Harold A. Riggs

Group-level duality in WZW models and Chern-Simons theory☆

Abstract We exhibit a duality of primary fields, conformal dimensions, braid matrices, and modular transformation matrix elements S0,λ/S0,0 (so also Weyl characters) for sequences of dual pairs (SU (N)K, SU(K)N), (SO(N)K, (SO(N)K, SO(K)N) and Sp(N)K, Sp(K)N. As a consequence an analogous duality of Chern-Simons skein relations and Wilson line observables is uncovered.

Group-level duality of WZW fusion coefficients and Chern-Simons link observables

Abstract We present the duality of the fusion rule multiplicities, modular tranformation matrices and conformal weights of the Wess-Zumino-Witten models based on any of the classical affine Lie algebras SU( N ) K , Sp( N ) K and SO( N ) K under the exchange of N and K . We also exhibit the transformation properties of these quantities under the symmetries of the extended Dynkin diagrams, which play a central role in this group-level duality. We interpret these results in Chern-Simons theory, and derive identities between link observables. Several surprising results involving spinor representations of SO( N ) are proven.

TWO-DIMENSIONAL YANG-MILLS THEORIES ARE STRING THEORIES

We show that two-dimensional SO(N) and Sp(N) Yang-Mills theories without fermions can be interpreted as closed string theories. The terms in the 1/N expansion of the partition function on an orientable or nonorientable manifold ℳ can be associated with maps from a string worldsheet onto ℳ. These maps are unbranched and branched covers of ℳ with an arbitrary number of infinitesimal worldsheet cross-caps mapped to points in ℳ. These string theories differ from SU(N) Yang-Mills string theory in that they involve odd powers of 1/N and require both orientable and nonorientable worldsheets.

Topological Landau-Ginzburg matter from Sp(N)-K fusion rings

We find and analyze the Landau-Ginzburg potentials whose critical points determine chiral rings which are exactly the fusion rings of SP(N)K WZW models. The quasihomogeneous part of the potential associated with SP(N)K is the same as the quasihomogeneous part of that associated with SU(N+1)K, showing that these potentials are different perturbations of the same Grassmannian potential. Twisted N=2 topological Landau-Ginzburg theories are derived from these superpotentials. The correlation functions, which are just the SP(N)K Verlinde dimensions, are expressed as fusion residues. We note that the SP(N)K and SP(K)N topological Landau-Ginzburg theories are identical, and that while the SU(N)K and SU(K)N topological Landau-Ginzburg models are not, they are simply related.

The quasi-rational fusion structure of SU(m/n) Chern-Simons and WZW theories

Abstract The SU( m | n ) K fusion ring is found. The existence of a positive trace on this ring, rather than integrability, is the common principle underlying the fusion rings of WZW models, ordinary and super. The argument proceeds by means of a combined examination of the ring of positive q -superdimensions, associated Chern-Simons observables, and several WZW correlation functions. The tensor product ring of U q ( su (m|n)) for q a root of unity admits a doubly truncated tensor product structure isomorphic to the SU( m | n ) K fusion ring in the cases we examine. The SU( m | n ) K WZW models are found to be quasi-rational (non-unitary) conformal field theories. Many important quantities in SU( m | n ) K theories are given exactly by analogous SU( m − n ) K quantities, including the conformal weights, fusion coefficients, Chern-Simons observables, and the matrix elements of modular transformations of the relevant affine supercharacters. These matrix elements are related to the fusion ring via a (slightly modified) Verlinde formula. The SU( n + N | n ) K and SU( k + K | k ) N Chern-Simons and WZW theories are related by group-level duality.

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

Abstract 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.

CP violating Yukawa couplings in the Skyrme model and the neutron electric dipole moment

Abstract We argue that the large- N c behaviour of the Yukawa couplings in the Skyrme model involves issues more subtle than the vanishing of linear fluctuations needed for classical stability of the skyrmion. The chiral fluctuations about the skyrmion must be quantized in order to reach a conclusion. An improved quantization procedure allows us to confront this question directly. The pion-nucleon coupling constants g πNN (CP conserving) and g πNN (CP violating) are calculated in the large- N c , three-flavour Skyrme model by direct evaluation of the leading matrix elements appearing in the LSZ reduction formula. We find that g π N N ∼ N c 3 2 , but that, at most g π N N ∼ m π 2 N c −1 2 . These results show that the leading contribution to the neutron electric dipole moment in large- N c Skyrme model is the direct one ( D n ∼ N c m π 2 ), rather than the pion loop contribution.

String calculation of QCD Wilson loops on arbitrary surfaces.

Compact string expressions are found for nonintersecting Wilson loops in SU([ital N]) Yang-Mills theory on any surface (orientable or nonorientable) as a weighted sum over covers of the surface. All terms from the coupled chiral sectors of the 1/[ital N] expansion of the Wilson loop expectation values are included.

Integrable N = 2 Landau-Ginzburg theories from quotients of fusion rings☆

Abstract The discovery of integrable N = 2 supersymmetric Landau-Ginzburg theories whose chiral rings are fusion rings suggests a close connection between fusion rings, the related Landau-Ginzburg superpotentials, and N = 2 quantum integrability. We examine this connection by finding the natural SO( N ) K analogue of the construction that produced the superpotentials with Sp( N ) K and SU( N ) K fusion rings as chiral rings. The chiral rings of the new superpotentials are not directly the fusion rings of any conformal field theory, although they are natural quotients of the tensor subring of the SO( N ) K fusion ring. The new superpotentials yield solvable (twisted N = 2) topological field theories. We obtain the integer-valued correlation functions as sums of SO( N ) K Verlinde dimensions by expressing the correlators as fusion residues. The SO(2 n + 1) 2 k + 1 and SO(2 k + 1) 2 n + 1 related topological Landau-Ginzburg theories are isomorphic, despite being defined via quite different superpotentials.

The String calculation of Wilson loops in two-dimensional Yang-Mills theory

We demonstrate that the large N expansion of Wilson loop expectation values in SO(N) and Sp(N) Yang-Mills theory on orientable and nonorientable surfaces has a natural description as a weighted sum over covers of the given surface. The sum takes the form of the perturbative expansion of an open string theory. The derivation makes contact with the classification of branched covers by Gabai and Kazez. Comparison with the analogous results for the chiral sectors of QCD2 is instructive for both cases.