, z- 1] denote the algebra of Laurent polynomials in z. We have a non-
degenerate pairing U [[z,z-1]] xC [z, z-1J ➔ U defined by (f,<p) = Resz f(z)<p(z),
hence the Laurent polynomials should be viewed as "test functions" for the formal distributions. Note that formal distributions a(z) and b(z) are equal iff (a,<p) (b,<p) for any test function <p E C[z,z-1].
17

18

2. CALCULUS OF FORMAL DISTRIBUTIONS

We introduce the formdl,delta-function 8(z -w) as the following formal distribution in z and w with values in (C1 :

(2.1.3)

In order to establish its properties, introduce one more notation. Given a

rational function R(z, w) with poles only at z = 0, w = 0 and izl = lwl, we denote

by iz,wR (resp. iw,zR) the power series expansion of R in the domain izl > lwl

(resp. lwl > lzl)- For example, we have for j E Z+:

(2.1.4a) (2.1.4b)

.

1

iz,w (z - w)i+l

.

1

Zw,z (z - w)i+l

f (rr:) 2 -m-lwm-j,

m=O J

_ _ (m) -

~
~

. z-m-1 wm-j .

m=-1 J

From (2.1.3) and (2.l.4a and b) we obtain the following important formula:

(2.l.5a) (2.l.5b)

.

1

.

1

= Zz,w (z - w)i+l - iw,z (z - w)i+l

'°' (m) = ~ . z -m-1wm-j . mEZ J

Here and further for an operator A we let

(2.1.6)

Note that (2.1.5a) is a formal distribution with integer coefficients. The formal delta-function has the usual properties listed below.

PROPOSITION 2.1. (a) For any formal distribution f(z) EU [[z, z-1]] one has:

(2.1.7)

Resz f(z)6(z - w) = f(w).

{The product f(z)6(z - w) always converges.)
(b) 6(z - w) = 6(w - z).
(c) Oz6(z - w) = -8w6(z - w). (d) (z - w)aY+ 1)6(z - w) = aY)6(z - w), j E Z+·
= (e) (z -w)i+1aYl6(z - w) 0, j E Z+·
1 This notation is very suggestive but somewhat misleading as o(z - w) is not a function of z - w. Purists may use notation o(z,w).

2.2. FORMAL FOURIER TRANSFORM

19

PROOF. It suffices to check (2.1.7) for f(z) = azn, which is straightforward.
Furthermore, we have:

proving (b). Since c>(z - w) = Lm z-m-lwm = Lm z-m-2 wm+l, we see that

82 c>(z - w) = -8wc>(z - w), proving (c). Finally, (d) and (e) follow from (2.l.5a

and b).

□

Note that Proposition 2.1 (c-e) can be also proved by comparing the values of both sides on test functions. Let us use this method in order to prove the following
useful formula (which is a generalization of Proposition 2.1 (e) for j = 0):

(2.1.8)

c>(z - w)a(z) = c>(z - w)a(w), where a(z) EU [[z, z-1]].

Indeed, by (2.1.7), the pairing of both sides of (2.1.8) with cp(z) E C[z,z- 1] is equal to a(w)cp(w).
Letting a(z) = c>(z - t), we obtain an important special case of (2.1.8), after
exchanging t and z:

(2.1.9)

c>(z - t)c>(w - t) = c>(w - t)c>(z - w).

Applying to both sides a-;a;:: and using Proposition 2.1 (c) we obtain

This formula is very useful for checking locality of formal distributions.

2.2. An expansion of a formal distribution a(z, w) and formal Fourier transform

Here we consider the question: when a formal distribution

L a(z,w) =

am,nZmwn EU [[z,z- 1 ,w,w-1]]

m,nEZ

has an expansion of the form

(2.2.1)

(X)
a(z,w) = Ld(w)aYlc>(z - w).
j=O

20

2. CALCULUS OF FORMAL DISTRIBUTIONS

Multiplying both sides of (2.2.1) by (z - wt and taking Res2 we obtain using Proposition 2.1 (a, d, and e)

(2.2.2)

Denote by U [[z, z-1 , w, w-1]] 0 the subspace consisting of formal U-valued distri-

butions a(z, w) for which the following series converges:

(2.2.3)

00
7ra(z, w) := L (Res2 a(z, w)(z - w)J) oS\5(z - w).
j=O

Let

(2.2.4)

a(z,w)+(z) := L am,nZmWn.
mEZ+ nEZ

A formal distribution a(z, w) is called holomorphic in z if a(z, w) = a(z, w )+(z).

PROPOSITION 2.2. (a) The map 7r is a projector (i.e., 7r2 = 7r) on
U [[z,z- 1 ,w,w-1]]°. (b) Ker7r = {a(z, w) E U [[z, z-1, w, w- 1]] 0 which are holomorphic in z} .

(c) Any formal distribution a(z,w) from U [[z,z- 1 ,w,w- 1]] 0 is uniquely repre-

sented in the form:

(2.2.5)

00
a(z,w) = Ld(w)oSlJ(z-w)+b(z,w)
j=O

where b(z, w) is a formal distribution holomorphic in z. The coefficients d (w) are

given by (2.2.2).

PROOF. (a) follows by the argument preceding formula (2.2.2). It is clear that

a(z, w) E Ker 7r if a(z, w) is holomorphic in z. Conversely, if a(z, w) E Ker 7r, writing

a(z,w) = LnEzan(w)zn, we see from (2.2.2) that c0 (w) = 0 implies a_1(w) = 0, c0 (w) = c1(w) = 0 implies a_1(w) = a_2 (w) = 0, etc., proving (b). (c) follows

from (a) and (b).

D

COROLLARY 2.2. The null space of the operator of multiplication by (z-w)N,
N 2:: 1, in U [[z,z-1,w,w-1]] is

(2.2.6)

LN-1 aSlJ(z -w)U [[w,w-1]].
j=O

2.2. FORMAL FOURIER TRANSFORM

21

Any element a(z, w) from (2.2.6) is uniquely represented in the form

(2.2.7)

LN-l
a(z,w) = d(w)8~)c5(z -w),
j=O

the d(w) being given by (2.2.2).

PROOF. That (2.2.6) lies in the null space of (z - w)N follows from Proposi-
tion 2.le.
Conversely, if (z-w)Na(z,w) = 0, then a(z,w) EU [[z,z-1,w,w-1]] 0 and we
have by (2.2.5) and Proposition 2.1 (d and e):
L00
0 = J+N (w)o~lc5(z - w) + (z - w)Nb(z, w).
j=O
By the uniqueness in Proposition 2.2c we conclude that d(w) = 0 for j 2:: N and that (z - w)Nb(z, w) = 0. The last equality implies b(z, w) = 0 since b(z, w) =
□

We shall often write a formal distribution in the form

L a(z) = La(n)Z-n-l, a(z,w) =

a(m,n)Z-m-lw-n-l,etc.

nEZ

m,nEZ

= This is a natural thing to do since a(n) Resz a(z)zn. Then the expansion (2.2.7)

is equivalent to

(2.2.8)

This follows by using (2.l.5b) and comparing coefficients.

•
DEFINITION

2. 2.

A formal

distribution

a(z, w)

is

called

local if

(z-w)Na(z,w)=0 for N»0.

Corollary 2.2 says that any local formal distribution a(z, w) has the expansion (2.2.7). This expansion is called the OPE expansion of a(z, w) and the cn(w) (given by (2.2.2)) are called the OPE coefficients of a(z,w).
In order to study the properties of the expansion (2.2.5), it is convenient to introduce the formal Fourier transform of a formal distribution a(z, w) by the formula:
F>z..w (a(z ' w)) = Resz e>.(z-w)a(z' w) •

22

2. CALCULUS OF FORMAL DISTRIBUTIONS

This is a C-linear map from U [[z,z-1,w,w-1]] to U [[w,w- 1]] [[.>.]]. It is immediate, by Proposition 2.l(d and e) and (2.1.7), that

Hence the formal Fourier transform of the expansion (2.2.5) is

(2.2.9)

L F;',w(a(z,w)) = >,(n)cn(w).
nEZ+

(As before, >,(n) stands for >,n /n!.) In other words, the formal Fourier transform of a formal distribution a(z, w) is the generating series of its OPE coefficients.
The following simple lemma is very useful.

LEMMA 2.2.

PROOF. It is straightforward using Proposition 2.1 (d) and (e).

□

Along with the operators Oz and Ow on the space of formal distribution
U [[z,z- 1,w,w-1]], consider the permutation operator a(z,w) = a(w,z). It is
clear that all three operators preserve the property of locality. The following formulas describe the behavior of the formal Fourier transform with respect to these operators:

(2.2.10) (2.2.11)

(The right-hand side of (2.2.11) means that the indeterminate.>. in (2.2.9) is replaced by the operator -A - Ow.) Formulas (2.2.10) follow from the definition of F:w using integration by parts (they hold without the assumption of locality). Due to locality of a(z, w), we can use expansion (2.2.7), hence it suffices to check (2.2.11)
for a(z, w) = c(w)a!<5(z - w). We use Proposition 2.l(c) and Lemma 2.2:
Fz~wa(w, z) = (-1/ Resz ( e-X(z-w)c(z)a!<5(z - w)) = (-1/(>- + awl Resz c(z)<5(z - w)
= (->- - aw/c(w).

2.2. FORMAL FOURIER TRANSFORM

23

REMARK 2.2a. Formulas (2.2.10) and (2.2.11) are equivalent to the following relations for the OPE coefficients c~ (w), c:;, (w) and en (w) of the formal distributions 82 a(z, w), 8wa(z, w) and ii(z, w) respectively:

c~(w) = -ncn-l(w),

c:(w) = 8wcn(w) + ncn-1 (w),

L = cn(w)

(-l)i+naff)cn+i(w).

jEZ+

A composition of two Fourier transforms, FtwFf.w is a C-linear map from U [ [z, z-1, w, w-1, x, x-1]] to U [[w, w- 1]] [[,\, µ]]. The following relation will simplify significantly our calculations:

(2.2.12)

The proof ofit is very easy. Indeed, the left-hand side applied to a(z, w, x) is equal to
Res2 Resx e>.(z-w)+µ(x-w)a(z, w, x) = Resx Res2 e>.(z-x)e(>.+µ)(x-w)a(z, w, x), which
is the right-hand side applied to a(z,w,x).

REMARK 2.2b. A language alternative to that of U-valued local formal distributions in z and w is the language of differential operators from U [w, w-1] to U [[w,w-1]]. Indeed, for a formal distribution a(z,w) the associated operator is
= (Da(z,w)f) (w) Resz a(z, w)f(z).

It is easy to see that Datc5(z-w) = 8! (k E Z+), hence for

I: = a(z, w)

ck(w)8ikl£5(z - w),

k

we have:

Note that we also have:

= Da(z,w) I:ck(w)8ik)_
k

= Da(w,z) 2)-8w)(k)ck(w).
k

24

2. CALCULUS OF FORMAL DISTRIBUTIONS

2.3. Locality of two formal distributions

Suppose now that the vector space U carries a structure of an associative super-
algebm. This simply means that U = U0 EEl U1 is a Z /2Z-graded associative algebra (i.e., UaUf3 C Ua+f3, a, /3 E Z/2Z).
The most important example of an associative superalgebra is the endomorphism algebra EndV of a superspace V (see Section 1.3) with the Z/2Z-grading given by;./,.....--

(EndV)a = {a E EndV IaVf3 c Va+f3}-

One defines the bmcket [, ] on an associative superalgebra U by letting

(2.3.1) [a, b] = ab - p(a, b)ba, where a E Ua, b E Uf3, p(a, b) = (-l)af3.

Here and further we adopt the convention of [Kl] that the bracket of an even element with any other element is the usual commutator and the bracket of two odd elements is the anti-commutator (physicists usually write [a, bJ+ in the latter case). Recall that the Z/2Z-graded space U with the bracket (2.3.1) is a basic example of a Lie superalgebra (see e.g. [Kl] for a definition).
We can define now the notion of locality of formal distributions, with values in a Lie superalgebra 9, hence in its universal enveloping algebra U(9).

DEFINITION 2.3. Two formal distributions a(z) and b(z) with values in a Lie superalgebra £l are called mutually local (or simply local, or form a local pair) if the formal distribution [a(z),b(w)] E 9 [[z,z-1,w,w-1]] is local, i.e. if

(2.3.2)

= (z - w)N [a(z), b(w)] 0 for N » 0.

We shall always assume that all coefficients of a formal distribution a(z) have the same parity, which will be denoted by p(a). We shall also use the following notation:
p(a, b) = (-l)P(a)p(b).

REMARK 2.3a. Differentiating both sides of (2.3.2) by z and multiplying by z-w, we see that the locality of a(z) and b(z) implies the locality of 8a(z) and b(z).

2.3. LOCALITY OF TWO FORMAL DISTRIBUTIONS

25

In order to state equivalent definitions of locality we need some notation. Given a formal distribution a(z) = Lna a(n)Z-n-1, let

(2.3.3)

a(Z)- = ~ ~ a(n)Z -n-1 , a(Z)+ = ~ ~ a(n)Z -n-1 .
n<O

This is the only way to break a(z) into a sum of "positive" and "negative" parts

such that

(2.3.4)

(oa(z))± = o(a(z)±)-

Given formal distributions a(z) and b(z), define the following formal distribution in z and w with values in the universal enveloping algebra U(fl):

(2.3.5)

: a(z)b(w) := a(z)+b(w) + p(a, b)b(w)a(z)_,

Note the following formulas:

(2.3.6a) (2.3.6b)

a(z)b(w) = [a(z)_, b(w)] +: a(z)b(w):

p(a, b)b(w)a(z)

- [a(z)+, b(w)] +: a(z)b(w) :

THEOREM 2.3. Each of the following properties (i)-(vii) is equivalent to
(2.3.2):
LN-1
(i) [a(z), b(w)] = o}jlo(z - w)d (w), where ci (w) E fl [[w, w- 1]] .
i=O

(ii) [a(z)_, b(w)]

} ; (iz.w (z _ ~)i+l) d(w),

- [a(z)+, b(w)] = } ; (i,L',z (z _ ~)J+l) d(w),

fo where ci(w) E fl [[w, w-1]].

(iii)

a(z)b(w)

1 (iz,w (z _ ~)J+l) d(w)+: a(z)b(w) :,

~ p(a, b)b(w)a(~)

} ; (iw,, (z _ ~)i+') d(w)+, a(z)b(w) ,,

where d(w) E fl [[w,w- 1]].

I: (iv) [a(m),b(n)] =

(~)cfm+n-j)' m,n E Z.

j=O J
I : (v) [acm),b(w)] = (~)d(w)wm-i, m E Z.

j=O J

26

2. CALCULUS OF FORMAL DISTRIBUTIONS

LN-1
(vi) [a(m),b(n)] = Pj(m)d~+n, m,n E Z,
j=O
for some polynomials Pj(x) and elements d{ of fl.

(vii)

a(z)b(w)

(iz,w (z _1w)N) c(z, w),

p(a, b)b(w)a(z)

(iw,z (z _1w)N) c(z, w)

for a formal distribution c(z, w).

PROOF. (i) is equivalent to (2.3.2) due to Corollary 2.2. (ii) is equivalent to (i)

by taking all terms in (i) with negative (resp. non-negative) powers of z. (iii) is

equivalent to (ii) due to ((2.3.6a) and b). (iv) and (v) are equivalent to (i) due to

(2.2.8). (vi) is equivalent to (iv) since any polynomial is a linear combination of

binomial coefficients. Finally, (iii) implies (vii) and (viO implies (2.3.2).

□

By abuse of notation physicists write the first of the relations.pf ':Qworem 2.3(iii)

as follows:

(2.3.7a)

a(z)b(w)

=

N~-1

(z

d(w) _ w)i+l

+:

a(z)b(w)

:,

1=0

or often write just the singular part:

(2.3.7b)

~ a(z)b(w)

~ N-1

(z

d(w) - w)Hl.

Formulas (2.3.7a) and (2.3.7b) are called the operator product expansion (OPE). By Theorem 2.3 the singular part of the OPE encodes all the brackets between all the coefficients of mutually local formal distributions a(z) and b(z). That is why it is important to develop techniques for the calculation of the OPE's. Most of the time we shall use the form (2.3.7b) of the OPE as typographically the most convenient.
For each n E Z+ introduce then-th product a(w)(n)b(w) on the space of formal distributions by the formula

(2.3.8)

a(w)(n)b(w) = Resz [a(z),b(w)] (z -wt.

Then, due to Corollary 2.2, the OPE (2.3.7a) becomes (for any two local formal

distributions a(z) and b(z)):

(2.3.9a)

ko a(z)b(w)

=

~

a(w)uib(w) (z _ w)i+l

+:

a(z)b(w):.

2.3. LOCALITY OF TWO FORMAL DISTRIBUTIONS

27

Equivalently:

(2.3.9b)

L [a(z), b(w)] =

(a(w)u)b(w)) aijl8(z - w).

jEZ+

As we have seen in the previous section, an efficient way to study the OPE is

to consider its formal Fourier transform.

