Holomorphic anomaly equations and the Igusa cusp form conjecture

Abstract

Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov–Witten theory of the Calabi–Yau threefold $S \times E$ for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture. The proof relies on new results in the Gromov–Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov–Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov–Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi–Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov–Witten theory of K3 surfaces in primitive classes.

Appendices

Appendix A. Elliptic functions and quasimodular forms

1.1 Overview

We prove that for certain multivariate elliptic functions F, the constant term of the Fourier expansion of F (in the elliptic parameter) is a quasimodular form. We also calculate the $C_2$-derivative of these quasimodular forms. In Sect. A.3 we treat the single variable case as a warm-up for the general case which appears in Sect. A.4. The main result of this appendix is Theorem 7.

1.2 Preliminaries

Let $z \in {\mathbb {C}}$ and $\tau \in {\mathbb {H}}$, where ${\mathbb {H}}= \{ z \in {\mathbb {C}}| \mathrm {Im}(z)>0 \}$ is the upper half plane. We will use the auxiliary variables

$$\begin{aligned} w = 2 \pi i z, \quad p = e^{2 \pi i z}, \quad q = e^{2 \pi i \tau }. \end{aligned}$$

The operator of differentiation with respect to z is denoted

$$\begin{aligned} \partial _z = \frac{1}{2 \pi i} \frac{d}{dz} = \frac{d}{dw} = p \frac{d}{dp} \end{aligned}$$

and for the kth derivative of a function f(z) we write

$$\begin{aligned} f^{(k)}(z) = \partial _z^k f(z). \end{aligned}$$

For any meromorphic function f(z) we let $[ f(z) ]_{(z-a)^{\ell }}$ denote the coefficient of $(z-a)^{\ell }$ in the Laurent expansion around a. The residue at a is

$$\begin{aligned} \mathrm {Res}_{z=a} f(z) \, = \, \big [ f(z) \big ]_{(z-a)^{-1}}. \end{aligned}$$

If $f(z) = g(z) h(z)$ where h(z) is regular at a we have

$$\begin{aligned} \mathrm {Res}_{z=a} f(z) = \sum _{k \ge 1} \big [ g(z) \big ]_{(z-a)^{-k}} \frac{(2 \pi i)^{k-1} h^{(k-1)}(a)}{(k-1)!}. \end{aligned}$$

(63)

1.3 Elliptic functions

Consider the Eisenstein series $C_{2k}(\tau )$ defined in (5) as functions on ${\mathbb {H}}$ under the change of variables $q= e^{2 \pi i \tau }$. Consider also the Weierstraß function $\wp (z)$ which has Laurent expansion

$$\begin{aligned} \wp (z) = \frac{1}{12} + \frac{p}{(1-p)^2} + \sum _{d \ge 1} \sum _{k | d} k (p^k - 2 + p^{-k}) q^{d} \end{aligned}$$

(64)

in the region $0< |q|< |p| < 1$, and has Laurent expansion

$$\begin{aligned} \wp (z) = \frac{1}{w^2} + \sum _{k \ge 2} (2k-1) 2k C_{2k}(\tau ) w^{2k-2} \end{aligned}$$

at $w=0$.

Let $\mathsf {E}$ be the ring generated by quasimodular forms and derivatives of the Weierstraß function,

$$\begin{aligned} \mathsf {E}= {\mathbb {Q}}\left[ C_2(\tau ), C_4(\tau ), C_6(\tau ), \wp ^{(k)}(z) \big | \, k \ge 0 \, \right] . \end{aligned}$$

The ring is graded by weight:

$$\begin{aligned} \mathsf {E}= \bigoplus _{k \ge 0} \mathsf {E}_{k}, \end{aligned}$$

where $C_{k}$ has weight k and $\wp ^{(k)}(z)$ has weight $2+k$. We also let

$$\begin{aligned} \frac{d}{dC_2} : \mathsf {E}\rightarrow \mathsf {E}\end{aligned}$$

be the formal differentiation with respect to the generator $C_2$.¹⁸

Every $F(z) \in \mathsf {E}$ admits a Fourier expansion in the region $0< |q|< |p|<1$,

$$\begin{aligned} F(z) = \sum _{n \in {\mathbb {Z}}} a_n(\tau ) p^n. \end{aligned}$$

The constant term in the expansion is denoted by

$$\begin{aligned} \big [ F(z) \big ]_{p^0} = a_0(\tau ). \end{aligned}$$

As a warm-up for the general case we prove the following proposition.

