A Lecture on Holomorphic Anomaly Equations and Extended Holomorphic Anomaly Equations

TL;DR

This paper introduces the BCOV holomorphic anomaly equations and Walcher's extended versions, providing foundational insights into their structure and implications in complex geometry and string theory.

Contribution

It offers a concise overview of the original and extended holomorphic anomaly equations, clarifying their mathematical formulation and significance.

Findings

Clarified the structure of BCOV equations

Explained Walcher's extensions to the anomaly equations

Provided insights into their applications in geometry and physics

Abstract

This is a brief introduction to the Bershadsky-Cecotti-Ooguri-Vafa (BCOV) holomorphic anomaly equations and Walcher's extended holomorphic anomaly equations.

Equations

$$\text{A-model }(X,\omega )\stackrel{\text{Mirror Symmetry}}{⟷}\text{B-model }(Y,\Omega ),$$

$$\begin{array}[]{ccccccc}&&&1&&&\\ &&0&&0&&\\ &0&&h^{2,2}(X)&&0\\ 1&&h^{2,1}(X)&&h^{1,2}(X)&&\quad 1\\ &0&&h^{1,1}(X)&&0\\ &&0&&0&&\\ &&&1.&&&\end{array}$$

$$\kappa :=h^{1,1}(X)=h^{2,2}(X)={\dim }_{\mathbb{C}}{H}^2(X;\mathbb{C})={\dim }_{\mathbb{C}}{\mathcal{M}}_A,$$

$$\kappa =h^{2,1}(Y)=h^{1,2}(Y)={\dim }_{\mathbb{C}}{H}^1(Y,T_Y)={\dim }_{\mathbb{C}}{\mathcal{M}}_B,$$

$$[{\overline{\mathcal{M}}}_g(X,\beta ){]}^{\mathrm{vir}}\in H_0({\overline{\mathcal{M}}}_g(X,\beta );\mathbb{Q}).$$

$$N_{g,\beta }^X:={\int }_{[{\overline{\mathcal{M}}}_g(X,\beta ){]}^{\mathrm{vir}}}1\in \mathbb{Q}$$

$${\overline{\mathcal{M}}}_g(X,0)={\overline{\mathcal{M}}}_g\times X,$$

$$N_{g,0}^X=\frac{(-1{)}^g}{2}{\int }_{{\overline{\mathcal{M}}}_g}{\lambda }_{g-1}^3{\int }_X{c}_3(X)=\frac{(-1{)}^g|B_{2g}|\cdot |B_{2g-2}|}{4g(2g-2)\cdot (2g-2)!}{\int }_X{c}_3(X),$$

$$t=\sum _{i=1}^{\kappa }{t}^i{H}_i,$$

$$F_g^X(t):=\{\begin{array}{ll}\frac{1}{6}{\int }_X{t}^3+\sum _{\beta \ne 0}{N}_{0,\beta }^X\exp ({\int }_{\beta }t),& g=0;\\ -\frac{1}{24}{\int }_X{c}_2(X)t+\sum _{\beta \ne 0}{N}_{1,\beta }^X\exp ({\int }_{\beta }t),& g=1;\\ N_{g,0}^X+\sum _{\beta \ne 0}{N}_{g,\beta }^X\exp ({\int }_{\beta }t),& g\ge 2.\end{array}$$

$$F_g^X=F_g^{X,\text{classical}}+F_g^{X,\text{quantum}},$$

$$F_g^{X,\text{quantum}}=\sum _{\beta \ne 0}{N}_{g,\beta }^X\exp ({\int }_{\beta }t)$$

$$\exp ({\int }_{\beta }t)=Q_1^{d_1}\cdots Q_{\kappa }^{d_{\kappa }},$$

$$\underset{Q_i\to 0}{\lim }{F}_g^{X,\mathrm{quantum}}(t)=0.$$

$$\frac{{\partial }^3{F}_0^X}{\partial t^i\partial t^j\partial t^k}=(H_i\star H_j,H_k)$$

$$H_{\ast }(L;\mathbb{Q})=H_{\ast }(S^3;\mathbb{Q}).$$

$$F_{(g,h)}^{X,L}(t)=\sum _{\beta \ne 0}{N}_{(g,h),\beta }^{X,L}\exp ({\int }_{\beta }t),$$

