Reduction cohomology on complex manifolds

A. Zuevsky

Historical introduction

There exists a natural problem of calculating continuous cohomologies ofholomorphic structures on complex manifolds [Bott-Segal, Feigin,Wagemann, Khoroshkin, Kawazumi]. The continuous cohomology of Liealgebras of C8-vector fields [Gelfand-Fuks, Feigin-Fuks] has proven to bea subject of great geometrical interest.

In 1990 Feigin has obtained various results concerning (co)homologies ofcertain Lie algebras associated to a complex curve M.

For the Hodge decomposition of the tangent bundle complexification ofM, corresponding Lie bracket in the space of holomorphic vector fieldsextends to a differential Lie superalgebra structure on the Dolbeaultcomplex. This is called the differential Lie superalgebra ΓpLiepMqq ofholomorphic vector fields on M. The Lie algebra of holomorphic vectorfields LieDpMq is defined as the cosimplicial object in the category of Liealgebras obtained from a covering of M by associating to any ordered seti1 ă i2 ă . . . ă il , the Lie algebra of holomorphic vector fieldsLie pUi1 X . . .X Uil q.

Feigin has calculated the continuous (co)homologies with coefficients incertain one-dimensional representations τcp,q of these Lie (super)algebraswhere cp,q denotes the value of the central charge for correspondingVirasoro algebra.

The main result states that H0pLieDpMq, τcp,qq is isomorphic toHpM, p, qq, where a representation τcp,q is derived from a vacuumrepresentation of Virasoro algebra.

[Wagemann] continued works of Feigin, Kawazumi on the Gelfand–Fukscohomology of the Lie algebra of holomorphic vector fields on complexmanifolds. To enrich the cohomological structure, he had to involvecosimplicial and differential graded Lie algebras well known inKodaira-Spencer deformation theory.

The idea to use cosimplicial spaces to study cohomology of mappingspaces goes back at to [Anderson], and it was further developed in[Bott–Segal]. In [Wagemann] they compute corresponding cohomologiesfor arbitrary complex manifolds up to calculation of cohomology ofspaces of sections for complex bundles on auxiliary manifolds.

Results obtained in [Wagemann] are very similar to results of [Haefliger]and [Bott–Segal] in the case of C8 vector fields. Following constructionsof [Feigin], applications in conformal field theory (for Riemann surfaces),deformation theory, and foliation theory were proposed in [Wagemann].

In addition to that, in [Wagemann] the Quillen functor scheme was usedfor the sheaf of holomorphic vector fields on a complex manifold, and itsfine resolution was given by the sheaf of dz̄-forms with values inholomorphic vector fields, the sheaf of Kodaira-Spencer algebras.

Let M be a smooth compact manifold and VectpMq be the Lie algebra ofvector fields on M. Bott and Segal proved that the Gelfand–Fukscohomology H˚pVectpMqq is isomorphic to the singular cohomologyH˚pE q of the space E of continuous cross sections of a certain fibrebundle E over M. Authors of [Patras–Thomas] continued to useadvanced topological methods for more general cosimplicial spaces ofmaps.

As it was demonstrated in [Wagemann], the ordinary cohomology ofvector fields on complex manifolds turns to be not the most effective andgeneral one. In order to avoid trivialization and reveal a richercohomological structure of complex manifolds cohomology, one has totreat [Feigin] holomorphic vector fields as a sheaf rather than takingglobal sections.

An important problem of understanding relations betweennon-commutative algebraic structures and geometrical objects oncomplex manifolds still remains underinvestigeted in the literature.

Our new algebraic and geometrical approach for computation ofcontinual cohomology involves Lie-algebraic formal series, andapplications of techniques used in surgery of spheres [Huang].

In contrast to more geometrical methods in ordinary cosimplicialcohomology for Lie algebras, the reduction cohomology we introducepays more attention to the analytical structure of elements (constructedthrough non-degenerate bilinear forms for Lie-algebraic modes withcomplex parameters) of chain complex spaces.

Computational methods involving recurrent relations for n-pointfunctions description proved their effectiveness in conformal field theory,geometrical description of intertwined modules for Lie algebras, anddifferential geometry of integrable models.

