Knots in $SU\left(M|N\right) $ Chern-Simons Field Theory

TL;DR

This paper investigates knot invariants within the $SU(M|N)$ Chern-Simons theory, deriving a polynomial invariant with skein relations for the fundamental representation of the Lie superalgebra.

Contribution

It introduces a new polynomial invariant for knots in $SU(M|N)$ Chern-Simons theory, extending previous invariants to the super gauge group context.

Findings

Derived the $S_{L}( ext{alpha}, ext{beta}, z)$ polynomial invariant.

Established skein relations for the invariant.

Applied the invariant to knots in the super gauge group setting.

Abstract

Knots in the Chern-Simons field theory with Lie super gauge group $SU\left(M|N\right) $ are studied, and the $% S_{L}\left(\alpha,\beta,z\right) $ polynomial invariant with skein relations are obtained under the fundamental representation of $\mathfrak{su}\left(M|N\right) $.