VERTEX ALGEBRAS AND QUANTUM MASTER EQUATION3. Vertex algebra and BV master equation 14 3.1. Vertex...

VERTEX ALGEBRAS AND QUANTUM MASTER EQUATION

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ABSTRACT. We contribute to study the two dimensional chiral conformal field theories that arisefrom chiral deformations of free theories such as the βγ-system, bc-system and chiral bosons. Ourmethod is based on the effective Batalin-Vilkovisky quantization theory. We establish an exact cor-respondence between renormalized quantum master equations for chiral deformations and Maurer-Cartan equations for the Lie algebra of modes of chiral vertex algebras. The generating functions onthe torus are proven to have modular property with mild holomorphic anomaly. As an application,we construct an exact solution of quantum B-model (BCOV theory) in complex one dimension thatsolves the full higher genus mirror symmetry conjecture on elliptic curves.

CONTENTS

1. Introduction 2

2. Batalin-Vilkovisky formalism and effective renormalization 6

2.1. Batalin-Vilkovisky algebras and the master equation 6

2.2. Odd symplectic space and the toy model 7

2.3. UV problem and homotopic renormalization 8

2.4. Example: Topological quantum mechanics 13

3. Vertex algebra and BV master equation 14

3.1. Vertex algebra 14

3.2. Free theory examples 17

3.3. Chiral deformation of βγ − bc system on flat spaces 19

3.4. Ultra-violet finiteness 20

3.5. Quantum master equation 22

3.6. Generating function and modularity 28

3.7. Example: Poisson σ-model 32

4. Application: quantum B-model on elliptic curves 34

4.1. BCOV theory on elliptic curves 34

4.2. Hodge weight and dilaton equation 37

4.3. Reduction to Fedosov’s equation 38

4.4. Exact solution of quantum BCOV theory 42

4.5. Higher genus mirror symmetry 44

Appendix A. Analytic results on Feynman graph integrals 451

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Ultra-violet finiteness 45

Quantum master equation at the UV 46

References 47

1. INTRODUCTION

Quantum field theory provides a rich source of mathematical thoughts. One important featureof quantum field theory that lies secretly behind many of its surprising mathematical predictionsis the role of infinite dimensionality. A famous example is the mysterious mirror symmetry con-jecture between symplectic and complex geometries, which can be viewed as a version of infinitedimensional Fourier transform. Typically, many quantum problems are formulated in terms of“path integrals”, which require measures that are mostly not yet known to mathematicians. Nev-ertheless, asymptotic analysis can always be performed with the help of the celebrated idea ofrenormalization.

Despite the great success of renormalization theory in physics applications, its use in mathemat-ics is relatively limited but extremely powerful when it does apply. One such example is Kont-sevich’s solution [29] to the deformation quantization problem on arbitrary Poisson manifolds.Kontsevich’s explicit formula of star product is obtained via graph integrals on a compactificationof configuration space on the disk, which can be viewed as a geometric renormalization of theperturbative expansion of Poisson sigma model (see [8]). Another recent example is Costello’s ho-motopic theory [12] of effective renormalizations in the Batalin-Vilkovisky formalism. This leadsto a systematic construction of factorization algebras via quantum field theories [14]. For example,a natural geometric interpretation of the Witten genus is obtained in such a way [13].

To facilitate geometric applications of effective renormalization methods, it would be crucial toconnect renormalized quantities to geometric objects. We will be mainly interested in quantumfield theory with gauge symmetries. The most general framework of quantizing gauge theories isthe Batalin-Vilkovisky formalism [7], where the quantum consistency of gauge transformations isdescribed by the so-called quantum master equation. There have developed several mathematicalapproaches to incorporate Batalin-Vilkovisky formalism with renormalizations since their birth.The central quantity of all approaches lies in the renormalized quantum master equation. In thispaper, we will mainly discuss the perturbative formalism in [12], which has developed a conve-nient framework that is rooted in the homotopic culture of derived algebraic geometry. A briefintroduction to the philosophy of this approach is discussed in Section 2.

The simplest nontrivial cases are quantum mechanical models, which can be viewed as quan-tum field theories in one dimension. An an example, consider the classical mechanical systemwith phase space a symplectic manifold X. Fedosov [19] gave a beautiful construction of defor-mation quantization on X, which amounts to constructing a flat connection (called an abelianconnection) on the Weyl bundle over X. It turns out that such geometric data has a precise quan-tum field theory interpretation in terms of effective renormalization [24, 25, 32]. It is shown in[32] that every Fedosov deformation quantization arises from a perturbative quantization of topo-logical mechanics. More precisely, every solution to the renormalized quantum master equationproduces a Fedosov’s abelian connection. The algebraic nature of this correspondence is reviewed

VERTEX ALGEBRAS AND QUANTUM MASTER EQUATION 3

in Section 2.4. Such a correspondence leads to a geometric approach to the algebraic index the-orem [20, 42], where the index formula follows from the homotopic renormalization group flowtogether with an exact semi-classical approximation via BV integration [24, 32].

In this paper, we study systematically the renormalized quantum master equation in two di-mensions. We will focus on quantum theories obtained by chiral deformations of free conformalfield theory(CFT)’s (see Section 3.3 for our precise set-up) on flat spaces. One important feature ofsuch two dimensional chiral theories is that they are free of ultra-violet divergence (see Theorem3.10). This greatly simplifies the analysis of quantization since singular counter-terms are not re-quired. However, the renormalized quantum master equation requires quantum deformations bychiral local functionals. Such quantum deformations could in principle be very complicated.

One of our main results in this paper (Theorem 3.14) is an exact description of the renormalizedquantum master equations for chiral deformations of free βγ− bc systems. Briefly speaking, The-orem 3.14 states that solutions to the renormalized quantum master equations (QME) correspondto solutions to the Maurer-Cartan (MC) equation of the modes Lie algebra of the chiral vertexalgebra (see Definition 3.4) associated to the free theory

renormalized QME ⇐⇒ MC equation for chiral modes Lie algebra

Our main Theorem 3.14 is stated for chiral deformations of βγ− bc systems, but it actually worksin parallel for chiral bosons as well. We do not present such a general discussion in this paper, butillustrate how to modify by a concrete example in Section 4 (see Theorem 4.4).

Geometrically, the MC element leads to a cohomological operator Q (degree 1 and squares zero)acting on the vertex algebra V , which can be identified with the BRST operator. The cohomology

H(V , Q)

gives a new chiral vertex algebra, which can be identified with local operators associated to thechiral deformed theory. In particular, this chiral deformation realizes the usual BRST constructionof vertex algebras.

Furthermore, we prove a general result on the modularity property of the generating functionson elliptic curves and their holomorphic anomaly (Theorem 3.25). This work is partially motivatedfrom understanding Dijkgraaf’s description [18] of chiral deformation of conformal field theories.The Maurer-Cartan equation serves as an integrability condition for chiral vertex operators, whichis often related to integrable hierarchies in concrete cases. We discuss one such example in Section4. See [26, 36] for further discussions from the perspective of integrable hierarchy.

The above correspondence can be viewed as the two dimensional vertex algebra analogue ofthe one dimensional result in [32]. In fact, one main motivation of the current work is to explorethe analogue of index theorem for chiral vertex operators in terms of the method of semi-classicalmethod proceeded in [24, 32]. A family version of the above correspondence should lead to theanalogue of Fedosov’s connection on vertex algebra bundles over geometric spaces.

As an application in Section 4, we construct an exact solution of quantum B-model on ellipticcurves, which leads to the solution of the corresponding higher genus mirror symmetry conjec-ture. Mirror symmetry is a famous duality between symplectic (A-model) and complex (B-model)geometries that arises from superconformal field theories. It has been a long-standing challengefor mathematicians to construct quantum B-model on compact Calabi-Yau manifolds. There is a

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categorical approach [10,11,30] to the quantum B-model partition function associated to a Calabi-Yau category based on a classification of two-dimensional topological field theories. Unfortu-nately, it is extremely difficult to perform this categorical computation (recently a first non-trivialcategorical computation is carried out in [3] for one-point functions on the elliptic curve ). Anotherapproach is through quantum field theory. In [15], we construct a gauge theory of polyvector fieldson Calabi-Yau manifolds (called BCOV theory) as a generalization of the Kodaira-Spencer gaugetheory [6]. It is proposed in [15] (as a generalization of [6]) that the Batalin-Vilkovisky quantizationof BCOV theory leads to quantum B-model that is mirror to the A-model Gromov-Witten theoryof counting higher genus curves. Our construction in Section 4 gives a concrete realization of thisprogram. This leads to the first mathematically fully established example of higher genus mirrorsymmetry on compact Calabi-Yau manifolds.

Our result in Section 4 also leads to an interesting result in physics. Quantum BCOV theory canbe viewed as a complete description of topological B-twisted closed string field theory in the senseof Zwiebach [45]. Zwiebach’s closed string field theory describes the dynamics of closed strings interm of the so-called string vertices. Despite the beauty of this construction, string vertices are verydifficult to compute and few concrete examples are known. Our exact solution in Section 4 canbe viewed as giving an explicit realization of Zwiebach’s string vertices for B-twisted topologicalstring on elliptic curves.

Acknowledgement: The author would like to thank Kevin Costello, Cumrun Vafa, Andrei Losev,Robert Dijkgraaf, Jae-Suk Park, Owen Gwilliam, Ryan Grady, Qin Li, and Brian Williams for dis-cussions on quantum field theories, and thank Jie Zhou for discussions on modular forms. Partof the work was done during visiting Perimeter Institute for theoretical physics and IBS center forgeometry and physics. The author thanks for their hospitality and provision of excellent work-ing enviroments. Special thank goes to Yunchen Li and Xinyi Li, whose birth and growth haveinspired and reformulated many aspects of the presentation of the current work. S.L. is partiallysupported by grant 11801300 of National Natural Science Foundation of China and grant Z180003of Beijing Municipal Natural Science Foundation.

Conventions

• Let V be a Z-graded k-vector space. We use Vm to denote its degree m component. Givena ∈ Vm, we let a = m be its degree.

– V[n] denotes the degree shifting of V such that V[n]m = Vn+m.– V∗ denotes its dual such that V∗m = Homk(V−m, k). Our base field k will mainly be R

or C.– Symm(V) and∧m(V) denote the graded symmetric product and graded skew-symmetric

product respectively. We also denote

Sym(V) :=⊕m≥0

Symm(V), Sym(V) := ∏m≥0

Symm(V).

Given I = ∑m≥0

Im ∈ Sym(V∗), and a1, · · · , am ∈ V, we denote its m-order Taylor

coefficient∂

∂a1· · · ∂

∂amI(0) := Im(a1, · · · , am)

where we have viewed Im as a multi-linear map V⊗m → k.


– Given P ∈ Sym2(V), it defines a “second order operator” ∂P on Sym(V∗) or Sym(V∗)by

∂P : Symm(V∗)→ Symm−2(V∗), I → ∂P I,

where for any a1, · · · , am−2 ∈ V,

∂P I(a1, · · · am−2) := I(P, a1, · · · am−2).

– V[z], V[[z]] and V((z)) denote polynomial series, formal power series and Laurentseries respectively in a variable z valued in V.

• Let A be a graded algebra. [−,−] always means the graded commutator, i.e., for elementsa, b with specific degrees,

[a, b] := a · b− (−1)abb · a.

We always assume Koszul sign rule in dealing with graded objects.• ⊗ without subscript means tensoring over the real numbers R.• Given a manifold X, we denote the space of real smooth forms by

Ω•(X) =⊕

k

Ωk(X)

where Ωk(X) is the subspace of k-forms. If furthermore X is a complex manifold, we denotethe space of complex smooth forms by

Ω•,•(X) =⊕p,q

Ωp,q(X) = Ω•(X)⊗C

where Ωp,q(X) is the subspace of (p, q)-forms.• Dens(X) denotes the density line bundle on a manifold X. When X is oriented, we natu-

rally identify Dens(X) with top differential forms on X.• Let E be a vector bundle on a manifold X. E = Γc(X, E) denotes the space of smooth

sections with compact supports, and E ′ = D′(X, E) denotes the distributional sections. IfE∗ is the dual bundle of E, then we have a natural pairing

E ′ ⊗ Γc(X, E∗ ⊗Dens(X))→ R.

• We will also work with infinite dimensional functional spaces that carry natural topologies.The above notions for V will be generalized as follows. We refer the reader to [43] orAppendix 2 of [12] for further details.

– All topological vector spaces we consider will be nuclear (or at least locally convex Haus-dorff) and we use ⊗ to denote the completed projective tensor product. For example,∗ Given two manifolds M, N, we have a canonical isomorphism

C∞(M)⊗C∞(N) = C∞(M× N);

∗ Similarly, if D(M) denotes distributions, then again we have

D(M)⊗D(N) = D(M× N).

• H denotes the upper half plane.

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2. BATALIN-VILKOVISKY FORMALISM AND EFFECTIVE RENORMALIZATION

In this section, we collect basics and fix notations on the quantization of gauge theories in theBatalin-Vilkovisky (BV) formalism. We explain Costello’s homotopic renormalization theory ofBatalin-Vilkovisky quantization and illustrate its geometric structures by a one-dimensional ex-ample of topological quantum mechanics. In the next section, we give a self-contained realizationof Costello’s framework in two dimensions based on heat kernel estimates.

2.1. Batalin-Vilkovisky algebras and the master equation.

Definition 2.1. A differential Batalin-Vilkovisky (BV) algebra is a triple (A, Q, ∆) where

• A is a Z-graded commutative associative unital algebra.• Q : A → A is a derivation of degree 1 such that Q2 = 0.• ∆ : A → A is a linear operator of degree 1 such that ∆2 = 0.• ∆ is a “second-order” operator. Precisely, define the binary operator −,− : A⊗A → A

a, b := ∆(ab)− (∆a)b− (−1)aa∆b, a, b ∈ A.

Then −,− satisfies the following graded Leibnitz rule

a, bc = a, b c + (−1)(a+1)bb a, c , a, b, c ∈ A.

• Q and ∆ are compatible: [Q, ∆] = Q∆+ ∆Q = 0.

Here ∆ is called the BV operator. −,− is called the BV bracket, which measures the failure of∆ being a derivation. It follows that −,− defines a Poisson bracket of degree 1 satisfying

• a, b = (−1)ab b, a.• a, bc = a, b c + (−1)(a+1)bb a, c.• ∆ a, b = −∆a, b − (−1)a a, ∆b.

The (Q, ∆)-compatibility condition implies the following Leibniz rule

Q a, b = −Qa, b − (−1)a a, Qb .

Definition 2.2. Let (A, Q, ∆) be a differential BV algebra. A degree 0 element I ∈ A0 is said tosatisfy classical master equation (CME) if

QI +12I, I = 0.

If I solves CME, then it is easy to see that Q + I,− defines a differential on A, which can beviewed as a Poisson deformation of Q. However, it may not be compatible with ∆. A sufficientcondition for the compatibility is the “divergence freeness” ∆I = 0. A slight generalization of thisis the following.

