12 problems

Holomorphic anomaly equations. Assuming the quasi-Jacobi form property,

Primitive conjecture.

Fiber-class conjecture. Up to a constant, $F^{S^{[n]}}_{g,0}$ is a meromorphic quasi-Jacobi form of weight $k$ and index $0$, with poles at torsion points. Modulo constants it also…

Modularity conjecture for refined BPS generating functions. The functions $h^{\rm ref}_{p,\mu}$ transform as vector-valued higher-depth Jacobi forms of the stated weight and index

Let $S$ be a K3 surface, let $g\geq 0$, let $m$ be a positive integer, let $\alpha\in R^*(\overline{M}_{g,n})$ be a tautological class, and let…

Yamaguchi–Yau's holomorphic anomaly equations. The potentials satisfy

Let $\mathsf{A}=E_1\times E_2$ be an abelian surface fibered over $E_1$, and let ${\mathcal A}_g(\gamma_1,\ldots,\gamma_n)$ be its reduced primitive Gromov–Witten class. Define…

Let $\pi:X\to B$ be an elliptic fibration, let $\mathsf{k}\in H_2(B,\mathbb{Z})$, and let ${\mathcal C}^{\pi}_{g,\mathsf{k}}(\gamma_1,\ldots,\gamma_n)$ be the associated relative G…

Let $S$ be an elliptic K3 surface with section and let ${\mathcal K}_g(\gamma_1,\ldots,\gamma_n)$ be the reduced relative generating series for the classes $\beta_h=B+hF$. Let…

Let $X\to B$ be an elliptic fibration with section, and let ${\mathcal C}^{\pi}_{g,\mathsf{k}}(\gamma_1,\ldots,\gamma_n)$ be the associated relative Gromov--Witten generating class…

Let $(X,X^\vee)$ be a mirror pair of Calabi–Yau threefolds, and choose a large complex structure limit on the complex moduli space $\mathcal{M}$ of $X$. Let $\mathcal{L}$ be the va…

Let $Z_{g;n}$ be the quasi-modular-form Gromov–Witten potentials satisfying the recursion relation … Let ${\mathcal F}^{(g)}$ denote the genus-$g$ BCOV free energy, and let the BCO…