Renormalization Method and Mirror Symmetry

This is a brief summary of our works [arXiv:1112.4063, arXiv:1201.4501] on constructing higher genus B-model from perturbative quantization of BCOV theory. We analyze Givental's symplectic loop space formalism in the context of B-model geometry on Calabi-Yau manifolds, and explain the Fock space construction via the renormalization techniques of gauge theory. We also give a physics interpretation of the Virasoro constraints as the symmetry of the classical BCOV action functional, and discuss the Virasoro constraints in the quantum theory.


Introduction
Mirror symmetry is a remarkable duality between symplectic geometry and complex geometry. It originated from string theory as a duality between superconformal field theories, and became a celebrated idea in mathematics since the physics work [6] on the successful prediction of the number of rational curves on the quintic 3-fold.
The symplectic side of mirror symmetry, which is called the A-model, is established in mathematics as the Gromov-Witten theory [20,27] counting the number of Riemann surfaces on smooth projective varieties. The complex side of mirror symmetry, which is called the B-model, is concerned with the deformation of complex structures. The B-model at genus 0 is described by the variation of Hodge structures on Calabi-Yau manifolds, which has been proven to be equivalent to the genus 0 Gromov-Witten theory on the mirror Calabi-Yau for a large class of examples [15,24]. However, for compact Calabi-Yau manifolds, the mirror symmetry at higher genus is not very well formulated, due to the difficulty of B-model geometry at higher genus.
Bershadsky, Cecotti, Ooguri and Vafa [5] proposed a gauge theory interpretation of the B-model via polyvector fields, which they called the Kodaira-Spencer gauge theory of gravity. BCOV suggested that the higher genus B-model could be constructed from the quantum theory of Kodaira-Spencer gravity. This point of view has remarkable consequences in physics [5,18,29], but is much less appreciated in mathematics due to the difficulty of rigorous quantum field theory.
In the long paper [10], we initiated a mathematical analysis of the quantum geometry of perturbative BCOV theory based on the effective renormalization method developed in [8]. In [10,21], we have shown that the quantum BCOV theory on elliptic curves are equivalent to Gromov-Witten theory on the mirror elliptic curves. This gives the first compact Calabi-Yau example where mirror symmetry is established at all genera. A related work on the finitedimensional toy model of BCOV theory has been discussed by Losev, Shadrin and Shneiberg [25] to avoid the issue of renormalization.
In this paper we will outline the main techniques used in [10] toward the construction of higher genus B-model. We will focus on the Fock space formalism, and give an interpretation of Virasoro constraints in the classical and quantum BCOV theory.
2 Classical BCOV theory 2.1 Polyvector f ields Let X be a compact Calabi-Yau manifold of dimension d. Follow [4] we consider the space of polyvector fields on X PV(X) = 0≤i,j≤d Here T X is the holomorphic tangent bundle of X, and A 0,j (X, ∧ i T X ) is the space of smooth (0, j)forms valued in ∧ i T X . PV(X) is a differential bi-graded commutative algebra; the differential is the operator and the algebra structure arises from wedging polyvector fields. The degree of elements of PV i,j (X) is i + j. The graded-commutativity says that where |α|, |β| denote the degree of α, β respectively. The Calabi-Yau condition implies that there exists a nowhere vanishing holomorphic volume form Ω X ∈ Ω d,0 (X), which is unique up to a multiplication by a constant. Let us fix a choice of Ω X . It induces an isomorphism between the space of polyvector fields and differential forms where is the contraction map. The holomorphic de Rham differential ∂ on differential forms defines an operator on polyvector fields via the above isomorphism, which we still denote by Obviously, the definition of ∂ doesn't depend on the choice of Ω X . It induces a bracket on polyvector fields which associates PV(X) the structure of Batalin-Vilkovisky algebra.
We define the trace map Tr : PV(X) → C by Let −, − be the induced pairing It's easy to see that∂ is skew self-adjoint for this pairing and ∂ is self-adjoint.

