Interacting fermions and N=2 Chern-Simons-matter theories

TL;DR

This work extends the Fermi gas reformulation from ${\cal N}=3$ to ${\cal N}=2$ Chern-Simons-matter theories, recasting sphere partition functions as interacting one-dimensional Fermi gases. In the large-${N}$ limit, Hartree/Thomas-Fermi theory provides a tractable mean-field description, reproducing known ${\cal N}=2$ large-${N}$ results and enabling detailed analysis of theories with a single node, both flavored and with long-range adjoint interactions. For one-node flavored theories, exchange corrections yield an Airy-function form for the partition function, while theories with long-range forces are governed by an integral equation that determines the large-${N}$ R-charge; these results connect to and strengthen prior matrix-model analyses. The framework offers a coherent route to compute leading large-${N}$ free energies, assess subleading corrections, and explore the qualitative differences between short-range and long-range interactions in the holographic context.

Abstract

The partition function on the three-sphere of N=3 Chern-Simons-matter theories can be formulated in terms of an ideal Fermi gas. In this paper we show that, in theories with N=2 supersymmetry, the partition function corresponds to a gas of interacting fermions in one dimension. The large N limit is the thermodynamic limit of the gas and it can be analyzed with the Hartree and Thomas-Fermi approximations, which lead to the known large N solutions of these models. We use this interacting fermion picture to analyze in detail N=2 theories with one single node. In the case of theories with no long-range forces we incorporate exchange effects and argue that the partition function is given by an Airy function, as in N=3 theories. For the theory with g adjoint superfields and long-range forces, the Thomas-Fermi approximation leads to an integral equation which determines the large N, strongly coupled R-charge.

Figures (4)

• Figure 1: The Fermi surface for the interacting Fermi gas associated to flavored one-node theories, in the Thomas--Fermi approximation.
• Figure 2: The exchange correction due to the two-body interaction (\ref{['twobody']}) (left) and the first ring diagram contributing to correlation effects (right).
• Figure 3: The graph of $f_{g,h}(x)$, the solution of (\ref{['int-sim']}), for $g=2$ and $h=0.2726$.
• Figure 4: The graph of $C|_{\epsilon=0}$ as a function of $h$ for $g=2$ (left) and $g=3$ (right).