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\({\mathcal {N}}=1\) Geometric Supergravity and Chiral Triples on Riemann Surfaces

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Abstract

We construct a global geometric model for the bosonic sector and Killing spinor equations of four-dimensional \(\mathcal {N}=1\) supergravity coupled to a chiral non-linear sigma model and a \(\mathrm {Spin}^{c}_0\) structure. The model involves a Lorentzian metric g on a four-manifold M, a complex chiral spinor and a map \(\varphi :M\rightarrow \mathcal {M}\) from M to a complex manifold \(\mathcal {M}\) endowed with a novel geometric structure which we call a chiral triple. Using this geometric model, we show that if M is spin then the Kähler-Hodge condition on \(\mathcal {M}\) suffices to guarantee the existence of an associated \(\mathcal {N}=1\) chiral geometric supergravity. This positively answers a conjecture proposed by D. Z. Freedman and A. V. Proeyen. We dimensionally reduce the Killing spinor equations to a Riemann surface X, obtaining a novel system of partial differential equations for a harmonic map with potential \(\varphi :X\rightarrow \mathcal {M}\). We characterize all Riemann surfaces X admitting supersymmetric solutions with vanishing superpotential, showing that such solutions \(\varphi \) are holomorphic maps satisfying a certain condition involving the canonical bundle of X and the chiral triple of the theory. Furthermore, we determine the biholomorphism type of all Riemann surfaces carrying supersymmetric solutions with complete Riemannian metric and finite-energy scalar map \(\varphi \).

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Notes

  1. We thank the anonymous referee for clarifying this point.

  2. We thank an anonymous referee of Communications in Mathematical Physics for writing the following detailed explanation.

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Acknowledgements

CSS would like to thank Mario García-Fernández for useful comments. The work of C. I. L. was supported by grants IBS-R003-S1 and IBS-R003-D1. The work of C.S.S. is supported by the Humboldt foundation through the Humboldt grant ESP 1186058 HFST-P. The work of V.C. and C.S.S. is supported by the German Science Foundation (DFG) under the Research Training Group 1670 Mathematics inspired by String Theory.

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Authors and Affiliations

  1. Department of Mathematics, University of Hamburg, Hamburg, Germany

    Vicente Cortés & C. S. Shahbazi

  2. Center for Geometry and Physics, Institute for Basic Science, Pohang, 37673, Republic of Korea

    C. I. Lazaroiu

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  1. Vicente Cortés
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  2. C. I. Lazaroiu
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Correspondence to C. S. Shahbazi.

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Communicated by N. Nekrasov

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Cortés, V., Lazaroiu, C.I. & Shahbazi, C.S. \({\mathcal {N}}=1\) Geometric Supergravity and Chiral Triples on Riemann Surfaces. Commun. Math. Phys. 375, 429–478 (2020). https://doi.org/10.1007/s00220-019-03476-7

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