Vertex algebras and quantum master equation
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.
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 [Kontsevich-DQ] 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 configuration space on the disk, which can be viewed as a geometric renormalization of the perturbative expansion of Poisson sigma model (see also [CF]). Another recent example is Costello’s homotopic theory [Kevin-book] of effective renormalizations in the Batalin-Vilkovisky formalism. This leads to a systematic construction of factorization algebras via quantum field theories [Kevin-Owen]. For example, a natural geometric interpretation of the Witten genus is obtained in such a way [Kevin-genus].
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 [BV], 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 in the renormalized quantum master equation. In this paper, we will mainly discuss the formalism in [Kevin-book], 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 [GG, LLG] for topological quantum mechanics. In particular, it is shown in [LLG] that the renormalized quantum master equation can be identified with the geometric equation of Fedosov’s abelian connection [Fedosov] on Weyl bundles over symplectic manifolds. The algebraic nature of this correspondence is reviewed in Section LABEL:section-example. Such a correspondence leads to a simple geometric approach to algebraic index theorem [Fedosov-index, Nest-Tsygan], where the index formula follows from the homotopic renormalization group flow together with an equivariant localization of BV integration [LLG].
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 LABEL:section-2d 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 LABEL:thm-defn). 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 LABEL:main-thm) is an exact description of the quantum corrections in terms of vertex algebras. Briefly speaking, Theorem LABEL:main-thm 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
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 LABEL:section-B. Furthermore, we prove a general result on the modularity property of the generating functions and their holomorphic anomaly (Theorem LABEL:thm-modularity). This work is also motivated from understanding Dijkgraaf’s description [Dijkgraaf-chiral] of chiral deformation of conformal field theories.
The above correspondence can be viewed as the two dimensional vertex algebra analogue of the one dimensional result in [LLG]. 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 [LLG]. It allows us to solve many quantization problems in terms of powerful techniques in vertex algebras.
As an application in Section LABEL:section-B, 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. There is a categorical approach [Partition, TCFT, KS] to the quantum B-model partition function associated to a Calabi-Yau category based on a classification of two-dimensional topological field theories. Unfortunately, it is extremely difficult to perform this categorical computation (recently a first non-trivial categorical computation is carried out in [CT] for one-point functions on the elliptic curve ). Another approach is through quantum field theory. In [Si-Kevin], we construct a gauge theory of polyvector fields on Calabi-Yau manifolds (called BCOV theory) as a generalization of the Kodaira-Spencer gauge theory [BCOV]. It is proposed in [Si-Kevin] (as a generalization of [BCOV]) 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 LABEL:section-B gives a concrete realization of this program. This leads to the first mathematically fully established example of higher genus mirror symmetry on compact Calabi-Yau manifolds.
Our result in Section LABEL:section-B 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 [Zwiebach]. 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 LABEL:section-B 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 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.
Let be a -graded -vector space. We use to denote its degree component. Given , we let be its degree.
denotes the degree shifting of such that .
denotes its dual such that . Our base field will mainly be or .
and denote the graded symmetric product and graded skew-symmetric product respectively. We also denote
Given , and , we denote its -order Taylor coefficient
where we have viewed as a multi-linear map .
Given , it defines a “second order operator” on or by
where for any ,
, and denote polynomial series, formal power series and Laurent series respectively in a variable valued in .
Let be a graded commutative algebra. always means the graded commutator, i.e., for elements with specific degrees,
We always assume Koszul sign rule in dealing with graded objects.
without subscript means tensoring over the real numbers .
Given a manifold , we denote the space of real smooth forms by
where is the subspace of -forms. If furthermore X is a complex manifold, we denote the space of complex smooth forms by
where is the subspace of -forms.
denotes the density bundle on a manifold . When is oriented, we naturally identify with top differential forms on .
Let E be a vector bundle on a manifold . denotes the space of smooth sections, and denotes the distributional sections. If is the dual bundle of , then we have a natural pairing
denotes 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
A differential Batalin-Vilkovisky (BV) algebra is a triple
is a -graded commutative associative unital algebra.
is a derivation of degree such that .
is a second-order operator of degree such that .
and are compatible: .
