Heavenly metrics, BPS indices and twistors
Sergei Alexandrov, Boris Pioline

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 satisfying a heavenly equation, or a potential satisfying an isomonodromy equation. After recasting the RH problem into a system of TBA-type equations, we obtain integral expressions for both and 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 as a formal series in the rational DT invariants. Furthermore, we produce similar solutions…
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