Toward $\mathrm {U}(N|M)$ knot invariant from ABJM theory

Abstract

We study $\mathrm {U}(N|M)$ 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 $\mathrm {U}(N|M)$ character expectation values in terms of $\mathrm {U}(1|1)$ averages for a particular type of character representations. This means that the $\mathrm {U}(1|1)$ character expectation value is a building block for the $\mathrm {U}(N|M)$ averages and also, by an appropriate limit, for the $\mathrm {U}(N)$ 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.

Appendix: Mirror of the torus knot ABJM partition function

In this appendix, we derive the mirror description of the torus knot ABJM partition function [25]. We first expand the determinants in (3.10) as summation over permutations

$$\begin{aligned} \mathcal {Z}_\mathrm{ABJM}^{(P,Q)}= & {} \sum _{\sigma ,\sigma ' \in \mathfrak {S}_N} (-1)^{\sigma +\sigma '} \frac{1}{N!^2} \int [\mathrm{d}x]^N [\mathrm{d}y]^N\nonumber \\&\quad \prod _{i=1}^N \left( 2 \cosh \frac{x_i - y_{\sigma (i)}}{2P} \, 2 \cosh \frac{x_i - y_{\sigma '(i)}}{2Q} \right) ^{-1}. \end{aligned}$$

(7.1)

Applying the formula (2.20), we have

$$\begin{aligned}&\sum _{\sigma ,\sigma ' \in \mathfrak {S}_N} (-1)^{\sigma +\sigma '} \frac{1}{N!^2} \int \! \frac{\mathrm{d}^N x}{(2\pi )^N} \frac{\mathrm{d}^N y}{(2\pi )^N} \frac{\mathrm{d}^N z}{(2\pi )^N} \frac{\mathrm{d}^N w}{(2\pi )^N} \, \prod _{i=1}^N \left( \cosh z_i \, \cosh w_i \right) ^{-1} \nonumber \\&\quad \times \exp \left[ \frac{ik}{4PQ\pi } \sum _{i=1}^N (x_i^2 - y_i^2) + \frac{i}{\pi } \sum _{i=1}^N \left( x_i \left( \frac{z_i}{P} + \frac{w_i}{Q} \right) - y_i \left( \frac{z_{\sigma ^{-1}(i)}}{P} + \frac{w_{\sigma '^{-1}(i)}}{Q} \right) \right) \right] \nonumber \\&\qquad = \sum _{\sigma ,\sigma ' \in \mathfrak {S}_N} (-1)^{\sigma +\sigma '} \frac{(PQ)^N}{N!^2 \, k^N} \int \! \frac{\mathrm{d}^N z}{(2\pi )^N} \frac{\mathrm{d}^N w}{(2\pi )^N} \, \prod _{i=1}^N \frac{\exp \left[ - \frac{2i}{k\pi } \left( z_i w_i - z_{\sigma ^{-1}(i)} w_{\sigma '^{-1}(i)} \right) \right] }{ \cosh z_i \, \cosh w_i }.\nonumber \\ \end{aligned}$$

(7.2)

At this moment, it is obvious that the partition function depends only on the composition of permutations $\sigma \cdot \sigma '^{-1}$. Thus, by fixing either of them $\sigma '$ as the trivial permutation, we obtain

$$\begin{aligned} \mathcal {Z}_\mathrm{ABJM}^{(P,Q)}&= \sum _{\sigma \in \mathfrak {S}_N} (-1)^N \frac{(PQ)^N}{N! \, k^N} \int \! \frac{\mathrm{d}^N z}{(2\pi )^N} \frac{\mathrm{d}^N w}{(2\pi )^N} \, \prod _{i=1}^N \frac{\exp \left[ - \frac{2i}{k\pi } \left( z_i - z_{\sigma (i)} \right) w_i \right] }{ \cosh z_i \, \cosh w_i } \nonumber \\&= \sum _{\sigma \in \mathfrak {S}_N} (-1)^N \frac{(PQ)^N}{N! \, k^N} \int \! \frac{\mathrm{d}^N z}{(2\pi )^N} \, \prod _{i=1}^N \left( \cosh z_i \cdot 2\cosh \frac{z_i - z_{\sigma (i)}}{k} \right) ^{-1} \nonumber \\&= \frac{(PQ)^N}{N! (2k)^N} \int \! \frac{\mathrm{d}^N z}{(2\pi )^N} \, \prod _{i<j}^N \left( \tanh \frac{z_i - z_j}{2k} \right) ^2 \, \prod _{i=1}^N \left( 2 \cosh \frac{z_i}{2} \right) ^{-1} \nonumber \\&= (PQ)^N \mathcal {Z}_\mathrm{ABJM}^{(1,1)}. \end{aligned}$$

(7.3)

This is the mirror expression of the partition function (3.10). Especially for $k=1$, the mirror theory turns out to be $\mathcal {N}=4$ SYM theory with a single fundamental and a single adjoint hypermultiplet. The dependence on the parameters (P, Q) becomes trivial in this mirror representation.