Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-22T15:28:47.828Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  11 May 2022

Christian Pauly
Laboratoire de Mathématiques J.A. Dieudonné, UMR 7351 CNRS, Université de Nice Sophia-Antipolis, 06108 Nice Cedex 02, France (
Johan Martens
School of Mathematics and Maxwell Institute, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom (
Michele Bolognesi
Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS, Université de Montpellier, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France (
Thomas Baier
CAMGSD, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal (


We give an algebro-geometric construction of the Hitchin connection, valid also in positive characteristic (with a few exceptions). A key ingredient is a substitute for the Narasimhan–Atiyah–Bott Kähler form that realizes the Chern class of the determinant-of-cohomology line bundle on the moduli space of bundles on a curve. As replacement we use an explicit realisation of the Atiyah class of this line bundle, based on the theory of the trace complex due to Beilinson–Schechtman and Bloch–Esnault.

Research Article
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Atiyah, M. F. and Bott, R., ‘The Yang–Mills equations over Riemann surfaces’, Philos. Trans. Roy. Soc. London Ser. A 308(1505) (1983), 523615. doi: 10.1098/rsta.1983.0017. Google Scholar
Axelrod, S., Della Pietra, S. and Witten, E., ‘Geometric quantization of Chern–Simons gauge theory’, J. Differential Geom. 33(3) (1991), 787902. url: Scholar
Andersen, J. E., Gammelgaard, N. L. and Lauridsen, M. R., ‘Hitchin’s connection in metaplectic quantization’, Quantum Topol. 3(3-4) (2012), 327357. doi: 10.4171/qt/31. CrossRefGoogle Scholar
Problem 1.04 in AimPL, ‘Spectral data for Higgs bundles’. url: Scholar
Atiyah, M. F., ‘Complex analytic connections in fibre bundles’, Trans. Amer. Math. Soc. 85(1957), 181207. doi: 10.2307/1992969. Google Scholar
Atiyah, M., The Geometry and Physics of Knots, Lezioni Lincee. [Lincei Lectures] (Cambridge University Press, Cambridge, 1990). doi: 10.1017/CBO9780511623868. CrossRefGoogle Scholar
Andersen, J. E. and Ueno, K., ‘Construction of the Witten–Reshetikhin–Turaev TQFT from conformal field theory’, Invent. Math. 201(2) (2015), 519559. doi: 10.1007/s00222-014-0555-7. CrossRefGoogle Scholar
Beĭlinson, A. and Bernstein, J., ‘A proof of Jantzen conjectures’, In I. M. Gel $^{\prime }$ fand Seminar, Adv. Soviet Math., Vol. 16 (Amer. Math. Soc., Providence, RI, 1993) 150.CrossRefGoogle Scholar
Bloch, S. and Esnault, H., ‘Relative algebraic differential characters’, In Motives, Polylogarithms and Hodge Theory, Part I (Irvine, CA, 1998), Int. Press Lect. Ser., Vol. 3 (Int. Press, Somerville, MA, 2002), 4773.Google Scholar
Beĭlinson, A. A., ‘Residues and adèles’, Funktsional. Anal. i Prilozhen. 14(1) (1980), 4445.Google Scholar
Belkale, P., ‘Strange duality and the Hitchin/WZW connection’, J. Differ. Geom. 82(2) (2009), 445465. url: Scholar
Beĭlinson, A. A. and Kazhdan, D., ‘Flat projective connections’, unpublished manuscript, 1990.Google Scholar
Brylinski, J.-L. and McLaughlin, D., ‘Holomorphic quantization and unitary representations of the Teichmüller group’, In Lie Theory and Geometry, Progr. Math. , Vol. 123 (Birkhäuser Boston, Boston, MA, 1994), 2164. doi: 10.1007/978-1-4612-0261-5-2. CrossRefGoogle Scholar
Boer, A. L., ‘A unitary structure for the graded quotient of conformal coblocks’, PhD thesis, University of Utrecht, 2008. url: Scholar
Braunling, O., ‘On the local residue symbol in the style of Tate and Beilinson’, New York J. Math. 24 (2018), 458513.Google Scholar
Beĭlinson, A. A. and Schechtman, V. V., ‘Determinant bundles and Virasoro algebras,’ Comm. Math. Phys. 118(4) (1988), 651701. url: Scholar
