数理解析研究所講究録904 巻 1995 年 87-115

87

Moonshine Cohomology
Bong H. Lian and Gregg J. Zuckerman

ABSTHACT. We construct a new cohomology functor from a certain category of quantum operator
algebras to the category of Batalin-Vilkovisky algebras. This Moonshine cohomology has, as a groupof natural automorphisms, the Fischer-Griess Monster finite group. We prove a general vanishingtheorem for this cohomology. For a certain commutative QOA attached to a rank two hyperbolic
lattice, we show that the degree one cohomology is isomorphic to the so-called Lie algebra ofphysical states. In the case of a rank two unimodular lattice, the degree one cohomology gives
a new construction of Borcherd’s Mollster Lie algebra. As applications, we compute the gradeddimensions and signatures of this cohomology as a hermitean Lie algebra graded by a hyperbolic
lattice. In the first half of this paper, we give as preparations an exposition of the theory of quantum
operator algebras. Some of the results here were announced in lectures given by the first author atin September 94.
the Research Institute for Mathematical Sciences in$\mathrm{I}\langle \mathrm{y}\mathrm{o}\mathrm{t}_{0}$Introduction

1, there took place the following developments, all of which are now understoodIn theto be connected with two-dimensional quantum field theory:$1980’ \mathrm{s}$1) Frenkel and Kac gave a construction of the simply-laced simple Lie algebras via thevertex operator representation of the corresponding affine Kac-Moody Lie algebras. The vertex
operators were already thought of as quantum fields that depend holomorphically on one complex
variable. These fields had their origin in the dual resonance model, which was central to thecreation of string theory. Frenkel later extended this vertex operator construction to obtain some
new infinite dimensional Lie algebras that contained Kac-Moody Lie algebras of hyperbolic type.
In the process, Frenkel gave a proof of the No Ghost Theorem in string theory.

2) Frenkel, Lepowsky and Meurman generalized the Frenkel-Kac vertex operator constructionin order to construct an infinite dimensional graded representation of the Monster finite group.
The existence of the FLM Moonshine module immediately explained the empirically observed
and the dimensions of irreducible representations
connection between the modular function
discoveredthat the Moonshine module possessed
afterwardsof the Monster. Borcherds soon
$j(\tau)$$0_{\mathrm{q}- \mathrm{a}}/9501015$$_{\mathrm{B}.\mathrm{H}}$.L. is supported by grant DE-FG02-88-ER-25065. G.J.Z. is supported by NSF Grant DMS-9307086 and

DOE Grant DE-FG02-92-ER-25121.

88

the structure of what he called a vertex algebra. The book by Frenkel, Lepowsky and Meurman
gave a complete treatment of vertex operator algebras and the Moonshine module.
3) Belavin, Polyakov and Zamolodchikov developed the so-called operator product expansion
into a powerful tool for the study of two dimensional conformal quantum fields. As a consequence, BPZ determined the possible central charges and critical dimensions of the minimal
conformal field theories. This analysis required the deep work by Kac, and later Feigin andFuchs on the structure of the highest weight representations of the Virasoro Lie algebra. It wassoon recognized (see for example [29]) that there was a close relationship between the theory of
the operator product expansion and the theory of vertex operator algebras.

4) Feigin invented the theory of semi-infinite cohomology for Virasoro and Kac-Moody Lie
algebras. The construction of the differential in Feigin’s complex was soon reinterpreted interms of some elementary holomorphic quantum fields arising in conformal field theory and the
BRST quantization construction in string theory $[18][36]$ . Moreover, Frenkel, Garland and thesecond author of the current paper were able to give a new proof of the No Ghost Theorem via
the analysis of a particular semi-infinite cochain complex. At the end of the eighties, the two
authors of the current paper began a long series of papers building on the earlier work of FGZ.

5) Koszul discovered a new relationship between graded commutative superalgebras andgraded Lie superalgebras. Specifically, Koszul found that under certain very general hypotheses,
the failure of a second order differential operator to be a derivation led to the existence of agraded Lie bracket on the underlying commutative algebra. An important special case of thesame relationship was found independently by the physicists Batalin and Vilkovisky. At thetime, there was no perceived connection between the work of Koszul, Batalin and Vilkovisky
and the rapidly developing study of conformal quantum fields.

, Borcherds proved the Conway-Norton conjectures for the FLM Moonshine
In the early
module. The first two developments above are fundamental for Borcherds. As a brief aside,
Borcherds claims in his paper that semi-infinite cohomology theory, as discussed in 4 above,
can be employed to obtain alternate constructions of the infinite dimensional Lie algebras thatprominently in his work. However, he presents no details to support his claim.
$1990’ \mathrm{s}$$\mathrm{f}\mathrm{i}\mathrm{g}\mathrm{u}\dot{\mathrm{r}}\mathrm{e}$The main purpose of the current paper, partly inspired by [28], is to forge a synthesis of
Borcherds work with all five of the above developments. We construct a new functor that wecall Moonshine cohomology and which fully justifies Borcherds claim. This new cohomology
theory is an outgrowth of the developments sketched in 3, 4 and 5 above. In particular, we
employ a new mathematical approach to the operator product expansion, and we work with arecent generalization of the notion of a VOA to the notion of a commutative quantum operator
algebra. Our paper on CQOAs [24] is designed to be a companion to the current paper on
Moonshine cohomology.

Degree one Moonshine cohomology provides a functor from the category of vertex operator
algebras to the category of Lie algebras that carry an action of the Monster finite group by automorphisms. The degree one Moonshine cohomology of a particular VOA yields the generalized

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Kac-Moody Lie algebra that is the key to Borcherds proof of the Conway-Norton conjectures.
The total Moonshine cohomology provides a functor with values in the category of BatalinVilkovsky algebras that carry an action of the Monster finite group by automorphisms. BV
algebras are odd Poisson algebras in which the graded Lie bracket is related to the graded
commutative product in the fashion first described by Koszul and Batalin-Vilkovisky (see 5
above.) The abstract notion of a BV algebra, though present in the paper of Koszul, did not
become well known until the recent work of Penkava-Schwarz and Getzler.

As in any cohomology theory, the total cohomology is better behaved and more fundamental
than the cohomology of any special degree. Moreover, the BV structure on the total cohomology
allows us to relate a complicated Lie algebra structure to a more elementary commutative algebra
structure. We hope that Moonshine cohomology will yield further insight into the structure of
the Moonshine module as well as into the proof of the Conway-Norton conjectures. We also hopethat our current paper unifies a number of seemingly disparate points of view in mathematics
and mathematical physics.
Here is a brief summary of the contents of this paper:
In section 2, we state the definitions of our main concepts: .uantum operators, matrix”
products, theelements, Wick product.s, iterated Wick products, the infin, itely many
operator product expansion, the notions of locality ari commutativity, and finally the corresponding notions of local and commutative quantum operator algebras. Finally, we discuss someelementary known facts about commutativity.
$\mathrm{q}$($‘ \mathrm{c}\mathrm{i}\mathrm{r}\mathrm{C}\mathrm{l}\mathrm{e}$$\mathrm{d}$In section 3, we introduce what we call the Wick calculus, which deals with operator productsof the form: $t(z)u(z)$ : $v(w)$ as well as $t(z)$ : $u(w)v(w)$ : under the assumption that the quantum
operators $t(z),$ $u(z)$ , and $v(z)$ are pairwise commutative. Here, : $t(z)u(Z)$ : denotes the Wick or
normal ordered product of $t(z)$ with $u(z)$ . The Wick calculus is essential for both computations
as well as for theoretical issues, such as the explicit construction of CQOAs. Section 3 continues
with the construction of the CQOA $O(b, c)$ , which acts in the ghost Fock space of the BRST
, which arose originally in theconstruction. Following this is a construction of the CQOA
seminal work of BPZ [3], and which acts in the state space of any conformal field theory having
central charge . This section concludes with a construction of a CQOA from a Lie algebra
equipped with an symmetric invariant form. All three examples are special, in that thesetives of the generating quantum operators. In
algebras are spanned by Wick products offact, we exhibit explicit bases consisting of such products in the first two examples.
$O_{\kappa}(L)$$\kappa$$\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{i}_{\mathrm{V}}\dot{\mathrm{a}}$In section 4, we discuss the BRST construction in the language of what we call conformal
QOAs. Given a conformal QOA $0$ with central charge , we form the tensor product $C^{*}(O)=$$O(b, c)\otimes O$ . We then construct the special quantum operator $J(z)$ , which we call the BRST
current. We also give a simple characterization of $J(z)$ . We then recall the famous result thatthe coefficientis square-zero if and only if $\kappa=26$ . After that we specialize tothis central charge.
$\kappa$$J(\mathrm{O})={\rm Res}_{z}J(z)$90

The first main result of section 4 is Theorem 4.6, which states that the Wick product induces
a graded commutative associative product on the cohomology of $C^{*}(O)$ with respect to the. This theorem first appeared in work of E. Witten [33], who called thederivation,
ghost number zero subalgebra the “ground ring of a string background”. An approach to thistheorem via VOA theory appears in [26]. The approach in the current paper is via CQOA
theory. Continuing section 4, we develop the theory of the ghost field, $b(z)$ , and its coefficient,$b(1)$ . As a preparation, we remind the reader of the definition of a Batalin-Vilkovisky
operator and BV algebra. The second main result of section 4 is Theorem 4.8, which states thatthe operator $b(1)$ induces a BV operator acting in the BRST cohomology algebra. Thus thecohomology becomes a BV algebra. This theorem was inspired by work of Witten and Zwiebach, where it was derive At the end of section 4 we state the precise[37], and first appeared
connection between BRST cohomology and semi-infinite cohomology.
$[J(\mathrm{O}), -]$$(\mathrm{B}\mathrm{V})$$\mathrm{i}!\mathrm{n}[26]$as a functor
In section 5, we finally present the construction of Moonshine cohomology
asserts that
5.2,
which
Theoremisresultmainfrom conformal QOAs to BV algebras. OurMoonshine cohomology vanishes for degrees less than zero and greater than three. In fact, wefirst prove Theorem 5.3, which states a vanishing theorem for semi-infinite cohomology; Theorem
5.2 follows immediately.
$\mathrm{M}^{*}$

We specialize in section 6 to the Moonshine cohomology of the conformal QOA attachedby FLM to a hyperbolic lattice of rank 2. Theorem 6.2 asserts that in this case, the degree
zero and degree three Moonshine cohomology groups are both one dimensional; moreover, thedegree one and degree two cohomology groups can both be canonically identified with the socalled space of physical states, whose definition dates back to the early days of string theory
from a semi(see $[8][9][5]$ ). We actually prove a more general result, Theorem 6.4,
infinite cohomology calculation, Theorem 6.5. As a consequence, we are able to compute the fullMoonshine cohomology as a -graded linear space, and determine all the graded dimensions of
this space. As a second application, we compute the signature of the hermitean form on degree
one cohomology and show that the form is positive definite.
$\mathrm{w}-\mathrm{h}-\mathrm{i}\Gamma_{--}\mathrm{h}_{-\mathrm{O}}\sim \mathrm{f}\underline{\mathrm{l}}\underline{\iota}\circ \mathrm{w}- \mathrm{s}$$\mathrm{t}\mathrm{r}\mathrm{i}$At the end of section 6 we discuss some open questions about Moonshine cohomology. In
is isomorphic
particular, we conjecture that the natural automorphism group of the functor
to the Monster finite group. This conjecture is suggested by the known theorem that the Monster
is the full automorphism group of the Moonshine VOA [11]. We hope that future study of thewill cast new light on a rich amalgam of mathematics and mathematical physics.
functor$\mathrm{M}^{*}$$\mathrm{M}^{*}$

Sections 2-4 of this paper are meant to be an exposition, and should be accessible to thereaders who are new to the theory of operator product expansions. Many useful (in the authors’opinion) exercises are given. Sections 5-6 are more advanced because they draw from several
different subjects. We hope that the included references will help the readers who are interested
in further details.
.L. would like to express special thanks to Prof. M. Miyamoto
Acknowledgments:
and Prof. H. Yamada for their invitation to lecture at the Research Institute for Mathematical
Sciences in Kyoto in September 94. He also thanks them for their hospitality during his stay in$\mathrm{B}.\mathrm{H}$91

Kyoto, and for their patience waiting for the final draft of this paper. We thank F. Akman for
carefully proofreading our manuscript.