For an arbitrary (not necessarily associative or Lie) algebra U define the .>.product a(w) >. b(w) of two U-valued formal distributions a(w) and b(w) as the formal Fourier transform of the formal distribution a(z)b(w):

(2.3.lOa)

L00
a(w)>.b(w) = F:w (a(z)b(w)) = >,(n) (a(w)nb(w)).
n=O

As before, we have the (allowing formula for n-th product:

(2.3.lGb)

In the case when U is a Lie (super)algebra we will use the bracket notation for

the .>.-product, will call it the .>.-bracket and will denote by a(w)(n)b(w) then-th

product (given by (2.3.8)), i.e.:

(2.3.11)

L00
[a(w)>.b(w)] = >,(m) (a(w)(m)b(w)).
m=O

The following formulas are very useful in studying associativity properties of the

.>.-product:

(2.3.12)

(2.3.13) The first of these two formulas is obvious, while the second is immediate by (2.2.12).
Now we can prove the basic properties of .>.-products and .>.-brackets.
PROPOSITION 2.3. (a) For any two U-valued formal distributions a(w) and b(w), where U is an arbitrary algebra, one has:
uta(w)h b(w) = ->.a(w)>.b(w),
a(w)>.8wb(w) = (>. + 8w) (a(w)>.b(w)).
In particular, 8w is a derivation of the >.-product.

28

2. CALCULUS OF FORMAL DISTRIBUTIONS

(b) Let a(w) and b(w) be U -valued formal distributions, where U is an arbitrary algebra, such that the formal distribution a(z) b(w) is local. Let a(w) o b(w) denote
>.
the >.-product of the algebra U0 P (which is U with the opposite multiplication aob =
ba). Then

(c) Let a(w) and b(w) beg-valued mutually local formal distributions, where g is a Lie superalgebra (i.e. (2.3.2) holds). Then

(d) Let a(w), b(w) and c(w) be g-valued formal distributions, where g is a Lie superalgebra. Then
[a(w)>. [b(w)µc(w)]] = [[a(w)>.b(w)h+µ c(w)] + p(a, b) [b(w)µ [a(w)>.c(w)]].
PROOF. (a) follows from (2.2.10) applied to the formal distribution a(z, w) =
a(z)b(w). (b) follows similarly from (2.2.11). (c) is immediate by (b). Finally (d) follows from (2.3.12) and (2.3.13) applied to the Jacobi identity:
[a(z), [b(x), c(w)]] = [[a(z), b(x)], c(w)] + p(a, b) [b(x), [a(z), c(w)]].

□

REMARK 2.3b. (i) Proposition 2.3 (a) in terms of n-th products means the following formulas (cf. Remark 2.2a):

(2.3.14a)

oa(w)nb(w) = -na(w)n-1b(w),

(2.3.14b)

a(w)nob(w) = 8 (a(w)nb(w)) + na(w)n-1b(w).

Hence Ow is a derivation of all n-th products. (ii) Proposition 2.3 (c) in terms of n-th products means (cf. Remark 2.2b):

(2.3.15)

00
a(w)(n)b(w) = -p(a,b) ~)-1)Hnay) (b(w)(n+j)a(w)),
j=O

provided that a(w) and b(w) are mutually local. (iii) Proposition 2.3 (d) in terms of n-th products means

t, (7) 2.4. TAYLOR'S FORMULA

(2.3.16) a(w)(m) (b(w)(n)c(w)) =

(a(w)(j)b(w)\m+n-j) c(w)

29

+ p(a, b)b(w)(n) (a(w)(m)c(w)) .

The following well-known statement has many important applications.

COROLLARY 2.3. Let g be a Lie superalgebra.
(a) If a(z) and b(z) are g-valued formal distributions, then [a(o), b(z)] = 0
= if! a(z)(0ib(z) 0.

(b) If a(z) is an odd g-valued formal distribution, then azo)
Resz a(z)(o)a(z) ·= 0.

0 if!

(c) Let A be a space consisting of mutually local formal g-valued distributions in

w which is 8-invdriant and closed with respect to all n-th products, n E Z+-

Then with respect to the 0-th product 8A is a 2-sided ideal of A and A/8A

is a Lie superalgebra. Moreover, the 0-th product defines on A a structure

of a left A/BA-module.

PROOF. Statements (a) and (b) are obvious by definitions. From (2.3.14a) and
(2.3.14b) for n = 0 we get

(2.3.17)

Hence 8A is a 2-sided ideal. Furthermore, (2.3.15) for n = 0 gives

(2.3.18)

= a(w)(o)b(w) -p(a, b)b(w)(o)a(w) mod 8A.

Hence the 0-th product induces a super skew-symmetric bracket on A/8A. The
super Jacobi identity in A/8A follows from (2.3.16) for m = n = 0. This proves

(c).

□

2.4. Taylor's formula
One of the devices in calculating the OPE is Taylor's formula. Here and further we shall adopt the following notational conventions. Given a formal distribution
a(z) = I:n anzn we may construct a formal distribution in z and w:

n

30

2. CALCULUS OF FORMAL DISTRIBUTIONS

In order to further simplify notation we shall often say instead that we consider the
formal distribution a(z - w) in z and win the domain lzl > lwl.

PROPOSITION 2.4 (Taylor's formula). Let a(z) be a formal distribution. Then one has the following equality of formal distributions in z and in w in the domain
lzl > lwl:

(2.4.1)

00
a(z+w) = LaUla(z)wi.
j=O

PROOF. Let a(z) = Ln an Zn' so that au) a(z) = Ln (1J) anzn-j. Comparing

coefficients of an in (2.4.1), we need to show that

(2.4.2)

f: (z + w)n = zn-jwj (~).

j=O

J

But (2.4.2) is the binomial expansion in the domain lzl > lwl'-

□

Replacing z by w and w by z - w in (2.4.1) we get another form of Tay-

lor's formula as an equality of formal distributions in w and z - w in the domain

lz-wl < lwl:

(2.4.3)

00
a(z) = L 8(j)a(w)(z - w)J.
j=O

The following, yet another version of Taylor's formula, shows that when cal-

culating the singular part of the OPE one can use Taylor's expansion up to the

required order.

THEOREM 2.4. Let a(z) be a formal distribution and N be a non-negative integer. Then one has the following equality of formal distributions in z and w:

(2.4.4)

I:N
a::c5(z - w)a(z) = a:: c5(z - w) a(j) a(w)(z - w)j.
j=O

PROOF. It suffices to check that for an arbitrary Laurent polynomial f(z) one has:

Resz 81; c5(z - w)a(z)f(z)
N
L aUla(w) Resz af c5(z - w)(z - w)i f(z). j=O

2.5. CURRENT ALGEBRAS

31

Integrating by parts N times transforms this to the equality
L N
Resz 6(z - w)8N (a(z)f(z)) = aUla(w) Resz 6(z - w)8;' ((z - w)i f(z))
j=O
which, due to (2. 1. 7) and Leibnitz rule, is

This holds by Leibnitz rule.

□

2.5. Current algebras
Here we discuss one. of the most important examples of algebras spanned by mutually local formal distributions-the current algebras.
First we consider the simplest case-the oscillator algebra .s. This is a Lie algebra with basis an (n E Z), Kand the following commutation relations:

(2.5.1)

Consider the following .s-valued formal distribution:
L a(z) = DnZ-n-l_
nEZ
Then it is straightforward to check that

(2.5.2)

[a(z), a(w)] = 8wi5(z - w)K

(this follows also from the equivalence of (i) and (iv) of Theorem 2.3). In other words, the formal distribution a(z) is local (with respect to itself) with the OPE, considered in the universal enveloping algebra of .s:

(2.5.3)

K a(z)a(w) ~ (z-w)2 •

The (even) formal distribution a(z) is usually called a free boson. The current algebra is a non-abelian generalization of the oscillator algebra.
Let g be a Lie superalgebra with an invariant supersymmetric bilinear form (-!-)"Invariant" means

([a, b] le) = (al [b, cl), a, b, c E g,

32

2. CALCULUS OF FORMAL DISTRIBUTIONS

and "supersymmetric" means

(alb)= (-l)P(a)(bla) (in particular, (9ol£lI) = 0).

The loop algebra associated to 9 is the Lie superalgebra

g = 9 [t, r 1] (= 9 Q9ic c [t, r 1])

over C with Z/2'1/., grading extending that of 9 by p(t) relations (m,n E Z;a,bE 9):

0, and commutation

Here and further am stands for a 18) tm. Note that g is the Lie superalgebra of
regular maps of ex to 9 (hence the name "loop algebra").
The affinization of the pair (9, (.I.)) is a central extension of the loop algebra g
by a 1-dimensional even center CK:

defined by the commutation relations (m,n E Z; a,b E 9):

(2.5.4)

The Lie superalgebra g is usually called by physicists a current algebra. Note that
loop algebra is a special case of a current algebra when the bilinear form (-1-) is zero. If 9 is a simple finite-dimensional Lie algebra with the (normalized) Killing
form (-1-), then g is known as the affine Kac-Moody algebra [K2]. If 9 is the 1-
dimensional Lie algebra with a non-degenerate bilinear form, then we recover the example of the oscillator algebra.
Introduce the following formal distributions with values in g which are usually
called currents:

a(z ) = ~ L....,; anz -n-1 , a E 9.
nEZ
Then by the equivalence of (i) and (iv) of Theorem 2.3, we see that

(2.5.5)

[a(z), b(w)] = c>(z - w) [a, b] (w) + 8wc>(z - w)(alb)K,

2.5. CURRENT ALGEBRAS

33

hence all the currents a(z) are mutually local with the OPE, considered in the

universal enveloping algebra U(:g) of g:

(2.5.6)

~ a (z )b( w)

[a,b](w) (alb)K z-w + (z-w)2 •

There exists a natural super extension of the affinization, called the superaf-

finization, which is a central extension of the super loop algebra (called a supercur-

rent algebra):

9super = g 0c C [t, r1,0] + CK,

where 02 = 0, p(0) = I and the remaining OPE are as follows. For a E g define the
supercurrent

where an+½ = a 0 tn0. Then the supercurrents a(z) are mutually local and also
local with respect to the currents, and the remaining OPE are given by

(2.5.7a) (2.5.7b)

~ a(z)b(w)

z-w

~ a(z)b(w)

(bla)K z-w

The supercurrents form a closed (under OPE) subalgebra. In view of its importance, we repeat its construction in a slightly different form. Let A be a superspace with a skew-supersymmetric bilinear form, i.e.,

(rpl¢) = -(-l)P(r,ol(¢lrp) (in particular, (A0 IA1) = 0).

The Clifford affinization of (A, (.I.)) is the Lie superalgebra

with commutation relations (m, n E ½+ Z; rp, ¢EA)

(2.5.8)

where rpm = <p 0 tm-½. The formal distributions rp(z) = LnEZ <pn+F-n-l are

mutually local with the OPE (in the universal enveloping algebra of CA):

(2.5.9)

~ 'P(z)¢(w) (rpl¢)K. z-w

34

2. CALCULUS OF FORMAL DISTRIBUTIONS

Two particularly important special cases of the Clifford affinization are the
following.
Let A be the odd I-dimensional superspace Op with the bilinear form (rplrp) = 1,
and let K = 1. Then CA turns into the algebra

(2.5.10)

m,nE

1
2

+z.

The (odd) formal distribution rp(z) = LnE½+z <pnz-n- 1/ 2 is called a neutral free
fermion; its OPE is

(2.5.11)

1 rp(z)rp(w) ~ -z--w.

In the second example let A be the odd 2-dimensional superspace rap+ E0 rap-
with the symmetric bilinear form (rp+lrp-) = 1, (rp±lrp±) = 0, and again let K = 1.
Then we obtain the algebra (m, n E ½+ Z):

(2.5.12)

<p±m<p±n + <p±n rp±m -- 0·

= The odd formal distributions rp±(z) LnE½+zrp!z-n-1/ 2 are called charged free
fermions; their OPE are:

(2.5.13)

These examples show that superalgebra is far from being a senseless generalization of the usual algebra.

2.6. Conformal weight and the Virasoro algebra Let H be a diagonalizable derivation of the associative algebra U, called a Hamiltonian. Then H acts on the space of all formal distributions with values in U in the obvious way (coefficient-wise). The following definition is motivated by (1.2.6b).
DEFINITION 2.6a. A formal U-valued distribution a= a(z,w, ... ) is called an eigendistribution for H of conformal weight ~ E C if
(H - ~ - z8z - w8w - •··)a = 0.
Here are some obvious properties of conformal weights.

2.6. CONFORMAL WEIGHT AND THE VIRASORO ALGEBRA

35

PROPOSITION 2.6. Suppose a and bare eigendistributions of conformal weights ~ and ~, respectively. Then
(a) Oza is an eigendistribution of conformal weight ~ + 1.
(b) : a(z)b(w): is an eigendistribution of conformal weight~+~'. (c) Then-th OPE coefficient of[a(z), b(w)] is an eigendistribution of conformal
weight~+~, - n - l(n E Z+)(d) If f is a homogeneous function of degree j then fa is an eigendistribution
of conformal weight ~ - j.

COROLLARY 2.6. If a(z) and b(z) are mutually local eigendistributions of con-

formal weights~ and~', then in the OPE

~ a(z)b(w)

~ N-1

(z

d(w) _ w)J+l

all the summands have the same con/ormal weight ~ + ~,.

If a(z) is an eigendistribution of conformal weight ~, one usually writes it in the form (without parenthesis around indices):
a(z) =

The condition of a(z) being an eigendistribution of conformal weight ~ is then equivalent to (2.6.1) As a result, the commutation relations given by Theorem 2.3(iv) take a graded form:
(2.6.2a)

or equivalently

(2.6.2b)

I: [am, b(z)] = N-1(m + ~ - l ) d (z)zm+~-j-l.

j=O

J

EXAMPLE 2.6. Choosing for the algebra of currents g (resp. supercurrents Osuper) the Hamiltonian H = -tat (resp. = -tat - ½080), we see that the cur-

rents a(z) (resp. supercurrents a(z)) have conformal weight 1 (resp. 1/2).

36

2. CALCULUS OF FORMAL DISTRIBUTIONS

Corollary 2.6 is a very useful bookkeeping device in calculating the OPE. In many examples (e.g., from the considerations of unitarity) the conformal weight is in ½Z+ and it is O iff the eigendistribution is a constant element commuting with all formal distributions of the theory.
If the above positivity condition holds, then due to Corollary 2.6, all mutually
local eigendistributions of conformal weight ½have the OPE of the form (2.5.7b),
all eigendistributions of conformal weight 1 have the OPE of the form (2.5.6) and the OPE between the latter and the former is given by (2.5.7a).
We consider now the next case-a local (i.e., local to itself) even eigendistribution L(z) of conformal weight 2:
L L(z) = Lnz-n-2 .
nEZ
As has been mentioned above, it is natural to assume that the OPE has the form

(2.6.3)

where C is a constant formal distribution.

THEOREM 2.6. Suppose that L(z) is an even local formal distribution with the OPE of the form (2.6.3). Then
(a) a(w) = 0 and c(w) = ab(w).
(b) If in addition [C, L(z )] = 0 and

(2.6.4)

[L_ 1, L(z )] = 8L(z), [Lo, L(z )] = (za + 2)L(z)

then (2.6.3) becomes

(2.6.5)

~ ( ) ½C

2L(w) 8L(w)

L(z)L(w) z-w 4 + (z-w)2 + -z--w,

or, equivalently, we have the Virasoro algebra (m, n E Z):

(2.6.6)

PROOF. Exchanging z and win (2.6.3), we obtain

~ L(w)L(z)

lC
(z ~ w) 4

a(z) + 2b(z) _ c(z)
(z-w) 3 (z-w)2 z-w

2.6. CONFORMAL WEIGHT AND THE VIRASORO ALGEBRA

37

By making use of Taylor's formula, this turns into:

(2.6.7)

~ ½C
L(w)L(z)

a(w) + 8a(w)(z - w) + a(2)a(w)(z - w) 2

(z-w)4

(z-w)3

+2b-(w)-+(-Wz--bw(w-))2(-z --w-) -z-c(-ww-) ·

Due to locality the right-hand sides of (2.6.3) and (2.6.7) must be equal. Matching the coefficients of (z - w)-3 and (z - w)-1 we get (a). Thus, we have:

(2.6.8)

~ L(z)L(w)

½C + 2b(w) + 8b(w).

(z-w)4 (z-w)2 z-w

Due to (2.6.2b) this implies, in particular:

[£_1,,L(z)] = 8b(z), [Lo, L(z)] = (z8 + 2)b(z).

Hence assumptions (2.6.4) imply that b(z) = L(z). This proves (2.6.5). The equa-

tion (2.6.6) is equivalent to this OPE due to (2.6.2a).

□

A local formal distribution L(z) with the OPE (2.6.5) is called a Virasoro formal distribution with central charge C.

In Table OPE we give a table of the most commonly used OPE of mutually local formal distributions and the equivalent commutation relations (all these are special cases of formula (2.6.2a)).

The definition given below singles out the most important for conformal field theory Lie superalgebras, which includes the (super)current algebra and the Virasoro algebra.

DEFINITION 2.6b. A Lie superalgebra fl is called a formal distribution Lie superalgebra if it is spanned over (C by coefficients of a family F of fl-valued mutually local formal distributions.

For example, the Virasoro algebra with F = {L(z), C} and the current algebra
g with F = {a(z) where a E fl,K} are formal distribution Lie (super)algebras. We
shall often write (fl, F) in order to emphasize the dependence on F. Note that formal distribution Lie superalgebras form a category with mor-
phisms (fl,F) ➔ (fl1,F1) being homomorphisms <p: fl ➔ fl1 such <p(F) C F1, where
F 1 is the closure of Fi, defined in the next section.

c,, 00

Table OPE.

!"

