A universal formula for the density of states in theories with finite-group symmetry

TL;DR

The paper addresses counting black hole microstates charged under a finite gauge group and, more broadly, the high-energy density of states in theories with finite-group symmetry. Using Euclidean gravity with a finite G and topological Wilson lines, they derive a simple irrep-resolved density formula $\rho_\alpha(E)=\frac{d_\alpha^2}{|G|}\rho(E)$, showing each irrep contributes with weight $d_\alpha^2/|G|$. They conjecture this relation extends to any QFT with a finite-group global symmetry at high energy, supported by a 1+1D modular-invariance (Cardy-like) argument and by gravity-based reasoning, with nonperturbative corrections exponentially suppressed. The result provides a new proof of the completeness hypothesis for finite gauge fields and suggests a universal counting principle with potential tests in holography and condensed matter.

Abstract

In this paper we use Euclidean gravity to derive a simple formula for the density of black hole microstates which transform in each irreducible representation of any finite gauge group. Since each representation appears with nonzero density, this gives a new proof of the completeness hypothesis for finite gauge fields. Inspired by the generality of the argument we further propose that the formula applies at high energy in any quantum field theory with a finite-group global symmetry, and give some evidence for this conjecture.

Figures (5)

• Figure 1: The Euclidean calculation of the twisted black hole partition function: we fill in a boundary temporal circle of length $\beta$ with the Euclidean Schwarzschild geometry, but with an asymptotic symmetry operator $U(g)$ inserted.
• Figure 2: The rule for moving an asymptotic symmetry operator past a Wilson line endpoint.
• Figure 3: Showing that $\langle W_{\alpha,ij}\rangle=C_\alpha \delta _{ij}$. By introducing a trivial factor of $U(g)U(g^{-1})$ and then moving $U(g)$ around the thermal circle using the rule from figure \ref{['wilsonfig']} (and its conjugate), we see that $\langle W_{\alpha,ij}\rangle$ must be invariant under conjugation. Since $\alpha$ is irreducible, by Schur's lemma $\langle W_{\alpha,ij}\rangle$ must therefore be proportional to the identity. We can interpret $\langle W_{\alpha,ij}\rangle \mathcal{Z}(\beta,e)$ as the norm of the unnormalized thermofield double state with a background charge inserted, so $C_\alpha$ is positive semi-definite. In fact it is strictly positive, as we are assuming that the gauge field is in a deconfined phase so this norm should be nonzero.
• Figure 4: Deriving \ref{['Zresult1']}. Since $G$ is a finite group the Wilson line $W_{\alpha,ij}$ is a topological operator, so we can move one endpoint around the thermal circle, picking up a group transformation along the way. In the first and last steps we use cluster decomposition/locality: when the Wilson line is arbitrarily small and far from $U(g)$, we can replace it by its expectation value $\langle W_{\alpha,ij}\rangle=C_\alpha \delta_{ij}$.
• Figure 5: Proving conjecture \ref{['conjecture1']} for $1+1$ dimensional CFTs. By modular invariance, the high-temperature limit of $\mathcal{Z}(\beta,g)$ is the same as the low-temperature limit of the thermal trace in a sector twisted by $g$.

Theorems & Definitions (1)

• Conjecture 1