# Regularized Integrals on Elliptic Curves and Holomorphic Anomaly Equations

@inproceedings{Li2022RegularizedIO, title={Regularized Integrals on Elliptic Curves and Holomorphic Anomaly Equations}, author={Si Li and Jie Zhou}, year={2022} }

We derive residue formulas for the regularized integrals (introduced in [LZ21]) on conﬁguration spaces of elliptic curves. Based on these formulas, we prove that the regularized integrals satisfy holomorphic anomaly equations, providing a mathematical formulation of the so-called contact term singularities. We also discuss residue formulas for the ordered A -cycle integrals and establish their relations with those for the regularized integrals.

## References

SHOWING 1-10 OF 17 REFERENCES

### Regularized Integrals on Riemann Surfaces and Modular Forms

- MathematicsCommunications in Mathematical Physics
- 2021

We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms…

### A Generalized Jacobi Theta Function and Quasimodular Forms

- Mathematics
- 1995

In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \( {\tilde…

### Tropical Mirror Symmetry for Elliptic Curves

- Mathematics
- 2017

Mirror symmetry relates Gromov-Witten invariants of an elliptic curve with certain integrals over Feynman graphs. We prove a tropical generalization of mirror symmetry for elliptic curves, i.e., a…

### Nearly Overconvergent Modular Forms

- Mathematics
- 2014

We introduce and study finite slope nearly overconvergent (elliptic) modular forms. We give an application of this notion to the construction of the Rankin-Selberg p-adic L-function on the product of…

### Holomorphic anomaly equations and the Igusa cusp form conjecture

- Mathematics
- 2017

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 $$S \times E$$S×E for all curve classes which are primitive in the K3…

### Counting Feynman-like graphs: Quasimodularity and Siegel–Veech weight

- MathematicsJournal of the European Mathematical Society
- 2019

We prove the quasimodularity of generating functions for counting torus covers, with and without Siegel-Veech weight. Our proof is based on analyzing decompositions of flat surfaces into horizontal…

### Mirror Symmetry and Elliptic Curves

- Mathematics
- 1995

I review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: (1) counting functions of holomorphic curves…

### Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes

- Mathematics
- 1994

We develop techniques to compute higher loop string amplitudes for twistedN=2 theories withĉ=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in…