1st distribution

2nd distribution

commutation relations

OPE

0
>t-<

0

C:

L L a(z) = amz-m-l b(w) = bnw-n-l

[am, bn] = Cm+n

c(w) = L CnW-n-l
z-w

t-< C:
00
0

>i:j

>i:j

L a(z) = Lamz-m-l b(w) = bnw-n-l

[am, bn] = m8m,-n

1 (z - w) 2

0
~
>t-<

L L(z) = LLmz-m-2 a(w) =

anw-n-b. [Lm,an] = ((~ - l)m - n)am+n

-8za--(w w+ )

~a(w) (z-w) 2

8
00 ,,J
~

tJi

L L(z) = LLmZ-m-2 L(w) = Lnw-n-2

[Lm, Ln] = (m - n)Lm+n

8L(w) 2L(w)
-z--w+ (z-w) 2

C:
,,J
0 z

m3 -m
+ 12 8m,-nC

+

(z

c/2 - w)4

00

2.7. LIE SUPERALGEBRAS AND CONFORMAL SUPERALGEBRAS

39

2. 7. Formal distribution Lie superalgebras and conformal superalgebras

This and the next two sections is an introduction to the theory of conformal (super)algebras. Though they are ideologically closely related to the theory of vertex algebras, the rest of the book may be read independently of them, except for the last Section 5.10.
Let g be an arbitrary Lie superalgebra. We denote by fd(g) the space of all g-valued formal distributions in z endowed with n-th products (2.3.8), n E Z+-
This is also a C(8]-module (8 = 8z).
Consider the subspace Rover C of fd(g) which is closed under all n-th products, n E Z+, and denote by g(R) the C-space of all coefficients of all formal distributions from R. Provided that ,all formal distributions from R are mutually local, g(R) is a subalgebra of g with the bracket

(2.7.1)

This follows from Theorem 2.3(iv). Clearly, g(R) is a formal distribution Lie superalgebra and all of them are thus obtained.
Let F be a collection of mutually local formal distributions from fd(g). We
denote by P the closure of F, defined as the minimal C[8]-submodule of fd(g) closed
under all n-th products, n E Z+· Due to Lemma 2.8 proved in Section 2.8 (applied
to the adjoint representation) and Remark 2.3a, P consists of mutually local formal
distributions and therefore we have a formal distribution Lie superalgebra g(F). In view of Proposition 2.3 (or rather Remark 2.3b), this leads us to the following definition.
DEFINITION 2. 7. A conformal superalgebra R is a left Z /2Z-graded C [8]-mod-
ule R = R 0 E9 R1 with a C-bilinear product a(n)b for each n E Z+ such that the
following axioms hold (a, b,c ER, m,n E Z+):
(CO) a(n)b = 0 for ngO ,

00
(C2) a(n)b = -p(a, b) L(-I)Hna(j) (b(n+j)a) ,
j=O

40

2. CALCULUS OF FORMAL DISTRIBUTIONS

Note that axioms (Cl) and (C2) imply

(Cl') hence{) is a derivation of all n-th products (cf. Proposition 2.3(a)).

REMARK 2.7a. The operator a(o) is a derivation of all n-th products (due ~o (C3)) and it commutes with {) (due to (Cl')). As in the proof of Corollary 2.~ it follows (using also (Cl) and (C2)) that, with respect to 0-th product, 8R is a 2-sided ideal of R such that R/8R is a Lie superalgebra, and that 0-th product defines on R a structure of a left R/8R-module for which R/8R commutes with <C[8].

The notions of a homomorphism, ideal and subalgebra of a conformal superal-

gebra Rare defined in the usual way. Conformal superalgebras form a category with
= morphisms being homomorphisms. An element a ER is called central if a(n)R 0
for all n E Z+ (and hence R(n)a = 0, n E Z+)- A conformal superalgebra is called
finite if it is finitely generated as a <C [8]-module. An efficient way to handle then-th products of a conformal superalgebra R is
to introduce the >..-bracket (cf. Section 2.3):

(2.7.2)

L00
[a>.b] = >,_(n) (a(n)b).
n=O

Here >.. is an indeterminate and, as before, >,_(n) stands for >,_n /n!. Due to axiom

(CO), the >..-bracket defines a <C-linear map

Due to Proposition 2.3 and Remark 2.3a axioms (Cl)-(C3) are equivalent respectively to

(Cl)>. (C2)>. (C3)>.

[8a>.b] = ->.. [a>.b], [a>.b] = -p(a, b) [L>.-aa],
[a>. [bµcl] - p(a, b) [bµ [a>.c]] = [[a>.bh+µ c].

2.7. LIE SUPERALGEBRAS AND CONFORMAL SUPERALGEBRAS

41

Axioms (Cl)>. and (C2).>- imply

(Cl')>-
a hence is a derivation of the .>.-bracket.
The first application of the .>.-product is the following corollary.

COROLLARY 2. 7. [DK] Any torsion element a of a finite conformal superalgebra R is central. In particular, if R is finite, then, as a C [8]-module, R is a direct
sum of a finite-dimensional (over CJ central subalgebra and a free C[8]-module of
finite rank.

PROOF. By definition, we have P(8)a = 0 for some polynomial P, hence

[P(8)a>-b] = 0 for any b E R, and P(-.>.) [a.>-b] = 0 by (Cl)>-- It follows that

[a.>-b] = 0 for any b ER, hence a is a central element.

□

Conformal superalgebras are an effective tool to study formal distribution Lie superalgebras. Indeed, if g is spanned by coefficients of a collection F of mutually
local formal distributions, then P is a conformal superalgebra, due to Proposi-
tion 2.3. Conversely, we may construct a formal distribution Lie superalgebra Lie R associated with a conformal superalgebra R as follows. Let Lie R be the quotient of the vector space with basis an (a E R, n E Z) by the subspace spanned over (C by elements:

(>.a)n - >-an, (a+ b)n - an - bn, (8a)n + nan-1, where a, b ER,>. E C, n E Z.

One can check that a formula similar to (2.7.1) gives a well-defined bracket on LieR:

(2.7.3)

Instead of doing this calculation, we shall use a more conceptual approach. The affinization of a conformal superalgebra R is the conformal superalgebra

42

2. CALCULUS OF FORMAL DISTRIBUTIONS

with 8 8 l8l 1 + 1 l8l 8t and the n-th product defined by (a, b E R, f, g E
C[t,c1], n E Z+):

(2. 7.4)

= L (a Q9 f)(n)(b ® g)

(a(n+j)b) Q9 ((aij) f)g).

jEZ+

(We shall see that the affinization of a conformal superalgebra is a straightforward generalization of a more naturally looking notion of affinization of a vertex algebra
= introduced by Borcherds; see Section 4.3.) Letting an a 18) tn, formula (2.7.4)
becomes (m,n E Z):

(2.7.5)

Letting

LieR = R/8R

with the bracket induced by the 0-th product on R, (and keeping the notation an for
its image in Lie R) we obtain, due to Remark 2.7a, a well-defined Lie superalgebra, which is obviously the same as the one introduced above.
It remains to check the axioms of conformal superalgebra for R. A simple
calculation shows that the corresponding >.-bracket is given by

(2.7.6)

[a l8l f>.. b 18) g] = [a.x+a,b] l8l J(t)g(t')lt'=t·

The verification of axioms is now straightforward. Let us check, for example, axiom (C2).x:
[a l8l f.xb 18) g] = [a.x+a,b] l8l J(t)g(t')lt'=t
= -p(a, b) [L.x-a,-aa] l8l J(t)g(t')lt'=t
= -p(a,b) [L.x-8-8,-8,,+8,,a] ®g(t')J(t)lt=t'
= -p(a, b) [b 18) g_ _x_8a 18) J] .
REMARK 2.7b. It is clear from (2.7.5) that -118) 8t is a derivation of the 0-th
product of the conformal superalgebra R. Since this operator commutes with 8, it
induces a derivation T of the Lie superalgebra Lie R, given by the formula:

2.7. LIE SUPERALGEBRAS AND CONFORMAL SUPERALGEBRAS

43

REMARK 2.7c. Let R be a conformal superalgebra and suppose that, as a <C [8]module,

where V is a vector space (over <C) and C consists of torsion elements. Then the vector space V [t, t- 1] EB C is complementary in R, to ail. Hence, as a vector space (over <C), LieR ~ V [t,t-1] EB C, where C is a central subalgebra of Lie R (by
Corollary 2.7). It suffices to check this in two cases: 1) dime V = 1 and C = 0,
2) V = 0 and dime C = 1, when it is straightforward. It follows, in particular, that
Tlc=8,
where T is the derivation of Lie R defined by Remark 2. 7b.
REMARK 2.7d. The construction of the Lie superalgebra Lie R can be generalized by taking an arbitrary commutative associative algebra A with a derivation 6 and letting

with the bracket

where 61 (fg) = 6(f)g. The operator -106 on R0A induces a derivation ofLieA R, giving it a structure of a differential Lie superalgebra, cf. [Rl].

= Each element a E R gives rise to a formal distribution a(z) LnEZ anz-n-l

with coefficients in Lie R. We denote this family of formal distributions by F(R).

They obviously span LieR and are mutually local since formula (2.7.5) fork= 0 is

equivalent to

(2.7.7)

L [a(z), b(w)] = (a(j)b) (w)a~lJ(z - w)

jEZ+

and au) b = 0 for j » 0. Hence (Lie R, F(R)) is a formal distribution Lie superalge-

bra. We thus constructed a functor, from the category of conformal superalgebras

to the category of formal distribution Lie superalgebras.

Note that F(R)(C Jd(LieR)) is a conformal superalgebra and that the map
= <p: R ➔ F(R) defined by <p(a) a(z) is a surjective homomorphism of conformal

44

2. CALCULUS OF FORMAL DISTRIBUTIONS

superalgebras. Indeed, r.p preservesj-th products due to (2.7.7), and r.p preserves the
C[8]-module structure since (8a)n = -nan-1, which means that 8za(z) = (8a)(z).

= LEMMA 2.7. If a E R and the element a_1 E LieR is 0, then a 0. In
particular, r.p is an isomorphism of conformal superalgebras.

PROOF. Define a linear map'¢ : R ➔ R of vector spaces over C by
'¢ (aci-l) = aUla, '¢ (ati) = 0, where j E Z+·
ail) Then'¢ ( = 0, hence,(/; induces a map 'lj;: Lie R ➔ R such that 'lj;(a_1) = a. □

Recall that to a formal distribution Lie superalgebra (g, F) one canonically
associates a conformal superalgebra Conf(g, F) = P. This gives us a functor from
the category of formal distribution Lie superalgebras to the category of conformal superalgebras, which we denote by Con£. We also have constructed a functor in the opposite direction that canonically associates to a conformal superalgebra R a formal distribution Lie superalgebra (LieR, F(R)); we denote this functor by Lie. Due to Lemma 2.7, we have:

Conf(LieR)::: R.

Furthermore, we have:
Lie(Conf(g,F)) = (LieF,F).

By the very definition, the Lie superalgebra g is a quotient of Lie F by an ideal that does not contain all the coefficients of a non-zero formal distribution from P. Such an ideal is called an irregular ideal of Lie F. Conversely, if g is obtained from Lie F as a quotient by an irregular ideal, then Con£ g ::: F. The formal distribution
Lie superalgebras (LieF,F) and ((LieF)/J,F) are called equivalent. Hence, it is natural to call (Lie R, F(R)) the maximal formal distribution Lie superalgebra associated to the conformal superalgebra R.
So, the functor Con£ induces a functor Con£' from the category of equivalence classes of formal distribution Lie superalgebras to the category of conformal superalgebras and the functor Lie induces a functor Lie' going in the opposite direction.
Thus we have proved the following result.

2.7. LIE SUPERALGEBRAS AND CONFORMAL SUPERALGEBRAS

45

THEOREM 2. 7. The functor Conf' and Lie' are inverse of each other and establish equivalence between the category of equivalence classes of formal distribution Lie super-algebras and the category of conformal super-algebras

A formal distribution Lie superalgebra (g, F) is called finite if F is a finitely
generated C[8]-module. Theorem 2.7 reduces the classification of (finite) formal distribution Lie superalgebras to the classification of (finite) conformal superalgebras.
Due to Corollary 2.7, the description of finite conformal superalgebras splits into two problems:
1. describe conformal superalgebras that are free (('. [8]-modules of finite rank; 2. find central extensions of conformal superalgebras from 1. with center being
in torsion.
The first problem is reduced to solution of a finite system of functional equations on a finite set of polynomials in two indeterminates as follows.
n
Let R = E9C[8]ai be a Z/2Z graded (('.[8]-module with p(ai), denoted by
j=l
p(i), and let [a\ai] = I:k Q~(>..,8)ak. These >..-brackets give rise to a structure of a conformal superalgebra on R if Q~ (i,j, k = 1, ... , n) are polynomials in >. and 8 subject to the following relations that are equivalent to axioms (C2).x and (C3).x respectively:

(2. 7.9)

L (n Q~k(µ, 8 + >..)Q;s (>.., 8) - (-I)P(i)p(j)Q!k(>.., 8 + µ)Qf' 8 (µ, 8))
s=l
L n
= Q!i(>.., ->.. - µ)Qfk(>.. + µ, 8).
s=l

Due to equivalence of (C3).x to the Jacobi identity in Lie R, it suffices to check (2.7.9) for all triples 1 ~ i ~ j ~ k ~ n and 1 ~ t ~ n.
It is clearly impossible to solve these equations directly for n 2: 2. Below a solution is presented for n = 1 and R = Ro (obtained jointly with Minoru Wakimoto).

46

2. CALCULUS OF FORMAL DISTRIBUTIONS

We have: R = C[8] a and [a>.a] = Q(>., 8)a, where Q(>., 8) is a polynomial in).. and 8 satisfying two equations:

(2.7.10)

Q(>.,8) = -Q(-8- >.,8),

(2.7.11) Q(µ, 8 + >.)Q(>., 8) - Q(>., 8 + µ)Q(µ, 8) = Q(>., ->. - µ)Q(>. + µ, 8).

Let Q(>., 8) = ~j=O Cj(>.)8i with Cr(>.) # 0. Comparing coefficients of 82r-l in (2.7.11) we obtain: r(>. - µ)cr(>.)cr(µ) = 0 if r > I, a contradiction. Hence
Q(>., 8) = a(>.)8 + b(>.). Letting)..=µ in (2.7.11), we get Q(>., -2>.)Q(2>., 8) = 0, hence Q(>., -2>.) = 0, which means that b(>.) = 2>.a(>.), hence Q(>., 8) = a(>.)(8 +
2>.). Plugging this in (2.7.10), we see that a(>.) is a constant. Since a can be chosen up to a non-zero constant factor, we arrive at two solutions: Q(>.,8) = 0
and Q(>., 8) = 8+ 2>.. In the first case we get a commutative conformal algebra (i.e.
all products are trivial), and in the second case we arrive at the Virasoro conformal algebra discussed below.
Now we discuss briefly the second problem, the construction of central exten-
sions: R = R EB C where Rand Care C[8]-submodules of Rand C,>.R = 0. The
>.-bracket [a>.bf" on RC R is given by

(2.7.12)

[a>.bf" = [a>.b] + a,>.(a, b),

where [a>.b] is the >.-product on R and a,>.(a, b) = ~n2'.0 >,.(n)an(a, b) is a C-linear
map R © R ➔ C[>.] ©ic C. The axioms (Cl)>., (Cl')>., (C2)>. and (C3)>. for Rare equivalent to the following properties of the 2-cocycle a,>. (a, b):

(2.7.13)

a>.(8a,b) = ->.a,>.(a,b), a,>.(a,8b) = (8+>.)a>.(a,b),

(2.7.14)

a,>.(a, b) = -p(a, b)a->.-a(b, a),

(2.7.15)
As above, these equations are equivalent to a system of functional equations on a
set of polynomials in two indeterminates. If we take another complement of C in R
by replacing a ER by a-f(a), where f: R ➔ C is a C[8]-module homomorphism,
then a,>.(a,b) gets replaced by a'.x(a,b) = a,>.(a,b) + f(a,>.b). The trivial 2-cocycle

2.7. LIE SUPERALGEBRAS AND CONFORMAL SUPERALGEBRAS

47

f(a>..b) defines a trivial extension, and equivalent 2-cocycles a~(a,b) and a>..(a,b) define isomorphic extensions.
One can develop a cohomology theory of conformal superalgebras similar to the Lie algebra cohomology (see Section 2.11). The central extensions of R by C are then parameterized by H 2 (R, C).
We consider now three main examples of finite conformal (super)algebras R. Due to (Cl) and (Cl') it suffices to define n-th products on the generators of the C [8]-module R.

EXAMPLE 2.7a. Let g be a finite-dimensional Lie superalgebra. Recall (see
Section 2.5) that the associated loop algebra g = g [t, t-1] is a formal distribution
Lie superalgebra with the family F consisting of currents a(z) = I':n (atn) z-n-l
where a E g. Recall that (cf. (2.5.5)):
[a(z), b(w)] = [a, b] (w)b(z - w).

Hence the conformal superalgebra associated to (g, F) is C[8] @cg with a structure of a conformal superalgebra defined on a, b E g by

(2.7.16)

This is called the current conformal superalgebra associated to g. It is denoted by Cur g.
The following formula defines a 2-cocycle on Cur g with values in the trivial C[8]-module C (a, b E 1@ g C Cur g):

(2.7.17)

a1 (a, b) = (alb), am(a, b) = 0 if m f 1,

where (-1-) is a supersymmetric invariant bilinear form on g. It is easy to see
that (2. 7.17) gives all 2-cocycles, up to taking for a 0 a 2-cocycle on g, provided
that [g, g] = g. In particular, if g is a simple finite-dimensional Lie algebra, then
(2. 7.17) gives all 2-cocycles, up to equivalence. The corresponding central extension
is the conformal superalgebra Confg associated to the current algebra g defined in Section 2.5. It follows from Remark 2.7c that Lie(Confg) = g and Lie(Cur g) = g, i.e. both fi and g are maximal formal distribution Lie superalgebras. Note that
I= g [t, t-1] P(t), where P(t) is a non-invertible Laurent polynomial, is an irregular

48

2. CALCULUS OF FORMAL DISTRIBUTIONS

ideal of g, hence the formal distribution Lie algebras g and g/I are equivalent (i.e.
give rise to the same conformal algebra).