Proposition 8

For every $F \in \mathsf {E}_k$ the series $\big [ F \big ]_{p^0}$ is a quasimodular form of weight k and we have

$$\begin{aligned} \frac{d}{dC_2} \Big [ \, F \, \Big ]_{p^0} = \left[ \frac{d}{dC_2} F \right] _{p^0} - 2 \big [ \, F \, \big ]_{w^{-2}}. \end{aligned}$$

Consider the function

$$\begin{aligned} \mathsf {A}(z)&= - \frac{1}{2} - \sum _{m \ne 0} \frac{p^m}{1-q^m} \\&= \frac{1}{w} - \sum _{\ell \ge 1} 2\ell C_{2\ell }(q) w^{2 \ell - 1}, \end{aligned}$$

where the expansion in p, q is taken in the region $0< |q|< |p| < 1$. For the proof of the Proposition we require the following Lemma.

Lemma 18

$\mathsf {A}(z + \lambda \tau + \mu ) = \mathsf {A}(z) - \lambda$ for every $\lambda , \mu \in {\mathbb {Z}}$.

Proof

We have $\mathsf {A}(z) = \partial _z \log \Theta (z)$ where $\Theta$ is the Jacobi theta function

$$\begin{aligned} \Theta (z) = (p^{1/2} - p^{-1/2}) \prod _{m \ge 1} \frac{ (1-pq^m) (1-p^{-1}q^m)}{ (1-q^m)^2 }. \end{aligned}$$

A direct check using this definition shows

$$\begin{aligned} \Theta (z + \lambda \tau + \mu ) = (-1)^{\lambda + \mu } p^{-\lambda } q^{-\lambda ^2/2} \Theta (z) \end{aligned}$$

for all $\lambda , \mu \in {\mathbb {Z}}$ which implies the claim. $\square$

Proof of Proposition 8

We have

$$\begin{aligned} \big [ F(z) \big ]_{p^0} = \int _{\mathsf {C}_a} F(z) \mathop {}\!\mathrm {d}{z} \end{aligned}$$

where $\mathsf {C}_a$ is the line segment from a to $a+1$ for some $a \in {\mathbb {C}}$ with $0< \mathrm {Im}(a) < \mathrm {Im}(\tau )$. Since F(z) is periodic, i.e.

$$\begin{aligned} F(z + \lambda \tau + \mu ) = F(z) \end{aligned}$$

for every $\lambda , \mu \in {\mathbb {Z}}$, we may instead assume $-\mathrm {Im}(\tau )< \mathrm {Im}(a) < 0$.

By Lemma 18 the function $f(z) = F(z) \cdot \mathsf {A}(z)$ satisfies

$$\begin{aligned} f(z+1) = f(z), \quad f(z+\tau ) = f(z) - F(z). \end{aligned}$$

Hence we may replace the integral of F over $\mathsf {C}_a$ by the integral of $F \cdot \mathsf {A}$ over the boundary of the fundamental domain $B_a$ depicted in Fig. 2,

$$\begin{aligned} \big [ F(z) \big ]_{p^0} = \oint _{B_a} F(z) \cdot \mathsf {A}(z) \mathop {}\!\mathrm {d}{z}. \end{aligned}$$

Fig. 2

The closed path $B_{a}$

Since both F and $\mathsf {A}$ have poles inside $B_a$ only at 0, an application of the residue theorem gives

$$\begin{aligned} \big [ F(z) \big ]_{p^0}&= \big [ F(z) \cdot \mathsf {A}(z) \big ]_{w^{-1}} \\&= [ F ]_{w^0} - \sum _{\ell \ge 1} 2 \ell C_{2\ell }(\tau ) [ F(z) ]_{w^{-2 \ell }}. \end{aligned}$$

An inspection of the Laurent series of F(z) yields now both claims. $\square$

1.4 Multiple variables

Let $n \ge 2$ and $z = (z_1, \ldots , z_n) \in {\mathbb {C}}^n$, and denote

$$\begin{aligned} w_a = 2 \pi i z_a, \quad p_a = e^{2 \pi i z_a}, \quad a \in \{ 1, \ldots , n \}. \end{aligned}$$

Every permutation $\sigma \in S_n$ determines a region $U_{\sigma } \subset {\mathbb {C}}^n$ by requiring

$$\begin{aligned} \mathrm {Im}(\tau )> \mathrm {Im}(z_a - z_b) > 0 \end{aligned}$$