Balaji, V. and Seshadri, C. S., ‘Moduli of parahoric $\mathbf{\mathcal{G}}$ -torsors on a compact Riemann surface’, J. Algebraic Geom. 24(1) (2015), 149. doi: 10.1090/S1056-3911-2014-00626-3. CrossRefGoogle Scholar
Ben-Zvi, D. and Frenkel, E., ‘Geometric realization of the Segal-Sugawara construction,’ In Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., Vol. 308 (Cambridge Univ. Press, Cambridge, 2004), 4697. doi: 10.1017/CBO9780511526398.006. CrossRefGoogle Scholar
Drezet, J.-M. and Narasimhan, M. S., ‘Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques’, Invent. Math. 97(1) (1989), 5394. doi: 10.1007/BF01850655. CrossRefGoogle Scholar
Esnault, H. and Tsai, I.-H., ‘Determinant bundle in a family of curves, after A. Beilinson and V’, Schechtman. Comm. Math. Phys. 211(2) (2000), 359363. doi: 10.1007/s002200050816. Google Scholar
Faltings, G., ‘Stable $G$ -bundles and projective connections’, J. Algebraic Geom. 2(3) (1993), 507568.Google Scholar
Gilmer, P. M., ‘Integrality for TQFTs’, Duke Math. J. 125(2) (2004), 389413. doi: 10.1215/S0012-7094-04-12527-8. CrossRefGoogle Scholar
Ginzburg, V., ‘Resolution of diagonals and moduli spaces], In The Moduli Space of Curves (Texel Island, 1994), Progr. Math., Vol. 129, (Birkhäuser Boston, Boston, MA, 1995), 231266. doi: 10.1007/978-1-4612-4264-2-9. CrossRefGoogle Scholar
Gilmer, P. M. and Masbaum, G., ‘Integral lattices in TQFT’, Ann. Sci. École Norm. Sup. 40(5) (2007), 815844. doi: 10.1016/j.ansens.2007.07.002. CrossRefGoogle Scholar
Gilmer, P. M. and Masbaum, G., ‘Irreducible factors of modular representations of mapping class groups arising in integral TQFT’, Quantum Topol. 5(2) (2014), 225258. doi: 10.4171/QT/51. CrossRefGoogle Scholar
Gilmer, P. M. and Masbaum, G., ‘An application of TQFT to modular representation theory’, Invent. Math. 210(2) (2017), 501530. doi: 10.1007/s00222-017-0734-4. Google Scholar
Grothendieck, A., ‘Éléments de géométrie algébrique I–IV’, Inst. Hautes Études Sci. Publ. Math., (4, 8, 11, 17, 20, 24, 28), 19601961.Google Scholar
Heinloth, J., ‘Uniformization of $\mathbf{\mathcal{G}}$ -bundles’, Math. Ann. 347(3) (2010), 499528. doi: 10.1007/s00208-009-0443-4. CrossRefGoogle Scholar
Hitchin, N., ‘Stable bundles and integrable systems’, Duke Math. J. 54(1) (1987), 91114. doi: 10.1215/S0012-7094-87-05408-1. CrossRefGoogle Scholar
Hitchin, N. J., ‘Flat connections and geometric quantization’, Comm. Math. Phys. 131(2) (1990), 347380. url: Scholar
Hitchin, N. J., ‘The symplectic geometry of moduli spaces of connections and geometric quantization’, Progr. Theoret. Phys. Suppl. 102 (1991), 159174, 1990. doi: 10.1143/PTP.102.159. Common trends in mathematics and quantum field theories (Kyoto, 1990).Google Scholar
Hoffmann, N., ‘The Picard group of a coarse moduli space of vector bundles in positive characteristic’, Cent. Eur. J. Math. 10(4) (2012), 13061313. doi: 10.2478/s11533-012-0064-0. CrossRefGoogle Scholar
Katz, N. M., ‘Algebraic solutions of differential equations ( $p$ -curvature and the Hodge filtration)’, Invent. Math. 18(1972), 1118. doi: 10.1007/BF01389714. CrossRefGoogle Scholar
Knudsen, F. F. and Mumford, D., ‘The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”’, Math. Scand. 39(1) (1976), 1955.Google Scholar
Laszlo, Y., ‘Hitchin’s and WZW connections are the same. J. Differ. Geom. 49(3) (1998), 547576.Google Scholar
Looijenga, E., ‘From WZW models to modular functors’, In Handbook of Moduli Vol. II, Adv. Lect. Math. (ALM), Vol. 25 (Int. Press, Somerville, MA, 2013), 427466.Google Scholar
Laszlo, Y., Pauly, C. and Sorger, C., ‘On the monodromy of the Hitchin connection’, J. Geom. Phys. 64 (2013), 6478. doi: 10.1016/j.geomphys.2012.11.003. CrossRefGoogle Scholar