Definition 2.3. Let (A, Q, ∆) be a differential BV algebra. A degree 0 element I ∈ A[[h]] is said tosatisfy quantum master equation (QME) if

QI + h∆I +12I, I = 0.

Here h is a formal variable representing the quantum parameter.


The “second-order” property of ∆ implies that QME is formally equivalent (without worryingabout the powers of h−1) to

(Q + h∆)eI/h = 0.

If we decompose I = ∑g≥0

Ighg, then the h→ 0 limit of QME is precisely CME

QI0 +12I0, I0 = 0.

We can rephrase the (Q, ∆)-compatibility as Q + h∆ squares to zero. It is direct to check thatQME implies that Q + h∆+ I,− squares to zero as well.

2.2. Odd symplectic space and the toy model. We discuss a toy model of differential BV algebravia (−1)-shifted symplectic space. This serves as the main motivation of our quantum field theoryexamples.

Let (V, Q) be a finite dimensional dg vector space. The differential Q : V → V induces adifferential on various tensors of V, V∗, still denoted by Q. Let

ω ∈ ∧2V∗, Q(ω) = 0,

be a Q-compatible symplectic structure such that deg(ω) = −1. The non-degeneracy of the sym-plectic pairingω implies that givenϕ ∈ V∗, there exists a unique vϕ ∈ V such that

ϕ(−) = ω(−, vϕ).

The condition deg(ω) = −1 says

deg(vϕ) = deg(ϕ) + 1.

This gives an isomorphism of graded vector spaces

V∗ ∼= V[1], ϕ→ vϕ[1].

Let K = ω−1 ∈ Sym2(V) be the Poisson kernel of degree 1 under

∧2V∗ ∼= Sym2(V)[2]

ω K[2]

where we have used the isomorphism

∧2(V[1]) ∼= Sym2(V)[2]

a[1] ∧ b[1]→ (−1)|b|a · b[2], a, b ∈ V.

Let

O(V) := Sym(V∗) = ∏n

Symn(V∗).

Then (O(V), Q) is a graded-commutative dga.

The degree 1 Poisson kernel K defines the following BV operator

∆K := ∂K : O(V)→ O(V) by

∆K(ϕ1 · · ·ϕn) = ∑i, j±(K,ϕi ⊗ϕ j)ϕ1 · · ·ϕi · · ·ϕ j · · ·ϕn, ϕi ∈ V∗.

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Here (K,ϕi ⊗ϕ j) denotes the natural paring between V⊗V and V∗⊗V∗. ± is the Koszul sign bypermuting ϕi’s. The following lemma is well-known (we follow the presentations in [12] whichwill be relevant for our later discussions on effective renormalization).

Lemma 2.4. (O(V), Q, ∆K) defines a differential BV algebra.

Sketch of proof. The compatibility ofω and Q implies Q(K) = 0, hence

[Q, ∆K] = [Q, ∂K] = [∂Q(K)] = 0.

Other properties of differential BV algebra are easily verified.

The above construction can be summarized as

(−1)-shifted dg symplectic =⇒ differential BV.

Remark 2.5. Lemma 2.4 holds for any K ∈ Sym2(V) of degree 1 such that Q(K) = 0. In particular,it applies to the case for (−1)-shifted dg Poisson structure where K may be degenerate (soω doesnot exist) but ∆K is still well-defined. We will see such an example in Section 4.

2.3. UV problem and homotopic renormalization. Let us now move on to discuss examples ofquantum field theory that we will be mainly interested in.

2.3.1. The ultra-violet problem. One important feature of quantum field theory is its infinite dimen-sionality. It leads to the main challenge in mathematics to construct measures on infinite dimen-sional space (called the path integrals). It is also the source of the difficulty of ultra-violet diver-gence and the motivation for the celebrated idea of renormalization in physics. Let us addresssome of these issues via the Batalin-Vilkovisky formalism.

In the previous section, we discuss the (−1)-shifted dg symplectic space (V, Q,ω). There V isassumed to be finite dimensional. This is the reason we call it “toy model”. Typically in quantumfield theory, V will be modified to be the space of smooth sections of certain vector bundles on asmooth manifold, while the differential Q and the pairingω come from something “local”. SuchV will be called the space of fields, which is evidently a very large space with delicate topology.Nevertheless, let us naively perform similar constructions as that in the toy model.

More precisely, let X be a smooth oriented manifold without boundary. Let E be a Z-gradedvector bundle on X of finite rank. Our space of fields E replacing V will be the space of smoothglobal sections with compact support

E = Γc(X, E)

equipped with a differential operator of degree 1

Q : E → E

such that Q2 = 0. We assume that (E , Q) is an elliptic complex. The symplectic pairing will be

ω(s1, s2) :=∫

X(s1, s2) , s1, s2 ∈ E ,

where(−,−) : E⊗ E→ Dens(X)

is a non-degenerate graded skew-symmetric bundle morphism of degree−1. Here the line bundleDens(X) sits at degree 0. The symplectic pairingω is required to be compatible with Q.

To perform the toy model construction, we need the following steps


(1) The dual vector space E∗ (analogue of V∗).This can be defined as the space of continuous linear duals of E (i.e. distributional sec-

tions of the bundle Hom(E, Dens(X))

E∗ := Hom(E ,R)

where Hom is the space of continuous maps.(2) The tensor space (E∗)⊗k (analogue of (V∗)⊗k).

This can be defined via the completed tensor product for distributions

(E∗)⊗k = E∗⊗ · · · ⊗E∗,

i.e. the distributional sections of the bundle Hom(E · · · E, Dens(X × · · · × X)) overX × · · · × X. Symk(E∗) is defined to be the subspace of (E∗)⊗k by taking invariants of thegraded permutations. Then we have a well-defined notion (via distributions)

O(E) := ∏k≥0

Symk(E∗)

as the analogue of O(V).(3) The Poisson kernel K0 =ω−1 (the analogue of K).

In the finite-dimensional setting, the Poisson kernel is simply the inverse matrix of thesymplectic pairing. In our infinite-dimensional setting, this amounts to asking the Pois-son kernel to be the integral kernel of the identity operator with respect to ω, that is, K0

satisfying

ω(K0(x, y),φ(y)) = φ(x) for all φ ∈ E .

Sinceω is given by integration, K0 behaves like a δ-function. We can view K0 as a distribu-tional section of E× E supported on the diagonal of X×X. Again K0 is graded symmetric.

Let Sym2(E) ⊂ E⊗E be the subspace of graded-symmetric elements. Unlike the finitedimensional case, K0 is not an element of Sym2(E), but in fact a distributional section

K0 ∈ Sym2(E ′).

See Conventions for E ′. It is at Step (3) where we get trouble. In fact, if we naively define the BVoperator

∆K0 : O(E) ?→ O(E),

then ∆K0 is ill-defined, since we can not pair a distribution K0 with another distribution fromO(E).This difficulty originates from the infinite dimensional nature of the problem.

2.3.2. Homotopic renormalization. The solution to the above problem requires the method of renor-malization in quantum field theory. There are several different approaches to renormalizations.We will work with Costello’s homotopic theory [12] in this paper that will be convenient for ourapplications.

The key observations are

(1) K0 is a Q-closed distribution: Q(K0) = (Q⊗ 1 + 1⊗Q)K0 = 0.(2) elliptic regularity: there is a canonical isomorphism (Atiyah-Bott Lemma [2])

H∗(smooth, Q) ∼= H∗(distribution, Q).

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It follows that we can find a distribution Pr ∈ Sym2(E ′) and a smooth element Kr ∈ Sym2(E) suchthat

K0 = Kr + Q(Pr).

Pr is the familiar notion of a parametrix. In other words, Kr is a smooth representative of thecohomology class of K0.

Definition 2.6. Kr will be called the renormalized BV kernel with respect to the parametrix Pr.

Since Kr is smooth, there is no problem to pair Kr with distributions. The same formula as inthe toy model leads to

Lemma/Definition 2.7. We define the renormalized BV operator ∆Kr

∆Kr : O(E)→ O(E)

via the smooth renormalized BV kernel Kr. The triple (O(E), Q, ∆Kr) defines a differential BV algebra,called the renormalized differential BV algebra (with respect to Pr).

Therefore (O(E), Q, ∆Kr) can be viewed as a homotopic replacement of the original naive prob-lematic differential BV algebra. As the formalism suggests, we need to understand relations be-tween difference choices of the parametrices.

Let Pr1 and Pr2 be two parametrices, Kr1 and Kr2 be the corresponding renormalized BV kernels.Let us denote

Pr2r1

:= Pr2 − Pr1 .

Since Q(Pr2r1 ) = Kr1 − Kr2 is smooth, Pr2

r1 is smooth itself by elliptic regularity.

Definition 2.8. Pr2r1 will be called the regularized propagator.

Example 2.9. Typically, suppose we have an adjoint operator Q† such that [Q, Q†] is a generalizedLaplacian. For example, we may pick a metric on E and define the adjoint Q† via the L2 innerproduct. In all the examples that we will be dealing with, such constructed [Q, Q†] will be ageneralized Laplacian.

Given t > 0, let Kt be the integral kernel for the heat operator e−t[Q,Q†] with respect to thesymplectic pairingω. In other words, Kt is a section of E E that is graded-symmetric and satisfies

ω(Kt(x, y),φ(y)) =(

e−t[Q,Q†]φ)(x) for all φ ∈ E .

By the theory of generalized Laplacian, such integral kernel Kt exists and is smooth any for t > 0.It approaches the δ-function distribution when t→ 0 (see for example [4]).

The operator equation (using the fact that [Q, Q†] graded commutes with Q and Q†)[Q,∫ t2

t1

Q†e−t[Q,Q†]dt]=∫ t2

t1

[Q, Q†

]e−t[Q,Q†]dt = e−t1[Q,Q†] − e−t2[Q,Q†]

is translated to the following kernel equation

Q(Pt2t1) := (Q⊗ 1 + 1⊗Q)Pt2

t1= Kt1 − Kt2 .

Here

Pt2t1=∫ t2

t1

(Q† ⊗ 1)Ktdt

is the integral kernel for the operator Q†e−t[Q,Q†] with respect to the symplectic pairingω.


We find that Kt can be viewed as a renormalized BV kernel for any t > 0. In this case, theparamatrix is Pt

0 and the regularized propagator is Pt2t1

. Such heat kernel renormalization will beour main applications.

Next we explain how to formulate quantum master equations. The key obervation is that Pr2r1

can be viewed as a homotopy linking two different renormalized differential BV algebras. In fact,analogous to the definition of renormalized BV operator,

Definition 2.10. We define ∂Pr2r1

: O(E) → O(E) as the second-order operator of contracting with

the smooth kernel Pr2r1 ∈ Sym2(E) (see also Conventions).

Lemma 2.11. The following equation holds formally as operators on O(E)[[h]](Q + h∆Kr2

)e

h∂P

r2r1 = e

h∂P

r2r1

(Q + h∆Kr1

),

i.e., the following diagram commutes

O(E)[[h]]Q+h∆Kr1 //

exp(

h∂P

r2r1

)

O(E)[[h]]

exp(

h∂P

r2r1

)

O(E)[[h]]Q+h∆Kr2

// O(E)[[h]]

Sketch of proof. Q(Pr2r1 ) = Kr1 − Kr2 implies[

Q, ∂Pr2r1

]= ∆Kr1

− ∆Kr2.

It follows that

exp(

h∂Pr2r1

)Q exp

(−h∂Pr2

r1

)= Q− h

[Q, ∂Pr2

r1

]= Q− h∆Kr1

+ h∆Kr2.

Since the operators ∆Kr1, ∆Kr2

, ∂Pr2r1

graded commute with each other, we find

exp(

h∂Pr2r1

) (Q + h∆Kr1

)exp

(−h∂Pr2

r1

)= Q + h∆Kr2

.

This proves the Lemma. See [12, Chapter 5, Section 9] for further discussions.

Definition 2.12. Let

O+(E)[[h]] := Sym≥3(E∗) + hO(E)[[h]] ⊂ O(E)[[h]]

be the subspace of those functionals which are at least cubic modulo h. Given any two parametri-ces Pr1 , Pr2 , we define the homotopic renormalization group (HRG) operator

W(Pr2r1

,−) : O+(E)[[h]]→ O+(E)[[h]]

W(Pr2r1

, I) := h log(

eh∂

Pr2r1 eI/h

).

The real content of the above formula is

W(Pr2r1

, I) = ∑Γ connected

WΓ (Pr2r1

, I)

where the summation is over all connected Feynman graphs with Pr2r1 being the propagator and I

being the vertex (see for example [5] for Feynman graph techniques). Wick’s Theoreom identifiesthe above two expressions. In particular, the graph expansion formula implies that HRG operator

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W(Pr2r1 ,−) is well-defined on O+(E)[[h]]. We refer to [12] for a thorough discussion in the current

context.

Lemma 2.11 motivates the following definition.

Definition 2.13 ([12]). A solution of effective quantum master equation is an assignment I[r] ∈O(E)[[h]] for each parametrix Pr satisfying

• Renormalized quantum master equation (RQME)

(Q + h∆Kr)eI[r]/h = 0.

• Homotopic renormalization group flow equation (HRG): for any two parametrices Pr1 , Pr2 ,

I[r2] = W(Pr2r1

, I[r1]).

RQME and HRG are compatible by Lemma 2.11. A solution of effective quantum master equa-tion is completely determined by its value at a fixed parametrix. Functionals at other paramatricesare obtained via HRG.

Remark 2.14. Here we adopt the name “homotopic RG flow” as opposed to the name ”RG flow”in [12]. If the manifold preserves a rescaling symmetry, it will induce a rescaling action on thesolution space of effective quantum master equations. Such flow equation will be called RG flowto be consistent with the physics terminology.

2.3.3. Locality and counter-term technique. In practice, we obtain solutions of renormalized quan-tum master equations via local functionals with the help of the method of counter-terms. Weexplain this construction in this subsection.

Definition 2.15. A functional I ∈ O(E) is called local if it can be expressed in terms of an integra-tion of a Lagrangian density L

I(φ) =∫

XL(φ), φ ∈ E .

L can be viewed as a Dens(X)-valued function on the jet bundle of E . The subspace of localfunctionals of O(E) is denoted by Oloc(E). We also denote

O+loc(E)[[h]] = O

+(E)[[h]] ∩Oloc(E)[[h]].

Let us assume we are in the situation of Example 2.9. Given L > 0, we have a smooth reg-ularized BV kernel KL in terms of the heat kernel. Let PL

ε be the regularized propagator for0 < ε < L < ∞.

Given any local functional I ∈ O+loc(E)[[h]], we can find anε-dependent local functional ICT(ε) ∈

hOloc(E)[[h]], such that the following limit exists

I[L] = limε→0

W(PLε , I + ICT(ε)) ∈ O+(E)[[h]].

Here ICT(ε) has a singular dependence onεwhenε→ 0, and this singularity exactly cancels thosethat come from the naive graph integral W(PL

ε , I). ICT(ε) is called the counter-term, which playsan important role in the renormalization theory of quantum fields. For a proof of the existence ofcounter-terms in the current context, see for example [12, Appendix 1].