$$\underset{Q_i\to 0}{\lim }{F}_{(g,h)}^{X,L}(t)=0.$$

$$H_2(X;\mathbb{Z})=\mathbb{Z}⟶H_2(X,L;\mathbb{Z})=\mathbb{Z}$$

$$F_g^{X,\text{quantum}}\in \mathbb{Q}[[Q]],F_{(g,h)}^{X,L}\in \mathbb{Q}[[Q^{1/2}]],$$

$$\begin{array}{ccc}\tilde{\mathcal{T}}& \to & {\tilde{\mathcal{M}}}_B\\ \downarrow & & \downarrow p\\ \mathcal{T}& \stackrel{\pi }{\to }& {\mathcal{M}}_B.\end{array}$$

$$\nabla :{\Omega }^0({\mathcal{M}}_B,{\mathbb{H}}_{\mathbb{C}})=C^{\infty }({\mathcal{M}}_B,{\mathbb{H}}_{\mathbb{C}})⟶{\Omega }^1({\mathcal{M}}_B,{\mathbb{H}}_{\mathbb{C}})=C^{\infty }({\mathcal{M}}_B,T^{\ast }{M}_B\otimes {\mathbb{H}}_{\mathbb{C}})$$

$${\int }_{Y_q}{\alpha }_i\cup {\alpha }_j={\int }_{Y_q}{\beta }^i\cup {\beta }^j=0,{\int }_{Y_q}{\alpha }_i\cup {\beta }^j=-{\int }_{Y_q}{\beta }^j\cup {\alpha }_i={\delta }_{ij}.$$

$$\nabla {\alpha }_i=\nabla {\beta }^i=0,i=0,1,\dots ,\kappa .$$

$$s=\sum _{i=0}^{\kappa }(a^i{\alpha }_i+b_i{\beta }^i),$$

$$\nabla s=\sum _{i=0}^{\kappa }(d{a}^i{\alpha }_i+d{b}_i{\beta }^i),$$

$$\nabla ={\nabla }^{\prime }+{\nabla }^{\prime \prime },$$

$$s=\sum _{i=0}^{\kappa }(a^i{\alpha }_i+b_i{\beta }^i),$$

$$0\subset {\mathbb{F}}^3\subset {\mathbb{F}}^2\subset {\mathbb{F}}^1\subset {\mathbb{F}}^0={\mathbb{H}}_{\mathbb{C}},$$

$${\mathbb{F}}_q^i=\underset{p\ge i}{⨁}{H}^{p,3-p}(Y_q;\mathbb{C}).$$

Full text

1. Mirror Symmetry for Compact Calabi-Yau threefolds

Mirror symmetry relates the A-model on a compact Calabi-Yau threefold $X$, defined in terms of the symplectic structure on $X$, to the B-model on a mirror compact Calabi-Yau threefold $Y$, defined in terms of the complex structure on $Y$,

[TABLE]

where $\omega$ is a Ricci flat Kähler form on $X$ and $\Omega$ is a holomorphic volume form (i.e. a nowhere vanishing holomorphic 3-form) on $Y$.

The Hodge diamond of $X$ is of the form

[TABLE]

The Hodge diamond of $Y$ is of the same form. Let

[TABLE]

where $\mathcal{M}_{A}$ is the complexified Kähler moduli of $X$. Then

[TABLE]

where $\mathcal{M}_{B}$ is the moduli of complex structures on $Y$.

2. A-model on compact Calabi-Yau threefolds

2.1. A-model topological closed strings: Gromov-Witten invariants

Given a non-negative integer $g$ and an effective curve class $\beta\in H_{2}(X;\mathbb{Z})$, let $\overline{\mathcal{M}}_{g}(X,\beta)$ be the moduli space of genus $g$ degree $\beta$ stable maps to $X$ (with no marked points). Note that $\overline{\mathcal{M}}_{g}(X,\beta)$ is empty if $(g,\beta)=(0,0)$ or $(1,0)$. The moduli $\overline{\mathcal{M}}_{g}(X,\beta)$ is a proper Deligne-Mumford stack equipped with a virtual fundamental class

[TABLE]

The genus $g$, degree $\beta$ Gromov-Witten invariant of $X$ is defined by

[TABLE]

when $(g,\beta)\neq(0,0),(1,0)$. The integral sign in the above equation stands for the natural pairing between $[\overline{\mathcal{M}}_{g}(X,\beta)]^{\mathrm{vir}}\in H_{0}(\overline{\mathcal{M}}_{g}(X,\beta);\mathbb{Q})$ and $1\in H^{0}(\overline{\mathcal{M}}_{g}(X,\beta);\mathbb{Q})$. For $g\geq 2$, we have

[TABLE]

and

[TABLE]

where $B_{2m}$ are Bernoulli numbers.