Here is called the BV operator. being “second-order” means the following: define the BV bracket as measuring the failure of being a derivation
Then defines a Poisson bracket of degree satisfying
The -compatibility condition implies the following Leibniz rule
Let be a differential BV algebra. A degree element is said to satisfy classical master equation (CME) if
If solves CME, then it is easy to see that defines a differential on , which can be viewed as a Poisson deformation of . However, it may not be compatible with . A sufficient condition for the compatibility is the “divergence freeness” . A slight generalization of this is the following.
Let be a differential BV algebra. A degree element is said to satisfy quantum master equation (QME) if
Here is a formal variable representing the quantum parameter.
The “second-order” property of implies that QME is equivalent to
If we decompose , then the limit of QME is precisely CME
We can rephrase the -compatibility as the nilpotency of . It is direct to check that QME implies the nilpotency of , 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 -shifted symplectic space. This serves as the main motivating resources of our quantum field theory examples.
Let be a finite dimensional dg vector space. The differential induces a differential on various tensors of , still denoted by . Let
be a -compatible symplectic structure such that . It identifies
Let be the Poisson kernel of degree under
where we have used the canonical identification . Let
Then is a graded-commutative dga.
The degree Poisson kernel defines the following BV operator
Here denotes the natural paring between and . is the Koszul sign by permuting ’s. The following lemma is well-known.
defines a differential BV algebra.
The above construction can be summarized as
Since we only use to define BV operator, the above process is well-defined for -shifted dg Poisson structure where 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 -shifted dg symplectic space . There is assumed to be finite dimensional. This is why we call it “toy model”. Typically in quantum field theory, will be modified to be the space of smooth sections of certain vector bundles on a smooth manifold, while the differential and the pairing come from something “local”. Such 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 be a smooth oriented manifold without boundary. Let be a complex of vector bundles on
where is the differential. We assume that is an elliptic complex. Our space of fields replacing will be the space of smooth global sections
with the induced differential, still denoted by . The symplectic pairing will be
is a non-degenerate graded skew-symmetric pairing of degree . To perform the toy model construction, we need the following steps
The dual vector space (analogue of ). This can be defined via the space of distributions on
where is the space of continuous maps.
The tensor space (analogue of ). This can be defined via the completed tensor product for distributions
where is the distributions on the bundle over . is defined similarly by taking care of the graded permutation. Then we have a well-defined notion (via distributions)
as the analogue of .
The Poisson kernel (the analogue of ). The pairing does not induce an identification between and its dual in this case. Since is defined via integration, the Poisson kernel is the -function representing integral kernel of the identity operator. Therefore is a distributional section of supported on the diagonal
See Conventions for . It is at Step (3) where we get trouble. In fact, if we naively define the BV operator
then is ill-defined, since we can not pair a distribution with another distribution from . 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 [Kevin-book] in this paper that will be convenient for our applications.
The key observations are
is a -closed distribution: .
elliptic regularity: there is a canonical isomorphism of cohomologies
It follows that we can find a distribution and a smooth element such that
is the familiar notion of a parametrix.
will be called the renormalized BV kernel with respect to the parametrix .
Since is smooth, there is no problem to pair with distributions. The same formula as in the toy model leads to
We define the renormalized BV operator
via the smooth renormalized BV kernel . The triple defines a differential BV algebra, called the renormalized differential BV algebra (with respect to ).
Therefore 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 and be two parametrices, and be the corresponding renormalized BV kernels. Let us denote
Since is smooth, is smooth itself by elliptic regularity.
will be called the regularized propagator.
Typically, suppose we have an adjoint operator such that is a generalized Laplacian. Then given , the integral kernel for the heat operator can be viewed as a renormalized BV kernel. In this case, the regularized propagator is given by
can be viewed as a homotopy linking two different renormalized differential BV algebras. In fact, similar to the definition of renormalized BV operator,
We define as the second-order operator of contracting with the smooth kernel (see also Conventions).
The following equation holds formally as operators on
i.e., the following diagram commutes