VERTEX ALGEBRAS AND QUANTUM MASTER EQUATION SI LI A

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SI LI

ABSTRACT. We study the effective Batalin-Vilkovisky quantization theory for chiral deformation of two dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators.

The generating functions 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 that solves the 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 14

3. Vertex algebra and BV master equation 16

3.1. Vertex algebra 16

3.2. Two dimensional chiral QFT 19

3.3. Quantum master equation and OPE 22

3.4. Generating function and modularity 26

3.5. Example: Poissonσ-model 31

4. Application: quantum B-model on elliptic curves 33

4.1. BCOV theory on elliptic curves 33

4.2. Hodge weight and dilaton equation 36

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4.3. Reduction to Fedosov’s equation 37

4.4. Exact solution of quantum BCOV theory 43

4.5. Higher genus mirror symmetry 46

References 47

1. INTRODUCTION

Quantum field theory provides a rich source of mathematical thoughts. One important feature of quantum field theory that lies secretly behind many of its surprising mathematical predictions is about its nature of infinite dimensionality. A famous example is the mysterious mirror symmetry conjecture between symplectic and complex geometries, which can be viewed as a version of infinite dimensional Fourier transform. Typically, many quantum problems are formulated in terms of “path integrals”, which require measures that are mostly not yet known to mathematicians. Nevertheless, asymptotic analysis can always be performed with the help of the celebrated idea of renormalization.

Despite the great success of renormalization theory in physics applications, its use in mathematics is relatively limited but extremely powerful when it does apply. One such example is Kontsevich’s solution [21] to the deformation quantization problem on arbitrary Poisson manifolds. Kontsevich’s explicit formula of star product is obtained via graph integrals on a compactification of configu- ration space on the disk, which can be viewed as a geometric renormalization of the perturbative expansion of Poisson sigma model (see also [5]). Another recent example is Costello’s homotopic theory [7] of effective renormalizations in the Batalin-Vilkovisky formalism. This leads to a systematic construction of factor- ization algebras via quantum field theories [9]. For example, a natural geometric interpretation of the Witten genus is obtained in such a way [8].

To facilitate geometric applications of effective renormalization methods, it would be key to connect renormalized quantities to geometric objects. We will be mainly interested in quantum field theory with gauge symmetries. The most general framework of quantizing gauge theories is the Batalin-Vilkovisky formalism [4], where the quantum consistency of gauge transformations is described by the so-called quantum master equation. There have developed several mathematical approaches to incorporate Batalin-Vilkovisky formalism with renormalizations since their birth. The central quantity of all approaches lies

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in the renormalized quantum master equation. In this paper, we will mainly discuss the formalism in [7], which has developed a convenient framework that is also rooted in the homotopic culture of derived algebraic geometry. A brief introduction to the philosophy of this approach is discussed in Section 2.

The simplest nontrivial example is given by quantum mechanical models, which can be viewed as quantum field theories in one dimension. The renormalized Batalin-Vilkovisky quantization in the above fashion is analyzed in [18, 23]

for topological quantum mechanics. In particular, it is shown in [23] that the renormalized quantum master equation can be identified with the geometric equation of Fedosov’s abelian connection [14] on Weyl bundles over symplectic manifolds. The algebraic nature of this correspondence is reviewed in Section 2.4. Such a correspondence leads to a simple geometric approach to algebraic index theorem [15, 31], where the index formula follows from the homotopic renormalization group flow together with an equivariant localization of BV integration [23].

In this paper, we study systematically the renormalized quantum master equation in two dimensions. We will focus on quantum theories obtained by chiral deformations of free CFT’s (see Section 3.2 for our precise set-up). One important feature of such two dimensional chiral theories is that they are free of ultra-violet divergence (see Theorem 3.9). This greatly simplifies the analysis of quantization since singular counter-terms are not required. However, the renormalized quantum master equation requires quantum corrections by chiral local functionals. Such quantum corrections could in principle be very complicated.

One of our main results in this paper (Theorem 3.11) is an exact description of the quantum corrections in terms of vertex algebras. Briefly speaking, Theorem 3.11 states that the renormalized quantum master equations (QME) is equivalent to quantum corrected chiral vertex operators that satisfies Maurer-Cartan (MC) equations. In other words, we have an exact description of the quantization of chiral deformation of two dimensional conformal field theories

renormalized QME ⇐⇒ MC equations for chiral vertex operators The Maurer-Cartan equation serves as an integrability condition for chiral vertex operators, which is often related to integrable hierarchies in concrete cases.

We discuss such an example in Section 4. Furthermore, we prove a general result on the modularity property of the generating functions and their holomorphic anomaly (Theorem 3.21). This work is also motivated from understanding Dijk- graaf’s description [13] of chiral deformation of conformal field theories.

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The above correspondence can be viewed as the two dimensional vertex algebra analogue of the one dimensional result in [23]. In fact, one main motivation of the current work is to explore the analogue of index theorem for chiral vertex operators in terms of the method of equivariant localization in BV integration as proceeded in [23]. It allows us to solve many quantization problems in terms of powerful techniques in vertex algebras.