2

Quantum Operator Algebras

be a doubly graded vector space $V=\oplus V^{n}[m]$ . The degrees of a homogeneousLetelement in $V^{n}[m]$ will be denoted by $|v|=n,$ $||v||=m$ respectively. In physical applications,will be the conformal
will be the fermion or ghost number of . In conformal field theory,
dimension or weight of . We say that is bounded if for each $n,$ $V^{n}[m]=0$ for $m<<0$ .
$V$$\mathrm{Z}$$v$$||v||$

$v$

$|v|$

$v$

$V$

Let be a formal variable with degrees $|z|=0,$ $||z||=-1$ . Then it makes sense to speak of
a homogeneous (biinfinite) formal power series
$z$

(2.1)

$u(z)=n \in\sum_{\mathrm{z}}u(n)z^{-}n-1$

withof degrees $|u(z)|,$ $||u(z)||$ where the coefficients $u(n)$ are homogeneous linear maps indegrees $|u(n)|=|u(\mathcal{Z})|,$ $||u(n)||=-n-1+||u(z)||$ . Note then that the terms $u(n)z-n-1$ indeed
have the same degrees $|u(z)|,$ $||u(z)||$ for all . We require that for every $v\in V_{f}u(n)v=0$ for$n>>0$ . (If is bounded, then this requirement is superfluous.) We call a finite sum of suchhomogeneous series $u(z)$ a quantum operator on , and we denote the linear space of quantum
operators as $QO(V)$ .
$V$

$n$

$V$

$V$

an $integer_{f}$ we usually mean its formal power
Notations: By the expression $(z-w)^{n},$series expansion in the region $|z|>|w|$ . Thus $(z-w)^{-2}$ and $(-w+z)^{-2}$ are different, $as$
power series. When such expressions are to be regarded as rational functions rather than formal
series, we will explicitly mention so. $W_{l}enA(z)= \sum A(n)z-n-1$ is a formal series with coefficients $A(n)$ in whatever linear space, we define ${\rm Res}_{z}A(z)=A(\mathrm{O}),$ $A(z)^{+}=\Sigma_{n\geq 0}A(n)Z-n-1$ ,$A(z)^{-}=\Sigma_{n<0}A(n)Z-n-1,$ $\partial A(z)=\Sigma-(n+1)A(n)Z-n-2$ . If $u(z),$belong to QOAs $O,$ $O’$
, simply as $u(z)u’(Z)$ . When norespectively, we abbreviate $u(z)\otimes u’(z)_{)}$ as an element of$||u(z)||$$|u(z)|,$. The restricted dual of a graded
simply as
ambiguity occurs, we denoteis denoted $V\#$ . If $A_{1}(z),A_{2(z}),\ldots$ are quantum operators, an arbitrary matrix
vector space. In thewith $x\in V^{\#},$ $y\in V$ , is denoted as
element$(-1)^{1}t||u|$ simply $as\pm$ .
only
is
used
This
convention
like
signs
interest of clarity, we often writewhen the sign arises from permutation of elements. When in doubt, the reader can easily recover
the correct sign from such a permutation. Given two homogeneous linear operators, $X,$ , wewrite [X, ] $=XY-(-1)^{|X|}|Y|YX.$ A similar notation applies to quantum operators when it
makes sense.
$n$$u’(\mathcal{Z})$$O\otimes O’$$|u|,$ $||u||$$V$$\langle x, A_{1}(Z_{1})A_{2}(z2)\cdots y\rangle$$\langle A_{1}(Z_{1})A_{2}(Z_{2})\cdots\rangle$

$Y$

$Y$

Given two quantum operators $u(z),$ $v(z)$ , we write$:u(z)v(w):=u(z)^{-}v(w)+(-1)^{|u|||}v(vw)u(z)^{+}$ .

(2.2)

92

Because $v(n)t=u(n)t=0$ for $n>>0$ , it’s easy to check that if we replace by , the right handside makes sense as a quantum operator and hence defines a nonassociative product on $QO(V)$ .
:, we define :It is called the Wick product. Similarly given
inductively as : $u_{1}(Z_{1})(:u2(Z2)\cdots u_{n}(Z_{n}) :)$ :.
$w$$u_{1}(Z),$$\cdot \mathrm{v}\cdot,$$z$$u_{1}(z_{1})\cdots u_{n}(Z_{n})$$u_{n}(z)$Exercise 1 Show that : $u_{1}(z)\cdots u_{n}(Z)$ : makes sense as an element of $QO(V)$ .
Definition 2.1 For each integer$n$

we define a product on $Qo(V)$ :$u(w)\mathrm{o}_{n}v(w)={\rm Res}_{z}u(z)v(w)(Z-w)^{n}-(-1)^{|u||v}|{\rm Res}_{z}v(w)u(z)(-w+z)^{n}$

.

(2.3)

Explicitly we have:$\frac{1}{(-n-1)!}$$u(z)\mathrm{o}_{n}v(Z)=\{$

if $n<0$: $\partial^{-n-1}u(z)v(z)$ :(2.4)

if $n\geq 0$ .

$[(\Sigma_{m=0}^{n}u(m)(-Z)n-m), v(Z)]$If is a homogeneous linear operator on , then it’s clear that the graded commutator $[A$ ,is a graded derivation of each of the products . Since $u(z)\mathrm{o}_{0}v(Z)=[u(0), v(z)]$ , we have
$V$

$A$

- $]$

$0_{n}$

Proposition 2.2

For any $t(z),$ $u(z),$ $v(z)$ in $QO(V)$ and$t(z)\mathrm{o}_{0}(u(_{Z)}\mathrm{o}_{n}v(Z))=[t(z)\mathrm{o}_{0}u(z)]$$ie$. $t(Z)0_{0}$ is a derivation of every product in

O n $v$ ($QO(\iota\nearrow)$

$Z$

)

$n$

integer, we have$\pm u(_{Z})0_{n}[t(_{Z)\mathrm{o}v}\mathrm{o}(z)]$,.

Proposition 2.3 For $u(z),$ $v(z)$ in $QO(V)_{J}$ the following equality of formal power series in two
variables holds:$.u(z)v(w)= \sum_{0n\geq}u(w)\mathrm{o}nv(w)(z-w)^{-n}-1:+u(Z)v(w)$

:.

(2.5)

Proof: We have $u(z)v(w)=[u(z)^{+}, v(w)]+:u(z)v(w)$ :. On the other hand by inverting the. in (2.4), we getsecond$\mathrm{e}\mathrm{q}\mathrm{n}$$[u(m), v(w)]= \sum_{n=0}^{m}u(w)$ o n$v$( ) $w^{m}-n$ .
$w$

(2.6)

Thus we have$[u(Z)^{\dagger}, v(w)]$$=$$\sum_{m\geq n\geq 0}u(w)\circ_{n}v(w)w^{m}-n-Zm-1$$=$$n \geq\sum_{0}u(w)\mathrm{o}_{n}v(w)^{\frac{1}{n!}}\partial_{w}n(_{Z}-w)^{-1}$$=$$\sum_{n\geq 0}u(w)\circ_{n}v(w)(Z-w)-n-1$.

$\square$

(2.7)

93

In the sense of the above Proposition, : $u(z)v(w)$ : is the nonsingular part of the operator(see
product expansion (2.5), while $u(w)\mathrm{o}_{n}v(w)(z-w)^{-}n-1$ is the polar part of$\frac{1}{2\pii}\int_{C}u(z)v(w)(z-w)^{n}dz$
where
is
as
writtenis often[3] . In physics literature,a small circle around . The above proposition clearly justifies this notation. The products
will become important for describing the algebraic and analytic structures of certain algebras ofquan.t um operators. Thus we introduce the following mathematical definitions:$\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}-n-1$$C$$u(w)\mathrm{o}_{n}v(w)$

$)$

$w$

$0_{n}$

Definition 2.4 A graded subspaceof $QO(V)$ containing the identity operator and closed
is called a quantum operator algebra. We say that $u(z)$ is local
with respect to all the products$u(z)\mathrm{o}_{n}v(z)=0$
to $v(z)$ if
for all but finitely many positive A $QOA$ is called local if its
elements are pairwise mutually local.
$A$$0_{n}$$A$$n.$

We observe that for any element $a(z)$ of a QOA, we haveclosed with respect to formal differentiation.

$a(\mathcal{Z})0_{-}21=\partial a(z)$. Thus a QOA is

and $N$ a nonnegative integer. If $u(z)0_{n}$Let $u(z),$ $v(z)$ be quantum
represents a rational function in $|z|>|w|$ with poles along$v(z)=0$ for $n\geq N$ , then$N$.
$z=w$ of order at most

Proposition 2.5$operat_{\mathit{0}}rs_{f}$$\langle u(z)v(w)\rangle$

Proof: By eqn (2.5), we have$\langle u(z)v(w)\rangle=\sum_{n\geq 0}\langle u(w)\mathrm{o}_{n}v(w)\rangle(z-w)-n-1+\langle:u(z)v(w):\rangle$

It is trivial to check that
immediately.

$\langle:u(z)v(w):\rangle,$$\langle u(w)\mathrm{o}_{n}v(w)\rangle\in \mathrm{C}[z^{\pm 1}, w]\pm 1$

(2.8)

.

. Thus our claim follows$\square$Lemma 2.6 Let $u(z)$ be local to $v(z)$ , andThen for $|w|>|z-w|$ ,$\langle u(z)v(w)\rangle$

represent the rational function $f(z, w)$ .

$f(z, w)= \sum_{\in n\mathrm{Z}}\langle u(w)\mathrm{o}nv(w)\rangle(z-w)^{-n}-1$

Proof: The Laurent polynomial$\langle:$\Sigma_{i\geq 0}\frac{1}{i!}\langle:. Now apply(\partial^{i}u(w))v(w):\rangle(zarrow w)^{i}$u(z)v(w):\rangle$

.

(2.9)

in the above region is just. (2.4).
$\square$$\mathrm{e}\mathrm{q}\mathrm{n}$is associative in general. However it clearly makes
We note that none of the products
sense to speak of the left, right or two sided ideals in a QOA as well as homomorphisms of
QOAs and they are defined in an obvious way. For example, a linear map : $Oarrow O’$ is ahomomorphism if $f(u(z)\mathrm{o}_{n}v(z))=fn(\mathcal{Z})\mathrm{o}_{n}fv(z)$ for all $u(z),$ $v(z)\in O$ , and $f(1)=1$ . An
-module is a graded space $M$ equipped with a homomorphism of QOAs $g:Oarrow QO(M)$ .
$0_{n}$

$f$

$O$

94

Exercise 2 Define the notions of a left, right, and two sided ideals for QOAs.
Definition 2.7
and
ally

Two quantum operators
are said to commute if they are muturepresent the same rational function. This is equivalent(Proposition 2.5) to the following: for some $N\geq 0,$ $(z-W)^{N}\langle u(z)v(w)\rangle=\pm(z-w)N\langle v(w)u(z)\rangle$
as Laurent polynomials. We call a $QOAO$ whose elements pairwise commute a commutative$loCal_{J}$$QOA$$u(z\mathrm{I}, v(Z)$$\langle u(z)v(w)\rangle,\pm\langle v(w)u(z)\rangle$.

Proposition 2.8

If $u(z),$ $v(z)$ commute, then for all$m$$[u(m), v(w)]= \sum_{n\geq 0}u(w)\mathrm{o}_{n}v(w)w^{m}-n$.

(2.10)

Proof: The case $m\geq 0$ is obtained by inverting the second. in (2.4). Since $u(z)v(w)=$$[u(z)^{+}, v(w)]+$ : $u(z)v(w)$ : and $v(w)u(z)=\mp[u(z)-, v(w)]\pm$ : $u(z)v(w)$ :, it follows from
commutativity that
represents the same rational function as $-\langle[u(z)+, v(w)]\rangle$
does, which is just. This gives$\mathrm{e}\mathrm{q}\mathrm{n}$$\langle[u(z)^{-}, v(w)]\rangle$$- \sum_{n\geq 0}\frac{u(w)\mathrm{o}_{n}v(w)}{(z-w)^{n+1}}$$[u(z)-, v(w)]=- \sum_{n\geq 0}u(w)\mathrm{o}_{n}v(w)(-w+z)^{-n-1}$

Taking ${\rm Res}_{z}[u(Z)-, v(w)]z^{m}$ for $m<0$ gives the desired result.

.

