Factorization and global symmetries in holography

Abstract

We consider toy models of holography arising from 3d Chern-Simons theory. In this context a duality to an ensemble average over 2d CFTs has been recently proposed. We put forward an alternative approach in which, rather than summing over bulk geometries, one gauges a one-form global symmetry of the bulk theory. This accomplishes two tasks: it ensures that the bulk theory has no global symmetries, as expected for a theory of quantum gravity, and it makes the partition function on spacetimes with boundaries coincide with that of a modular-invariant 2d CFT on the boundary. In particular, on wormhole geometries one finds a factorized answer for the partition function. In the case of non-Abelian Chern-Simons theories, the relevant one-form symmetry is non-invertible, and its gauging corresponds to the condensation of a Lagrangian anyon.

Figures (9)

• Figure 1: Basic properties of the Abelian Lagrangian algebra object $\mathcal{A}$.
• Figure 2: Line configuration for the gauging on $Y_n$. We draw $S^2 \smallsetminus n \mathring{D}_2$, while the transverse $S^1$ is implicit. The $n-1$ lines denoted as $\mathcal{A}$ wrap one of the boundary circles, while $\mathcal{A}^\perp$ (drawn as a broken line) wraps the transverse $S^1$.
• Figure 3: Surgery performed on $Y_2 = T^2 \times I$ (the transverse $S^1$ is kept implicit) in order to reduce it to ${\sf{S}} T^2 \times {\sf{S}} T^2$ and show factorization of the partition function of theory $\mathcal{T}$.
• Figure 4: Surgery around a boundary component for a trivial bulk TQFT.
• Figure 5: Left: a basis of line insertions in a genus-$g$ handlebody that generate the Hilbert space of states on the boundary Riemann surface $\Sigma_g$. Right: example of insertions for $g=2$.