Keywords
Hodge bundles
moduli spaces
stable triples
vector bundles
fibrados de Higgs
fibrados de Hodge
espacios móduli
triples estables
fibrados vectoriales
How to Cite
Abstract
Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.
References
M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. 308 (1982), no. 1505, 523-615.
A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973), 480–497.
S.B. Bradlow, O. García-Prada, P.B. Gothen, What is a Higgs bundle?, Notices of the American Mathematical Society 54 (2007), no. 8, 980-981.
S.B. Bradlow, O. García-Prada, P.B. Gothen, Homotopy groups of moduli spaces of representations, Topology 47 (2008), no. 4, 203-224.
P.B. Gothen, R.A. Zúñiga-Rojas, Stratifications on the moduli space of Higgs bundles, Portugaliae Mathematica 74 (2017), 127-148.
P.B. Gothen, R.A. Zúñiga-Rojas, Stratifications on the Nilpotent Cone of the Hitchin Map, in progress.
G. Harder, M.S. Narasimhan, On the Cohomology Groups of Moduli Spaces of Vector Bundles on Curves, Math. Ann. 212 (1975), 215-248.
T. Hausel, Geometry of the moduli space of Higgs bundles, Ph.D. Thesis, Univ. of Cambridge, 1998.
N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), no. 3, 59-126.
N.J. Hitchin, Gauge theory on Riemann surfaces (M. Carvalho, X. Gomez- Mont and A. Verjovsky, editors), Lectures on Riemann surfaces: proceedings of the college on Riemann surfaces, Italy, 99-118, 1989.
S. Kobayashi, Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, Vol. 15, Iwanami Shoten, Publishers and Princeton Univ. Press, 1987.
M. Lübke, A. Teleman, The Kobayashi-Hitchin Correspondence, World Scientific Publishing Co., 1995.
D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Springer, 1994.
M.S. Narasimhan, C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Annals of Mathematics, Second Series, Vol. 82, No. 3 (Nov., 1965), 540-567.
N. Nitsure, Moduli space of semistable pairs on a curve, Proc. London Math. Soc. Vol. s3-62, Issue 2 (March, 1991), 275-300.
S.S. Shatz, The Decomposition and Specialization of Algebraic Families of Vector Bundles, Compositio Mathematica 35 (1977), no. 2., 163-187.
C.T. Simpson, Higgs bundles and local systems, Publ. Math. de l’IHÉS, Tome 75, (1992) 5-95.
C.N. Yang, R.L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), no. 1, 191-195.
R.A. Zúñiga-Rojas, Homotopy groups of the moduli space of Higgs bundles, Ph.D. Thesis, Universidade do Porto, 2015.
R.A. Zúñiga-Rojas, Stabilization of the homotopy groups of the moduli spaces of k-Higgs bundles, Revista Colombiana de Matemáticas 52 (2018), no. 1, 9-31.
R.A. Zúñiga-Rojas, Variations of Hodge structures of rank three k-Higgs bundles. Preprint, available at arXiv:1803.01936v3 [math.AG].
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