Accidental Symmetries and the Conformal Bootstrap

TL;DR

The paper investigates a three-dimensional ${\cal N}=2$ supersymmetric generalization of the critical $O(N)$ vector model with superpotential $W = \frac{g_1}{2} X \sum_i Z_i^2 + \frac{g_2}{6} X^3$, combining conformal bootstrap with supersymmetric localization to argue that IR symmetry enhancement occurs as $g_2$ flows to zero. The authors show that the naive IR fixed point with both couplings nonzero is inconsistent with unitarity for $N>2$ via bootstrap bounds on the $O(N)$ current central charge $c_J^{O(N)}$, and they further bound the $N>5$ case with the $F$-theorem. They provide evidence that the $g_2=0$ fixed points, possessing enhanced $O(N)\times U(1)$ symmetry, nearly saturate the bootstrap bounds, and extend their analysis to fractional dimensions using the $4-\varepsilon$ expansion. The work demonstrates a nonperturbative mechanism for accidental symmetry in IR fixed points and highlights the power of combining localization with the conformal bootstrap to constrain IR dynamics in supersymmetric theories.

Abstract

We study an ${\cal N} = 2$ supersymmetric generalization of the three-dimensional critical $O(N)$ vector model that is described by $N+1$ chiral superfields with superpotential $W = g_1 X \sum_i Z_i^2 + g_2 X^3$. By combining the tools of the conformal bootstrap with results obtained through supersymmetric localization, we argue that this model exhibits a symmetry enhancement at the infrared superconformal fixed point due to $g_2$ flowing to zero. This example is special in that the existence of an infrared fixed point with $g_1,g_2\neq 0$, which does not exhibit symmetry enhancement, does not generally lead to any obvious unitarity violations or other inconsistencies. We do show, however, that the $F$-theorem excludes the models with $g_1,g_2\neq 0$ for $N>5$. The conformal bootstrap provides a stronger constraint and excludes such models for $N>2$. We provide evidence that the $g_2=0$ models, which have the enhanced $O(N)\times U(1)$ symmetry, come close to saturating the bootstrap bounds. We extend our analysis to fractional dimensions where we can motivate the nonexistence of the $g_1,g_2\neq 0$ models by studying them perturbatively in the $4-ε$ expansion.

Figures (5)

• Figure 1: RG flow lines obtained from the one-loop beta functions---see \ref{['beta']}, where these beta functions are given to two-loop order. The green square, red circle, blue triangle, and black diamond correspond to the fixed points in \ref{['Solutions']}. Note that the last fixed point only exists for $N=1$.
• Figure 2: $\delta{\tilde{F}}\equiv \tilde{F}_{UV} - \tilde{F}_{IR}$ as a function of $2< d < 4$ for $N=1,2,3,4,5,6,9$ (dark to light). Here, $\tilde{F}_{UV}$ is the generalized free energy defined in Giombi:2014xxa corresponding to the theory \ref{['SuperPot']} with $O(N)\times \mathbb{Z}_3$ symmetry, while $\tilde{F}_{IR}$ corresponds to the infrared fixed point obtained by deforming that theory by $X^2$.
• Figure 3: Lower bounds on central charge $c_{J}^{O(N)}$ for $\mathcal{N}=2$ SCFTs with $O(N)$ symmetry in $d=3$ for $N=2,...,10$, computed using the conformal bootstrap. The black dotted line denotes the $N$ independent analytical value of $c_{J}^{O(N)}$ computed from localization for SCFTs with super potential \ref{['SuperPot']}. For $N\geq 3$, the black dotted line falls outside of the region allowed by the bootstrap, making model \ref{['SuperPot']} a disallowed SCFT.
• Figure 4: Conformal bootstrap lower bounds on $c_{J}^{O(N)}$ for $\mathcal{N}=2$ SCFTs with $O(N)$ symmetry in 3d, for $N=3, 6, 9$. The black diamond denotes the $N$-independent analytical value of $(\Delta_{Z_i},c_{J}^{O(N)})=(2/3,0.521)$, computed from localization for SCFTs with super potential \ref{['SuperPot']}. The diamond falls outside of the orange shaded region allowed by the bootstrap. The blue triangles and the dotted blue line correspond to the interacting SCFT with $O(N) \times U(1)$ flavor symmetry---see Table \ref{['RCharges']}. The blue triangles correspond to $N=3, 6, 9$, from right to left.
• Figure 5: The left plot shows lower bounds on central charge $c_{J}^{O(N)}$ for $\mathcal{N}=2$ SCFTs with $O(N)$ symmetry in dimensions $2 < d<4$ for $N = 2,\, 3,\, 4,\, 6,\, 9$. The black dotted curve denotes the $N$ independent analytical value of $c_{J}^{O(N)}$ computed from localization for SCFTs with super potential \ref{['SuperPot']} in $2<d<4$. The right plot shows the difference between the $c_J^{O(N)}$ bound determined by the bootstrap and the results from localization, focusing on $N=2,\,3,\,4$.