Null Kähler Geometry and Isomonodromic Deformations

Dunajski, Maciej

doi:10.1007/s00220-021-04270-0

Cited by 5 publications

(5 citation statements)

References 33 publications

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“…which preserve the form g, and which are parallel with respect to the holomorphic Levi-Civita connection. Such structures have appeared before in the literature, often under different names [13,27,42,43].…”

Section: Given a Symplectic ν-Pencil Hmentioning

confidence: 99%

Joyce structures on spaces of quadratic differentials I

Bridgeland¹

2022

Preprint

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Consider the space parameterising curves of genus g equipped with a quadratic differential with simple zeroes. We use the geometry of isomonodromic deformations to construct a complex hyperkähler structure on the total space of its tangent bundle. This gives a new class of examples of the Joyce structures introduced in [17] in relation to Donaldson-Thomas theory. Our results relate to recent work in mathematical physics on supersymmetric gauge theories of class S[A 1 ].

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Section: Given a Symplectic ν-Pencil Hmentioning

confidence: 99%

Joyce structures on spaces of quadratic differentials I

Bridgeland¹

2022

Preprint

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“…As explained in [13,17], a solution of (2.20) defines a complex hyperkähler metric on M, the total space of the holomorphic tangent bundle over S,…”

Section: Plebanski Potential and Heavenly Metricsmentioning

confidence: 99%

“…As explained in [13,17], the main step in constructing the complex hyperkähler metric on M = T S is to produce a solution W (z, θ) of the partial differential equations 3…”

Section: Introductionmentioning

confidence: 99%

“…In real dimension 4, (1.1) is the 'second heavenly equation' governing generic HK manifolds [1], in a coordinate system with reality conditions mixing z 1 , z 2 with θ 1 , θ 2 . In higher dimensions, the HK metrics constructed from solutions of (1.1) are distinguished by the existence of a covariantly constant nilpotent endomorphism of the tangent bundle [17]. The solution W (z, θ) constructed by Bridgeland is homogeneous of degree −1 with respect to the coordinates z a , leading to a conical metric on M = T S. As shown in [13], its derivative…”

Section: Introductionmentioning

confidence: 99%

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Heavenly metrics, BPS indices and twistors

Alexandrov¹,

Pioline²

2021

Preprint

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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, θ) satisfying a heavenly equation, or a potential F (z, θ) 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 τ function for arbitrary values of the fiber coordinates θ, in terms of a suitable two-variable generalization of Barnes' G function.1 Since hyperkähler (HK) and quaternion-Kähler (QK) spaces are parametrized by solutions of non-linear differential equations often known as heavenly equations, we refer to such quaternionic metrics as 'heavenly', irrespective of their precise nature. See [1] and [2] for the original descriptions in the four-dimensional HK and QK cases,and [3] for a recent discussion of the higher dimensional case.2 In the context of class S field theories, the full space of stability conditions Stab(D) [16] can be interpreted as the total space of the Coulomb branch fibration over the conformal manifold, but the significance of the complex HK metric on M is yet to be understood.

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“…Then, from the Laurent expansion ω = t −1 ω + + iω 3 + O(t) near t = 0, one deduces the complex HK structure on T S in the usual way. It turns out that the resulting HK metric is encoded in a single function W (z, θ), called Plebanski potential, which must satisfy a system of non-linear differential equations, known as heavenly equations [5,26]. Furthermore, this function determines a connection on P 1 such that X γ are its flat sections, which is equivalent to the following horizontal section condition [4]…”

Section: Rh Problem In the Uncoupled Casementioning

confidence: 99%

Conformal TBA for resolved conifolds

Alexandrov,

Pioline

2021

Preprint

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We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the τ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture a new integral representation for the double sine function.

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Null Kähler Geometry and Isomonodromic Deformations

Cited by 5 publications

References 33 publications

Joyce structures on spaces of quadratic differentials I

Joyce structures on spaces of quadratic differentials I

Heavenly metrics, BPS indices and twistors

Conformal TBA for resolved conifolds

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