As an application in Section 4, we construct an exact solution of quantum B- model on elliptic curves, which leads to the solution of the corresponding higher genus mirror symmetry conjecture. 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 challenge for mathematicians to construct quantum B-model on compact Calabi-Yau manifolds. In [10], we construct a gauge theory of polyvector fields on Calabi-Yau manifolds (called BCOV theory) as a generalization of the Kodaira-Spencer gauge theory [3]. It is proposed in [10] (as a generalization of [3]) that the Batalin-Vilkovisky quantization of BCOV theory leads to quantum B-model that is mirror to the A- model Gromov-Witten theory of counting higher genus curves. Our construction in Section 4 gives a concrete realization of this program. This leads to the first mathematically fully established example of quantum B-model on compact Calabi-Yau manifolds.

Our result in Section 4 also leads to an interesting result in physics. Quantum BCOV theory can be viewed as a complete description of topological B-twisted closed string field theory in the sense of Zwiebach [33]. Zwiebach’s closed string field theory describes the dynamics of closed strings in term of the so-called string vertices. Despite the beauty of this construction, string vertices are very difficult to compute and few concrete examples are known. Our exact solution in Section 4 can be viewed as giving an explicit realization of Zwiebach’s string vertices for B-twisted topological string 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 discussions on quantum field theories, and thank Jie Zhou for discussions on modular forms. Part of the work was done during visiting Perimeter Institute for theoretical physics and IBS center for geometry and physics. The author thanks for their hospitality and provision of excellent working enviroments. Special thank goes to Xinyi Li, whose birth and growth

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have inspired and reformulated many aspects of the presentation of the current work. S. L. is partially supported by Grant 20151080445 of Independent Research Program at Tsinghua university.

Conventions

• Let V be aZ-gradedk-vector space. We use Vm to denote its degreem component. Givena ∈Vm, we let ¯a =mbe its degree.

– V[n]denotes the degree shifting ofVsuch thatV[n]_m =V_n+m. – V^∗denotes its dual such thatV_m^∗ =_Hom_k(V−m,k). Our base fieldk

will mainly beR^orC^.

– Sym^m(V) and ∧^m(V) denote the graded symmetric product and graded skew-symmetric product respectively. We also denote

Sym(V):= ^M

m≥0

Sym^m(V), Sym[(V):=

∏

m≥0

Sym^m(V). Given I = _∑

m≥0

I_m ∈ _Sym[(V^∗)_{, and} a₁,· · · _,a_m ∈ V, we denote its m-order Taylor coefficient

∂

∂a₁ · · · ^∂

∂am

I(₀)_:= I_m(a₁,· · · _,a_m)

where we have viewedI_m as a multi-linear mapV^⊗^m →k.

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

∂_P : Sym^m(V^∗)→Sym^m⁻²(V^∗), I →∂_PI, where for anya₁,· · · ,a_m−2 ∈V,

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

– V[_z]_,_V[[_z]]_and_V((_z))denote polynomial series, formal power series and Laurent series respectively in a variablezvalued inV.

• LetAbe a graded commutative algebra.[−,−]always means the graded commutator, i.e., for elementsa,bwith specific degrees,

[a,b]:=a·b−(−1)^a^¯^b^¯b·a.

We always assume Koszul sign rule in dealing with graded objects.

• ⊗without subscript means tensoring over the real numbersR.

• Given a manifoldX, we denote the space of real smooth forms by Ω^•(X) =^M

k

Ω^k(X)

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whereΩ^k(X)is the subspace ofk-forms. If furthermore X is a complex manifold, we denote the space of complex smooth forms by

Ω^•^,^•(X) =^M

p,q

Ω^p,q(X) =Ω^•(X)⊗C whereΩ^p,q(X)is the subspace of(p,q)-forms.

• Dens(X)denotes the density bundle on a manifold X. When X is oriented, we naturally identify Dens(X)with top differential forms onX.

• Let E be a vector bundle on a manifold X. E = Γ(X,E) denotes the space of smooth sections, andE⁰ = D⁰(X,E)denotes the distributional sections. IfE^∗is the dual bundle ofE, then we have a natural pairing

E⁰⊗Γ(X,E^∗⊗Dens(X))→R.

• Hdenotes the upper half plane.

2. BATALIN-VILKOVISKY FORMALISM AND EFFECTIVE RENORMALIZATION

In this section, we collect basics and fix notations on the quantization of gauge theories in the Batalin-Vilkovisky (BV) formalism. We explain Costello’s homotopic renormalization theory of Batalin-Vilkovisky quantization and present a one-dimensional example to motivate our discussions in two dimensions.

2.1. Batalin-Vilkovisky algebras and the master equation.

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

• Ais aZ-graded commutative associative unital algebra.

• Q:A → Ais a derivation of degree 1 such thatQ² =0.

• ∆:A → Ais a second-order operator of degree 1 such that∆² =_0.

• Qand∆are compatible:[Q,∆] =Q∆+∆Q=0.

Here ∆is called the BV operator. ∆being “second-order” means the following: define theBV bracket{−,−}as measuring the failure of∆being a derivation

{a,b}:=∆(ab)−(∆a)b−(−1)^a^¯a∆b.

Then{−,−} :A ⊗ A → Adefines a Poisson bracket of degree 1 satisfying

• {a,b}= (−1)^a^¯^b^¯{b,a}.

