Automorphism-weighted ensembles from TQFT gravity

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.

• The paper establishes a holographic duality where each CFT in the ensemble is weighted by a symmetry factor derived from its categorical symmetry within a 3D TQFT gravity framework.
• It employs genus reduction to simplify the sum over topologies, making the computational approach more tractable in the high genus limit.
• The study extends its implications to non-compact TQFTs, including cases with the Virasoro algebra, thereby advancing our understanding of quantum gravity ensembles.

Automorphism-Weighted Ensembles from TQFT Gravity

Introduction

The study of holographic duality between three-dimensional gravity and ensembles of two-dimensional conformal field theories (CFTs) addresses intriguing questions about the nature of quantum gravity, especially in the context of the AdS/CFT correspondence. Previous explorations have suggested that in multiboundary scenarios, gravity might correspond to an ensemble rather than a unique CFT on the boundary. This paper builds upon that framework by examining a specific proposal involving topological quantum field theories (TQFTs).

Main Concepts and Results

The paper explores a holographic duality where the bulk gravitational theory is a three-dimensional TQFT summed over all topologies, and the boundary is a unitary ensemble of CFTs. The main result establishes that each CFT in this ensemble is weighted by a symmetry factor derived from its categorical symmetry, relative to the bulk TQFT as its symmetry topological field theory (SymTFT). This weighting is calculated via the order of the group of invertible symmetries of the categorical symmetry, effectively representing a "uniform" measure in an automorphism group context.

Holographic Duality and Genus Reduction

The duality is realized by summing over topologies using genus reduction, starting from a boundary with large genus and reducing it progressively. In this framework, duality simplifies in the high genus limit where the sum over topologies becomes a sum over handlebodies, making computational approaches to the ensemble weights more tractable.

Furthermore, the connection to the Siegel-Weil formula is highlighted, showing that this particular approach to duality serves as a generalization by considering not just handlebodies but equivalence classes of manifold topologies.

Implications for Non-Compact TQFTs

The paper extends its implications to non-compact TQFTs, particularly for the Virasoro case, suggesting an ensemble of all CFTs at a given central charge. Here, CFTs are assessed based on their invertible symmetry, with the Virasoro algebra playing a pivotal role. This provides a framework to potentially develop an understanding of the baby universe Hilbert space in TQFT gravity, crucial for resolving aspects of the holographic principle.

Conclusion

The investigation into TQFT gravity and its automorphism-weighted ensembles yields a compelling framework for considering holographic duality beyond the traditional single-CFT paradigm. By incorporating symmetry considerations, the study advances a uniform ensemble averaging approach to quantum theories of gravity, highlighting a novel mechanism to address disorder averaging through categorical symmetries. The insights from this work not only contribute to a deeper theoretical understanding of TQFTs but also open avenues for exploring quantum gravity's fundamental aspects in higher dimensions and other TQFT contexts.