EXAMPLE 2.7b. Let Vect ex denote the Lie algebra of regular vector fields on
ex. It has a basis Ln = -tn+l8t (n E Z) with commutation relations

This is a formal distribution Lie algebra with the family F consisting of a single formal distribution
n
Either directly (cf. Theorem 2.6) or using (2.1.10) we obtain:
[L(z),L(w)] = 8wL(w)c5(z - w) + 2L(w)8wc5(z - w).
Hence the conformal algebra associated to (Vect ex, { L(z)}) is Conf (Vect ex) =
e[8] L, with products:
(2.7.18)
This is called the Virasoro conformal algebra and is denoted by Vir. It has been
already encountered above in terms of the ,\-bracket: [L.xL] = (8 + 2,\)L.
One can show that this conformal algebra has a unique, up to equivalence, 2-cocycle, which is given by
(2.7.19)
The corresponding central extension is the conformal algebra Conf(Vir) associated to the Virasoro algebra (see Section 2.6). Note that both Vect ex and Vir are maximal formal distribution Lie algebras. Both have no irregular ideals.
EXAMPLE 2.7c. The obvious semidirect sum (Vect ex) + g defined by
[f(t)8t, a 18) g(t)] = a 18) f (t)8tg(t) is a maximal formal distribution Lie algebra with
no irregular ideals. One has:
[L(z), a(w)] = (8wa(w)) c5(z - w) + a(w)8wc5(z - w).

2.7. LIE SUPERALGEBRAS AND CONFORMAL SUPERALGEBRAS

49

Hence the associated to (Vect ex ) + g conformal algebra is the semidirect sum Conf(Vect ex) + Cur g, defined by (a E g):

(2.7.20)

In conclusion of this section we state without proofs the results of [DK] on classification of finite conformal algebras.
A conformal (super)algebra is called simple if it is not commutative and it contains no nontrivial ideals. The paper [DK] contains a proof of Conjecture 2. 7 stated in the first edition of this book:
Any simple finite conformal algebra is isomorphic either to a current conformal
algebra Cur g, where g is a simple finite-dimensional Lie algebra, or to the Virasoro conformal algebra.
Of course, translating this into the language of formal distribution Lie algebras, we obtain the following result: Any finite formal distribution Lie algebra which is simple (i.e. any its non-trivial ideal is irregular) is isomorphic either to (VectCx,{L(z)}) or to a quotient of (g,{a(z)la E g}) where g is a simple finitedimensional Lie algebra.
The C-span of all elements of the form a(m)b of a conformal (super)algebra R, m E Z+, is called the derived algebra of R and is denoted by R'. It is easy to see that R' is an ideal of R such that R/R' is commutative. We have the derived series R ::) R' :J R" :J .... A conformal (super)algebra is called solvable if the n-th member of this series is zero for ngO. A conformal (super)algebra is called semisimple if it contains no non-zero solvable ideals. The second main result of the paper [DK] states that any finite semisimple conformal algebra is a direct sum of conformal algebras of the following types:
(i) current conformal algebra Cur g, where g is a semisimple finite-dimensional Lie algebra,
(ii) Jlirasoro conformal algebra, (iii) the semidirect sum of (i} and (ii}.
The proof of these results uses heavily Cartan's theory of filter~d Lie algebras. As we shall see in Sections 5.9 and 5.10, the list of simple finite conformal superalgebras is much richer than that of conformal algebras.

50

2. CALCULUS OF FORMAL DISTRIBUTIONS

2.8. Conformal modules and modules over conformal superalgebras

Let g be a Lie superalgebra and let V be a g-module. We say that formal distributions a(z) E g [[z,z-1]] and v(z) E V [[z,z-1]] form a local pair if the formal distribution a(z)v(w) EV [[z,z-1,w,w-1]] is local, i.e.

(2.8.1)

(z-w)Na(z)v(w) =0 for NgO.

It follows from Corollary 2.2 that (2.8.1) is equivalent to

(2.8.2)

N-1
a(z)v(w) = L (a(w)u)v(w)) a}j><5(z - w),
j=O

where a(w)(j)v(w) EV [[w,w-1]] is defined by

(2.8.3)

a(w)(j)v(w) = Resz(z - w)ia(z)v(w).

DEFINITION 2.8a. Let (g, F) be a formal distribution Lie superalgebra and let V be a g-module spanned over <C by coefficients of a family E of formal distributions such that all pairs (a(z), v(z)), where a(z) E F and v(z) EE, are local. Then (V, E) is called a conformal module over (g, F).
The following is a representation-theoretic analogue of Dong's lemma proved in Section 3.2.
LEMMA 2.8. Let V be a module over a Lie superalgebra g, let a(z), b(z) E g [[z,z-1]] andv(z) EV [[z,z-1]]. (a) If (a(z), v(z)) is a local pair, then both pairs (8a(z), v(z)) and (a(z), 8v(z)) are local. (b) If all three pairs (a(z),b(z)), (a(z),v(z)) and (b(z),v(z)) are local, then the pairs (a(z)(j)b(z),v(z)) and (a(z),b(z)(j)v(z)) are local for eachj E Z+.
PROOF. (a) is clear. In order to prove the first part of (b) we may assume that all three pairs satisfy (2.3.2) and (2.8.1) respectively for some N E Z+ Then we

2.8. CONFORMAL MODULES AND MODULES OVER CONFORMAL SUPERALGEBRAS 51
have:

(z - w) 3N (a(z)u)b(z)) v(w)

(2~) = (z - w)N Resu ~2N

(z - u)i(u - w) 2N-i(u - z)i[a(u), b(z)]v(w).

The summation over i in the right-hand side may be replaced by that from Oto N since a(u) and b(z) are mutually local. Hence it can be written as follows:
(z - w)N Resu(u - w)NP(z,u,w)(u - z)i(a(u)b(z)v(w) - b(z)a(u)v(w))
for some polynomial P. But this is zero since both pairs (b,v) and (a,v) are local, which proves that the pair (a(j) b, v) is local.
Next, using the first part of lemma, we may find N for which all pairs (b(j)a, v) and (a,v) satisfy (2.8.1). Then we have:
a(z) (b(w)(j)v(w)) = Resu a(z)b(u)v(w)(u - w)i
= - Resu([b(u), a(z)]v(w) - b(u)a(z)v(w))(u - w)i = - Resu ( ~ (b(z)(i)a(z)) v(w)8ii)8(u - z) - b(u)a(z)v(w) }u - w)i,

hence (z - w)N a(z) (b(w)(j)v(w)) = 0.

□

Lemma 2.8 shows that the family E of a conformal module (V, E) over (9, F) can always be included in its closure, i.e. the minimal family E which is still local
with respect to F and such that C[8].E C E and a(j)E C E for all a E F and
j E Z+- The same lemma shows that Eis local with respect to P. Thus, we obtain
the following corollary.

COROLLARY 2.8. (a) If a Lie superalgebra 9 is generated (as an algebra) by coefficients of a\family of mutually local formal distributions F, then (9, F) is a formal distribution Lie superalgebra. (b) If V is a module over a formal distribution Lie superalgebra (9, F), generated (as a module} by coefficients of a family E of formal distributions- such that all pairs (a(z), v(z)), where a(z) E F, v(z) EE, are local, then (V, E) is a conformal module over (9, F).

52

2. CALCULUS OF FORMAL DISTRIBUTIONS

Note that conformal modules over a formal distribution Lie superalgebra (fl, F) form a category with morphisms r.p: (V,E) ➔ (Vi,E1) being fl-module homomorphisms r.p: V ➔ Vi such that r.p(E) C E1.
The same calculation as in the proof of Proposition 2.3 gives for all a(w), b(w) E
fl [[w,w-1]] and v(w) EV [[w,w-1]] the following relations:

(2.8.4) (2.8.5)

t (7) [a(w)(m),b(w)(n)] v(w) =

(a(w)(j)b(w)\m+n-j) v(w).

(here [,] is the bracket of operators on E.) It follows from (2.8.5) by induction on m (or from (2.8.9) below) that a(j)E EE for all a E F and j E Z+
Thus, any conformal module (V, E) over a formal distribution Lie superalgebra
(fl, F) gives rise to a conformal module M(V) = E over the conformal superalgebra
P, defined as follows.

DEFINITION 2.8b. A module Mover a conformal superalgebra Risa left Z/2Zgraded C[8]-module with Clinear maps a i--+ a1(;.) of R to EndicM for each n E Z+ such that the following properties hold for a, b ER, v EM, m, n E Z+:

(Ml) (M2)

t (7) (8a)1(;.)v = [aM,at;.)] V = -na1(;._1)V,

[a~)' bt';.)] v =

(au)b)~+n-j) v.

An R-module M is called conformal if it satisfies the property

(MO)

at';.) v = 0 for n » 0.

REMARK 2.8a. We have in an arbitrary module Mover a conformal superalge-
bra R: (8R)(o)M = 0, hence the map R(o)M ➔ M endows M with the structure of
a module over the Lie superalgebra R/8R with respect to the 0-th product (cf. Remark 2.7a) which commutes with aM. Thus, we get the R/8R-module M/8M M.

Using this remark, conversely, as in Section 2.7, we canonically associate to a conformal module M over a conformal superalgebra Ra conformal module V(M) over the formal distribution Lie superalgebra Lie R as follows. First, we construct
the affinization module M = M [t, t- 1] over the conformal superalgebra R by

2.8. CONFORMAL MODULES AND MODULES OVER CONFORMAL SUPERALGEBRAS 53

letting f)M = + aM ® 1 1 ® at and defining for a E R, v E M, !, g E C [t, C 1],
nE Z+:

(2.8.6)

L ((a! (a@f)t;,/v®g) =

(at;.+j)v) ®

1l f ) g ) .

jEZ+

Then we let V (M) = M/ fJM M with the action of Lie R induced by the 0-th action:

Letting, as before, an = a® tn and Vn = v ® tn, we obtain from (2.8.6) an explicit formula for the action of LieR on V(M) (a ER, v EM, m, n E Z):
(2.8.7)

The (Lie R, R)-module (V (M), E (M)), where
L Iv E(M) = { v(z) = VnZ-n-l EM},
n
is conformal, E(M) being canonically isomorphic to the R-module M. Proofs of these facts are similar to those in Section 2. 7. The calculations, as
before, are greatly simplified by introduction of the >.-action (a E R, v E M):
L00
arv = >,.(n)at;,)v EM[[>.]].
n=O
Then axioms (Ml) and (M2) become respectively:

(Ml)>. (M2)>.

(8a)rv = [8M, ar] = -AarV,
[ar,b:] v = [a>.b]~µv

Axiom (MO) means that arv EC[>.] ®ic M.

REMARK 2.8b. Replacingµ byµ - ).. in (M2)>., we invert (M2)>.:

(2.8.8)
\
Equivalently:

(2.8.9)

54

2. CALCULUS OF FORMAL DISTRIBUTIONS

Let R be a conformal superalgebra. We have constructed a functor from the category of conformal (Lie R)-modules to the category of conformal R-modules by
sending (V, E) to M(V) = E, and a functor in the opposite direction by send-
ing M to (V(M),E(M)). As in Section 2.7, it is easy to see that these functors induce equivalence between the category of equivalence classes of conformal (Lie R)modules and the category of conformal R-modules.
The following proposition is proved in the same way as Corollary 2.7.

PROPOSITION 2.8. Let M be a module over a conformal superalgebra R. Then (a) Any torsion element of R acts trivially on M.
(b) Any torsion element v E M is an invariant of R, i.e. Rt:,) v = 0 for all n E Z+.

An R-module M is called finite if it is finitely generated as a C[8]-module. An example of a conformal R-module, is, of course, the adjoint module R given
by a f--+ a~) = a(n). It is finite iff R is finite.
We consider now the basic examples of finite conformal modules over finite conformal algebras.

EXAMPLE 2.8a. Let g be a finite-dimensional Lie algebra and let Ube a finite-
dimensional g-module. Then U := U [t, t-1] is naturally a ii-module. Letting
I E = { u(z) := LnEZ (utn) z-n-l = u6(z - t) u E U}, we obtain a conformal mod-
ule (U, E) over the current formal distribution Lie algebra (ii, F). Indeed, using
(2.1.9), we obtain (a E g, u EU):

= a(z)u(w) (au)(w)6(z - w).

Hence the associated to U module over the current conformal algebra Cur g is M(U) = C[o] 181c U defined by (a E g, u EU):

(2.8.10)

a(o)u = au, a(j)u = 0 for j > 0.

The module M (U) is finite and conformal, and it is irreducible iff U is a nontrivial
irreducible g-module.

EXAMPLE 2.8b. Let ~ and a be complex numbers. Consider the representation of the Lie algebra Vect ex on the following space of densities:

2.8. CONFORMAL MODULES AND MODULES OVER CONFORMAL SUPERALGEBRAS 55
The action is defined as follows (f(t) E C[t,t-1], g(t) E C[t,t-1] e-°'t):
f(t)8t (g(t)(dt) 1-Ll) = (f(t)8tg(t) + (1- A)g(t)8tf(t)) (dt) 1-Ll.
Introduce the F(A, a)-valued formal distribution
I : m(z) := (tne-°''(dt) 1-Ll) z-n-l = c5(t - z)e-°''(dt) 1-Ll.
nEZ
Recalling that L(z) = -c5(t-z)8t (see Example 2.7b), and using (2.1.9), we obtain:
L(z)m(w) = ((8w + a)m(w)) c5(z - w) + 6.m(w)8wc5(z - w).
Hence (F(A, a), {m(z)}) is a conformal module over (Vect ex, {L(z)}), and the as-
sociated finite conformal module over the Virasoro conformal algebra is M(A, a) =
C[8]m defined by
(2.8.11)
It is clear from the formula:
L>.(P(8)m) = P(8 + .\)(8 +a+ 6.-\)m, P(8) E C[8],
that the module M(A, a) is irreducible if A =f. 0, and that (8 + a)M(O, a) is a nontrivial submodule of M(O, a).
EXAMPLE 2.8c. The formal distribution Lie algebra Vect ex + 9 considered in Example 2.7c acts naturally on the space F(tl.,a) ®ic U (cf. Examples 2.8a,
2.8b). This is a conformal module. The associated finite conformal module over
the correspoding conformal algebra Conf (Vect ex )+Cur g is M(A, a, U) = C[8]®U
defined by (a E 9, u E U):
(2.8.12)
This module is irreducible iff th"-9-module U is irreducible and U is non-trivial if A=O.

56

2. CALCULUS OF FORMAL DISTRIBUTIONS

2.9. Representation theory of finite conformal algebras

Let R be a conformal superalgebra and let (Lie R, R) be the associated maximal formal distribution Lie superalgebra (see Section 2.7). Recall that the Lie superalgebra LieR admits a (even) derivation T defined by (see Remark 2.7b):

(2.9.1)

T(an) = -nan-l, a ER, n E Z.

It is clear from (2.7.3) that
(LieR)- = C-span of {anla ER, n E Z+}

is a subalgebra of the Lie superalgebra Lie R. It is called the annihilation algebra. (This subalgebra will annihilate the vacuum vector in the future vertex algebra, cf. Section 4.7, hence the name.) It is clear from (2.9.1) that (LieR)_ is T-invariant,
hence we may form the semi-direct sum (Lie R)- = CT + (Lie R) _, called the
extended annihilation algebra. Comparing formulas (2.7.4), (2.9.1) and definition of LieR with the definition
of an R-module, we arrive at the following simple (but important) observation.

REMARK 2.9a. A module Mover a conformal superalgebra R is the same as a module over the extended annihilation algebra (Lie R)-. This R-module is conformal iff the following property holds:

(2.9.2)

anv = 0 for a E R, v E M, n90.

A module over (Lie R)- satisfying (2.9.2) is called a conformal (Lie R)--module. A (Lie R)- -module is called finite if it is finitely generated as a C[T]-module.

REMARK 2.9b. Let M be a module over a conformal superalgebra R and let V(M)_ be the C-span of {Vn E V(M)ln E Z+}- This is a (LieR)--submodule of the LieR-module V(M), called the annihilation submodule. It follows from definitions that the R-module Mis isomorphic to the (LieR)--module V(M)/V(M)_.

Now, choose a system of generators {a°'} of i! := (LieR)- viewed as a C[T]module. Then we may define a descending system of subspaces

(2.9.3)

2.9. REPRESENTATION THEORY OF FINITE CONFORMAL ALGEBRAS

57

by letting .Cn = C-span of {aj /a E R, j ~ n}. It clearly has the property

(2.9.4)

This leads us to the following lemma.

LEMMA 2.9. [CK2] Let£ be a Lie superalgebra with a descending system of subspaces (2.9.3) and an element T satisfying (2.9.4). Let M be an £-module and let

(a) Provided that U is a subspace of Mn such that Un Mn-I = 0 and n ~ 1, one
has: C[T]U = C[T] 181c U. In particular, dim U < oo if M is a finitely generated
C[T]-module. (b)Suppose that Mn -:f. 0 for some n E Z+ and let N denote the minimal such n. Suppose that N ~ l. Then provided that£= CT+ £ 0 , that .Co is a subalgebra of £ and [£0 , £N] C £N (so that .CoMN C MN), the irreducibility of the £-module M implies
(2.9.5)
hence the irreducibility of the .Co-module MN. Conversely, if the £ 0 -module MN is irreducible and has no non-zero vectors annihilated by £N-I, then the £-module (2. 9. 5) is irreducible.
PROOF. Let La and Ra denote the operator of left and right multiplication by
an element a of an associative algebra A. Using Ra = La - ad a and the binomial
formula, we get the following well-known formula in A:

(2.9.6)

a,g EA.