• {a,bc}={a,b}c+ (−1)⁽^a^¯⁺¹⁾^b^¯b{a,c}.

• ∆{a,b}=− {∆a,b} −(−1)^a^¯{a,∆b}.

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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 ∈ A₀is said to satisfyclassical master equation(CME) if

QI+ ¹

2{I,I}=0.

If I solves CME, then it is easy to see that Q+{I,−} defines a differential on A, which can be viewed as a Poisson deformation of Q. However, it may not be compatible with ∆. A sufficient condition for the compatibility is the

“divergence freeness”∆I =0. A slight generalization of this is the following.

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

QI+¯h∆I+¹

2{I,I}=0.

Here ¯his a formal variable representing the quantum parameter.

The “second-order” property of∆implies that QME is equivalent to (Q+h∆¯ )e^I^/^¯^h =0.

If we decomposeI = _∑

g≥0

I_gh¯^g, then the ¯h→0 limit of QME is precisely CME QI₀+ ¹

2{I₀,I₀}=0.

We can rephrase the (Q,∆)-compatibility as the nilpotency of Q+¯h∆. It is direct to check that QME implies the nilpotency ofQ+¯h∆+{I,−}, which can be viewed as a compatible deformation.

2.2. Odd symplectic space and the toy model. We discuss a toy model of differential BV algebra via(−1)-shifted symplectic space. This serves as the main motivating resources of our quantum field theory examples.

Let(V,Q)be a finite dimensional dg vector space. The differentialQ:V→V induces a differential on various tensors ofV,V^∗, still denoted byQ. Let

ω∈ ∧²V^∗, Q(ω) =0,

be aQ-compatible symplectic structure such that deg(ω) =−1. It identifies V^∗ 'V[1].

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LetK=ω⁻¹∈ _Sym²(V)be the Poisson kernel of degree 1 under

∧²V^∗ 'Sym²(V)[2]

ω K

where we have used the canonical identification∧²(V[₁])'_Sym²(V)[₂]_{. Let} O(V)_:=[_Sym(V^∗) =

∏

n

Symⁿ(V^∗)_. Then(O(V),Q)is a graded-commutative dga.

The degree 1 Poisson kernelKdefines the following BV operator

∆_K:O(V)→ O(V) by

∆_K(ϕ₁· · ·ϕ_n) =

∑

i,j

±(K,ϕ_i⊗ϕ_j)ϕ₁· · ·ϕˆ_i· · ·ϕˆ_j· · ·ϕ_n, ϕ_i ∈V^∗. Here(K,ϕ_i⊗ϕ_j)denotes the natural paring betweenV⊗VandV^∗⊗V^∗. ±is the Koszul sign by permutingϕ_i’s. The following lemma is well-known.

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

The above construction can be summarized as

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

Remark2.5. Since we only useKto define BV operator, the above process is well- defined for(−1)-shifted dg Poisson structure whereK may be degenerate. We will see such an example in Section 4.

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

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

In the previous section, we discuss the(−1)-shifted dg symplectic space(V,Q,ω). ThereVis assumed to be finite dimensional. This is why we call it “toy model”.

Typically in quantum field theory,Vwill be modified to be the space of smooth sections of certain vector bundles on a smooth manifold, while the differentialQ and the pairingωcome from something “local”. SuchVwill be called thespace

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of fields, which is evidently a very large space with delicate topology. Neverthe- less, let us naively perform similar constructions as that in the toy model.

More precisely, let Xbe a smooth oriented manifold without boundary. Let E^• be a complex of vector bundles onX

· · ·E⁻¹ →^Q E⁰ →^Q E¹· · ·

whereQis the differential. We assume that(E^•,Q)is an elliptic complex. Our space of fieldsE replacingVwill be the space of smooth global sections

E = Γ(X,E^•),

with the induced differential, still denoted byQ. The symplectic pairing will be ω(s₁,s₂):=

Z

X

(s₁,s₂), s₁,s₂ ∈ E, where

(−_,−)_:E^•⊗E^• →_Dens(X)

is a non-degenerate graded skew-symmetric pairing of degree−1. To perform the toy model construction, we need the following steps

(1) The dual vector spaceE^∗ (analogue ofV^∗). This can be defined via the space of distributions onE

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^∗⊗ · · ·ˆ ⊗Eˆ ^∗

where(E^∗)^⊗^k is the distributions on the bundleE^•· · ·^E^• ^over ^X×

· · · ×X. Sym^k(E^∗)is defined similarly by taking care of the graded per- mutation. Then we have a well-defined notion (via distributions)

O(E)_:=

∏

k≥0

Sym^k(E^∗) as the analogue ofO(V).

(3) The Poisson kernelK₀ = ω⁻¹ (the analogue of K). The pairingωdoes not induce an identification between E[1] and its dual E^∗ in this case.

Sinceωis defined via integration, the Poisson kernelK₀is theδ-function

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representing integral kernel of the identity operator. Therefore K₀ is a distributional section ofE⊗E^ˆ supported on the diagonalX,→X×X

K₀ ∈Sym²(E⁰).

See Conventions forE⁰. It is at Step (3) where we get trouble. In fact, if we naively define the BV operator

∆_K₀ :O(E)→ O(E^? ),

then∆_K₀ isill-defined, since we can not pair a distributionK₀with another distribution from O(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 renormalization in quantum field theory. There are several different approaches to renormalizations, and we will adopt Costello’s homotopic theory [7] in this paper that will be convenient for our applications.