Laszlo, Y. and Sorger, C., ‘The line bundles on the moduli of parabolic $G$ -bundles over curves and their sections’, Ann. Sci. École Norm. Sup. (4) 30(4) (1997), 499525. doi: 10.1016/S0012-9593(97)89929-6. CrossRefGoogle Scholar
Martinengo, E., ‘Higher brackets and Moduli space of vector bundles’, PhD thesis, Università degli Studi di Roma, La Sapienza, 2009.Google Scholar
Masbaum, G., ‘An element of infinite order in TQFT-representations of mapping class groups’, In Low-Dimensional Topology (Funchal, 1998), Contemp. Math., Vol. 233 (Amer. Math. Soc., Providence, RI, 1999), 137139. doi: 10.1090/conm/233/03423. CrossRefGoogle Scholar
Mehta, V. B. and Ramadas, T. R., ‘Moduli of vector bundles, Frobenius splitting, and invariant theory’, Ann. of Math. (2) 144(2) (1996), 269313. doi: 10.2307/2118593. CrossRefGoogle Scholar
Narasimhan, M. S., ‘Elliptic operators and differential geometry of moduli spaces of vector bundles on compact Riemann surfaces, In Proc. Internat. Conf. on Functional Analysis and Related Topics (Tokyo, 1969), (Univ. of Tokyo Press, Tokyo, 1970) 6871.Google Scholar
Narasimhan, M. S. and Ramanan, S., ‘Deformations of the moduli space of vector bundles over an algebraic curve’, Ann. Math. (2) 101(1975), 391417. doi: 10.2307/1970933. Google Scholar
Pappas, G. and Rapoport, M., ‘Some questions about $\mathbf{\mathcal{G}}$ -bundles on curves’, In Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), Adv. Stud. Pure Math., Vol. 58 (Math. Soc. Japan, Tokyo, 2010), 159171. doi: 10.2969/aspm/05810159. Google Scholar
Quillen, D., Determinants of Cauchy–Riemann operators over a Riemann surface’, Functional Analysis and Its Applications 19(1) (1985), 3134. doi: 10.1007/BF01086022. CrossRefGoogle Scholar
Ramadas, T. R., Faltings’ construction of the K-Z connection, Comm. Math. Phys. 196(1) (1998) 133143. doi: 10.1007/s002200050417. CrossRefGoogle Scholar
Ran, Z., ‘Jacobi cohomology, local geometry of moduli spaces, and Hitchin connections’, Proc. London Math. Soc. (3) 92(3) (2006), 545580. doi: 10.1017/S0024611505015704. CrossRefGoogle Scholar
Sernesi, E., Deformations of Algebraic Schemes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 334 (Springer-Verlag, Berlin, 2006).Google Scholar
Scheinost, P. and Schottenloher, M., ‘Metaplectic quantization of the moduli spaces of flat and parabolic bundles’, J. Reine Angew. Math. 466(1995), 145219.Google Scholar
Sun, X. and Tsai, I.-H., ‘Hitchin’s connection and differential operators with values in the determinant bundle’, J. Differential Geom. 66(2) (2004), 303343. url: Scholar
Schechtman, V. and Varchenko, A., ‘Solutions of KZ differential equations modulo $p$ ’, Ramanujan J. 48(3) (2019), 655683. doi: 10.1007/s11139-018-0068-x. CrossRefGoogle Scholar
Tate, J., ‘Residues of differentials on curves’, Ann. Sci. École Norm. Sup. 4(1) (1968), 149159. URL Scholar
The Stacks Project Authors, Stacks Project, 2019. url: Scholar
Tsuchimoto, Y., ‘On the coordinate-free description of the conformal blocks’, J. Math. Kyoto Univ. 33(1) (, 1993), 2949. doi: 10.1215/kjm/1250519338. Google Scholar
Tsuchiya, A., Ueno, K. and Yamada, Y., ‘Conformal field theory on universal family of stable curves with gauge symmetries’, In Integrable Systems in Quantum Field Theory and Statistical Mechanics, Adv. Stud. Pure Math., Vol. 19 (Academic Press, Boston, MA, 1989), 459566.Google Scholar
Vakil, R., ‘The rising sea—foundations of algebraic geometry’, 2017. url: Scholar
van Geemen, B. and de Jong, A. J., ‘On Hitchin’s connection,’ J. Amer. Math. Soc. 11(1) (1998), 189228. doi: 10.1090/S0894-0347-98-00252-5. CrossRefGoogle Scholar
Welters, G. E., ‘Polarized abelian varieties and the heat equations’, Compositio Math. 49(2) (1983), 173194. url: Scholar
Witten, E., ‘Quantum field theory and the Jones polynomial’, Comm. Math. Phys. 121(3) (1989), 351399. url: Scholar