By construction, I[L] satisfies HRG for all heat kernel regularizations:

I[L2] = W(PL2L1

, I[L1]), ∀0 < L1, L2 < ∞.


It remains to analyze the quantum master equation.

Firstly we observe that although the BV operator ∆L becomes singular as L → 0, its associatedBV bracket is in fact well-defined on local functionals. This is because only a single delta-functionappears in the limit L→ 0 of the bracket and it can be naturally integrated within local functionals.

Definition 2.16. We define the classical BV bracket on Oloc(E) by

I1, I2 := limL→0I1, I2L , I1, I2 ∈ Oloc(E).

Definition 2.17. A local functional I ∈ Oloc(E) is said to satisfy classical master equation (CME) if

QI +12I, I = 0.

Therefore classical master equation does not require renormalization. In practice, any localfunctional with a gauge symmetry can be completed into a local functional that satisfies the clas-sical master equation. This fact lies in the heart of the Batalin-Vilkovisky formalism.

Remark 2.18. Given I0 ∈ Oloc(E) satisfying the classical master equation, Q + I0,− defines asquare-zero vector field on E . The physical meaning is that it generates the infinitesimal gaugetransformation. In mathematical terminology, it defines a (local) L∞ structure on E .

To proceed to construct solutions of effective quantum master equations, let us naively usecounter-terms to construct a family I[L] as above from I0. It can be shown that I[L] satisfies renor-malized quantum master equation modulo h

QI[L] + h∆L I[L] +12I[L], I[L]L = O(h).

Then we need to find quantum corrections deforming I0 via higher order terms in h

I0 + hI1 + h2 I2 + · · · ∈ O+loc(E)[[h]]

such that the renormalized quantum master equation holds true at all orders of h. This can beformulated as a deformation problem. The quantum corrections may become very complicated ingeneral, and intrinsic obstructions for solving the quantum master equation could exist at certainh-order (gauge anomalies). Nevertheless, one of our main purposes here is to understand thegeometric meaning of such quantum corrections and find their solutions.

2.4. Example: Topological quantum mechanics. The simplest nontrivial example is when X isone-dimensional. This corresponds to quantum mechanical models. We explain the one dimen-sional Chern-Simons theory that is studied in detail in [32]. In such a geometric situation, therenormalized BV master equation is related to Fedosov’s abelian connection on Weyl bundles[19]. In the next section, we generalize this analysis to two dimensional models.

Let X = S1. Let V be a graded vector space with a degree 0 symplectic pairing

(−,−) : ∧2V → R.

The space of fields will beE = Ω•(S1)⊗V.

Let us denote XdR by the super-manifold whose underlying topological space is X and whosestructure sheaf is the de Rham complex on X. We can identify ϕ ∈ E with a map ϕ betweensuper-manifolds

ϕ : XdR → V

14 SI LI

whose underlying map on topological spaces is the constant map to 0 ∈ V.

The differential Q = dS1 on E is the de Rham differential on Ω•(S1). The (−1)-shifted symplec-tic pairing is the pairing

ω(ϕ1,ϕ2) :=∫

S1(ϕ1,ϕ2).

The induced differential BV structure is just the AKSZ-formalism [1] applied to the one-dimensionalσ-model. Given I ∈ O(V) of degree k, it induces an element I ∈ O(E) of degree k− 1 via

I(ϕ) :=∫

S1ϕ∗(I), ∀ϕ ∈ E .

We choose the standard flat metric on S1 and use the heat kernel regularization as in Example2.9. The following Theorem is established in [32].

Theorem 2.19 ([32]). Given I ∈ Sym≥3(V∗) + hO(V)[[h]] of degree 1, the limit

I[L] = limε→0

W(PLε , I)

exists as an degree 0 element of O+(E)[[h]]. The family I[L]L>0 solves the effective quantum masterequation if and only if

[I, I]? = 0.

Here O(V)[[h]] inherits a natural Moyal-Weyl product ? from the linear symplectic form (−,−). [−,−]?is the commutator with respect to the Moyal-Weyl product.

In [32], the above theorem is further extended to a family version of V parametrized by a sym-plectic manifold. Then the effective quantum master equation is equivalent to a flat connectiongluing the Weyl bundle (i.e. the bundle of Weyl algebra with fiberwise Moyal-Weyl product). Asbeautifully shown by Fedosov [19], flat sections of the Weyl bundle leads to a deformation quanti-zation of the Poisson algebra of smooth functions on X. A further analysis of the partition functionleads to a geometric formulation of the algebraic index theorem [24, 32].

3. VERTEX ALGEBRA AND BV MASTER EQUATION

In this section we study Batalin-Vilkovisky quantization of two dimensional chiral deforma-tions of free theories on flat surfaces. We give a self-contained treatment of Costello’s homotopicrenormalization framework in this case using heat kernel estimates of graph integrals establishedin [34]. We show that such two dimensional chiral theories are free of ultra-violet divergence(Theorem 3.10), and establish our main theorem on the correspondence between solutions of therenormalized quantum master equation and Maurer-Cartan elements of the modes Lie algebra ofchiral vertex operators (Theorem 3.14). We prove the modularity property of generating functionsof chiral theories on elliptic curves and establish the polynomial nature of their anti-holomorphicdependence on the moduli (Theorem 3.25).

3.1. Vertex algebra. In this section we collect some basics on vertex algebras that will be used inthis paper. We refer to [22, 27] for details.


3.1.1. Definition of vertex algebras. In this section V will always denote a Z/2Z-graded superspaceover C. It has two components with different parities

V = V0 ⊕V1, Z/2Z = 0, 1.

We say v ∈ V has parity p(a) ∈ Z/2Z if v ∈ Vp(v).

Definition 3.1. A field on V is a power series

A(z) = ∑k∈Z

A(k)z−k−1 ∈ End(V)[[z, z−1]]

such that for any v ∈ V , A(z)v ∈ V((z)). We will denote

A(z)+ = ∑k<0

A(k)z−k−1, A(z)− = ∑

k≥0A(k)z

−k−1.

A(k)’s will be called Fourier modes of A.

• Given two fields A(z), B(z), we define their normal ordered product

: A(z)B(w) := A(z)+B(w) + (−1)p(A)p(B)B(w)A(z)−.

Note that End(V) is naturally a superspace. p(A), p(B) are the parities.• Two fields A(z), B(z) are called mutually local if

A(z)B(w) =N−1

∑k=0

Ck(w)

(z− w)k+1 + : A(z)B(w) : .

for some N ∈ Z≥0 and fields Ck. The first term on the right is called the singular part, andwe shall write

A(z)B(w) ∼N−1

∑k=0

Ck(w)

(z− w)k+1 .

The above two formulae are called the operator product expansion (OPE).

Definition 3.2. A vertex algebra is a collection of data

• (space of states) a superspace V ;• (vacuum) a vector |0〉 ∈ V0;• (translation operator) an even linear operator T : V → V ;• (state-field correspondence) an even linear operation (vertex operators)

Y(·, z) : V → EndV [[z, z−1]]

taking each A ∈ V to a field Y(A, z) = ∑n∈Z

A(n)z−n−1;

satisfying the following axioms:

• (vacuum axiom) Y(|0〉, z) = IdV . Furthermore, for any A ∈ V we have

Y(A, z)|0〉 ∈ V [[z]], and limz→0

Y(A, z)|0〉 = A.

• (translation axiom) T|0〉 = 0. For any A ∈ V , [T, Y(A, z)] = ∂zY(A, z);• (locality axiom) All fields Y(A, z), A ∈ V , are mutually local.

16 SI LI

Let A, B ∈ V . Their OPE can be expanded as

Y(A, z)Y(B, w) = ∑n∈Z

Y(A(n) · B, w)

(z− w)n+1 ,

where

A(n) · B

n∈Zcan be viewed as defining an infinite tower of products.

Definition 3.3. A vertex algebra V is called conformal, of central charge c ∈ C, if there is a vectorωvir ∈ V (called a conformal vector) such that under the state-field correspondence Y(ωvir, z) =

∑n∈Z

Lnz−n−2:

• L−1 = T, L0 is diagonalizable on V;• Lnn∈Z span the Virasoro algebra with central charge c.

Let V be a conformal vertex algebra. The field Y(ω, z) will often be denoted by T(z), called theenergy momentum tensor. V can be decomposed as

V =⊕α

Vα , L0|Vα = α.

A ∈ Vα is said to have conformal weightα. Its field will often be expressed by

Y(A, z) = ∑k

Akz−k−α , Ak : V• → V•−k,

where Ak has conformal weight −k. In previous notations, Ak = A(k+α−1).

3.1.2. Modes Lie algebra. Following the presentation in [22], we associate a canonical Lie algebra toa vertex algebra via Fourier modes of vertex operators.

Definition 3.4. Given a vertex algebra V , we define a Lie algebra, denoted by∮V , as follows. As

a vector space, the Lie algebra∮V has a basis given by A(k)’s∮

V := SpanC

∮dzzkY(A, z) := A(k)

A∈V ,k∈Z

.

The Lie bracket is determined by the OPE (Borcherds commutator formula)[A(m), B(n)

]= ∑

j≥0

(mj

)(A( j)B

)(m+n− j)

.

Here we use a different notation∮

than U(·) in [22, Section 4.1] to emphasize its nature of modeexpansion. The commutator relations are better illustrated formally in terms of residues[∮

dzzmY(A, z),∮

dwwnY(B, w)

]=∮

dwwn∮

wdzzm

∑j∈Z

Y(A( j) · B, w)

(z− w) j+1 ,

where∮

w dz :=∫

Cwdz

2π i is the integration over a small loop Cw around w.

Equivalently, the vector space∮V can be described as∮

V = V [z, z−1]/ im ∂

where ∂(A ⊗ zk) := T(A) ⊗ zk + kA ⊗ zk−1, A ∈ V. Then A ⊗ zk represents the Fourier mode∮dzzkY(A, z) and im ∂ represents the space of total derivatives.


3.2. Free theory examples. In this subsection, we describe several free theory examples of vertexalgebras given by βγ − bc systems and chiral bosons (Heisenberg vertex algebra). Our main goalin this paper is to study chiral deformations of such free theories. The basic reference are [21, 22].

3.2.1. βγ − bc system. Let h = ⊕α∈Qhα, where hα = hα0 ⊕ hα1 , be a Q-graded superspace. Herethe Q-grading is the conformal weight and we assume for simplicity only finitely many weightsappear in h. The data of conformal weight will not be used for our main theorem in Section 3.3. Itgives important information on various selection rules.

Let h be equipped with an even symplectic pairing

〈−,−〉 : ∧2h→ C

which is of conformal weight −1, that is, the only nontrivial pairing is

〈−,−〉 : hα ⊗ h1−α → C.

For each a ∈ hα, we associate a field

a(z) = ∑r∈Z−α

arz−r−α .

We define their singular part of OPE by

a(z)b(w) ∼(

ihπ

)〈a, b〉

(z− w), ∀a, b ∈ h.

This is equivalent to the commutator relations

[ar, bs] =ihπ〈a, b〉 δr+s,1, ∀a, b ∈ h, r ∈ Z−α, s ∈ Z+α.

The vertex algebra V [h] that realizes the above OPE relations is given by the Fock representationspace. The vacuum vector satisfies

ar|0〉 = 0, ∀a ∈ hα , r +α > 0,

and V [h] is freely generated from the vacuum by the operators arr+α≤0, a ∈ hα. For any a ∈ h,a(z) becomes a field acting naturally on the Fock space V [h]. V [h] is a conformal vertex algebra,with energy momentum tensor

T(z) =12 ∑

i, jωi j : ∂ai(z)a j(z) :

where ai is a basis of h, ωi j is the inverse matrix of⟨

ai, b j⟩ so that ∑kωik⟨

ak, a j⟩ = δji . The

central charge is dim h0 − dim h1.

Remark 3.5. In quantum field theory language, this system is described by the free chiral theory

12 ∑

i, jωi j

∫d2zai

∂a j

and ai’s are fundamental fields. When h = h0 is purely bosonic, the associated vertex algebra isthe βγ system (or the chiral differential operators [23, 40]). When h = h1 is purely fermionic, theassociated vertex algebra is the bc system.

18 SI LI

Under the state-field correspondence, the vertex algebra V [h] can be identified with the poly-nomial algebra

V [h] ∼= C[∂

kai]

, ai is a basis of h, k ≥ 0.

We also denote its formal completion by

V [[h]] := C[[∂kai]].

V [h],V [[h]] can be viewed as the chiral analogue of the Weyl algebra.

3.2.2. Heisenberg vertex algebra. Let h = h0 be a finite dimensional vector space (of pure even type)equipped with an inner product 〈−,−〉. For each a ∈ h, we associate a field

a(z) = ∑r∈n

anz−n−1.

We define their singular part of OPE by

a(z)b(w) ∼(

ihπ

)〈a, b〉

(z− w)2 , ∀a, b ∈ h.

This is equivalent to the commutator relations

[an, bm] =ihπ〈a, b〉 nδn+m,0, ∀a, b ∈ h, m, n ∈ Z.

The Heisenberg vertex algebra V [h] that realizes the above OPE relations is also given by the Fockrepresentation space. The vacuum vector satisfies

an|0〉 = 0, ∀a ∈ h, n ≥ 0,

and V [h] is freely generated from the vacuum by the operators ann<0, a ∈ h. For any a ∈ h, a(z)becomes a field acting naturally on the Fock space V [h]. V [h] is a conformal vertex algebra, withenergy momentum tensor

T(z) =12 ∑

i: ∂ai(z)∂ai(z) :

where ai is an orthonormal basis of h. The central charge is dim h.

Remark 3.6. In quantum field theory language, this system is described by free chiral bosons

∑i

∫d2z∂zϕ

i∂zϕ

i.

The fields ai can be identified with ai = ∂zφi which has conformal weight 1.

Under the state-field correspondence, the vertex algebra V [h] can be identified with the poly-nomial algebra

V [h] ∼= C[∂

kai]

, ai is a basis of h, k ≥ 0.

Under this isomorphism, the vacuum |0〉 corresponds to 1, the operator ai−n−1 corresponds to

multiplication by ∂nai

n! , and the translation T corresponds to the differentiation ∂.

We also denote its formal completion by

V [[h]] := C[[∂kai]].

This example will play an important role in our application in Section 4.


3.3. Chiral deformation of βγ − bc system on flat spaces. Let Σ be a complex curve, KΣ be itscanonical line bundle. We are interested in the following data as a BV set-up:

• (E, δ) is a Z-graded holomorphic bundle E on Σ with a square-zero holomorphic differen-tial operator δ of degree 1;• A degree 0 symplectic pairing of bundles

(−,−) : E⊗ E→ KΣ.

Here KΣ is viewed as a line bundle concentrated at degree 0.

The space of quantum fields associated to the above data will be compactly supported Dolbeaultcomplex

E = Ω0,•c (Σ, E).

The (−1)-symplectic pairingω on E is obtained via

E ⊗ E

ω!!

(−,−)// Ω0,•

c (Σ, KΣ) = Ω1,•c (Σ)

∫ΣwwC

and we require its compatibility with δ.

