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8 - Identification of the Space of Conformal Blocks with the Space of Generalized Theta Functions

Published online by Cambridge University Press:  19 November 2021

Shrawan Kumar
Shrawan Kumar
Affiliation:
University of North Carolina, Chapel Hill
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Summary

We prove that the ind-scheme ? consisting of regular maps from an affine curve to a simple, simply-connected group G as closed points is an irreducible and reduced ind-projective variety. By taking the Laurent series expansion at any point at infinity, we can view ? as a subgroup of the loop group. We prove that the central extension of the loop group splits over ? if the affine curve has a single point at infinity. We prove that the space of vacua for any s-pointed smooth curve ? is identified with the space of global sections of the moduli stack of quasi-parabolic G-bundles over ? with respect to a certain line bundle. We also determine the Picard group of this moduli stack. We introduce the notion of theta bundle for a family of G-bundles over ? and determine it for the tautological family of G-bundles over ? parameterized by the infinite Grassmannian in terms of the Dynkin index. The main result of this chapter asserts that there is a canonical identification between the space of global sections of a line bundle over the coarse moduli space of parabolic G-bundles over any s-pointed smooth projective curve ? and the space of vacua.

Keywords

determinant bundletheta bundleDynkin indexspace of vacua and its identification with generalized theta functionsmoduli stack of quasi-parabolic G-bundlesmoduli space of parabolic G-bundlesGrothendieck--Riemann--Roch theorem
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Conformal Blocks, Generalized Theta Functions and the Verlinde Formula , pp. 328 - 378
Publisher: Cambridge University Press
Print publication year: 2021

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References

Abe, T. Strange duality for parabolic symplectic bundles on a pointed projective line, Int. Math. Res. Not., Art. ID rnn121 (2008).Google Scholar
Abe, T. Deformation of rank 2 quasi-bundles and some strange dualities for rational surfaces, Duke Math. J. 155, 577–620 (2010).Google Scholar
Abe, T. Strange duality for height zero moduli spaces of sheaves on P2, Michigan Math. J. 64, 569–586 (2015).CrossRefGoogle Scholar
Agnihotri, S. and Woodward, C. Eigenvalues of products of unitary matrices and quantum Schubert calculus, Math. Res. Lett. 5, 817–836 (1998).CrossRefGoogle Scholar
Agrebaoui, B. Standard modules and standard modules of level one, J. Pure Appl. Alg. 102, 235–241 (1995).Google Scholar
Alekseev, A., Meinrenken, E. and Woodward, C. Formulas of Verlinde type for non-simply connected groups, ArXiv: math/0005047 (2000).Google Scholar
Altschuler, D., Bauer, M. and Itzykson, C. The branching rules of conformal embeddings, Comm. Math. Phys. 132, 349–364 (1990).CrossRefGoogle Scholar
Anchouche, B., Azad, H. and Biswas, I. Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold, Math. Ann. 323, 693–712 (2002).CrossRefGoogle Scholar
Arbarello, E., Cornalba, M., Griffiths, P. and Harris, J. Geometry of Algebraic Curves, vol. I. Springer-Verlag, Berlin, 1985.Google Scholar
Arbarello, E., Cornalba, M., Griffiths, P. and Harris, J. Geometry of Algebraic Curves vol. II. Grundlehren der Mathematischen Wissenschaften, vol. 268. Springer, New York, 2011.CrossRefGoogle Scholar
Atiyah, M. F. Vector bundles over an elliptic curve, Proc. London Math. Soc. (Third Series) 7, 412–452 (1957).Google Scholar
Atiyah, M. F. and Bott, R. The Yang–Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A 308, 523–615 (1982).Google Scholar
Atiyah, M.F. and Macdonald, I.G. Introduction to Commutative Algebra. Addison– Wesley, Boston, MA, 1969.Google Scholar
Baier, T., Bolognesi, M., Martens, J. and Pauly, , C, . The Hitchin connection in arbitrary chracteristic, ArXiv: 2002-12288 (Math. AG) (2020).Google Scholar
Bais, F.A. and Bouwknegt, P.G. A classification of subgroup truncations of the Bosonic string, Nuclear Phys. B 279, no. 3–4, 561–570 (1987).Google Scholar
Bakalov, B. and Kirillov, Jr., A. Lectures on Tensor Categories and Modular Functors. University Lecture Series, vol. 21. American Mathematical Society, Providence, RI, 2001.CrossRefGoogle Scholar
Balaji, V., Biswas, I. and Nagaraj, D.S. Principal bundles over projective manifolds with parabolic structure over a divisor, Tohoku Math. J. 53, 337–367 (2001).Google Scholar
Balaji, V., Biswas, I. and Pandey, Y. Connections on parahoric torsors over curves, Publ. RIMS 53, 551–585 (2017).Google Scholar
Balaji, V. and Parameswaran, A.J. Semistable principal bundles-II (positive characteristics), Transformation Groups 8, 3–36 (2003).Google Scholar
Balaji, V. and Seshadri, C.S. Moduli of parahoric G-torsors on a compact Riemann surface, J. Alg. Geom. 24, 1–49 (2015).Google Scholar
Baldoni, V., Boysal, A. and Vergne, M. Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces, J. Symbolic Computation 68, 27–60 (2015).Google Scholar
Barlet, D. and Magnússon, J. Cycles Analytiques complexes I: Théorèmes de Préparation des Cycles, Publication Société Mathématique de France, Cours Spécialisés 22 (2014).Google Scholar
Beauville, A. Conformal blocks, fusion rules and the Verlinde formula, Israel Math. Conf. Proc.,vol.9, 75–96 (1996).Google Scholar
Beauville, A. The Verlinde formula for PGLp, Adv. Ser. Math. Phys., vol.24, 141–151 (1997).Google Scholar
Beauville, A. Orthogonal bundles on curves and theta functions, Ann. Inst. Fourier (Grenoble) 56, 1405–1418 (2006).Google Scholar
Beauville, A. and Laszlo, Y. Conformal blocks and generalized theta functions, Commun. Math. Phys. 164, 385–419 (1994).Google Scholar