(65)

whenever $\sigma (a) > \sigma (b)$, or equivalently by

$$\begin{aligned} \mathrm {Im}(z_{\sigma ^{-1}(n)})> \cdots> \mathrm {Im}(z_{\sigma ^{-1}(1)}) > \mathrm {Im}(z_{\sigma ^{-1}(n)} - \tau ). \end{aligned}$$

(66)

Consider the ring of multivariate elliptic functions

$$\begin{aligned} \mathsf {ME}= {\mathbb {Q}}\left[ \, C_2(\tau ), C_4(\tau ), C_6(\tau ), \wp ^{(k)}( z_a - z_b )\ \big |\ k \ge 0,\, 1 \le a < b \le n \, \right] . \end{aligned}$$

We assign $\wp ^{(k)}$ and $C_k$ the weights $2+k$ and k respectively and let

$$\begin{aligned} \mathsf {ME}= \bigoplus _{k \ge 0} \mathsf {ME}_k \end{aligned}$$

be the induced grading by weight k. Let

$$\begin{aligned} \frac{d}{dC_2} : \mathsf {ME}\rightarrow \mathsf {ME}\end{aligned}$$

be the formal differentiation with respect to the generator $C_2$.

Every $F \in \mathsf {ME}$ has a well-defined Fourier expansion in the region $U_{\sigma }$,

$$\begin{aligned} F = \sum _{k_1, \ldots , k_n \in {\mathbb {Z}}} a_{k_1 \ldots k_n}(\tau ) p_1^{k_1} \ldots p_n^{k_n}, \quad (z_1, \ldots , z_n) \in U_{\sigma }. \end{aligned}$$

The constant coefficient in this expansion, i.e. the coefficient of $\prod _i p_i^0$, is denoted

$$\begin{aligned} \left[ F \right] _{p^0, \sigma } = a_{0\ldots 0}(\tau ). \end{aligned}$$

Define the constant coefficient of F averaged over all permutation $\sigma$,

$$\begin{aligned} \left[ F(z_1, \ldots , z_n) \right] _{p^0} = \frac{1}{n!} \sum _{\sigma \in S_n} \left[ F(z_1, \ldots , z_n) \right] _{p^0, \sigma }. \end{aligned}$$

(67)

The following is the main result of this appendix.¹⁹

Theorem 7

Let $F \in \mathsf {ME}_k$. Then the following holds.

• (1) $\big [ F(z) \big ]_{p^0, \sigma } \in \mathsf {QMod}_{\le k}$ for every permutation $\sigma$.

• (2) $\big [ F(z) \big ]_{p^0} \in \mathsf {QMod}_k$.

• (3) We have

  $$\begin{aligned} \frac{d}{dC_2} \Big [ F(z) \Big ]_{p^0} = \left[ \frac{d}{dC_2} F \right] _{p^0} - \sum _{\begin{array}{c} a,b = 1 \\ a \ne b \end{array}}^{n} \left[ (2 \pi i)^2 \mathrm {Res}_{z_a = z_b}\Big ( (z_a - z_b) \cdot F \Big ) \right] _{p^0}. \end{aligned}$$

1.5 Preparations for the proof

We prove a series of results leading up to the proof of Theorem 7 in Sect. A.6.

Lemma 19

Let $F \in \mathsf {ME}$ and $\sigma \in S_n$. Then

$$\begin{aligned}{}[ F ]_{p^0, \sigma } = [ F ]_{p^0, \widetilde{\sigma }} \end{aligned}$$

for every cyclic permutation $\widetilde{\sigma }$ of $\sigma$.

Proof

Let $(a_1, \ldots , a_n) \in U_{\sigma }$ and let $\mathsf {C}_{a_i}$ be the line segment from $a_i$ to $a_i + 1$ in the $z_i$-plane. Then

$$\begin{aligned}{}[ F ]_{p^0, \sigma } = \int _{\mathsf {C}_{a_1}} \cdots \int _{\mathsf {C}_{a_n}} F(z_1, \ldots , z_n) \mathop {}\!\mathrm {d}{z_n} \cdots \mathop {}\!\mathrm {d}{z_{1}}. \end{aligned}$$

Since F is periodic, i.e. $F(z + \lambda \tau + \mu ) = F(z)$ for every $\lambda , \mu \in {\mathbb {Z}}^n$, we may replace the integral over $\mathsf {C}_{a_{\sigma ^{-1}(n)}}$ by the integral over $\mathsf {C}_{a_{\sigma ^{-1}(n)}-\tau }$. But comparing with (66) this corresponds to taking the constant coefficient of F with respect to a cyclic permutation of $\sigma$. $\square$

