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Deformation Theory of Deformed Hermitian Yang–Mills Connections and Deformed Donaldson–Thomas Connections

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Abstract

A deformed Donaldson–Thomas (dDT) connection is a Hermitian connection of a Hermitian line bundle over a \(G_2\)-manifold X satisfying a certain nonlinear PDE. This is considered to be the mirror of a (co)associative cycle in the context of mirror symmetry. The dDT connection is an analogue of a deformed Hermitian Yang–Mills (dHYM) connection which is extensively studied recently. In this paper, we study the moduli spaces of dDT and dHYM connections. In the former half, we prove that the deformation of dDT connections is controlled by a subcomplex of the canonical complex, an elliptic complex defined by Reyes Carrión, by introducing a new coclosed \(G_2\)-structure. If the deformation is unobstructed, we also show that the connected component of the moduli space is a \(b^{1}\)-dimensional torus, where \(b^{1}\) is the first Betti number of X. A canonical orientation on the moduli space is also given. We also prove that the obstruction of the deformation vanishes if we perturb the \(G_2\)-structure generically under some assumptions. In the latter half, we prove that the moduli space of dHYM connections, if it is nonempty, is a \(b^{1}\)-dimensional torus, especially, it is connected and orientable. We also prove the existence of a family of moduli spaces along a deformation of underlying structures if two cohomology classes vanish.

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Acknowledgements

The authors would like to thank Naichung Conan Leung for his helpful comments to the idea of this paper when they met at Gakushuin University and thank Spiro Karigiannis and Henrique N. Sá Earp for answering our questions on dDT connections. They also would like to thank to Hiroshi Konno for his comments to the previous version, which strengthened our main theorems and simplified the proofs. The authors thank the anonymous referee for carefully reading the previous version of this paper and providing useful comments.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Gakushuin University, 1-5-1 Mejiro, Toshima-ku, Tokyo, 171-8588, Japan

    Kotaro Kawai

  2. Department of Mathematics, Faculty of Pure and Applied Science, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 305-8577, Japan

    Hikaru Yamamoto

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  1. Kotaro Kawai
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  2. Hikaru Yamamoto
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Correspondence to Hikaru Yamamoto.

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The first named author is supported by JSPS KAKENHI Grant Number JP17K14181 and Research Grants of Yoshishige Abe Memorial Fund, and the second named author is supported by JSPS KAKENHI Grant Number JP18K13415 and Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics)

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Kawai, K., Yamamoto, H. Deformation Theory of Deformed Hermitian Yang–Mills Connections and Deformed Donaldson–Thomas Connections. J Geom Anal 32, 157 (2022). https://doi.org/10.1007/s12220-022-00898-z

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