The key observations are

(1) K₀is aQ-closed distribution:Q(K₀) = (Q⊗₁+₁⊗Q)K₀ =_0.

(2) elliptic regularity: there is a canonical isomorphism of cohomologies H^∗(smooth,Q)∼= H^∗(distribution,Q).

It follows that we can find a distributionP_r ∈ Sym²(E⁰)and a smooth element K_r ∈Sym²(E)such that

K0= Kr+Q(Pr). P_ris the familiar notion of a parametrix.

Definition 2.6. K_rwill be called the renormalized BV kernel with respect to the parametrixPr.

SinceK_ris smooth, there is no problem to pairK_rwith distributions. The same formula as in the toy model leads to

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

∆_K_r : O(E)→ O(E)

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

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

Let Pr₁ andPr₂ be two parametrices,Kr₁ andKr₂ be the corresponding renormalized BV kernels. Let us denote

P_r^r₁² :=Pr2−Pr1.

SinceQ(P_r^r₁²) =K_r₁ −K_r₂ is smooth,P_r^r₁² is smooth itself by elliptic regularity.

Definition 2.8. Pr^r₁² will be called the regularized propagator.

Example 2.9. Typically, suppose we have an adjoint operatorQ^†such that[Q,Q^†] is a generalized Laplacian. Then givent > 0, the integral kernelK_tfor the heat operatore⁻^t^[^Q,Q^†^] can be viewed as a renormalized BV kernel. In this case, the regularized propagator is given by

P_t^t₁² =

Z _t₂

t1

(Q^†⊗1)Ktdt.

P_r^r₁² can be viewed as a homotopy linking two different renormalized differential BV algebras. In fact, similar to the definition of renormalized BV operator, Definition 2.10. We define∂_P^r2

r1

:O(E)→ O(E)as the second-order operator of contracting with the smooth kernelP_r^r₁² ∈Sym²(E)(see also Conventions).

Lemma 2.11. The following equation holds formally as operators onO(E)[[h¯]]

Q+¯h∆_K_r

2

e^¯^h∂^P^r^r¹² =e^¯^h∂^P^r^r¹²

Q+h¯∆_K_r

1

, i.e., the following diagram commutes

O(E)[[h¯]]

Q+¯h∆_Kr

1 //

exp

¯ h∂Pr2

r1

O(E)[[h¯]]

exp

¯ h∂Pr2

r1

O(E)[[h¯]]

Q+¯h∆_Kr₂

// O(E)[[¯h]]

Sketch. This follows from the observation that h

Q,∂_P^r2 r1

i

=∆_K_r

1 −∆_K_r

2.

See for example [7] for further discussions.

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Definition 2.12. Let

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

be the subspace of those functionals which are at least cubic modulo ¯h. Given any two parametricesP_r₁,P_r₂, we define the homotopic renormalization group (HRG) operator

W(P_r^r₁²,−)_:O⁺(E)[[_¯h]]→ O⁺(E)[[_¯h]]

W(P_r^r₁²,I):=h¯ log

e^¯^h∂^P^r^r¹²e^I^/^¯^h

. The real content of the above formula is

W(P_r^r₁²,I) =

∑

Γ connected

W_Γ(P_r^r₁²,I)

where the summation is over all connected Feynman graphs withP_r^r₁² being the propagator andI being the vertex (see for example [2] for Feynman graph techniques). Wick’s Theoreom identifies the above two formula. In particular, the graph expansion formula implies that HRG operatorW(P_r^r₁²,−)is well-defined onO⁺(E)[[¯h]]. We refer to [7] for a thorough discussion in the current context.

Lemma 2.11 motivates the following definition.

Definition 2.13([7]). A solution of effective quantum master equation is an as- signementI[r]∈ O(E)[[h¯]]for each parametrixP_rsatisfying

• Renormalized quantum master equation (RQME) (Q+¯h∆_K_r)e^I^[^r^]/^¯^h= 0.

• Homotopic renormalization group flow equation (HRG): for any two parametricesP_r₁,P_r₂,

I[r₂] =W(P_r^r₁²,I[r₁]).

RQME and HRG are compatible by Lemma 2.11. A solution of effective quantum master equation is completely determined by its value at a fixed parametrix.

Functionals at other paramatrices are obtained via HRG.

Remark2.14. Here we adopt the name “homotopic RG flow” as opposed to the name ”RG flow” in [7]. If the manifold preserves a rescaling symmetry, it will induce a rescaling action on the solution space of effective quantum master equations. Such flow equation will be called RG flow to be consistent with the physics terminology.

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

Definition 2.15. A functional I ∈ O(E) is calledlocalif it can be expressed in terms of an integration of a Lagrangian densityL

I(φ) =

Z

X

L(φ), φ∈ E.

L can be viewed as a Dens(X)-valued function on the jet bundle of E. The subspace of local functionals ofO(E)is denoted byO_loc(E). We also denote

O⁺_loc(E)[[¯h]] =O⁺(E)[[h¯]]∩ O_loc(E)[[h¯]].