BCOV theory and Givental formalism
Following Givental's symplectic formulation [16,17] of Gromov-Witten theory in the A-model and the parallel Barannikov's works [1,3] in the B-model, we consider the dg symplectic vector space which is the space of Laurent polynomials in t with coefficient in PV(X). Here t has cohomology degree two, and the differential is given by Q =∂ + t∂, [2] is the conventional shifting of degree by two such that PV 1,1 (X), which describes fields associated to the classical deformation of complex structures as in BCOV theory [5], is of degree zero. The symplectic pairing is Bershadsky, Cecotti, Ooguri and Vafa [5] introduce a gauge theory for polyvector fields on Calabi-Yau three-folds. This is further extended to arbitrary Calabi-Yau manifolds in [10]. The space of fields of the BCOV theory is which is a Lagrangian subspace of S(X). The classical action functional of the BCOV theory can be constructed from the Lagrangian cone describing the period map of semi-infinite variation of Hodge structures [1]. Let The geometry of L X can be described by the following Lemma 1 ( [10]). L X is a formal Lagrangian submanifold of S(X), preserved by the differential Q =∂ + t∂. Moreover, L X − t is a Lagrangian cone preserved by the infinitesimal symplectomorphism of S(X) given by multiplying by t −1 .

Consider the polarization
The generating functional F L X is a formal function on S + (X) such that It's easy to see that F L X has cohomology degree 6−2d, where d = dim C X. The explicit formula is worked out in [10].
The lowest component is cubic: it's non-zero only on t 0 PV(X) ⊂ E(X) which is literally called the Yukawa-coupling.
). The classical BCOV action functional is the formal local functional on polyvector fields E(X) given by F L X .
Our definition of BCOV action functional extends the original BCOV action in [5] by including the "gravitational descendants" t.
We can transfer the geometry of the Lagrangian L X into properties of F L X .

Proposition 2 ([10]
). The BCOV action functional satisfies the classical master equation where Q is the induced derivation on the functionals of E(X), and {−, −} is the Poisson bracket on local functionals induced from the distribution representing the operator ∂ (see Remark 3).
This is equivalent to that L X is preserved by Q. The appearance of the bracket term follows from the fact that the splitting S(X) = S + (X) ⊕ S − (X) doesn't respect Q, or more precisely, S − (X) is not preserved by Q. We refer to [10,22] for more careful treatment.
The classical master equation implies that Q + {F L X , −} is a nilpotent operator acting on local functionals. In physics terminology, it generates the gauge symmetry, and defines the gauge theory in the Batalin-Vilkovisky formalism.
This follows from Givental's interpretation [16]: the dilaton equation is equivalent to the statement that L X − t is a cone, and the string equation is that L X − t is preserved by the infinitesimal symplectomorphism of S(X) given by multiplying by t −1 .

Remark 2.
The BCOV action functional is equivalent to that used by Losev-Shadrin-Shneiberg [25] in the discussion of finite-dimensional models of Hodge field theory, where the dilaton equation and string equation have also been obtained. Lemma 1 is purely an algebraic property, which is essentially used in [28] in a different context.

Fock space and renormaliztion
According to Givental and Coates [7], the generating function for the Gromov-Witten invariants of a variety X is naturally viewed as a state in a Fock space constructed from the cohomology of X. We will describe the Fock space construction in the B-model from the renormalization and effective field theory method [8,10].