(2.11)$\square$

The notion of commutativity here is closely related to the physicists’ notion of duality inconformal field
Frenkel-Lepowsky-Meurman have reformulated the axioms of a VOA
in terms of what they call rationality, associativity and commutativity. The notion of commutativity in Definition 2.7 is essentially the same as FLM’s. This notion has also been reformulated
in the language of formal variables in [6].
$\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}[29].$

3

Wick’s calculus

In this section, we derive a number of useful formulas relating various iterated products of threequantum operators. Most of these
are well-known to physicists who are familiar withthe calculus of operator product expansions. We will also include a lemma on commutativity.
$\mathrm{f}_{0\Gamma \mathrm{n}}1\mathrm{U}\mathrm{l}\mathrm{a}\mathrm{S}$Let $t(z),$ $u(Z),$ $v(z)$ be$\mathrm{h}\mathrm{o}\mathrm{m}o$geneous quantum operators which pairwise commute.

Lemma 3.1 (see [22]) For all$n,$$t(z)\mathrm{o}_{n}u(z)$and $v(z)$ commute.

95

Proof: We include Li’s proof here for completeness. For a positive integer $N,$ $(z-w)^{2N}$ is abinomial sum of terms $(z-x)^{i}(X-w)^{2Ni}-,$ $i=1,$ $2N$ . So $(z-w)^{N2}+N(t(Z)\mathrm{o}_{n}u(Z))v(w)$ is abinomial sum of terms$..,$${\rm Res}_{x}((z-w)^{N}(z-x)i(X-w)^{2N}-\mathrm{i}(t(X)u(z)(x-Z)^{n}\mp u(z)t(X)(-z+x)^{n})v(w))$

We want to show that for large enough $N$ , and$\mathrm{f}\mathrm{o}1^{\cdot}0\leq i\leq 2N$.

(3.12), term by term we have$(z-w)^{N}(Z-X)^{i}(x-w)^{2N-}i(t(x)u(\mathcal{Z})(X-Z)n\mp u(z)t(X)(-z+x)^{n})v(w)$$=\pm(z-w)^{N}(z-X)^{i}(x-w)^{2}N-iv(w)(t(X)u(Z)(x-z)^{n}\mp u(z)t(X)(-z+x)^{n})$ . $(3.13)$

Consider two cases: $i\geq N$ and $i<N$ . By assumption, $(z-x)^{k}(t(X)u(z)(X-z)^{n}\mp u(Z)t(x)(-z+$for all large enough . So for large enough $N$ , (3.13) holds for $i\geq N$ . Similarly
for $i<N,$ $(z-w)^{N}(x-w)^{2N-i}(t(x)u(z)(X-z)^{n}\mp u(Z)t(X)(-z+x)^{n})v(w)$ coincides with$\pm(z-w)^{N}(x-w)^{2}N-iv(w)(t(X)u(Z).(x-Z)^{n}\mp u(Z)t.(X)(-z+x)^{n})$ . This shows that (3.13) holds
for each .
$x)^{n})=0$

$i$

$k$

$\square$This lemma is useful for showing existence of commutative QOAs: it says that given a setof pairwise commuting quantum operators, the QOA generated by the set is commutative. We
now develop some abstract tools for studying the structure of commutative QOAs.
Applying (2.5), we have

: $t(z)u(_{Z)}$ : $v(w)$$=$$(t(Z)^{-_{u}}(\mathcal{Z})\pm u(z)t(z)^{+}\mathrm{I}^{v}(w)$$=$$t(z)^{-_{u()(w)}}Zv\pm u(z)v(w)t(z)^{+}\pm u(z)[t(Z)^{+}, v(w)]$$=$$n \geq\sum_{0}$: $t(z)(u(w)\mathrm{o}_{n}v(w))$ : $(z-w)^{-}n-1:+t(Z)u(_{Z)v}(w)$ :$\pm\sum_{0n,m\geq}u(w)\circ_{m}$$\pm\sum_{n\geq 0}$( $t(w)$ o n: $u(z)$ ( $t(w)$ o n

$v$

$(w)$

$v($ $w)$) $(_{Z}-w)^{-}n-m-2+$) : $(z-w)-n-1$ .

(3.14)

Similarly,$t(z)$$=$$=$: $u(w)v(w)$ :$\pm[u(w)^{-}, t(z)]v(w)\pm u(w)^{-_{t()(w)}}Zv\pm t(z)v(w)u(w)+$$\pm\sum_{n,m\geq 0}(-1)n+1u(w)\circ n(t(w)\mathrm{o}mv(w))(_{Z}-w)^{-n-}m-2$$\pm\sum_{n\geq 0}(-1)^{n}+1$$\pm\sum_{n\geq 0}$:$u(Z)\mathrm{o}_{n}t(\mathcal{Z})v(w)$: $(z-w)-n-1$: $u(w)t(w)\circ_{n}v(w):(z-w)^{-}n-1\pm:u(w)t(z)v(w)$ :(3.15)

96

Lemma 3.2

The following equalities hold in $|w|>|z-w|$ :.

(i)$\sum_{k\in \mathrm{Z}}.\frac{\langle(.t(w)u(w).)\mathrm{o}_{k}v(w)\rangle}{(z-w)^{k+1}}$.
$= \sum_{n,m\geq 0}.\frac{\langle\cdot\partial^{m}t(w)u(w)\mathrm{o}v(nw)\cdot\rangle\pm\langle.\partial^{m}u(w)t(w)\mathrm{o}_{n}v(w).\rangle}{m!(z-w)n-m+1}.\cdot$$\pm\sum_{n,m\geq 0}\frac{\langle u(w)\circ_{n}(t(w)\mathrm{o}mv(w))\rangle}{(z-w)n+m+2}$(3.16)$+ \sum_{m\geq 0}\frac{\langle.\partial^{m}(t(w)u(w))v(w)\cdot\rangle}{m}$(ii).
$\pm\sum_{k\in^{\mathrm{z}}}\frac{\langle t(w)0_{k}.u(w)v(w)\cdot\rangle}{(z-w)k+1}.$$= \sum_{0n,,m\geq}(-1)^{n+}1\frac{\langle u(w)\mathrm{o}_{n}(t(w)\circ vm(w))\rangle}{(z-w)n+m+2}$.
$+ \sum_{0n,,m\geq}(-1)^{n+1}.\frac{\langle.\partial^{m}(u(w)\mathrm{o}_{n}t(w))v(w)\cdot\rangle}{m!(_{Z}-w)^{n-m}+1}$.
$+ \sum_{n\geq 0}.\frac{\langle.u(w)t(w)\mathrm{o}_{n}v(w)\cdot\rangle}{(z-w)^{n+1}}$(3.17)$+ \sum_{m\geq 0}\frac{\langle.u(w)(\partial^{m}t(w))v(w)\cdot\rangle}{m}$Proof: To prove (i), consider matrix coefficients on both sides of. (3.14). By assumptionof commutativity these matrix coefficients represent rational functions. Expanding both sides. . The.
using Lemma 2.6, we get the firstis derived similarly from (3.15).
$\mathrm{e}\mathrm{q}\mathrm{n}$$(\mathrm{i})$$\mathrm{e}\mathrm{q}\mathrm{n}$$\square$$(\mathrm{i}\mathrm{i})$$\mathrm{e}\mathrm{q}\mathrm{n}$By reading off coefficients of the $(z-w)^{i}$ , we can use this lemma to simultaneously compute
all products : $t(w)u(w)$ :, and $t(w)0_{k}$ : $u(w)v(w)$ : in terms of other products among theconstituents $t(w),$ $u(w),$ $v(w)$ . Thus it is a kind of recursion relation for the products. In theexamples below, we will see how it allows us to understand the structure of commutative QOAs.
$\mathrm{o}_{k}v(w)$Lemma 3.3 If $t(z)^{\pm_{u}}(w)^{\pm}=(-1)^{|t||u}|u(w)^{\pm}t(z)\pm$ , then
: $t(z)u(w)v(x):=(-1)^{1}t||u|$ : $u(w)t(z\mathrm{I}v(x)$ :.
Proof: Applying the definition of the Wick product (and surpressing$z,$ $w,$ $x$):: $tuv:-(-1)^{||}t|u|$ : $utv$ :$=t^{-}(u^{-}v+(-1)^{|u||v}|)vu^{+}+(-1)^{|t|(|u}|+|v|)(u^{-}v+(-1)^{1u}||v|vu+)t+$$-(-1)^{||}t|u|(u^{-}(t^{-}v+(-1)^{|t|||}vt^{+})v+(-1)^{|u|(|y}|+|v|)(t^{-}v+(-1)^{|t||v}|vt+)u^{+})$

$=0$

.

$\square$(3.18)

97

Examples3.1

Let $Qo(V)^{-}=\{u(z)^{-}|u(z)\in QO(V)\}$ . This space is obviously closed under differentiation
negative. Alsoand the Wick product. It follows that the space is also closed under all$u(z)^{-_{v}(}w)-=:u(z)^{-_{v}(}w)-:$. It follows thatobserve that for any $u(z),$ $v(z)\in QO(V)$ , we have
$n=0,1,$ , restricted to $Qo(V)^{-}$ , all vanish. Thus $Qo(V)^{-}$ is a local QOA.
the products$0_{n},$$0_{n},$$n$$\ldots$

Let $LO(V)$ be the algebra spanned by homogeneous linear operators on . We can regard
each operator as a formal series with just the constant term. This makes $LO(V)$ a subspace
restricted to $LO(V)$ vanishes except for $n=-1$ , in which
of $QO(V)$ . It is obvious that every
is the usual product on $LO(V)$ . Thus $LO(V)$ is a very degenerate example of a QOA.
case
Obviously, any commutative subalgebra of $LO(V)$ is a commutative QOA.
$V$

$A$

$0_{n}$$0_{-1}$be the Clifford algebra with the generators $b(n),$ $C(n)$ ( $n\in$ Z) and the relations
Let[13] [17] [1]$C$$b(n)C(m)+c(m)b(n)$$=$$\delta_{n,-m-1}$$b(n)b(m)+b(m)b(n)$

$=$

$0$

$c(n)C(7n)+c(r’ ?)c(n)$

$=$

$0$

(3.19)-bigraded if we define the degrees $|b(n)|=$-modulebe the graded irreducible$-|C(n)|=-1,$ $||b(n)||=\lambda-n-1,$ $||C(n)||=-\lambda-n$ .
with generator 1 and relations
Let$\lambda$

be a fixed integer. The algebra$C$

becomes$\mathrm{Z}$$C^{*}$$\mathrm{L}\mathrm{e}\mathrm{t}\wedge^{*}$$b(m)1=c(\uparrow\eta)1=0$,(3.20)$m\geq 0$

Let $b(z),$ $c(Z)$ be the quantum operators$b(z)$$=$$\sum_{m}b(?n)Z-m-1$$c(z)$$=$$\sum_{m}c(m)_{Z}-m-1$

Let $O(b, c)$ be the smallest QOA containing $b(z),$Proposition 3.4$c(\mathcal{Z})$(3.21).

The $QOAO(b, C)$ is commutative. It has a basis consisting of the monomials:$\partial^{n_{1}}b(z)\cdots\partial n\mathfrak{i}b(Z)\partial^{m_{1}}c(Z)\cdots\partial^{m_{g}}C(Z)$:(3.22)

with $n_{1}>\ldots>n_{i}\geq 0,$ $m_{1}>\ldots>m_{j}\geq 0$ .
Proof: Computing the OPE of $b(z),$ $C(w)$ , we have$b(z)C(w)$$=$$(z-w)^{-}1+:b(z)c(w)$ :$c(w)b(z)$$=$$(w-z)^{-}1+:C(w)b(z)$ :: $b(z)c(w)$ :$=$-: $c(w)b(z):$ .

