The Partition Function of ABJ Theory

TL;DR

This work develops a complete framework to compute the partition function of the ABJ theory $U(N_1)_k\times U(N_2)_{-k}$ via localization. By solving the lens space matrix model exactly and performing a carefully prescribed analytic continuation $N_2\to -N_2$, the authors obtain an ABJ partition function described by min$(N_1,N_2)$-dimensional integrals, extending the ABJM ’mirror description’ to unequal ranks. The construction reproduces perturbative expansions, yields a nonzero partition function only when $|N_1-N_2|\le k$, and passes nontrivial Seiberg duality checks (notably for $N_1=1$). These results pave the way for ABJ generalizations of the Fermi gas approach and offer insights into AdS$_4$/higher-spin dualities and potential M-theory microscopic structures behind ABJ(M).

Abstract

We study the partition function of the N=6 supersymmetric U(N_1)_k x U(N_2)_{-k} Chern-Simons-matter (CSM) theory, also known as the ABJ theory. For this purpose, we first compute the partition function of the U(N_1) x U(N_2) lens space matrix model exactly. The result can be expressed as a product of q-deformed Barnes G-function and a generalization of multiple q-hypergeometric function. The ABJ partition function is then obtained from the lens space partition function by analytically continuing N_2 to -N_2. The answer is given by min(N_1,N_2)-dimensional integrals and generalizes the "mirror description" of the partition function of the ABJM theory, i.e. the N=6 supersymmetric U(N)_k x U(N)_{-k} CSM theory. Our expression correctly reproduces perturbative expansions and vanishes for |N_1-N_2|>k in line with the conjectured supersymmetry breaking, and the Seiberg duality is explicitly checked for a class of nontrivial examples.

Figures (7)

• Figure 1: "The integration contour" $C_j=I+C_j^{\infty}$ for the perturbative ABJ partition function: The only perturbative (P) poles are indicated by red "$+$". See text for detail.
• Figure 2: The nonperturbative (NP) poles are added and indicated by blue "$\times$". The left panel corresponds to the complex $g_s$ case. The right panel is the actual case of our interest $g_s=2\pi i/k$. (Shown is the case $k=3$ and $M=3$.)
• Figure 3: The integration contour $C=I\cap I_{i\infty}\cap I_k\cap I_{-i\infty}$ (clockwise) and poles, for various values of $k,N_2$. (a) and (b) are Seiberg duals of each other and so are (c) and (d). The contour $I_k$ is parallel to $I$ and shifted by $k$, and the contours $I_{i\infty}$ and $I_{-i\infty}$ are at infinity. "$+$" (red) denotes the P pole and "$\times$" (blue) the NP pole. Some poles and zeros are shown slightly above or below the real $s$ axis, but this is for the convenience of presentation and all poles and zeros are on the real $s$ axis. The choices of the parameter $\eta$ for the contour $I$ are $\eta=0_+$ for (a) and (c), $\eta={1\over 2}+0_+$ for (b), and $\eta=1+0_+$ for (d).
• Figure 4: The regions that can contribute to $\Phi(1,N_2)$. (a), (b): For $(n_1,n_2)\in{\mathbb{Z}}^2$ in the shaded regions (denoted by dots), the summand in $\Phi(1,N_2)$ in \ref{['gyiq5Sep12']} is ${\cal O}(1)$. Outside the shaded regions, the summand is ${\cal O}(\epsilon,\eta)$ and vanishes as $\epsilon,\eta\to 0$. (c): the $N_2=-1$ case special and the summand is non-vanishing only on the dots.
• Figure 5: The regions that can contribute to $\Phi(1,N_2)$. These are the same as Figure \ref{['nonvanishing_n1n2']}, but plotted for $(s_1,s_2)$ instead.