Let {vi}iEI be a IC-linearly independent set of vectors in U generating the
C[T]-module C[T]U. Suppose that °I:iPi(T)vi = 0, where Pi(T) E C[T], and not all Pi(T) = 0. Let m be the maximal degree of the Pi(T)'s. We write Pi(T) = 'I:']:0 CijTi, Cij EC, so that we have Cim -:f. 0 for some i. Using (2.9.6) and (2.9.4),

58

2. CALCULUS OF FORMAL DISTRIBUTIONS

we have, since n 2: 1:

We have therefore

~ 0 = £n+m-l ~Pi(T)vi = Cim£n-IVi = £n-l ( :~:::>imVi) •

i

i

i

Since Li CimVi f 0, we arrive at a contradiction, proving (a).
Under the assumptions of (b), if M is an irreducible £-module, then M =

C[T]MN, hence by (a), (2.9.5) holds and MN must be an irreducible £ 0-module.

Conversely, if the £ 0-module MN is irreducible, but the £-module (2.9.5) is re-

ducible, then a non-trivial quotient of the latter would contradict (a) for U

□

Now it is easy to classify all finite conformal irreducible modules over the most
important finite conformal algebras. An R-module Mis called trivial if at';,)m = 0
for all a E R, m E M, n E Z+

THEOREM 2.9. Let R be a conformal algebra of one of the three types described by Examples 2.7a-2.7c. Then a complete list of non-trivial conformal finite irreducible R-modules M is as follows. (a) If R is the current conformal algebra Cur g, where g is a finite-dimensional
semisimple Lie algebra, then M ~ M (U), where U is a non-trivial finite-dimensional
irreducible g-module (see Example 2.8a). (b) If R is the Virasoro conformal algebra, then M ~ M(-6., a) with .6. f O (see Example 2.8b). (c) If R is the semi-direct sum of the Virasoro conformal algebra and the current conformal algebra Cur g, where g is a finite-dimensional Lie algebra, then M ~ M(-6., a, U), where U is a finite-dimensional irreducible g-module which must
be non-trivial if .6. = 0 (see Example 2.8c).

PROOF. Let R = Cur g. Then we have:
£ := (LieR)- =CT+ g[t], T = -8t,

2.9. REPRESENTATION THEORY OF FINITE CONFORMAL ALGEBRAS

59

with the filtration .Cn = 9[t]tn, n E Z+. Let M be an irreducible R-module. Then,
by Remark 2.9a, it is a conformal £-module and we may apply Lemma 2.9. If N ~ 1, we have (2.9.5), where MN is an irreducible £ 0 /,CN-module. If, in addition,
M is finite module, then dime MN < oo, and we may apply a well-known result
from Lie algebra theory (see e.g. [Se]) to show that MN is an irreducible 9[t]-module
with trivial action of 9[t]t.
If N = 0, then MN is a trivial 9[t]-module, hence an £-submodule of M, hence M = MN and M is a trivial R-module. This proves (a).
Let Vir be the Virasoro conformal algebra. Then we have:

(Lie Vir)- = CT + Vect C,

where Vect C = ffinEZ+etnat and Tacts on it as - ad8t, It follows that (Lie Vir)-

is a direct sum (as ideals) of the commutative Lie algebra C(T + 8t) and the Lie

algebra ,C := Vect C. Let M be an irreducible Vir-module. By Remark 2.9a, it is

an irreducible (Lie Vir)- -module. Hence T + 8t acts as a scalar, which we denote
by a and ,C acts irreducibly on M. Define the following filtration on ,C : .Cn =

EBj~n(Cti+I 8t and apply Lemma 2.9. If N ~ 1, we argue in the same way as in the

case (a) to show (2.9.5) with MN irreducible and to show that M ~ M(D..., a) with

6.. f O if M is finite. If N = 0, then it is easy to see that M is the 1-dimensional

trivial £-module, proving (b).

The proof of (c) is similar.

□

A conformal (9, F)-module (V, E) is called finite if Eis a finitely-generated (('.[8]module and is called irreducible if it contains only irregular non-zero submodules. (As before a submodule I C V is called irregular if it does not contain all coefficients of a non-zero formal distribution from E.) A conformal (9, F)-module (V, E) is
called trivial if 9V = 0. In view of the discussion in Section 2.8, Theorem 2.9 is
equivalent to the following corollary.

COROLLARY 2.9. All non-trivial finite irreducible conformal modules over the
loop algebra ii, where 9 is a finite-dimensional semisimple Lie algebra, over the Lie
algebra Vect ex , and over their semi-direct sum are respectively; quotients of loop
modules U, where U is a non-trivial finite-dimensional irreducible 9-module; the

60

2. CALCULUS OF FORMAL DISTRIBUTIONS

density modules F(Ll,a), where Li¥ O; and the modules Oe-at(dt) 1-b.., where U
is a finite-dimensional irreducible g-module which must be non-trivial if Li = 0.

In conclusion of this section, we show how one uses Lemma 2.9 in order to prove an analogue of classical Lie theorem on representations of solvable Lie algebras.
CONFORMAL ANALOGUE OF LIE THEOREM [DK]. Let M be a finite conformal
module over a finite solvable conformal algebra R. Then there exists a common eigenvector (with eigenvalues in CJ of all the operators af;,), where a E R, n E Z+·
PROOF. We prove the theorem by induction on the lexicographically ordered pair (rank R, dim tor R) of non-negative integers.
Let S C R be the last non-zero member of the derived series of R. Then S is commutative and R(j)S CS for all j E Z+ Hence we have a representation of R/S in S. By the inductive assumption applied to the conformal algebra R/S, we may deduce that there exists a non-zero element b E S such that:

(2.9.7)

R(j)b E Cb for j E Z+.

Consider the Lie algebras of operators on M

L fl = (Lie R)1! =

<eafl)

aER,jEZ+

and

with the filtration

L b = aJM + Cbtf)
jEZ+

bn = LCbfJ)·
j"2_n

(2.9.8)
Let Mn= {v E Mlbnv = O}, and let N be the minimal n E Z+ such that Mn¥ 0.
Case 1. N = 0. Then, due to (2.9.8), Mo is a non-zero R-submodule, hence a
R/C[a]b submodule, and we may apply the inductive assumption.

2.10. ASSOCIATIVE CONFORMAL ALGEBRAS

61

Case 2. N ;:=: 1. Then, by Lemma 2.9 applied to the b-module M, dime MN< oo. By (2.9.8), 9MN C MN, and since fJ is a solvable Lie algebra, we may apply the classical Lie theorem (see e.g. [Se]) to find an eigenvector for fJ in MN. □

2.10. Associative conformal algebras and the general conformal algebra
Some of the main examples of Lie algebras are associative algebras with the Lie bracket. Here we discuss a similar construction in the "conformal" framework.
Let A be an associative algebra over <C. Two A-valued formal distributions a(z) and b(w) are called local (or form a local pair) if the formal distribution a(z)b(w) is local. Due to Corollary 2.2, we have the expansion into a finite sum for any local pair of A-valued formal distributions:

(2.10.1) where

a(z)b(w) = L (a(w) 1b(w)) afjlJ(z - w),
jEZ+

(2.10.2)

a(w)1b(w) = Resz(z - w)1a(z)b(w).

Suppose that the algebra A is spanned by coefficients of a family F of pairwise local formal distributions. Then (A, F) is called a formal distribution associative algebra.
As before, we consider the closure F of F, which is the minimal <C[a)-module
containing F and closed under all products (2.10.2). All pairs from Fare local due to an "associative" analogue of Lemma 2.8 (which is easy to prove).
As before, the properties of the products a(w)1b(w) on Fare neatly described in terms of the >.-product

a(whb(w) = L >.Ula(w)1b(w).
jEZ+
As in the Lie algebra case, this leads us to the notion of an associative conformal algebra. This is a <C[a)-module R endowed with the >.-product R@cR ➔ <C[>.] @ic R, denoted by a;,..b, satisfying the following axioms:

(Al);,..

(aa);,..b = ->.a;,..b, a;,..ab = (8 + >.)(a;,..b),

(A2);,..

62

2. CALCULUS OF FORMAL DISTRIBUTIONS

= Of course, writing a>,.b ~jEZ+ >.Jilajb, one may write equivalent axioms for the
products ajb.
As in the Lie algebra case, we have an associative conformal algebra P associ-
ated to any formal distribution associative algebra (A, F). Conversely, introducing
the affinization R of an associative conformal algebra R, we may construct the for-
mal distribution associative algebra AssR = R/8R, in the same way as we did in

Section 2.7 for Lie algebras. As in the Lie algebra case, this establishes a bijective

correspondence between associative conformal algebras Rand families of formal distribution associative algebras obtained from Ass R as quotients by irregular ideals.

Similarly, one defines conformal modules over formal distribution associative alge-

bras and establishes their correspondence to conformal modules over associative

conformal algebras as in Section 2.8. A conformal module over an associative conformal algebra Risa C[8]-module M endowed with a C-linear map R ➔ C[.X] 18lc M, denoted by a t-+ af, satisfying the properties:

(8a)f = [aM,af] = -.Xaf, a ER,
a1{ b~ = (a>.b)~µ, a, b ER.

REMARK 2.10a. Let (A,F) be a formal distribution associative algebra. Then (A0 P, F), where A0 P is the associative algebra with the opposite multiplication, is a formal distribution associative algebra as well. Translating into the language of associative conformal algebras, we see, using Proposition 2.3(b), that, given an associative conformal algebra R with .\-product a>.b, its opposite associative conformal algebra R0 P has ,\-product L>.-aa. In particular, the .\-bracket
(2.10.3)
makes Ra conformal algebra (satisfying axioms (Cl)>.-(C3)>.).
REMARK 2.10b. An associative conformal superalgebra R is simply a 'll/2'llgraded associative conformal algebra. The .\-bracket (cf. Section 2.3)

turns R into a conformal superalgebra.

2.10. ASSOCIATIVE CONFORMAL ALGEBRAS

63

An associative conformal algebra A is called commutative if

Obviously, commutative associative conformal algebras correspond to formal distribution commutative associative algebras. It would be very interesting to develop an algebraic geometry based on commutative associative conformal algebras.
Now we turn to examples.

EXAMPLE 2.10a. If A is an arbitrary associative algebra, then the corresponding current algebra A [t, t-1] is a formal distribution associative algebra with the
family F = {a(z) = Ln (atn)z-n-l / a E A} of local formal distributions. Indeed,
we have:
a(z)b(w) = (ab)(w)6(z - w). The corresponding associative conformal algebra is R = C[a] 18lic A with >.-product
a>.b = ab, a,b EA.
A much more interesting example is the following.
EXAMPLE 2.10b. Let DifH.::x be the associative algebra of regular differential operators on ex. It has a basis t18'f', j E Z, m E Z+· Introduce the formal distributions (m E Z+):
Jm(z) = L)1(-8t)mz-j-l = 6(t- z)(-8t)m.
jEZ
Using (2.1.10) (for n = 0), we obtain:

Jm(z)r(w) =ft(~) ({)al-iJm+n-i(w)8~6(z-w). 1=0i=O J
It follows that the family F = {Jm(z)/m E Z+} consists of pairwise local formal

distributions with products:

(2.10.4)

ti{;) Jm(w)ir(w) =

({)al-iJm+n-j(w).

Hence (Diff ex , F) is a formal distribution associative algebra. The corresponding associative conformal algebra is

64

2. CALCULUS OF FORMAL DISTRIBUTIONS

with the .>.-product, derived from (2.10.4) being as follows (m, n E Z+):

(2.10.5)

f (n:) Jm).r =

(>- + a)i Jm+n-j.

j=O J

Consider the obvious representation of the algebra Diff ex on the space e [t, t-1].
Letting v(z) = EnEZtnz-n-l = 8(z - t), we obtain, using (2.1.10):

(2.10.6)

L m
Jm(z)v(w) = m(m - 1) ... (m - j + l)a:;:-iv(w)a}jl8(z - w).
j=O

Hence (e [t, c 1] , {v(w)}) is a conformal module over (Diff ex, F). The associated

conformal module over Conf(Diff ex, F) is C[8]v with the >.-action obtained from

(2.10.6) to be given by

(2.10.7)

(A simpler way to derive formulas (2.10.5) and (2.10.7) is to use Lemma 2.2.)

A matrix generalization of this example is also important.

EXAMPLE 2.10c. The associative algebra

of all N x N matrix valued regular differential operators on ex is a formal distribution associative algebra with the family of pairwise local formal distributions

The associated associative conformal algebra is

with .>.-products (2.10.8)

:E (n:) JA).JB =

(>- + a)i J";,:n-j.

j=O J

The obvious representation of DiffNex on the space e[t,t-1] 0 eN is an irre-
ducible conformal module with the family E = {Va (w) = v(w) 0 ala E eN}. The

associated (conformal) module over Conf(DiffN ex, F) is C[8] 0ic eN with the >.-

action

(2.10.9)

2.10. ASSOCIATIVE CONFORMAL ALGEBRAS

65

A more conceptual understanding of Example 2.10c is given by Proposition 2.10 below.

DEFINITION 2.10. Let U and V be two C[a]-modules. A conformal linear map from U to V is a C-linear map a : U-+ C[>.] 0c V, denoted by a>. : U-+ V, such that

(This equation means: av a,>. - a,>.au = ->.a,>..) Denote the vector space (over q
of all such maps by Chom(U, V). It has a canonical structure of a C[a]-module:

REMARK 2.10c. Let U and V be modules over a conformal algebra R. Then the C[a]-module H := Chom(U, V) carries an R-module structure defined by (a E R, cp E H, u E U):
Hence one may define the contragredient conformal R-module U* = Chom(U, q,
where (('. is the trivial R-module and C[a]-module, and the tensor product of R-
modules: U 0 V = Chom(U*, V). It is easy to see that the R-module Chom(U, V)
is conformal iff both U and V are finite conformal R-modules.
In the special case U = V we let Cend V = Chom(V, V). Provided that V is a finite C[a]-module, the C[a]-module Cend V has a canonical structure of an associative conformal algebra defined for a, b E Cend V by
(2.10.10)
Indeed, axiom (Al)>. is immediate, while axiom (A2)>. is obtained from (2.10.10) by replacingµ byµ+>.. Finally, it is easy to show that a,>.b depends polynomially on >. using that V is a finite C[a]-module.
REMARK 2.10d. By the very definition, a structure of a conformal module over an associative conformal algebra R in a finite C[a]-module V. is the same as a homomorphism of R to the associative conformal algebra Cend V.

66

2. CALCULUS OF FORMAL DISTRIBUTIONS

The >.-bracket (2.10.3) on Cend V, where V is a finite C[8]-module, makes it a

conformal algebra, which we denote by gc V and call the general conformal algebra.

The second term of the bracket (2.10.3) can be simplified:

L ( - (b->.-aa): V = -

(->. - a)(n) (bna)): V = - = (bµ->.a): V -br-A (arV) .

n2::0

Thus the >.-bracket of gc V looks as follows:

(2.10.11)

REMARK 2.lOe. Formula (2.10.11) shows that a structure of a conformal module over a conformal algebra Rina finite C[8]-module V is the same as a homomorphism of R to the conformal algebra gc V.

For a positive integer N we let CendN = Cend<C[8]N, gcN = gc<C[8]N (where C[8]N is the free C[8]-module of rank N). Recall that we have a representation of the associative conformal algebra Conf(Diff ex) in qaJN defined by (2.10.9). By Remark 2.10d this gives us a homomorphism r.p : Conf (DiffN ex) ➔ CendN of associative conformal algebras. Likewise, by Remark 2.lOe we get a confomal algebra homomorphism 'P- : Conf (DiffN ex, F) ➔ gcN, where DiffN ex stands for DiffN ex with the usual Lie bracket.

PROPOSITION 2.10. [DK] The homomorphisms r.p and 'P- are isomorphisms.

PROOF. We have by (2.10.9):

This formula shows that r.p and 'P- are injective. The same formula shows that

r.p and 'P- are surjective since a conformal linear map is determined by its values

on a set of the generators of a <C[8]-module, but the polynomials >.k (>. + a)mv

(k, m E Z+, v E CN) span over (C the space <C[>., 8] l8l (CN.

□

REMARK 2.lOf. The associative conformal algebra CendN and the general conformal algebra gcN are interesting examples of simple algebras which are not finite (but have finite Gelfand-Kirillov dimension). It is an interesting open problem to classify such algebras. A related open problem is to classify infinite subalgebras of CendN and gcN which act irreducibly on <C[8]N. (For a classification of such finite algebras see [DK].)

2.11. COHOMOLOGY OF CONFORMAL ALGEBRAS

67

2.11. Cohomology of conformal algebras

This section is an exposition of some of the results of the paper [BKV]. (A generalization to the super case is straightforward by making use of the usual sign rule.)

DEFINITION 2.lla. An n-cochain (n E Z+) of a conformal algebra R with coefficients in an R-module over it is a C-linear map

where M[A1, ... , An] denotes the space of polynomials with coefficients in M, satisfying the following conditions:

(2.11.1) 'Y>- 1 , ...,.x,. (a1, ... , aai, ... , an)= -Ai'Y>-1 , ... ,.x,. (a1, ... , ai, ... , an),

(2.11.2)

'Y is skew-symmetric with respect to simultaneous permutations of ai 's and Ai 's.