We choose a basis $H_{1},\ldots,H_{k}$ of $H^{1,1}(X)=H^{2}(X;\mathbb{C})$ such that each $H_{i}$ is in $H^{2}(X;\mathbb{Z})$ and also in the nef cone (which is the closure of the Kähler cone of $X$). A complexified Kähler class is of the form

[TABLE]

where $t^{1},\ldots,t^{\kappa}\in\mathbb{C}$ are complexified Kähler parameters. The genus $g$ Gromov-Witten potential of $X$, $F_{g}^{X}$, is a generating function of genus $g$ Gromov-Witten invariants of $X$:

[TABLE]

The sum $\displaystyle{\sum_{\beta\neq 0}}$ is over all non-zero effective classes. We may write

[TABLE]

where

[TABLE]

is the contribution from non-constant genus $g$ stable maps to $X$. We have

[TABLE]

where $Q_{i}=\exp(t^{i})$ and $d_{i}=\int_{\beta}H_{i}\in\mathbb{Z}_{\geq 0}$. So $F_{g}^{X,\mathrm{quantum}}$ is a formal power series in $Q_{1},\ldots,Q_{\kappa}$ with rational coefficients; it tends to zero at the large radius limit $Q_{i}\to 0$:

[TABLE]

We have

[TABLE]

where $\star$ is the quantum product, where $\cup$ is the classical cup product, and where $(\ ,\ )$ is the Poincaré pairing on $H^{*}(X)$.

2.2. A-model topological open strings: open Gromov-Witten invariants

Let $L\subset X$ be a closed oriented Lagrangian submanifold. Then the tangent bundle of $L$ is trivial – recall that the tangent bundle of any orientable 3-manifold is trivial. In this paper, we further assume that $L$ is a rational homology 3-sphere:

[TABLE]

Then the map $H_{2}(X;\mathbb{Q})\to H_{2}(X,L;\mathbb{Q})$ is an isomorphism, so $H_{2}(X;\mathbb{Z})\to H_{2}(X,L;\mathbb{Z})$ has finite kernel and cokernel. Since $H_{1}(L;\mathbb{Z})$ is torsion and $H_{0}(L;\mathbb{Z})=\mathbb{Z}$, we have $H^{1}(L;\mathbb{Z})=0$ by the universal coefficient theorem. In particular, the Maslov class $\mu(L)\in H^{1}(L;\mathbb{Z})$ (which is defined for any Lagrangian submanifold in a Calabi-Yau manifold) is zero. Given a pair $(g,h)$, where $g$ is a nonnegative integer and $h$ is a positive integer, let $\overline{\mathcal{M}}_{(g,h)}(X,L,\beta)$ be the stable compactification of the moduli $\mathcal{M}_{g,h}(X,L,\beta)$, parametrizing holomorphic maps $u:({\Sigma},\partial{\Sigma})\to(X,L)$, where ${\Sigma}$ is a bordered Riemann surface with $g$ handles and $h$ holes, and $u_{*}[\Sigma]=\beta\in H_{2}(X,L;\mathbb{Z})$, where the domain ${\Sigma}$ is oriented by its complex structure. Then $\overline{\mathcal{M}}_{g,h}(X,L,\beta)$ is a (usually singular) orbifold whose virtual dimension is zero. In some cases it is possible to define a virtual number $N_{(g,h),\beta}^{X,L}$ of points in $\overline{\mathcal{M}}_{g,h}(X,L,\beta)$; in general, $N_{(g,h),\beta}^{X,L}$ is a rational number (instead of an integer) due to the existence of orbifold points. We define generating functions of open Gromov-Witten invariants of the pair $(X,L)$ by

[TABLE]

where the sum is over nonzero relative homology classes $\beta\in H_{2}(X,L;\mathbb{Z})$. By assumption, for any $\beta\in H_{2}(X,L;\mathbb{Z})$ there exists a positive integer $r$ such that $r\beta$ lies in the image of $H_{2}(X;\mathbb{Z})\to H_{2}(X,L;\mathbb{Z})$, so $F_{(g,h)}^{X,L}(t)$ is a formal power series in $Q_{1}^{1/r_{1}},\ldots,Q_{\kappa}^{1/r_{\kappa}}$ for some positive integers $r_{1},\ldots,r_{\kappa}$, and it tends to zero at the large radius limit:

