Duality in osp(1|2) Conformal Field Theory and link invariants

TL;DR

This paper explores the duality between osp(1|2) conformal field theory and link invariants, establishing connections with su(2) algebra through free field realization, and analyzing topological Chern-Simons theory for knot and link invariants.

Contribution

It provides a new framework linking osp(1|2) conformal blocks, fusion, and braiding matrices to su(2) counterparts via free field realization and relates osp(1|2) link invariants to su(2) polynomials through Chern-Simons quantization.

Findings

Derived fusion and braiding matrices for osp(1|2) conformal blocks.

Established a simple relation between osp(1|2) and su(2) link invariants.

Verified the relation for arbitrary knots and links via Chern-Simons theory.

Abstract

We study the crossing symmetry of the conformal blocks of the conformal field theory based on the affine Lie superalgebra osp(1|2). Within the framework of a free field realization of the osp(1|2) current algebra, the fusion and braiding matrices of the model are determined. These results are related in a simple way to those corresponding to the su(2) algebra by means of a suitable identification of parameters. In order to obtain the link invariants corresponding to the osp(1|2) conformal field theory, we analyze the corresponding topological Chern-Simons theory. In a first approach we quantize the Chern-Simons theory on the torus and, as a result, we get the action of the Wilson line operators on the supercharacters of the affine osp(1|2). From this result we get a simple expression relating the osp(1|2) polynomials for torus knots and links to those corresponding to the su(2) algebra. Further, this relation is verified for arbitrary knots and links by quantizing the Chern-Simons theory on the punctured two-sphere.