Refinement and modularity of immortal dyons

TL;DR

This paper explores the universal structure of the holomorphic anomaly equation for refined BPS indices in string theory, unifying known modularity results across different supersymmetry levels and extending to non-trivial torsion cases.

Contribution

It proposes a universal holomorphic anomaly equation for refined BPS indices that applies across various supersymmetry settings and generalizes existing modularity results.

Findings

Recovered known modularity of quarter-BPS dyons in ${ m N}=4$ string theory

Extended the holomorphic anomaly to cases with non-trivial torsion

Unified the structure of BPS state modularity across different ${ m N}$ levels

Abstract

Extending recent results in ${\cal N}=2$ string compactifications, we propose that the holomorphic anomaly equation satisfied by the modular completions of the generating functions of refined BPS indices has a universal structure independent of the number ${\cal N}$ of supersymmetries. We show that this equation allows to recover all known results about modularity (under $SL(2,\mathbb{Z})$ duality group) of BPS states in ${\cal N}=4$ string theory. In particular, we reproduce the holomorphic anomaly characterizing the mock modular behavior of quarter-BPS dyons and generalize it to the case of non-trivial torsion invariant.