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Heavenly metrics, BPS indices and twistors. (English) Zbl 1491.53063

Summary: Recently, T. Bridgeland [“Geometry from Donaldson-Thomas invariants”, Preprint, arXiv:1912.06504] 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.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
34M56 Isomonodromic deformations for ordinary differential equations in the complex domain
35Q15 Riemann-Hilbert problems in context of PDEs
53C28 Twistor methods in differential geometry
53D55 Deformation quantization, star products
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14J42 Holomorphic symplectic varieties, hyper-Kähler varieties
81T60 Supersymmetric field theories in quantum mechanics
33B15 Gamma, beta and polygamma functions
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References:

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