Beauville, A. and Laszlo, Y. Un lemme de descente, C.R. Acad. Sci. Paris 320, 335–340 (1995).Google Scholar
Beauville, A., Laszlo, Y. and Sorger, C. The Picard group of the moduli of G-bundles on a curve, Compositio Math. 112, 183–216 (1998).CrossRefGoogle Scholar
Beauville, A, Narasimhan, M.S. and Ramanan, , S. Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398, 169–179 (1989).Google Scholar
Behrend, K. A. The Lefschetz trace formula for the moduli stack of principal bundles, PhD thesis, University of California at Berkeley, 1991.Google Scholar
Behrend, K. A. Semi-stability of reductive group schemes over curves, Math. Ann. 301, 281–305 (1995).Google Scholar
Beilinson, A., Bloch, S. and Esnault, H. ɛ-factors for Gauss–Manin determinants, Moscow Mathematical Journal 2, 477–532 (2002).Google Scholar
Beilinson, A. and Drinfeld, V. Quantization of Hitchin’s integrable system and Hecke eigensheaves, Preprint (1994).Google Scholar
Beilinson, A., Feigin, B. and Mazur, B. Introduction to algebraic field theory on curves, Preprint (1990).Google Scholar
Beilinson, A. and Ginzburg, V. Infinitesimal structure of moduli spaces of G-bundles, International Math. Res. Notices,Issue 4, 63–74 (1992).Google Scholar
Beilinson, A. and Kazhdan, D. Flat projective connections, Preprint (1990).Google Scholar
Beilinson, A. and Schechtman, V. Determinant bundles and Virasoro algebras, Commun. Math. Phys. 118, 651–701 (1988).CrossRefGoogle Scholar
Belkale, P. Local systems on P1 − S for S a finite set, Compositio Math. 129, 67–86 (2001).Google Scholar
Belkale, P. Invariant theory of GL(n) and intersection theory of Grassmannians, Int. Math.Res.Not. 2004, 3709–3721 (2004a).Google Scholar
Belkale, P. Transformation formulas in quantum cohomology, Compos. Math. 140, 778–792 (2004b).Google Scholar
Belkale, P. Quantum generalization of the Horn conjecture, J. Amer. Math. Soc. 21, 365–408 (2008a).Google Scholar
Belkale, P. The strange duality conjecture for generic curves, J. Amer. Math. Soc. 21, 235–258 (2008b).Google Scholar
Belkale, P. Strange duality and the Hitchin/WZW connection, J. Differential Geom. 82, 445–465 (2009).CrossRefGoogle Scholar
Belkale, P. Orthogonal bundles, theta characteristics and symplectic strange duality, Contemp. Math. 564, 185–193 (2012a).Google Scholar
Belkale, P. Unitarity of the KZ/Hitchin connection on conformal blocks in genus 0 for arbitrary Lie algebras, J. Math. Pures Appl. 98, 367–389 (2012b).Google Scholar
Belkale, P. and Fakhruddin, N. Triviality properties of principal bundles on singular curves, Algebr. Geom. 6, 234–259 (2019).CrossRefGoogle Scholar
Belkale, P., Gibney, A. and Mukhopadhyay, S. Vanishing and identities of conformal blocks divisors, Algebr. Geom. 2, 62–90 (2015).Google Scholar
Belkale, P., Gibney, A. and Mukhopadhyay, S. Nonvanishing of conformal blocks divisors on ¯M0,n, Transformation Groups 21, 329–353 (2016).Google Scholar
Belkale, P. and Kumar, S. The multiplicative eigenvalue problem and deformed quantum cohomology, Advances in Math. 288, 1309–1359 (2016).CrossRefGoogle Scholar
Beltrametti, M. and Robbiano, L. Introduction to the theory of weighted projective spaces, Expo. Math. 4, 111–162 (1986).Google Scholar
Berline, N., Getzler, E. and Vergne, M. Heat Kernels and Dirac Operators. Grundlehren Text Editions. Springer. New York, 2004.Google Scholar
Bernshtein, I.N. and Shvartsman, O.V. Chevalley’s theorem for complex crystallographic Coxeter groups, Functional Analysis and its Applications 12, 308–310 (1978).Google Scholar
Bertram, A. Generalized SU(2) theta functions, Invent. Math. 113, 351-372 (1993).Google Scholar
Bertram, A. and Szenes, A. Hilbert polynomials of moduli spaces of rank 2 vector bundles II, Topology 32, 599-609 (1993).Google Scholar
Bhosle, U. and Ramanathan, A. Moduli of parabolic G-bundles on curves, Math. Z. 202, 161–180 (1989)Google Scholar
Bialynicki-Birula, A. and Swiecicka, J. A reduction theorem for existence of good quotients, Amer. J. Math. 113, 189–201 (1991).Google Scholar
Bismut, J.-M. and Labourie, F. Symplectic geometry and the Verlinde formulas, Surveys in Differential Geometry 5, 97–311 (1999).Google Scholar
Biswas, I. Parabolic bundles as orbifold bundles, Duke Math. J. 88, 305–325 (1997).CrossRefGoogle Scholar
Biswas, I. A criterion for the existence of a parabolic stable bundle of rank two over the projective line, International Journal of Mathematics 9, 523–533 (1998).Google Scholar
Biswas, I. and Holla, Y. I. Harder–Narasimhan reduction of a principal bundle, Nagoya Math. J. 174, 201–223 (2004).CrossRefGoogle Scholar
Biswas, I. and Holla, Y. I. Principal bundles whose restriction to curves are trivial, Math. Z. 251, 607–614 (2005).Google Scholar
Biswas, I. and Ramanan, S. An infinitesimal study of the moduli of Hitchin pairs, J. London Math. Soc. 49, 219–231 (1994).CrossRefGoogle Scholar
Biswas, I. and Raghavendra, N. Determinants of parabolic bundles on Riemann surfaces, Proc. of the Indian Academy of Sciences - Mathematical Sciences 103, 41–71 (1993).Google Scholar
Boden, H. Representations of orbifold groups and parabolic bundles, Comment. Math. Helvetici 66, 389–447 (1991).Google Scholar
Borel, A. Linear Algebraic Groups, 2nd enlarged edn. Graduate Texts in Mathematics, vol. 126. Springer, New York, 1991.Google Scholar
Bott, R. An application of the Morse theory to the topology of Lie-groups, Bull. Soc. Math. France 84, 251–281 (1956).Google Scholar
Bott, R. and Samelson, H. Applications of the theory of Morse to symmetric spaces, Am.J.Math. 80, 964–1029 (1958).Google Scholar
Bourbaki, N. Lie Groups and Lie Algebras, Chap. 4–6, Springer, New York, 2002.Google Scholar
Bourbaki, N. Lie Groups and Lie Algebras, Chap. 7–9, Springer, New York, 2005.Google Scholar
Boutot, J.-F. Singularités rationnelles et quotients par les groupes ŕeductifs, Invent. Math. 88, 65-68 (1987).Google Scholar