Results of reduction cohomology on complex manifolds, and ofgeneralizations of the Bott–Segal theorem have their consequences inconformal field theory, deformation theory, non-commutative geometry,modular forms, and the theory of foliations.

Example: Cohomology of Jacobi forms

For Riemann surfaces, and even for higher dimension complex manifolds,the classical cohomology of holomorphic vector fields is often trivial[Kawazumi, Wagemann].

Vertex algebra theory of modular forms goes back to celebratedMoonshine problem. Most of n-point characteristic functions for vertexalgebras deliver examples of modular forms with respect to appropriategroups attached to geometry of corresponding underlying manifolds.Also, n-point functions are subject to action of differential operators withspecific analytical behavior [Bringmann-Krauel-Tuite, Gaberdiel–Keller,Gaberdiel–Neitzke, Oberdieck].

In this lecture we develop ideas and previous results on cohomology ofJacobi forms originating from algebraic and geometrical procedures inconformal field theory.

We aim at developing algebraic, differential geometry, and topologicalmethods for the investigation of cohomology theories of Jacobi formsgenerated by vertex algebras, with applications in algebraic topology,number theory and mathematical physics.

In most cases, at least for lower genus Riemann surfaces, there existalgebraic formulas relating n-point functions with n´ 1-point functions ina linear way [Zhu, Mason-Tuite-Z]. Coefficients in the reduction formulasare expressed in terms of quasi-modular forms. The reductioncohomology is defined via such recursion formulas.

We consider specific examples of coboundary operators for spaces ofJacobi forms subject to various conditions on vertex algebra elements.

Quasi-Jacobi forms have found applications in vertex algebra theory in[Heluani–Van Ekeren], for characteristic functions of topological N “ 2vertex algebras, Gromov-Witten potentials [Kawai–Yoshioka],computation of elliptic genera [Libgober] related to Jacobi zero-pointfunctions, Landau-Ginzburg orbifolds [Kawai, Yamada, Yang].

Chain complex of n-point Jacobi functionsHere we give definition of the chain complex associated to a space ofJacobi forms. Let us fix a vertex algebra V .

We denote by vn “ pv1, . . . , vnq, vi P V vertex algebra elements. Denoteby zn “ pz1, . . . , znq local coordinates around n points on T . Let usintroduce the notation: xn “ pvn, znq. For a vertex operator algebra Vwith Virasoro vector ω of central charge c we formulate

DefinitionConsider an element J P V1 such that Jp0q acts semisimply on V . Forvn, and a V -module W , the Jacobi n ě 0-point function is

ZJW pxn;Bq “ TrW

´

Y´

ez LV p0qv, ez¯

nζJp0qqLp0q

¯

, (0.1)

where B denotes parameters of ZJW including z and τ , and

ζ “ qz “ e2πiz ,

with z P C, and τ being the modular parameter of T .

Quasi-Jacobi forms

Here we recall definitions and properties of Jacobi and quasi-Jacobi forms[Bringmann-Krauel-Tuite].

First, we provide the definition of ordinary Jacobi forms [Eichler–Zagier].Let H be the upper-half plane. Let k , m P N0, and χ be a rationalcharacter for a one dimensional representation of the Jacobi groupSLp2,Zq ˙ Z2. Let γ P SL2pZq,

γ “

ˆ

a bc d

˙

.

Define

γ.τ “aτ ` b

cτ ` d,

A holomorphic Jacobi form of weight k and index m on SL2pZq withrational multiplier χ is a holomorphic function

φ : CˆHÑ C.

which satisfies the following conditions. Then, for pλ, µq P Zˆ Z,

φˇ

ˇ

ˇ

k,mpγ, pλ, µqq “ χ pγ, pλ, µqqφ,

where for a functionφ : CˆHÑ C,

φˇ

ˇ

ˇ

k,mpγ, pλ, µqq pz , τq

“ pcτ ` dq´ke

ˆ

´cmpz ` λτ ` µq2

cτ ` d`m

`

λ2τ ` 2λz˘

˙

¨ φ

ˆ

z ` λτ ` µ

cτ ` d, γ.τ

˙

.

with epwq “ e2πiw .

