Abstract
We review some recent results in the theory of affine manifolds and bundles on them. Donaldson–Uhlenbeck–Yau type correspondences for flat vector bundles and principal bundles are shown. We also consider flat Higgs bundles and flat pairs on affine manifolds. A bijective correspondence between polystable flat Higgs bundles and solutions of the Yang–Mills–Higgs equation in the context of affine manifolds is shown. Also shown, in the context of affine manifolds, is a bijective correspondence between polystable flat pairs and solutions of the vortex equation.
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Amari S., Nagaoka H.: Methods of Information Geometry. American Mathematical Society, Providence (2000)
Biswas I.: Stable Higgs bundles on compact Gauduchon manifolds. C. R. Math. Acad. Sci. Paris 349, 71–74 (2011)
Biswas, I.; Loftin, J.: Hermitian–Einstein connections on principal bundles over flat affine manifolds. Int. J. Math. 23(4) (2012). doi:10.1142/S0129167X12500395
Biswas, I.; Loftin, J.: Stemmler, M.: Affine Yang–Mills–Higgs metrics. J. Symp. Geom. (to appear). arXiv:1208.1578v1
Biswas, I.; Loftin, J.; Stemmler, M.: The vortex equation on affine manifolds (2012, preprint)
Borel, A.: Linear Algebraic Groups, 2nd edn. Graduate Texts in Mathematics, vol. 126. Springer, New York (1991)
Bradlow S.B.: Vortices in holomorphic line bundles over closed Kähler manifolds. Commun. Math. Phys. 135, 1–17 (1990)
Bradlow S.B.: Special metrics and stability for holomorphic bundles with global sections. J. Differ. Geom. 33, 169–213 (1991)
Bruasse L.: Harder–Narasimhan filtration on non Kähler manifolds. Int. J. Math. 12, 579–594 (2001)
Cheng, S.-Y.; Yau, S.-T.: The real Monge–Ampère equation and affine flat structures. In: Differential Geometry and Differential Equations Proc. 1980 Beijing Symp. 1, 339–370 (1982)
Corlette K.: Flat G-bundles with canonical metrics. J. Diff. Geom. 28, 361–382 (1988)
Delanoë P.: Remarques sur les variétés localement hessiennes. Osaka J. Math. 26, 65–69 (1989)
Donaldson S.K.: Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. Lond. Math. Soc. 50, 1–26 (1985)
Donaldson S.K.: Infinite determinants, stable bundles and curvature. Duke Math. J. 54, 231–247 (1987)
Donaldson S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. 55, 127–131 (1987)
Fried D., Goldman W.M.: Three-dimensional affine crystallographic groups. Adv. Math. 47, 1–49 (1983)
Fried D., Goldman W.M., Hirsch M.W.: Affine manifolds with nilpotent holonomy. Comment. Math. Helv. 56, 487–523 (1981)
García-Prada O.: Invariant connections and vortices. Commun. Math. Phys. 156, 527–546 (1993)
García-Prada O.: A direct existence proof for the vortex equation over a compact Riemann surface. Bull. Lond. Math. Soc. 26, 88–96 (1994)
García-Prada O.: Dimensional reduction of stable bundles, vortices and stable pairs. Int. J. Math. 5, 1–52 (1994)
Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987)
Humphreys, J.E.: Linear algebraic groups. Graduate Texts in Mathematics, vol. 21. Springer, New York (1981)
Jacob, A.: Stable Higgs bundles and Hermitian–Einstein metrics on non-Kähler manifolds. arXiv:1110.3768v1 [math.DG] (2011)
Jost J., Yau S.-T.: A nonlinear elliptic system for maps from Hermitian to Riemannian manifolds and rigidity theorems in Hermitian geometry. Acta Math. 170, 221–254 (1993)
Li, J.; Yau, S.-T.: Hermitian–Yang–Mills connection on non-Kähler manifolds. In: Mathematical Aspects of String Theory Proc. Conf., San Diego/Calif. 1986. Adv. Ser. Math. Phys. 1, 560–573 (1987)
Loftin J.: Affine Hermitian–Einstein metrics. Asian J. Math. 13, 101–130 (2009)
Lübke M., Teleman A.: The Kobayashi–Hitchin Correspondence. World Scientific Publishing Co., River Edge (1995)
Markus, L.: Cosmological models in differential geometry. Mimeographed Notes, p. 58. Univ. of Minnesota (1962)
Ramanathan A.: Stable principal bundles on a compact Riemann surface. Math. Ann. 213, 129–152 (1975)
Shima H.: The Geometry of Hessian Structures. World Scientific, Singapore (2007)
Simpson C.T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)
Simpson C.T.: Higgs bundles and local systems. Inst. Hautes Études Sci. Publ. Math. 75, 5–95 (1992)
Strominger A., Yau S.-T., Zaslow E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243–259 (1996)
Uhlenbeck K., Yau S.-T.: On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Commun. Pure Appl. Math. 39, 257–293 (1986)
Uhlenbeck K., Yau S.-T.: A note on our previous paper: on the existence of Hermitian Yang–Mills connections in stable vector bundles. Commun. Pure Appl. Math. 42, 703–707 (1989)
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