Lectures on BCOV Holomorphic Anomaly Equations

Abstract

The present article surveys some mathematical aspects of the BCOV holomorphic anomaly equations introduced by Bershadsky, Cecotti, Ooguri and Vafa [8,9]. It grew from a series of lectures the authors gave at the Fields Institute in the Thematic Program of Calabi-Yau Varieties in the fall of 2013.A candidate of the higher genus B-model was provided by Bershadsky, Cecotti, Ooguri and Vafa in the seminal papers [8,9] (BCOV theory). Among other things, they derived a set of equations, now called the BCOV holomorphic anomaly equa-In this section, we give a brief summary of the basics of special Kähler geometry that we need throughout this article. Special Kähler geometry is a basic computational tool used in the calculations in mirror symmetry. This section also serves to set conventions and notations. Standard references are [37,8,19]. Special Coordinates and PrepotentialLet M be the moduli space of complex structures of a smooth Calabi-Yau threefold X of dimension n := dim M = h 2,1 (X). The vector bundle H := R 3 π * C ⊗ O M of rank 2n + 2 comes equipped with the Gauss-Manin connection ∇ and the natural Hodge filtration F • of weight 3. The Hodge filtration F • yields the smooth decompositionis called the vacuum bundle. We also fix a reference point [X] ∈ M and smoothly identify 1 the fibers of H with H 3 (X, C). We endow H 3 (X, C) with the symplectic pairing (α, β ) := √ −1´X α ∪ β . Then the period domain D is defined by3 (X, C). More concretely, by fixing a symplectic basis {α I , β J } n I,J=0 of H 3 (X, Z) and its dual basis {A I , B J } n I,J=0 of H 3 (X, Z), the period map P is written in terms of a section Ω = {Ω z } z∈M of the vacuum bundle L as 2 P(z) := φ I (z)α I + F J (z)β J , where φ I (z) :=´A I Ω z and F J (z) :=´BJ Ω z . Proposition 1. With the notation above, the following hold:

Holomorphic anomaly of 2d Yang-Mills theory on a torus revisited

“…for the 2d QCD example it was argued in[24] that such a recursion does not exist." Also, in[30] it is stated that "There is, however, no known explicit holomorphic anomaly equations of higher genus for elliptic curves. "…”

Virasoro blocks and quasimodular forms

“…where the F (n,g) are amplitudes of an N = 2 topological string on a Calabi-Yau background, see [31] and references therein. These amplitudes satisfy a holomorphic anomaly equation which allows for them to be constructed recursively.…”

The Universal Coefficient Theorem and Quantum Field Theory

“…This is the theory of topological strings and the recursive relation is known as the holomorphic anomaly equation. For this introduction I follow mainly [235] and [246]. However, it appears that the universal coefficient theorem is also capable of encoding general anomalies that appear when an algebraic variety degenerates into another one.…”

“…The most important differential equation in this case is the holomorphic anomaly equation of Bershadsky, Cecotti, Ooguri and Vafa [246], [247] which controls the amplitudes as functions over the coupling space. The origin of the holomorphic anomaly can be related to the unitarity and CPT invariance of the underlying conformal field theory [235].…”