Spin structures and baby universes

TL;DR

This work extends a 2D topological gravity model by summing over spin structures, introducing NS and R boundaries and revealing a topological obstruction when Ramond boundaries are odd. The bulk path integral becomes a non-factorizing correlator of baby-universe operators, which the authors interpret as an ensemble average over boundary theories with Poisson-distributed dimensions and boson/fermion content, further enriched by End-of-the-World branes. They propose two prescriptions to realize a single-duality description: a geometric rule adding spin-structure-summed boundaries and an algebraic rule using spacetime D-branes to project to eigenstates, and they extend the construction to Jackiw–Teitelboim gravity, where the spin-structured ensemble maps to matrix models with bosonic/fermionic sectors or to eigenbranes. Additionally, they explore the JT+EOW setup’s compatibility with ensemble averages, show how alternative spin-structure sums affect factorization, and discuss potential topological obstructions via spin cobordism and implications for holography and black-hole microstate descriptions.

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.

Figures (5)

• Figure 1: A Euclidean spacetime with a "wormhole" handle which contributes to the gravitational path integral. If we slice the path integral open along the bottom blue dotted line, we get a state in a universe with a connected Cauchy slice. If we instead slice along the top blue dotted line, we see the parent universe has "emitted" a baby universe, which appears as a disconnected component of the Cauchy slice.
• Figure 2: A genus $g=2$ contribution to the bulk path integral with one NS and two R boundaries. With these boundary conditions the gravitational path integral computes $\langle \hat{Z} \hat{\tilde{Z}}^2 \rangle$. The depicted diagram's contribution to this expectation is $64e^{-5S_0+3S_\partial}$. Asymptotic boundaries with NS (R) spin structure are shown as solid (dashed) blue circles. In the free energy \ref{['eq:ramond-free-energy']}, this connected manifold contributes $32e^{-5S_0+3S_\partial}$ to the coefficient of $u\tilde{u}^2$ (the only difference compared to its contribution to $\langle \hat{Z}\hat{\tilde{Z}}^2 \rangle$ is the additional symmetry factor $(n_{NS}! n_R!)^{-1}$ which arises in the usual way by passing from the generating function to the free energy). In the alternative sum over spin structures discussed in section \ref{['alt']}, the contribution from this manifold to both $\langle \hat{Z} \hat{\tilde{Z}}^2 \rangle$ and the free energy \ref{['eq:alt-ramond-free-energy']} vanishes identically since $n_R \neq 0$.
• Figure 3: A contribution to the expectation $\langle \hat{\tilde{S}}_{ij} \hat{\tilde{S}}_{ji} \hat{\tilde{Z}} \hat{S}_{kl} \hat{\tilde{S}}_{lk} \rangle$. We do not sum over indices in such expressions; if we had not matched the outgoing EOW brane label of one boundary segment with the incoming EOW brane label of the subsequent boundary segment, the bulk path integral on this manifold would have been zero. Red segments are EOW branes and solid (dashed) blue segments are asymptotic NS (R) boundary segments. We can of course still have circular NS or R boundaries as before. There can also be closed EOW branes (red circle) floating in the bulk, which are not part of the boundary data.
• Figure 4: A contribution to the expectation $\langle Z=3,\tilde{Z}=-1 | \hat{Z} \hat{\tilde{Z}} | Z=3,\tilde{Z}=-1 \rangle$. Since we wish to compute the expectation in an eigenstate, we must modify the usual rules for the gravitational path integral. Since we want $Z=3$, the geometry must have exactly 3 connected components. To select the $\tilde{Z}$ eigenstate, we choose specific sums over spin structures on the one additional boundary (green) on each component. Choosing two $NS-R$ sums (dashed) and one $NS+R$ sum (solid) fixes $\tilde{Z} = -1$. Finally, since the operator we want is $\hat{Z}\hat{\tilde{Z}}$, as usual we must fix the boundary conditions to have one NS boundary (solid blue) and one R boundary (dashed blue).
• Figure 5: An $N=2$ example of the identity in JT gravity relating the path integral with EOW branes and boundary segments forming a circular boundary to the path integral with only boundary segments forming a circular boundary. The red EOW brane segments which have tension $\mu$ and lengths $l_i$ turn into NS boundary segments of length $\tau_i'$. For the equality to hold, we must integrate over the EOW brane lengths and transformed NS boundary segment lengths with the weights shown, given in \ref{['eq:brane-weights']}. This transformation leaves the spin structure unchanged; here it is Ramond, since we have an odd number of dashed segments.