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Heavenly metrics, BPS indices and twistors

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-03206137
Contributor : L2c Aigle <>
Submitted on : Friday, April 23, 2021 - 3:40:08 AM
Last modification on : Sunday, April 25, 2021 - 3:26:06 AM

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  • HAL Id : hal-03206137, version 1
  • ARXIV : 2104.10540

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Sergey Alexandrov, Boris Pioline. Heavenly metrics, BPS indices and twistors. 2021. ⟨hal-03206137⟩

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