(3.23)

98

It follows that $b(z)$ and $c(z)$ commute. Also $b(z),$ $c(Z)$ each commutes with itself, hence they
form a pairwise commuting set. By Lemlna 3.1, they generate a commutative QOA.

is of the form
, let’s call : $u_{1}(z)\cdots u_{k}(z)$ : a monoor. We claim that it’s proportional to some monomial (3.22) with $n_{1}>\ldots>$$n_{i}\geq 0,$ $m_{1}>\ldots>m_{j}\geq 0$ .
If $t(z),$ $u(z)$ each is of the formor
, it is easy
to check that $t(z)^{\pm_{u(}}z)\pm=-u(z)^{\pm}t(z)\pm$ . It follows from Lemma 3.3 that : $t(z)u(Z)v(z)$ $:=$$-u(z)t(Z)v(z)$ : for any element $v(z)\in O(b, c)$ . This shows that : $u_{1}(z)\cdots u_{k}(z)$ : is equal to:: for any permutation of 1, ..., .
If eachmial of degree$u_{1}(z),$$\ldots,$$uk(z)$$\partial^{n}b(z)$$\partial^{m}c(z)$$k$$\partial^{n}b(z)$$(-1)^{\sigma}$$u_{\sigma(1)}(Z)\cdots u_{\sigma(}k)(Z)$$\partial^{m}c(z)$$k$$\sigma$Let $O’$ be the linear span of the monomials (3.22). We now show thatfor any
and any two monomials $A,$ ; hence $O(b, c)=O’$ . We will do a double induction on the degrees
of and $B$ . Case 1: let $A=t(z),$ $B=:u(Z)v(z)$ : with $t(z),$ $u(z)$ each monomial of degree 1,
and $v(z)$ of any degree. If $v(z)=1$ , then by (3.23) $t(w)0_{k}$ : $u(w)v(z):\in 0’$ . By induction on
the degree of $v(z)$ and applying Lemma, we see that $t(w)0_{k}$ : $u(w)v(w):\in O’$ . This showsthatforof degree 1,of any degree. Now case 2: suppose $A=:t(z)u(z)’.$ ,$B=v(z)$ , where $t(z)$ is of degree 1 and $u(z),$ $v(Z)$ of any degree. By induction on the degree of$u(z)$ , it’s clear from Lemma
that this case reduces to case 1.
$A$ $\mathrm{o}_{k}B\in O’$

$k$

$B$

$A$

$3.2(\mathrm{i}\mathrm{i})$$A$ $\mathrm{o}_{k}B\in O’$

$A$

$B$

$3.2(\mathrm{i})$Finally we must show that the mononlials (3.22) are linearly independent. Define a map
by $u(z)\vdash\Rightarrow u(-1)1$ . We see that this map gives a 1-1 correspondence between theset of monomials (3.22) and a basis. This completes the proof. .
$O(b, c)arrow\wedge$$\square$$\mathrm{o}\mathrm{f}\wedge$

Exercise 3 Let $j(z)=:c(z)b(z)$ :. Show that $j(z)j(w)=(z-w)^{-2}+:j(z)j(w)$ : by direct computation (Hint: $Eqn$ . (3.14) is a good guide.). Use Lemma 2.8 to conclude that$[j(m),j(n)]=n\delta_{n+m,0}$ . Construct a canonical basis of $O(j)$ , the $QOA$ generated by $j(z)$ in$O(b_{C},)$.

Let $M(\kappa, 0)$ be the Verma module of the Virasoro algebra with highest weightvacuum vector . Let
be the quotient ofby the submodule generated byLet
be the QOA generated by $L(z)=\Sigma L_{n}z^{-n-2}$ in $QO(M(\kappa))$ .

$(\kappa, 0)$$M(\kappa)$$v_{0}$$l\mathcal{V}I(\kappa, \mathrm{o})$and$L_{-1}v_{0}$.

$O_{\kappa}(L)$Proposition 3.5
monomials

(see

) The

$QOAO_{\mathrm{i}},.(L)$$[\mathit{3}][l\mathit{2}]$:is commutative. It has a basis consisting of$\partial^{n_{1}}L(z)\cdots\partial^{n_{i}}L(Z)$:(3.24)

with $n_{1}\geq\ldots\geq n_{i}\geq 0$ .
Proof: A direct computation gives$[L(z)+, L(w)]$$=$$\frac{\kappa}{2}(z-w)^{-}4+2L(w)(z-w)^{-}2\partial L(+w)(z-w)^{-1}$$[L(z)^{-}, L(w)]$$=$$- \frac{\kappa}{2}(w-z)^{-4}-2L(w)(w-z)-2+\partial L(w)(w-Z)-1$.

(3.25)

99

But we also have $L(z)L(w)=[L(z)^{+}, L(w)]+:L(z)L(w)$ :, and $L(w)L(z)=-[L(z)^{-}, L(w)]+$ :and$L(z)L(w)$ :. Combining these with (3.25), it is obvious that
$L(z)$
operator.
quantumaaswithitselfcommutesThusfunction.
rational
represent the same
is commutative.
By Lemma 3.1,$\langle L(w)L(z)\rangle$$\langle L(z)L(w)\rangle$$O_{\kappa}(L)$unrestricted. To show
Let $O’$ be the linear span of the monomials (3.24) with
$O’$ is closed under all the products (hence $O_{\kappa}(L)=O’$ ), we apply induction and Lemma 3.2 asin the case of $O(b, c)$ above. We now show that we can restrict to those monomials (3.24) with, and that the resulting monomials form a basis. First by direct computation,
is a $Vir$ -module defined by the action $(L(n)=L_{n-1})$
we see (see Lemma 4.2 of [24]) that
$L(n)\cdot u(Z)=L(Z)\mathrm{o}_{n}u(z)$ .
(3.26)$n_{1},$$..,$$n_{i}\geq 0$$n_{1}\geq,$ $\ldots\geq n_{i}\geq 0$$O_{\kappa}(L)$:is spanned by the monomials (3.24), and because $L(-n-1) \cdot u(z)=\frac{1}{n!}$ :Since
for $n\geq 0$ , it follows that the module is cyclic. Thus we have a unique onto map of Vir-modules
has a PBW basis consisting of $L(-n_{1}-1)\cdot\cdot,$ $L(-ni-$
sending to 1. But
span. This shows that the monomials (3.24) with.
.
$u(z)rightarrowu(-1)v_{0}$. This is the lnverse to theby. Now define a map
basis.
to
a
map
a
basis
must
it
hence
previous map,$\partial^{n}L(\mathcal{Z})u(z)$$O_{\kappa}(L)$$M(\kappa)arrow O_{\kappa}(L)$$1)v_{0},$$n_{1}\geq,$$M(\kappa)$$v_{0}$$n_{1}\geq,$ $..,$$\geq n_{i}\geq 0$$.,$$\geq$$n_{i}\geq 0$$O_{\kappa}(L)arrow M(\kappa)$$O_{\kappa}(L)$$\square$Let
erate. Let

be any Lie algebra with an invariant symmetric bilinear form
with bracket:
be the affinization of$B$$(\mathrm{g}, B)$$\hat{\mathrm{g}}$$(\mathrm{g}, B),$, possibly degen-$\mathrm{i}\mathrm{e}.\hat{\mathrm{g}}=\mathrm{g}[t, t^{-1}]\oplus \mathrm{C}$(3.27)$[Xt^{n}, Yt^{m}]=[X, Y]t^{n}+m+n\delta_{n+m,0}B(x, Y)$

acts by the
locally finite -module in whichbeing central. Let $M$ be anyand, and define the currents:
scalar 1. Denote by $X(n)$ the operator representing(3.28)$X(z)= \sum_{n\in \mathrm{Z}}x(n)_{Z}-n-1$$t\mathrm{g}[t]$$\mathrm{C}$$1\in\hat{\mathrm{g}}$$\hat{\mathrm{g}}$$Xt^{n}$for$X\in \mathrm{g}$

. Let

$O$

be the QOA generated by all the currents in $QO(M)$ .

Proposition 3.6 The $QOAO$ is commutative. It is spanned by the monomials.
:, with:$\partial^{n_{1}}X_{1}(Z)\cdots\partial nixi(Z)$Proof: For $X,$

$Y\in \mathrm{g}$$n_{1},$$n_{i}\geq 0_{\rangle}X_{1},$$\ldots,$$X_{i}\in \mathrm{g}$$\ldots,$, we have$[X(z)^{+}, Y(w)]$$=$$B(x, Y)(z-w)^{-}2+[x, Y](w)(Z-w)^{-}1$$[X(\mathcal{Z})^{-,Y(}w)]$$=$$-B(X, Y)(w-Z)-2+[x, Y](w)(w-z)-1$ .

(3.29)

It follows that the currents are pairwise commuting quantum operators, and hence
mutative.

$O$

is com-To prove the second statement, it’s enough to show that for any integer and any two
is a linear sum of those monomials. This follows by induction
monomials $A,$ $A’$ above,$A’$$X$ occuring in $A,$ $A’$ ) and by applying Lemma 3.2.
$A,$
.
(
the
numberofdegreeson the$n$$A$$\mathrm{o}_{n}A’$$\square$$\mathrm{i}\mathrm{e}$There is a vast literature closely related to the study of the algebra
sample, see for example $[20][35][32][12][23][22]$ .

$O$

above. For a small

100

4

BRST cohomology algebras

Definition 4.1 A conformal $QOA$ with central charge
tative $QOA$ equipped with a homomorphism :$u(z)\in O$ ,$f$$\kappa$

is a commuis a pair $(O, f)$ , where
homogeneous
such that for every$O$$O_{\mathrm{i}},.(L)arrow O$(4.30)$fL(z)u(w)=\cdots+||u||u(w)(Z-w)-2+\partial u(w)(Z-w)^{-}1+:fL(\mathcal{Z})u(w)$ :where “ ” denotes the higher order polar terms. In other words, $fL(z)\circ_{1}u(z)=||u||u(w)$or simply:
and $fL(z)\mathrm{o}_{0^{u}(}Z)=\partial u(z)$ . For simplicity, we sometimes write$O_{\kappa}(L)arrow O$ , to denote a
conformal $QOA.$ A homomorphism $(O, f)\prec^{h}(O’, f’)$ of conformal
QOAs is a homomorphism of QOAs $h:Oarrow O’$ such that $h\mathrm{o}f=f’$ .
$\ldots$$O_{\kappa}(L)arrow O_{f}$$f$Recall that $M$ is a positive energy $Vir$ -module of central charge if for every $v\in M$ , weacts diagonalizably. It’s easy to show that $M$ is a positive
have $L_{n}\cdot v=0$ for $n>>0$ , andenergy $Vir$ -module iff there’s a quantum operator $X(z)\in QO(M)$ which commutes with itself
and has the OPE$\kappa$$L_{0}$$X(z)x(w)= \frac{\kappa}{2}(Z-w)-4+2X(w)(\mathcal{Z}-w)-2+\partial X(w)(z-w)^{-}1+:X(z)X(w):$.

(4.31)

Lemma 4.2 Let $X(z)\in QO(M)$ define a positive energy $Vir$ -module. Then every subalgebra-module defined by $L_{n}\cdot u(z)=$of $QO(M)$ containing $X(z)$ is naturally a positive energy$V_{i}r$$X(z)0_{n+}1u(z)$.

Lemma 4.3 Let be a commutative $QOA$ generated by a set $S\subset QO(M)$ , and let $X(z)\in O$
have $OPE$ (4.31). Suppose for all $u(z)\in S$ ,$O$

$X(z)u(w)=\cdots+||u||u(w)(z-w)^{-}2+\partial u(w)(z-w)^{-}1+:X(z)u(w)$ :.

(4.32)

such that $fL(z)=X(z)$ . Moreover $(O, f)$
Then there’s a unique homomorphism :is a conformal $QOA$ with central charge . In particular, every positive energy $Vir$ -module $M$-module structure.
has a canon ical$f$$O_{\kappa}(L)arrow O$$\kappa$$O_{\kappa}(L)$We refer the interested readers to section 4 of [24]Consider, as an example, $O(b, c)$ . For a fixed$\lambda$$\mathrm{f}\mathrm{o}1$

the complete proofs.

, let$X(z)=(1-\lambda):\partial b(Z)c(z)-\lambda$:$b(Z)\partial c(Z)$:(4.33)

and $S=\{b(z), C(z)\}$ . Then we have, by direct computation [13],:$X(z)b(w)$$=$$\lambda b(w)(_{Z}-w)-2+\partial b(w)(Z-w)^{-}1+:x(_{Z)(w)}b$$X(_{Z})C(w)$$=$$(1-\lambda)c(w)(z-w)^{-2}+\partial C(w)(Z-w)-1+:x(_{Z)_{C(w)}}$$X(z)x(w)$$=$$\frac{\kappa}{2}(z-w)^{-}4+2x(w)(z-w)^{-}2+\partial x(w)(z-w)-1+:x(_{Z)X}(w):(4.34)$

:

101

where $\kappa=-12\lambda^{2}+12\lambda-2$ . It follows that we have a conformal QOA$f_{\lambda}$: $O_{\kappa}(L)arrow O(b, c)$ .

