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The algebra of \(\mathrm{SL}_3(\mathbb{C})\) conformal blocks. (English) Zbl 1327.14055

Summary: We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal \(\mathrm{SL}_{3}(\mathbb{C})\) bundles on a smooth, marked curve \((C, \vec{p})\): Elements of this algebra have a well known interpretation as conformal blocks, from the Wess-Zumino-Witten model of conformal field theory. For the genus 0, 1 cases we find the level of conformal blocks necessary to generate the algebra. In the genus 0 case we also find bounds on the degrees of relations required to present the algebra. As a consequence we obtain a toric degeneration for the projective coordinate ring of an effective divisor on the moduli \(\mathcal{M}_{C,\vec{p}}({\text{SL}_3}(\mathbb{C})\) of quasi-parabolic principal \(\mathrm{SL}_{3}(\mathbb{C})\) bundles on \((C, \vec{p})\). Along the way we recover positive polyhedral rules for counting conformal blocks.

MSC:

14D06 Fibrations, degenerations in algebraic geometry
14D23 Stacks and moduli problems
14C20 Divisors, linear systems, invertible sheaves
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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[1] A. Jensen, D. Speyer, B. Sturmfels, R. Thomas, Computing tropical varieties, J. Symb. Comput. 42 (2007), no. 1-2, 54-73. · Zbl 1121.14051
[2] T. Abe, Projective normality of the moduli space of rank 2 vector bundles on a generic curve, Trans. AMS 362 (2010), 477-490. · Zbl 1197.14033 · doi:10.1090/S0002-9947-09-04816-8
[3] Beauville, A., Conformal blocks, fusion rules, and the Verlinde formula, No. 9, 75-96 (1996), Ramat Gan · Zbl 0848.17024
[4] W. Bruns, J. Herzog, Cohen−Macaulay Rings, Cambridge Studies in Advanced Mathematics, Vol. 39, Cambridge University Press, Cambridge, 1993. · Zbl 0788.13005
[5] A. Beauville, Y. Laszlo, Conformal blocks and generalized theta functions, Comm. Math. Phys. 164 (1994), no. 2, 385-419. · Zbl 0815.14015 · doi:10.1007/BF02101707
[6] A. Beauville, Y. Laszlo, C. Sorger, The Picard group of the moduli of G-bundles over curves, Compositio Math. 112 (1998), no. 2, 183-216. · Zbl 0976.14024 · doi:10.1023/A:1000477122220
[7] A. Berenstein, A. Zelevinksy, Triple multiplicities for sl(r+1) and the spectrum of the algebra of the adjoint representation, J. Algebraic Combinatorics 1 (1992), 7-22 · Zbl 0799.17005 · doi:10.1023/A:1022429213282
[8] A. Berenstein, A. Zelevinsky, Canonical bases for the quantum group of type Arand piecewise-linear combinatorics, Duke Math. J. 82 (1996), no. 3, 473-502. · Zbl 0898.17006 · doi:10.1215/S0012-7094-96-08221-6
[9] V. Buczynska, Phylogenetic toric varieties on graph, arXiv:1004.1183. · Zbl 1376.14050
[10] W. Buczynska, J. Buczynski, K. Kubjas, M. Michalek, Degrees of generators of phylogenetic semigroups on graphs, arXiv:1105.5382v1. · Zbl 1282.14087
[11] W. Buczynska, J. Wiesniewski, On the geometry of binary symmetric models of phylogenetic trees, J. European Math. Soc. 9 (2007), 609-635. · Zbl 1147.14027 · doi:10.4171/JEMS/90
[12] N. Fakhruddin, Chern classes of conformal blocks on \[{{\bar{\mathcal{M}}}_{0,n }} \], arXiv:0907.0924v2. · Zbl 1244.14007
[13] G. Faltings, A proof of the Verlinde formula, J. Algebraic Geometry 3 (1994), 347-374. · Zbl 0809.14009
[14] E. Frenkel, D. Ben-Zvi, Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, American Mathematical Society, Providence, RI, 2001. · Zbl 0981.17022 · doi:10.1090/surv/088
[15] A. Gibney, N. Giansiracusa, The cone of type A, level one conformal blocks divisors, arXiv:1105.3139v2. · Zbl 1316.14051
[16] F. D. Grosshans, Algebraic Homogeneous Spaces and Invariant Theory, Springer Lecture Notes, Vol. 1673, Springer, Berlin, 1997. · Zbl 0886.14020
