Heavenly metrics, BPS indices and twistors

TL;DR

This paper explores complex hyperk"ahler metrics linked to stability conditions, expressing them via Riemann-Hilbert problems, TBA equations, and integrals, connecting to Donaldson-Thomas invariants, BPS states, and D-instantons.

Contribution

It recasts Bridgeland's hyperk"ahler metric construction into integral formulas using TBA equations, extends solutions to deformed equations, and introduces a generalized tau function.

Findings

Derived integral expressions for the metric functions W and F.

Reproduced Joyce's series construction of F from DT invariants.

Established a generalized tau function involving Barnes' G function.

Abstract

Recently T. Bridgeland defined a complex hyperk\"ahler metric on the tangent bundle over the space of stability conditions of a triangulated category, based on a Riemann-Hilbert problem determined by the Donaldson-Thomas invariants. This metric is encoded in a function $W(z,\theta)$ satisfying a heavenly equation, or a potential $F(z,\theta)$ satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both $W$ and $F$ in terms of solutions of that system. These expressions are recognized as conformal limits of the `instanton generating potential' and `contact potential' appearing in studies of D-instantons and BPS black holes. By solving the TBA equations iteratively, we reproduce Joyce's original construction of $F$ as a formal series in the rational DT invariants. Furthermore, we produce similar solutions to deformed versions of the heavenly and isomonodromy equations involving a non-commutative star-product. In the case of a finite uncoupled BPS structure, we rederive the results previously obtained by Bridgeland and obtain the so-called $\tau$ function for arbitrary values of the fiber coordinates $\theta$, in terms of a suitable two-variable generalization of Barnes' $G$ function.