%0 Journal Article
%T Heavenly metrics, BPS indices and twistors
%+ Laboratoire Charles Coulomb (L2C)
%+ Laboratoire de Physique Théorique et Hautes Energies (LPTHE)
%A Alexandrov, Sergey
%A Pioline, Boris
%< avec comité de lecture
%Z L2C:21-034
%@ 0377-9017
%J Letters in Mathematical Physics
%I Springer Verlag
%V 111
%P 116
%8 2021-09-03
%D 2021
%Z 2104.10540
%R 10.1007/s11005-021-01455-5
%Z Physics [physics]/High Energy Physics - Theory [hep-th]
%Z Physics [physics]/Mathematical Physics [math-ph]
%Z Mathematics [math]/Algebraic Geometry [math.AG]
%Z Mathematics [math]/Mathematical Physics [math-ph]
%Z Mathematics [math]/Number Theory [math.NT]Journal articles
%X 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.
%G English
%L hal-03206137
%U https://hal.science/hal-03206137
%~ CNRS
%~ LPTHE
%~ L2C
%~ TDS-MACS
%~ MIPS
%~ UNIV-MONTPELLIER
%~ SORBONNE-UNIVERSITE
%~ SORBONNE-UNIV
%~ SU-SCIENCES
%~ TEST-HALCNRS
%~ SU-TI
%~ ALLIANCE-SU
%~ UM-2015-2021