[17] R. Howe, E.-C. Tan, J. Willenbring, A basis for the GLntensor product algebra, Adv. Math. 196 (2005), no. 2, 531-564. · Zbl 1072.22007 · doi:10.1016/j.aim.2004.09.007
[18] K. Kaveh, Crystal bases and Newton−Okounkov bodies, arXiv:1101.1687v1. · Zbl 1428.14083
[19] K. Kaveh, M. Harada, Integrable systems, toric degenerations and Okounkov bodies, arXiv:1205.5249. · Zbl 1348.14122
[20] A. N. Kirillov, P. Mathieu, D. Senechal, M. A. Walton, Crystalizing the depth rule and WZNW fusion coefficients, in: Proceedings of the XIXth International Colloquium on Group Theoretical Physics, Salamanca, Spain, 1992. · Zbl 0799.17005
[21] T. Kohno, Conformal Field Theory and Topology, Translations of Mathematical Monographs, Vol. 210, American Mathematical Society, Providence, RI, 2002. · Zbl 1024.81001
[22] S. Kumar, M. S. Narasimhan, A. Ramanathan, Infinite Grassmannians and moduli spaces of G-bundles, Math. Annalen 300 (1994), 41-75. · Zbl 0803.14012 · doi:10.1007/BF01450475
[23] A. Knutson, T. Tao, C. Woodward The honeycomb model of GLn \[( \mathbb{C} )\] tensor products II: Puzzles determine facets of the Littlewood−Richardson cone, J. Amer. Math. Soc. 17 (2004), 19-48. · Zbl 1043.05111 · doi:10.1090/S0894-0347-03-00441-7
[24] Y. Laszlo, C. Sorger, The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Sci. Ecole Norm. Sup. (4) 30 (1997), 499-525. · Zbl 0918.14004
[25] E. Looijenga, Conformal blocks revisited, arXiv:math/0507086v1. · Zbl 1165.32304
[26] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), 356-421. · doi:10.1090/S0894-0347-1991-1088333-2
[27] C. Manon, Presentations of semigroup algebras of weighted trees, J. Algebraic Combinatorics 31 (2010), no. 4, 467-489. · Zbl 1230.05159 · doi:10.1007/s10801-009-0195-y
[28] C. Manon, The algebra of conformal blocks, arXiv:0910.0577v4. · Zbl 1327.14055
[29] C. Manon, Toric degenerations and tropical geometry of branching algebras, arXiv:1103.2484.
[30] C. Manon Coordinate rings for the moduli of SL \[2( \mathbb{C} )\] quasi-parabolic principal bundles on a curve and toric fiber products, J. Algebra 365 (2012), no. 1, 163-183. · Zbl 1262.14011 · doi:10.1016/j.jalgebra.2012.05.007
[31] C. Manon, Z. Zhou, Semigroups of sl \[3( \mathbb{C} )\] tensor product invariants, arXiv:1206.2529. · Zbl 1298.17013
[32] C. Sorger, La formule de Verlinde, in: Séminaire Bourbaki, Vol. 1994/95, Astérisque 237 (1996), Exp. no. 794, 3, 87-114.
[33] B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, Vol. 8, American Mathematical Society, Providence, RI, 1996. · Zbl 0856.13020
[34] B. Sturmfels, Z. Xu, Sagbi Bases of Cox−Nagata Rings, J. Eur. Math. Soc. 12 (2010), no. 2, 429-459. · Zbl 1202.14053 · doi:10.4171/JEMS/204
[35] B. Sturmfels, M. Velasco, Blow-ups of \[{{\mathbb{P}}^{n-3 }}\] at n points and spinor varieties, arXiv:0906.5096. · Zbl 1237.14025
[36] A. Tsuchiya, K. Ueno, Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Adv. Studies in pure Math. 19 (1989), 459-566.
[37] C. Teleman, C. Woodward, Parabolic bundles, products of conjugacy classes, and Gromov−Witten invariants, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 3, 713-748. · Zbl 1041.14025 · doi:10.5802/aif.1957
[38] Ueno, K., Introduction to Conformal Field Theory with Gauge Symmetry, No. 184, 603-745 (1997), New York · Zbl 0873.32022
[39] E. Verlinde, Fusion rules and modular transformations in 2d conformal field theory, Nuclear Physics B300 (1988), 360-376. · Zbl 1180.81120
[40] D. P. Zhelobenko, Compact Lie Groups and Their Representations, American Mathematical Society, Providence, RI, 1973. · Zbl 0228.22013
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