For every $a \ne b$ let $R_{ab}$ denote the operation of taking the residue in $z_a = z_b$ written as a right operator,

$$\begin{aligned} f(z_1, \ldots , z_n) R_{ab} := 2 \pi i \cdot \mathrm {Res}_{z_a = z_b} f(z_1, \ldots , z_n). \end{aligned}$$

We also write

$$\begin{aligned} \mathsf {A}_{ab} = \mathsf {A}(z_a - z_b). \end{aligned}$$

Lemma 20

Let $F(z) \in \mathsf {ME}_k$, and let $i_1, \ldots , i_m \in \{ 1, \ldots , n \}$ be pairwise distinct. Then for any $r \ge 0$ we have

$$\begin{aligned} \big ( F(z) \mathsf {A}_{i_1 i_{m}}^{r} \big ) R_{i_1 i_2} R_{i_2 i_3} \cdots R_{i_{m-1} i_{m}} \in \mathsf {ME}_{k+r-(m-1)}. \end{aligned}$$

Proof

Let $F(z) \in \mathsf {ME}_k$ be a monomial in the generators and consider the splitting

$$\begin{aligned} F(z) = F_{ab}(z_a - z_b) \cdot \widetilde{F}_{ab}(z), \end{aligned}$$

where $F_{ab}$ is the product of all factors in F of the form $\wp ^{(s)}(z_a - a_b)$ for some s. In particular, $\widetilde{F}_{ab}(z)$ is regular at $z_a = z_b$.

Consider the action of $R_{ab}$ on $F(z) \mathsf {A}_{ac}^r$. If $b \ne c$ we have

$$\begin{aligned} (F \mathsf {A}_{ac}^r) R_{a b} = \sum _{\ell \ge 1} \left[ F_{ab} \right] _{(w_a - w_b)^{-\ell }} \frac{1}{(\ell - 1)!} \partial _{z_a}^{\ell -1} \left( \widetilde{F}_{ab} \mathsf {A}^r_{ac} \right) \Big |_{z_a = z_b}, \end{aligned}$$

(68)

where we have used (63). Since

$$\begin{aligned} \partial _z \mathsf {A}(z) = - \wp (z) - 2 C_2(\tau ) \end{aligned}$$

the right hand side of (68) can be written as a sum of terms

$$\begin{aligned} F'(z) \cdot \mathsf {A}^{r'}_{bc}, \end{aligned}$$

where $F' \in \mathsf {ME}_{k'}$ with $k' + r' = k+r-1$. Similarly, if $b = c$ we have

$$\begin{aligned} ( F(z) \mathsf {A}_{ac}^r ) R_{a c} \in \mathsf {ME}_{k+r-1}. \end{aligned}$$

The claim follows from the steps above and an induction argument. $\square$

Let $\sigma \in S_n$ be a permutation, let

$$\begin{aligned} g_{ab} = {\left\{ \begin{array}{ll} 1 &{} \text { if } \sigma (a) > \sigma (b), \\ 0 &{} \text { otherwise}, \end{array}\right. } \end{aligned}$$

and for all $x \in {\mathbb {C}}$ and non-negative integers a define

$$\begin{aligned} \left( {\begin{array}{c}x\\ a\end{array}}\right) = \frac{x \cdot (x-1) \cdots (x-a+1)}{a!}. \end{aligned}$$

Proposition 9

Let $F(z) \in \mathsf {ME}$. Then

$$\begin{aligned}&\big [ F(z) \big ]_{p^0, \sigma } = \sum _{\ell \ge 1} \sum _{i_1, i_2, \ldots , i_{\ell }}\\&\quad \times \left[ F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + \ell - 2 - g_{i_1 i_2} - \ldots - g_{i_{\ell -1} i_{\ell }} \\ \ell -1\end{array}}\right) R_{i_1 i_2} \cdots R_{i_{\ell -1} i_{\ell }} \right] _{p^0, \sigma }, \end{aligned}$$

where the inner sum is over all non-recurring²⁰ sequences $i_1, \ldots , i_{\ell } \in \{ 1, \ldots , n \}$ with endpoints $i_1=1$ and $i_{\ell }=n$.