Boysal, A. Nonabelian theta functions of positive genus, Proc. A.M.S. 136, 4201–4209 (2008).Google Scholar
Boysal, A. and Kumar, S. Explicit determination of the Picard group of moduli spaces of semistable G-bundles on curves, Math. Ann. 332, 823–842 (2005).Google Scholar
Boysal, A. and Kumar, S. A conjectural presentation of fusion algebras. Advanced Studies in Pure Math. vol. 54 (Algebraic Analysis and Around), 95–107 (2009).Google Scholar
Boysal, A. and Pauly, C. Strange duality for Verlinde spaces of exceptional groups at level one, Int. Math. Res. Not., Issue 4, 595–618 (2010).Google Scholar
Boysal, A. and Vergne, M. Multiple Bernoulli series, an Euler–Maclaurin formula, and wall crossings, Ann. Inst. Fourier, Grenoble 62, 821–858 (2010).CrossRefGoogle Scholar
Bremner, M. R., Moody, R.V. and Patera, J. Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras, Marcel Dekker, New York, 1985.Google Scholar
Bröcker, T. and tom, Dieck, T. Representations of Compact Lie Groups. Graduate Texts in Mathematics, vol. 98. Springer, New York, 1985.CrossRefGoogle Scholar
Bruguieres, A. Filtration de Harder-Narasimhan et stratification de Shatz, in: Module des Fibrés Stables sur les Courbes Algébriques. Progress in Mathematics, vol. 54. Birkhäuser, Basel, pp. 83–106 (1985).Google Scholar
Cartan, H. Quotient d’un espace analytique par un groupe d’automorphismes, in: Algebraic Geometry and Topology (A Symposium in Honor of S. Lefschetz, Princeton, NJ), 90–102 (1957).Google Scholar
Cartan, H. and Eilenberg, S. Homological Algebra. Princeton University Press, Princeton, NJ, 1956.Google Scholar
Cellini, P., Kac, V.G., Möseneder, F.P. and Papi, P. Decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of Z2-graded Lie algebras, Adv. Math. 207, 156–204 (2006).Google Scholar
Chevalley, C. and Eilenberg, S. Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63, 85–124 (1948).Google Scholar
Chriss, N. and Ginzburg, V. Representation Theory and Complex Geometry.Birkhäuser, Basel, 1997.Google Scholar
Conrad., B., Gabber, O. and Prasad, G. Pseudo-reductive Groups, 2nd edn. Cambridge University Press, Cambridge, 2015.Google Scholar
Damiolini, C. Conformal blocks attached to twisted groups, Math. Z. 295, 1643–1681 (2020).Google Scholar
Damiolini, C., Gibney, A. and Tarasca, N. On factorization and vector bundles of conformal blocks from vertex algebras, ArXiv: 1909.04683 v2 (2019).Google Scholar
Damiolini, C., Gibney, A. and Tarasca, N. Vertex algebras of cohFT-type, ArXiv: 1910.01658 v2 (2020).Google Scholar
Daskalopoulos, G. and Wentworth, R. Local degeneration of the moduli space of vector bundles and factorization of rank two theta functions. I, Math. Ann. 297, 417–466 (1993).Google Scholar
Daskalopoulos, G. and Wentworth, R. Factorization of rank two theta functions. II: proof of the Verlinde formula, Math. Ann. 304, 21–51 (1996). Deligne, P. Cohomologie Etale, SGA 4 12 , Springer Lecture Notes in Mathematics, vol. 569. Springer, New York, 1977.Google Scholar
Deligne, P. and Mumford, D. The irreducibility of the space of curves of a given genus, Publications Math. IHES 36, 75–109 (1969).Google Scholar
Demazure, M. and Gabriel, P. Introduction to Algebraic Geometry and Algebraic Groups. North-Holland Mathematics Studies, vol. 39. North-Holland, Amsterdam, 1980.Google Scholar
Demazure, M. and Grothendieck, A. Schémas en groupes, SGA III. Lecture Notes in Mathematics, vol. 153. Springer, New York, 1970.Google Scholar
Dolgachev, I. Weighted projective varieties, in: Group Actions and Vector Fields. Springer Lecture Notes in Mathematics, vol. 956, 34–71. Springer, New York, 1982.Google Scholar
Donagi, R. and Tu, L. W. Theta functions for SL(n) versus GL(n), Math. Res. Lett. 1, 345–357 (1994).Google Scholar
Donaldson, S.K. A new proof of a theorem of Narasimhan and Seshadri, J. Diff. Geom. 18, 269–277 (1983).Google Scholar
Douady, A. Le problème des modules pour les variétés analytiques complexes, Séminaire Bourbaki, Exposé 277, anneés 1964–65 (1966).Google 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, 53–94 (1989).Google Scholar
Drinfeld, V. and Simpson, C. B-structures on G-bundles and local triviality, Math. Res. Letters 2, 823–829 (1995).CrossRefGoogle Scholar
Dynkin, E.B. Semisimple subalgebras of semisimple Lie algebras, Am. Math. Soc. Transl. (Ser. II) 6, 111–244 (1957).Google Scholar
Edixhoven, B. Néron models and tame ramification, Compositio Math. 81, 291–306 (1992).Google Scholar
Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York, 1995.Google Scholar
Eisenbud, D. and Harris, J. The Geometry of Schemes. Graduate Texts in Mathematics, vol. 197. Springer, New York, 2000.Google Scholar
Fakhruddin, N. Chern classes of conformal blocks, Contemp. Math. 564, 145–176 (2012).Google Scholar
Faltings, G. Stable G-bundles and projective connections, J. Alg. Geom. 2, 507–568 (1993).Google Scholar
Faltings, G. A proof for the Verlinde formula, J. Alg. Geom. 3, 347–374 (1994).Google Scholar
Faltings, G. Algebraic loop groups and moduli spaces of bundles, J. Eur. Math. Soc. 5, 41–68 (2003).Google Scholar
Faltings, G. Theta functions on moduli spaces of G-bundles, J. Alg. Geom. 18, 309–369 (2009).Google Scholar
Fantechi, B. Stacks for everybody, in: Progress in Mathematics vol. 201. Birkhäuser, Basel, 2001.Google Scholar
Feigin, B.L., Schechtman, V.V. and Varchenko, A.N. On algebraic equations satisfied by correlators in Wess–Zumino–Witten models, Letters in Mathematical Physics 20, 291–297 (1990).CrossRefGoogle Scholar
Fontaine, Jean-Marc. Groupes p-divisible sur les corps locaux, Astérisque Bd. 47/48, Publication Société Mathématique de France, 1977.Google Scholar
Friedman, R., Morgan, J.W. and Witten, E. Principal G-bundles over elliptic curves, Mathematical Research Letters 5, 97–118 (1998).Google Scholar
Fuchs, J. and Schweigert, C. A representation theoretic approach to the WZW Verlinde formula, ArXiv: hep-th/9707069 (1997).Google Scholar