[TABLE]

Example 2.1**.**

Let $X$ be a quintic Calabi-Yau threefold with real coefficients, and let $L$ be the real quintic. Then $L$ is a Lagrangian submanifold of $X$, which is diffeomorphic to $\mathbb{R}\mathbb{P}^{3}$, so it is orientable and is a rational homology sphere. The group homomorphism

[TABLE]

is injective with cokernel $\mathbb{Z}/2\mathbb{Z}$. In this case $\kappa=1$, and

[TABLE]

where $Q=Q_{1}$.

3. Preliminaries on moduli of complex structures

3.1. The complex moduli and the vacuum line bundle

Recall that $\mathcal{M}_{B}$ is the moduli space of complex structures on $Y$, and that $\dim_{\mathbb{C}}\mathcal{M}_{B}=\kappa$. Let $q=(q_{1},\ldots,q_{\kappa})$ be the local holomorphic coordinates on $\mathcal{M}_{B}$ such that $q=0$ corresponds to a maximal unipotent monodromy point in the boundary of a (partial) compactification $\overline{\mathcal{M}}_{B}$ of $\mathcal{M}_{B}$. (In particular, $q=0$ is in $\overline{\mathcal{M}}_{B}\setminus\mathcal{M}_{B}$.) Let $\mathcal{L}^{\vee}\to\mathcal{M}_{B}$ be the complex line bundle over $\mathcal{M}_{B}$ whose fiber over $q\in\mathcal{M}_{B}$ is $\mathcal{L}^{\vee}_{q}=H^{0}(Y_{q},\Omega^{3}_{Y_{q}})\cong\mathbb{C}$; then $\mathcal{L}^{\vee}$ is a holomorphic line bundle over $\mathcal{M}_{B}$, and its dual $\mathcal{L}$ is the vacuum line bundle in the physics literature such as [13] and [4]. The extended moduli space $\widetilde{\mathcal{M}}_{B}$ is the total space of the frame bundle of $\mathcal{L}^{\vee}$; it parametrizes pairs $(Y_{q},\Omega)$, where $Y_{q}$ corresponds to a point $q\in\mathcal{M}_{B}$ and $\Omega$ is a nonzero holomorphic 3-form on $Y_{q}$. So $p:\widetilde{\mathcal{M}}_{B}\to\mathcal{M}_{B}$ is a principal $\mathbb{C}^{*}$-bundle, and $\dim_{\mathbb{C}}\widetilde{\mathcal{M}}_{B}=\kappa+1$.

3.2. The Torelli space

Let $H$ denote the rank 2 lattice $\mathbb{Z}\oplus\mathbb{Z}$ equipped with the symplectic form \Big{(}\begin{array}[]{cc}0&1\\ -1&0\end{array}\Big{)}. The Torelli space of $Y$ is the moduli of the marked Calabi-Yau threefold $(Y_{q},\gamma)$, where $Y_{q}$ corresponds to a point $q\in\mathcal{M}_{B}$ and the marking $\gamma$ is an isometry from $H^{\oplus(\kappa+1)}$ to $H^{3}(X_{q};\mathbb{Z})/\mathrm{Tor}$. Forgetting the marking $\gamma$ defines a covering map $\pi:\mathcal{T}\to\mathcal{M}_{B}$, which is a principal $Sp(2\kappa+2;\mathbb{Z})$-bundle. Let $\widetilde{\mathcal{T}}$ be the fiber product:

[TABLE]

Then $\widetilde{\mathcal{T}}\to\mathcal{T}$ is a principal $\mathbb{C}^{*}$-bundle that is the frame bundle of $\pi^{*}\mathcal{L}^{\vee}$, and $\widetilde{\mathcal{T}}\to\widetilde{\mathcal{M}}_{B}$ is a covering map that is a principal $Sp(2\kappa+2;\mathbb{Z})$ bundle.

