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The moduli space of $SU(3)$-flat connections
and the fusion rules

Author: Masato Hayashi
Journal: Proc. Amer. Math. Soc. 127 (1999), 1545-1555
MSC (1991): Primary 58D29, 57M05; Secondary 17B81, 17B10
Published electronically: January 29, 1999
MathSciNet review: 1476136
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Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is to determine the existence condition of the moduli space of $SU(3)$-flat connections on $3$-holed $2$-sphere $D$, the so-called pair of pants, and to study its relationship to the $\widehat {\mathfrak{sl}}(3;\mathbb{C})$ fusion rules. The existence condition can be expressed by a system of inequalities with the entries of highest weights with respect to the fundamental weights. This gives a necessary condition for the fusion coefficents to be nontrivial. We also find that the fusion coefficient of a triplet of extremal highest weights equals one. This can be considered a quantum counterpart of the PRV-conjecture.

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  • [B] H. U. Boden, Unitary representations of Brieskorn spheres, Duke Math. 75(No.1) (1994), 193-220. MR 95f:57033
  • [BL] A. Beauville and Y. Laszlo, Conformal blocks and generalized theta function, Comm. Math. Phys. 164 (1994), 385-419. MR 95k:14011
  • [BZ] A. D. Berenstein and A. Z. Zelevinsky, Cornell preprint-technical report 90-60 (1990).
  • [F] G. Faltings, A proof of the Verlinde formula, J. Alg. Geom. 3 (1994), 347-374. MR 95j:14013
  • [Fu] J. Fuchs, Affine Lie algebras and quantum groups, Cambridge, 1995. MR 96a:17018
  • [GN] F. M. Goodman and T. Nakanishi, Fusion algebras in integrable systems in two dimensions, Phys. Lett. B262 (1991), 259-264. MR 92i:81281
  • [K] V. G. Kac, Infinite dimensional Lie algebras, Cambridge, 1990. MR 92k:17038
  • [KMSW] A. N. Kirillov, P. Mathieu, D. Sénéchal and M. A. Walton, Can fusion coefficients be calculated from the depth rule?, Nuclear Phys. B391 (1993), 651-674. MR 94b:81110
  • [KNR] S. Kumar, M. S. Narasimhan and A. Ramanathan, Infinite Grassmannians and moduli spaces of $G$-bundles, Math. Ann. 300 (1994), 41-75. MR 96e:14011
  • [Ku] S. Kumar, Proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, Invent. Math. 93 (1988), 117-130. MR 89j:17009
  • [MS] V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structure, Math. Ann. 248 (1980), 205-239. MR 81i:14010
  • [PRV] K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan, Representations of complex semi-simple Lie groups and Lie algebras, Ann.of Math. 85 (1967), 383-429. MR 37:1526
  • [S] T. A. Springer, Linear algebraic groups, Birkhäuser, 1981. MR 84i:20002
  • [TK] A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on $P^{1}$ and monodromy representations of braid group, Adv. Stu. Pure Math. 85 (1988), 297-372. MR 89m:81166
  • [W] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, IASSNS-HEP-93/41. CMP 96:03

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Additional Information

Masato Hayashi
Affiliation: Department of Mathematical Sciences, University of Tokyo, Komaba, Meguro-ku, Tokyo

Keywords: Representations of the fundamental group of a surface, Bruhat decomposition, fusion rules, PRV-conjecture
Received by editor(s): May 13, 1996
Received by editor(s) in revised form: September 3, 1997
Published electronically: January 29, 1999
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 1999 American Mathematical Society

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