Rank-level duality for conformal blocks of the linear group
Author:
Rémy Oudompheng
Journal:
J. Algebraic Geom. 20 (2011), 559-597
DOI:
https://doi.org/10.1090/S1056-3911-2010-00529-2
Published electronically:
December 16, 2010
MathSciNet review:
2786666
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Abstract |
References |
Additional Information
Abstract: We generalise the rank-level duality (also known as strange duality) proved by P. Belkale [J. Amer. Math. Soc. 21 (2008), no. 1, 235–258] for generic curves and A. Marian and D. Oprea [Invent. Math. 168 (2007), no. 2, 225–247] for every smooth curve to the case of spaces of conformal blocks related to moduli spaces of parabolic bundles on a smooth pointed projective curve.
References
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- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Alina Marian and Dragos Oprea, The level-rank duality for non-abelian theta functions, Invent. Math. 168 (2007), no. 2, 225–247. MR 2289865, DOI https://doi.org/10.1007/s00222-006-0032-z
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- Alina Marian and Dragos Oprea, A tour of theta dualities on moduli spaces of sheaves (2007), available at 0710.2908"> arXiv:0710.2908.
- V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), no. 3, 205–239. MR 575939, DOI https://doi.org/10.1007/BF01420526
- Tomoki Nakanishi and Akihiro Tsuchiya, Level-rank duality of WZW models in conformal field theory, Comm. Math. Phys. 144 (1992), no. 2, 351–372. MR 1152377
- W. M. Oxbury and S. M. J. Wilson, Reciprocity laws in the Verlinde formulae for the classical groups, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2689–2710. MR 1340183, DOI https://doi.org/10.1090/S0002-9947-96-01563-2
- Christian Pauly, Espaces de modules de fibrés paraboliques et blocs conformes, Duke Math. J. 84 (1996), no. 1, 217–235 (French). MR 1394754, DOI https://doi.org/10.1215/S0012-7094-96-08408-2
- Christian Pauly, Fibrés paraboliques de rang 2 et fonctions thêta généralisées, Math. Z. 228 (1998), no. 1, 31–50 (French). MR 1617983, DOI https://doi.org/10.1007/PL00004601
- Bernd Siebert and Gang Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679–695. MR 1621570, DOI https://doi.org/10.4310/AJM.1997.v1.n4.a2
- Carlos T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), no. 3, 713–770. MR 1040197, DOI https://doi.org/10.1090/S0894-0347-1990-1040197-8
- C. Teleman and C. Woodward, Parabolic bundles, products of conjugacy classes and Gromov-Witten invariants, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 3, 713–748 (English, with English and French summaries). MR 2008438
- Akihiro Tsuchiya, Kenji Ueno, and Yasuhiko Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math., vol. 19, Academic Press, Boston, MA, 1989, pp. 459–566. MR 1048605, DOI https://doi.org/10.2969/aspm/01910459
- Erik Verlinde, Fusion rules and modular transformations in $2$D conformal field theory, Nuclear Phys. B 300 (1988), no. 3, 360–376. MR 954762, DOI https://doi.org/10.1016/0550-3213%2888%2990603-7
- Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772
- Edward Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 357–422. MR 1358625
- Don Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 445–462. MR 1360519
References
- Takeshi Abe, Strange duality for parabolic symplectic bundles on a pointed projective line, Int. Math. Res. Not. IMRN (2008), Art. ID rnn121, 47. MR 2448083
- M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806 (85k:14006)
- Arnaud Beauville, Conformal blocks, fusion rules and the Verlinde formula, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 75–96. MR 1360497 (97f:17025)
- Arnaud Beauville and Yves Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385–419. MR 1289330 (95k:14011)
- Arnaud Beauville, Yves Laszlo, and Christoph Sorger, The Picard group of the moduli of $G$-bundles on a curve, Compositio Math. 112 (1998), no. 2, 183–216. MR 1626025 (99i:14011)
- Prakash Belkale, Strange duality and the Hitchin/WZW connection (2007), available at 0705.0717v2"> arXiv:0705.0717v2.
- Prakash Belkale, The strange duality conjecture for generic curves, J. Amer. Math. Soc. 21 (2008), no. 1, 235–258. MR 2350055 (2009c:14059)
- Aaron Bertram, Generalized $\textrm {SU}(2)$ theta functions, Invent. Math. 113 (1993), no. 2, 351–372. MR 1228129 (95g:14011)
- Aaron Bertram, Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian, Internat. J. Math. 5 (1994), no. 6, 811–825, available at alg-geom/9403007"> arXiv:alg-geom/9403007. MR 1298995 (96h:14070)
- Usha N. Bhosle and Indranil Biswas, Maximal subbundles of parabolic vector bundles, Asian J. Math. 9 (2005), no. 4, 497–522. MR 2216243 (2007j:14048)
- Aaron Bertram, Georgios Daskalopoulos, and Richard Wentworth, Gromov invariants for holomorphic maps from Riemann surfaces to Grassmannians, J. Amer. Math. Soc. 9 (1996), no. 2, 529–571, available at alg-geom/9306005"> arXiv:alg-geom/9306005. MR 1320154 (96f:14066)
- N. Bourbaki, Éléments de mathématique, Groupes et algèbres de Lie, Actualités Scientifiques et Industrielles, Hermann, Paris (French).