3.3. The Hodge bundle $\mathbb{H}_{\mathbb{C}}$ and the Gauss-Manin connection

Let $\mathbb{H}_{\mathbb{Z}}$ be the local system of lattices on $\mathcal{M}_{B}$ whose fiber over $q\in\mathcal{M}_{B}$ is $H^{3}(Y_{q};\mathbb{Z})\cong\mathbb{Z}^{2k+2}$. Then $\mathbb{H}_{\mathbb{R}}=\mathbb{H}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{R}$ (resp. $\mathbb{H}_{\mathbb{C}}=\mathbb{H}_{\mathbb{Z}}\otimes_{\mathbb{Z}}\mathbb{C}$) is a flat real (resp. complex) vector bundle of rank $2\kappa+2$ whose fiber at $q\in\mathcal{M}_{B}$ is $H^{3}(Y_{q};\mathbb{R})$ (resp. $H^{3}(Y_{q};\mathbb{C})$); the flat connection is known as the Gauss-Manin connection.

More explicitly, let

[TABLE]

be the Gauss-Manin connection. Let $U$ be an open subset on $\mathcal{M}_{B}$ such that $\mathbb{H}_{\mathbb{Z}}|_{U}$ is trivial; we choose a trivialization of $\mathbb{H}_{\mathbb{Z}}|_{U}$, or equivalently, a symplectic basis $\{\alpha_{i},\beta^{i}:i=0,1,\ldots,\kappa\}$ of $H^{3}(Y_{q};\mathbb{Z})\cong H^{\oplus(\kappa+1)}$ for $q\in U$:

[TABLE]

Then $\{\alpha_{i},\beta^{i}\}$ is a frame of $\mathbb{H}_{\mathbb{C}}|_{U}$ and

[TABLE]

Any $C^{\infty}$ section $s\in C^{\infty}(U,\mathbb{H}_{\mathbb{C}})$ is of the form

[TABLE]

where $a^{i},b_{i}$ are complex-valued $C^{\infty}$ functions on $U$. Then

[TABLE]

where $da^{i},db_{i}\in\Omega^{1}(U,\mathbb{C})$ are $C^{\infty}$ 1-forms on $U$.

We may write

[TABLE]

where $\nabla^{\prime}:\Omega^{0}(\mathcal{M}_{B},\mathbb{H}_{\mathbb{C}})\to\Omega^{1,0}(\mathcal{M}_{B},\mathbb{H}_{\mathbb{C}})$ is a $(1,0)$-connection on the $C^{\infty}$ complex vector bundle $\mathbb{H}_{\mathbb{C}}$ and where $\nabla^{\prime\prime}:\Omega^{0}(\mathcal{M}_{B},\mathbb{H}_{\mathbb{C}})\to\Omega^{0,1}(\mathcal{M}_{B},\mathbb{H}_{\mathbb{C}})$ is a $(0,1)$-connection on $\mathbb{H}_{\mathbb{C}}$. The $(0,1)$-connection $\nabla^{\prime\prime}$ defines a holomorphic structure on $\mathbb{H}_{\mathbb{C}}$: a section $s\in C^{\infty}(U,\mathbb{H}_{\mathbb{C}})$ is holomorphic iff $\nabla^{\prime\prime}s=0$ iff

[TABLE]

where $a^{i},b_{i}$ are holomorphic functions on $U$.

3.4. Hodge filtration and the holomorphic polarization

We have the Hodge filtration

[TABLE]

where

[TABLE]

The complex vector bundles $\mathbb{F}^{1},\mathbb{F}^{2},\mathbb{F}^{3}$ are (non-flat) holomorphic subbundles of $\mathbb{F}^{0}=\mathbb{H}_{\mathbb{C}}$ of ranks $2\kappa+1$, $\kappa+1$, $1$, respectively. In particular, $\mathbb{F}^{3}=\mathcal{L}^{\vee}$ is the dual of the vacuum line bundle. We also have

[TABLE]

where

[TABLE]

For each $q\in U$,

[TABLE]

is the holomorphic polarization of the complex symplectic space $H^{3}(Y_{q};\mathbb{C})$.