Then the triple (E , Q = ∂ + δ,ω) defines an infinite dimensional (−1)-symplectic structurein the sense of section 2. Our main goal in this section is to study the associated renormalizedquantum master equation, which can be viewed as chiral deformations of βγ − bc systems.

We will mainly consider the flat space in this paper when Σ = C,C∗ or the elliptic curve Eτ =

C/(Z⊕Zτ) (τ ∈ H). We assume this from now on.

Definition 3.7. We will fix a coordinate z on Σ: this is the linear coordinate on C; z ∼ z + 1 on C∗;and z ∼ z + 1 ∼ z + τ on Eτ . Our convention of the volume form is

d2z :=i2

dz ∧ dz.

We also use the following notations. Let z = x + iy where x, y are real coordinates. Let −→z denotes(x, y). Then f (−→z ) means a smooth function on z while f (z) means a holomorphic function.

Since Σ is flat, we will focus on the situation in this paper when (E, (−,−)) comes from lineardata (h, 〈−,−〉) as follows:

• h =⊕

m∈Zhm is a Z-graded vector space (the grading is the cohomology degree) and each

hm is of finite dimension;• 〈−,−〉 : ∧2h→ C is a degree 0 symplectic pairing;• E = Σ× h, and E = Ω0,•

c (Σ)⊗ h;• δ ∈ C

[∂

∂z

]⊗⊕

mHom(hm, hm+1). HereC

[∂

∂z

]represents translation invariant holomorphic

differential operators on Σ. Moreover, δ2 = 0;• ω (−,−) is induced from fiberwise 〈−,−〉 via the identification KΣ

∼= OΣdz

ω (ϕ1,ϕ2) :=∫

dz ∧ 〈ϕ1,ϕ2〉 .

We require that δ is compatible withω (−,−).

20 SI LI

We consider the dual Z-graded vector space h∗ with the induced symplectic pairing still de-noted by 〈−,−〉. The Z/2Z-grading associated to the Z-grading defines the parity on h∗. Theextra Z-grading of conformal weight will not be explicit at this stage. At this stage, we assume forsimplicity that the pairing 〈−,−〉 on h∗ has conformal weight −1 and obtain the vertex algebraV [[h∗]] as in Section 3.2.1. The same discussion can be easily generalized when the conformalweight 〈−,−〉 is different from −1 or examples of chiral bosons. We describe such an example ofHeisenberg vertex algebra in Section 4.

3.3.1. chiral local functional. The relevant chiral local functionals will be described by elements of∮V [[h∗]].

Definition 3.8. Given I ∈ V [[h∗]][[h]], we extend it Ω0,•c [[h]]-linearly to a map

I : E → Ω0,•c [[h]].

Explicitly, if I = ∑ ∂k1 a1 · · · ∂km am ∈ V [[h∗]], where ai ∈ h∗,ϕ ∈ E , then

I(ϕ) = ∑±∂k1z a1(ϕ) · · · ∂km

z am(ϕ).

Here ai(ϕ) ∈ Ω0,•c comes from the natural pairing between h∗ and h. ∂z is the holomorphic

derivative with respect to our prescribed linear coordinate z on Σ. ± is the Koszul sign. Weassociate a local functional I ∈ Oloc(E)[[h]] on E by

I(ϕ) := i∫Σ

dz I(ϕ), ϕ ∈ E .

Note that deg( I) = deg(I) − 1 for each homogenous element I = ∂k1 a1 · · · ∂km am. It differs by−1 because the integration map

∫dz(−) is of cohomology degree−1 (only non-vanishing on Ω0,1

component).

The following definition is analogous to Definition 2.12.

Definition 3.9. We define the subspace V+[[h∗]][[h]] ⊂ V [[h∗]][[h]] by saying that I ∈ V+[[h∗]][[h]]if and only if lim

h→0I is at least cubic when V [[h∗]] is viewed as a (formal) polynomial ring.

3.4. Ultra-violet finiteness.

3.4.1. Regularization. We work with heat kernel regularization as in Example 2.9.

We will fix the standard flat metric on Σ. Let ∂∗ denote the adjoint of ∂, and ht ∈ C∞(Σ×Σ), t >0, be the heat kernel function of the Laplacian operator H =

[∂, ∂∗

]. It is normalized by

(e−tH f )(−→z 1) =∫Σ

d2z2 ht(−→z 1,−→z 2) f (−→z 2), t > 0.

For Σ = C, we have ∂∗ = −2∂zι∂zand H = −2∂z∂z = 1

2

(∂2

x + ∂2y

)where z = x + iy, and ι

∂zis

the contraction with the vector field ∂

∂z . Then explicitly

ht(−→z 1,−→z 2) =

12π t

e−|z1−z2|2/2t.

When Σ = C∗ or Eτ , ht is obtained from the above heat kernel on C by a further summation overthe relevant lattices.


Note that δ commutes with ∂ and ∂∗, hence H = [Q, ∂∗] = [∂+ δ, ∂∗]. The regularized BV kernelKL ∈ Sym2(E) is given by

KL(−→z 1,−→z 2) = i hL(

−→z 1,−→z 2)(dz1 ⊗ 1− 1⊗ dz2)Ch.

Here Ch = ∑i, jωi j(ai ⊗ a j) is the Casimir element where ai is a basis of h, ωi j is the inversematrix of

⟨ai, b j⟩. The normalization constant is chosen such that

(e−LHϕ)(−→z 1) = −12

∫Σ

dz2 ∧⟨KL(−→z 1,−→z 2),ϕ(

−→z 2)⟩

, ∀ϕ ∈ E

where 〈−,−〉 inside the integral is the pairing in the z2-component. The factor 12 is the symmetry

factor, while the extra minus sign comes from passing KL through dz2. Then K0 is precisely thedesired singular Poisson tensor. The regularized propagator is

PLε =

∫ L

ε(∂∗ ⊗ 1)Kudu, 0 < ε < L < ∞.

3.4.2. UV finiteness. The first important property of our 2d chiral deformed theory is the absenceof ultra-violet divergences.

Theorem 3.10. Assume the sep-up of βγ − bc system in Section 3.3. Let I ∈ V+[[h∗]][[h]] of degree 1and I be the associated chiral local functional defined in Definition 3.8. Let W(−,−) be the homotopy RGflow operator defined in Definition 2.12. Then the limit

I[L] := limε→0

W(PLε , I), L > 0

exists as a degree 0 element of O+(E)[[h]]. I[L]L>0 defines a family of O+(E)[[h]] satisfying the homo-topic RG flow equation, and lim

L→0I[L] = I.

Proof. When Σ = C, the existence of the limit ε → 0 follows from Prop A.1. The other casesΣ = C∗, Σ = Eτ follow from the result on C since the corresponding propagators have the samesingular behavior (see the argument of [34, Lemma 3.1] on this formal reasoning). The homotopicRG flow equation follows by construction (see also the discussion in Section 2.3.3).

In terms of the discussion in Section 2.3.3, it says that singular ε-dependent counter terms arenot needed. In other words, the UV divergence is absent and we say the theory is UV finite.

We remark that the order of limit is important in the proof of Theorem 3.10. The chiral natureof the problem implies that all potential singularities in W(PL

ε , I) in fact vanish upon integrationby parts before taking the limit ε→ 0 [34].

Remark 3.11. Theorem 3.10 is stated only for βγ − bc system for convenience. It in fact works alsofor chiral deformations of chiral bosons. This is because the relevant propagator (regularizationof 1

(z1−z2)2 ) is given by a further holomorphic derivative of PLε described as above, and Prop A.1

still holds in this case. We choose not to state a general effective BV renormalization theory forchiral bosons in this paper. It involves further complications by introducing degenerate BV the-ory. Instead, we explain by a concrete example in Section 4 (see Theorem 4.4) and illustrate themodifications in practice.

Remark 3.12. The UV property for chiral deformations is known to physicists via the method ofpoint-splitting regularization. In [38], we will give a sheaf theoretical interpretation of our heatkernel estimate in terms of meromoprhic differential forms. The basic idea is illustrated in [36].

22 SI LI

3.5. Quantum master equation. In this subsection, we study the quantization problem for chiraldeformations of βγ− bc systems and analyze the renormalized quantum master equation for I[L]constructed in Theorem 3.10. We assume the set-up in Section 3.3.

The differential δ induces naturally a differential on the formal polynomial ring V [[h∗]] (denotedby the same symbol)

δ : V [[h∗]]→ V [[h∗]].Let us write δ = D⊗φ, where D represents a holomorphic differential operator andφ ∈ Hom(h, h).Letφ∗ ∈ Hom(h∗, h∗) be the dual ofφ. Then in terms of generators,

δ(∂kza) := ∂

kzD(φ∗(a)), a ∈ h∗.

Since δ commutes with ∂z, it induces a differential on the Fourier modes

δ :∮V [[h∗]]→

∮V [[h∗]].

Recall we have a natural Lie bracket defined on∮V [[h∗]][[h]] as in Section 3.1.2.

Lemma 3.13. The differential δ and the Lie bracket [, ] define a structure of differential graded Lie algebraon∮V [[h∗]][[h]].

Proof. We leave this formal check to the interested reader.

The main theorem in this paper is the following, which can be viewed as the two dimensionalchiral analogue of Theorem 2.19.

Theorem 3.14. Assume the sep-up of βγ − bc system in Section 3.3. Let I ∈ V+[[h∗]][[h]] of degree 1and I be the associated chiral local functional defined in Definition 3.8. Let W(−,−) be the homotopy RGflow operator defined in Definition 2.12. Let PL

ε be the regularized propagator (ε, L > 0) defined in Section3.4. Define

I[L] = limε→0

W(PLε , I)

where the limit exists by Theorem 3.10. Let∮

dzI = I(0) be the corresponding mode of the vertex operator Ias an element of the modes Lie algebra

∮V [[h]][[h]].

Then the family I[L]L>0 satisfies renormalized quantum master equation if and only if

δ

∮dzI +

12

(ihπ

)−1 [∮dzI,

∮dzI]= 0,

i.e..∮

dzI is a Maurer-Cartan element of∮V [[h]][[h]].

Remark 3.15. As in Remark 3.11, this Theorem also holds for chiral deformations of chiral bosons.We will not present a general theory for this, but illustrate how to set things up and modify inpractice by a concrete example in Section 4 (Theorem 4.4).

Proof. It is enough to work with Σ = C, since C∗, Eτ are quotients by translations and our δ andI are translation invariant. The quantization on C is translation invariant and will descent toquotient spaces by translations due to the locality of the obstruction for solving the renormalizedquantum master equation [12, Chapter 5, Section 13]. In the following we assume Σ = C.

Since the renormalized quantum master equations

(†) QI[L] + h∆L I[L] +12

I[L], I[L]

L = 0, or (Q + ∆L) e I[L]/h = 0,


are equivalent for each L under homotopy RG flow, it is enough to analyze the limit L→ 0.

By Theorem 3.10,

(Q + h∆L) e I[L]/h = limε→0

(Q + h∆L)(

eh∂PLε e I/h

)= limε→0

eh∂PLε

((Q + h∆ε) e I/h

)=

1h

limε→0

eh∂PLε

(QI + h∆ε I +

12

I, Iε

)e I/h

=1h

limε→0

eh∂PLε

(δ I +

12

I, Iε

)e I/h.

Here we have observed

• ∆ε I = 0, since I is local and Kε becomes zero when restricted to the diagonal (the factordz1 − dz2 vanishes when z1 = z2 = z).• ∂ I = 0, since it contributes to a total derivative.

Therefore formally

he− I/h limL→0

(Q + h∆L) e I[L]/h = δ I +12

limL→0

limε→0

e− I/heh∂PLε

(I, Iε

e I/h)

.

The first term matches with that in the theorem up to a factor of i (recall Definition 3.8). It remainsto analyze the second term involving effective BV bracket. We proceed in several steps.

Step 1: Reduction to two-vertex diagram. We claim

limL→0

limε→0

e− I/heh∂PLε

(I, Iε

e I/h)= lim

L→0limε→0

∑m≥0

WΓm( I, PLε , Kε).

Here WΓm( I, PLε , Kε) is the Feynman graph integral whose diagram contains

• two vertices where we put I,• m propagators where we put PL

ε (draw by solid lines),• and one extra propagator by Kε (draw by dashed line):

This says that only diagrams with two vertices contribute to the limit L→ 0.

To see this, we first observe that the limit

limε→0

WΓm( I, PLε , Kε)

exists as a chiral local functional which does not depend on L. This follows from the special caseof Prop A.2 when k = E− 1. Let us denote it by

Om = limε→0

WΓm( I, PLε , Kε) = lim

L→0limε→0

WΓm( I, PLε , Kε)

andO = ∑

m≥0Om.

24 SI LI

Next, let us analyze the combinatorial structure of the expression

eh∂PLε

(I, Iε

e I/h)

.

The analogue of the Feynman graph explanation of Definition 2.12 leads to

eh∂PLε

(I, Iε

e I/h)=

1h ∑

(Γ ,e) connected

WΓ( I, PL

ε , Kε)

exp

(1h ∑

Γ connectedWΓ (PL

ε , I)

)

where the summation (Γ , e) is over all connected graphs with a choice of a distinguished edgee of Γ . And W

Γ( I, PL

ε , Kε) is the graph integral where we put I on each vertex, put Kε on thedistinguished edge e, and put PL

ε on all the other edges. Here is an example for illustration.

I I

I I

PLε

PLε

Kε

PLε

PLε

e

Γ :

FIGURE 1. An example of (Γ , e)

Γ contains a maximal subgraph of the type Γm of two-vertex graph described above. It is theread part of the graph in the above example. Then Prop A.2 can be pictured by

I I

I I

PLε

PLε

Kε

PLε

PLε

PLε PL

εe

limε→0

= limε→0

I I

PLε

PLε

PLε PL

ε

O

In other words, the limit limε→0

WΓ( I, PL

ε , Kε) is simply computed by the limε→0

of the new graph

integral obtained by collapsing all the edges with the same ends as the edge e in Γ and put thechiral local functional Om = lim

ε→0WΓm( I, PL

ε , Kε) on the new vertex. We conclude that

limε→0

eh∂PLε

(I, Iε

e I/h)= limε→0

eh∂PLε

(Oe I/h

).


Note that the limit on the RHS exists, again by Prop A.1, since both I and O contain only holomor-phic derivatives. Furthermore, if we take the limit L → 0, all graphs containing propagators willvanish. We find

limL→0

limε→0

eh∂PLε

(I, Iε

e I/h)= lim

L→0limε→0

eh∂PLε

(Oe I/h

)= Oe I/h.

This establishes our claim for Step1

limL→0

limε→0

e− I/heh∂PLε

(I, Iε

e I/h)= O = lim

L→0limε→0

∑m≥0

WΓm( I, PLε , Kε).

Step 2: It remains to show

limL→0

limε→0

∑m≥0

WΓm( I, P(ε, L), Kε) =π

ih

[∮dzI,

∮dzI]

.