Let us assume we are in the situation of Example 2.9. Given L > _{0, we have} a smoothly regularized BV kernelK_L in terms of the heat kernel. LetP^Lbe the regularized propagator for 0<< L<_∞.

Given any local functional I ∈ O⁺_loc(E), we can find an-dependent local functionalI^CT()∈h¯O_loc(E)[[h¯]], such that the following limit exists

I[L] =lim

→0W(P^L,I+I^CT())∈ O⁺(E)[[¯h]].

Here I^CT()has a singular dependence on when → 0, and this singularity exactly cancels those that come from the naive graph integralW(P^L,I). I^CT() is called thecounter-term, which plays an important role in the renormalization theory of quantum fields. For a proof of the existence of counter-terms in the current context, see for example [7, Appendix 1].

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

I[L₂] =W(P_L^L²

1,I[L₁]), ∀0< L₁,L₂ <_∞.

It remains to analyze the quantum master equation.

Firstly, although the BV operator∆_Lbecomes singular asL→0, its associated BV bracket is in fact well-defined on local functionals.

Definition 2.16. We define the classical BV bracket onO_loc(E)by {I₁,I₂}:= lim

L→0{I₁,I₂}_L, I₁,I₂∈ O_loc(E).

The reason that the above limit exists lies in the observation that the delta- function can be naturally paired with integration.

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Definition 2.17. A local functionalI ∈ O_loc(E)is said to satisfy classical master equation (CME) if

QI+ ¹

2{I,I}=0.

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

Remark2.18. Given I0 ∈ O_loc(E)satisfying the classical master equation, Q+ {I₀,−} defines a nilpotent vector field on E. The physical meaning is that it generates the infinitesimal gauge transformation. In mathematical terminology, it defines a (local)L_∞structure onE.

To proceed to construct solutions of effective quantum master equations, let us naively use counter-terms to construct a familyI[L]as above from I₀. It can be shown thatI[L]satisfies renormalized quantum master equation modulo ¯h

QI[L] +¯h∆_LI[L] +¹

2{I[L],I[L]}_L=O(¯h). Then we need to find quantum corrections

I₀ → I₀+¯hI₁+h¯²I₂+· · · ∈ O_loc⁺(E)[[¯h]]

such that the renormalized quantum master equation holds true at all orders of

¯

h. This can be formulated as a deformation problem. The quantum corrections may become very complicated in general, and intrinsic obstructions for solving the quantum master equation could exist at certain ¯h-order (gauge anomalies).

Nevertheless, one of our main purposes here is to understand the geometric meaning of such quantum corrections and find their solutions.

2.4. Example: Topological quantum mechanics. The simplest nontrivial example is when X is one-dimensional. This corresponds to quantum mechanical models. We explain the one dimensional Chern-Simons theory that is studied in detail in [23]. In such a geometric situation, the renormalized BV master equation is related to Fedosov’s abelian connection on Weyl bundles [14]. In the next section, we generalize this analysis to two dimensional models.

LetX=S¹. LetVbe a graded vector space with a degree 0 symplectic pairing (−,−):∧²V→R^.

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The space of fields will be

E =Ω^•(S¹)⊗V.

Let us denoteX_dR by the super-manifold whose underlying topological space isXand whose structure sheaf is the de Rham complex on X. We can identify ϕ ∈ E with a map ˆϕbetween super-manifolds

ˆ

ϕ :X_dR →V

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

The differential Q = d_S1 on E is the de Rham differential on Ω^•(S¹). The (−1)-shifted symplectic pairing is the pairing

ω(ϕ₁,ϕ₂):=

Z

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 ˆI ∈ O(E)of degreek−1 via

Iˆ(ϕ):=

Z

S¹ϕˆ^∗(I), ∀ϕ∈ E.

We choose the standard flat metric onS¹and use the heat kernel regulariza- tion as in Example 2.9. The following Theorem is a consequence of [23].

Theorem 2.19([23]). Given I ∈Sym^≥³(V^∗) +¯hO(V)[[h¯]]of degree1, the limit Iˆ[L] =lim

→0W(P^L, ˆI)

exists as an degree0element of O⁺(E)[[h¯]]. The family{I^ˆ[L]}_L_>₀ solves the effective quantum master equation if and only if

[I,I]_? =_0.

HereO(V)[[h¯]] inherits a natural Moyal product ? from the linear symplectic form (−,−).[−,−]_?is the commutator with respect to the Moyal product.

In [23], the above theorem is formulated as a family version ofVparametrized by a symplectic manifold. Then the effective quantum master equation is equivalent to a flat connection gluing the Weyl bundle (i.e. the bundle of associative algebra with fiberwise Moyal product). A further analysis of the partition function leads to a simple formulation of the algebraic index theorem.

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3. VERTEX ALGEBRA ANDBVMASTER EQUATION

In this section we establish our main theorem on the correspondence between solutions of the renormalized quantum master equation for two dimensional chiral theories and Maurer-Cartan elements for chiral vertex operators (Theo- rem 3.11). We prove the modularity property of generating functions of chiral theories on elliptic curves and establish the polynomial nature of their anti- holomorphic dependence on the moduli (Theorem 3.21).

3.1. Vertex algebra. In this section we collect some basics on vertex algebras that will be used in this paper. We refer to [16, 19] for details.