Fock space and quantization
Let's recall the construction of the Fock module for a finite-dimensional dg symplectic vector space (V, ω, d), where ω is the symplectic form on V , and d is the differential which is skew self-adjoint with respect to ω.
Let W(V ) be the Weyl algebra of V , which is the pro-free dg algebra generated by V ∨ and a formal parameter , subject to the relation that where ω −1 ∈ ∧ 2 V is the inverse of ω. Let L be a Lagrangian sub-complex of V , and Ann(L) ⊂ V ∨ is the annihilator of L. Then the Fock module Fock(L) is defined to be the quotient Since L is preserved by the differential, Fock(L) naturally inherits a dg structure from d. We will denote it byd. Let us choose a complementary Lagrangian L ⊂ V , V = L ⊕ L . L may not be preserved by the differential. It allows us to identify be the space of formal functions on L. L defines a splitting of the map V ∨ → L ∨ , hence a map which identifies the Fock module with the algebra O(L)(( )). The differentiald can be described as follows. Let π : V → L be the projection corresponding to the splitting and it's easy to see that P ∈ Sym 2 (L). Let where d L is the induced differential on O(L) from that on L.
If we run the machine to construct the Fock space as in the previous section, we immediately run into trouble: PV(X) is infinite-dimensional! This is a well-known phenomenon in quantum field theory, which is related to the difficulty of ultra-violet divergence. The standard way of solving this is to use the renormalization technique. We'll follow the approach developed in [8].

Functionals on the f ields
Let E(X) ⊗n be the completed projective tensor product of n copies of E(X). It can be viewed as the space of smooth polyvector fields on X n with a formal variable t for each factor. Let denote the space of continuous linear maps (distributions), and the subscript S n denotes taking S n coinvariants. O (n) (E(X)) will be the space of homogeneous degree n functionals on the space of fields E(X), playing the role of Sym n (V ∨ ) in the case of finite-dimensional vector space V . We will also let be the subspace of local functionals, i.e. those of the form given by the integration of a Lagrangian density µ ∈ E(X).
Definition 2. The algebra of functionals O(E(X)) on E(X) is defined to be the product and the space of local functionals is defined to be the subspace loc (E(X)).

Ef fective Fock space
Let g be a Kähler metric on X. Let be the heat kernel for the operator e −L[∂,∂ * ] , where∂ * is the adjoint of∂ with respect to the metric g. It's a smooth polyvector field on X × X defined by the equation where we have chosen coordinates (x, y) on X × X, and we integrate over the second copy of X using the trace map.
Definition 3. The effective inverse ω −1 g,L for the symplectic form ω is defined to be the kernel Note that lim L→0 K g L is the delta-function distribution, which is no longer a smooth polyvector field, hence not an element of S(X) ⊗ S(X). ω −1 g,L can be viewed as the regularization of ω −1 in the infinite-dimensional setting.
Definition 4. The effective Weyl algebra W(S(X), g, L) is the quotient of the completed tensor algebra by the topological closure of the two-sided ideal generated by Here is the natural pairing between S(X) and its dual.
Similarly, the Fock space can also be defined using the regularized kernel ω −1 g,L .
Definition 5. The Fock space Fock (S + (X), g, L) is the quotient of W(S(X)) by the left ideal generated topologically by the subspace Similar to the finite-dimensional case, the polarization S(X) = S + (X) ⊕ S − (X) gives the identification We refer to [10] for detailed discussions.

Ef fective BV formalism
We would like to understand the quantized operatorQ L for Q acting on the Fock space represented by the above identification. This is completely similar to the finite-dimensional construction. Let be the kernel for the operator ∂e −L[∂,∂ * ] . It can be viewed as the projection of (Q ⊗ 1)ω −1 L,g ∈ Sym 2 (S(X)) to Sym 2 (E(X)).
Definition 6. The effective BV operator is the operator of contracting with the smooth kernel (∂ ⊗ 1)K g L .

S. Li
Since ∆ L : O (n) (E(X)) → O (n−2) (E(X)), it could be viewed as an order two differential operator on the infinite-dimensional vector space E(X). Note that ∆ L has odd cohomology degree, therefore It defines a Batalin-Vilkovisky structure on O(E(X)), with the Batalin-Vilkovisky bracket defined by The proof is similar to the finite-dimensional case.