Similarly $id:O_{\hslash}(L)arrow O_{\kappa}(L)$ is itself a conformal QOA. Thus by definition, it is the initial
object in the category of conformal QOAs with central charge .
$\kappa$

Exercise 4 In the previous exercise, we define $j(z)=:c(z)b(z)$ : which has $||j||=1$ . Compute
the generating function $\sum_{n}dimO(j)[n]q^{n}$ .
Exercise 5 Use Lemma 4.3 to classify the homomorphisms $f:O_{\kappa}(L)arrow O(j)$ .
Exercise 6 Show that $O(j)$ coincides with the subalgebra $O(b, c)^{0}$ of all elements of $u(z)\in$$O(b, c)$ with $|u|=0$ .
(Hint: Compute the generating function for the graded dimensions of$O(b, c)0$ . Alternatively for the readers who know it, you can use the so-called boson-fermion
correspondence.)

Exercise 7 Use the last two exercises to classify the (for each ) the homomorphisms$\lambda$$O_{\kappa}(L)arrow O(b, C)$

4.1

$f$

:

.

The BRST construction

It is evident that if $(O, f),$ $(O’, fJ)$ are conformal QOAs on the respective spaces $V,$ $V’$ with, then $(O\otimes O’, f\otimes f’)$ is a conformal QOA on $V\otimes V’$ with central charge
central charges
now has central charge-26.
. From now on we fix $\lambda=2$ which means that $(O(b, C),$
consider
and
$(O,
f)$
chargewithcentral
QOA
be any conformalLet$\kappa’$$\kappa,$$\kappa+\kappa’$$f\lambda)$$\kappa$(4.35)$C^{*}(O)=O(b, c)\otimes O$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}*\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{s}$the total first degree. For simplicity, we write$\hat{f}=f_{\lambda}\otimes f$.

Proposition 4.4 For every $(O, f)$ , there is a unique homogeneous element $J_{f}(z)\in C^{*}(O)$ withthe following properties:(i) (Cartan identity) $J_{f}(Z)b(w)=\hat{f}L(w)(z-w)^{-}1+:J_{f}(z)b(w):$ .
(ii) (Universality) If $(O, f)arrow(O’, f’)$ is a homomorphism of conformal QOAs, then the induced
to.
homomorphism $C^{*}(O)arrow C^{*}(O’)$ sends$J_{f}(Z)$$J_{f’}(Z)$as the initial
Proof: Since the category of conformal QOAs with central charge hassatisfying property (i), then (ii) implies that
object, if we can show that there is a unique
the same holds for every other object in that category.
$\kappa$$(O_{\kappa}(L), id)$

$J_{id}$Property (i) implies $|J_{id}|=1=||J_{id}||$ . Let’s list a basis of the degree $(1, 1)$ subspace of. Take a linear
given by Propositions 3.4, 3.5: : $c(z)L(z)$ :, : $b(z)C(z)\partial_{C}(Z)$ :,$C(O_{\kappa}(L))$$\partial^{2}c(z)$102

combination of these elements and compute its OPE with $b(z)$ . Requiring property (i), we
determine the coefficients of the linear combination and get$J_{id}(z)=:c(Z)L(Z):+:b(Z)C(Z)\partial c(\mathcal{Z}):$.

(4.36)

Now given a conformal QOA $(O, f)$ , the induced map: $C^{*}(o_{\kappa}(L))arrow C^{*}(O)$ sends$J_{j}(z)=:c(z)fL(z)$ $:+:b(z)C(z)\partial_{C}(Z)$ :. This completes our proof.
.
$f^{*}$$J_{id}(z)$to$\square$

It follows from property (i) that$\hat{f}L(z)=J_{f}(_{\mathcal{Z}\mathrm{I}^{0}(}\mathrm{o}bz)=[Q_{f}, b(Z)]$(4.37)

where $Q_{f}={\rm Res}_{z}J_{f}(z)$ .
Lemma 4.5 [$\mathit{1}\mathit{8}\mathit{1}[7\mathit{1}[\mathit{9}\mathit{1}$

Given

$f$

:

$O_{\kappa}(L)arrow O$, we have$J_{f}(w) \mathrm{o}0Jf(w)=\frac{3}{2}\partial(\partial^{2}C(w)c(w))+\frac{\kappa-26}{12}\partial^{3}c(w)c(w)$(4.38)

Thus $Q_{f}^{2}=0$ iff $\kappa=26$ .
Proof:
drop the subscripts forand write $fL(z)$ as $L(z)$ . Since $J(w)0_{0^{J(w}})$ is thecoefficient of $(z-w)^{-1}$ in the OPE $J(\approx)J(w)$ , we can extract this term from the OPE. Now$J(z)J(w)$ is the sum of 4 terms:$\mathrm{W}\mathrm{e}’ 11$$J_{f},$$Q_{f}$(i)$c(z)L(_{Z)}c(w)L(w)$$(i_{i}.)$$c(_{Z})L(_{Z}):b(w)C(w)\partial c(w)$:(iii) : $b(\approx)C(Z)\partial C(z):c(w)L(w)$(iv) : $b(z)C(Z)\partial C(z)::b(w)C(w)\partial c(w):$ .

(4.39)

Extracting the coefficient of $(z-w)^{-1}$ (which is done by applying Lemma 3.2 repeatedly) ineach of these 4 OPEs, we get respectively (surpressing ):
$w$

(i)
(ii)
(iii)

(iv)

$2 \partial ccL+\frac{\kappa}{12}\partial^{3}cC$$c\partial cL$$c\partial cL$$\underline{\frac{3}{9}}\partial(\partial^{2}Cc)-\frac{13}{6}\partial^{3}Cc$(4.40)

Thus (4.38) follows. Now $2Q^{2}=[Q, Q]={\rm Res}_{w}[Q, J(w)]={\rm Res}_{w}J(w)\mathrm{o}0J(w)$, which is zero iff$\kappa=26$.

$\square$Given: $O_{26}(L)arrow O,$ $[Q_{f}, -]=J_{f}(z)0_{0}$ is a derivation of the QOA $C^{*}(O)$ (Lemma 2.2).
For, which we assume from now on,,
becomes a differential on $C^{*}(O)$ and we havea cochain complex$[Q_{f}, -]$ : $c*(O)arrow C^{*+1}(o)$ .
(4.41)$f$$\kappa=26$$[Q_{f}$

- $]$

103

It is called the BRST complex associated to : $O_{26}(L)arrow O$ . Its cohomology will be denoted
on $C^{*}(O)$ descend to the cohomology. However, all but one isas $H^{*}(O)$ . All the operations
trivial.
$f$$0_{n}$

induces a graded commutative associative
product
The
Theorem 4.6
product on $H^{*}(O)$ with unit element represented by the identity operator. Moreover, every cohomology class is represented by a quantum operator $u(z)$ with $||u||=0$ .
$\mathrm{T}/Vick$$[\mathit{3}\mathit{3}]l\mathit{3}7\mathit{1}[\mathit{2}\mathit{6}\mathit{1}$$0_{-1}$

Exercise 8 Check that 1 represents the unit of the commutative algebra $H^{*}(O)$ . Show that for
all $n\neq-1_{f}0_{n}$ is homologically trivial, . if $u(z),$ $v(z)$ represent two cohomology classes, then$u(z)\mathrm{o}_{n}v(z)=[Q_{f}, t(z)]$for some $t(z)$ . (Hint: Recall that $\partial A(w)=[Q_{f}, b(w)]\mathrm{o}_{0}A(w)$ and$ie$$||A||A(w)=[Q_{f}, b(w)]\mathrm{o}_{1}A(Z).)$

Batalin-Vilkovisky Algebras

4.2

be a graded commutative associative algebra. For every $a\in A$ , let denote the linearLet
map ongiven by the left multiplication by . Recall that a (graded) derivation on is ahomogeneous linear operator such that $[d, l_{a}]-l_{da}=0$ for all . A BV operator $[34][31][15]\triangle$onis a linear operator of degree-l such that:;(i).
is a second order derivation.
is a derivation on for all(ii)$A^{*}$$l_{a}$$\mathrm{Z}$

$A$

$d$

$a$

$A$

$a$

$A^{*}$$\triangle^{2}=0$$A$$[\triangle, l_{a}]-l_{\Delta a}$$a,$$\mathrm{i}\mathrm{e}$$\triangle$

is a graded commutative algebra andwhere
A BV algebra is a pair
operator on . The following is an elementary but fundamental lemma:$A$$(A, \triangle)$$\triangle$

is a BV$A$

Lemma 4.7$[\mathit{2}\mathit{1}]l\mathit{1}\mathit{5}\mathit{1}[\mathit{3}\mathit{0}]$Given a $BV$ algebra$(A, \triangle)$, define the $BV$ bracket ,$\{$$(-1)^{|u|\{\}}u,$$v=[\triangle, l_{u}]v-l_{\Delta}uv$

Then ,$\{$$\}$

is a graded Lie bracket on$A$of degree$- \mathit{1}_{f}ie$$\}$on A by:.

.

$\{u, v\}+(-1)^{\mathrm{t}|u|}-1)(|v|-1)\{v,u\}=0$$(-1)^{(||1)(|t}u-|-1)\{u, \{v,t\}\}+(-1)^{(|t|)(||)}-1v-1\{t, \{u, v\}\}+(-1)^{\langle 1v}|-1)(|u|-1)\{v, \{t, u\}\}=0$,
is a derivation on
By property (ii) above, it follows immediately that for every $u\in A,$
which,
mathematics, isinalgebraan
odd
Poissonspecial
a
kind
of
is
a
. Thus BV algebrais canonically a Liealso known as a Gerstenhaber algebra [14]. It’s important to note thatis an-module.
algebra and that each$\{u$- $\}$$A$$A^{1}$$A^{p}$$A^{1}$

its exterior algebra and the Chevalley-Eilenberg
be an Lie algebra,$BV$
is canonically isomorphic
algebra. Showis a
. Check thatdifferential
to as a Lie algebra.

Exercise 9 Let$\delta$$\wedge^{*}\mathrm{g}$$\mathrm{g}$$on\wedge^{*}\mathrm{g}$$\mathrm{g}$$(\wedge^{*}\mathrm{g}, \delta)$$that\wedge^{\mathrm{l}}\mathrm{g}$104

be any commutative algebra and : $\mathrm{g}arrow DerB$ be a Lie
a -module). Consider the Lie algebra homology complex
algebra homomorphism (making. Show that the Chevalley-Eilenberg differential is a $BV$ operator on this complex.

Exercise 10 More generally, let

$B$

$f$

$B$

$\mathrm{g}$$\wedge^{*}\mathrm{g}\otimes B$

Given

$f$

$b(z)\circ 1u(z)$: $O_{26}(L)arrow O$ , consider the linear operator.

$C^{*}(O)arrow C^{*-1}(O),$$\triangle_{f}*$.

$u(z)\vdash\Rightarrow$descends to the cohomology $H^{*}(O)$ . Morever, it is a $BV$
Theorem 4.8 [26] The operator
operator on the commutative algebra $H^{*}(O)$ . Thus $H^{*}(O)$ is naturally a $BV$ algebra.
$\triangle_{J}$For complete proofs of the two theorems above, see section 4 of [24]. The theorems were originallyproved in [26] in the context of vertex operator algebras. (For related versions of Theorem 4.8,
see $[15][31][19][16].)$
as a bifunctor from pairs of conformal QOAs to $BV$ algebras.
Program 4.9 Studyand study $H(O\otimes-)$ as a functor from conformal QOAs to $BV$ algebras.
In particular, fix
An automorphism group of acts by natural automorphisms on the functor $H(O\otimes -)$ .
$H^{*}(O\otimes \mathit{0}^{J})$

$O$

$O$

4.3

Modules: $O_{26}(L)arrow O$ , and an -module $M$ equipped with the structure homomorphism$g:Oarrow QO(M)$ . The homomorphism induces): $C^{*}(O)arrow QO(\wedge\otimes M)$ . We write, and theas $C^{*}(O, M)$ . By Lemma 4.5,
, its residue as
asturns $C^{*}(O, M)$ into a complex whose cohomology is denoted as $H^{*}(O, M)$ . It turns out that$H^{*}(O, M)$ is a module over the BV algebra $H^{*}(O)$ in a suitable sense. This will be the topic ofa future paper.
Consider$O$$f$$J_{f,g}(z)$$Q_{f,g}$$g^{*}\overline{J}_{f(}Z$$g^{*}$$g$$Q_{f,g}$$C^{*}(O)-\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{e}\wedge\otimes M$-module of central charge 26. By Lemma 4.3, we have a
Let $N$ be a positive energy
into adule
canonical homomorphism $g:O_{26}(L)arrow QO(N)$ . This
. On the other hand,is by definition the space of semi-infinite
with BRST differential
cochains
of $Vir$ with coefficients in $N$ (see $[7][9]$ ). The differential of this cochain, and its cohomology ascomplex is denoted by. It’s easily seen that we have$d_{N}=Q_{id,g}[9]$ . It follows that we have$V_{i}r$$\mathrm{m}\mathrm{a}\mathrm{k}\mathrm{e}\mathrm{s}\wedge\otimes N$$C^{*}(o_{26}(L))- \mathrm{m}o$$\wedge\otimes N$$Q_{id,g}$$C^{\frac{\infty}{2}+*}(Vir, N)$$H^{\frac{\infty}{2}+*}(Vir, N)$$d_{N}$$H^{*}(O_{26}(L\mathrm{I}, N)\cong H^{\frac{\infty}{2}}+*(Vir, N)$.