Fulton, W. Intersection Theory, 2nd edn. Springer, New York, 1998.Google Scholar
Fulton, W. and Pandharipande, R. Notes on stable maps and quantum cohomology, in: Algebraic Geometry—Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Part 2, 45–96 (1997).Google Scholar
Furuta, M. and Steer, B. Seifert fibred homology 3-spheres and the Yang–Mills equations on Riemann surfaces with marked points, Adv. Math. 96, 38–102 (1992).Google Scholar
Garland, H. The arithmetic theory of loop groups, Publ. Math. IHES 52, 5–136 (1980).Google Scholar
Garland, H. and Lepowsky, J. Lie algebra homology and the Macdonald–Kac formulas, Invent. Math. 34 , 37–76 (1976).Google Scholar
Garland, H. and Raghunathan, M.S. A Bruhat decomposition for the loop space of a compact group: A new approach to results of Bott, Proc. Natl. Acad. Sci. USA 72, 4716–4717 (1975).Google Scholar
van Geemen, B. and de Jong, A.J. On Hitchin’s connection, J. Am. Math. Soc. 11, 189–228 (1998).Google Scholar
Ginzburg, V. Resolution of diagonals and moduli spaces, in: The Moduli Space of Curves. Progress in Mathematics, vol. 129. Birkhäuser, Basel, 231–266 (1995).Google Scholar
Goddard, P., Kent, A. and Olive, D. Virasoro algebras and coset space models, Physics Letters B152, 88–92 (1985).CrossRefGoogle Scholar
Goodman, R. and Wallach, N.R. Symmetry, Representations, and Invariants. Graduate Texts in Mathematics, vol. 255. Springer, New York, 2009.Google Scholar
Goodman, F.M. and Wenzl, H. Littlewood–Richardson coefficients for Hecke algebras at roots of unity, Adv. Math. 82, 244–265 (1990).CrossRefGoogle Scholar
Grauert, H. and Riemenschneider, O. Verschwindungssätze für analytische Kohomologiegruppen auf Komplexen Raümen, Lecture Notes in Mathematics, vol. 155. Springer, New York, 1970.Google Scholar
Grégoire, C. and Pauly, C. The space of generalized G2-theta functions of level 1, Michigan Math. J. 62, 857–867 (2013).Google Scholar
Griffiths, P. and Harris, J. Principles of Algebraic Geometry. Wiley, Chichester, 1978.Google Scholar
Grothendieck, A. Sur le mémoire de Weil. “généralisation des fonctions abéliennes”, Séminaire Bourbaki, Exposé 141, pp. 57–71, 1956–57.Google Scholar
Grothendieck, A. Techniques de construction et théorèmes d’existence en géometérie algébrique IV: Les schémas de Hilbert, Séminaire Bourbaki, Exposés 205–222, talk 221, pp. 249–276, 1960–61.Google Scholar
Grothendieck, A. Éléments de géométrie algébrique: I, Publications Math. IHES 4, pp. 5–228 (1960).Google Scholar
Grothendieck, A. Éléments de géométrie algébrique II, Publications Math. IHES 8, pp. 5–222, (1961a).Google Scholar
Grothendieck, A. Éléments de géométrie algébrique III (Première Partie), Publications Math. IHES 11, pp. 5–167 (1961b).Google Scholar
Grothendieck, A. Éléments de géométrie algébrique: IV (Seconde Partie), Publications Math. IHES 24, pp. 5–231(1965).Google Scholar
Grothendieck, A. Éléments de géométrie algébrique: IV (Quatrième Partie), Publications Math. IHES 32, pp. 5–361 (1967).Google Scholar
Grothendieck, A. Revêtements Étales et Groupe Fondamental (SGA 1). Lecture Notes in Mathematics, vol. 224. Springer, New York, 1971.Google Scholar
Harder, G. Halbeinfache gruppenschemata über Dedekindringen, Invent. Math. 4, 165–191 (1967).Google Scholar
Harder, G. Halbeinfache gruppenschemata über vollständigen kurven, Invent. Math. 6, 107–149 (1968).Google Scholar
Harder, G. and Narasimhan, M.S. On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212, 215–248 (1975).Google Scholar
Harris, J. and Morrison, I. Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, New York, 1998.Google Scholar
Hartshorne, R. Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer-Verlag, New York, 1977.Google Scholar
Hasegawa, K. Spin module versions of Weyl’s reciprocity theorem for classical Kac– Moody Lie algebras—an application to branching rule duality, Publ. Res. Inst. Math. Sci. 25, 741–828 (1989).Google Scholar
Hatcher, A. Algebraic Topology. Cambridge University Press, Cambridge, 2001.Google Scholar
Heinloth, J. Bounds for Behrend’s conjecture on the canonical reduction, Int. Math. Res. Not., Issue 9 (2008).Google Scholar
Heinzner, P. and Kutzschebauch, F. An equivariant version of Grauert’s Oka principle, Invent. Math. 119, 317–346 (1995).Google Scholar
Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, 1978.Google Scholar
Hilton, P.J. and Stammbach, U. A. Course in Homological Algebra, 2nd edn. Graduate Texts in Mathematics, vol. 4. Springer, New York, 1997.Google Scholar
Hitchin, N. Flat connections and geometric quantization, Commun. Math. Phys. 131, 347–380 (1990).Google Scholar
Hochschild, G. and Serre, J-P. Cohomology of group extensions, Trans. Amer. Math. Soc. 74, 110–134 (1953).Google Scholar
Hong, J. and Kumar, S. Conformal blocks for Galois covers of algebraic curves, ArXiv: 1807.00118 v3 (2019).Google Scholar
Hotta, R., Takeuchi, K. and Tanisaki, T. D-Modules, Perverse Sheaves, and Representation Theory. Progress in Mathematics, vol. 236. Birkhäuser, Basel, 2008.Google Scholar
Huang, Y.-Z. Vertex operator algebras and the Verlinde conjecture, Comm. Contemp. Math. 10, 103–154 (2008).Google Scholar
Humphreys, J.E. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9. Springer, New York, 1972.Google Scholar
Humphreys, J.E. Conjugacy Classes in Semisimple Algebraic Groups. American Mathematical Society, Providence, RI, 1995.Google Scholar
Hurtubise, J.C. Holomorphic maps of a Riemann surface into a flag manifold, J. Diff. Geom. 43, 99–118 (1996).Google Scholar
Illusie, L. Grothendieck’s existence theorem in formal geometry, in: Fundamental Algebraic Geometry- Grothendieck’s FGA Explained (edited by B. Fantechi et al.), Mathematical Surveys and Monographs, vol. 123. American Mathematical Society, Providence, RI, 181–233 (2005).Google Scholar