Proof

We argue by induction on L that for every $L \ge 1$ we have

$$\begin{aligned}&\big [ F(z) \big ]_{p^0, \sigma } =\sum _{\ell = 1}^{L} \sum _{i_1, i_2, \ldots , i_{\ell }}\nonumber \\&\quad \times \left[ F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + \ell - 2 - g_{i_1 i_2} - \ldots - g_{i_{\ell -1} i_{\ell }} \\ \ell -1\end{array}}\right) R_{i_1 i_2} \cdots R_{i_{\ell -1} i_{\ell }} \right] _{p^0, \sigma },\nonumber \\ \end{aligned}$$

(69)

where the inner sum runs over all non-recurring sequences $(i_1, \ldots , i_{\ell })$ such that $i_1 = 1$ and the following holds:

• if $\ell < L$ then $i_{\ell } = n$,

• if $\ell = L$ and $i_r = n$ then $r=\ell$.

If $L = 1$ equality (69) holds by definition. Hence we may assume the claim holds for $L \ge 1$ and we show the case $L+1$. Every summand on the right hand side of (69) with $i_{\ell } \ne n$ is equal to the $p^0$-coefficient (in $U_{\sigma }$) of

$$\begin{aligned} \int _{\mathsf {C}_{a}} F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + \ell - 2 - g_{i_1 i_2} - \cdots - g_{i_{\ell -1} i_{\ell }} \\ \ell - 1\end{array}}\right) R_{i_1 i_2} \cdots R_{i_{\ell -1} i_{\ell }} \mathop {}\!\mathrm {d}{z_{i_{\ell }}} \end{aligned}$$

(70)

for some $a \in {\mathbb {C}}$ such that

$$\begin{aligned} (z_1, \ldots , z_{i_{\ell }-1}, a, z_{i_{\ell }+1}, \ldots , z_n) \in U_{\sigma } \end{aligned}$$

and $\mathsf {C}_a$ is the line segment from a to $a+1$ in the $z_{i_{\ell }}$-plane. Define the function

$$\begin{aligned} \mathsf {H}(z) = F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + \ell - 1 - g_{i_1 i_2} - \cdots - g_{i_{\ell -1} i_{\ell }} \\ \ell \end{array}}\right) R_{i_1 i_2} \cdots R_{i_{\ell -1} i_{\ell }} \end{aligned}$$

Using Lemma 18 and $\mathrm {Res}_{z=r+s} f(z) = \mathrm {Res}_{z=r}f(z+s)$ repeatedly we find

$$\begin{aligned}&\mathsf {H}(z_1, \ldots , z_{i_{\ell }} + \tau , \ldots , z_n) = \mathsf {H}(z_1, \ldots , z_n) \\&\quad - F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + \ell - 2 - g_{i_1 i_2} - \cdots - g_{i_{\ell -1} i_{\ell }} \\ \ell - 1\end{array}}\right) R_{i_1 i_2} \cdots R_{i_{\ell -1} i_{\ell }}. \end{aligned}$$

Hence arguing as in the proof of Proposition 8 we may replace (70) by an integral of $\mathsf {H}(z)$ over the box $B_{a}$ depicted in Fig. 2. The function $\mathsf {H}$ has possible poles inside $B_{a}$ only at the points²¹

$$\begin{aligned} z_{i_{\ell }} = z_{i_{\ell +1}} + g_{i_{\ell } i_{\ell +1}} \tau \end{aligned}$$

for some $i_{\ell +1} \notin \{ i_1, \ldots , i_{\ell } \}$. By the residue theorem (70) is therefore

$$\begin{aligned} 2 \pi i \sum _{i_{\ell +1} \notin \{ i_1, \ldots , i_{\ell } \}} \mathrm {Res}_{z_{i_{\ell }} = z_{i_{\ell +1}} + g_{i_{\ell } i_{\ell +1}} \tau } \mathsf {H}(z), \end{aligned}$$

which after moving the shift by $g_{i_{\ell } i_{\ell +1}} \tau$ inside simplifies to

$$\begin{aligned} \sum _{i_{\ell +1} \notin \{ i_1, \ldots , i_{\ell } \}} F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + \ell - 1 - \sum _{a=1}^{\ell } g_{i_a i_{a+1}} \\ \ell \end{array}}\right) R_{i_1 i_2} \cdots R_{i_{\ell -1} i_{\ell }} R_{i_{\ell } i_{\ell +1}}. \end{aligned}$$

Plugging back into (69) we obtain the case $L+1$. The induction is complete. $\square$

Averaging Proposition 9 over all permutations $\sigma$ yields the following.

