F-theorem, duality and SUSY breaking in one-adjoint Chern-Simons-Matter theories

TL;DR

The work analyzes three-dimensional ${\cal N}=2$ Chern-Simons-matter theories with a single adjoint using localization to compute the $S^3$ partition function and the $F$-maximization principle. It tests 3D Seiberg-like dualities, investigates the conjectured $F$-theorem in the large-$N$ limit, and proposes a decoupling-aware modification of $F$-maximization to account for free fields that appear at strong coupling. A novel non-perturbative constraint on spontaneous SUSY breaking is formulated via $Q$-deformed $S^3$ partition functions, with illustrative analytic results from the Chern-Simons matrix model. The paper provides extensive numerical and analytic evidence for duality invariance of the free energy, confirms several RG flow inequalities predicted by the $F$-theorem, and offers a framework to understand SUSY-breaking patterns and decoupling phenomena in 3D gauge theories. Overall, it advances non-perturbative tests of duality and RG monotonicity in simple yet rich 3D CS-matter systems, with potential implications for the broader 3D-4D duality landscape and SUSY breaking diagnostics.

Abstract

We extend previous work on N=2 Chern-Simons theories coupled to a single adjoint chiral superfield using localization techniques and the F-maximization principle. We provide tests of a series of proposed 3D Seiberg dualities and a new class of tests of the conjectured F-theorem. In addition, a proposal is made for a modification of the F-maximization principle that takes into account the effects of decoupling fields. Finally, we formulate and provide evidence for a new general non-perturbative constraint on spontaneous supersymmetry breaking in three dimensions based on Q-deformed S^3 partition functions. An explicit illustration based on the known analytic solution of the Chern-Simons matrix model is presented.

Figures (6)

• Figure 1: Plot $(a)$: A typical one-cut distribution of the eigenvalues $t_i$$(i=1,\ldots, N)$ in the complex plane. This particular plot has been obtained for $N=100$, $\lambda=1$ and $R=0.28$. Plot $(b)$: A typical two-cut solution depicted here for $N=100$, $\lambda=1.5$ and $R=0.5$. The specific example crosses the imaginary axis at $\pm \frac{\tt i}{2}$.
• Figure 2: Plot $(a)$ depicts the numerically determined renormalized free energy $f(\lambda)$ for the topological ${\bf A}_2$ theory (blue data points) in the supersymmetric interval $[0,1]$. The blue curve depicts the analytic result \ref{['saddleai']}. Plot $(b)$ provides the analogous numerical result for the ${\bf A}_4$ theory (blue data points) for $\lambda\in[0,3]$. The dashed red curve is a fit provided by the expression \ref{['saddleaj']}.
• Figure 3: A plot of the free energy of a free field $f(R)=-\ell(1-R)$.
• Figure 4: The numerically determined R-charge curve in the very strong coupling regime for $10<\lambda<100$ and $N=100$. The upper blue points represent the result of the standard $F$-maximization recipe. The lower black square points represent the result of the modified $F$-maximization recipe \ref{['modifyab']} which is designed to take into account the effects of the decoupling fields. The exact result is required by consistency to lie within the bounds of the lower dashed black curve $\frac{1}{2(\lambda+1)}$ and the upper dotted red curve $\frac{2}{2\lambda+1}$ (the lower bound holds for $\lambda \in \mathbb{N}$ and the upper bound for $\lambda \in \mathbb{N}/2$) Niarchos:2011sn.
• Figure 5: Plots of the function $D(\lambda)$, defined in terms of the free energies in eq. \ref{['matchac']}, for the ${\bf A}_4$ theory (plot (a)), the ${\bf A}_5$ theory (plot (b)) and the ${\bf A}_6$ theory (plot (c)). The blue points represent the numerically computed results and the solid curves the duality-predicted functions $\lambda^2(n-\lambda)^{-2}$.