$\mathrm{SL}(2,\mathbb{Z})$ modular forms and anomaly cancellation formulas for almost complex manifolds

Yong Wang

Vol. 333 (2024), No. 1, 181–196

DOI: 10.2140/pjm.2024.333.181

Abstract

Building on a kind of elliptic genus for almost complex manifolds introduced by Ping Li and its various properties established by him, we define a generalized elliptic genus where an extra complex bundle is involved. This generalized elliptic genus is a generalized Jacobi form. By this generalized Jacobi form, we can get some $\mathrm{SL}<!--FUNCTION\; APPLICATION-->(2,\mathbb{Z})$ modular forms. By these $\mathrm{SL}<!--FUNCTION\; APPLICATION-->(2,\mathbb{Z})$ modular forms, we get some interesting anomaly cancellation formulas for an almost complex manifold. As corollaries, we get some divisibility results of the holomorphic Euler characteristic number.

Keywords

generalized Jacobi forms, $\mathrm{SL}(2,\mathbb{Z})$ modular forms, anomaly cancellation formulas, divisibility of the holomorphic Euler characteristic number

Mathematical Subject Classification

Primary: 58C20

Milestones

Received: 11 April 2023

Revised: 10 October 2024

Accepted: 30 November 2024

Published: 19 December 2024