We let R®0 = C, so that a 0-cochain 'Y is an element of Mand (d'Y).x(a) = a.x'YSometimes, when the module M is not conformal, one may consider formal power series instead of polynomials in this definition.
We define a differential d of an n-cochain 'Y as follows:

n+l

=

"~ (-l

)

i

+

i

'

Y

, '·,
Al,•••,Ai,•••,An+l

(

a1

,

•

•

·,

a

i

,

•·

·,

a

n

+

l

)

i=l

n+l

+ "L.(.J-

l

)

i

+

i

'

Y,. A1,

+'·, A3 ,A

l

,

•

•

•

,

'·
Ai,

••

•,

'
A

·
3

,

•

•

,,

A,

n

+

l

([ai_x.aj],a1, t

...

,ai,···,aj,•··,an+l),

i,j=l
i<j

where 'Y is extended linearly over the polynomials in Ai-

REMARK 2.lla. Property (2.11.1) implies the following relation for an n-cochain '}':

LEMMA 2.11. (b) d2 =0.

(a) The operator d preserves the space of cochains.

68

2. CALCULUS OF FORMAL DISTRIBUTIONS

PROOF. (a) Property (2.11.1) obviously holds for d'Y if it holds for 'Y· The only non-trivial point in checking (2.11.2) of d'Y amounts to the equation

which follows from Remark 2.lla and the skew-symmetry (C2)>. of [a>.b]. (b) We have for an n-cochain 'Y

n+2

",L.(...i-

l

)

i

+

1

a

i

>.· t

(d'Y), '· , Al,, .. ,At,•••,An+2

(a1,

..

·, ai,

•

..

,

an+2)

i=l

i,j=l i<j

L ( n+2

-

l)

i+j+sign{j,i}

a

i

>.

•
t

{ a

1

,
A

.
3

'Y, A

l

, . .
,,,,,A1,,3 , ..

,
,,An+2

( a1,

...

,

~ai '1·,

...

,

an+2

))

i,j=l

if.j

+
i,j,k=l j<k,i,f-j,k

([a1>.. ak], a1, ... , lli,j,k, ••. , an+2)

'

+

'°'n+2

~

(-l)i+j+k+sign{k,i,j}a 'Y

_

k).k >.;+>.;,>-1, .. ,,>.i,j,k, ...,>.,,+2

i,j,k=l

i<j,k,f-i,j

([ai>.i a1], a1, ... , lli,j,k, . , . , an+2)

n+2

+ ",L(.-..lt)

i

+

i

[

a

i

>

-

•

a

1

·
t

l

>

-

i

+

>

-

·' J

Y

,
A

l

,

.

.

,

,

'
A

·
1,

.
13

,

•

•

•

,

A,

n

+

2

(a1,

..

-,ai1' ',

•..

,an+2)

i,j=l

i<j

+ '°'n+2

(-l)i+i+k+l+sign{i,#k,!},v

_

~

1Ak+>-1 ,>.;+>.; ,Al ,.. ,,Ai,j,k,l ,.. ,,A,,+2

i,j,k,l=l

i<#k<l

i,j,k=l i<j,k,f-i,j
where sign{i1 , ... ,ip} is the sign of the permutation putting the indices in the increasing order and ai,j, ... means that ai, a1, ... are omitted. Notice that each term

2.11. COHOMOLOGY OF CONFORMAL ALGEBRAS

69

i(n ithe _su:m~)tion over i, j, k, l is skew-symmetric with respect to the permutation

J

. Therefore, the terms of that summation will cancel pairwise. The

k l i j

first and the forth summations cancel each other, because M is a conformal algebra

module:

The second summation becomes equal to the third one after the substitution (ikj), except they differ by a sign. Thus, they cancel each other, as well. Finally, the
sixth summation can be rewritten as summation over i < j < k of the sum of three
permutations of the initial summand. Precisely, in the first entry of 'Y, we will have

Using Remark 2.lla, we can transform this sum inside 'Y into

which vanishes by the Jacobi identity (C3)>. and skew-symmetry (C2)>. in R. Thus,

we see that all of the terms in ,12,y cancel.

□

Thus the cochains of a conformal algebra R with coefficients in an R-module

M form a complex:

EB C(R,M)= cn(R,M),
nEZ+

where {Jn(R, M) denotes the space of all n-cochains. This complex is called the

basic complex for the R-module M. This is not yet the complex defining the right cohomology of a conformal algebra: we need to consider a certain quotient complex.
Define the structure of a (('.[8]-module on C(R, M) by letting

(2.11.3)

n
(o"()>.1,... ,>.,,(a1, ... ,an) = (oM + I:Ai)"/>-1,... ,>.,,(a1, ... ,an),
i=l

where aM denotes the action of a on M.

REMARK 2.llb. do = 8d, and therefore the graded subspace 8C(R, M) of

C(R, M) is a subcomplex. Indeed, the first summation in the differential transforms

r:;= r:;,;;/ the factor aM + 1 Ai into aM +

Ai, because of the properties (Cl)>. and

70

2. CALCULUS OF FORMAL DISTRIBUTIONS

(Cl')>. of the >.-bracket. The second summation does the same for more obvious reasons.

Define the quotient complex

EB C(R,M) = C(R,M)/fJC(R,M) =

cn(R,M),

called the reduced complex.

= DEFINITION 2.llb. The basic cohomology H(R, M) EBnEZ+ fin(R, M) is the = cohomology of the basic complex C(R, M). The reduced cohomology H(R, M)
EBnEZ+ Hn(R, M) of a conformal algebra R with coefficients in a module M is the
cohomology of the reduced complex C(R, M).

REMARK 2.llc. The exact sequence O➔ fJC(R, M) ➔ C(R, M) ➔ C(R, M) ➔

0 gives the long exact sequence of cohomology:

(2.11.4)

ii 0 ➔ H 0 (8C(R,M)) ➔ 0 (R,M) ➔ H 0 (R,M) ➔

➔ H 1(8C(R,M)) ➔ H 1(R,M) ➔ H 1(R,M) ➔

➔ H 2 (8C(R, M)) ➔ H 2 (R, M) ➔ H 2(R, M) ➔ •••

This cohomology theory describes extensions and deformations, just as any cohomology theory.

PROPOSITION 2.11. (a) ii0 (R, M) = {m EM I a>-.m = 0 for all a ER}.
(b) The isomorphism classes of extensions
O➔M➔E➔C➔O
of the trivial R-module C (8 and R act by zero) by a conformal R-module M correspond bijectively to H 0 (R, M).
(c) The isomorphism classes of C[fJ]-split extensions
O➔ M ➔ E➔ N➔O
of conformal modules over a conformal algebra R correspond bijectively to
H 1 (R, Chom(N, M)),
where M and N are assumed to be finite. If, in particular, N = C is the trivial
module, then there exist no non-trivial C[fJ]-split extensions.

2.11. COHOMOLOGY OF CONFORMAL ALGEBRAS
(d) Let C be a conformal R-module, considered as a conformal algebra with respect to the zero >.-bracket. Then the equivalence classes of C[a]-split "abelian" extensions

0--tC-+R-tR-+0

of the conformal algebra R correspond bijectively to H 2 (R, C). (e) The equivalence classes of first-order deformations of a conformal algebra
R correspond bijectively to H 2 (R, R).

PROOF. (a) The computation of ii0 (R, M) follows directly from the definition:
form EM= C0 (R,M) and a ER, (dm).x(a) = a_xm.
(b) Given an extension 0 -+ M -+ E -+ <C ➔ 0 of modules over a conformal algebra R, pick a splitting of the short exact sequence over <C, i.e., assume that as
a complex vector space, E ~ M EB <C = {(m, n) Im EM, n E <C}. Define f EM by
writing down the action of 8 on the pair (m, 1) E E:

(2.11.5)

a(m, 1) = (am+ f, 0).

We claim that f E M = 6°(R, M) defines a 0-cocycle in the reduced complex C(R, M) and thereby a class in H 0 (R, M).
To see that, define a 1-cochain 'YE C1 (R,M) using the action'of Ron E:

(2.11.6)

a_x(m, 1) = (a.xm + 'Y.x(a), 0)

for a E R. The property (2.11.1) of "(: 'Y.x (aa) = ->-.'Y.x (a), follows from the fact that (aa).x(m, 1) = ->-.(a.x(m, 1)). The property a_x(a(m, 1)) = (>-.+ a)(a.x(m, 1)) of
the action of R on E expands as

(2.11.7)
which means that df = 0 in the reduced complex.
If we choose another splitting (m,n)' of the extension E, it will differ by an element g E M:

(m, 1)' = (m + g, 1),
so that the new 0-cocycle becomes f' = f + ag, therefore defining the same cochain
in the reduced complex.

72

2. CALCULUS OF FORMAL DISTRIBUTIONS

If we have two isomorphic extensions and choose a compatible splitting over C, we will get the same 0-cocycles corresponding to them. This proves that isomorphism classes of extensions give rise to elements of H 0 (R, M).
Conversely, given a cocycle in C0 (R, M), we can choose a representative f EM
of it to alter the natural C[8]-module structure on M EB C by adding f to the action of 8 on M EB C as in (2.11.5). This will obviously extend to an action of C[8]. We can also alter the natural R-module structure by adding 'Y to the action of a E R as in (2.11.6), where 'Y is a solution of equation (2.11.7), which means that f is a cocycle in the reduced complex. This action satisfies (Ml)>. because of (2.11.7)
and the property (2.11.1) of 'Y, and it satisfies (M2)>. because d'Y = 0, which follows
from (2.11.7) and the fact that C[8] acts freely on basic 2-cochains. By construction the natural map M -+ M EB C and M EB C -+ C will be
morphisms of C[8]- and R-modules. This construction of a new conformal module structure on M EB C involved a
number of choices. The choice of a different representative J1 = f + 8g defines
an isomorphism of the two C[8]-module structures on M EB C, which automatically becomes an isomorphism of the corresponding R-module structures, because the corresponding 'Y's are unique. The 1-cochain 'Y is uniquely determined by f {because
C[8] acts freely on the space C1 (R, M) of basic 1-cochains.
(c) We will adjust the proof of (b) to the new situation. Given a C[8]-split extension O -+ M -+ E -+ N -+ 0 of modules over a conformal algebra R, pick a splitting of the short exact sequence over C[8], i.e., assume that as a C[8]-module,
E '.::'. M EB N = {(m,n) I m E M,n E N}. We are going to construct a re-
duced 1-cochain with coefficients in Chom(N, M) out of this data. Note that such
cochains are linear maps 'Y = 'Y.x(a)µ from R 18) N to M depending on two vari-
ables ,X and µ, considered modulo ,X - µ. Note that 'Y>- (a)µ mod (,X - µ) is fully
determined by the restriction 'Y.x(a).x to the diagonal ,X = µ. Define a 1-cochain
'YE C 1 (R, Chom(N, M)) using the action of Ron E:

(2.11.8)

for a ER. The property (2.11.1) of 'Y: 'Y.x(8a)>. = -A'Y.x(a).x, follows from the fact
that (8a).x(m,n) = -.Xa.x(m,n). The property a.x(8(m,n)) = (,X + 8)(a.x(m,n)) of

2.11. COHOMOLOGY OF CONFORMAL ALGEBRAS

73

the action of R on E expands as

(2.11.9)

which means that 'Y>.(a)>. is a conformal linear map N ➔ M. Finally, the module property (M2)>. for elements in E implies that d'Y = 0.
If we choose another C[8]-splitting (m, n)1 of the extension E, it will differ by an element /3 E HomqaJ (N, M):

(m, n)1 = (m + /3(n), n).

HomqaJ (N, M) may be identified with the degree zero part ofChom(N, M), so that
the new 1-cocycle becomes ")'1 = 'Y + d/3, therefore defining the same cohomology
class. If we have two isomorphic extensions and choose a compatible splitting over
q 8], we will have exactly the same 1-cocycles 'Y corresponding to them. This proves
that isomorphism classes of extensions give rise to elements of H 1 (R, Chom(N, M)). Conversely, given a cohomology class in H 1(R, Chom(N, M)), we can choose a
representative 'Y E C1 (R, Chom(N, M)) of it to alter the natural R-module structure on M ffiN by adding 'Y to the action of Ron M ffiN as in (2.11.8). This action will satisfy (Ml)>. because of (2.11.9) and (2.11.1). This action will define an R-
module structure on M ffi N, because d'Y = 0 after the restriction to µ = >.1 + >.2 in C2 (R, Chom(N, M)).
By construction the natural mappings M ➔ M ffi N and M ffi N ➔ N will be morphisms of C[8]- and R-modules.
This construction of a new conformal module structure on M ffi N is indepen-
dent of the choice of a different representative ")'1 = 'Y + d/3, because it defines an
isomorphic structure of an R-module on M ffi N. Finally, if N = C, then Chom(C, M) = 0, and therefore, there are no split
extensions. (d) Given a C[8]-split extension of a conformal algebra R by a module C, choose
a splitting R= C ffi R. Then the bracket in R .

for a, b ER

74

2. CALCULUS OF FORMAL DISTRIBUTIONS

defines a map c: R®R ➔ C[>.], satisfying (Cl)>. and (Cl')>. which we may combine with the natural mapping

to get the composite mapping, denoted C>.ih· It defines a 2-cochain, because it is
obviously skew-symmetric and (c>.(8a, b), a,>.b) = [(O, 8a)>.(0, b)] = [8(0, a)>.(0, b)] =
-[>.(O,a)>.(0,b)] = ->.(c>.(a,b),a>.b), which implies C>.1 h(8a,b) = -A1C>.1 ,>.2 (a,b),
= and similarly, C>.1 h(a,8b) -A2C>.1,>.2 (a,b) mod (8 + >-1 + >-2). In fact, this 2-
cochain c is a cocycle:

This is just because the Jacobi identity (C3)>., is satisfied in R. The construction of c assumed the choice of a splitting R= C E9 R. A different
splitting would differ by a mapping f: R ➔ C, which can be thought of as f: R ➔ C[.>.]/(8 + >.), which would contribute by df to c.
Thus, any extension determines a cohomology class in H 2 (R, C). The above arguments can be reversed to show that a class in the cohomol~ group defines an extension.
(e) Let D = IC[t]/(t2) be the algebra of dual numbers. Then a first-order
deformation of a conformal algebra R is the structure of a conformal algebra over D on R ® D, so that the mapping R ® D ➔ R, a® p(t) f--+ p(O)a, is a morphism of conformal algebras. This means that classes of first-order deformations are in bijection with classes of <C[8]-split abelian extensions of R with the R-module R in the sense of part (d) of this theorem. Therefore, they are classified by H 2(R, R). □
Now I shall explain how the basic and reduced cohomology of a conformal algebra R with coefficients in a conformal R-module M is related to Lie algebra cohomology. Recall that M is canonically a module over the annihilation Lie algebra g_ = (LieR)_ (see Remark 2.9a). Let C(g-,M) = EBnEZ+ cn(g-,M) be the Chevalley-Eilenberg complex defining the cohomology of g_ with coefficients in M. Recall that, by definition (see e.g. [Fl), cn(g_, M) is the space of skew-symmetric

2.11. COHOMOLOGY OF CONFORMAL ALGEBRAS

75

linear maps 'Y: (9- )®n ➔ M such that

for all but a finite number of m1, ... , mn E Z+, where a1, ... , an E R, and aim, E
9- = (LieR)- = R[t]/(8 + 8t)R[t] is the image of the element aitm•. Note that
C(9-,M) has the following structure of a C[8]-module:

(2.11.10)

(8'Y)(a1 ®···®an)
L n
= 8('Y(a1 ®···®an)) - 'Y(a1 ® ••• ® 8ai ®···®an).
i=l

THEOREM 2.11. Let R be a conformal algebra, let 9- denote its annihilation algebra and let M be a conformal R-module. Then

(a) There is a canonical isomorphism of complexes C(R, M) and C(9-, M), compatible with the action of C[8]. Consequently, H(R, M) ~ H(9-, M) and the complex C(R, M) is isomorphic to C(9-, M)/8C(9_, M).
(b) Provided that Mis a free C[8]-module, the complex C(R, M) is isomorphic to the subcomplex C(9-, V(M)-) 8 of 8-invariant cochains in C(9-, V(M)-).

PROOF. (a) For a cochain 'Y E cn(R, M), we write

In terms of the linear maps .
'Y(m1,...,mn)·• R®n ➔ M '
the definition of C(R, M) translates as follows. (i) for any a1, ... ,an ER, 'Y(m1,...,mn)(a1, ... ,an) is non-zero for only a finite number ofm1,• .. ,mn,
(il) 'Y(m1,,..,m;,...,mn) (a1,••·, 8ai,·••,an )
= -mi'Y(m1,...,m;-1,...,mn) (a1, · · •, ai, ··•,an),
(iii) 'Y is skew-symmetric with respect to simultaneous permutations of Oi 'sand mi's.