For a multiplier χ,

χ

ˆ

a bc d

˙

“ χ

ˆˆ

a bc d

˙

, p0, 0q

˙

, χpλ, µq “ χ

ˆˆ

1 00 1

˙

, pλ, µq

˙

,

and N1,N2 P N uniquely defined by

χ

ˆ

1 10 1

˙

“ e2πi

a1N1 , χp0, 1q “ e

2πia2N2 ,

where aj P N. The function φ has a Fourier expansion of the form withq “ epτq, ζ “ epzq,

φ pz , τq “ÿ

nPN0`ρ1

ÿ

rPZ`ρ2r2ď4nm

cpn, rqqnζr ,

where ρj “ajNjpmod Zq with 0 ď ρj ă 1.

We next consider quasi-Jacobi forms as introduced in [Libgober].

DefinitionAn almost meromorphic Jacobi form of weight k , index 0, and depthps, tq is a meromorphic function in Ctq, ζurz´1, z2

τ2, 1τ2s with

z “ z1 ` iz2, τ “ τ1 ` iτ2) satisfying the main condition for φ, andwhich has degree at most s, t in z2

τ2, 1τ2

, respectively.

DefinitionA quasi-Jacobi form of weight k , index 0, and depth ps, tq is defined bythe constant term of an almost meromorphic Jacobi form of index 0considered as a polynomial in z2

τ2, 1τ2

.

Modular and elliptic functions

For a variable x , set

Dx “1

2πi

B

Bx,

and qx “ e2πix . Define for

m P N “ t` P Z : ` ą 0u,

the elliptic Weierstrass functions

P1pw , τq “ ´ÿ

nPZzt0u

qnw1´ qn

´1

2,

Pm`1pw , τq “p´1qm

m!Dmw pP1pw , τqq “

p´1qm`1

m!

ÿ

nPZzt0u

nmqnw1´ qn

.

Next, we have

DefinitionThe modular Eisenstein series Ekpτq, defined by Ek “ 0 for k for odd andk ě 2 even

Ekpτq “ ´Bk

k!`

2

pk ´ 1q!

ÿ

ně1

nk´1qn

1´ qn,

where Bk is the k-th Bernoulli number defined by

pez ´ 1q´1 “ÿ

kě0

Bk

k!zk´1.

It is convenient to define E0 “ ´1. Ek is a modular form for k ą 2 and aquasi-modular form for k “ 2. Therefore,

Ekpγτq “ pcτ ` dqkEkpτq ´ δk,2cpcτ ` dq

2πi.

DefinitionFor w , z P C, and τ P H let us define

rP1pw , z , τq “ ´ÿ

nPZ

qnw1´ qzqn

.

We also have

Definition

rPm`1pw , z , τq “p´1qm

m!Dmw

´

rP1pw , z , τq¯

“p´1qm`1

m!

ÿ

nPZ

nmqnw1´ qzqn

.

It is thus useful to give

DefinitionFor m P N0, let

Pm`1,λ pw , τq “p´1qm`1

m!

ÿ

nPZzt´λu

nmqnw1´ qn`λ

.

On notes thatP1,λ pw , τq “ q´λw pP1pw , τq ` 1{2q,

with

Pm`1,λ pw , τq “p´1qm

m!Dmw pP1,λ pw , τqq .

We also consider the expansion

P1,λpw , τq “1

2πiw´

ÿ

kě1

Ek,λpτqp2πiwqk´1,

where we find [Zagir]

Ek,λpτq “kÿ

j“0

λj

j!Ek´jpτq.

DefinitionWe define another generating set rEkpz , τq for k ě 1 together with E2pτqgiven by [Oberdieck]

rP1pw , z , τq “1

2πiw´

ÿ

kě1

rEkpz , τqp2πiwqk´1,

where we find that for k ě 1,

rEkpz , τq “ ´δk,1qz

qz ´ 1´

Bk

k!

`1

pk ´ 1q!

ÿ

m,ně1

´

nk´1qmz ` p´1qknk´1q´mz

¯

qmn,

and rE0pz , τq “ ´1.

DefinitionFor a V -module W , we consider the spaces of n-point Jacobi forms

CnpW q “

!