First, the vertex operator [∮

dzI,∮

dzI] is computed by the residue form of Borcherds commuta-tor formula as described in Section 3.1.2. This formula has a nice interpretation in terms of Wickcontractions using normal ordered product and OPE’s (see [21, Chapter 4] and [22, Section 3.3.10]).Explicitly, assume I = ∑ ∂k1 a1 · · · ∂kn an is expressed in terms of holomorphic derivatives of fields.Then the commutator [

∮dzI,

∮dzI] is a sum[∮

dzI,∮

dzI]= ∑

m≥0Jm

where each Jm consists of arbitrary m contractions of fundamental fields between the two copiesof∮

dzI, replacing them by the singular part of their OPE, and them compute the residue via thecontour integral as described in Section 3.1.2. Since the singular part of the OPE of fundamentalfields in the current step-up is

a(z1)b(z2) ∼(

ihπ

)〈a, b〉

(z1 − z2), ∀a, b ∈ h,

Jm can be pictured by

ihπ(z1−z2)

ihπ(z1−z2)

ihπ(z1−z2)

I IJm =

FIGURE 2. There are m+ 1 internal edges for Wick contractions. External edges arenon-contracted fields

In other words, we can write Jm as a sum of terms of the form (up to some normalization factor)

π

(m + 1)!

∫C

d2z2B(−→z 2)∮

z2

dz1 A(−→z 1)m

∏i=0

(∂

kiz1

iπ(z1 − z2)

)where A, B collects all those fields which are not contracted. The factor (m + 1)! is the symmetryfactor of the graph since it has m + 1 identical edges. The derivatives ∂

kiz1 come from those in I.

26 SI LI

Similarly, we can explicitly write limε→0

WΓm( I, PLε , Kε) as a sum of terms of the form (up to some

normalization factor)

limε→0

1(m + 1)! ∑

σ∈Sm+1

∫C

d2z1

∫C

d2z2 A(−→z 1)B(−→z 2)∂kσ(0)z1 Kε(

−→z 1,−→z 2)1

m!

m

∏i=1

∂kσ(i)z1 PL

ε (−→z 1,−→z 2)

where again A, B collects the external inputs. m! is the symmetry factor indicating m identicaledges where we put PL

ε . The average over permutations σ of 0, 1, · · · , m is due to the fact thatthe holomorphic derivatives are distributed symmetrically because the local functional I (whichcontains only holomorphic derivatives) is a (graded) symmetric multi-linear map.

The Theorem follows by showing that the above two expressions are the same for all m. After atedious match of the normalization factor, it is sufficient to prove the following identity

limε→0

1(m + 1)! ∑

σ∈Sm+1

∫C

d2z1

∫C

d2z2 A(−→z 1)B(−→z 2)∂kσ(0)z1 Kε(

−→z 1,−→z 2)1

m!

m

∏i=1

∂kσ(i)z1 PL

ε (−→z 1,−→z 2)

=π

(m + 1)!

∫C

d2z2B(−→z 2)∮

z2

dz1 A(−→z 1)m

∏i=0

(∂

kiz1

iπ(z1 − z2)

).

(†)

for any k0, · · · , km ∈ Z≥0, and smooth functions A, B with compact support. Here

Kε(−→z 1,−→z 2) =

i2πε

e−|z1−z2|2/2ε

PLε (−→z 1,−→z 2) = (−2i)

∫ L

εdt∂z1 ht(

−→z 1,−→z 2) = (−2i)∫ L

ε

dt2π t

z2 − z1

2te−|z1−z2|2/2t

where we have used the same symbols but only keep factors of the BV kernel and the regularizedpropagator that are relevant in this computation. A, B are test functions viewed as collectingall inputs on external edges of the two vertices.

∮z2

dz1 means a loop integral of z1 around z2

normalized by ∮z2

dz1

z1 − z2= 1.

Our notation∮

z2dz1 A(−→z 1)

m∏

i=0

(∂

kiz1

1z1−z2

)means only picking up the holomorphic derivative of A

according to the pole condition, i.e.,∮z2

dz1 A(−→z 1)1

(z1 − z2)m+1 :=1

m!∂

mz2

A(−→z 2).

The required identity (†) follows from Prop A.2. In fact, Prop A.2 implies

limε→0

∫C

d2z1

∫C

d2z2 A(−→z 1)B(−→z 2)∂k0z1

Kε(−→z 1,−→z 2)

1m!

m

∏i=1

∂kiz1

PLε (−→z 1,−→z 2)

=im+1(−2)m2mC(k0; k1, · · · , km)

(4π)mm!

∫d2zA(−→z )

∂

k0+m∑

i=1(ki+1)

z B(−→z )

=

im+1(−2)m2mC(k0; k1, · · · , km)

(4π)mm!

∫d2zB(−→z )

((−∂z)

k0+m∑

i=1(ki+1)

A(−→z )

)

=

im+1(−1)m+1C(k0; k1, · · · , km)

(k0 +

m∑

i=1(ki + 1)

)!

πmm!

∫d2z2B(−→z 2)

(∮z2

dz1A(−→z 1)

∏mi=0(z2 − z1)ki+1

)


where

C(k0; k1, · · · , km) =∫ 1

0· · ·

∫ 1

0

m

∏i=1

dui

m∏

i=1uki

i(1 +

m∑

i=1ui

) m∑

j=0(k j+1)

.

To simplify the coefficient, we use the identity

n!an+1 =

∫ ∞0

duune−ua

to write

C(k0; k1, · · · , km)

(k0 +

m

∑i=1

(ki + 1)

)!

=∫ 1

0· · ·

∫ 1

0

m

∏i=1

dui

m

∏i=1

ukii

∫ ∞0

du0 uk0+

m∑

i=1(ki+1)

0 e−u0(1+

m∑

i=1ui)

ui→uiu−10=

∫0≤ui≤u0 ,1≤i≤m

m

∏i=0

dui

m

∏i=0

(uki

i e−ui)

.

Since we eventually need to arerage over the permutations of 0, 1, · · · , m, we find

1(m + 1)! ∑

σ∈Sm+1

C(kσ(0); kσ(1), · · · , kσ(m))

(kσ(0) +

m

∑i=1

(kσ(i) + 1)

)!

=1

m + 1

∫ ∞0

m

∏i=0

dui

m

∏i=0

(uki

i e−ui)=

∏mi=0 ki!

m + 1.

It follows that

limε→0

∑σ∈Sk+1

1(m + 1)!

∫C

d2z1

∫C

d2z2 A(−→z 1)B(−→z 2)∂kσ(0)z1 Kε(

−→z 1,−→z 2)1

m!

m

∏i=1

∂kσ(i)z1 PL

ε (−→z 1,−→z 2)

=

im+1(−1)m+1m∏

i=0ki!

πm(m + 1)!

∫d2z2B(−→z 2)

∮z2

dz1A(−→z 1)

m∏

i=0(z2 − z1)ki+1

=π

(m + 1)!

∫d2z2B(−→z 2)

∮z2

dz1 A(−→z 2)m

∏i=0

∂kiz1

iπ(z1 − z2)

.

This proves the Theorem.

An another approach to Step 2: The above proof of Step 2 relies on A.2, which hides heavy analyticestimates established in [34]. Here below we give a different but more direct computation (thoughstill need a different computation in [34]) in order to give reader a feeling on what is happening.

Let us change coordinates by

(z1, z2)→ (z = z1 − z2, z2), z = reiθ .

Let us first focus on the term when σ is the trivial permutation.∫C

d2z1

∫C

d2z2 A(−→z 1)B(−→z 2)∂k0z1

Kε(−→z 1,−→z 2)

1m!

m

∏i=1

∂kiz1

PLε (−→z 1,−→z 2)

=− im+1(−2)m

m!

∫C

d2z2B(−→z 2)∫C

d2zA(−→z 2 +

−→z )

(−z)k0+···+km+m

(∫ L

ε

m

∏i=1

dti

2π ti

)e− r2

2ε−m∑

i=1

r22ti 1

2πε

(r2

2ε

)k0 k

∏i=1

(r2

2ti

)ki+1

28 SI LI

=− im+1(−2)m

m!

∫C

d2z2B(−→z 2)∫ ∞0

rdr

(∫ L

ε

m

∏i=1

dti

2π ti

)e− r2

2ε−m∑

i=1

r22ti 1

2πε

(r2

2ε

)k0 k

∏i=1

(r2

2ti

)ki+1 ∫|z|=r

dθA(−→z 2 +

−→z )

(−z)k0+···+km+m .

To compute∫|z|=r dθ A(−→z 2+

−→z )

zk0+···+km+m , we need to do Taylor expansion of A(−→z 2 +−→z ) around z = 0 to

get the Taylor series expansion

A(−→z 2 +−→z ) ∼ A(−→z 2) + ∂z2 A(−→z 2)z + ∂z2 A(−→z 2)z + · · · .

However, as shown by [34, Proposition B.2], only terms involving holomorphic derivatives willsurvive in the ε→ 0 limit. Therefore the θ integral can be replaced by∫

|z|=rdθ

A(−→z 2 +−→z )

(−z)k0+···+km+m → 2π∮

z2

dz1

z1 − z2

A(−→z 1)

(z2 − z1)k0+···+km+m ,

where the meaning of∮

z2dz1 is explained above. In particular, its value does not depend on r.

Therefore the r-integral can be evaluated as

limε→0

∫ ∞0

rdr

(∫ L

ε

m

∏i=1

dti

2π ti

)e− r2

2ε−m∑

i=1

r22ti 1

2πε

(r2

2ε

)k0 k

∏i=1

(r2

2ti

)ki+1

r2=2εu0====

ti=εu0/ui

1(2π)m+1

∫ ∞0

du0

∫ u0

0

k

∏i=1

duie−

m∑

i=0ui

m

∏i=0

ukii

=1

(2π)m+1

∫0≤ui≤u0 ,1≤i≤m

m

∏i=0

dui

m

∏i=0

(uki

i e−ui)

.

Areraging over the permutations of 0, 1, · · · , m, this integral leads to

1(2π)m+1

1(m + 1)

∫ ∞0

m

∏i=0

dui

m

∏i=0

(uki

i e−ui)=

1(2π)m+1

m∏

i=0ki!

(m + 1).

It follows by combining the above computations (we arrive that the same final computation as inthe above proof of Step 2) that

limε→0

∑σ∈Sk+1

1m!

∫C

d2z1

∫C

d2z2 A(−→z 1)B(−→z 2)∂kσ(0)z1 hε(

−→z 1,−→z 2)m

∏i=1

∂kσ(i)z1 PL

ε (−→z 1,−→z 2)

=−im+1(−2)m

m∏

i=0ki!

(2π)m(m + 1)!

∫d2z2B(−→z 2)

∮z2

dz1A(−→z 1)

m∏

i=0(z2 − z1)ki+1

=π

(m + 1)!

∫d2z2B(−→z 2)

∮z2

dz1 A(−→z 2)m

∏i=0

∂kiz1

iπ(z1 − z2)

.

3.6. Generating function and modularity. In this subsection we focus on the chiral deformedtheory on the flat torus. We show that in the limit L → 0, the Taylor components of the effectiveinteraction I[∞] exhibit modularity properties with prescribed anti-holomorphic dependence. InSection 4, this will be applied to explain why the generating function of Gromov-Witten invariantson the elliptic curve are quasi-modular forms in terms of the mirror B-model.


3.6.1. Generating function. We describe the generating function when Σ = Eτ = C/(Z+Zτ) is anelliptic curve.

Let I ∈ V [[h∗]][[h]] satisfy the Maurer-Cartan equation in Theorem 3.14. I[L] be the associatedfamily solving the renormalized quantum master equation

(Q + h∆L) e I[L]/h = 0.

Definition 3.16. We define the generating function I [Eτ ] = ∑g≥0 hg Ig [Eτ ] ∈ O(h[ε])[[h]] as aformal function on h[ε] where

• ε is an odd element of degree 1, representing the generator dzIm τ of harmonics H0,1(Eτ ).

• Under the identification H•(E , ∂) ∼= h⊗H0,•(Eτ ) ∼= h[ε], I[Eτ ] is the restriction of I[∞] toharmonic elements h⊗H0,•(Eτ ).• Given a1, · · · , am ∈ h[ε], we denote its m-th Taylor coeffeicient by

〈a1, · · · , am〉g =∂

∂a1· · · ∂

∂amIg[Eτ ](0).

Remark 3.17. Elements of h⊗H0,•(Eτ ) are ofter called zero modes. Our definition of I[Eτ ] is just theeffective theory on zero modes following physics terminology.

The space h[ε] carries naturally a (−1)-symplectic structure. The nontrivial pairing is betweenh and hε where

ω(a, bε) = 〈a, b〉 , a, b ∈ h.

Let ∆ denote the associated BV operator on O(h[ε]). It is not hard to see that ∆ can be identifiedwith ∆∞. The differential δ also induces a differential on h[ε]. Here we only need to keep theconstant part of the differential operator in δ, since ∂

∂z will annihilate the harmonics H0,•(Eτ ).

Proposition 3.18. The triple (O(h[ε]), δ, ∆) is a differential BV algebra. The generating function I[Στ ]satisfies the BV master equation

(δ+ h∆) e I[Eτ ]/h = 0.

Proof. We observe that the BV kernel KL lies in Sym2(h[ε]) when L → ∞, which defines the BVoperator ∆ as that on h[ε]. The proposition is just the quantum master equation

(Q + h∆∞)e I[∞]/h = 0

restricted to harmonic subspaces h[ε].

3.6.2. Modularity. Now we analyze the dependence of I[Eτ ] on the complex structure τ . We con-sider the modular group SL(2,Z), which acts on the upper half plane H by

τ → γτ :=Aτ + BCτ + D

, for γ ∈(

A BC D

)∈ SL(2,Z), τ ∈ H.

Recall that a function f : H→ C is said to have modular weight k under the modular transforma-tion SL(2,Z) if

f (γ−→τ ) = (Cτ + D)k f (−→τ ), for γ ∈(

A BC D

)∈ SL(2,Z).

30 SI LI

Definition 3.19. We extend the SL(2,Z) action to Cn ×H by

γ : (z1, · · · , zn, τ)→ (γz1, · · · ,γzn,γτ) :=(

z1

Cτ + D, · · · ,

zn

Cτ + D,

Aτ + BCτ + D

),

for γ ∈(

A BC D

)∈ SL(2,Z).

It is easy to see that this defines a group action, in other words,

γ1(γ2(z1, · · · , zn, τ)) = (γ1γ2)(z1, · · · , zn, τ), γ1,γ2 ∈ SL(2,Z).

Definition 3.20. A differential form Ω on Cn × H is said to have modular weight k under theabove SL(2,Z) action if

γ∗Ω = (Cτ + D)kΩ.

When n = 0 and Ω being a 0-form on H, this reduces to the above modular function of weightk.

Let hL be the heat kernel function on Eτ as before. Pulled back to the universal cover C of Eτ ,hL gives rise to a function hL on C×C×H by

hL(−→z 1,−→z 2;−→τ ) =

12πL ∑

λ∈Λτe−|z1−z2+λ|2/2L, Λτ = Z⊕Zτ .

