Heavenly metrics, BPS indices and twistors

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