The Hermite-Einstein equation and stable principal bundles (an updated survey)

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This is a survey of work done in the past two decades relating a basic equation of general relativity and quantum field theory with the theory of stable vector bundles and stable principal bundles, as well as providing moduli spaces for such objects, and compactifying them.
4th Iberoamerican Conference on Complex Geometry Location: Ouro Preto, BRAZIL Date: AUG 12-18, 2007.
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Atiyah M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85, 181–207 (1957) Balaji V.: Principal bundles on projective varieties and the Donaldson-Uhlenbeck compactification. J. Differ. Geom. 76(3), 351–398 (2007) Balaji V., Parameswaran A.J.: Semistable principal bundles.II. Positive characteristics. Transform. Groups 8, 3–36 (2003) Biswas, I., Gómez, T.: Higgs fields and flat connections on a principal bundle over a compact Kähler manifold. In: Algebraic Groups and Homogeneous Spaces, pp. 65–82. Tata Institute of Fundamental Research Studies in Mathematics, Narosa Publishing House, (2007) Biswas, I., Gómez, T.: Connections and Higgs fields on a principal bundle. Ann. Glob. Anal. Geom. 33, 19–46 (2008) Corlette K.: Flat G-bundles with canonical metrics. J. Differ. Geom. 28(3), 361–382 (1988) Donaldson S.K.: A new proof of a theorem of Narasimhan and Seshadri. J. Differ. Geom. 18, 269–277 (1983) Donaldson S.K.: Twisted harmonic maps and the self-duality equations. Proc. Lond. Math. Soc. (3) 55(1), 127–131 (1987) Donaldson, S.K., Kronheimer, P.B.: The geometry of four-manifolds. Oxford Mathematical Monographs. Oxford University Press, New York (1990) Faltings G.: Moduli-stacks for bundles on semistable curves. Math. Ann. 304, 489–515 (1996) Faltings G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. 5, 41–68 (2003) Gieseker D.: On the moduli of vector bundles on an algebraic surface. Ann. Math. 106, 45–60 (1977) Gómez, T., Langer, A., Schmitt, A., Sols, I.: Moduli spaces for principal bundles in arbitrary characteristic. Preprint Math. AG/0506511 (2005) Gómez, T., Langer, A., Schmitt, A., Sols, I.: Moduli spaces for principal bundles in large characteristic. Proceedings of International Workshop on Teichmüller Theory and Moduli Problems. Allahabad, India (2006) Gómez T., Sols I.: Stability of conic bundles. Int. J. Math. 11, 1027–1055 (2000) Gómez, T., Sols, I.: Stable tensors and moduli space of orthogonal sheaves. Math. AG/0103150 Gómez, T., Sols, I.: Projective moduli space of semistable principal sheaves for a reductive group. Le Matematiche 15, 437–446 (2000). Conference in Honor of Silvio Greco (April 2001) Gómez T., Sols I.: Moduli space of principal sheaves over projective varieties. Ann. Math. (2) 161(2), 1037–1092 (2005) Gómez, T. Sols, I.: Stable Higgs G-sheaves. Rev. Mat. Iberoam. 24, 703–719 (2008) Heinloth, J.: Semistable reduction for G-bundles on curves. J. Algebraic Geom. 17, 167–183 (2008) Hitchin N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55, 59–126 (1987) Huybrechts D., Lehn M.: The geometry of moduli spaces of sheaves, Aspects of Mathematics E31. Vieweg, Braunschweig (1997) Hyeon D.: Principal bundles over a projective scheme. Trans. Am. Math. Soc. 354, 1899–1908 (2002) Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication, xvi+468 pp. Wiley, New York (1996) Kobayashi S.: Differential Geometry of Complex Vector Bundles. Princeton University Press (1987) Langer, A.: Semistable sheaves in positive characteristic. Ann. Math. (2) 159, 251–276 (2004). Addendum: Ann. Math. (2) 160, 1211–1213 (2004) Langer A.: Moduli spaces of sheaves in mixed characteristic. Duke Math. J. 124, 571–86 (2004) Langton S.G.: Valuative criteria for families of vector bundles on algebraic varieties. Ann. Math. 101(2), 88–110 (1975) Maruyama, M.: Moduli of stable sheaves, I and II. J. Math. Kyoto Univ. (17), 91–126 (1977); 18 (1978), 557–614 Misner, Ch., Thorne, K., Wheeler J.A.: Gravitation. Freeman and Co (1973) Mumford, D.: Projective invariants of projective structures and applications. 1963 Proc. Internat. Congr. Mathematicians (Stockholm, 1962), pp. 526–530. Inst. Mittag-Leffler, Djursholm (1963) Narasimhan M.S., Seshadri C.S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math. 82(2), 540–567 (1965) Nitsure N.: Moduli space of semistable pairs on a curve. Proc. Lond. Math. Soc. (3) 62(2), 275–300 (1991) Ramanathan A.: Stable principal bundles on a compact Riemann surface. Math. Ann. 213, 129–152 (1975) Ramanathan, A.: Moduli for principal bundles over algebraic curves: I and II. Proc. Indian Acad. Sci. (Math. Sci.), 106 301–328, 421–449 (posthumous publication of the 1976 PhD thesis) (1996) Ramanathan A., Subramanian S.: Einstein-Hermitian connections on principal bundles and stability.J. Reine Angew. Math. 390, 21–31 (1988) Schmitt, A.H.W.: Projective moduli for Hitchin pairs. Int. J. Math. 9(1) 107–118 (1998). Erratum: 11(no. 4)589 (2000) Schmitt A.H.W.: Singular principal bundles over higher-dimensional manifolds and their moduli spaces. Int. Math. Res. Notes 23, 1183–1209 (2002) Schmitt A.H.W.: A universal construction for moduli spaces of decorated vector bundles over curves. Transform. Groups 9(2), 167–209 (2004) Schmitt A. H.W.: A closer look at semistability for singular principal bundles. Int. Math. Res. Notes 62, 3327–3366 (2004) Schmitt, A.H.W.: Geometric invariant theory and decorated principal bundles. In: Zürich Lectures in Advanced Mathematics, viii +389 pp. European Mathematical Society, Zürich (20008) Seshadri C.S.: Space of unitary vector bundles on a compact Riemann surface. Ann. Math. 85(2), 303–336 (1967) Simpson C.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Am. Math. Soc. 1(4), 867–918 (1988) Simpson C.: Higgs bundles and local systems. Publ. Math. I.H.E.S. 75, 5–95 (1992) Simpson C.: Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math. I.H.E.S. 79, 47–129 (1994) Simpson C.: Moduli of representations of the fundamental group of a smooth projective variety II. Publ. Math. I.H.E.S. 80, 5–79 (1995) Sorger C.: Thêta-charactéristiques des courbes tracées sur une surface lisse. J. Reine Angew. Math. 435, 83–118 (1993) Tian G.: Gauge theory and calibrated geometry, I. Ann. Math. 151(2), 193–268 (2000) Uhlenbeck K.K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys. 83, 11–29 (1982) Uhlenbeck K.K.: Connections with L p bounds on curvature. Commun. Math. Phys. 83, 31–42 (1982)