SL(2,Z) transforms the lattice Λ by

Λγτ =1

Cτ + DΛτ , γ ∈

(A BC D

)∈ SL(2,Z).

It follows that the heat kernel hL transforms under SL(2,Z) as

hL(γ−→z 1,γ−→z 2;γτ) = |Cτ + D|2 h|Cτ+D|2 L(

−→z 1,−→z 2;−→τ ), ∀γ ∈ SL(2,Z).

The regularized BV kernel KL and propagator PLε are

KL(−→z 1,−→z 2;−→τ ) = i hL(

−→z 1,−→z 2;−→τ ) (dz1 ⊗ 1− 1⊗ dz2)Ch

and

PLε (−→z 1,−→z 2;−→τ ) = −2i

∫ L

εdu∂z1 hu(

−→z 1,−→z 2;−→τ )Ch.

We define similarly KL, PLε as hL on the universal cover C of Eτ . The following lemma is straight-

forward.

Lemma 3.21. We have the following modular transformation properties of the regularized propagators andeffective BV kernel

γ∗PLε = (Cτ + D)P|Cτ+D|2 L

|Cτ+D|2ε, γ∗KL = (Cτ + D)K|Cτ+D|2 L.

In particular, P∞0 , K∞ have weight 1 in the sense of Definition 3.20. More generally, the k-th holomorphic

derivative ∂kz1

P∞0 (−→z 1,−→z 2;−→τ ) has modular weight k + 1.

Remark 3.22. K∞(−→z 1,−→z 2;−→τ ) = dz1⊗1−1⊗dz22 Im τ Ch.

Definition 3.23. A quantization I[L] defined by I = ∑g≥0

Ighg ∈∮V [[h∗]][[h]] in Theorem 3.10 is

called modular invariant if each Ig contains exactly g holomorphic derivatives.


Remark 3.24. This definition may vary according to the conformal weight of the pairing 〈−,−〉 onh∗. See Remark 4.7 for a specific example.

Theorem 3.25. Let I[Eτ ] = ∑g≥0

Ig[Eτ ]hg be the generating function of a modular invariant quantization,

Ig[Eτ ] ∈ O(h[ε]). Then for any a1, · · · , ak ∈ h, b1, · · · , bm ∈ hε, the Taylor coefficient of Ig[Eτ ]

〈a1, · · · , ak, b1, · · · , bm〉gis modular of weight m + g− 1 as a function on H. Moreover, It has the following expansion

〈a1, · · · , ak, b1, · · · , bm〉g =N

∑i=0

fi(τ)

(Im τ)i

where fi(τ)’s are holomorphic functions in τ and N < ∞ is an integer.

Proof. Let us writeI[∞] = ∑

Γ connectedW(Γ , I).

Let Γ be a Feynman graph which contributes to 〈a1, · · · , ak, b1, · · · , bm〉g. Due to the type of dif-ferential forms used (the propagator only contains 0-forms on Eτ ), Γ contains m vertices v jm

j=1each of which has an external input given by b j, j = 1, · · ·m.

Assume the vertex v j has genus g j, i.e., given by the local functional Ig j . Let E be the set ofpropagators in Γ . Then

m + g =m

∑j=1

g j + |E|+ 1.

The graph integral W(Γ , I) can be written as

W(Γ , I) = limε→0L→∞ A

m

∏i=1

∫Eτ

d2zi

Im τ∏e∈E

∂nezh(e)

PLε (−→z h(e),

−→z t(e);−→τ )

where A is a combinatorial coefficient not depending on τ .

h, t : E→ 1, · · · , m

denote the head and the tale of the edge, where we have chosen an arbitrary orientation on edges.ne denotes the number of holomorphic derivatives applied to propagator at the edge e. Since allthe external inputs ai, b j’s are harmonics, all holomorphic derivatives in the vertex Ig j will go to thepropagators. By the modular invariance of the quantization, Ig j contains exactly g j holomorphicderivatives, hence

∑e∈E

ne =m

∑j=1

g j.

32 SI LI

It follows from the modular property of the measure d2zIm τ and the propagator P∞

0 that W(Γ , I) is amodular function on H of weight

∑e∈E

(ne + 1) = |E|+m

∑j=1

g j = m + g− 1.

The polynomial dependence on 1Im τ follows from [34, Proposition 5.1].

Given a function f on H of the form f =N∑

i=0

fi(τ)(Im τ)i , we denote

limτ→∞ f := f0(τ).

It is shown in [28] that f is determined by the leading term f0 and modular property. In partic-ular, the operation lim

τ→∞ identifies the space of almost holomorphic modular forms with the space

of quasi-modular forms [28]. In general, the τ → ∞ limit of the generating function I[Eτ ] will bereduced to certain characters on vertex algebras. This is argued in [18] by the method of contactterms, and realized in [35] by a method of cohomological localization via the quantum masterequation. This pheonomenon will be systematically studied in [38].

3.7. Example: Poisson σ-model. We illustrate the application of Theorem 3.14 by the example ofthe AKSZ formalism of Poisson σ-model as described in [9].

Let V = Rn and P be a Poisson bi-vector field on V. Let Σ be a flat surface as before. We considerthe BV formalism of Poisson sigma model in the formal neighborhood of constant maps from Σ tothe origin of V. The space of fields is

E = Ω•(Σ)⊗ (V ⊕V∗[1]).

The differential on E is the de Rham differential dΣ on Ω•(Σ). The (−1)-shifted symplectic pairingis

ω(ϕ, η) :=∫Σ(ϕ, η), whereϕ ∈ Ω•(Σ)⊗V, η ∈ Ω•(Σ)⊗V∗[1].

Let us choose linear coordinates xi on V and P = ∑i, j

Pi j(x)∂xi ∧ ∂x j . The above fields ϕ, η in

coordinate components are

ϕ = ϕini=1, η = ηin

i=1, whereϕi ∈ Ω•(Σ), ηi ∈ Ω•(Σ)[1].

The action functional is given by [9]

S = ∑i

∫ΣηidΣϕ

i +∑i, j

∫Σ

Pi j(ϕ)ηiη j.

Let us split the differential dΣ = Q + δ, where Q = ∂ and δ = ∂ are the (0, 1)-differential and(1, 0)-differential on Σ. Then the above theory falls into the setting of Section 3.3, and we can applyTheorem 3.14 to study its quantization via chiral deformations. In terms of notations in Section3.3, we have

h = V[dz]⊕V∗[dz][1],

where V[dz] = V ⊗C[dz], V∗[dz] = V∗ ⊗C[dz]. The relevant vertex algebra is generated byϕi, ηi:ϕi represent components in V[dz] and ηi represent components in V∗[dz][1]. The nontrivial OPEs


are given by

ϕi(z)η j(w) ∼ δi, j

(ihπ

)dz− dw

z− w.

Here it is understood that we have to match the corresponding components in dz, dw in the aboveformula. For example, if we write ϕi(z) = ϕi

0(z) +ϕi1(z)dz and ηi(z) = ηi0(z) + ηi1(z)dz, then

matching the dw component we find

ϕi0(z)η j1(w) ∼ δi, j

(ihπ

)−1

z− w.

The classical interaction is represented by the vertex operator∮I, where I = ∑

i, jPi j(ϕ)ηiη j.

Here it is understood (due to the type of differential forms used) that only the dz-component of Icontributes to

∮I when we expand the fieldsϕi, ηi into forms in z.

The classical master equation is satisfied because it satisfies two independent equations

δ

∮I = 0,

∮I,∮

I

= 0.

The first term vanishes since it produces a total derivative while the vanishing of the second termfollows from Jacobi identity for P [9].

Proposition 3.26. The classical interaction∮

I satisfies the quantum master equation of Theorem 3.14.

Sketch of proof. We only need to prove [∮

I,∮

I] = 0 since δ∮

I = 0. By definition, [∮

I,∮

I] iscomputed by Wick contractions and OPE’s (see [27]). A single contraction gives rise to the classicalbracket

∮I,∮

I. If we have two or more contractions between the fields, the form of I impliesthat each contraction contributes a factor of dz − dw. However, the product of two copies ofdz− dw vanishes by the type of the differential forms used. This implies [

∮I,∮

I] = ∮

I,∮

I =0.

This proposition says that no quantum correction is needed at all! This remarkable fact is dueto cancellations between bosons and fermions, which is just a incidence of supersymmetry. Thisresult gives a natural interpretation of Kontsevich’s graph formula of star product [29], which isargued in [8] by the requirement of BV quantum master equation. We remark that the tadpolediagrams (i.e. with edges that start and end at the same vertex) that appeared in [8] vanish by ourrenormalization scheme on flat spaces: the regularized BV kernel Kε and propagator PL

ε on thetadpole are zero before we take the limit ε → 0 (see the proof of Theorem 3.14). In particular, ourconstruction here should lead to a rigorous formulation of [8] by gluing the above linear case toPoisson manifolds.

In general, when the surface Σ is a compact Riemann surface which is no longer flat, furtherobstructions may exist for quantization. One such example is the topological B-model from agenus g surface Σ to a complex manifold X. The perturbative BV quantization in the currentsense is analyzed in [31]. It is found that the tadpole diagram gives rise to the obstruction class(anomaly) by (2g− 2)c1(X), requiring for a Calabi-Yau geometry to be quantizable. It would bevery interesting to extend the results in this paper systematically to arbitrary surfaces and to thestring-theoretical formulation of coupling with 2d gravity.

34 SI LI

4. APPLICATION: QUANTUM B-MODEL ON ELLIPTIC CURVES

In this section, we apply our theory to solve the higher genus B-model on elliptic curves. Partof the results are presented in [35] based mainly on symmetry argument. Using the techniquewe have developed in this paper, we present a self-contained description of quantum B-model onelliptic curves in terms of chiral vertex algebra and give stronger results on the exact solution ofthe full system. Combining with the A-model results in [44], it leads to a manifest interpretationof higher genus mirror symmetry on elliptic curves in terms of boson-fermion correspondence.

4.1. BCOV theory on elliptic curves. We consider topological B-model on Calabi-Yau geometry,which concerns with the geometry of complex structures. In [6], Bershadsky, Cecotti, Ooguri, andVafa proposed Kodaira-Spencer gauge theory on Calabi-Yau 3-folds as the leading approximationof B-twisted closed string field theory. This is fully generalized in [15] to arbitrary Calabi-Yaumanifolds, giving rise to a complete description of B-twisted closed string field theory in the senseof Zwiebach [45]. We shall call this BCOV theory. An earlier related work on the finite dimensionaltoy model of BCOV theory appeared in [39] in the absence of the issue of renormalization.

In this section, we describe BCOV theory on elliptic curves [35] and study its quantum geometryusing the tools we have developed in this paper.

Let Eτ = C/(Z⊕ Zτ) as before. Let z denote the linear holomorphic coordinate. The space offields of BCOV theory [15] specializing to Eτ is given by

E = Ω0,•(Eτ ,OEτ )[[t]]⊕Ω0,•(Eτ , TEτ [1])[[t]].

Here TEτ [1] is the holomorphic tangent bundle sitting at degree −1. Let t be a formal variableof cohomology degree 0 that represents “gravitational descendants”. Note that we have used adifferent grading from [15] 1 so our interaction will have degree 0. The differential is given by

Q = ∂ + δ, δ = t∂,

where ∂ : TEτ → OEτ is the divergence operator with respect to the holomorphic volume form dz.In terms of the set-up in section 3.3, we have

• h = C[[t,θ]], deg(t) = 0, deg(θ) = −1. Here θ represents the global vector field ∂z. ThenE ∼= Ω0,•(Eτ )⊗ h.• δ = ∂

∂z ⊗ t ∂

∂θ.

• However, E is (−1)-shifted Poisson instead of symplectic. Let

hL(−→z 1,−→z 2) = ∑

λ∈Λ

12πL

e−|z1−z2+λ|2/2L, Λ = Z+Zτ ,

be the heat kernel function on Eτ . The regularized BV kernel is given by

KL = i∂z1 hL(−→z 1,−→z 2)(dz1 ⊗ 1− 1⊗ dz2)Ch

where Ch := t0 ⊗ t0 ∈ h⊗ h. The regularized propagator is

PLε = −2i

∫ L

εdu ∂

2z1

hu(−→z 1,−→z 2)Ch.

The situation differs a bit from our set-up in Section 3.3. There is one more holomorphic deriv-ative for our BV kernel and the factor Ch is highly degenerate. Nevertheless, techniques in section3 can be applied to the Poisson case without much change (see also Remark 2.5). In particular,

1In [15], the grading is E = Ω0,•(Eτ ,OEτ )[[t]]⊕Ω0,•(Eτ , TEτ [−1])[[t]] and t has cohomology degree 2


• KL defines a regularized BV operator ∆L such that (O(E), Q, ∆L) is a differential BV alge-bra.• K0 well-defines a BV-bracket on local functionals as in Definition 2.16

−,− : Oloc(E)⊗Oloc(E)→ Oloc(E).

Introduce

〈−〉0 : Sym•(C[[t]])→ C,⟨

tk1 ⊗ · · · ⊗ tkn⟩

0=

(n− 3

k1, · · · , kn

).

These coefficients represents intersection numbers of ψ-classes on moduli space of stable rationalcurves. We extend it Ω0,•[θ]-linearly to

〈−〉0 : Sym•(E)→ Ω0,•[θ].

Definition 4.1 ([15]). We define the classical BCOV interaction IBCOV ∈ Oloc(E) by the local func-tional

IBCOV(ϕ) = i∫

Eτdz∫

dθ 〈eϕ〉0 , ϕ ∈ E .

Here 〈eϕ〉0 is understood as

〈eϕ〉0 = ∑k≥3

⟨ϕ⊗k

k!

⟩0

.

The fermionic integral∫

dθ means taking the coefficient of the term with θ∫dθ(a +θb) := b.

Proposition 4.2. IBCOV satisfies the classical master equation

QIBCOV +12

IBCOV , IBCOV

= 0.

This is proved in [15] for any Calabi-Yau. See [26, Theorem 3.5] for another proof and general-ization. To be self-contained, we remark that this proposition also follows from Corollary 4.19.

Remark 4.3. By Remark 2.18, Q + IBCOV ,− defines a L∞-structure on E . It is shown in [15] (see[37] for a geometric presentation) that this L∞-structure is quasi-isomorphic to the standard dgLie algebra structure with differential Q and Schouten-Nijenhuis bracket.

Since the Poisson kernel is degenerate and contains one more holomorphic derivative, the ap-plication of Theorem 3.14 to our situation requires slight modification. Let us parametrize

ϕ = ∑k≥0

bktk + ηkθtk, ϕ ∈ h.

We introduce mutually local fields bk(z), ηk(z) with OPE relations

b0(z)b0(w) ∼ ihπ

1(z− w)2 , bk(z)bm(w) ∼ 0, k + m > 0.

b•(z)η•(w) ∼ 0, η•(z)η•(w) ∼ 0.