Proposition 10

Let $F(z) \in \mathsf {ME}$. Then

$$\begin{aligned} \big [ F(z) \big ]_{p^0} = \sum _{m \ge 1} \sum _{i_1=1, i_2, \ldots , i_{m+1}=n} \left[ \left( F \cdot \frac{\mathsf {A}_{1n}^{m}}{m!}\right) R_{i_1 i_2} R_{i_2 i_3} \cdots R_{i_m i_{m+1}} \right] _{p^0}, \end{aligned}$$

where the inner sum runs over all non-recurring sequences $i_1, \ldots , i_{\ell +1} \in \{ 1, \ldots , n \}$ with endpoints $i_1=1$ and $i_{\ell +1}=n$.

Proof

Setting $\ell = m+1$ in Proposition 9 yields

$$\begin{aligned}&\big [ F(z) \big ]_{p^0, \sigma } = \sum _{m \ge 1} \sum _{i_1, i_2, \ldots , i_{m+1}} \\&\quad \times \left[ F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + m - 1 - g_{i_1 i_2} - \cdots - g_{i_{m} i_{m+1}} \\ m\end{array}}\right) R_{i_1 i_2} \cdots R_{i_{m} i_{m+1}} \right] _{p^0, \sigma }, \end{aligned}$$

where the non-recurring sequence $(i_1, \ldots , i_{m+1})$ satisfies $i_1 = 1$, $i_{m+1} = n$.

We sum the previous equation over all permutations $\sigma \in S_n$. By Lemmas 19 and 20 it is enough to sum over all $\sigma$ with $\sigma (n) = n$. It follows $g_{i_{m} i_{m+1}} = 0$ above. We then split the sum over all such $\sigma$ into a sum over orderings $\rho$ of the variables $z_{i}, i \notin \{ i_1, \ldots , i_{m}, n \}$, a sum over orderings $\tau \in S_{m}$ of the variables $z_{i_1}, \ldots , z_{i_{m}}$ and the $\left( {\begin{array}{c}n-1\\ m\end{array}}\right)$ refinements of both orderings. Since

$$\begin{aligned} F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + m - 1 - g_{i_1 i_2} - \cdots - g_{i_{m-1} i_{m}} \\ m\end{array}}\right) R_{i_1 i_2} \cdots R_{i_{m} i_{m+1}} \end{aligned}$$

depends only on the variables $z_i$ with $i \notin \{ i_1, \ldots , i_{m} \}$ we find

$$\begin{aligned}&\big [ F(z) \big ]_{p^0} = \sum _{m \ge 1} \sum _{i_1, i_2, \ldots , i_{m+1}} \sum _{\rho \in S_{n-m-1}} \frac{1}{(n-1)!} \cdot \left( {\begin{array}{c}n-1\\ m\end{array}}\right) \\&\quad \cdot \left[ \sum _{\tau \in S_{m}} F \cdot \left( {\begin{array}{c} \mathsf {A}_{1n} + m - 1 - g_{i_1 i_2} - \cdots - g_{i_{m-1} i_{m}} \\ m\end{array}}\right) R_{i_1 i_2} \cdots R_{i_{m} i_{m+1}} \right] _{p^0, \tilde{\rho }} \end{aligned}$$

where $\tilde{\rho }$ is any fixed refinement of the ordering $\rho$. The proposition follows now by an application of Worpitzky’s identity

$$\begin{aligned} \sum _{\tau \in S_{m}} \left( {\begin{array}{c}x+m-1-a_{\tau }\\ m\end{array}}\right) = x^m, \end{aligned}$$

where $a_{\tau }$ is the number of ascents of $\tau$, i.e. the number of $i \in \{ 1, \ldots , \ell -1 \}$ with $\tau (i+1) > \tau (i)$. $\square$

Lemma 21

The action of the residue operators $R_{ab}$ on meromorphic functions of variables $z_1,\ldots ,z_n$ with poles only along $z_i-z_j = 0$ for $i<j$ satisfy

$$\begin{aligned} R_{ab} R_{cb} = R_{cb} R_{ab} + R_{ca} R_{ab}, \ \quad \ R_{ab} R_{bc} = -R_{ba} R_{ac} \end{aligned}$$

for all pairwise distinct a, b, c.