3.5. Real polarization

We choose a symplectic basis $\{A^{i},B_{i}:i=0,1,\ldots,\kappa\}$ of $(H_{3}(Y;\mathbb{Z}),\cap)$, where $\cap$ is the intersection form, and let $\{\alpha_{i},\beta^{i}:i=0,1,\ldots,\kappa\}$ be the dual symplectic basis of $(H^{3}(Y;\mathbb{Z}),Q)$, where

[TABLE]

We have

[TABLE]

[TABLE]

For any $\phi\in H^{3}(X;\mathbb{R})$ we have

[TABLE]

where

[TABLE]

Then $\{x^{i},p_{i}:i=0,1,\ldots,k\}$ are Darboux coordinates of the real linear symplectic space $(H^{3}(Y;\mathbb{R}),Q)$. The linear symplectic form

[TABLE]

on $H^{3}(Y;\mathbb{R})$ is independent of the choice of a symplectic basis. The integral symplectic basis $\{\alpha_{i},\beta^{i}:i=0,1,\ldots,\kappa\}$ extends to a flat frame of $\mathbb{H}_{\mathbb{R}}$ on an open neighborhood $U$ of $q_{0}$ in $\mathcal{M}_{B}$, and $\{x^{i},p_{i}:i=0,1,\ldots,\kappa\}$ are flat fiber coordinates of $\mathbb{H}_{\mathbb{R}}|_{U}\cong U\times\mathbb{R}^{2\kappa+2}$.

The Gauss-Manin connection is compatible with the symplectic structure $Q$ on $\mathbb{H}_{\mathbb{C}}$: given two $C^{\infty}$ sections $s_{1},s_{2}$ of $\mathbb{H}_{\mathbb{C}}$, and a $C^{\infty}$ vector field $\xi$ on $\mathcal{M}_{B}$, we have

[TABLE]

where

[TABLE]

is the Lie derivative on $C^{\infty}$ functions, and where

[TABLE]

is the covariant derivative defined by the Gauss-Manin connection.

3.6. Special homogeneous coordinates and the period map

Let $s:\widetilde{\mathcal{M}}_{B}\to p^{*}\mathcal{L}^{\vee}$ be the tautological section. Write

[TABLE]

Then $x^{0}(\xi),\ldots,x^{\kappa}(\xi)$ are the local holomorphic coordinates on the extended complex moduli $\widetilde{\mathcal{M}}_{B}$ and the local homogeneous coordinates on the complex moduli $\mathcal{M}_{B}$; they are called “special homogeneous coordinates” in [13].

Let $V=\left(H^{\oplus(\kappa+1)}\right)\otimes_{\mathbb{Z}}\mathbb{C}$, which is a complex symplectic vector space of dimension $2(\kappa+1)$. The isometry $\gamma:H^{\oplus(\kappa+1)}\to H^{3}(X_{q};\mathbb{Z})/\mathrm{Tor}$ extends to an isomorphism $\gamma:V\to H^{3}(X_{q};\mathbb{C})$ of complex symplectic vector spaces. There is a period map

[TABLE]

More explicitly,

[TABLE]

where $\xi\in\widetilde{\mathcal{T}}$ is any point in the fiber of $\widetilde{\mathcal{T}}\to\mathcal{T}$ over $q\in\mathcal{T}$.

Let $\mathcal{O}_{\mathbb{P}(V)}(-1)$ be the tautological line bundle over $\mathbb{P}(V)$. Then

[TABLE]

Recall the Euler sequence:

[TABLE]

Pulling back the above sequence under $P_{\mathcal{T}}$, we obtain

[TABLE]

Here, $\pi:\mathcal{T}\to\mathcal{M}_{B}$ is the projection from the Torelli space to the complex moduli as before, and

[TABLE]

3.7. The Hodge metric

The symplectic form $Q$ on $H^{3}(Y;\mathbb{R})$ extends to $V=H^{3}(Y;\mathbb{C})$. For $\alpha,\beta\in H^{3}(Y,\mathbb{C})$, define

[TABLE]

By the Hodge-Riemann bilinear relation,

• •

$(\alpha,\alpha)=0$ if $\alpha\in H^{3,0}(Y)\oplus H^{2,1}(Y)$;

• •

$(\alpha,\bar{\alpha})>0$ if $\alpha\in H^{3,0}(Y)$ and $\alpha\neq 0$;

• •

$(\alpha,\bar{\alpha})<0$ if $\alpha\in H^{2,1}(Y)$ and $\alpha\neq 0$.

Define a Hermitian metric on the holomorphic line bundle $\mathcal{L}^{\vee}=\mathbb{F}^{3}$ by

[TABLE]

This is known as the Hodge metric on $\mathbb{F}^{3}$.