ZJW pxn, ;Bq , n ě 0,

)

.

The coboundary operator δnpxn`1q on CnpW q-space is defined accordingto the reduction formula [Zhu, Bringmann-Krauel-Tuite, Mason-Tuite-Z]for V -module W Jacobi forms.

DefinitionFor n ě 0, and any xn`1 P V ˆ C, define

δnpxn`1q : CnpW q Ñ Cn`1pW q, (0.2)

with non-commutative operators Tjpv rms.q, j ě 0, given by the reductionformulas

δn pxn`1qZJW pxn;Bq “

nÿ

k“0mě0

fk,mpxn`1;Bq Tkpvn`1rms.qZJW pxn;Bq .

The operators Tkpv rms.q are insertion operators of vertex algebra modesv rms., m ě 0, into ZJ

W pxn;Bq at the k-th entry:

Tkpv rms.q ZJW pxn;Bq “ ZJ

W pTk pv rmsq .xn;Bq ,

where we use the notation

pΓ.qk xn “ px1, . . . , Γ.xk , . . . , xnq .

For n ě 0, let us denote by Bn the subsets of all xn, such that the chaincondition

δn`1pxn`1q δnpxnq ZJ

W pxn;Bq “ 0,

for the coboundary operators is satisfied.

Explicitly, the chain condition leads to an infinite n ě 0 set of equationsinvolving functions fk,m pxn`1;Bq and ZJ

W pxn;Bq:

¨

˚

˝

n`1,nÿ

k1,k“0m1,mě0

fk 1,m1 pxn`1q fk,m pxnqTk 1pvn`2rm1s.qTkpvn`1rms.q

˛

‹

‚

ZJW pxn;Bq “ 0.

DefinitionThe CnpW q spaces with chain conditions constitute a semi-infinite chaincomplex

0 ÝÑ C 0 δ0px1qÝÑ C 1 δ1px2q

ÝÑ . . .δn´2pxn´1qÝÑ Cn´1 δn´1pxnq

ÝÑ Cn δnpxn`1qÝÑ . . . .

For n ě 1, we call corresponding cohomology

HnJ pW q “ Ker δnpxn`1q{Im δn´1pxnq,

the n-th reduction cohomology of a vertex algebra V -module W on T .

Reduction formulas for Jacobi forms

We formulate

DefinitionFor vn`1, with Jp0qvn`1 “ αvn`1, α P C, the coboundary operator isdefined above with

f0pxn`1;αz , τq T0pvn`1rmsq “ δαz,λτ`µPZτ`Z e´zn`1λ T0popvλpvn`1qq,

fk,mpzn`1;λ, k , αz , τq “ T 1´δαz,λτ`µPZτ`Z .

Pm`1,λ

ˆ

zn`1 ´ zk2πi

, p1´ δαz,λτ`µPZτ`Zq αz , τ

˙

,

with rPm`1,λ pzn`1, αz , τq. Here oλpv1q “ v1pwt v1 ´ 1` λq, and

Pm`1,λ pw , τq “p´1qm

m!Dmw pP1,λ pw , τqq .

Cohomology

Now we compute the reduction cohomology in the case of Jacobi forms.Our main result is

Proposition

Under assumptions above, the n-th reduction cohomology of the space ofJacobi forms for a V -module W is given by the space of analyticalcontinuations of solutions ZJ

W pxn;Bq to a analogue of theKnizhnik-Zamolodchikov equation

nÿ

k“0

ÿ

mě0

fk,m pxn;BqTkpvn`1rms.q ZJM pxn;Bq “ 0,

with xi R Bi , for 1 ď i ď n. These are given by the space ofquasi-modular forms in terms of series of deformed Weierstrass functions,recursively generated by reduction/recursion formulas.

RemarkOne can make connection with the first cohomology of vertex algebras[Huang] in terms of derivations, and with the second cohomology interms of certain extensions of V by W .