The OPEs exactly respect the structure of our Poisson kernel: 1(z−w)2 represents the derivative of

the delta-function, and only b0 appears in the propagator. The associated vertex algebra V [[h∗]]is the tensor product of a Heisenberg vertex algebra (generated by the field b0(z)) with several

36 SI LI

copies of commutative vertex algebra (generated by the fields (b>0(z), η•(z))). Equivalently, if wecollect the fields with parameter t

b(z, t) = ∑k≥0

bk(z)tk, η(z, t) = ∑k≥0ηk(z)tk,

then the OPE’s can be simply written as

b(z1, t1)b(z2, t2) ∼ihπ

1(z1 − z2)2 , b(z1, t1)η(z2, t2) ∼ 0, η(z1, t1)η(z2, t2) ∼ 0.

The differential δ is then dually expressed as

δb(z, t) = t∂zη(z, t), δη(z, t) = 0.

In terms of components,

δbk+1 = ∂zηk, δηk = 0.

We adapt notations in section 3.3. The classical BCOV interaction can be expressed as IBCOV = I0,where I0 ∈ V [[h∗]] is defined similarly to Definition 4.1

I0(b, η) =⟨

eb ⊗ η⟩

0.

Classical master equation implies that it satisfies the following Maurer-Cartan equation modulo h

δ

∮dzI0 +

12π

ih

[∮dzI0,

∮dzI0

]= O(h).

Our goal is to find I = ∑g≥0

hg Ig ∈ V [[h∗]][[h]] as a quantum correction of the classical BCOV

interaction I0 satisfying the exact Maurer-Cartan equation

δ

∮dzI +

12π

ih

[∮dzI,

∮dzI]= 0.

This will give a solution of the effective BV quantum master equation for our BCOV theory by thenext theorem, which is the analogue of Theorem 3.14 but for chiral bosons.

Theorem 4.4. Let I = ∑g≥0

hg Ig ∈ V [[h∗]][[h]] of degree 1 and I be the associated chiral local functional for

BCOV theory on elliptic curves. Let W(−,−) be the homotopy RG flow operator defined in Definition 2.12.Let PL

ε = −2i∫ Lε du ∂2

z1hu(−→z 1−→z 2)Ch be the regularized propagator and KL = i∂z1 hL(

−→z 1,−→z 2)(dz1 ⊗1− 1⊗ dz2)Ch be the regularized BV kernel for BCOV theory on elliptic curves. Let

∮dzI = I(0) be the

corresponding mode of the vertex operator I as an element of the modes Lie algebra∮V [[h]][[h]].

Then the limit

I[L] := limε→0

W(PLε , I)

exists. The family I[L]L>0 satisfies the renormalized quantum master equation

(Q + h∆L) I[L] +12

I[L], I[L]

L = 0

if and only if

δ

∮dzI +

12

(ihπ

)−1 [∮dzI,

∮dzI]= 0,

i.e..∮

dzI is a Maurer-Cartan element of∮V [[h]][[h]]. Here the BV operator ∆L is associated to the BV

kernel KL as defined in Definition 2.7.


Proof. The existence of the limit I[L] = limε→0

W(PLε , I) follows by Remark 3.11. For the statement

on quantum master equaton, the proof of Theorem 3.14 carries over here word by word. Theonly difference is that our propagator and BV kernel are the holomorphic derivative of those inTheorem 3.14. This implies that at the end of the two vertex computation in Step 2 there, eachcontraction leads to a factor (up to some normalization)

∂z2

1(z1 − z2)

=1

(z1 − z2)2

which is precisely the singular part of the OPE in the BCOV theory.

In the rest of this section, I will explain how to find a canonical solution of the Maurer-Cartanequation. It will be related to Gromov-Witten theory on elliptic curves by mirror symmetry.

4.2. Hodge weight and dilaton equation. The first simplification we will make is to use rescalingsymmetries of quantum master equation to restrict the possible form of the action functional.

We assigne the following gradings in V [[h∗]][[h]] and∮V [[h∗]][[h]].

∂mz bk ∂m

z ηk h δ z∮

dz [−,−]cohomology degree (deg) 0 1 0 1 0 0 0

conformal weight (cw) -k+m+1 -k+m 0 1 -1 -1 0dilaton dimension (dim) m m -2 1 0 0 -1

Hodge weight (hw=cw-dim) -k+1 -k 2 0 0 -1 1

They are all compatible with quantum master equation. The classical BCOV interaction has

deg(I0) = 1, cw(I0) = 2, dim(I0) = 0.

Definition 4.5. A solution I ∈ V [[h∗]][[h]] of quantum master equation for which

deg(I) = 1, cw(I) = 2, dim(I) = 0,

will be called an equivariant quantization.

Remark 4.6. For an equivariant quantization,

hw(Ig) = cw(Ig)− dim(Ig) = 2− 2g.

This is exactly the Hodge weight condition for elliptic curves described in [15]. The conditiondim(I) = 0 is essentially equivalent to the dilaton equation [35]. Therefore the equivariancecondition is naturally viewed as imposing the Hodge weight condition and the dilaton equation.

Remark 4.7. The dilaton dimension condition implies that the genus g correction Ig contains exactly2g holomorphic derivatives. This is the modification of modular invariant quantization (Defini-tion 3.23). It changes the number of holomorphic derivatives from g to 2g exactly because thereexists an extra holomorphic derivative on our propagator. In particular, an analogue of Theorem3.25 holds in this situation. See also [35].

Definition 4.8. Let us denote the homogeneous component

V [[h∗]]dw := a ∈ V [[h∗]]|deg(a) = d, cw(a) = w.

38 SI LI

We will focus on equivariant quantizations, which are given by elements I ∈ V [[h∗]]12 satisfyingthe Maurer-Cartan equation. In this case, we can use the rescaling symmetry to set h = π/i. Thepower of h can be recovered from counting the number of derivatives.

From now on, we will work with the normalized OPE

b0(z)b0(w) ∼ 1(z− w)2 , bk(z)bm(w) ∼ 0, k + m > 0.

b•(z)η•(w) ∼ 0, η•(z)η•(w) ∼ 0.

4.3. Reduction to Fedosov’s equation. In this section, we use boson-fermion correspondence tofurther reduce quantum master equation to a deformation quantization problem of Fedosov’sequation [19].

Definition 4.9. Let us package the fields b>0, η• into new series denoted by

b(z) = ∑k≥1

tk

k!bk(z), η(z) = ∑

k≥1

tk

k!ηk−1(z).

These fields do not appear in the propagator, and will be called the background fields. b0 doesappear in the propagator, and will be called the dynamical field.

The deformation quantization problem arises from viewing the linear coordinate z and the de-scendant variable t as a Darboux system of a holomorphic symplectic structure on C2:

ω = dz ∧ dt.

We consider the following differential ring freely generated by two generators b, η and twoderivatives ∂z, ∂t:

B = C[[∂•z ∂•t b, ∂

•z ∂•t η]].

Here b has even parity of cohomology degree 0, η has odd parity of cohomology degree 1. Herewe have confused ourselves to use the same symbols b, η for our generators in B. Later on, wewill plug into expressions in terms of the background fields in Definition 4.9 when the meaning isclear from the context.

The symplectic form induces a Poisson structure on B by

F, G = ∂tF∂zG− ∂zF∂tG, F, G ∈ B.

We also introduce a differential δ as an analogue of that in our BCOV theory

δ : B → B, b→ ∂zη.

It is easy to check that δ is compatible with the Poisson bracket, hence B becomes a dg Poissonalgebra. It has a natural deformation quantization in terms of the Moyal product ?

B ? B → B, F ? G = ∑k1 ,k2≥0

(−1)k2

2k1+k2 k1!k2!

(∂

k1t ∂

k2z F) (

∂k2t ∂

k1z G)

.

The following lemma is straight-forward.

Lemma 4.10. The triple (B, ?, δ) defines an associative differential graded algebra. In particular, (B, δ, [−,−]?)is a DGLA, where [−,−]? is the commutator with respect to the Moyal product.


We also introduce the analogue grading of conformal weight by

cw(∂mz ∂

kt b) = m− k + 1, cw(∂m

z ∂kt η) = m− k + 1.

Note that η has now cw = 1 by the shift in our Definition 4.9. Then

cw(?) = 0, cw(δ) = 1.

We denote the homogeneous component

Bdw = u ∈ B|deg(u) = d, cw(u) = w.

Our goal in this section is to construct a morphism of DGLA preserving the conformal weight

Φ : (B, δ, [−,−]?)→(∮

(V [[h∗]]), δ, [−,−])

and construct a canonical solution of Maurer-Cartan equation in B. This leads to a solution ofquantum master equation for our BCOV theory.

4.3.1. boson-fermion correspondence. We will construct Φ in terms of boson-fermion correspondence.Let us first fix our notations here and refer details to [27,41]. We introduce a pair of fermionsψ,ψ†

with OPE

ψ(z)ψ†(w) ∼ 1z− w

.

We introduce a free boson with OPE

φ(z)φ(w) ∼ log(z− w), φ(z) = ∑k 6=0

αnz−n

−n+α0 log z + p.

p is the momemtum creation operator as a conjugate of α0. The boson-fermion correspondencesays that a free boson is equivalent to a pair of fermions, under the following correspondence rule

ψ =: eφ :B, ψ† =: e−φ :B, and ∂φ =: ψψ† :F.

Here : − :B, : − :F denote the normal ordering for bosonic fields and fermionic fields respectively.The following fundamental relation holds

: ψ(z)ψ†(w) :F=1

z− w

(: eφ(z)−φ(w) :B −1

).

Let us expand by

: eφ(z)−φ(w) :B= 1 + ∑k≥1

(z− w)k

k!W(k)(w), W(k)(z) = ∑

n∈Zz−n−kW(k)

n ,

then W(k)n generate the so-called W1+∞ algebra. In terms of bosons,

W(k)(∂zφ) = ∑∑i≥1 iki=k

k!∏i ki!

: ∏i

(1i!

∂iφ

)ki

:B .

Note that W(k) only depends on ∂zφ. We can also express it in terms of fermions

W(k)(ψ,ψ†) = k : (∂k−1ψ)ψ† :F.

It follows from the fermionic expression that the Fourier modes∮

dzzmW(k) generates a centralextension of the Lie algebra of differential operators on the circle

kzm∂

k−1z

∮dzzmW(k), k ≥ 1, m ∈ Z.

40 SI LI

If we only look at non-negative modes, then we have a Lie algebra isomorphism

W : C [z, ∂z]∼=→ SpanC

∮dzzmW(k)

k≥1,m≥0

kzm∂

k−1z →

∮dzzmW(k).

Our field b0 of Heisenberg vertex algebra can be identified via free boson by

b0(z) = ∂zφ(z).

The other fields b>0, η• generate holomorphic (commutative) vertex algebra (i.e., all OPE’s be-tween them have no singular terms). Under the boson-fermion correspondence,

V [[h∗]][[e±p]] ∼= C[[∂mz ψ, ∂

mz ψ†, ∂

mz b>0, ∂

mz η•]]m≥0.

If we introduce the charge grading,

charge(ψ) = 1, charge(ψ†) = −1, charge(others) = 0,

then V [[h∗]ekp corresponds the homogenous component of charge k on the fermionic side. We willbe mainly interested in the charge 0 component. The operator δ does not involve b0. The onlynontrivial part of δ is

δ(∂mz bk+1) = ∂

m+1z ηk, ∀k, m ≥ 0.

4.3.2. Reduction to Fedosov’s equation. Now we construct the map Φ.

Definition 4.11. We define Φ : B →∮(V [[h∗]]) by

Φ(J) = ∑k≥0

1k + 1

∮dzW(k+1)(b0)

∮dtt−k−1e

12

∂

∂z∂

∂t J(b, η), J ∈ B.

In this expression, we need to substitute the generators b, η in terms of the background fields as inDefinition 4.9.

∮dt is the same as taking the residue at t = 0. W(k+1)(b0) is defined in the previous

subsection under b0 = ∂zφ.

It is easy to see that Φ preserves the conformal weight. Moreover,

Proposition 4.12. Φ is a morphism of DGLA’s

Φ : (B, δ, [−,−]?)→(∮

(V [[h∗]]), δ, [−,−])

.

Proof. In terms of notations in the previous subsection, Φ can be written as

Φ(J) = W

(∑k≥0

(∮dtt−k−1e

12 ∂z∂t J

)∂

kz

).

To clarify the meaning of this formula, let

ρ : C[z, t]→ C[z, ∂z], ρ(zmtk) = zm∂

kz.

An equivalent formal description is

ρ( f ) = ∑k≥0

(∮dtt−k−1 f (t, z)

)∂

kz, f ∈ C[z, t].

Let ? denote the Moyal product on C[z, t]

f ? g = e12 (∂t1 ∂z2−∂z1 ∂t2 )( f (z1, t1)g(z2, t2))

∣∣∣zi=z,ti=t

, f , g ∈ C[z, t].


Consider

e12 ∂z∂t( f ? g) = e

12 ∂z∂t

(e

12 (∂t1 ∂z2−∂z1 ∂t2 )( f (z1, t1)g(z2, t2))

∣∣∣zi=z,ti=t

)= e

12 (∂z1+∂z2 )(∂t1+∂t2 )e

12 (∂t1 ∂z2−∂z1 ∂t2 )( f (z1, t1)g(z2, t2))

∣∣∣zi=z,ti=t

= e∂t1 ∂z2 (e12 ∂z1 ∂t1 f (z1, t1)e

12 ∂z2 ∂t2 g(z2, t2))

∣∣∣zi=z,ti=t

.

Comparing with the associative composition of differential operators, we find

ρ(e12 ∂z∂t( f ? g)) = ρ(e

12 ∂z∂t f ) ρ(e 1

2 ∂z∂t g).

In particular,

ρ(e12 ∂z∂t [ f , g]?) =

[ρ(e

12 ∂z∂t f ),ρ(e

12 ∂z∂t g)

].

It follows from this algebraic fact and W being a Lie algebra morphism that

Φ([J1, J2]?) = [Φ(J1), Φ(J2)] , ∀J1, J2 ∈ B.

The compatibility of Φ with δ is easy to verify.

To construct an equivariant solution of quantum master equation, we only need to find J ∈ B11

satisfying

δ J +12[J, J]? = 0.

This can be viewed as a version of Fedosov’s abelian connection [19].

Lemma 4.13.H•(B, δ) = C[[∂k

t η]].

Proof. This follows from the observation that the complex (B, δ) can be identified (as a cochaincomplex) with the de Rham complex of C[[∂•z ∂•t b]] with coefficient in C[[∂k

t η]]. The Lemma followsby Poincare’s Lemma.

Corollary 4.14.H1(B, δ)1 = Cη, H2(B, δ)2 = 0.

Here Hk(B, δ)w denotes the component of Hk(B, δ) of conformal weight w.

Proof. H1(B, δ)1 = Cη is obvious. The only possible term in C[[∂kt η]] with deg = 2 and cw = 2 is

η2, which vanishes by the odd parity of η.

