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Mathematics Subject Classification: Primary 14H60, Secondaries 14D07, 55Q52.
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Revista de Matemática Teoría y Aplicaciones
versión impresa ISSN 1409-2433
Rev. Mat vol.26 no.2 San José jul./dic. 2019
http://dx.doi.org/10.15517/rmta.v26i2.38315
Artículo
A brief survey of higgs bundles
Un estudio conciso de fibrados de higgs
1Universidad de Costa Rica, Escuela de Matemática, Centro de Investigaciones Matemáticas y Metamatemáticas CIMM, San José, Costa Rica. E-Mail: ronald.zunigarojas@ucr.ac.cr
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 the- ory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle to- gether with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror sym- metry, 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.
Keywords: Higgs bundles; Hodge bundles; moduli spaces; stable triples; vector bundles
Considerando una superficie compacta de Riemann de género mayor o igual que dos, un fibrado de Higgs es un par compuesto por un fibrado holomorfo sobre la superficie de Riemann, junto con un campo vectorial auxiliar, llamado campo de Higgs. Esta teoría inició hace unos treinta años, con el trabajo de Hitchin, cuando él reduce las ecuaciones de auto- dualidad de dimensión cuatro a dimensión dos, y así, estudiar esas ecua- ciones sobre superficies de Riemann. Hitchin bautizó esos campos como campos de Higgs pues en el contexto de la física y de la teoría de gauge, describen partículas similares a las descritas por el bozón de Higgs. Más tarde, Simpson usó el nombre fibrado de Higgs para un fibrado holomorfo junto con un campo de Higgs. Hoy, los fibrados de Higgs son objeto de investigación en varias áreas tales como la teoría de Hodge no abeliana, Langlands, simetría de espejo, sistemas integrables, teoría cuántica de campos (QFT), entre otros. Los propósitos principales aquí son introducir estos objetos y presentar una breve pero completa construcción del espacio móduli de los fibrados de Higgs y algunas de sus estratificaciones.
Palabras clave: fibrados de Higgs; fibrados de Hodge; espacios móduli; triples estables; fibrados vectoriales
Acknowledgements
We thank Peter B. Gothen for introducing us to the beautiful subject of Higgs bundles. We thank the referees for their comments and suggestions. We thank organizers of XXI-SIMMAC-2018 for the opportunity of share our work with the scientific community. Financial support from Vicerrectoría de Investigación, Universidad de Costa Rica, is acknowledged.
Research supported by Universidad de Costa Rica through Escuela de Matemática and through CIMM (Centro de Investigaciones Matemáticas y Metamatemáticas), Project 820-B8-224. This work is partly based on the Ph.D. Project [19] called “Homotopy Groups of the Moduli Space of Higgs Bundles”, supported by FEDER through Programa Operacional Factores de Competitividade-COMPETE, and also supported by FCT (Fundação para a Ciência e a Tecnologia) through the projects PTDC/MAT-GEO/0675/2012 and PEstC/MAT/UI0144/2013 with grant reference SFRH/BD/51174/2010.
References
M.F. Atiyah, R. Bott; The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. 308 (1982), no. 1505, 523-615. [ Links ]
A. Białynicki-Birula, Some theorems on actions of algebraic groups; Ann. of Math. 98 (1973), 480-497. [ Links ]
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. [ Links ]
S.B. Bradlow, O. García-Prada P.B. Gothen; Homotopy groups of moduli spaces of representations, Topology 47 (2008), no. 4, 203-224. [ Links ]
P.B. Gothen, R.A. Zúñiga-Rojas; Stratifications on the moduli space of Higgs bundles, Portugaliae Mathematica 74 (2017), 127-148. [ Links ]
P.B. Gothen, R.A. Zúñiga-Rojas; Stratifications on the Nilpotent Cone of the Hitchin Map, in progress. [ Links ]
G. Harder,M.S. Narasimhan; On the Cohomology Groups of Moduli Spaces of Vector Bundles on Curves, Math. Ann. 212 (1975), 215-248. [ Links ]
T. Hausel; Geometry of the moduli space of Higgs bundles, Ph.D. Thesis, Univ. of Cambridge, 1998. [ Links ]
N.J. Hitchin; The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), no. 3, 59-126. [ Links ]
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. [ Links ]
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. [ Links ]
M. Lübke, A. Teleman; The Kobayashi-Hitchin Correspondence, World Scientific Publishing Co., 1995. [ Links ]
D. Mumford, J. Fogarty, F. Kirwan; Geometric Invariant Theory, Springer, 1994. [ Links ]
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. [ Links ]
N. Nitsure; Moduli space of semistable pairs on a curve, Proc. London Math. Soc . Vol. s3-62, Issue 2 (March, 1991), 275-300. [ Links ]
S.S. Shatz; The Decomposition and Specialization of Algebraic Families of Vector Bundles, Compositio Mathematica 35 (1977), no. 2., 163-187. [ Links ]
C.T. Simpson; Higgs bundles and local systems, Publ. Math. de l’IHÉS, Tome 75, (1992) 5-95. [ Links ]
C.N. Yang, R.L. Mills; Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), no. 1, 191-195. [ Links ]
R.A. Zúñiga-Rojas; Homotopy groups of the moduli space of Higgs bundles, Ph.D. Thesis, Universidade do Porto, 2015. [ Links ]
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. [ Links ]
R.A. Zúñiga-Rojas; Variations of Hodge structures of rank three k-Higgs bundles. Preprint, available at arXiv:1803.01936v3. [math.AG]. [ Links ]
Received: September 9, 2018; Revised: May 5, 2019; Accepted: June 6, 2019
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