In certain cases of coboundary operators, we are able to compute then-th cohomology even more explicitly by using recursion formulas interms of generalized elliptic functions. In particular, for orbifold n-pointJacobi functions associated to a vertex operator superalgebra we obtainusing [Mason-Tuite-Z]

Corollary

For vn R Bn, the n-th cohomology is given by the space of determinantsof n ˆ n-matrices containing deformed elliptic functions depending onzi ´ zj , 1 ď i , j ď n, for all possible combinations of vn-modes.

Proof

The n-th reduction cohomology is defined by the subspace of CnpW q offunctions ZJ pxn;Bq satisfying Knizhnik-Zamolodchikov equation,modulo the subspace of CnpW q n-point Jacobi forms ZJ

W px1n;Bqresulting from:

ZJW

`

x1n;B˘

“

˜

n´1ÿ

k“1

ÿ

mě0

fk,m pxn;Bq Tpgqk pv 1nrmsβq

¸

ZJW

`

x1n´1;B˘

.

Subject to other fixed parameters, n-point Jacobi forms are completelydetermined by all choices xn which do not belong to B.

Consider a non-vanishing solution ZJW pxn;Bq to the

Knizhnik-Zamolodchikov equation for some xn. Let us use the reductionformulas recursively for each xi , 1 ď i ď n of xn, in order to expressZJW pxn;Bq in terms of the Jacobi partition function ZJ

W pBq, i.e., weobtain

ZJW pxn;Bq “ Dpxn;Bq ZJ

W pBq .

as in [Mason-Tuite-Z].

Thus, for xi R Bi for 1 ď i ď n, i.e., at each stage of the recursionprocedure reproducing ZJ

W pxi ,Bq, otherwise ZJW pxn;Bq is zero.

Therefore, ZJW pxn;B, q is explicitly known and is represented as a series

of auxiliary functions Dpxn;Bq depending on V .

Consider now ZJW px1n;Bq given by the reduction formula. It either

vanishes when vn´i P Vn´i , 2 ď i ď n, or given by correspondingrecursion formula with x1n arguments.

The way the reduction relations were derived in [Yamada] is exactly thesame as for the vertex algebra derivation in [Knizhnik-Zamolodchikov,Tsuchiya-Kanie] of the Knizhnik-Zamolodchikov equations.

Namely, one considers a double integration of ZJW pxn;Bq along small

circles around two auxiliary variables with the action of appropriatereproduction kernels inserted. Then, these procedure leads to recursionformulas relating ZJ

W pxn`1;Bq and ZJW pxn;Bq with functional

coefficients depending on the nature of a vertex algebra V .

In [Zhu, Mason-Tuite-Z, Bringmann-Krauel-Tuite] formulas for n-pointfunctions and Dpxn;Bq in various specific examples of V , andconfiguration of Riemann surfaces were explicitly obtained.

Namely, in the Knizhnik-Zamolodchikov equation above the operatorsTkpvn`1rmsβ.q act by certain modes vn`1rms. of a vertex algebraelement vn`1 on elements vn.

Using vertex algebra associativity property we express the action of ofoperators Tkpvn`1rms.q in terms of modes vn`1rms inside vertexoperators in actions of V -modes on the whole vertex operator at expenseof a shift of their formal parameters zn by zn`1, i.e., z1n “ zn ` zn`1.Note that under such associativity transformations v -parts of xn, i.e., vndo not change.

Thus, the n-th reduction cohomology of a V -module W is given by thespace of analytical continuations of n-point functions ZJ

W pxn;Bq withxn´1 R Vn´1 that are solutions to the Knizhnik-Zamolodchikov equations.

Geometrical meaning of reduction formulas

Now let us explain the geometrical meaning of Jacobi forms reductionformulas as multipoint connections on a vector bundle over Tgeneralizing ordinary holomorphic connections on complex curves.

Let us recall the notion of a multipoint connection. Motivated by thedefinition of a holomorphic connection for a vertex algebra bundle[Ben-Zvi–Frenkel, Gunning] over a smooth complex curve, we introducethe definition of the multiple point connection over T .

