The algebra of $SL_3(\C)$ conformal blocks

We construct and study a family of toric degenerations of the algebra of conformal blocks for a stable marked curve $(C, \vec{p})$ with structure group $SL_3(\C).$ We find that this algebra is Gorenstein. 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. Along the way we recover polyhedral rules for counting conformal blocks originally due to Senechal, Mathieu, Kirillov, and Walton.

[1]  Noah Giansiracusa,et al.  The cone of type A, level one conformal blocks divisors , 2011, 1105.3139.

[2]  Christopher Manon.  Coordinate rings for the moduli stack of SL2(C) quasi-parabolic principal bundles on a curve and toric fiber products , 2011, 1105.2045.

[3]  T. Abe.  Projective normality of the moduli space of rank 2 vector bundles on a generic curve , 2009 .

[4]  Christopher Manon.  Presentations of semigroup algebras of weighted trees , 2008, 0808.1320.

[5]  B. Sturmfels,et al.  Sagbi Bases of Cox-Nagata Rings , 2008, 0803.0892.

[6]  Weronika Buczýnska,et al.  On the geometry of binary symmetric models of phylogenetic trees , 2007 .

[7]  Rekha R. Thomas,et al.  Computing tropical varieties , 2005, J. Symb. Comput..

[8]  R. Howe,et al.  A basis for the GLn tensor product algebra , 2004, math/0407468.

[9]  T. Tao,et al.  The honeycomb model of _{}(ℂ) tensor products II: Puzzles determine facets of the Littlewood-Richardson cone , 2001, math/0107011.

[10]  E. Frenkel,et al.  Vertex Algebras and Algebraic Curves , 2000, math/0007054.

[11]  A. Beauville,et al.  The Picard group of the moduli of G-bundles on a curve , 1996, Compositio Mathematica.

[12]  Andrei Zelevinsky,et al.  Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics , 1996 .

[13]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[14]  A. Beauville.  Conformal blocks, fusion rules and the Verlinde formula , 1994, alg-geom/9405001.

[15]  A. Beauville,et al.  Conformal blocks and generalized theta functions , 1993, alg-geom/9309003.

[16]  George Lusztig,et al.  Canonical bases arising from quantized enveloping algebras , 1990 .

[17]  E. Verlinde,et al.  Fusion Rules and Modular Transformations in 2D Conformal Field Theory , 1988 .

[18]  D. P. Zhelobenko.  Compact Lie Groups and Their Representations , 1973 .

[19]  C. Woodward,et al.  Parabolic Bundles, Products of Conjugacy Classes, and Gromov-witten Invariants , 2022 .

[20]  河野 俊丈.  Conformal field theory and topology , 2002 .

[21]  F. Grosshans.  Algebraic homogeneous spaces and invariant theory , 1997 .

[22]  Christoph Sorger,et al.  The line bundles on the moduli of parabolic G-bundles over curves and their sections , 1997 .

[23]  D. Zagier,et al.  Infinite Grassmannians and Moduli Spaces of G–bundles , 1994 .

[24]  Edgar E. Enochs,et al.  On Cohen-Macaulay rings , 1994 .

[25]  K. Ueno,et al.  Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries , 1989 .