A Hyperkähler geometry associated to the BPS structure of the resolved conifold

Murad Alim; Arpan Saha; Iván Tulli

doi:10.1016/j.geomphys.2022.104618

A Hyperkähler geometry associated to the BPS structure of the resolved conifold

Murad Alim, Arpan Saha, Iván Tulli

Research output: Working paper › Preprint

2 Citations (Scopus)

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Abstract

We associate to the resolved conifold an affine special K\"{a}hler (ASK) manifold of complex dimension 1, and an instanton corrected hyperk\"{a}hler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.

Original language	Undefined/Unknown
Publisher	arXiv
DOIs	https://doi.org/10.1016/j.geomphys.2022.104618
Publication status	Published - 22 Jun 2021

Keywords

math.DG
hep-th
math-ph
math.AG
math.MP

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Cite this

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title = "A Hyperk{\"a}hler geometry associated to the BPS structure of the resolved conifold",

abstract = " We associate to the resolved conifold an affine special K\{"}{a}hler (ASK) manifold of complex dimension 1, and an instanton corrected hyperk\{"}{a}hler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland. ",

keywords = "math.DG, hep-th, math-ph, math.AG, math.MP",

author = "Murad Alim and Arpan Saha and Iv{\'a}n Tulli",

note = "49 pages, references added and typos fixed. Previous theorem 43 and corollary 44 revised, and the rest of the results unchanged",

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T1 - A Hyperkähler geometry associated to the BPS structure of the resolved conifold

AU - Alim, Murad

AU - Saha, Arpan

AU - Tulli, Iván

N1 - 49 pages, references added and typos fixed. Previous theorem 43 and corollary 44 revised, and the rest of the results unchanged

PY - 2021/6/22

Y1 - 2021/6/22

N2 - We associate to the resolved conifold an affine special K\"{a}hler (ASK) manifold of complex dimension 1, and an instanton corrected hyperk\"{a}hler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.

AB - We associate to the resolved conifold an affine special K\"{a}hler (ASK) manifold of complex dimension 1, and an instanton corrected hyperk\"{a}hler (HK) manifold of complex dimension 2. We describe these geometries explicitly, and show that the instanton corrected HK geometry realizes an Ooguri-Vafa-like smoothing of the semi-flat HK metric associated to the ASK geometry. On the other hand, the instanton corrected HK geometry associated to the resolved conifold can be described in terms of a twistor family of two holomorphic Darboux coordinates. We study a certain conformal limit of the twistor coordinates, and conjecture a relation to a solution of a Riemann-Hilbert problem previously considered by T. Bridgeland.

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KW - math.MP

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