Publication: On the rational homotopy type of a moduli space of vector bundles over a curve
Loading...
Official URL
Full text at PDC
Publication Date
2008
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Int press co ltd
Citations
Exportar
Abstract
We study the rational homotopy of the moduli space N-X that parametrizes the isomorphism classes of all stable vector bundles of rank two and fixed determinant of odd degree over a compact connected Riemann surface X of genus g, with g >= 2. The symplectic group Aut(H-1(X, Z)) congruent to Sp(2g, Z) has a natural action on the rational homotopy groups pi(n)(N-X)circle times(Z)Q. We prove that this action extends to an action of Sp(2g, C) on pi(n)(N-X)circle times C-Z. We also show that pi(n)(N-X)circle times C-Z is a non-trivial representation of Sp(2g, C) congruent to Aut (H-1(X, C)) for all n >= 2g - 1. In particular, N-X is a rationally hyperbolic space. In the special case where g = 2, for each n is an element of N, we compute the leading Sp(2g, C) representation occurring in pi(n)(N-X)circle times C-Z.
Description
UCM subjects
Unesco subjects
Keywords
Citation
M.F. Atiyah and R. Bott, The Yang–Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523–615.
I. Biswas, Determinant bundle over the universal moduli space of vector bundles over the Teichm¨uller space, Ann. Inst. Fourier 47 (1997), 885–914.
A. Borel, Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. 72 (1960), 179–188.
P. Deligne, Equations differentielles `a points singuliers reguliers, Lecture Notes in Mathematics 163, Springer-Verlag, Berlin, NY, 1970.
P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kahler manifolds, Invent. Math. 29 (1975), 245–274.
Y. Felix, La dichotomie elliptique-hyperbolique en homotopie
rationnelle, Asterisque 179 Soci´et´e Mathematique de France, Paris,1989.
W. Fulton and J. Harris, Representation theory. A first course, Graduate Texts in Mathematics 129, Springer-Verlag, Berlin, NY, 1991.
P. Griffiths and J.W. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics 16,Birkhauser, Boston, MA, 1981.
A.D. King and P.E. Newstead, On the cohomology ring of the moduli space of rank 2 vector bundles on a curve, Topology 37 (1998),407–418.
J.E. Humphreys, Introduction to Lie algebras and representation theory,Graduate Texts in Mathematics 9, Springer-Verlag, Berlin, NY, 1978.
V. Muñoz, Quantum cohomology of the moduli space of stable bundles over a Riemann surface, Duke Math. J. 98 (1999),525–540.
D. Mumford and P.E. Newstead, Periods of a moduli space of bundles on curves, Amer. J. Math. 90 (1968), 1200–1208.
M.S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. 89 (1969), 14–51.
[14] M.S. Narasimhan and C.S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. Math. 82 (1965), 540–567.
V. Navarro Aznar, Sur la connexion de Gauss–Manin en homotopie rationnelle, Ann. Sci. ´Ecole Norm. Sup. 26 (1993), 99–148.
P.E. Newstead, Topological properties of some spaces of stable bundles,Topology 6 (1967), 241–262.
P.E. Newstead, Stable bundles of rank 2 and odd degree over a curve of genus 2, Topology 7 (1968), 205–215.
P.E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Am. Math. Soc. 169 (1972), 337–345.
P.E. Newstead, Introduction to moduli problems and orbit spaces,T.I.F.R. Lectures on Mathematics and Physics, 51,Narosa Publishing House, New Delhi, 1978.
B. Siebert and G. Tian, Recursive relations for the cohomology ring of moduli spaces of stable bundles, Turkish J. Math. 19 (1995), 131–144.
M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Diff. Geom. 35(1992), 131–150.