Rank-level Duality Research Articles

Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves

This paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel–Whitney class and the associated space of twisted spin bundles. In particular, we prove a Verlinde type formula and a dimension equality that was conjectured by Oxbury–Wilson. Modifying Hitchin’s argument, we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection. We furthermore address the issue of strange duality for odd orthogonal bundles, and we demonstrate that the naive conjecture fails in general. A consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the Hitchin connection for these moduli spaces. Finally, we answer a question of Nakanishi–Tsuchiya about rank-level duality for conformal blocks on the pointed projective line with spin weights.

Rank-level duality of conformal blocks for odd orthogonal Lie algebras in genus $0$

Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r+1) on the pointed projective line conjectured by T. Nakanishi and A. Tsuchiya.

Rank-level duality and conformal block divisors

We describe new relations among conformal block divisors in Pic(M‾0,n). These relations appear from various rank-level dualities of conformal blocks on P1 with n-marked points. We also give a coordinate free description of rank-level duality maps on M‾g,n.

Three Combinatorial Models for $$ _ $$ Crystals, with Applications to Cylindric Plane Partitions

We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight. Cylindric plane partitions actually parametrize a basis for the tensor product of an irreducible representation with the space spanned by all partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of rank level duality. Finally, we use an explicit bijection to relate our work to the Kyoto path model.

Rank-level duality for conformal blocks of the linear group

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.

Topological closed-string interpretation of Chern-Simons theory.

The exact free energy of SU($N$) Chern-Simons theory at level $k$ is expanded in powers of $(N+k)^{-2}.$ This expansion keeps rank-level duality manifest, and simplifies as $k$ becomes large, keeping $N$ fixed (or vice versa)---this is the weak-coupling (strong-coupling) limit. With the standard normalization, the free energy on the three-sphere in this limit is shown to be the generating function of the Euler characteristics of the moduli spaces of surfaces of genus $g,$ providing a string interpretation for the perturbative expansion. A similar expansion is found for the three-torus, with differences that shed light on contributions from different spacetime topologies in string theory.

Simple-current symmetries, rank-level duality, and linear skein relations for Chern-Simons graphs

A previously proposed two-step algorithm for calculating the expectation values of arbitrary Chern-Simons graphs fails to determine certain crucial signs. The step which involves calculating tetrahedra by solving certain non-linear equations is repaired by introducing additional linear equations. The step which involves reducing arbitrary graphs to sums of products of tetrahedra remains seriously disabled, apart from a few exceptional cases. As a first step towards a new algorithm for general graphs we find useful linear equations for those special graphs which support knots and links. Using the improved set of equations for tetrahedra we examine the symmetries between tetrahedra generated by arbitrary simple currents. Along the way we describe the simple, classical origin of simple-current charges. The improved skein relations also lead to exact identities between planar tetrahedra in level K G( N) and level N G( K) Chern-Simons theories, where G( N) denotes a classical group. These results are recast as WZW braid-matrix identities and as identities between quantum 6- jsymbols at appropriate roots of unity. We also obtain the transformation properties of arbitrary graphs, knots, and links under simple-current symmetries and rank-level duality. For links with knotted components this requires precise control of the braid eigenvalue permutation signs, which we obtain from plethysm and an explicit expression for the (multiplicity-free) signs, valid for all compact gauge groups and all fusion products.

Strange duality revisited

We give a proof of the strange duality or rank-level duality of the WZW models of conformal blocks by extending the genus-0 result, obtained by Nakanishi-Tsuchiya in 1992, to higher genus curves via the sewing procedure. The new ingredient of the proof is an explicit use of the branching rules of the conformal embedding of the affine Lie algebras sl(r) x sl(l) in sl(rl). We recover the strange duality of spaces of generalized theta functions obtained by Belkale, Marian-Oprea, as well as by Oudompheng in the parabolic case.