DefinitionLet V be a holomorphic vector bundle over T , and T0 Ă T be itssubdomain. Denote by SV the space of sections of V. A multi-pointconnection G on V is a C-multi-linear map

G : T ˆn ˆ Vbn Ñ C,

such that for any holomorphic function f , and two sections φppq andψpp1q of V at points p and p1 on T0 Ă T correspondingly, we have

ÿ

q,q1PT0ĂTG`

f pψpqqq.φpq1q˘

“ f pψpp1qq G pφppqq ` f pφppqq G`

ψpp1q˘

,

where the summation on left hand side is performed over locuses ofpoints q, q1 on T0. We denote by Conn the space of n-point connectionsdefined over T .

Geometrically, for a vector bundle V defined over T , a multi-pointconnection relates two sections φ and ψ at points p and p1 with anumber of sections on T0 Ă T .

DefinitionWe call

G pφ, ψq “ f pφppqq G`

ψpp1q˘

` f pψpp1qq G pφppqq

´ÿ

q,q1PT0ĂTG`

f pψpq1qq.φpqq˘

,

the form of a n-point connection G. The space of n-point connectionforms will be denoted by Gn.

We prove the following

LemmaJacobi n-point forms generated by recursion formulas are n-pointconnections on the space of automorphisms g P AutpV q deformedsections of the vertex algebra bundle V associated to V . For n ě 0, then-th reduction cohomology of Jacobi forms is given by

HnJ pW q “ Hn

J pSVg q “ Conn{Gn´1, (0.3)

and it is isomorphic to the cohomology of the space of deformedV-sections.

RemarkThis lemma is a vertex algebra version for deformed sections of the mainproposition of [Bott-Segal, Wagemann], i.e., the Bott–Segal theorem forRiemann surfaces.

ProofIn [Ben-Zvi–Frenkel] the vertex operator bundle V was explicitlyconstructed. It is easy to see that n-point connections are holomorphicconnection on the bundle V with the following identifications. Fornon-vanishing f pφppqq let us set

G “ ZJW pxn;Bq ,

ψpp1q “ pxn`1q ,

φppq “ pxnq ,

G`

f pψpqqq.φpq1q˘

“ Tkpv rmsβ.q ZJW pxn;Bq ,

´f pψpp1qq

f pφppqqG pφppqq “ f0 pxn`1;Bq T0poλpvn`1qq ZJ

W pxn;Bq ,

f ´1pφppqqÿ

qn,q1nPT0ĂT

G`

f pψpqqq.φpq1q˘

“

nÿ

k“1mě0

fk,m pxn`1;BqTkpv rmsβ.q ZJW pxn;Bq .

Recall [Ben-Zvi–Frenkel] the construction of the vertex algebra bundle V.Here we use a g -twisted version of it. According to Proposition 6.5.4 of[Ben-Zvi–Frenkel], one canonically (i.e., coordinate independently)associates End V-valued sections Yp of the g -twisted bundle V˚ (thebundle dual to V). The intrinsic, i.e., coordinate independent, vertexalgebra operators are defined by

xu,`

Y˚p pipvnqq˘

ng vy “ xu,Ypxnqvy.

to matrix elements of a number of vertex operators on appropriatepunctured disks around points with local coordinates zn on T . Thespaces of such V-sections for each n of is described by identificationsabove.

Taking into account the construction of Section 6 (subsection 6.6.1, inparticular, construction 6.6.4, and Proposition 6.6.7) of[Ben-Zvi–Frenkel], we see that n-point Jacobi functions are connectionson the space of sections of V. ˝

The geometrical meaning of the chain conditionSince in the reduction formulas above operators act on vertex algebraelements only, we can interpret it as a relation on modes of V withfunctional coefficients. In particular, all operators T change vertexalgebra elements by action either of opvq “ vpwtv ´ 1q, or positivemodes of v rms., m ě 0.

Recall that for n-point Jacobi forms are quasi-modular forms. Moreover,the reduction formulas can be used to prove modular invariance forhigher n-point Jacobi forms.

Due to automorphic properties of n-point functions, the chain conditionscan be also interpreted as relations among modular forms. It also definesa complex variety in zn P Cn with non-commutative parameters vn.

As most identities (e.g., trisecant identity [Fay, Mumford] and tripleproduct identity [Kac, Mason-Tuite-Z]) for n-point functions (??) has itsalgebraic-geometrical meaning. The chain conditions relate finite seriesof vertex algebra correlations functions on T with elliptic functions [Zhu,Mason-Tuite-Z]. Since n-point Jacobi forms are quasi-modular forms, wetreat chain conditions as a source of new identities on such forms.