Renormalization group f low and homotopy equivalence
We need to specify a choice of the metric g and a positive number L > 0 to construct the Fock space Fock (S + (X), g, L). However, we are in a bit better situation. The general machinery of renormalization theory in [8] allows us to prove that the effective Fock spaces are independent of the choice of g and L up to homotopy. This is discussed in detail in [10]. We'll discuss here the homotopy between different choices of the scale L, which is related to the renormalization group flow in quantum field theory.
is the operator of contraction by the smooth kernel P L .
It follows from this lemma that e ∂ P L defines the homotopy between Fock spaces defined at scales and L. It defines a flow on the space of functionals on the fields, which is called the renormalization group flow in [8] following the physics terminology.

Quantization and higher genus B-model
We have discussed the classical BCOV action functional F L X , and the effective Fock space in the B-model. In this section, we will make the connection between these two. We will discuss the perturbative quantization of BCOV theory from the effective field theory point of view [8], which is modeled on the usual Feynman graph integrals. Any such quantization will give rise to a state in the Fock space. In the case of elliptic curves, we find a canonical quantization, and the corresponding higher genus B-model is mirror to the Gromov-Witten theory. 3. The locality axiom, as in [8]. This says that F[L] has a small L asymptotic expansion in terms of local functionals.

The classical limit condition
6. Hodge weight axiom. We will give E(X) an additional grading, which we call Hodge weight, by saying that elements in t m Ω 0, * ∧ k T X = t m PV k, * (X) have Hodge weight k + m − 1. Then the functional F g [L] must be of Hodge weight .

Remark 5.
The space of quantizations of the BCOV theory has a simplicial enrichment, where the above definition gives the 0-simplices. The higher simplices are given by families of metrics, and homotopies of theories, which allow us to identify the quantizations with different choices of the metric. We refer to [8,10] for the general discussion.

Higher genus B-model
Given a quantization {F[L]} L>0 of the BCOV theory, we obtain a state e F[L]/ in the Fock space Fock (H * (S + (X))) [ −1 ] by Proposition 5. We will denote it by Z F , the partition function.

Definition 9. A polarization of H * S(X) is a Lagrangian subspace L ⊂ H * S(X) preserved by
such that This definition of polarization is used by Barannikov [2] in the study of semi-infinite variation of Hodge structures. We refer to [10] for more detailed discussion about the relation with BCOV theory.
There's a natural bijection between polarizations of H * S(X) and splittings of the Hodge filtration on H * (X). Moreover, the choice of a polarization induces isomorphisms In particular, it induces a natural identification Definition 10. Let F be a quantization of the BCOV theory on X, and L be a polarization of H * (S(X)). Let α 1 , . . . , α n ∈ H * ∂ (X, ∧ * T X ). The correlation functions associated to F, L is defined by Here the superscript "B" refers to the B-model. We can further decompose F B,L X = g≥0 g F B,L g,X .
Then F B,L g,X will be the candidate for the higher B-model invariants on X. It's natural to conjecture that there's a unique quantization F (up to homotopy) of the BCOV theory on X satisfying natural symmetry constraints. Then Z X will be the mirror of the Gromov-Witten invariants on the mirror Calabi-Yau manifold X ∨ . This proves to be the case for X being an elliptic curve.