3.1.1. Definition of vertex algebras. In this section V will always denote aZ/2Z^- graded superspace overC. It has two components with different parities

V =V_¯0⊕ V_¯1, Z/2Z={¯0, ¯1}. We sayv∈ Vhas parity p(a)∈Z/2Z^if^v∈ V_p₍_v₎. Definition 3.1. AfieldonV is a power series

A(z) =

∑

k∈Z

A₍_k₎z⁻^k⁻¹ ∈End(V)[[z,z⁻¹]]

such that for anyv∈ V_,A(z)v ∈ V((z)). We will denote A(z)+ =

∑

k<0

A₍_k₎z⁻^k⁻¹, A(z)₋ =

∑

k≥0

A₍_k₎z⁻^k⁻¹.

• Given two fields A(z),B(z), we define theirnormal 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 fieldsA(z),B(z)are calledmutually localif A(z)B(w) =

N−1

∑

k=0

C_k(w)

(z−w)^k⁺¹+: A(z)B(w): .

for someN ∈ Z^≥⁰ ^{and fields}^Ck. The first term on the right is called the singular part, and we shall write

A(z)B(w)∼

N−1

∑

k=₀

C_k(w) (z−w)^k⁺¹^.

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

Definition 3.2. Avertex algebrais a collection of data

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• (space of states) a superspaceV;

• (vacuum) a vector|0i ∈ V_¯0;

• (translation operator) an even linear operatorT:V → V;

• (state-field correspondence) an even linear operation (vertex operators) Y(·,z):V →EndV[[z,z⁻¹]]

taking eachA∈ Vto a fieldY(A,z) = _∑

n∈Z

A₍_n₎z⁻ⁿ⁻¹; satisfying the following axioms:

• (vaccum axiom)Y(|0i,z) =IdV. Furthermore, for anyA∈ Vwe have Y(A,z)|0i ∈ V[[z]], and lim

z→0Y(A,z)|0i= A.

• (translation axiom)T|0i= 0. For anyA∈ V,[T,Y(A,z)] =∂_zY(A,z);

• (locality axiom) All fieldsY(A,z),A∈ V, are mutually local.

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 ^, wheren

A₍_n₎·Bo

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

Definition 3.3. A vertex algebraV is calledconformal, ofcentral charge c ∈ C^{, if} there is a vectorω_vir ∈ V (called a conformal vector) such that under the state- field correspondenceY(ω_vir,z) = ∑

n∈Z

Lnz⁻ⁿ⁻²:

• L−1 =_T,_L₀is diagonalizable onV;

• {Ln}_n_∈_Zspan the Virasoro algebra with central chargec.

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

V =^M

α

V^α, L₀|_V_α =α.

A∈ V^αis said to have conformal weightα. Its field will often be expressed by Y(A,z) =

∑

k

A_kz⁻^k⁻^α, A_k :V^• → V^•−^k,

whereA_khas conformal weight−k. In previous notations,A_k = A₍_k₊_α₋₁₎.

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3.1.2. Modes Lie algebra. Following the presentation in [16], we associate a canonical Lie algebra to a vertex algebra via Fourier modes of vertex operators.

Definition 3.4. Given a vertex algebraV, we define a Lie algebra, denoted by H V, as follows. As a vector space, the Lie algebra^H V has a basis given byA₍_k₎’s

I

V :=Span

C

_I

dzz^kY(A,z):= A₍_k₎

A∈V,k∈Z

.

The Lie bracket is determined by the OPE (Borcherds commutator formula) h

A₍_m₎,B₍_n₎i

=

∑

j≥0

m j

A₍_j₎B

m+n−j.

Here we use a different notation^H thanU(·)in [16, Section 4.1] to emphasize its nature of mode expansion. The commutator relations are better illustrated formally in terms of residues

_I

dzz^mY(A,z), I

dwwⁿY(B,w)

=

I dwwⁿ

I

wdzz^m

∑

j∈Z

Y(A₍_j₎·B,w) (z−w)^j⁺¹ ^, where^H_wdz:=^R_C

w

dz

2πi is the integration over a small loopC_waroundw.

Equivalently, the vector space^H V can be described as I

V =V[z,z⁻¹]/_im∂

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

3.1.3. Examples. Let h = ⊕_α_∈_Qh^α, where h^α = h^α_¯0 ⊕h^α_¯1, be aQ-graded superspace. Here theQ-grading is the conformal weight and we assume for simplicity only finitely many weights appear inh. Lethbe equipped with an even symplectic pairing

h−,−i:∧²h→C

which is of conformal weight−1, that is, the only nontrivial pairing is h−,−i:h^α⊗h¹⁻^α→C^.

For eacha ∈_h^α, we associate a field a(z) =

∑

r∈Z−α

arz⁻^r⁻^α. We define their OPE by

a(z)b(w)∼ i¯h

π

ha,bi

(z−w)^, ∀a,b∈_h.