Vertex operator (super)algebrasIn this subsection we recall the notion of vertex operator (super)algebras[Borcherds, Frenkel-Huang-Lepowski, Kac]. Let V be a superspace, i.e.,a complex vector space

V “ V0̄ ‘ V1̄ “ ‘αVα,

with index label α in Z{2Z so that each a P V has a parity ppaq P Z{2Z.An C-graded vertex operator superalgebra is defined by pV ,Y , 1V , ωqwhere V is a superspace with a C-grading where

V “ ‘rěr0Vr ,

for some r0 and with parity decomposition

Vr “ V0̄,r ‘ V1̄,r .

1V P V0̄,0 is the vacuum vector and ω P V0̄,2 the conformal vector withproperties described below. The vertex operator Y is a linear map

Y : V Ñ pEndV qrrz , z´1ss,

for formal variable z , so that for any vector x “ pa, vq P V ˆ C,

Y pxq “ÿ

nPZapnqz´n´1.

The component operators (modes) apnq P EndV are such that

apnq1V “ δn,´1a,

for n ě ´1 andapnqVα Ă Vα`ppaq,

for a of parity ppaq.The vertex operators satisfy the locality property for allxi “ pvi , zi q P V ˆ C, i “ 1, 2,

pz1 ´ z2qN rY px1q,Y px2qs “ 0,

for N " 0, where the commutator is defined in the graded sense, i.e.,

rY px1q,Y px2qs “ Y px1qY px2q ´ p1qppv1qppv2qY px2qY px1q.

The vertex operator for the vacuum is

Y p1V , zq “ IdV .

The conformal vector is defined by ω is

Y pω, zq “ÿ

nPZLpnqz´n´2,

where Lpnq forms a Virasoro algebra for central charge c

rLV pmq, LV pnqs “ pm ´ nqLV pm ` nq `c

12pm3 ´mqδm,´nIdV .

LV p´1q satisfies the translation property

Y pLV p´1qxq “d

dzY pxq.

LV p0q describes the C-grading with

LV p0qa “ wtpaqa,

for weight wtpaq P C and

Vr “ ta P V |wtpaq “ ru.

We quote the standard commutator property of vertex operatorsuperalgebra, for x1 “ pa, z1q, x “ pb, z2q

rapmq,Y pxqs “ÿ

jě0

ˆ

m

j

˙

Y papjq.xqzm´j1 .

Taking a “ ω this implies for b of weight wtpbq that

rLV p0q, bpnqs “ pwtpbq ´ n ´ 1qbpnq,

so thatbpnqVr Ă Vr`wtpbq´n´1.

In particular, we define for a of weight wtpaq the zero mode

oλpaq “

"

apwtpaq ´ 1` λq, for wtpaq P Z0, otherwise,

which is then extended by linearity to all a P V .

Square bracket formalismDefine the square bracket operators for V by

Y rxs “ Y´

ez Lp0qv , ez ´ 1¯

“ÿ

nPZv rnsz´n´1.

For v of weight wtpvq and k P Z, [Zhu], we have

ÿ

jě0

ˆ

k ` wtpvq ´ 1

j

˙

vpjq “ÿ

mě0

km

m!v rms.

The square bracket operators form an isomorphic vertex operator algebrawith Virasoro vector

rω “ ω ´c

241V .

Let us now introduce [Dong–Mason] the shifted Virasoro vector

ωh “ ω ` hp´2q1V ,

where

h “ ´λ

αJ,

for λ P Z.

Then the shifted grading operator is

Lhp0q “ Lp0q ´ hp0q “ Lp0q `λ

αJp0q.

Denote the square bracket vertex operator for the shifted Virasoro vectorby

Y rxsh “ Y´

ez Lhp0qv , ez ´ 1¯

“ÿ

nPZv rnsh z´n´1,

Therefore,Y ra, zsh “ ezλY ra, zs,

or equivalently,

arnsh “ÿ

mě0

λm

m!arn `ms.