76

2. CALCULUS OF FORMAL DISTRIBUTIONS

The differential is given by:

n+l
= ,m :~:)-1/+lai(m;)'Y(m1,... 1,...,mn+l) (a1, • • ·, Ui, •· ·, Un+I)
i=l

Define linear maps </P: cn(R,M) ➔ cn(g_,M) by the formula
They are well-defined due to above condition (ii). Clearly, </P are bijective and, using (2.7.3), it is easy to see that ¢n+1 o d =do </Jn. Moreover, </Jn o a= a o </Jn,
where a acts on C(R, M) via (2.11.3) and on C(g_, M) via (2.11.10). (b) Now we assume that Mis a free C[a]-module: M = qa] ©c U for some
vector space U. Then the g_-module v_ = V(M)_ is just U[t] with
for u E U, a E R, see Section 2.9. In terms of the usual generating series a,x =
Em2::o >,(m)am, this can be rewritten as
Recall also that V_ is an R-module where C(a] acts by a= -at.
Let /3 E cn(g_, V_). As in the proof of (a), consider the generating series
= I: By (2.11.10), a acts on /3.xl,···,An;t as -at+ Ai. Hence /3 is a-invariant iff
(2.11.12)

2.11. COHOMOLOGY OF CONFORMAL ALGEBRAS

77

= where 'Y>-1 ,...,>-n /h,, ...,>-n;tlt=O takes values in U. Identifying U with 1 ©UC M,
we can consider 'Y as an element of cn(R, M). It is easy to check that /3 t-t 'f := 'Y
mod (8 + E Ai) is a chain map from C(g_, V_) to C(R, M).

Conversely, for 'f E cn(R, M) choose a representative 'Y E cJn(R, M) such
that 'f = 'Y mod (8 + E Ai)- Define f3 E cn(g_, V_) 8 by (2.11.11) and (2.11.12) = with 8 substituted by -8t in 'Y>.1 ,...,>-n (a1, ... , an) E M U[8]. Then clearly, /3 is

independent of the choice of 'Y·
The correspondence /3 ++ "i establishes an isomorphism between C(g_, V_ )8

and C(R,M).

D

EXAMPLE 2.11. Here we will compute the cohomology of the conformal algebra
Vir = C[8]L, [L.xL] = (8 + 2A)L with trivial coefficients M = C, where both 8 and

L act by zero. The answer is as follows:

dim

-
Hq

(Vir,

C)

= {

1

if q = 0 or 3,

0 otherwise,

~ dimH'(Vi,,C) = { if q = 0, 2, or 3, otherwise.

An n-cochain 'Yin this case is determined by its value on L®n:

Obviously, P(A1 , ... , An) is a skew-symmetric polynomial with values in C. The differential is then determined by the following formula:
Ln+l
(dP)(A1, ... , An+1) = (-l)i+j(Ai-Aj)P(Ai+Aj, A1, ... , Xi, ... , xj, ... , An+i)-
ii,<j=jl
This describes the complex C(Vir, C). The complex C(Vir, C) producing cohomology of Vir in degree n is nothing but the quotient of cn(Vir, C) by the ideal
generated by E7=1 Ai- In other words, cn(Vir, q is the space of regular (polynomial) functions on the hyperplane E7=1 Ai = 0 in en which are skew-symmetric
in the variables A1, ... , An· (This complex appeared as an intermediate step in [GFl] of the calculation of cohomology of the Virasoro algebra, and its cohomology was computed there.) Consider the following homotopy operator k: Cq(Vir, <C) -+ cq-l(Vir, <C):

78

2. CALCULUS OF FORMAL DISTRIBUTIONS

A straightforward computation shows that (dk + kd)P = (degP - q)P for P E
Cq(Vir,IC), where degP is the total degree of Pin A1,, .. ,Aq- Thus, only those

homogeneous cochains whose degree as a polynomial is equal to its degree as a
cochain contribute to the cohomology of C(Vir, IC). These polynomials must be
skew-symmetric and therefore divisible by Aq = IL<i(Ai - Aj), whose polynomial
degree is q(q - 1)/2. The quadratic inequality q(q - 1)/2 ~ q has q = 0, 1, 2,
and 3 as the only integral solutions. For q = 0, the whole 8° = C contributes
to fI0 . For q = 1, the only polynomial of degree 1 up to a constant factor is A1.
Next, d>.1 = A~ - >.~, which is the only skew-symmetric polynomial of degree 2 in
two variables. This shows that H1 = H2 = 0. Finally, for q = 3, the only skew-
symmetric polynomial of degree 3 in 3 variables is A3. It is easy to see that this polynomial represents a non-trivial class in the cohomology. Indeed, it is closed,
because a skew-symmetric function in four variables has a degree at least 6, which
is greater than deg(dA3) = 4, and A3 is not a coboundary, because it can be the
coboundary of a two-cochain of degree 2, which must be a constant multiple of
= A~ - >.~ d>.1 , whose coboundary is zero.
The computation of the cohomology of the quotient complex C (Vir, IC) is based
= on the long exact sequence (2.11.4). By definition, 8C0 0. To find the cohomology of 8C(Vir,IC), define a homotopy k1: 8Cq-+ acq-I as k1(8P) = 8k(P), where
8 = I:i Ai and PE Cq. Then (dk1 + k1d)8P = (degP- q)8P. As in the previous
paragraph, this implies that deg P = q = 0, 1, 2, or 3. Up to constant factors,
the only polynomials in 8C with this property are P1 = A~ for q = 1, P2 = (A1 + A2)(A~ - A~) for q = 2, and P3 = (A1 + A2 + A3)A3 for q = 3. One computes: dPi = -P2 and dP3 = 0. Therefore Hq(8C) = 0 for all q but q = 3, where it is
1-dimensional with the generator P3. From the long exact sequence (2.11.4) we see
that H 0 (Vir,IC) = C and Hq(Vir,IC) = 0 for q = 1,4,5,6, ... ; H 3(Vir,IC) = CA3 and H 2(Vir, IC) = C(>.f - >.~), because d(Af - >.~) = P3.
In a similar fashion one can show that if 8 acts on C non-trivially, then
flq(Vir, IC), remains the same, but Hq(Vir, IC) becomes Ofor all q.

COROLLARY 2.lla. The conformal algebra Vir has a non-trivial central exten-
sion by C iff 8C = 0; it is unique and is given by the 2-cocycle (2. 7.19).

2.11. COHOMOLOGY OF CONFORMAL ALGEBRAS

79

For the calculation of basic and reduced cohomology of Cur g with coefficients in

C as well as of Vir and Cur g with coefficients in M(A, a) and M(U) respectively

the reader is refered to [BKV]. One of the open problems is the calculation of

cohomology of the conformal algebra gcN. In order to demonstrate how beautiful

the results are, let me state, in conclusion of this section, the answer for the Vir-

modules M(A, a):

(a) H(Vir, M(~,a)) = 0 if a =I- 0 or if a= 0 and A =/- 1 - (3r2 + r)/2 for any

r E Z.

2 for q = r + 1,

= = (b) dimHq(Vir, M(l-(3r2±r)/2,o)) { 1 for q r or r + 2,

0 otherwise. Proof uses results of [FeF], [F] on cohomology of the Lie algebra of vector fields

on C vanishing at O (see [BKV], Theorem 7.2 for details).

This theorem, along with Proposition 2. lld, implies the following corollary:

COROLLARY 2.llb. There exists a non-trivial abelian extension O ➔ M(A, a) ➔ R ➔ Vir ➔ 0 iff a= 0 and A= 1, 0, -1, -4 or -6.

This corollary (obtained earlier by M. Wakimoto and myself by a direct, but very lengthy, calculation) shows that a Levi splitting theorem does not hold in general. It is closely related to the calculations of [R2].

CHAPTER 3
Local fields
3.1. Normally ordered product
Fix a vector superspace V = Va+ Vi (the space of states). Recall that a formal
distribution
L a(z) = ll(n)Z-n-1
nEZ
with values in the ring EndV (i.e., a(n) E EndV) is called a field if for any v E V one has:

This means that a(z)v is a formal Laurent series in z (i.e., a(z)v E V[[z]][z-1]).

(3.1.1)

00
= [a(z), b(w)] Ld(w)8!j)c5(z -w) + b(z,w)+(z)
j=O

all the coefficients d (w) are fields provided that b(w) is a field, due to formula

(2.2.2):

(3.1.2)

d(w) = Resz[a(z), b(w)](z -w)i.

The normally ordered product of two fields a(z) and b(z) is defined by

(3.1.3)

= : a(z)b(z): a(z)+b(z) + (-l)p(a)p(b)b(z)a(z)_.

Since

(3.1.4)

L L -oo

00

: a(z)b(z) :(n)=

a(j)b(n-j-1) + (-l)p(a)p(b) b(n-j-l)ll(j)

j=-1

j=O

we see that when applied to v E V each of the two sums gives only a finite number

of non-zero summands, hence : a(z )b(z) : is a well defined formal distribution. Here

we use that both a(z) and b(z) are fields; for general formal distribution one is able

to define only the normally ordered product (2.3.5) in two indeterminates.

81

82

3. LOCAL FIELDS

Moreover, it is clear from (3.1.3) that : a(z)b(z): is a field, since given v EV, b(z)v (resp. a(z)_v) is a formal Laurent series (resp. a Laurent polynomial) in z, hence a(z)+b(z)v (resp. b(z)a(z)_v) is a formal Laurent series in z.
Thus, the space of fields forms an algebra with respect to the normally ordered product (which is in general neither commutative nor associative).
Incidentally, it is straightforward to check that : a(z)b(z) : -p(a, b) : b(z)a(z) : is a Lie superalgebra bracket (in spite of the non-associativity of the normally ordered product) .1
The derivative 8a(z) of a field a(z) is again a field and, thanks to (2.3.4), 8 is a derivation of the normally ordered product:

(3.1.5)

8: a(z)b(z) :=: 8a(z)b(z) : +: a(z)8b(z) : .

Due to the existence of the normally ordered product, one can define then-th product between fields not only for n positive (see (2.3.8)), but also for n negative:

(3.1.6)

a(z)(-n-l)b(z) =: a(nla(z)b(z) :,

It is tempting now, using these products and Taylor's formula (2.4.3), to rewrite

the OPE (2.3.9a) of mutually local fields a(z) and b(z) in a "complete" form:

(3.1. 7)

L a(z)b(w) =

a(w)u)b(w) (z-wJ) ·+1 •

jEZ

However, (3.1.7) makes no sense as an equality of formal distributions since different parts of it are expanded in different domains. (In the "graded" case one can give a meaning to (3.1.7) using analytic continuation.) Still, formula (3.1.7) can be used, up to an arbitrary order of z - w.
In order to state the result we need the notion of a field in z and w. This is a formal EndV-valued distribution a(z,w) such that a(z,w)v E V[[z,w]][z-1 ,w-1].
For example, : a(z)b(w) : is a field if a(z) and b(w) are fields. Note that a partial derivative of a field is a field and that a(w,w) is a well defined field in the indeterminate w. The following is yet another version of Taylor's formula.

1This was pointed out to me by A. Radul. •

3.1. NORMALLY ORDERED PRODUCT

83

PROPOSITION 3.1. For any field a(z,w) and any positive integer N there exist fields d (w) (0::; j ::; N - 1) and a field dN (z, w) such that

(3.1.8)

N-1
a(z,w) = L d(w)(z - w)i + (z - w)N dN (z,w).
j=O

The coefficients d (w) are uniquely determined by this expansion and are given by

the usual formula:

(3.1.9)

= d(• w) 8(/') a(z, w) Jz=w.

PROOF. The uniqueness of the d (w) is proved in the usual way: differentiate j
times (3.1.8) by z and let z = w. It suffices to prove existence of (3.1.8) for N = 1:

(3.1.10)

= a(z,w) - a(w,w) (z - w)d(z,w) for some field d(z,w),

= since applying it again to d(z, w) gives (3.1.8) for N 2, etc. The proof of (3.1.10)

is straightforward.

□

THEOREM 3.1. Let a(z) and b(z) be mutually local fields and let N be a positive

integer. Then there exists a field dN (z, w) such that in the domain JzJ > JwJ one has:

(3.1.11)

_ '°' a(z)b(w) -

L.,

a(w)u)b(w) ( _ )i+l

+ (z -

N N w) d (z,w).

j'?:_-N Z W

The coefficients of (z-w)-i-l (j ~ -NJ in this expansion are uniquely determined.

PROOF. In view of (2.3.9a) and (3.1.5), the theorem is a consequence of

Proposition 3.1 applied to the field : a(z)b(w) :.

□

Proposition 3.1 and Theorem 3.1 show that when calculating the OPE of local fields one can use Taylor's expansions up to the required order.
The following lemma will be used in the sequel.
= = LEMMA 3.1. Let a(z) Lna(n)Z-n-l and b(z) Lnb(n)Z-n-l be EndV-
valued fields and let JO) E V be a vector such that
a(n) JO) = 0 and b(n) [O) = 0 for n E Z+.
Then (a(z)(n)b(z))JO) is a holomorphic V-valued formal distribu(ion for all n E Z with constant term a(n)b(-1)JO).

84

3. LOCAL FIELDS

PROOF. Let k E Z+ and consider separately two cases. The first case:

(a(z)(-k-l)b(z))IO) = : a(k)a(z)b(z): IO)= a(k)(a(z))+b(z)IO)
= (8(k)a(z))+b(z)+IO).

We have used here (2.3.4). The second case:

I: (k.) . = k

(-zl-3 a(j)b(z)+IO).

j=O J

Thus we see that in both cases lemma holds.

□

It turns out that there is a nice unified formula for all the n-th products of fields (n E Z):
(3.1.12)
a(w)(n)b(w) = Resz ( a(z)b(w)iz,w(z - w)n - (-l)P(a)p(b)b(w)a(z)iw,z(z - w)n).

Indeed, for n ~ 0 formula (3.1.12) obviously coincides with (2.3.8). For n < 0,
(3.1.12) follows from the following formal Cauchy formulas for any formal distribution a(z) and k E Z+:

(3.l.13a) (3.l.13b)

Resz a(z)iz,w (z - ~)k+l = a(k)a(w)+'

.

1

Resz a(z)iw,z (z _ w)k+l

-8(k)a(w)_.

It is immediate to check these formulas for k = O; the general case follows by
differentiating both sides by w k times.

3.2. Dong's lemma Now we are in a position to prove the following important lemma (see [Lil).

LEMMA 3.2 (Dong). If a(z), b(z) and c(z) are pairwise mutually local fields (resp. formal distributions), then a(z) (n)b(z) and c(z) are mutually local fields (resp. formal distributions) for all n E Z {resp. n E Z+J- In particular: a(z)b(z) : and c(z) are mutually local fields provided that a(z), b(z) and c(z) are.

3.2. DONG'S LEMMA

85

PROOF. It suffices to show that for M » 0:

(3.2.1)

where

(3.2.2a) A

iz1 ,z2 (z1 - z2)na(z1)b(z2)c(z3) - (-l)P(a)p(b)iz2 ,z1 (z1 - z2)nb(z2)a(z1)c(z3),

(3.2.2b)

= B

(-l)P(c)(p(a)+p(b)) (iz1 ,z2 (z1 - z2)nc(z3)a(z1)b(z2)

- (-l)P(a)p(b)iz2 ,z1 (z1 - z2)nc(z3)b(z2)a(zi)).

Indeed, taking Resz1 of both sides of (3.2.1) and letting z2 = z, z3 = w gives the
result due to (3.1.12).
The pairwise locality means that for r » 0:

(3.2.3a) (z1 - z2r a(z1)b(z2)

(z1 - z2r (- l)p(a)p(b) b(z2)a(z1),

(3.2.3b) (3.2.3c)

(z2 - Z3rb(z2)c(z3)

(z2 - Z3r (- l)p(b)p(c) c(z3)b(z2),

(z1 - z3ra(z1)c(z3) = (z1 - Z3r(-l)p(a)p(c)c(z3)a(z1),

t (3;) Taking r sufficiently large, we may assume that n ?: -r. Take M = 4r and use

(z2 - z3) 3r =

(z2 - z1) 3r-s(z1 - Z3) 8 •

Then the left-hand side of (3.2.1) becomes

(3.2.4)

-t, (3:) (z2 - z1)3r-s(z1 - Z3) 8 (Z2 - z3rA.

If 3r - s + n?: r, then (z1 - z2) 3r-siz1 ,z2 (z1 - z2)n = (z1 - z2y' where r' ?: r, hence
due to (3.2.3a) the s-th summand in (3.2.4) is Ofor O~ s ~ r. Hence the left-hand

side of (3.2.1) equals

(3.2.5a)

f: (3:) (z2 - z1)3r-s(z1 - z3)8(z2 - z3rA.
s=r+l

Similarly the right-hand side of (3.2.1) equals

(3.2.5b)

f: (3;) (z2 - z1)3r-s(z1 - Z3) 8 (Z2 - Z3rB,
s=r+l

Due to (3.2.3b and c), (3.2.5a) is equal to (3.2.5b).

□

86

3. LOCAL FIELDS

Let gff(V) denote the space (over C) of all fields with values in EndV. As we have seen, gff(V) is closed under all the products a(z)(n)b(z), n E Z. This is called the general linear field algebra.

DEFINITION 3.2. A subspace F of gff(V) containing the identity operator Iv and closed under all the products a(z)(n)b(z) (then automatically OzF C F) is called a linear field algebra. 2 A linear field algebra is called local if it consists of mutually local fields.

REMARK 3.2. A subspace F of gff(V) is a linear field algebra iff Iv E F, 8F C F, F is closed under normally ordered product and F is closed under OPE (i.e., all the OPE coefficients given by (3.1.2) are in F).

One says that a collection of fields generates a field algebra F if F is the minimal field algebra containing these fields. Dong's lemma implies

COROLLARY 3.2. A linear field algebra generated by a collection of mutually local fields is local.

Let F C glf(V) be a linear field algebra. Then we may associate to any a E F a formal distribution with values in Endrr:,F:
= Y(a, x) L x-n-la(n).
nEZ
Explicitly, using (3.1.12), this formal distribution can be written as follows:

(3.2.6)

Y(a(w),x)b(w)

= Resz(a(z)b(w)iz,wt5((z - w) - x) - p(a, b)b(w)a(z)iw,zt5((z - w) - x)).