Let $\mathbb{H}^{2,1}$ be the complex vector bundle over $\mathcal{M}_{B}$ whose fiber over $q\in\mathcal{M}_{B}$ is $H^{2,1}(Y_{q})$. Then $\mathbb{H}^{2,1}$ is a $C^{\infty}$ complex subbundle of $\mathbb{H}_{\mathbb{C}}$ but not a holomorphic subbundle of $\mathbb{H}_{\mathbb{C}}$. Define the Hodge metric $h$ on $\mathbb{H}^{2,1}$ by

[TABLE]

Then $(\mathbb{H}^{2,1},h)$ is a Hermitian vector bundle of rank $\kappa$ over $\mathcal{M}_{B}$.

3.8. The Chern connection on $\mathcal{L}^{\vee}=\mathbb{F}^{3}$

Let

[TABLE]

be the Chern connection determined by the holomorphic structure and the Hodge metric on $\mathcal{L}^{\vee}=\mathbb{F}^{3}$, and let $D^{\prime}$ and $D^{\prime\prime}$ be the $(1,0)$ and $(0,1)$ parts of $D$, so that

[TABLE]

where $D^{\prime\prime}$ depends on the holomorphic structure. Any $C^{\infty}$ section $s$ of $\mathcal{L}^{\vee}$ is also a $C^{\infty}$ section of $\mathbb{H}_{\mathbb{C}}$, and we have

[TABLE]

since $\mathcal{L}^{\vee}$ is a holomorphic subbundle of $\mathbb{H}_{\mathbb{C}}$.

If $(Y_{q},\Omega_{q})$ is a local holomorphic frame of $\mathcal{L}^{\vee}$ over an open neighborhood $U$ then $D^{\prime\prime}\Omega=0$, and for any tangent vector $v\in T_{q}\mathcal{M}_{B}$, where $q\in U$, we have

[TABLE]

The connection 1-form is

[TABLE]

The curvature form 2-form is

[TABLE]

The right-hand side is independent of the choice of the local holomorphic frame, so $F$ is a global $(1,1)$ form on $\mathcal{M}_{B}$. More explicitly, we define

[TABLE]

by

[TABLE]

Write

[TABLE]

where $(z^{1},\ldots,z^{\kappa})$ are the local holomorphic coordinates on $U\subset\mathcal{M}_{B}$. Define

[TABLE]

Then

[TABLE]

where the fourth equality follows from Equation (3.2), and the fifth (and last) equality follows from the identity $\nabla_{k}\Omega=D_{k}\Omega+\mathbb{D}_{k}\Omega$ where $D_{k}\Omega$ is a (3,0)-form and $\mathbb{D}_{k}\Omega$ is a (2,1)-form.

The above computation shows that the first Chern form

[TABLE]

is a positive $(1,1)$-form.

3.9. The Weil-Petersson metric

The Weil-Petersson metric on $\mathcal{M}_{B}$ is defined by the Kähler form

[TABLE]

In local holomorphic coordinates, the Weil-Petersson metric is given by

[TABLE]

where $g_{i\bar{j}}$ is the Hodge metric on $\mathbb{H}^{2,1}$ and $g_{0\bar{0}}$ is the Hodge metric on $\mathcal{L}^{\vee}=\mathbb{F}^{3}$.

We have an isomorphism of Hermitian vector bundles:

[TABLE]

where the tangent bundle $T\mathcal{M}_{B}$ is equipped with the Weil-Petersson metric $G_{i\bar{j}}$, while $\mathbb{H}^{2,1}$ is equipped with the Hodge metric $g_{i\bar{j}}$, and $\mathcal{L}^{\vee}$ is equipped with the Hodge metric $g_{0\bar{0}}$. The restriction of (3.3) to a point $q\in\mathcal{M}_{B}$ can be identified with

[TABLE]

4. B-model on compact Calabi-Yau threefold

4.1. Genus zero free energy

[TABLE]

are holomorphic sections of the line bundle

[TABLE]

over the Torelli space $\mathcal{T}$. Note that

[TABLE]

is a holomorphic section on $p^{*}\mathcal{L}^{2}\to\mathcal{T}$ and is a multi-valued holomorphic section of $\mathcal{L}^{2}\to\mathcal{M}_{B}$. Define the local holomorphic functions

[TABLE]

where $i=1,\ldots,\kappa$. In particular,

[TABLE]

The functions $(z^{1},\ldots,z^{\kappa})$ are known as special coordinates (or the B-model flat coordinates), which are local holomorphic coordinates on $\mathcal{T}$, defined in an open neighborhood of large complex structure with $A^{0}$ a vanishing cycle. The function $\check{F}_{0}$ is the B-model genus zero free energy.