Iwahori, N. and Matsumoto, H. On some Bruhat decomposition and the structure of the Hecke rings of p–adic Chevalley groups, Publ. Math. IHES 25, 5–48 (1965).Google Scholar
Jantzen, J.C. Representations of Algebraic Groups, 2nd edn. American Mathematical Society, Providence, RI, 2003.Google Scholar
Jeffrey, L. and Kirwan, F. Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. 148, 101–196 (1998).Google Scholar
Jones, G.A. and Singerman, D. Complex Functions. Cambridge University Press, Cambridge, 1987.Google Scholar
Kac, V.G. Highest weight representations of infinite-dimensional Lie algebras, in: Proceeding of ICM, Helsinki, 299–304 (1978).Google Scholar
Kac, V.G. Infinite Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge, 1990.Google Scholar
Kac, V.G. and Peterson, D.H. Spin and wedge representations of infinite-dimensional Lie algebras and groups, Proc. Nat. Acad. Sci. U.S.A. 78, 3308–3312 (1981).Google Scholar
Kac, V.G., Raina, A.K. and Rozhkovskaya, N. Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras, 2nd edn. Advanced Series in Mathematical Physics, vol. 29. World Scientific, Singapore, 2013.Google Scholar
Kac, V.G. and Sanielevici, M.N. Decompositions of representations of exceptional affine algebras with respect to conformal subalgebras, Phys. Rev. D 37, 2231–2237 (1988)Google Scholar
Kac, V.G. and Wakimoto, M. Modular and conformal invariance constraints in representation theory of affine algebras, Adv. Math. 70, 156–236 (1988).Google Scholar
Kazhdan, D. and Lusztig, G. Schubert varieties and Poincaré duality, Proc. Symp. Pure Math. (A.M.S.) 36, 185–203 (1980).Google Scholar
Kazhdan, D. and Lusztig, G. Tensor structures arising from affine Lie algebras. I, J. Am. Math. Soc. 6, 905–947 (1993).Google Scholar
Kempf, G. The Grothendieck–Cousin complex of an induced representation, Advances in Math. 29, 310–396 (1978).Google Scholar
Kirwan, F. The cohomology rings of moduli spaces of bundles over Riemann surfaces, J. Am. Math. Soc. 5, 853–906 (1992).Google Scholar
Knapp, A.W. Lie Groups Beyond an Introduction, 2nd edn. Progress in Mathematics, vol. 140, Birkhäuser, Basel, 2002.Google Scholar
Knudsen, F. The projectivity of the moduli space of stable curves, II: The stacks Mg,n, Math. Scand. 52, 161–199 (1983a).Google Scholar
Knudsen, F. The projectivity of the moduli space of stable curves, III: The line bundles on Mg,n, and a proof of the projectivity of ¯Mg,n in charactristic 0, Math. Scand. 52, 200–212 (1983b).Google Scholar
Knudsen, F. and Mumford, D. The projectivity of the moduli space of stable curves I: preliminaries on “det” and “div”, Math. Scand. 39, 19–55 (1976).Google Scholar
Kobayashi, S. and Nomizu, K. Foundations of Differential Geometry, vol. II. Wiley-Interscience, New York 1969.Google Scholar
Kodaira, K. and Spencer, D.C. On deformations of complex analytic structures, I, Annals of Math. 67, 328–401 (1958a).Google Scholar
Kodaira, K. and Spencer, D.C. A theorem of completeness for complex analytic fiber spaces, Acta Math. 100, 281–294 (1958b).Google Scholar
Kollár, J. Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 32. Springer-Verlag, Berlin, 1996.Google Scholar
Kontsevich, M. and Manin, Yu. Gromov–Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys. 164, 525–562 (1994).Google Scholar
Kostant, B. Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra, Invent Math. 158, 181–226 (2004).Google Scholar
Koszul, J.L. Lectures on Fibre Bundles and Differential Geometry. Tata Institute of Fundamental Research Lecture Notes, Bombay, 1960.Google Scholar
Kraft, H. Algebraic automorphisms of affine space, in: Proceedings of the Hyderabad Conference on Algebraic Groups (edited by S. Ramanan), Manoj Prakashan, 251–274 (1991).Google Scholar
Krepski, D. and Meinrenken, E. On the Verlinde formulas for SO(3)-bundles, Quarterly J. Math. 64, 235–252 (2013).Google Scholar
Kresch, A. Algebraic Stacks http://home.kias.re.kr/MKG/h/AlgebraicStacks/ (2007)Google Scholar
Kumar, S. Rational homotopy theory of flag varieties associated to Kac–Moody groups, in: Infinite-dimensional Groups with Applications. MSRI Publication 4. Springer-Verlag, Berlin, 233–273 (1985).Google Scholar
Kumar, S. Demazure character formula in arbitrary Kac–Moody setting, Invent. Math. 89, 395–423 (1987).Google Scholar
Kumar, S. Infinite Grassmannians and moduli spaces of G–bundles, in: Vector Bundles on Curves - New Directions. Lecture Notes in Mathematics, vol. 1649. Springer-Verlag, Berlin, 1–49, 1997a.Google Scholar
Kumar, S. Fusion product of positive level representations and Lie algebra homology, in: Geometry and Physics (edited by J. E. Andersen et al.). Lecture Notes in Pure and Applied Mathematics, vol. 184. Marcel Dekker, New York, 253–259, 1997b.Google Scholar
Kumar, S. Kac–Moody Groups, their Flag Varieties and Representation Theory. Progress in Mathematics, vol. 204. Birkhäuser, Basel, 2002.Google Scholar
Kumar, S. and Narasimhan, M.S. Picard group of the moduli spaces of G-bundles, Math. Ann. 308, 155–173 (1997).Google Scholar
Kumar, S., Narasimhan, M.S. and Ramanathan, A. Infinite Grassmannians and moduli spaces of G–bundles, Math. Ann. 300, 41–75 (1994).Google Scholar
Kuniba, A. and Nakanishi, T. Level-rank duality in fusion RSOS models, Modern quantum field theory, Bombay Quantum Field Theory, 344–374 (1991).Google Scholar
Kuranishi, M. Two elements generations on semi-simple Lie groups, Kôdai Math. Sem. Report 5–6, 9–10 (1949).Google Scholar
Lang, S. Algebra. Addison-Wesley, New York, 1965.Google Scholar
Lang, S. Introduction to Arakelov Theory. Springer, New York, 1988.Google Scholar
Laszlo, Y. A propos de l’espace des modules de fibrés de rang 2 sur une courbe, Math. Ann. 299, 597–608 (1994).Google Scholar
Laszlo, Y. Local structure of the moduli space of vector bundles over curves, Comm. Math. Helv. 71, 373–401 (1996).Google Scholar