Proof

We may assume that

$$\begin{aligned} f(z) = \prod _{1 \le i< j < n} (z_i - z_j)^{m_{ij}} \end{aligned}$$

for some $m_{ij} \in {\mathbb {Z}}$. The claim follows then from a direct calculation. $\square$

1.6 Proof of Theorem 7

We prove the quasimodularity of $[F]_{p^0, \sigma }$, the homogeneity of $[F]_{p^0}$, and the formula

$$\begin{aligned} \begin{aligned} \frac{d}{dC_2} \Big [ F(z) \Big ]_{p^0} =&\, \left[ \frac{d}{dC_2} F \right] _{p^0} - 2\sum _{a< b = n} \left[ (w_a - w_b) F R_{ab} \right] _{p^0} \\&- 2 \sum _{a< b < n} \left[ (w_b - w_a) F R_{ba} \right] _{p^0}, \end{aligned} \end{aligned}$$

(71)

which implies the formula in the Theorem by symmetrization over $S_n$. We argue by induction on n, the number of variables $z_i$ on which F depends.

If $n = 1$, then F is a quasimodular form and all three statements hold by inspection. Assume the statement is known for all functions which depend on a smaller number of variables. By Proposition 10, we have

$$\begin{aligned} \Big [ F(z) \Big ]_{p^0} = \sum _{m \ge 1} \sum _{i_1=1, i_2, \ldots , i_{m+1}=n} \left[ \left( F \frac{\mathsf {A}_{1n}^{m}}{m!}\right) R_{i_1 i_2} R_{i_2 i_3} \cdots R_{i_m i_{m+1}} \right] _{p^0}. \end{aligned}$$

Each summand on the right side depends on fewer variables than F and is therefore a quasi-modular form of weight k by Lemma 20 and induction. To obtain (71) we apply the $\frac{d}{dC_2}$ operator, use induction on the right side, and use Lemma 21 to commute the resulting $R_{ab}$ operators past the $R_{i_k i_{k+1}}$ operators. This yields (71) also for F. The quasimodularity of $[F]_{p^0, \sigma }$ (and the weight bound) follows similarly from Lemma 20 and Proposition 9. $\square$

Appendix B. Elliptic fibrations

1.1 Overview

We present a refinement of Conjecture B by weight, and give evidence in the case of elliptic Calabi–Yau threefolds in fiber classes.

1.2 Weight refinement

Let $\pi : X \rightarrow B$ be an elliptic fibration with a section and integral fibers. The holomorphic anomaly equation of Conjecture B and the argument used in the proof of Corollary 1 yield a refinement of Conjecture A by weight as follows.

Recall the divisor class W defined in Sect. 0.5. The endomorphisms of $H^{*}(X)$ defined by

$$\begin{aligned} T_+(\alpha ) = ( \pi ^{*} \pi _{*} \alpha ) \cup W, \quad T_-(\alpha ) = \pi ^{*} \pi _{*} ( \alpha \cup W ) \end{aligned}$$

satisfy $T_+^2 = T_+$ and $T_-^2 = T_-$ as well as $T_+ T_- = T_- T_+ = 0$. Hence the cohomology of X splits as

$$\begin{aligned} H^{*}(X) = \mathrm {Im}(T_+) \oplus \mathrm {Im}(T_-) \oplus \big ( \mathrm {Ker}( T_+ ) \cap \mathrm {Ker}(T_-) \big ) . \end{aligned}$$

Define a modified degree function $\underline{\deg }(\gamma )$ by the assignment

$$\begin{aligned} \underline{\deg }(\gamma ) = {\left\{ \begin{array}{ll} 2 &{} \text {if } \gamma \in \mathrm {Im}(T_+) \\ 1 &{} \text {if } \gamma \in \mathrm {Ker}( T_+ ) \cap \mathrm {Ker}(T_-) \\ 0 &{} \text {if } \gamma \in \mathrm {Im}(T_-). \\ \end{array}\right. } \end{aligned}$$

If X is an elliptic curve and B is a point then $\underline{\deg }$ specializes to the real cohomological degree $\deg _{{\mathbb {R}}}$.

Corollary* 3

Assume Conjectures A and B hold. Then for any $\underline{\deg }$-homogeneous classes $\gamma _1, \ldots , \gamma _n \in H^{*}(X)$ we have

$$\begin{aligned} {\mathcal C}^{\pi }_{g, \mathsf {k}}( \gamma _1, \ldots , \gamma _n ) \in H_{*}({\overline{M}}_{g,n}(B, \mathsf {k})) \otimes \frac{1}{\Delta (q)^{m}} \mathsf {QMod}_{\ell }, \end{aligned}$$

where $m = -\frac{1}{2} c_1(N_{\iota }) \cdot \mathsf {k}$ and $\ell = 2g - 2 + 12m + \sum _i \underline{\deg }(\gamma _i)$.

