Automorphism-weighted ensembles from TQFT gravity

TL;DR

This work clarifies how a 3d TQFT summed over all topologies is holographically dual to an ensemble of boundary CFTs arising from Lagrangian algebras, with each boundary member weighted by the inverse size of its invertible symmetry group. The authors derive these weights from SymTFT considerations, showing that in the large-genus limit the overlap of boundary states is governed by invertible symmetry data, leading to a natural uniform-up-to-isomorphism measure over the groupoid of Lagrangian algebras. They connect the sum over topologies to Heegaard-splitting decompositions and the baby universe Hilbert space, yielding a Siegel-Weil-type structure for boundary theories and providing concrete examples across Abelian, SU(2)$_k$, orbifold, and ADE cases, including Virasoro minimal models. The framework extends to noncompact TQFTs and Virasoro TQFT, suggesting a gravity-compatible ensemble perspective in which highly symmetric theories are suppressed, consistent with a maximal ignorance principle and potential regularizations via conformal-manifold data and Goeritz groups.

Abstract

We study the recent proposal of arXiv:2405.20366 which poses a precise holographic duality between a 3d TQFT summed over all topologies and a unitary ensemble of boundary 2d CFTs. In that proposal, the sum over topologies is obtained via genus reduction from topologies with a large genus boundary Riemann surface, while the boundary ensemble is given by all CFTs described by Lagrangian condensations of the bulk TQFT. The main result of this work is to show that each member of this ensemble is weighted by a symmetry factor given by the invertible symmetry group of its categorical symmetry relative to the bulk TQFT as its SymTFT. This is the natural $\unicode{x2014}$ uniform up to isomorphism $\unicode{x2014}$ measure on the groupoid of Lagrangian algebras that describe the boundary theories. We also write the sum over topologies more explicitly in terms of equivalence classes of Heegaard splittings of 3-manifolds with a given boundary and comment on their weights. The holographic duality in this framework can then be viewed as a generalization of the Siegel-Weil formula. We discuss the implications of the main result for non-compact TQFTs. In particular, for the Virasoro case, this implies an ensemble of all CFTs at a given central charge in which CFTs are weighted by their full invertible symmetry. Finally, we show how this TQFT gravity framework gives a natural construction of the baby universe Hilbert space.

Figures (6)

• Figure 1: A choice of basis states in $\mathcal{H}_{g}$ corresponding to inserting the shown network of lines inside in a handlebody $\mathsf{S}\Sigma_{g}$.
• Figure 2: Condensation of $\mathcal{A}$ can be reduced to inserting the shown network of $\mathcal{A}$ lines in a handlebody. This prepares the state $\left|Z_{\mathcal{A}}\right\rangle$.
• Figure 3: A depiction of the sandwich construction where the bulk TQFT is placed on an interval. The red boundary has a topological boundary condition with the green line as a defect living on the topological boundary. The blue boundary represents the physical boundary where the boundary QFT lives. We can have lines in the bulk extending between the two boundaries giving rise to local or defect operators in the boundary QFT. Upon collapsing the interval, we get the boundary QFT with possible insertions of topological defect lines and local or defect operators
• Figure 4: An illustration of the club sandwich construction. The middle figure shows a gapped interface $G$ between theory $\mathcal{T}$ and a condensed phase $\mathcal{T}/\xi$. Collapsing $G$ to the right gives us a sandwich construction for $\mathcal{T}$, while collapsing it to the left gives us a sandwich for $\mathcal{T}/\xi$. The theory at the physical boundary is the same for all these sandwiches, and the above diagram shows the relations between boundary conditions in the different scenarios.
• Figure 5: Illustration of projection via $\left\langle 0_{g-\tilde{g}},\cdot\right|$ in a simple case for $g=3$ and $\tilde{g}=1$. The resulting state can be viewed as a state in $\mathcal{H}_{\tilde{g}=1}$.