2d TQFTs and baby universes

Abstract

In this work, we extend the 2d topological gravity model of [1] to have as its bulk action any open/closed TQFT obeying Atiyah’s axioms. The holographic duals of these topological gravity models are ensembles of 1d topological theories with random dimension. Specifically, we find that the TQFT Hilbert space splits into sectors, between which correlators of boundary observables factorize, and that the corresponding sectors of the boundary theory have dimensions independently chosen from different Poisson distributions. As a special case, we study in detail the gravity model built from the bulk action of 2d Dijkgraaf-Witten theory, with or without end-of-the-world branes, and for arbitrary finite group G. The dual of this Dijkgraaf-Witten gravity model can be interpreted as a 1d topological theory whose Hilbert space is a random representation of G and whose aforementioned sectors are labeled by the irreducible representations of G.These holographic interpretations of our gravity models require projecting out negative-norm states from the baby universe Hilbert space, which in [1] was achieved by the (only seemingly) ad hoc solution of adding a nonlocal boundary term to the bulk action. In order to place their solution in the completely local framework of a TQFT with defects, we couple the boundaries of the gravity model to an auxiliary 2d TQFT in a non-gravitational (i.e. fixed topology) region. In this framework, the difficulty of negative-norm states can be remedied in a local way by the introduction of a defect line between the gravitational and non-gravitational regions. The gravity model is then holographically dual to an ensemble of boundary conditions in an open/closed TQFT without gravity.

Integrality, duality and finiteness in combinatoric topological strings

“…The relation between the projector basis and the conjugacy class basis of Z(C(G)) plays a key role in the algorithms as well as the geometrical gluing properties that define G-TQFT2. Summing the amplitudes of G-TQFT2 weighted by a string coupling defines combinatoric topological string theory [20,21], which we call G-CTST. We studied S-duality and the analytic structure of G-CTST as a function of the string coupling, connecting these to group theoretic combinatoric data.…”

“…Considering the sums of G-TQFT2 amplitudes over all genera which define G-CTST as in [20,21] we investigate the stringy property of S-duality. We then study the finiteness properties of G-TQFT2/G-CTST and their physical implications.…”

“…We will refer to this topological field theory based on G as G-TQFT2. Recent work in connection with wormhole physics and baby universes [20,21] has introduced sums over surfaces weighted by a string coupling g st , where each surface supports a G-TQFT2, thus defining topological string theories based on G-TQFT2. We will refer to these string theories as combinatoric topological string theories or G-CTST.…”

Comments on Summing over bordisms in TQFT

“…A long-standing question in quantum theories of gravity has been the question of whether, and how, one should sum over topologies of spacetime in the path integral [4,5,11,14,19]. Some recent literature in quantum gravity has considered a modification of topological field theory wherein the amplitudes, between fixed in and outgoing boundaries are defined by the sum over all the bordisms that fill in those boundaries [3,13,17,22]. While it is a standard idea in physics to sum over topologies, the sum in the context of topological field theory has not been considered before in the mathematical literature on topological field theory, as far as we are aware.…”

“…This combinatorial rule is extremely important. It is implicitly assumed in [3,13,17,22], so we will work with it. See Figure 1.…”