This formal distribution is a field if F is a local field algebra. The following proposition will be used in the sequel.

PROPOSITION 3.2. If a(z), b(z) are elements of a linear field algebra F and
= N > 0 is such that (z - w)N [a(z), b(w)] 0, then

(3.2.7)

(x - y)N [Y(a(w), x), Y(b(w), y)) = 0.

2Lian and Zuckerman [LZ] use the term "quantum operator algebra."

3.3. WICK'S THEOREM AND A "NON-COMMUTATIVE" GENERALIZATION

87

PROOF. It is straightforward to see from (3.2.6) that

[Y(a(w), x), Y(b(w), y)]c(w)

x8((z1 - w) - x)8((z2 - w) - y).

Since x - y = ((z2 - w) - y) - ((z1 - w) - x) + (z1 - z2), (3.2.7) follows using

Proposition 2.le (for j = 0).

□

3.3. Wick's theorem and a "non-commutative" generalization The normally ordered product of more than two fields a1(z), a2(z), ... , aN(z) is defined inductively "from right to left":

(3.3.1)

This is a sum of 2N terms of the form

(3.3.2)
where i1 < i2 < · · ·, i1 > i2 > · · · is a permutation of the index set {1, ... , N} and
± is the sign of this permutation from which the indices of even fields are removed.
REMARK 3.3. It is clear from (3.3.2) that if [ai(z)±, ai(z)±] = 0 for all i and j, then :a1(z)···aN(z): = ±: ai1 (z)···aiN(z): where± is the sign of
the permutation of i1, ••• , iN from which the indices of even fields are removed. It follows that in this case the normally ordered product is (super)commutative. However, it is not associative, as Example 4.8 in Section 4.8 shows.

The following well-known simple theorem is extremely useful for calculating the OPE of two normally ordered products of "free" fields.

THEOREM 3.3 (Wick theorem). Leta1 (z), ... ,aM(z) andb1 (z), ... ,bN(z) be two collections of fields such that the following properties hold:
(i) [[ai(z)-, bi (w)], ck (z)±] = 0 for all i, j, k, and c = a o~ b,
(ii) [ai(z)±,bi(w)±] =Oforalli andj.

88

3. LOCAL FIELDS

= Let [aibi] [ai(z)-,bi(w)] denote the "contraction" of ai(z) and bi(w). Then one

has:

(3.3.3)

L L min(M,N)
:a1(z)···aM(z)::b 1(w)···bN(w):=
s=O i1 <· .. <i.
ii#- ..#j.

(±[ai•1 b3'1] •·· [ai••b3'•] :a1 (z)···a M (z)b1 (w)···b N (w):(i1, ... ,i.;ji, ... ,j.))

where the subscript (i1···i 8 ;j1··•j8 ) means that the fields ai1 (z), ... , ai•(z),
bi1(w), ... , bi• (w) are removed, and the sign ± is obtained by the usual super
rule: each permutation of the adjacent odd fields changes the sign.

PROOF. The typical term on the left-hand side of (3.3.3) is

and we have to move the ai(z)_ across the bi(w)+ in order to bring this product to the normally ordered form (3.3.2). But due to the condition (ii) of the theorem,

(3.3.4)

Due to condition (i) the contractions commute with all fields, hence can be moved

to the left. This proves (3.3.3).

□

DEFINITION 3.3. A collection of fields {a°' (z)} is called a free field theory if all of these fields are mutually local and all the coefficients of the singular parts of the OPE are multiples of the identity.

By Remark 3.3, normally ordered products of free fields are, up to the sign, independent of the order. The OPE between these normally ordered products can be calculated using Wick's formula (3.3.3) and Taylor's formula (3.1.8).
Now we turn to a generalization of Wick's formula for arbitrary fields. First, we prove an analogue of Proposition 2.3 for all n-th products of fields.

PROPOSITION 3.3. (a) For any two fields a(w) and b(w) and any n E Zone has:

(3.3.5a) (3.3.5b)

= 8a(w)(n)b(w) -na(w)(n-l)b(w),
= a(w)(n)8b(w) 8w(a(w)(n)b(w)) + na(w)(n-1)b(w).

a Hence, is a derivation of all n-th products.

3.3. WICK'S THEOREM AND A "NON-COMMUTATIVE" GENERALIZATION

89

(b) For any mutually local fields a(w) and b(w), and for any n E Z one has:

(3.3.6)

00
a(w)(n)b(w) = -p(a, b) L(-l)i+na~f) (b(w)(n+j)a(w)).
j=O

(c) For any three fields a(w), b(w), and c(w) and for any m E Z+, n E Z

one has: (3.3.7)

t. (7) = a(w)(m) (b(w)(n)c(w))

(a(w)(j)b(w))(m+n-j) c(w)

+ p(a, b)b(w)(n) (a(w)(m)c(w)).

PROOF. The proof of (a) is straightforward.

We have by (3.1.11) in the domain lzl > lwl:

(3.3.8)

= b(z)a(w)

~ ~

b(w)(k)a(w) (z _ )k+l

+ (z -

N N
w) d (z,w).

k?_-N

W

Using locality (see Theorem 2.3(iii)) and exchanging z and w we obtain from (3.3.8)

in the domain lzl > lwl:

= p(a, b)a(z)b(w)

~ ~

b(z)(n)a(z) (w _ z)n+l

+ (w -

N
z) d(w, z).

n?_-N

Applying Proposition 3.1 to a(z,w) = b(z)(n)a(z) we rewrite this as:

(3.3.9)

p(a,b)a(z)b(w)-

~ ~

n+l
(-1)

~ ~

a

(")
J

(b(w)(n)a(w))

(z-w)n+l-j

N
+(z-w) d1 (w,z).

n?_-N

j?_O

Comparing the coefficients of (z -w)-k-i in (3.3.8) (where a and bare exchanged)

and in (3.3.9) we get (b).

An equivalent form of (3.3.7) is the following formula:

(3.3.10)

[a(w)>. (b(w)(n)c(w) )] = p(a, b)b(w)(n) [a(w )>.c(w)]

L00
+ _x(k) [a(w)>.b(w)](n+k)c(w)
k=O
(where [a(w)>.b(w)] is defined by (2.3.11)). The proof of (3.3.10) is straightforward

using the identity

[a, be] = [a, b]c + p(a, b)b[a, c].

Indeed, the left-hand side of (3.3.10) is

90
where

3. LOCAL FIELDS

A

Resz Resu e>-(z-w)[a(z), b(u)c(w)]iu,w(u - wt,

B

Resz Resue>-(z-w)[a(z),c(w)b(u)]iw,u(u - wt.

We have:

A = Resz Resu(e>-(z-w)[a(z), b(u)]c(w)iu,w(u - w)n
+ p(a, b)b(u)[a(z), c(w)]iu,w(u - wt)
= Resz Resu(e>-(z-u)e>-(u-w)[a(z), b(u)]c(w)iu,w(u - wt
+ p(a, b) Resu b(u)[a(w)>.c(w)]iu,w(u - w)n)
= Resu([a(u)>.b(u)]e>-(u-w) c(w)
+ p(a, b)b(u)[a(w)>.c(w)])iu,w(u - wt.
Similarly we obtain:

B = Resu([a(w)>.c(w)]b(u) + p(a, c)c(w)e>-(u-w) [a(u)>.b(u)])iw,u(u - w)n.

These two equations give (3.3.10).

□

The special case of (3.3.7) for n = -1 is called the "non-commutative" Wick formula (m E Z+):

(3.3.11) a(z)(m) : b(z)c(z) :=: (a(z)(m)b(z)) c(z):
+p(a, b): b(z) (a(z)(m)c(z)) : +}; (;) (a(z)(j)b(z)\m-l-j) c(z). •

Note that for free fields the "correcting" sum in (3.3.11) vanishes and we recover the usual Wick formula.
Formulas (3.3.6) and (3.3.11) allow one to calculate OPE of arbitrary normally ordered products of pairwise local fields knowing the OPE of these fields if they form a closed system under n-th products for n E Z+· In fact there is a Mathematica package [T] which provides a computer program for these calculations. The earliest known to me reference where formula (3.3.11) is explicitly written down and systematically used is the paper [BBSS].

3.4. BOUNDED AND FIELD REPRESENTATIONS OF LIE SUPERALGEBRAS

91

In the case of n = -1 formula (3.3.10) can be written in the following beautiful form (equivalent to (3.3.11)):

(3.3.12)

[a(w)>.: b(w)c(w) :] =: [a(w)>.b(w)]c(w) : +p(a, b) : b(w)[a(w),xc(w)] :
+ 1>. [[a(w)>.b(w)]µc(w)] dµ.

3.4. Bounded and field representations of formal distribution Lie superalgebras
DEFINITION 3.4a. Let g be a formal distribution Lie superalgebra, i.e. a Lie superalgebra spanned by coefficients of a family of mutually local formal distributions {a°'(z)}aEA (A an index set). A representation of gin a vector space Vis called a field representation if all the a°' (z) are represented by fields, i.e. for each v E V and a E A one has

An important problem of quantum field theory is the construction of local linear field algebras. The usual way of doing this is to take a field representation of a formal distribution Lie superalgebra; then the fields representing the a°'(z) generate a local linear field algebra.
Field representations are usually constructed by means of induced modules. Recall that for a Lie superalgebra g and a representation 1r of its subalgebra p in a vector space W the induced g-module is the vector space

Ind~1r ·- U(g) 18lu(p) W
= (U(g) l8l W) /U(g) (p l8l w - 1 l8l 1r(p)w Ip E p, w E W)

on which g E g acts by left multiplication on the 1st factor. Let g be a Lie superalgebra spanned by coefficients of mutually local formal
distributions {a°' (z)} aEA and assume that the IC[8]-span of the a°' (z) is closed under all n-th'products, n E Z+ (cf. Corollary 4.7). Let

(3.4.l)

Z+} . g_ = C-span of { a(n) la E A, n E

92

3. LOCAL FIELDS

Due to Theorem 2.3(iv), g_ is a subalgebra of g. It is called the annihilation subalgebra (cf. Section (2.9)). Let 1r be a representation of g_ in a vector space W such that for any w E W:

= 7r (afn)) w 0 for n » 0.

Then the induced g-module IndL 1r is a field representation. Indeed, one proves by induction on k (using Theorem 2.3(iv)) that

for n » 0.

Unfortunately, even the oscillator algebra has a lot of pathological irreducible field representations. The additional requirement of "boundedness" removes these pathologies.
We shall now assume that the formal distribution Lie superalgebra g is graded. This means that we have a diagonalizable derivation H of the Lie superalgebra g such that for some ~o: E IR:

(3.4.2)
= i.e., a°'(z) is an eigendistribution for Hof conformal weight ~o:- Writing a°'(z)
EnE-~a+za~z-n-~a we have, due to (2.6.1):

Ha~= -na~.

Hence g is a JR-graded Lie superalgebra by eigenspaces of H:

(3.4.3)

= g EBn9n, [gm, 9n] C 9m+n•

Let
g>- = EBn~09n, 9>O -_ wm n>09n , 9<O -_ wm n<09n ·

We have the triangular decomposition:
g = g< + 9o + g> •

DEFINITION 3.4b. A representation in a vector space V of graded formal distribution Lie superalgebra g is called bounded3 if the subalgebra g>0 acts locally
3This terminology differs from that of [K2], where field modules are called "restricted" and bounded modules are more or less the "category 0" modules.

3.5. FREE (SUPER)BOSONS

93

= nilpotently on V, i.e., for any v E V there exists n > 0 such that g1 • • • gn v 0 for
any n elements g1 , ... , gn of g>0 .

= Recall that a g-module Vis called graded if V ½ EBjER. and gmVn C Vm+n•
Consider a representation 71' of the subalgebra go, extend it to g:C:: by letting
= 71' (g>0 ) 0, and let

The g-module V(11') is called the (generalized) Verma module associated to 71'. Note that this is a graded module, the gradation being induced by JR-gradation (3.4.3):

(3.4.4)

V(11') = EB V(11')n,
n:C::O

so that the representation of go in V(11')o is 71'. It follows from (3.4.4) that the
representation of g in V(71') is a bounded field representation.
Denote by J(11') the sum of all g-submodules contained in EBn>O V(11')n, and let

It is clear that J(11') is a graded submodule, hence V(11') is a graded module.
A vector v of a g-module V is called singular if g>0v = 0.
The proof of the following proposition is straightforward.

PROPOSITION 3.4. (a) A graded bounded g-module V = ffii ½ is irreducible

iff all its singular vectors have minimal grade d and the representation of g0 in Vd

is irreducible.

(b) The map 71' f-+ V(11') gives us a bijection between the set of all (up to iso-

morphism} irreducible go-modules and the set of all (up to isomorphism and shift

of grade} irreducible bounded g-modules.

□

3.5. Free (super)bosons
Let 1J be a finite-dimensional superspace with a non-degenerate supersymmetric
bilinear form (.I.)- Viewing IJ as a commutative Lie superalgebra, we may consider
its affinization (see Section 2.5):

94

3. LOCAL FIELDS

with commutation relations (m, n E Z; a, b E Q):

(3.5.1)

where am stands for atm. Then the currents

= a(z) 'L°a' nz -n-1 , a E Q,
nEZ

are mutually local with the OPE (cf. (2.5.6)):

(3.5.2)

( )b( ) ~ (alb)K
a z w (z - w)2

6 It is natural to call the Weyl affinization of Q(vs. the Clifford affinization ClA

discussed in Section 2.5 and in the next section). The different nature of notation

stems from the difference of the generalizations of these two affinizations to the

non-commutative case discussed in Section 2.5.
6 Consider a field representation of the Lie superalgebra in a vector space V.

Then we get a set of mutually local fields with the OPE (3.5.2), called a system

of free bosons (sometimes called free superbosons if QI -::/- 0). Note that these fields

satisfy the conditions of Wick's theorem.

Choose bases {ai} and {bi} of Qconsistent with the Z 2-gradation such that

(3.5.3)

Such bases are called dual. Then for any h E Qwe have:

(3.5.4)

Consider now the field (3.5.5)

Using Wick's theorem, calculate the following OPE:

Using (3.5.4), we obtain (a E Q):

(3.5.6)

S(z)a(w) ~ a(z) K ~ ( a(w) + 8a(w)) K.

(z-w) 2

(z-w)2 z-w

In the last part of (3.5.6) we used Taylor's formula.

3.5. FREE (SUPER)BOSONS

95

Suppose now that K = klv where the affine central charge k is a non-zero

number. Let

(3.5.7)

L(z)

=

1 kS(z).

Then (3.5.6) gives us (a E ~):

(3.5.8)

~ L(z)a(w)

a(z) (z _ w) 2

a(w) oa(w)
---'--'---+--
(z - w) 2 z -w •

Writing L(z) = I":nez Lnz-n-2 , we obtain, due to Table OPE (Sec. 2.6):

(3.5.9)

Noting that

where

6, and that the elements ao lie in the center of we see from (3.5.9), in particular,
that

(3.5.10)

In other words, adH is a Hamiltonian and all fields a(z) have conformal weight 1. (Of course, it is even easier to check (3.5.9) and (3.5.10) directly.)
Note that (3.5.9) form= -1 and m = 0 means

[L-1,a(z)] = oa(z), [Lo, a(z)] = (zo + l)a(z).

It follows easily that L(z) satisfies (2.6.4). Since also L(z) is a local field whose OPE with itself, by Wick's theorem, has the form (2.6.3) we obtain by Theorem 2.6b that L(z) is a Virasoro field. (Of course, it is easy to see this directly using Wick's theorem.) In order to compute the central charge, we need to compute the s = 2
term of L(z)L(w) in Wick's formula (3.3.3), which is ½sdim~/(z - w) 4 . Thus we
obtain

(3.5.11)

central charge of L(z) = sdim ~-

96

3. LOCAL FIELDS

Since oa(z) has conformal weight 2 we can construct the following family of local fields on conformal weight 2:

= As usual, we let Lb(z) I:nL~z-n- 2 . It follows from (3.5.2) and (3.5.8) that

(3_5_12)

~ Lb(z)a(w) a(w) + oa(w) _ 2(alb)k . (z-w)2 z-w (z-w)3

Hence (using (2.6.3)) we obtain:

(3.5.13)
In particular, [L~ 1,an] = -nan-1, hence [L~ 1,a(z)] = oa(z) and, as above, we
deduce that Lb(z) is a Virasoro field. Using (3.5.2), (3.5.8) and (3.5.11), we see that the central charge of Lb(z) is equal to dimQ0 - dimQ1 -12(blb)k. Thus we have proved the following

PROPOSITION 3.5. For each b E Qo the field Lb(z) is a Virasoro field with central charge

(3.5.14)

sdim Q- 12(blb)k.

6. We apply now formula (3.5.10) to representation theory of the algebra Since 6is a direct sum of the abelian Lie superalgebra Qand the Heisenberg superalgebra

61 = Ef)Q ®tn +CK,
n;eO
it suffices to study representations of the latter. It is a Z-graded Lie superalgebra
with the triangular decomposition:
E9 6' = 6< + CK + 6>, where 6~ = (Q ® t=Fn) .
n>O
The following lemma is immediate from the definitions.

6 LEMMA 3.5. If v is a singular vector of a field representation of (i.e.,

6>v = 0), then Hv = 0.

□

6~ 6> Let = +CK. Given k E C, denote by 1rk the 1-dimensional representation of 6~ defined by: