Special geometry, quasi-modularity and attractor flow for BPS structures

TL;DR

This paper explores the mathematical structures of BPS moduli spaces in $ =2$ theories, linking special geometry, quasi-modularity, and attractor flows, especially for rank two charge lattices and their associated modular curves.

Contribution

It introduces a BPS variation of Hodge structure framework that connects special K"ahler geometry, Picard-Fuchs equations, and quasi-modular forms for specific rank two BPS structures.

Findings

Central charges expressed as quasi-modular forms.

Identified moduli spaces with modular curves.

Attractor flows determine BPS spectra.

Abstract

We study mathematical structures on the moduli spaces of BPS structures of $\mathcal{N}=2$ theories. Guided by the realization of BPS structures within type IIB string theory on non-compact Calabi-Yau threefolds, we develop a notion of BPS variation of Hodge structure which gives rise to special K\"ahler geometry as well as to Picard-Fuchs equations governing the central charges of the BPS structure. We focus our study on cases with complex one dimensional moduli spaces and charge lattices of rank two including Argyres-Douglas $A_2$ as well as Seiberg-Witten $SU(2)$ theories. In these cases the moduli spaces are identified with modular curves and we determine the expressions of the central charges in terms of quasi-modular forms of the corresponding duality groups. We furthermore determine the curves of marginal stability and study the attractor flow in these examples, showing that it provides another way of determining the complete BPS spectrum in these cases.