Laszlo, Y. Linearization of group stack actions and the Picard group of the moduli of SLr /μs -bundles on a curve, Bull. Soc. Math. France 125, 529–545 (1997).Google Scholar
Laszlo, Y. Hitchin’s and WZW connections are the same, J. Diff. Geom. 49, 547–576 (1998a).Google Scholar
Laszlo, Y. About G-bundles over elliptic curves, Ann. Inst. Fourier, Grenoble 48, 413–424 (1998b).Google Scholar
Laszlo, Y. and Sorger, C. The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Scient. Éc. Norm. Sup. 30, 499–525 (1997).Google Scholar
Laumon, G. and Moret-Bailly, L. Champs Algébriques. Springer-Verlag, New York, 1999.Google Scholar
Le Potier, J. Fibrés Vectoriels sur les Courbes Algébriques. Cours de DEA, Université Paris 7, 1991.Google Scholar
Le Potier, J. Lectures on Vector Bundles. Cambridge University Press, Cambridge, 1997.Google Scholar
Looijenga, E. Root systems and elliptic curves, Invent. Math. 38, 17–32 (1976).Google Scholar
Looijenga, E. Conformal blocks revisited, ArXiv:math/0507086 (2005).Google Scholar
Looijenga, E. The KZ system via polydifferentials, Adv. Stud. in Pure Math. 62, 189–231 (2012).Google Scholar
Looijenga, E. From WZW models to modular functors, in: Handbook of Moduli, vol. II. Advanced Lectures in Mathematics, vol. 25. International Press, Boston, MA, pp. 427–466 (2013).Google Scholar
Marian, A. and Oprea, D. The level-rank duality for non-abelian theta functions, Invent. Math. 168, 225-247 (2007).Google Scholar
Marian, A. and Oprea, D. Sheaves on abelian surfaces and strange duality, Math. Ann. 343, 1-33 (2009).Google Scholar
Marian, A. and Oprea, D. On the strange duality conjecture for abelian surfaces, J. Euro. Math. Soc. 16, 1221–1252 (2014).Google Scholar
Marian, A., Oprea, D., Pandharipande, R., Pixton, A. and Zvonkine, D. The Chern character of the Verlinde bundle over ¯Mg,n, J. Reine Angew. Math. 732, 147–163 (2017).Google Scholar
Mathieu, O. Formules de caractères pour les algèbres de Kac–Moody générales, Astérisque 159–160, 1–267 (1988).Google Scholar
Matsumura, H. Commutative Ring Theory. Cambridge University Press, Cambridge, 1989.Google Scholar
Maunder, C.R.F. Algebraic Topology. Van Nostrand Reinhold Company, London, 1970.Google Scholar
Mehta, V. B. and Ramadas, T.R. Moduli of vector bundles, Frobenius splitting, and invariant theory, Ann. of Math. 144, 269–313 (1996).Google Scholar
Mehta, V. B. and Ramanathan, A. Restriction of stable sheaves and representations of the fundamental group, Invent. Math. 77, 163–172 (1984).Google Scholar
Mehta, V. B. and Seshadri, C. S. Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248, 205–239 (1980).Google Scholar
Mehta, V.B. and Subramanian, S. On the Harder–Narasimhan filtration of principal bundles, in: Algebra, Arithmetic and Geometry Part 1(editedbyR.Parimala). Narosa Publishing House, New Delhi, pp. 405–415 (2002).Google Scholar
Meinrenken, E. and Woodward, C. Hamiltonian loop group actions and Verlinde factorization, J. Diff. Geom. 50, 417–469 (1998).Google Scholar
Milne, J.S. Chap. VII – Jacobian varieties, in: Arithmetic Geometry (edited by G. Cornell et al.). Springer-Verlag, Berlin, pp. 167–212 (1986).Google Scholar
Milne, J.S. Lectures on Étale Cohomology. Online version 2.21, www.jmilne.org/math/CourseNotes/LEC.pdf (2013).Google Scholar
Milnor, J. Morse Theory. Annals of Mathematics Studies, vol. 51. Princeton University Press, Princeton, NJ, 1969.Google Scholar
Milnor, J.W. and Stasheff, J.D. Characteristic Classes. Annals of Mathematics Studies, vol. 76. Princeton University Press, Princeton, NJ, 1974.Google Scholar
Mitchell, S.A. Quillen’s theorem on buildings and the loops on a symmetric space, L’Enseignement Math. 34, 123–166 (1988).Google Scholar
Mlawer, E.J., Naculich, S.G., Riggs, H.A. and Schnitzer, H.J. Group-level duality of WZW fusion coefficients and Chern–Simons link observables, Nuclear Phys. B 352, 863–896 (1991).Google Scholar
Moore, G. and Seiberg, N. Polynomial equations for rational conformal field theories, Phys. Lett. B 212, 451–460 (1988).Google Scholar
Moore, G. and Seiberg, N. Classical and quantum conformal field theory, Commun. Math. Phys. 123, 177–254 (1989).Google Scholar
Mukhopadhyay, S. Diagram automorphisms and rank-level duality, ArXiv: 1308.1756 (2013).Google Scholar
Mukhopadhyay, S. Strange duality of Verlinde spaces for G2 and F4, Math. Z. 283, 387–399 (2016a).Google Scholar
Mukhopadhyay, S. Rank-level duality of conformal blocks for odd orthogonal Lie algebras in genus 0, Trans. Amer. Math. Soc. 368, 6741–6778 (2016b).Google Scholar
Mukhopadhyay, S. Rank-level duality and conformal block divisors, Adv. Math. 287, 389–411 (2016c).Google Scholar
Mukhopadhyay, S. and Wentworth, R. Generalized theta functions, strange duality, and odd orthogonal bundles on curves, Commun. Math. Phys. 370, 325–376 (2019).Google Scholar
Mukhopadhyay, S. and Zelaci, H. Conformal embedding and twisted theta functions at level one, Proc. Amer. Math. Soc. 148, 9–22 (2020).Google Scholar
Mumford, D. Abelian Varieties, 2nd edn. Tata Institute of Fundamental Research, Bombay. Oxford University Press, Oxford, 1985.Google Scholar
Mumford, D. The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol. 1358. Springer-Verlag, Berlin, 1988.Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F. Geometric Invariant Theory, 3rd enlarged edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer-Verlag, Berlin, 2002.Google Scholar
Mumford, D. and Oda, T. Algebraic Geometry II. Texts and Readings in Mathematics, vol. 73. Hindustan Book Agency, Haryana, 2015.Google Scholar
Naculich, S.G. and Schnitzer, H.J. Duality relations between SU(N)k and SU(k)N WZW models and their braid matrices, Phys. Lett. B, 244, 235–240 (1990).Google Scholar
Nakanishi, T. and Tsuchiya, A. Level-rank duality of WZW models in conformal field theory, Commun. Math. Phys. 144, 351–372 (1992).Google Scholar
Narasimhan, M.S. and Ramadas, T.R. Factorisation of generalised theta functions. I, Invent. Math. 114, 565–623 (1993).Google Scholar