The polarizations
There are two natural polarizations of H * (S(X)).
The first one is given by the complex conjugate splitting of the Hodge filtration, which we denote by LX . In this case the correlation function F B,LX X can be realized explicitly as follows. Consider the limit which is well-defined since X is compact, hence P ∞ L is smooth. The quantum master equation at L = ∞ says that On the other hand, the choice of the metric induces isomorphisms [2] via Hodge theory, hence defining a polarization which is easily seen to be precisely LX . Therefore The second choice of the polarization is relevant for mirror symmetry, which is defined near a large complex limit in the moduli space of complex structures on X. Near any such large complex limit point, there's an associated limiting weight filtration W which splits the Hodge filtration. Then the correlation function will be the mirror of the descendant Gromov-Witten invariants on the mirror Calabi-Yau X ∨ − GW g,n,X ∨ : Sym n H * (X ∨ , C) → C under the mirror map. Note that F B,LX X doesn't vary holomorphically due to the complex conjugate splitting LX . This is the famous holomorphic anomaly discovered in [5]. Given a large complex limit point, the natural way to retain holomorphicity is to consider F B,W g,X , which is usually denoted in physics literature by as the "τ → ∞-limit" [5] near the large complex limit.

Virasoro equations
The "Virasoro conjecture" was invented by Eguchi, Hori, Jinzenji, Xiong and Katz [12,13] and axiomatized in the context of topological field theory by Dubrovin and Zhang [11]. Givental [17] gave a geometric formulation of Virasoro equations on Fock spaces from quantization of symplectic vector spaces. For Calabi-Yau manifolds, the Virasoro equations greatly simplify the calculation of Gromov-Witten invariants on elliptic curves [26], but do not provide much information if the dimension is bigger than two [14]. However, the effective version of the Virasoro equations as below is useful to analyze the obstruction of the quantization of BCOV theory. We will follow Givental's approach and discuss the Virasoro equations in the context of BCOV theory at genus 0. The formalism generalizes in fact to the Landau-Ginzburg twisted version of BCOV theory.

Virasoro operators in Givental formalism
Let H : S(X) → S(X) be the grading operator andt : S(X) → S(X) be the linear operator given by multiplication by t. The Virasoro operators are defined by the linear maps given by the compositions [17] L n = Ht n H : S(X) → S(X), n ≥ −1, where it's understood that L −1 =t −1 . Since both H andt are skew self-adjoint for the symplectic pairing, L n is also skew self-adjoint, hence defining linear symplectic vector field on S(X). They satisfy the Virasoro relations which follows from H,t =t.
Lemma 3. The Lagrangian cone L X − t is perserved by L n , n ≥ −1.
Proof . Given α = −te f /t ∈ L X − t, f ∈ S + (X), the tangent space at α is given by The vector field of L n at α is L n α. We observe that where U respects the product structure ∀ α, β ∈ S(X).

It follows that
Ht : S + (X)e f /t → S + (X)e f /t .
In fact, given any g ∈ S + (X), which lies in S + (X)e f /t . Hence L n (α) = − Ht n+1 e f /t ∈ S + (X)e f /t . This proves the lemma.
It follows that the Lagrangian L X is invariant under the symplectomorphism generated by the Virasoro vector fields L n . This geometric interpretation can be equivalently stated for the generating functional F L X . We have seen that the L 0 -symmetry is equivalent to the dilaton equation (if dim C X = 3), and the L −1 symmetry is equivalent to the string equation. For L n , n > 0, we have Proposition 6. The classical BCOV action functional F L X satisfies the classical Virasoro equations Here, : is the operator of contracting with L n (t) = t n+1 n k=0 k − dim C X−3

2
, and L n acting on O(E(X)) is the induced derivation from the linear map L n on E(X). The classical Virasoro bracket {−, −} Ln is defined in the next section. The proof of the proposition is purely a translation between Lagrangian submanifold and generating functional, similar to the classical master equation. The appearance of ∂ ∂Ln(t) comes from the dilaton shift. Therefore the classical BCOV action functional F L X has the Virasoro transformations L n , n ≥ −1 as classical symmetries. If the Virasoro symmetries are preserved in the quantization (see the discuss below), then the quantum correlation functions will also satisfy the Virasoro symmetries. This gives a natural geometric interpretation of "Virasoro constraints" [12] in the context of B-model.

