Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 048, 23 pages      arXiv:1903.03264      https://doi.org/10.3842/SIGMA.2020.048
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Triply Periodic Monopoles and Difference Modules on Elliptic Curves

Takuro Mochizuki
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan

Received October 29, 2019, in final form May 18, 2020; Published online June 03, 2020

Abstract
We explain the correspondences between twisted monopoles with Dirac type singularity and polystable twisted mini-holomorphic bundles with Dirac type singularity on a 3-dimensional torus. We also explain that they are equivalent to polystable parabolic twisted difference modules on elliptic curves.

Key words: twisted monopoles; twisted difference modules; twisted mini-holomorphic bundles; Kobayashi-Hitchin correspondence.

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References

1. Charbonneau B., Hurtubise J., Singular Hermitian-Einstein monopoles on the product of a circle and a Riemann surface, Int. Math. Res. Not. 2011 (2011), 175-216, arXiv:0812.0221.
2. Kontsevich M., Soibelman Y., Riemann-Hilbert correspondence in dimension one, Fukaya categories and periodic monopoles, Preprint.
3. Kronheimer P.B., Monopoles and Taub-NUT metrics, Master Thesis, Oxford, 1986.
4. Mochizuki T., Periodic monopoles and difference modules, arXiv:1712.08981.
5. Mochizuki T., Doubly-periodic monopoles and $q$-difference modules, arXiv:1902.08298.
6. Mochizuki T., Yoshino M., Some characterizations of Dirac type singularity of monopoles, Comm. Math. Phys. 356 (2017), 613-625, arXiv:1702.06268.
7. Simpson C.T., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918.
8. Yoshino M., A Kobayashi-Hitchin correspondence between Dirac-type singular mini-holomorphic bundles and HE-monopoles, arXiv:1902.09995.