(4.42)

Now given a conformal QOA $(O, f)$ and an -module $M$ equipped with the homomorphism: $Oarrow QO(M)$ , we can regard $M$ as an $O_{26}(L)$ -module via: $O_{26}(L)arrow QO(M)$ . It
follows that $(C^{*}(O, M),$ $Q_{J,g})=(C^{*}(O_{26}(L), M),$ $Q_{id,0}gf)=(C^{\frac{\infty}{2}+*}(Vir, M),$as complexes.
$O$$g\mathrm{o}f$$g$$d_{M})$$u(z)rightarrow u(-1)1$ (see proof of Lemma 3.4). Given
Recall the linear isomorphism$C^{*}(O)=O(b,c)\otimes
Oarrow\sim\wedge\otimes O$ . Call this map
a conformal QOA $(O, f)$ , we have
. We claim$O(b, c)\sim\prec\wedge,$

$k$

105

via coincides with the semi-infinite differentialthat the differential induced$J_{f}(z)\mathrm{o}_{0}(u(z)\otimes v(z))\vdash^{k}+d_{O}(u(-1)1\otimes v(z))$ .
must check that$\mathrm{o}\mathrm{n}\wedge\otimes O$$k$$d_{O}$. We

Let $a(z)=:b(z)C(z)\partial_{C}(Z)$ :. It acts only on , and hence $a(z)\circ_{0}(u(z)\otimes v(z))=[a(0), u(z)]\otimes$$v(z)rightarrow^{k}a(\mathrm{o})u(-1)1\otimes v(z)$ .
Use Lemma 2.6 to compute the OPE $c(z)fL(Z)(u(w)\otimes v(w))$ andget $c(z)fL( \mathcal{Z})\mathrm{o}_{0}(u(z)\otimes v(z))=\sum(c(Z)\mathrm{o}_{n}u(z))\otimes(fL(z)0_{-n}-1v(z))$ . It’s also easy to check,. $(2.4)$ , that under $O(b, c)arrow\wedge$ , we have $c(z)\mathrm{o}_{n}u(Z)\vdash+c(n)u(-1)1$ . It follows thatusing$\wedge$$\mathrm{e}\mathrm{q}\mathrm{n}$$J_{f}(z)\circ_{0}(u(z)\otimes v(z))rightarrow^{k}$$\sum c(n$I $u(-1)1\otimes L(-n-1)\cdot v(z)+a(0)u(-1)1\otimes v(z)$$=d_{O}(u(-1)1\otimes v(z))$

The last equality follows from the definition of$d_{O}$.

. ThusLemma 4.10 Given a conformal $QOA(O, f)$ and an
phism $g:Oarrow QO(M)$ , we have(i) $(C^{*}(O), Qf)\cong(C^{\frac{\infty}{2}+*}(V_{i}r, O),$ do).
.
$(ii)(c*(O, M),$$\mathrm{w}\mathrm{e}’ \mathrm{v}\mathrm{e}$$O$(4.43)

shown that-module $M$ equipped with homomor-$Q_{f},g)=(C^{\frac{\infty}{2}+*}(Vir, l\nu I),$ $d_{M})$5

Moonshine Cohomology

using the Moonshine VOA of rank 24 and the BRST construcnow construct a functor
tion above. This will be a functor from the category of conformal QOAs of central charge 2 toto
. In particular, it assigns a Lie algebra
the category of BV algebras
every such conformal QOA . As a special case, if $0$ is the conformal QOA corresponding to(see below), thenis Borcherds’ Monster
the unimodular rank 2 hyperbolic lattice
Lie algebra.
$\mathrm{M}$$\mathrm{W}\mathrm{e}’ 11$$\mathrm{M}^{1}(O)$$(\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{g}\mathrm{r}\mathrm{a}\ln 4.9)$$O$$\mathrm{M}^{1}(O)$$II_{1,1}$-Meurman andLet (V, 1, $Y(-,$ $z)$ ) be the Moonshine VOA as studied byBorcherds $[10][4][11]$ (see Definitions 8.10.1-8.10.18 of [11]). It’s now known thatis the automorphism group of the VOA .
(i) The Fischer-Griess Monster finite group$j(q)$$\sum
dimV[n]q^{n-1}=j(q)-744$
isthe Dedekind-Klein j-function.
where
(ii)
(iii) $Y(\omega, z)$ defines a unitarizable $Vir$ -module structure on .
$\mathrm{F}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{e}1- \mathrm{L}\mathrm{e}\mathrm{p}_{\mathrm{o}\mathrm{w}\mathrm{S}}\mathrm{k}\mathrm{y}$$\omega,$$V$$F_{1}$$V$) Let $(U, 1,\omega, Y(-, Z))$ be a $VOA$ of rank . Let $O(U)$ be the linear
Proposition 5.1 (seespace of vertex operators, . $O(U)=\{Y(a, z)|a\in U\}$ . Then $O(U)\subset QO(U)$ is a conformal$QOA$ with $O_{\kappa}(L)arrow O(U),$ $L(z)\vdash*Y(\omega, z).$ IVIoreover, $O(U)$ has an $O(U)$ -module structure\^u $(n)\cdot v(Z)=defu(z)\mathrm{o}_{n}v(Z)$ .
$O(U)arrow QO(O(U))$ defined by$[\mathit{2}\mathit{5}f$$\kappa$$\dot{i}e$$u(Z)-\# 1^{\wedge}\mathrm{t}(Z),$Proof: By Lemma 2.6 above and Proposition 8.10.5 of [11], we have$(z-w)-n-1=\langle Y(Y(u, z-w)v, w)\rangle$ . Thus $O(U)$ is closed under all the$Y(u,$ $w)$ o n $Y($$\sum\langle$$v,$ $w)\rangle$

106

products . We also see that $O(U)$ has an $O(U)$ -module structure as claimed. By Proposition
8.10.3 of [11], $O(U)$ is commutative. By definition, $Y( \omega, z)=\sum\omega(n)z-n-1$ satisfies, for all ,$Y(\omega(\mathrm{o})u, z)=\partial Y(u, z),$ $\omega(1)u=||u||u,$. By Lemma 4.3, this means that $O(U)$ is
a conformal QOA as claimed.
$0_{n}$$u$$\omega(\underline{9})\omega=\frac{\kappa}{2}\omega$$\square$

It follows immediately that the linear bijection $Uarrow O(U),$ $arightarrow Y(a, z)$ is an isomorphism
It’s also clear that any automorphism of the VOA yields an automorphismof the conformal QOA $O(U)$ . In the case of the Moonshine VOA , it follows thatacts by automorphisms.
(iv) $O(V)$ is a conformal QOA of central charge 24, in which
of $O(U)$ -modules.

$U$

$V$

$F_{1}$

Thus for any conformal QOA $0$ of central charge 2, we can consider the BRST QOA$C^{*}(O(V)\otimes O)$ .
, which by Theorem 4.8 is a BV algeWe denote its cohomology as
the Moonshine cohomology of . If $M$ is an -module, then $V\otimes M$ isbra. We callnaturally an $O(V)\otimes O$ -module. It follows from the preceeding section that $C^{*}(O(V)\otimes^{o}, V\otimes M)$, which we call the Moonshine
is a cochain complex. We denote its cohomology as
is a group of automorphisms of bothcohomology of $(O, M)$ . It follows from (iv) above that
(Program 4.9).
and$\mathrm{M}^{*}(O)$$O$$\mathrm{M}^{*}(O)$$O$$\mathrm{M}^{*}(O, M)$$F_{1}$$\mathrm{M}^{*}(O, M)$$\mathrm{M}^{*}(O)$Vanishing Theorem

5.1

A $Vir$-module is tame if it’s graded dimensions are finite. A $Vir$ -module is hermitean if it’sa direct sum of a tame positive energy modules equipped with an invariant nondegeneratehermitean form. A hermitean $Vir$ -module is unitarizable if its hermitean form is positive definite.
Unless specified otherwise, $Vir$ -modules and conformal QOAs from now on are assumed to havethe first degree.
$|\cdot|\equiv 0$

Theorem 5.2 For any conformal $QOAO$ of central charge 2, and any
vanish for all $p\neq 0,1,2$ , or 3.

$O$-module $M,$$\mathrm{M}^{p}(O)$,$\mathrm{M}^{p}(O, M)$

Theorem 5.3 Let be a real number with $1<r<25$ . Let and $N$ be positive energy Virmodules of central charges $26-r,$ respectively. Assume is unitarizable. Then
$N)$ vanishes
for all $p\neq 0,1,2$ , or 3.
$P$

$r$

$P$

$r$

$H^{\frac{\infty}{2}+p}(Vir,$ $P\otimes$

Proof: By the unitarizability of , it’s a direct sum of irreducible modules $L(26-r, h),$ $h\geq 0$ ,with suitable multiplicities. Thus it’s enough to do the case $P–L(26-r, h)$ .
$P$

Recall that $N$ is a $Vir$ -module of central charge , in whichacts diagonalizably. Since
every cohomology class inis represented by an element ofzero weight, we may assume, without loss of generality, that
only has real eigenvalues in$N$ .
Thus any irreducible module $L(r, k)$ occuring in the composition series of $N$ must havethe structure of the Verma modules, $L(r, k)=M(r, k)$ unless $k=0$ , andreal .
$L(26-r, h)=M(26-r, h)$ for $h>0$ .
$r$$L_{0}$$H^{\frac{\infty}{2}+*}(V_{\dot{i}\Gamma}, L(26-r, /\mathrm{t})\otimes N)=0$$L_{0}$$k$$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$107

By our reduction theorem on semi-infinite cohomology [27], for $k\neq 0$ or $h>0$ , we have$H^{\frac{\infty}{2}+p}(Vir, L(26-r, h)\otimes L(r, k))=0$ for $p\neq 1,2$ . It’s easy to verify that$H^{p}(O_{26}(L), L(26-r, \mathrm{O})\otimes L(r, 0))$ is zero if $p\neq 0,3$ , and one dimensional if $p=0,3$ . Thus if $N$. Every finitely
is a module of finite length, then $H^{\frac{\infty}{2}+p}(Vir, L(24, \mathrm{o})\otimes N)=0$ for $p\neq 0,$
dule is a
generated positive energy $Vir$ -module of central charge has finite length. Now any
torespect
limit
is
exact
with
cohomology.
direct limit of finitely generated modules, and direct. This completes the proof.
for $p\neq 0,$
If follows that
$..,$

$r$

$H^{\frac{\infty}{2}+p}(Vir, L(24, \mathrm{o})\otimes N)=0$$3$$\mathrm{m}o$$3$$\square$$..,$

Proof of Theorem 5.2: Specialize Theorem 5.3 to the case $r=2,$ $P=O(V)$ (which isvanishes for all $p\neq$
unitarizable), $N=O$ , and applying Lemma 4.10, we see that
$N=M$
for
.
$0,1,2$ , or 3. For, we have a similar statement$\mathrm{M}^{p}(O)$$\mathrm{M}^{p}(o, M)$$\square$Moonshine cohomology and the Monster Lie algebra

6

be respectively the subalgebras spanned by
Let $M$ be a positive energy $Vir$ -module. Let
$\pm
n>0$
. Define the physical space associated to $M$ :the$Vir_{\pm}$$L_{n},$(6.44)$\mathrm{P}(M)=M[1]^{Vir}+/N(M)$

where $N(M)=(Vir_{-}\cdot M)\cap M[1]^{Vir}+$ . If $M$ is a hermitean module of central charge 26, then$i=1,2$ , given respectively by
there are two natural linear maps l ノ i :$vrightarrow c(-1)v,$ $v\}\Rightarrow C(-2)c(-1)v$ (see [26] section 2.4 for details). To emphasize their dependence
on $M,$
refer to these maps as
for the module $M$ . Let be a conformal QOA. Supposethe-module structure : $O_{\kappa}(L)arrow QO(O)$ on , given by Lemma 4.2, is hermitean. Thenfor .
it makes sense to consider the maps$\mathrm{P}(M)arrow H^{\frac{\infty}{2}+i}(Vir, M),$$O$$\mathrm{w}\mathrm{e}’ 11$$\nu_{1},$$V_{i}r$$\nu_{2}$

$O$

$f$

$\nu_{1},$If$u(z)\in O^{Vi_{\Gamma}}+\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$$O$$\nu_{2}$it’s easy to show using commutativity that
. This implies that$u(z)00fL(z)=fL(z)\mathrm{o}_{0(Z)}u-\partial u(\mathcal{Z})=0$$u(z)\mathrm{o}0(fL(z)\mathrm{o}_{n}v(z))=fL(z)\mathrm{o}_{n}(u(z)\mathrm{o}0v(z))$.