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This is equivalent to the commutator relations [a_r,b_s] = ^i¯^h

π ha,biδ_r₊_s,0, ∀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 representation space. The vaccum vector satisfies

ar|0i=0, ∀a∈h^α,r+α >0,

andV[h]is freely generated from the vaccum by the operators{ar}_r+α≤0,a ∈h^α. For anya ∈h,a(z)becomes a field acting naturally on the Fock spaceV[h].V[h] is a conformal vertex algebra structure, with energy momentum tensor

T(_z) = ¹ 2

∑

i,j

ω_{i j} :∂aⁱ(_z)_a^j(_z)_: where{aⁱ}is a basis ofh,ω_{i j}is the inverse matrix of

aⁱ,b^j

:∑_kω_ik a^k,a^j

=δ_i^j. The central charge is dimh_¯0−dimh_¯1.

Remark3.5. Whenh= h_¯0is purely bosonic, the associated vertex algebra is the β−γsystem (or the chiral differential operators [17,29]). Whenh=h_¯1is purely fermionic, the associated vertex algebra is theb−csystem.

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

V[h]∼=C h

∂^kaⁱi

, aⁱis a basis ofh,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. Two dimensional chiral QFT. LetΣbe a complex curve,KΣbe its canonical line bundle. We will be interested in the following data as a BV set-up:

• (E^•,δ)is a differential complex of holomorphic bundles onΣ;

• A degree 0 symplectic pairing of complexes (−,−): E^•⊗E^• →KΣ.

HereKΣis viewed as a complex concentrated at degree 0.

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The space of quantum fields associated to the above data will be E =Ω^0,^•(Σ,E^•).

The(−1)-symplectic pairingωonE is obtained via E ⊗ E

ω ""

(−,−)

// Ω^0,^•(Σ,KΣ)

R

zz Σ

C

Then the triple(E,Q=∂^¯+δ,ω)defines an infinite dimensional(−1)-symplectic structure in the sense of section 2. Our main goal in this section is to study the associated renormalized quantum master equation.

We will be mainly concerning the flat situation in this paper whenΣ=C,C^∗ or the elliptic curveEτ =C/(Z⊕Z^τ)(τ ∈H). We assume this from now on.

Definition 3.6. We will fix a coordinatez onΣ: this is the linear coordinate on C, z ∼ z+₁ ∼ z+τ on E_τ, and e^2πiz parametrizesC^∗. Our convention of the volume form is

d²z := ⁱ

2dz∧dz.¯

We also use the following notation: f(−→_z )means a smooth function onzwhile f(z)means a holomorphic function.

SinceΣis flat, we will focus on the situation in this paper when(E^•,(−,−)) comes from linear data(h,h−,−i)as follows:

• h = ^L

m∈Z

h_m is aZ-graded vector space (the grading is the cohomology degree);

• h−,−i:∧²h→Cis a degree 0 symplectic pairing;

• E^• =O_Σ⊗h, andE =Ω^0,^•(Σ)⊗h;

• δ ∈ C h∂

∂z

i⊗^L

m Hom(h^m,h^m⁺¹). Here C h∂

∂z

i

represents the space of translation invariant holomorphic differential operators onΣ;

• (−,−)is induced from fiberwiseh−,−ivia the identificationKΣ∼=O_Σdz (ϕ₁,ϕ₂) =

Z

dz∧ hϕ₁,ϕ₂i.

We consider the dualZ-graded vector spaceh^∗ with the induced symplectic pairing still denoted byh−,−i. TheZ/2Z-grading associated to theZ^-grading defines the parity onh^∗. The extraZ-grading of conformal weight will not be explicit at this stage. At this stage, we assume for simplicity that the pairing

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h−,−i on h^∗ has conformal weight −1 and obtain the vertex algebra V[[h^∗]]

as in Section 3.1.3. The same discussion can be easily generalized when the conformal weighth−,−iis different from−1. We describe such an example in Section 4.

The relevant chiral local functionals will be described by^H V[[h^∗]]. Precisely, Definition 3.7. GivenI ∈ V[[h^∗]][[h¯]], we extend itΩ^0,^•[[h¯]]-linearly to a map

I :E → Ω^0,^•[[¯h]].

Explicitly, ifI =_{∑ ∂}^k¹a₁· · ·∂^k^ma_m ∈ V[[_h^∗]]_{, where}a_i ∈_h^∗_,ϕ∈ E_{, then} I(ϕ) =

∑

^±^∂^kz¹a₁(ϕ)· · ·∂^k_z^ma_m(ϕ),

Herea_i(ϕ) ∈ Ω^0,^• comes from the natural pairing betweenh^∗ andh. ∂_z is the holomorphic derivative with respect to our prescribed linear coordinatezonΣ.

±is the Koszul sign. We associate a local functional ˆI ∈ O_loc(E)[[¯h]]onE by Iˆ(ϕ):=i

Z

Σdz I(ϕ), ϕ ∈ E. Note that deg(I^ˆ) =deg(I)−1.

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

. It is normalized by (e⁻^tHf)(−→

z ₁) =

Z

Σd²z₂ht(−→ z ₁,−→

z ₂)f(−→

z ₂), t>0.

For Σ = C, we have ¯∂^∗ = −2∂_zι_∂_¯

¯

z and H = −2∂_z∂¯_z_¯ = ¹₂∂²_x+∂²_y

where z= x+iy, andι_∂_¯

¯

z is the contraction with the vector field _∂^∂_z_¯. Then explicitly h_t(−→

z ₁,−→

z ₂) = ¹

2πte^−|^z¹⁻^z²^|²^/^2t.

