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

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 the U (N| M) averages and also, 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.