To solve the above equation, let us introduce an auxiliary grading by

T(∂mz ∂

kt b) = k, T(∂m

z ∂kt η) = k,

i.e., T counts the number of ∂t’s. Let us introduce the operator

δ∗ : B → B, ∂m+1z ∂

kt η→ ∂

mz ∂

kt b, (m ≥ 0), ∂

kt η→ 0.

Let N = δδ∗ + δ∗δ. We define

δ−1 : α →

1mδ∗α if Nα = mα

0 if Nα = 0

Then δ−1 can be viewed as a homotopic inverse of δ, where 1− [δ, δ−1] is the projection toC[[∂kt η]]∼=

H•(B, δ).

42 SI LI

Lemma 4.15. There exists a unique JB ∈ B11 satisfying

(1) δ JB + 12

[JB, JB] = 0;

(2) limλ→0 λT(JB) = η, i.e., the leading degree 0 component of JB under the T-grading is η;

(3) δ−1 JB = 0.

Proof. Let us decompose JB = ∑k≥0

J(k), where T(J(k)) = kJ(k). The initial condition (2) is J(0) = η.

We show that other J(k)’s can be uniquely solved satisfying conditions (1) and (3). Let us denoteJ(<k) = ∑

0≤i<kJ(i). Suppose we have solved J(<k). The J(k) we are looking for satisfies

δ J(k) = −12

[J(<k), J(<k)

]∣∣∣∣(k)

.

Here the subscript |(k) on the right hand side means the component containing k ∂t’s (i.e. T-

eigenvalue k). By the standard deformation theory argument, − 12

[J(<k), J(<k)

]∣∣∣(k)

is annihilated

by δ and is of conformal weight 2. Then the existence of J(k) ∈ B11 follows from H2(B, δ)2 = 0. In

particular,

J(k) = δ−1

(−1

2

[J(<k), J(<k)

]∣∣∣∣(k)

).

solves equation (1) and (3) up to J<(k+1).

Assume J(k) + U is another solution, then U satisfies

deg(U) = 1, cw(U) = 1, T(U) = k > 0, and δU = δ−1U = 0.

Since H1(B, δ)1 is spanned by η, while T(η) = 0. It follows that U = 0.

4.4. Exact solution of quantum BCOV theory.

Definition 4.16. Let JB be in Lemma 4.15. We denote Φ(JB) =∮

dzIB.

By Proposition 4.12 and Lemma 4.15,∮

dzIB defines a Maurer-Cartan element of V [[h∗]]. Tojustify that

∮dzIB indeed defines a quantization of our BCOV theory, we are left to check the

following two properties:

(1) [Integrality]: only terms with even number of derivatives contributes to∮

dzIB.(2) [Classical limit]: the term in

∮dzIB containing no derivatives coincide with our classical

BCOV interaction.

Property (1) on integrality comes from our discussion in Section 4.2 on dilaton dimension. Aterm with m holomorphic derivatives contributes to hm/2, while we only allow integer powers ofh to appear in our quantization.

Property (2) is just about the classical limit.

We will explicitly check (1) and (2) below.

Remark 4.17. Half integer powers of h appear naturally when open strings are included. In [16],we have also developed an open-closed BCOV theory. The terms in JB with odd number of holo-morphic derivatives will be total derivatives. They vanish upon integration, but may couple non-trivially with open string sectors. It would be extremely interesting to see how open string wouldplay into a role here.


4.4.1. Integrality. Let us consider the following transformation

R : B → B, ∂kz∂

mt b→ (−∂z)

k∂

mt b, ∂

kz∂

mt η→ (−∂z)

k∂

mt η,

i.e., R is the reflection ∂z → −∂z. It is easy to check that R(JB) also satisfies (1)(2)(3) in Lemma4.15. It follows from the uniqueness that

R(JB) = JB.

Let us identify b0 = ∂zφ as in Section 4.3.1. Then

Φ(JB) = ∑k≥0

1k + 1

∮dzW(k+1)(b0)

∮dtt−k−1e

12 ∂z∂t JB

=∮

dz ∑k≥0

W(k+1)(b0)

(k + 1)!

∮ dtt

∂kt e

12 ∂z∂t JB

=∮

dz∮ dt

t: eφ(z+∂t)−φ(z) :B −1

∂te

12 ∂z∂t JB

=∮

dz∮ dt

te

12 ∂z∂t

(: eφ(z+ 1

2 ∂t)−φ(z− 12 ∂t) :B −1

∂tJB

)

=∮

dz∮ dt

t: eφ(z+ 1

2 ∂t)−φ(z− 12 ∂t) :B −1

∂tJB.

Here we have formally identified φ(z + ∂t) = ∑k≥0

∂kzφ

∂kt

k! in the above manipulation and used the

fact that the operator e12 ∂t∂z amounts to shifting z→ z + 1

2 ∂t. In the last line, we have thrown awayterms which are total derivatives in z. Now φ(z + 1

2 ∂t)−φ(z− 12 ∂t) contains only even number

of ∂z’s in terms of the field b0 = ∂zφ. From R(JB) = JB, we know that JB also contains only evennumber of ∂z’s. Therefore

∮dzIB = Φ(JB) satisfies Property (1) on integrality.

4.4.2. classical limit. We check that the classical limit∮

dzIB0 of

∮dzIB coincides with our classical

BCOV interaction. By dilaton dimension, the classical limit is related to the component JB0 of JB

which does not involve any ∂z and satisfies

δ JB0 +

12JB

0 , JB0 = 0,

where −,− is the Poisson bracket A, B = (∂t A∂zB− ∂z A∂tB) . Smilarly, JB is uniquely deter-mined by further imposing conditions (2)(3) in Lemma 4.15.

Lemma 4.18.

JB0 = η+ ∑

k≥1

∂k−1tk!

(bk

∂tη)

.

Proof. Let JB0 = η+ ∑

k≥1

∂k−1tk!

(bk∂tη

). Let us first rewrite the above formula as

∂t JB0 = ∑

k≥0

∂kt

k!(bk

∂tη) =∮

0

dλλ

eλ∂t1

1− λ−1b∂tη

=∮

0dλ

∂tη(z, t + λ)λ− b(z, t + λ)

=∂tη(z, t + λ)

1− ∂tb(z, t + λ)

∣∣∣∣λ=b(z,t+λ)

.

44 SI LI

Here the substitution λ = b(z, t + λ) means solving λ = ∑k≥0

λk

k! ∂kt b for λ as a power series in ∂k

t b,

then plugging λ into ∂tη(z, t + λ) = ∑k≥0

λk

k! ∂k+1t η and ∂tb(z, t + λ) = ∑

k≥0

λk

k! ∂k+1t b. This implies via a

simple chain rule computation

JB0 = η(z, t + λ)|λ=b(z,t+λ) .

Similar computations lead to

δ JB0 =

∂zη(t + λ)∂tη(t + λ)1− ∂tb(t + λ)

∣∣∣∣λ=b(t+λ)

∂z JB0 =

(∂zη(t + λ) +

∂zb(t + λ)1− ∂tb(t + λ)

∂tη(t + λ))∣∣∣∣

λ=b(t+λ).

It follows that

δ JB0 +

12

JB0 , JB

0

= δ JB

0 + ∂t JB0 ∂z JB

0 =∂zb(t + λ)(

1− ∂tb(t + λ))2 (∂tη(t + λ))

2

∣∣∣∣∣λ=b(t+λ)

= 0,

where we have used the odd parity of η. Also, JB0 = 0 satisfies condition (2)(3) in Lemma 4.15. It

follows from the uniqueness that JB0 = JB

0 .

Corollary 4.19.∮

dzIB0 coincides with the classical BCOV interaction. In particular,

∮dzIB is a quantiza-

tion of the classical BCOV theory satisfying the Hodge weight condition and dilaton equation.

Proof. The above lemma allows us to compute∮

dzIB0 explicitly∮

dzIB0 = ∑

k>0

1k

∮dzbk

0

∮dtt−k JB

0

=∮

dz ∑k,l≥0,ki>0

(k− 1)!k1! · · · km!l!

bk0bk1 · · · bkmηl

k!

∮dtt−k

∂m−1t

(tk1+···+km+l

)=∮

dz ∑k,l≥0,ki>0

k1+···+km+l=k+m−2

(k1 + · · ·+ km + l)!k1! · · · km!l!

bk0bk1 · · · bkmηl

k!

=∮

dz⟨

eb ⊗ η⟩

0.

4.5. Higher genus mirror symmetry. We briefly explain how our exact solution of quantum BCOVtheory on elliptic curves is related to the Gromov-Witten theory on mirror elliptic curves. For sim-plicity, let us consider the following subsector of our fields in BCOV theory on elliptic curves

b>0 = 0, ηk = constant.

We call this the stationary sector. This is a mirror notion (introduced in [35]) of that in [44] whichdetermines the full generating function with the help of Virasoro constraint.

From our construction, the restriction of∮

dzIB to our stationary sector, denoted by∮

dzIS, isgiven by ∮

dzIS = Φ(η) = ∑k≥0

∮dz

W(k+2)

k + 2ηk.


The quantum master equation in the stationary sector becomes[∮dzIS,

∮dzIS

]= 0.

Expanding the coefficients ηk’s, this is equivalent to[∮dz

W(k+2)

k + 2,∮

dzW(m+2)

m + 2

]= 0, k, m ≥ 0,

which represents infinite number of commuting Hamiltonians.

Remark 4.20. The classical limit of∮

dz Wk+2

k+2 gives rise to dispersionless KdV integrable hierarchy.This observation motivates a general proposal [36] to study integrable hierarchy from reducingtopological B-model on X× E to chiral algebras on E for general Calabi-Yau geometry X.

Consider the following charactor, which plays the role of generating function of stationary sec-tor of our quantum BCOV theory on elliptic curves

TrH qL0− 124 e

1h ∑

k≥0

∮dzηk

W(k+2)k+2

, q = e2π iτ .

HereH is the Heisenberg vertex algebra generated by b0, and L0 is the conformal weight operator.Then it is observed that this trace coincides with the generating function of A-model Gromov-Witten computation [44] under the boson-fermion correspondence. This gives a simple explana-tion of the explicit computations in [35] and can be viewed as a full generalization of [17] whoconsiders the cubic interaction W(3).

APPENDIX A. ANALYTIC RESULTS ON FEYNMAN GRAPH INTEGRALS

In this appendix, we collect several analytic results for Feynman graph integrals on C that areestablished in [34] and essentially used in this paper.

Let z be the linear holomorphic coordinate on C. Let

kt(z, z) =1

4π te−|z|

2/4t

be the standard heat kernel on C. We denote the following time integration by

HLε (z, z) =

∫ L

ε

dt4π t

e−|z|2/4t, 0 < ε < L < ∞.

The regularized propagators that appear in this paper are related to holomorphic derivatives ofHLε . For example, the analytic part of the regularized propagator for the βγ and bc systems in 3.4

is essentially ∂zHLε , and that for chiral bosons (BCOV theory) in Section 4.1 is essential ∂2

z HLε .

Ultra-violet finiteness. We will be mainly interested in Feynman graph integrals for translationinvariant chiral interactions. Such a graph integral involves propagators as described above, andseveral holomorphic derivatives that appears in the vertex of the chiral interaction. To treat themuniformly, let us define the following decorated graph integral.

Let Γ be a directed graph. Let V(Γ) be the set of vertices, E(Γ) be the set of edges, and

t, h : E→ V

be the assignments of tail and head to each directed edge. We will consider the decorated graph

(Γ , n) ≡ (Γ , nee∈E)

46 SI LI

where the decoration is given by

n : E(Γ)→ Z≥0, e→ ne

which associates each edge a non-negative integer. Each ne indicates how many holomorphicderivatives of ∂z are put on the propagator associated to the edge e.

We will assume that Γ is connected without self-loops (tadpoles, i.e. edges that connect a vertexto itself). Thess are the only graphs that appear in this paper. If we work with a curved model,self-loops will be relevant.

We consider the following graph integral on C

W(Γ ,n)(HLε , Φ) ≡ ∏

v∈V(Γ)

∫C

d2zv

(∏

e∈E(Γ)∂

neze

HLε (ze, ze)

)Φ, where ze = zh(e) − zt(e)

here Φ is a smooth function on C|V(Γ)| with compact support. In the above integral, we viewHLε (ze, ze) as propagators associated to the edge e ∈ E, and we have only holomorphic derivatives

on the propagators.

Proposition A.1. [34, Prop B.1] The following limit exists for the above graph integral

limε→0

W(Γ ,n)(HLε , Φ).

This proposition shows that chiral deformations of combinations of freeβγ, bc and chiral bosonsare free of ultra-violet divergence.

Quantum master equation at the UV. Once we know that the theory is ultra-violet finite, we candeduce the meaning of the renormalized quantum master equation in the ultra-violet limit L→ 0.We next discuss the relevant analytic results.

Let us denote in this subsection

RLε (z, a) = ∂zHL

ε (z, z).

Let (Γ , n) be a connected decorated graph without self-loops. We index the set of vertices by

v : 1, 2, · · · , V → V(Γ)

and index the set of edges by

e : 0, 1, 2, · · · , E− 1 → E(Γ)

such that e(0), e(1), · · · , e(k) ∈ E(Γ) are all the edges connecting v(1), v(V). We consider thefollowing Feynman graph integral by putting Kε on e(0), putting HL

ε on all the other edges, andputting a smooth function Φ on C|V(Γ)| with compact support for the vertices. We would like tocompute the following limit of the graph integral

limε→0

V

∏i=1

∫d2zi∂

n0ze(0)

kε(ze(0), ze(0))

(E−1

∏i=1

∂nize(i)

RLε (ze(i), ze(i))

)Φ

where we use the notation that

ze ≡ zi − z j, if h(e) = v(i), t(e) = v( j)


Proposition A.2. The above limit exists and we have the identity

limε→0

V

∏i=1

∫d2zi∂

n0ze(0)

kε(ze(0), ze(0))

(E−1

∏i=1

∂nize(i)

RLε (ze(i), ze(i))

)Φ

= limε→0

C(n0; n1, · · · , nk)

(4π)k

V

∏i=2

∫d2zi ∂

n0+k∑

i=1(ni+1)

z1

((E−1

∏i=k+1

∂ni RL

ε (ze(i), ze(i))

)Φ

)∣∣∣∣∣∣z1=zV

where the constant C(n0, n1, · · · , nk) is a rational number given by

C(n0; n1, · · · , nk) =∫ 1

0· · ·

∫ 1

0

k

∏i=1

dui

k∏

i=1uni

i(1 +

k∑

i=1ui

) k∑

j=0(n j+1)

Proof. [34, Prop B.2] proves the same statement when kε is replaced by ∂zkε (which was called Uεthere), and RL

ε is replaced by ∂zRLε (which was called PL

ε there). Such cases correspond to thosegraph integrals when each edge has decoration ne ≥ 1. However, literally the same proof as in[34, Prop B.2] shows that the argument is valid for any decorations. This proves the proposition.

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S. LI: YAU MATHEMATICAL SCIENCES CENTER, TSINGHUA UNIVERSITY, BEIJING, CHINA;

E-mail address: [email protected]

http://arxiv.org/abs/1911.11173

http://arxiv.org/abs/1910.05665

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