Towards U(N|M) knot invariant from ABJM theory

Bertrand Eynard (1, 2) , Taro Kimura (2, 3)

1 CRM - Centre de Recherches Mathématiques [Montréal]
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681
3 RIKEN - RIKEN - Institute of Physical and Chemical Research [Japon]

We study U(N|M) character expectation value with the supermatrix Chern-Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives U(N|M) character expectation values in terms of U(1|1) averages for a particular type of character representations. This means that the U(1|1) character expectation value is a building block for all the U(N|M) averages, and in particular, by an appropriate limit, for the U(N) invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern-Simons matrix model. We obtain the Rosso-Jones-type formula and the spectral curve for this case.

Domains: Physics [physics]; High Energy Physics - Theory [hep-th]; Physics [physics]; Mathematical Physics [math-ph]

Bertrand Eynard, Taro Kimura. Towards U(N|M) knot invariant from ABJM theory. 2016. ⟨cea-01277951⟩