We may write $\Omega=x^{0}\Omega_{0}$, where

[TABLE]

For $i=1,\ldots,\kappa$,

[TABLE]

Here, $\Omega_{0}$ is a local holomorphic section of $\mathbb{F}^{3}$, and $\nabla_{i}\Omega_{0}$ is a local holomorphic section of $\mathbb{F}^{2}$ by Griffiths transversality, so Hodge-Riemann bilinear relations imply

[TABLE]

To summarize:

• •

The B-model flat coordinates are given by

[TABLE]

• •

The B-model genus zero free energy $\check{F}_{0}$ is defined by

[TABLE]

which satisfies

[TABLE]

4.2. Yukawa coupling

[TABLE]

[TABLE]

[TABLE]

We also have

[TABLE]

So

[TABLE]

Define $C\in\Gamma((T^{*}M)^{3}\otimes\mathcal{L}^{2})$ by

[TABLE]

Then $C_{ijk}$ is symmetric in $i,j,k$, so

[TABLE]

Define

[TABLE]

Then

[TABLE]

4.3. Genus one free energy

The genus one free energy $\check{F}_{1}$ is a linear combination of Ray-Singer torsions. A mathematical definition of $\check{F}_{1}$ is given in [8].

4.4. Genus $g\geq 2$ free energies and the Holomorphic Anomaly Equations

The Weil-Petersson metric is Kähler, so the Chern connection on $T\mathcal{M}_{B}$ defined by the holomorphic structure and the Weil-Petersson metric is also torsion free. We equip $\mathcal{L}^{\vee}=\mathbb{F}^{3}$ with the connection $D$ in Section 3.8. These two connections induce connections on tensor bundles

[TABLE]

for any integers $m,k$. Let $D$ be the covariant derivative defined by these connections.

The special homogeneous coordinates $x^{0},x^{1},\ldots,x^{\kappa}$ are local holomorphic sections of the vacuum line bundle $\mathcal{L}$. For $g\geq 2$,

[TABLE]

The limit

[TABLE]

is a holomorphic function on $U$. The non-holomorphic section $\mathcal{F}_{g}(z,\bar{z})$ satisfies the following Holomorphic Anomaly Equation (BCOV [4]):

[TABLE]

More precisely, we have

[TABLE]

[TABLE]

where $\mathcal{C}$ is defined by Equation (4.2). Using the natural pairing between $T\mathcal{M}_{B}$ and $T^{*}\mathcal{M}_{B}$, we obtain

[TABLE]

With the above notation, the Holomorphic Anomaly Equation (4.3) can be rewritten in the following coordinate-free form:

[TABLE]

4.5. B-model topological open string and the Extended Holomorphic Anomaly Equation

Let

[TABLE]

be the second covariant derivatives of the disk potential, and define

[TABLE]

Then

[TABLE]

If $h>0$ and $2g-2+h>0$, then

[TABLE]

The limit

[TABLE]

is a holomorphic function on $U$. The non-holomorphic section $\mathcal{F}_{(g,h)}(z,\bar{z})$ satisfies the following Extended Holomorphic Anomaly Equation (Walcher [15])

[TABLE]

where the sum $\displaystyle{\sum^{\prime}}$ excludes the unstable case $(g_{i},h_{i})=(0,0),(0,1)$, and the last term on the RHS corresponds to the cases $(g_{1},h_{1})=(0,1)$ or $(g_{2},h_{2})=(0,1)$.

4.6. Mirror symmetry

Under the mirror map

[TABLE]

we have the following mirror conjectures:

• •

For any $i,j,k\in\{1,\ldots,\kappa\}$,

[TABLE]

• •

For $i=1,\ldots,\kappa$,

[TABLE]

or equivalently,

[TABLE]

• •

For $g\geq 2$,

[TABLE]

• •

If $h>0$ and $2g-2+h>0$ then

[TABLE]

Acknowledgments

This note is based on the author’s talks at the 6th Workshop on Combinatorics of Moduli Spaces, Cluster Algebras, and Topological Recursion in Moscow on June 4–9, 2018, and the conference on Crossing the Walls in Enumerative Geometry in Snowbird, Utah on May 21–June 1, 2018. The author sincerely thanks the organizers of these events for the invitation to participate as a speaker. The author also wishes to thank Bohan Fang, Sheldon Katz, Zhengyu Zong, and the anonymous referee for their helpful comments on an earlier version of this note.