Narasimhan, M.S. and Ramanan, S. Moduli of vector bundles on a compact Riemann surface, Ann. of Math. 89, 14–51 (1969).Google Scholar
Narasimhan, M.S. and Seshadri, C.S. Holomorphic vector bundles on a compact Riemann surface, Math. Ann. 155, 69–80 (1964).Google Scholar
Narasimhan, M.S. and Seshadri, C.S. Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82, 540–567 (1965).Google Scholar
Newstead, P.E. Introduction to Moduli Problems and Orbit Spaces. Narosa Publishing House, New Delhi, 2012.Google Scholar
Nitsure, N. Theory of Descent and Algebraic Stacks. KIAS, Seoul, 2005a.Google Scholar
Nitsure, N. Construction of Hilbert and Quot schemes, in: Fundamental Algebraic Geometry- Grothendieck’s FGA Explained (edited by B. Fantechi et al.), Mathematical Surveys and Monographs, vol. 123. American Mathematical Society, Providence, RI, pp. 107–137 (2005b).Google Scholar
Nitsure, N. Deformation theory for vector bundles, in: Moduli Spaces and Vector Bundles (edited by L. Brambila-Paz et al.), London Mathematical Society Lecture Note Series 359, 128–164 (2009).Google Scholar
Olsson, M. Hom-stacks and restriction of scalars, Duke Math. J. 134, 139–164 (2006).Google Scholar
Oort, F. Algebraic group schemes in characteristic zero are reduced, Invent. Math. 2, 79–80 (1966).Google Scholar
Oprea, D. The Verlinde bundles and the semihomogeneous Wirtinger duality, J. Reine Angew. Math. 654, 181–217 (2011).Google Scholar
Oudompheng, R. Rank-level duality for conformal blocks of the linear group, J. Alg. Geom. 20, 559–597 (2011).Google Scholar
Oxbury, W.M. and Wilson, S.M.J. Reciprocity laws in the Verlinde formulae for the classical groups, Trans. Amer. Math. Soc. 348, 2689–2710 (1996).Google Scholar
Pantev, T. Comparison of generalized theta functions, Duke Math. J. 76, 509–539 (1994).Google Scholar
Pappas, G. and Rapoport, M. Some questions about G -bundles on curves, in: Algebraic and Arithmetic Structures of Moduli Spaces. Advanced Studies in Pure Mathematics, vol. 58. Cambridge University Press, Cambridge, pp. 159–171 (2010).Google Scholar
Parthasarathi, P. On parabolic bundles on algebraic surfaces, J. Ramanujan Math. Soc. 28, 379–413 (2013).Google Scholar
Pauly, C. Espaces de modules de fibrés praboliques et blocs conformes, Duke Math. J. 84, 217–235 (1996).Google Scholar
Pauly, C. La dualité étrange [d’après P. Belkale, A. Marian et D. Oprea] Astérisque 326, Exp. No. 994, 363–377 (2009).Google Scholar
Pauly, C. Strange duality revisited, Math. Res. Lett. 21, 1353–1366 (2014).Google Scholar
Pauly, C. and Ramanan, S. A duality for spin Verlinde spaces and Prym theta functions, J. London. Math. Soc. 63, 513–532 (2001).Google Scholar
Popov, V.L. and Vinberg, E.B. Invariant Theory, Algebraic Geometry IV. Encyclopaedia of Mathematical Sciences. vol. 55. Springer, New York, pp. 123–278 (1994).Google Scholar
Pressley, A. and Segal, G. Loop Groups. Clarendon Press, Oxford, 1988.Google Scholar
Quillen, D. Determinants of Cauchy–Riemann operators over a Riemann surface, Funct. Anal. Appl. 19, 31–34 (1985).Google Scholar
Raghunathan, M. S. Principal bundles on affine space, in: C.P. Ramanujam– A Tribute. Springer-Verlag, Berlin, pp. 223–244 (1978).Google Scholar
Ramanan, S. A note on C.P. Ramanujam, in: C.P. Ramanujam – A Tribute. Springer-Verlag, Berlin, pp. 13–15 (1978).Google Scholar
Ramanan, S. and Ramanathan, A. Some remarks on the instability flag, Tôhoku Math. J. 36, 269–291 (1984).Google Scholar
Ramanathan, A. Stable principal bundles on a compact Riemann surface, Math. Ann. 213, 129–152 (1975).Google Scholar
Ramanathan, A. Moduli of principal bundles, Lecture Notes in Mathematics, vol. 732, 527–533. Springer, Berlin, 1979.Google Scholar
Ramanathan, A. Deformations of principal bundles on the projective line, Invent. Math. 71, 165–191 (1983).Google Scholar
Ramanathan, A. Moduli for principal bundles over algebraic curves: I and II, Proc. Indian Acad. Soc. (Math. Sci.) 106, 301–328 and 421–449 (1996).Google Scholar
Ramanathan, A. and Subramanian, S. Einstein–Hermitian connections on principal bundles and stability, J. Reine Angew. Math. 390, 21–31 (1988).Google Scholar
Remmert, R. Holomorphe und meromorphe abbildungen komplexer räume, Math. Ann. 133, 328–370 (1957).Google Scholar
Rudin, W. Real and Complex Analysis. McGraw-Hill, New York, 1966.Google Scholar
Safarevic, I.R. On some infinite–dimensional groups. II, Math. USSR Izvestija 18, 185–194 (1982).Google Scholar
Schellekens, A.N. and Warner, N.P. Conformal subalgebras of Kac–Moody algebras, Phys. Rev. D 34, 3092–3096 (1986).Google Scholar
Schieder, S. The Harder–Narasimhan stratification of the moduli stack of G-bundles via Drinfeld’s compactifications, Selecta Math. 21, 763–831 (2015).Google Scholar
Selberg, A. On discontinuous groups in higher-dimensional symmetric spaces, in: Internat. Colloquium on Function Theory, Bombay, 1960. Tata Institute of Fundamental Research, Bombay, pp. 147–164 (1960).Google Scholar
Serre, J.-P. Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6, 1–42 (1956).Google Scholar
Serre, J.-P. Espaces fibrés algébriques, in: Anneaux de Chow et applications, Séminaire C. Chevalley, 1958.Google Scholar
Serre, J.-P. Algèbres de Lie Semi-simples Complexes. W.A. Benjamin, Inc., New York, 1966.Google Scholar
Serre, J.-P. Topics in Galois Theory. Jones and Bartlett Publishers, Boston, MA, 1992.Google Scholar
Serre, J.-P. Galois Cohomology. Springer Monographs in Mathematics, Springer, New York (1997).Google Scholar
Seshadri, C.S. Space of unitary vector bundles on a compact Riemann surface, Annals of Math. 85, 303–336 (1967).Google Scholar
Seshadri, C.S. Moduli of vector bundles on curves with parabolic structures, Bull. Amer. Math. Soc. 83, 124–126 (1977).Google Scholar
