Heavenly metrics, BPS indices and twistors
arXiv:2104.10540 · doi:10.1007/s11005-021-01455-5
Abstract
Recently T. Bridgeland defined a complex hyperkähler 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 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 function for arbitrary values of the fiber coordinates , in terms of a suitable two-variable generalization of Barnes' function.
21+16 pages; version accepted for publication in Letters in Mathematical Physics