(6.45)For $v(z)\in O^{Vi_{\Gamma}}+$ , this shows that
. For $v(z)\in N(O)$ , it shows that$u(z)\mathrm{o}0v(Z)\in N(O)$ . Using commutativity, we show that $u(z)\mathrm{o}_{0}v(Z)+v(Z)00u(z)=\partial A(z)$ for, and it also factors
some $A(z)$ . Thus
is a skew symmetric product onsays exactly thatthrough $N(O)$ . The fact that $u(Z)0_{0}$ is a derivation of the productthe skew symmetric operation on
satisfies that Lie algebra Jacobi identity. Thus
is a Lie algebra with bracket .
use the convention that $-u(z)\mathrm{o}0v(Z)$ is the Lie bracket
of $u(z)$ with $v(z)$ .
$u(z)\mathrm{o}_{0^{v}}(z)\in O^{Vi\mathrm{r}}+$$O^{Vi\Gamma}+\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{o}N(O)$$0_{0}$$v(z)\mathrm{o}_{0}t(Z)$$0_{0}$$\mathrm{p}(O)$$\mathrm{p}(O)$$\mathrm{W}\mathrm{e}’ 11$$0_{0}$

If
has central charge 26, then the maps
together with Lemma 4.10 yield two new$\mathrm{P}(O)arrowH^{i}(O)$maps (which we also call ),. The bracket in $H^{1}(O)$ can be written as:$\{A(z), B(z)\}=(-1)^{|A|}(b(Z)0_{0^{A(z}}))\mathrm{o}_{0}B(Z)$ . (see [24] section 5 for details). Thus,$O$$\nu_{i}$$\nu_{i}$$\nu_{i}$$\{\nu_{1}u(Z), \nu_{1}v(z)\}$$=$$\{c(\mathcal{Z})u(_{Z)}, C(_{Z)}v(z)\}$

108

$=$

$-u(z)00(_{C}(\mathcal{Z})v(z))$$=$$\nu_{1}(-u(Z)\mathrm{o}_{0^{v}}(z))$.

(6.46)

Thus is a Lie algebra homomorphism. Since $H^{2}(o)$ is canonically a $H^{1}(O)$ -module, it becomes-module via . But we also havea$\nu_{1}$$\mathrm{p}(O)$$\nu_{1}$$\{\nu_{1}u(Z), \nu 2v(Z)\}=\nu 2(-u(z)\mathrm{o}_{0}v(Z))$(6.47).

to $H^{2}(O)$ . To em-module homomorphism from the adjoint module
is a
Thus
refer to those homomorphisms as
for thephasize their dependence on the QOA $O,$$QOAO$ . To summarize,$\nu_{2}$$\mathrm{p}(O)$$\mathrm{p}(O)$$\mathrm{w}\mathrm{e}’ 11$

Lemma 6.1$\nu_{1\prime}\nu_{2}$Given a hermitean $Vir$ -module $M$ of central charge 26, $we’ ve$ two linear maps$\nu_{i}$:$\mathrm{P}(M)arrow H^{\frac{\infty}{2}+i}(Vi.r, M)=H^{i}(O_{26}(L), M)\mathrm{z}\dot{i}=1,2\rangle g_{\dot{i}}ven$ by $u-,$ $c(-1)u,$ $u\vdash+c(-2)c(-1)v$a Lie algebra
respectively. Given a hermitean conformal $QOAO$ of central charge: $\mathrm{p}(O)arrow H^{i}(O),$ $i=1,2$ , given byhomomorphism and a module homomorphism$c(z)u(Z)\prime u(z)-t\partial c(z)c(z)u(z)$ respectively.
$\mathit{2}\mathit{6}_{2}we’ ve$$u(z)\vdasharrow$$\nu_{i}$. lattice, andbe the canoniLet be any rank 2 hyperbolic evenis a direct sum of the so-called. The $Vir$ -module
cal rank 2 VOA associated to:
have
above,
we
Fock modules, which are hermitean. By the lemma$H^{i}(O(V)\otimes o(V\Lambda))=\mathrm{M}i(O(V_{\Lambda})),$ $i=1,2$ .
$(V_{\Lambda}, 1_{\Lambda},\omega_{\Lambda,\Lambda}Y(-, Z))$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}_{1\mathrm{a}1}$$\Lambda$$O(V_{\Lambda})\cong V\Lambda$$\Lambda[11]$$\nu_{i}$Theorem 6.2

The homomorphisms$\nu_{1},$$\nu_{2}$for the$QOAO(V)\otimes O(V_{\Lambda})$$we’ ve$$\mathrm{C}1$$\nu_{1}\mathrm{P}(O(V)\otimes^{o}(V\Lambda))$$\mathrm{M}^{p}(O(V_{\Lambda}))=\{$$\nu_{2}\mathrm{P}(O(V)\otimes o(V\Lambda))$$\mathrm{p}(O(V)\otimes O(V_{\Lambda}))arrow$are isomorphisms; andif $p=0$
if $p=1$ .
if $p=2$ .

$\mathrm{C}\partial^{2_{C}}(_{Z)(Z)C(z}\partial c)$$\dot{i}fp=3$$0$

otherwise.

(6.48)

is canonically isomor. Thenbe the unimodular lattice
Corollary 6.3 Let
module.
adjoint
thetophic to the Monster Lie algebra, and$II_{1,1}$$\Lambda$$\mathrm{M}^{1}(O(V_{\Lambda}))$$\mathrm{M}^{2}(O(V_{\Lambda}))$, and its
Proof: By definition [5], the Monster Lie algebra has as its underlying space
follows from Proposition
bracket $[u, v]=-{\rm Res}_{z}Y(u, z)v$ . Now
5.1.
$\mathrm{P}(V\otimes V_{\Lambda})$$\mathrm{p}(O(V)\otimes O(l/_{\Lambda}^{\mathit{7}}))\cong \mathrm{p}(V\otimes V_{\Lambda})$$\square$

The rest of this paper is devoted to proving and generalizing the theorem above.

109

Hyperbolic lattices

6.1

Let be a rank $r\leq 26$ even integral Lorentzian lattice, and. This VOA has rank . Let
rank VOA associated to-module.
charge $26-r$ such that it’s unitarizable as a$\Lambda[11]$

$r$

$r$

the canonicalbe any conformal QOA of central$(V_{\Lambda}, 1_{\Lambda},\omega_{\Lambda,\Lambda}Y(-, Z))$$\Lambda$$O$$V_{i}r$

Theorem 6.4 Under the above assumptions the homomorphisms
are $isomorphi_{S}mS$ , and$\nu_{1},$$\nu_{2}$for the$QOAO\otimes O(V\Lambda)$$H^{p}(o\otimes o(V_{\Lambda}))=$(6.49)

Theorem 6.2 is clearly an immediate consequence when $r=2$ and $O=O(V)$ . By Lemma 4.10,
Theorem 6.4 is equivalent toUnder the above assumptions the linear maps

Theorem 6.5
bijections, and$\nu_{1},$$\nu_{2}$for the module’$Hom_{V}ir(L(26-r, \mathrm{o}),$$\mathit{0})$$\nu_{1}\mathrm{P}(o\otimes V_{\Lambda}\mathrm{I}$$H^{\frac{\infty}{2}+p}(Vir, \mathit{0}\otimes V_{\Lambda})=$$\nu_{2}\mathrm{P}(O\otimes V_{\Lambda})$$c(-3)c(-2)c(-1)H_{\mathit{0}}m_{V}ir(L(26-r, \mathrm{o}),$$\mathit{0})$$O\otimes V_{\Lambda}$

if $p=0$
if $p=1$ .
if $p=2$ .
if $p=3$are(6.50)

otherwise.

$0$$\mathrm{Y}$be the standard pseudo-euclidean space of signature
We now proceed to prove this. Let
be the standard, let$(k, l)$ .
. Given
The inner product is written as
representation of the Heisenberg algebra with generators $j^{a}(n)$ and relations $[j^{a}(n),jb(m)]=$-). Here, -, ..-) with$\eta=$ diag( $+,$
( $b=1,$ $k+l,$$j^{a}(0)$
by
. We
is
denotedmoduleofthecanonical
generator
The.
scalar
by
the
acts
the
$j^{a}(z)jb(Z):\eta^{ab}$$Vir$. This:acts byas a $Vir$ -module in which
now regard each
[9].
module has the standard hermitean folm$\mathrm{R}^{k,l}$$\alpha\cdot\alpha$$n\delta_{n+0}m,\eta^{ab}id$$a,$$n,$ $m\in \mathrm{Z},$$..,$$\alpha\in \mathrm{R}^{k,l}$$..,$$F_{k,l}(\alpha)$$k+\mathrm{a}\mathrm{n}\mathrm{d}l$$+$$\alpha^{a}$$|\alpha\rangle$$L(z)= \frac{1}{2}$$F_{k,l}(\alpha)$

Theorem 6.6
bijections, and] For any$f\mathit{9}$$\alpha\in \mathrm{R}^{25,1}$, the linear maps$\nu_{1},$$\nu_{2}$for the module’$\delta_{\alpha,0}\mathrm{C}1\otimes|0\rangle$$\nu_{1}\mathrm{P}(F_{25,1}(\alpha))$$H^{\frac{\infty}{2}+p}(Vir, F25,1(\alpha))=$$\nu_{2}\mathrm{P}(F_{25,1}(\alpha))$$\delta_{\alpha,0}\mathrm{C}c(-3)_{C}(-2)C(-1)1\otimes|0\rangle$$0$

if $p=0$
if $p=1$ .
if $p–2$ .
if $p=3$otherwise.

$F_{25,1}(\alpha)$are(6.51)

110

For a proof, see the original reference.
Denote the highest vector of the $Vir$ -modules
For any $h\geq 0$ and$L(26-r, h)\otimes F_{r-1,1}(\beta)$ are bijections, andLemma 6.7$\beta\in \mathrm{R}^{\gamma\cdot-1,1}$$L(\prime_{1^{\wedge}},, h)$orby$l\mathrm{t}’I(\kappa, h)$, the linear maps$\delta_{h,0}\delta_{\beta},0\mathrm{C}1\otimes|26-r,$$\nu_{1},$$|\kappa,$$h\rangle$.

for the module$\nu_{2}$if $p=0$
if $p=1$ .
if $p=2$ .
if $p=3$$0\rangle\otimes|\beta\rangle$$\nu_{1}\mathrm{P}(L(26-r, h)\otimes F_{\Gamma-1,1}(\beta))$$H^{\frac{\infty}{2}+p}(Vir, L(26-r, h)\otimes F_{\Gamma}-1,1(\beta))=\{$$\nu_{2}\mathrm{P}(L(26-r, h)\otimes F_{r-1,1}(\beta))$$\delta_{h,0}\delta_{\beta},0\mathrm{C}c(-3)c(-2)c(-1)1\otimes|26-r,$ $0\rangle\otimes|\beta\rangle$

otherwise(6.52)$0$

Proof: The case $r=26$ is just Theorem 6.6. So let’s assume $r<26$ . The case $h=0,$ $\beta=0$ canbe easily checked by hand. So let’s assulne that either or is nonzero. We claim that any,. Choose so that
irreducible module $L(26-r, h)$ is direct sulnmand in some$M(26-r,/l)arrow
F_{26-r,\mathrm{o}(\gamma)}$.
$|26-r,$Since
witha homomorphism
andis unitarizable, the image must also be unitarizable. Thus it must also be an irreducible direct
summand.
$h$$\beta$$F_{26-\Gamma,0}(\alpha)$$\gamma$$h\rangle$$\mathrm{w}\mathrm{e}’ \mathrm{v}\mathrm{e}$Now observe that bothis isomorphic to(6.52).