1.3 An example

Let X be a Calabi–Yau threefold and let $\pi : X \rightarrow B$ be an elliptic fibration with section and integral fibers over a Fano surface B. We consider the genus g Gromov–Witten potentials in fiber classes

$$\begin{aligned} F_g(q) = \sum _{d = 0}^{\infty } q^d \int _{[ {\overline{M}}_{g,0}(X,dF) ]^{\text {vir}}} 1 \end{aligned}$$

with the convention that the summation starts at $d=1$ if $g \in \{ 0, 1 \}$. By Toda’s calculation [46, Thm 6.9], the Pandharipande–Thomas invariants $\mathsf {P}_{n,\beta }$ of X in fiber classes form the generating series

$$\begin{aligned} \sum _{d = 0}^{\infty } \sum _{n \in {\mathbb {Z}}} \mathsf {P}_{n, dF} y^n q^d = \prod _{\ell , m \ge 1} (1- (-y)^{\ell } q^{m} )^{-\ell \cdot e(X)} \cdot \prod _{m \ge 1} (1-q^m)^{-e(B)} . \end{aligned}$$

Assuming X satisfies the Gromov–Witten/Pairs correspondence [41, 42], we therefore obtain

$$\begin{aligned} F_0(q)&= - e(X) \sum _{m, a \ge 1} \frac{1}{a^3} q^{ma} \\ F_1(q)&= \left( e(B) - \frac{1}{12} e(X) \right) \sum _{m,a \ge 1} \frac{1}{a} q^{ma} \\ F_g(q)&= e(X) \frac{(-1)^g B_{2g}}{4g} C_{2g-2}(q), \quad g \ge 2. \end{aligned}$$

If $g \ge 2$ the series

$$\begin{aligned} \int {\mathcal C}^\pi _{g,0}() = F_g(q) \end{aligned}$$

is quasimodular of weight $2g-2$ in agreement with Corollary* 3.

In genus $g \le 1$ the series $F_0$ and $F_1$ are not quasimodular forms. However, this does not contradict Corollary* 3 since the moduli spaces ${\overline{M}}_{g,0}(\mathbb {P}^1,0)$ are unstable here and ${\mathcal C}^\pi _g()$ is not defined. Instead, we need to add additional insertions to stabilize the moduli space. In genus 0 we obtain

$$\begin{aligned} \int {\mathcal C}^\pi _{0,0}(W,W,W)&= \int _X W^3 + \left( q \frac{d}{dq} \right) ^3 F_0(q) = -12 e(X) C_4(q), \\ \int {\mathcal C}^\pi _{0,0}(\pi ^{*}D,W,W)&= \int _X \pi ^{*} D \cup W^2 = 0,\\ \int {\mathcal C}^\pi _{0,0}(\pi ^{*}D, \pi ^{*} D',W)&= \int _X \pi ^{*} D \cup \pi ^{*} D' \cup W = \int _B D \cdot D', \end{aligned}$$

for any $D, D' \in H^{2}(B)$, where in the first equality we used

$$\begin{aligned} e(X) = -60 \int _{B} K_B^2. \end{aligned}$$

All three evaluations are in perfect agreement with Corollary* 3.

In genus 1 we obtain agreement with Corollary* 3 by

$$\begin{aligned} \int {\mathcal C}^{\pi }_{1,0}(W)&= \int _{{\overline{M}}_{1,1}(X,0)} {\text {ev}}_1^{*}(W) + \left( q \frac{d}{dq} \right) F_1(q) \\&= \left( e(B) -\frac{1}{12} e(X) \right) C_2(q), \end{aligned}$$

where we used

$$\begin{aligned} c_2(X) = \pi ^{*} c_2(B) + 11 \pi ^{*} c_1(B)^2 + 12 \iota _{*} c_1(B). \end{aligned}$$

A direct check shows that all evaluations above are also compatible with the conjectured holomorphic anomaly equation. For example, in genus 1 Conjecture B predicts correctly

$$\begin{aligned} \frac{d}{dC_2} \int {\mathcal C}^{\pi }_{1,0}(W)&= \int {\mathcal C}_{0,0}^{\pi }(W, \Delta _B) - 2 \int {\mathcal C}_{1,0}(\mathsf {1}) \psi _1 \\&= e(B) - 2 \int _{{\overline{M}}_{1,1}} \psi _1 \int _X c_3(X) \\&= e(B) - \frac{1}{12} e(X). \end{aligned}$$