On Hitchin’s connection

van Geemen, Bert; de Jong, Aise Johan

doi:10.1090/S0894-0347-98-00252-5

On Hitchin’s connection
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by Bert van Geemen and Aise Johan de Jong

J. Amer. Math. Soc. 11 (1998), 189-228

DOI: https://doi.org/10.1090/S0894-0347-98-00252-5

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Abstract:

The aim of the paper is to give an explicit expression for Hitchin’s connection in the case of stable rank 2 bundles on genus 2 curves. Some general theory (in the algebraic geometric setting) concerning heat operators is developed. In particular the notion of compatibility of a heat operator with respect to a closed subvariety is introduced. This is used to compare the heat operator in the nonabelian rank 2 genus 2 case to the abelian heat operator (on theta functions) for abelian surfaces. This relation allows one to perform the computation; the resulting differential equations are similar to the Knizhnik-Zalmolodshikov equations.

References

Jean-Luc Brylinski and Dennis McLaughlin, Holomorphic quantization and unitary representations of the Teichmüller group, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 21–64. MR 1327530, DOI 10.1007/978-1-4612-0261-5_{2}
Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174, DOI 10.1007/BFb0061194
Igor Dolgachev and David Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988), 210 pp. (1989) (English, with French summary). MR 1007155
U. V. Desale and S. Ramanan, Classification of vector bundles of rank $2$ on hyperelliptic curves, Invent. Math. 38 (1976/77), no. 2, 161–185. MR 429897, DOI 10.1007/BF01408570
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
Bert van Geemen, Schottky-Jung relations and vectorbundles on hyperelliptic curves, Math. Ann. 281 (1988), no. 3, 431–449. MR 954151, DOI 10.1007/BF01457155
B. van Geemen and E. Previato, On the Hitchin system, Duke Math. J. 85 (1996) 659–683.
Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
N. J. Hitchin, Flat connections and geometric quantization, Comm. Math. Phys. 131 (1990), no. 2, 347–380. MR 1065677, DOI 10.1007/BF02161419
Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
Toshitake Kohno, Linear representations of braid groups and classical Yang-Baxter equations, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 339–363. MR 975088, DOI 10.1090/conm/078/975088
Toshitake Kohno, Topological invariants for $3$-manifolds using representations of mapping class groups. I, Topology 31 (1992), no. 2, 203–230. MR 1167165, DOI 10.1016/0040-9383(92)90016-B
David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776, DOI 10.1007/978-0-8176-4578-6
Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177–254. MR 1002038, DOI 10.1007/BF01238857
M. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2) 89 (1969), 14–51. MR 242185, DOI 10.2307/1970807
Carlos T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95. MR 1179076, DOI 10.1007/BF02699491
Gerald E. Welters, Polarized abelian varieties and the heat equations, Compositio Math. 49 (1983), no. 2, 173–194. MR 704390
Gretchen Wright, The Reshetikhin-Turaev representation of the mapping class group, J. Knot Theory Ramifications 3 (1994), no. 4, 547–574. MR 1304402, DOI 10.1142/S021821659400040X