Heavenly metrics, BPS indices and twistors

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 \(W(z,\theta )\) 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.