Seshadri, C.S. Vector Bundles on Curves. Contemporary Mathematics, vol. 153. American Mathematical Society, Providence, RI (1993).Google Scholar
Seshadri, C.S. Moduli of π-vector bundles over an algebraic curve, in: Collected Papers of C.S. Seshadri, vol. 250. Hindustan Book Agency, 2011.Google Scholar
Shatz, S. The decomposition and specialization of algebraic families of vector bundles, Compositio Math. 35, 163–187 (1977).Google Scholar
Sheinman, O.K. Current Algebras on Riemann Surfaces (New results and applications). De Gruyter Expositions in Mathematics, vol. 58. De Gruyter, Berlin, 2012.Google Scholar
Simpson, C. Moduli of representations of the fundamental group of a smooth projective variety II, Publ. Math. I.H.E.S. 80, 5–79 (1994).Google Scholar
Simpson, C. Algebraic (Geometric) n-Stacks, ArXiv.alg-geom/9609014 (1996).Google Scholar
Slodowy, P. On the geometry of Schubert varieties attached to Kac–Moody Lie algebras, in: Can. Math. Soc. Conf. Proc. on ‘Algebraic Geometry’ (Vancouver), vol. 6, 405–442 (1984).Google Scholar
Sorger, C. La formule de Verlinde, Séminaire Bourbaki,47ème année, no 794 (1994).Google Scholar
Sorger, C. La semi-caractéristique d’Euler–Poincaré des faisceaux ω-quadratiques sur un schéma de Cohen-Macaulay, Bull. Soc. Math. France 122, 225–233 (1994).Google Scholar
Sorger, C. On moduli of G-bundles on a curve for exceptional G, Ann. Scient. Éc. Norm. Sup. 32, 127–133 (1999).Google Scholar
Sorger, C. Lectures on moduli of principal G-bundles over algebraic curves, School on Algebraic Geometry, Trieste, pp. 1–57 (1999).Google Scholar
Spanier, E.H. Algebraic Topology. McGraw-Hill, New York, 1966.Google Scholar
Stacks. The Stacks project, Online version, https://stacks.math.columbia.edu/ (2019).Google Scholar
Steenrod, N. The Topology of Fibre Bundles. Princeton University Press, Princeton, NJ, 1951.Google Scholar
Steinberg, R. Lectures on Chevalley Groups. University Lecture Series, vol. 66. American Mathematical Society, Providence, RI, 2016.Google Scholar
Sun, X. Degeneration of moduli spaces and generalized theta functions, J. Alg. Geom. 9, 459–527 (2000).Google Scholar
Sun, X. Factorization of generalized theta functions in the reducible case, Ark. Mat. 41, 165–202 (2003).Google Scholar
Sun, X. and Tsai, I-H. Hitchin’s connection and differential operators with values in the determinant bundle, J. Diff. Geom. 66, 303–343 (2004).Google Scholar
Sun, X. and Zhou, M. Globally F -regular type of moduli spaces and Verlinde formula, ArXiv: 1802.08392 (2018).Google Scholar
Suzuki, T. A new proof of the finite-dimensionality of the space of conformal blocks, Preprint (1990).Google Scholar
Szenes, A. Hilbert polynomials of moduli spaces of rank 2 vector bundles I, Topology 32, 587-597 (1993).Google Scholar
Szenes, A. The combinatorics of the Verlinde formula, in: Vector Bundles in Algebraic Geometry (edited by N. J. Hitchin et al.). London Mathematical Society Lecture Note Series, vol. 208. Cambridge University Press, Cambridge, pp. 241– 254 (1995).Google Scholar
Teleman, C. Lie algebra cohomology and the fusion rules, Commun. Math. Phys. 173, 265–311 (1995).Google Scholar
Teleman, C. Verlinde factorization and Lie algebra cohomology, Invent. Math. 126, 249–263 (1996).Google Scholar
Teleman, C. Borel–Weil–Bott theory on the moduli stack of G-bundles over a curve, Invent. Math. 134, 1–57 (1998).Google Scholar
Teleman, C. and Woodward, C. Parabolic bundles, products of conjugacy classes and Gromov–Witten invariants, Ann. Inst. Fourier (Grenoble) 51, 713–748 (2001)Google Scholar
Thaddeus, M. Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117, 317-353 (1994).Google Scholar
Tsuchimoto, Y. On the coordinate-free description of the conformal blocks, J. Math. Kyoto Univ. 33, 29–49 (1993).Google Scholar
Tsuchiya, A. and Kanie, Y. Vertex operators in Conformal field theory on P1 and monodromy representations of braid group, Adv. Stud. Pure Math. 16, 297–372 (1988).Google Scholar
Tsuchiya, A., Ueno, K. and Yamada, Y. Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Stud. Pure Math. 19, 459–566 (1989).Google Scholar
Tu, L.W. Semistable bundles over an elliptic curve, Adv. in Math. 98, 1-26 (1993).Google Scholar
Ueno, K. Conformal Field Theory with Gauge Symmetry. Fields Institute Monographs. American Mathematical Society, Providence, RI, 2008.Google Scholar
Verlinde, E. Fusion rules and modular transformations in 2D conformal field theory, Nucl. Phys. B 300, 360–376 (1988).Google Scholar
Vistoli, A. Grothendieck topologies, fibered categories and descent theory, in: Fundamental Algebraic Geometry– Grothendieck’s FGA Explained (edited by B. Fantechi et al.). Mathematical Surveys and Monographs, vol. 123. American Mathematical Society, Providence, RI, pp. 1–104 (2005).Google Scholar
Wang, J. The moduli stack of G-bundles, arXiv:1104.4828 [math.AG] (2011).Google Scholar
Waterhouse, W.C. Introduction to Affine Group Schemes. Graduate Texts in Mathematics, vol. 66. Springer-Verlag, Berlin, 1979.Google Scholar
Weil, A. Généralisation des fonctions abéliennes, J. Math. Pures Appl. 17, 47–87 (1938).Google Scholar
Witten, E. On quantum gauge theories in two dimensions, Commun. Math. Phys. 141, 153–209 (1991).Google Scholar
Zagier, D. On the cohomology of moduli spaces of rank two vector bundles over curves, in: The Moduli Space of Curves. Progress in Mathematics, vol. 129. Birkhäuser, Basel, pp. 533–563 (1995).Google Scholar
Zagier, D. Elementary aspects of the Verlinde formula and of the Harder–Narasimhan– Atiyah–Bott formula, IsraelMath.Conf.Proc.,vol.9, Bar-Ilan Univ., pp. 445–462 (1996).Google Scholar
Zhu, X. An introduction to affine Grassmannians and the geometric Satake equivalence, in: Geometry of Moduli Spaces and Representation Theory. IAS/Park City Mathematics Series, vol. 24. American Mathematical Society, Providence, RI, pp. 59–154 (2017).Google Scholar

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