- Ron Donagi and Loring W. Tu, Theta functions for $\textrm {SL}(n)$ versus $\textrm {GL}(n)$, Math. Res. Lett. 1 (1994), no. 3, 345–357, available at alg-geom/9303004"> arXiv:alg-geom/9303004. MR 1302649 (95j:14012)
- J.-M. Drézet and M. S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), no. 1, 53–94 (French). MR 999313 (90d:14008)
- Gerd Faltings, A proof for the Verlinde formula, J. Algebraic Geom. 3 (1994), no. 2, 347–374. MR 1257326 (95j:14013)
- R. Fossum and B. Iversen, On Picard groups of algebraic fibre spaces, J. Pure Appl. Algebra 3 (1973), 269–280. MR 0357396 (50 \#9864)
- Jürgen Fuchs, Bert Schellekens, and Christoph Schweigert, From Dynkin diagram symmetries to fixed point structures, Comm. Math. Phys. 180 (1996), no. 1, 39–97, available at hep-th/9506135"> arXiv:hep-th/9506135. MR 1403859 (98i:17021)
- Jürgen Fuchs and Christoph Schweigert, The action of outer automorphisms on bundles of chiral blocks, Comm. Math. Phys. 206 (1999), no. 3, 691–736, available at hep-th/9805026"> arXiv:hep-th/9805026. MR 1721887 (2001b:81049)
- W. Fulton, On the quantum cohomology of homogeneous varieties, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 729–736, available at math/0309436"> arXiv:math/0309436. MR 2077592 (2005d:14077)
- William Fulton and Piotr Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, vol. 1689, Springer-Verlag, Berlin, 1998. Appendix J by the authors in collaboration with I. Ciocan-Fontanine. MR 1639468 (99m:14092)
- Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638 (96e:13006)
- Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
- Yves Laszlo and Christoph Sorger, The line bundles on the moduli of parabolic $G$-bundles over curves and their sections, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 4, 499–525 (English, with English and French summaries). MR 1456243 (98f:14007)
- Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927 (2001f:14006)
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144 (96h:05207)
- Alina Marian and Dragos Oprea, The level-rank duality for non-abelian theta functions, Invent. Math. 168 (2007), no. 2, 225–247, available at math/\space0605097"> arXiv:math/\space0605097. MR 2289865 (2007k:14070)
- Alina Marian and Dragos Oprea, Virtual intersection on the $\mathrm {Quot}$ scheme and Vafa-Intriligator formulas, Duke Math. J. 136 (2007), no. 1, 81–113, available at math/0505685"> arXiv:math/0505685. MR 2271296 (2007k:14117)
- Alina Marian and Dragos Oprea, A tour of theta dualities on moduli spaces of sheaves (2007), available at 0710.2908"> arXiv:0710.2908.
- V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), no. 3, 205–239. MR 575939 (81i:14010)
- Tomoki Nakanishi and Akihiro Tsuchiya, Level-rank duality of WZW models in conformal field theory, Comm. Math. Phys. 144 (1992), no. 2, 351–372. MR 1152377 (93a:81181)
- W. M. Oxbury and S. M. J. Wilson, Reciprocity laws in the Verlinde formulae for the classical groups, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2689–2710. MR 1340183 (96i:14012)
- Christian Pauly, Espaces de modules de fibrés paraboliques et blocs conformes, Duke Math. J. 84 (1996), no. 1, 217–235 (French). MR 1394754 (97h:14022)
- Christian Pauly, Fibrés paraboliques de rang 2 et fonctions thêta généralisées, Math. Z. 228 (1998), no. 1, 31–50 (French). MR 1617983 (99m:14030)
- Bernd Siebert and Gang Tian, On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997), no. 4, 679–695, available at alg-geom/9403010"> arXiv:alg-geom/9403010. MR 1621570 (99d:14060)
- Carlos T. Simpson, Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), no. 3, 713–770. MR 1040197 (91h:58029)
- C. Teleman and C. Woodward, Parabolic bundles, products of conjugacy classes and Gromov-Witten invariants, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 3, 713–748 (English, with English and French summaries). MR 2008438 (2004g:14053)
- Akihiro Tsuchiya, Kenji Ueno, and Yasuhiko Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Integrable systems in quantum field theory and statistical mechanics, Adv. Stud. Pure Math., vol. 19, Academic Press, Boston, MA, 1989, pp. 459–566. MR 1048605 (92a:81191)
- Erik Verlinde, Fusion rules and modular transformations in $2$D conformal field theory, Nuclear Phys. B 300 (1988), no. 3, 360–376. MR 954762 (89h:81238)
- Edward Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), no. 3, 351–399. MR 990772 (90h:57009)
- Edward Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 357–422. MR 1358625 (98c:58016)
- Don Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, (Ramat Gan, 1993) Israel Math. Conf. Proc., vol. 9, Bar-Ilan Univ., Ramat Gan, 1996, pp. 445–462. MR 1360519 (96k:14005)
Additional Information
Rémy Oudompheng
Affiliation:
Laboratoire J.-A. Dieudonné, UMR CNRS 6621, Université de Nice-Sophia- Antipolis, Parc Valrose, 06108 NICE Cedex 2, France
Email:
oudomphe@math.unice.fr
Received by editor(s):
November 5, 2008
Received by editor(s) in revised form:
January 29, 2009
Published electronically:
December 16, 2010