$H^{\frac{\infty}{2}+p}(Vir, -)$$F_{26-\mathrm{r},0}(\gamma)\otimes F_{\mathrm{r}-1},1(\beta)$

and$F_{25,1}(\alpha)$$\vdasharrow|\gamma\rangle$$\frac{\gamma\cdot\gamma}{2}=h$$F_{26-\mathrm{r},0}(\gamma)$

are exact with respect to direct sum. Since, we see that Theorem 6.6 implies
for some$\mathrm{P}(-)$$\alpha\neq 0$$\square$is unitarizable and hence is a direct sum ofof of Theorem 6.5: By assumption,$h\geq0$
as $Vir$ -modules, if we choose. On the other hand$L(26-r, h),$R. Now the theorem follows from Lemma 6.7 and the fact
an identification
are exact with respect to direct sum.
andthat both$O$$\mathrm{P}\mathrm{r}o$$\dagger^{\gamma_{\Lambda}}=\oplus_{\beta\in\Lambda 1,1}F_{\mathrm{r}-}(\beta)$$\mathrm{R}^{\Gamma-1,1}=\Lambda\otimes \mathrm{z}$$H^{\frac{\infty}{2}+p}(Vir, -)$6.2$\square$$\mathrm{P}(-)$

Some applicationsin Theorem 6.4 is graded by because as a $Vir$ -module,
a decomposition of the Moonshine
is graded by . In particular,
is tame as $Vir$ -module, and. Since eachcohomology(see Theorem 6.2) are insince $O(V)$ is also tame, the graded dimensions
are finite dimensional representations of the group .
fact finite. Thus thecompute the dimensions using our results above together with well-known techniques in semiinfinite cohomology theory (see $[9][27]$ ).

The BV algebra$\Lambda$$H^{*}(O\otimes^{o}(V_{\Lambda}))$$\Lambda$$V_{\Lambda}\cong\oplus_{\alpha\in\Lambda}F_{1},1(\alpha)$$O(V_{\Lambda})\cong$$\mathrm{w}\mathrm{e}’ \mathrm{v}\mathrm{e}$$F_{1,1}(\alpha)$$\mathrm{M}^{*}(O(V_{\Lambda}))=\oplus_{\alpha\in\Lambda}\mathrm{M}^{*}(O(V_{\Lambda}))_{\alpha}$$di\uparrow n\mathrm{M}^{*}(O(V_{\Lambda}))\alpha$$F_{1}$$\mathrm{M}^{*}(O(V_{\Lambda}))\alpha$$\mathrm{W}\mathrm{e}’ 11$

is hermitean, there is an induced nondegenerate hermitean form
(see below). We will compute the signature of this hermiteanon the cohomology
form and show that it’s positive definite for nonzero (“no-ghost theorem”). We will restrict

Since each

$O(V)\otimes F_{1,1}(\alpha)$$\mathrm{M}^{1}(O(V_{\Lambda}))\alpha$$\alpha$111

ourselves to the case when is a rank 2 hyperbolic lattice. How to generalize our computations
to other Lorentzian lattices will becolne clear, and is left as an exercise to the readers.
$\Lambda$

and signatures of cohomology, it’s enough to work
only interested in
Sincewith the additive version of our results Theorem 6.5 and Lemma 6.7. We begin by introducing
one other tool: the notion of relative semi-infinite cohomology. We refer to readers to originalreferences for details.
$\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{S}\mathrm{i}_{0}\mathrm{n}\mathrm{s}$$\mathrm{w}\mathrm{e}’ \mathrm{r}\mathrm{e}$annihilated by $b(1)$ andbe the subspace of the $Vir$ -module
Let. Call this subspace. Because $[d_{M}, b(1)]=L_{0}$ , this subspace is a complex with differential
the relative cohomology. Note that if
the relative complex, and its cohomology$M$ is tame and its weightis finite dimensional.
is bounded from below, thencohomology. For
semi-infinite
Relative cohomology is an important tool for studying (absolute)
have to first prove a similar vanishing theorem
example, technically to prove Theorem 6.6,
for relative cohomology. In this papel, we manage to prove all our results so far without usingit. However for computing dimensions and signatures, relative cohomology is indispensable.
$C^{\frac{\infty}{2}+*}(Vir, M)$$C^{\frac{\infty}{\Delta^{2}}+*}(Vi_{\Gamma}, M)$$d_{M}$$L_{0}$$H^{\frac{\infty}{\triangle 2}+*}(Vir, M)$$C^{\frac{\infty}{\Delta 2}+*}(Vir, M)$$||\cdot||$$\mathrm{w}\mathrm{e}’ \mathrm{d}$. For simplicity,
be interested in
In this section,. Similar notationsassimplyabbreviate the absolute complex
apply to the absolute cohomology, the relative complex, and the relative cohomology.
$H^{\frac{\infty}{\Delta 2}+*}(Vir, V\otimes F_{1,1}(\alpha)),$$\mathrm{w}\mathrm{e}’ 11$$C^{\frac{\infty}{2}+*}(Vir, V\otimes F_{1,1}(\alpha))$$\alpha\in\Lambda$$\mathrm{w}\mathrm{e}’ 11$$C^{\frac{\infty}{2}+*}(\alpha)$

Lemma 6.8
(6.53)$H^{\frac{\infty}{\Delta 2}+p}(_{\mathit{0}})\cong$

Proof: The case $\alpha=0$ is a trivial exercise. There’s a long exact sequence (see [27] for details):.. .

$arrow H^{\frac{\infty}{\Delta^{2}}+p}(\alpha)arrow H^{\frac{\infty}{2}+p}(\alpha)arrow H^{\frac{\infty}{\Delta^{2}}+p-1}(\alpha)arrow H^{\frac{\infty}{\Delta^{2}}+p+1}(\alpha)arrow\cdots$.

(6.54)

in terms of its irreducible submodules $L(24, h)$ andnonzero. By decomposing
Assumeapplying Lemma 6.7 (for $r=2$ ), we see that the long exact sequence above degenerates andyields the desired result.
$V$$\alpha$$\square$Corollary 6.9$\mathrm{M}^{1}(O(V_{\Lambda}))\alpha\cong H^{\frac{\infty}{\Delta^{2}}}(a)+1$.

Thus to compute the graded dimensions of the Moonshine cohomology in degree 1 (which is a
, it’s enough to compute the corresponding-graded Lie algebra) for the conformal QOA-graded vector space $M$ , let $ch_{q}M=$
degree 1 relative cohomology. For a tame hermitian$\Sigma dimM[n]q^{n},$ $sign_{q}M=\Sigma SignM[n]q^{n}$ .
$O(1/_{\Lambda})’$$\Lambda$$||\cdot||$

By the Euler-Poincar\’e Principle, we have$\sum_{i}(-1)^{i}d_{im}H^{\frac{\infty}{\Delta 2}}(\alpha)+i=\sum_{i}(-1)^{i}dimC^{\frac{\infty}{\Delta 2}}(\alpha+i)$.

(6.55)

112

Note that $I\iota er\nearrow b(1)=Imb(1)$ . Thus the RHS of (6.55) is just the constant term of the following
q-series:$ch_{q}V \cdot Ch_{q}F1,1(\alpha)\cdot\sum_{i}(-1)ihb(C1)q\wedge^{i+1}$$=$$q(j(q)-744) \cdot q\frac{\alpha\circ}{2}\varphi(q)^{-2}\cdot(-1)q-1\varphi(q)^{2}$$=$$-q^{\frac{\circ\alpha}{2}}(j(q)-744)$(6.56)

where $\varphi(q)=\Pi_{n>0}(1-q^{n})$ . For $\alpha=0$ , the constant term on this RHS is zero–consistent with
$i=0,1,2$ , are respectively 1,2,1. For nonzero, combining Lemma
the fact that
. (6.55), we get
6.8, Corollary 6.9, and$dimH^{\frac{\infty}{\Delta^{2}}+i}(0),$$\alpha$$\mathrm{e}\mathrm{q}\mathrm{n}$$dim \mathrm{M}^{1}(o(1/’)\Lambda)_{\alpha}={\rm Res}_{q}q\frac{\alpha 0}{2}-1(j(q)-744)$.

(6.57)

such that $\langle 1, c(-3)c(-1)1\rangle=1,$
a unique hermitean form
The spacea hermitian space$b(-n+2)$ , and $c(n)\mathrm{t}=c(-n-4)[9]$ . This makes each complex
such that the semi-infinite differential (dropping the subscript) is self-adjoint. By the EulerPoincar\’e Principle for signature, we have$b(n)^{\uparrow}=$$\langle\cdot, \cdot\rangle$$b(1)\wedge^{*}\mathrm{h}\mathrm{a}\mathrm{S}$$c^{\frac{\infty}{\Delta^{2}}+*}(\alpha)$$d$.

$\sum_{1}$sign$H^{\frac{\infty}{\triangle 2}+i}( \alpha)=\sum_{i}$sign$c^{\frac{\infty}{\Delta 2}+i}(\alpha)$.

(6.58)As before, the RHS here is the constant term of the following q-series:$S \dot{i}gn_{q}V\cdot signF1,1(q)\alpha\cdot\sum_{*}$$sign_{q}b(1)\wedge^{i}+1$$=$$q(j(q)-744) \cdot q\frac{\mathfrak{a}\alpha}{2}\lambda(q)^{-}1.-q1\lambda(q)$$=$$q^{\frac{\alpha\alpha}{2}}(j(q)-744)$.

where $\lambda(q)=\Pi_{n>0}(1-q^{n})(1+q^{n})$ . Thus we conclude that forsign$\alpha$(6.59)

nonzero:(6.60)$H^{\frac{\infty}{\Delta 2}+1}(\alpha)=dimH^{\frac{\infty}{\Delta^{2}}+1}(\alpha)$

which is the statement of the no-ghost theorem. The interested readers should compare ourresults with the results of Borcherds [5] on graded dimensions and positive definiteness for the
Monster Lie algebra.

6.3

Remarks

In the light of our results, a number of interesting questions come to mind. We discuss two ofthem.

1. Recall that the Virasoro element of the Moonshine VOA is an -invariant element
is an -invariant quantum operator, and hence generates anof . This means that-invariant subalgebra in $O(V)$ . It’s easy to show that this subalgebra is isomorphic to $O_{24}(L)$ .
By unitarizability, $O_{24}(L)$ is a direct sumlnand in $O(V)$ as $Vir$ -module. Thus for any conformal$\omega$$V$$F_{1}$$Y(\omega, z)$$F_{1}$$V$$F_{1}$

113, as a BV algebra, has a canonical
QOA of central charge 2, the Moonshine cohomology. In particular, the vanishing theorem 5.2 holds for this algebra. Also,
subalgebra$H^{2}(o_{24}(L)\otimes O)$ is the adjoint module over the Lie algebra $H^{1}(o_{24}(L)\otimes O)$ .
$\mathrm{M}^{*}(O)$$O$$H^{*}(O_{24}(L)\otimes \mathit{0})$Problem: For any rank two hyperbolic lattice.
of$H^{1}(o_{24()}L\otimes O(V_{\Lambda}))$$\Lambda$, study the$\Lambda$-graded Lie subalgebra$\mathrm{M}^{1}(O(V_{\Lambda}))$acts as a group; thus,acts naturally on
2. Recall that the Monster finite groupas a functor. Since the full automorphism group ofof invertible natural transformations of
the Moonshine VOA , hence of the conformal QOA $O(V)$ , is isomorphic to , the following
seems quite plausible:$\mathrm{M}^{*}(O)$$F_{1}$$F_{1}$$\mathrm{M}^{*}$$V$

Conjecture:is isomorphic to$F_{1}$The group of natural automorphisms of the Moonshine cohomology functor$F_{1}$.

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DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY CAMBRIDGE, MA 02138.
lian@math.harvard.edu
DEPARTMENT OF MATHEMATICS, YALE UNIVERSITY NEW HAVEN, CT 06520.
gregg@math.yale.edu