WhenΣ=C^∗^or^Eτ,htis obtained from the above heat kernel onCby a further summation over the relevant lattices.

The regularized BV kernelK_L∈ Sym²(E)is given by K_L(−→

z₁,−→

z ₂) =i h_L(−→ z ₁,−→

z₂)(dz¯₁⊗1−1⊗dz¯₂)C_h.

HereC_h = ∑_i,_jω_{i j}(aⁱ⊗a^j)is the Casimir element where{aⁱ}is a basis ofh,ω_{i j} is the inverse matrix of

aⁱ,b^j

. The normalization constant is chosen such that (e⁻^LHϕ)(−→

z ₁) =−¹ 2

Z

Σdz₂∧K_L(−→ z ₁,−→

z ₂),ϕ(−→

z ₂), ∀ϕ ∈ E

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whereh−,−iinside the integral is the pairing in thez₂-component. The factor

1

2 is the symmetry factor, while the extra minus sign comes froming passing K_L through dz₂. ThenK₀ is precisely the desired singular Poisson tensor. The regularized propagator is

P^L=

Z _L

(∂^¯^∗⊗1)Kudu, 0<< L<_∞.

Definition 3.8. We define the subspace V⁺[[h^∗]][[¯h]] ⊂ V[[h^∗]][[¯h]] by saying thatI ∈ V⁺[[h^∗]][[¯h]]if and only if lim

¯

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

Theorem 3.9. Given I∈ V⁺[[h^∗]][[h¯]]of degree1and L> 0, the limit Iˆ[L]_:=_lim

→0W(P^L, ˆI)

exists as an element ofO⁺(E)[[¯h]].{I^ˆ[L]}_L_>₀defines a family ofO⁺(E)[[¯h]]satisfying the homotopic RG flow equation, andlim

L→0

Iˆ[L] = I. Here^ˆ I is defined in Definition 3.7.ˆ Proof. The limit behavior of and L is a consequence of [25, Lemma 3.1 and Proposition B.1]. The homotopic RG flow equation is similar to the discussion

in Section 2.3.3.

We remark that the order of limit is important in the proof of Theorem 3.9. The chiral nature of the problem implies that all potential singularities inW(P^L,I) in fact vanish upon integration by parts before taking the limit→0 [25].

It remains to analyze the renormalized quantum master equation for ˆI[L]. 3.3. Quantum master equation and OPE. The differentialδinduces naturally a differential on the formal polynomial ringV[[h^∗]](denoted by the same symbol)

δ :V[[h^∗]]→ V[[h^∗]]. Let us writeδ =D⊗φ, whereD∈C

h∂

∂z

i

,φ∈Hom(h,h). Letφ^∗ ∈Hom(h^∗,h^∗) be the dual ofφ. Then in terms of generators,

δ(∂^k_za):=∂^k_zD(φ^∗(a)), a∈ h^∗. This further induces a differential

δ: I

V[[h^∗]]→

I

V[[h^∗]].

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

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Lemma 3.10. The differentialδand the Lie bracket[,]define a structure of differential graded Lie algebra on^HV[[_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 dimensional chiral analogue of Theorem 2.19.

Theorem 3.11. LetIˆ[L]be defined in Theorem 3.9. Then the familyIˆ[L]satisfies renormalized quantum master equation if and only if

δ I

dzI+ ¹ 2

i¯h π

−1_I dzI,

I dzI

=0, i.e..^H dzI is a Maurer-Cartan element of^H V[[h]][[h¯]].

Proof. It is enough to work withΣ=C^{, since}C^∗^,^Στare quotients by translations and ourδandI are translation invariant. Moreover, the renormalized quantum master equations

(†) QIˆ[L] +¯h∆_LIˆ[L] +¹ 2

Iˆ[L], ˆI[L]

L =0, or (Q+∆_L)e^I^ˆ^[^L^]/^h^¯ =0,

are equivalent for eachL, therefore it is enough to analyze the limitL → 0. By Theorem 3.9,

(Q+¯h∆_L)e^I^ˆ^[^L^]/^¯^h =lim

→0(Q+h¯∆_L)e^h∂^¯ ^PLe^I^ˆ^/^¯^h

=lim

→0e^¯^h∂^PL

(Q+¯h∆)e^I^ˆ^/^¯^h

= ¹

¯ h lim

→0e^¯^h∂^PL

QIˆ+h¯∆Iˆ+ ¹ 2

I, ˆˆ I

e^I^ˆ^/^¯^h

= ¹ h¯ lim

→0e^¯^h∂^PL

δIˆ+¹ 2

Iˆ, ˆI

e^I^ˆ^/^h^¯. Here we have observed

• ∆Iˆ = _{0, since ˆ}_I is local and K becomes zero when restricted to the diagonal (the factordz¯₁−dz¯₂vanishes whenz₁= z₂ =z).

• ∂^¯Iˆ=0, since it contributes to a total derivative.

Therefore formally

¯

he⁻^I^ˆ^/^¯^hlim

L→0(Q+¯h∆_L)e^I^ˆ^[^L^]/^¯^h =δIˆ+¹ 2 lim

L→0lim

→0e⁻^I^ˆ^/^h^¯e^¯^h∂^PL

I, ˆˆ I e^I^ˆ^/^h^¯ .