Spin structures and baby universes

TL;DR

This paper extends a 2D topological gravity model to include spin structures, explores their effects on path integrals and baby universes, and proposes modifications to interpret the theory as a single dual system.

Contribution

It introduces methods to incorporate spin structures into 2D topological gravity and proposes prescriptions to interpret the resulting non-factorizing path integrals as dual to a single theory.

Findings

Path integral vanishes with odd R boundaries.

Dual interpretation involves partition functions and Witten indices.

Proposed geometric and algebraic modifications restore single dual theory.

Abstract

We extend a 2d topological model of the gravitational path integral to include sums over spin structure, corresponding to Neveu-Schwarz (NS) or Ramond (R) boundary conditions for fermions. The Euclidean path integral vanishes when the number of R boundaries is odd. This path integral corresponds to a correlator of boundary creation operators on a non-trivial baby universe Hilbert space. The non-factorization necessitates a dual interpretation of the bulk path integral in terms of a product of partition functions (associated to NS boundaries) and Witten indices (associated to R boundaries), averaged over an ensemble of theories with varying Hilbert space dimension and different numbers of bosonic and fermionic states. We also consider a model with End-of-the-World (EOW) branes: the dual ensemble then includes a sum over randomly chosen fermionic and bosonic states. We propose two modifications of the bulk path integral which restore an interpretation in a single dual theory: (i) a geometric prescription where we add extra boundaries with a sum over their spin structures, and (ii) an algebraic prescription involving "spacetime D-branes". We extend our ideas to Jackiw-Teitelboim gravity, and propose a dual description of a single unitary theory with spin structure in a system with eigenbranes.