Quantum Virasoro operators on Fock spaces
The classical Virasoro operators L n are linear symplectic, and commute with the heat operator e −L[∂,∂ * ] . The usual method of Weyl quantization gives rise to operators acting on the Fock space, similar to the discussion in Section 3.1. We will describe the explicit result in this section. Let be the kernel for the operator L n e −L[∂,∂ * ] acting on S(X) with respect to the symplectic pairing ω. Let denote the projection associated with the polarization S(X) = S + (X) ⊕ S − (X). Let (L n ⊗ 1) ω −1 L,g + ∈ Sym 2 (S + (X)) be the image of (L n ⊗ 1) ω −1 L,g under the projection π + ⊗ π + . We will denote by the operator of contracting with the kernel (L n ⊗ 1) ω −1 L,g + . Note that V n,L = 0 for n = −1, 0. Similar to the effective BV bracket, we can define the effective Virasoro bracket by and the classical Virasoro bracket for local functionals which is used in Proposition 6 to describe the Virasoro symmetry of the classical BCOV action functional F L X .
Definition 11. For n ≥ 0, L > 0, the effective Virasoro operator is defined by The effective operator for L −1 needs extra care since L −1 doesn't preserve E(X). Let's define a derivation Y [L] : O(E(X)) → O(E(X)) as the derivation associated to the linear map ) as the derivation associated to the linear map Definition 12. The effective Virasoro operator L −1 [L] is defined by loc (PV(X)) is seen to be the trace pairing. Proof . This is a straight-forward check. . Note that at the quantum limit, V n,∞ consists of only harmonic elements. This is the Virasoro equations on cohomology classes in the B-model, mirror to the Virasoro equations for Gromov-Witten theory.

Quantization with Virasoro symmetries
Remark 6. The reason we formulate the homotopic version of Virasoro equations is that this allows us to analyze the obstruction/anomaly of the quantization of BCOV theory preserving the Virasoro symmetry via homological algebra method. The obstructions are typical to the ultraviolet difficulty in quantum field theory due to the necessary introduction of counter-terms for Feynman integrals, which may break the classical symmetries at quantum level. The Virasoro equations above are described at the form level, which would help us to analyze the quantum corrections locally. See [10] for the details on the obstruction analysis.

Quantum BCOV theory on elliptic curves
We consider the simpliest example of Calabi-Yau manifolds: elliptic curves. Let E τ be the elliptic curve where Λ τ = Z ⊕ Zτ , and τ ∈ H represents the complex structure.
We will denote such quantization by F Eτ . Note that in the one-dimensional case, the operators V n,L = 0 for all n ≥ 0. The effective Virasoro operators become derivations on the space of functionals.
The proof of both theorems are based on the deformation-obstruction theory developed in [8] for the renormalization of gauge theories. This can be viewed as the mirror theorem of the Virasoro constraints for Gromov-Witten theory on elliptic curves established in [26].

Higher genus mirror symmetry
The mirror symmetry for elliptic curves is easy to describe. Let E represent an elliptic curve.
In the A-model, we have the moduli of (complexified) Kähler class [ω] ∈ H 2 (E, C) parametrized by the symplectic volume q = Tr ω, where the trace map in the A-model is given by the integration Tr = E . The mirror in the B-model is the elliptic curve E τ , with complex structure τ related by the mirror map q = e 2πiτ . Let Φ τ : be the unique isomorphism of commutative bigraded algebras which is compatible with the trace on both sides. (Φ τ (α 1 ), . . . , Φ τ (α n )) , where the large complex limit is taken to be Im τ → ∞ on the upper half plane H.
It's proved in [21,23] that the correlation functions for F B,LĒ τ Eτ , before taking theτ → ∞ limit, are almost holomorphic modular forms exhibiting mild anti-holomorphic dependence onτ . On the other hand, the correlation functions of Gromov-Witten theory are given by quasi-modular forms [26]. In this example, theτ → ∞ limit is the well-known identification between almost holomorphic modular forms and quasi-modular forms [19].