Summing over Geometries in String Theory

TL;DR

This work addresses how string theory implements a sum over bulk geometries with fixed boundary conditions, using a tensionless string on $\mathcal{M}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ (with NS-NS flux) that is dual to $\mathrm{Sym}^N(\mathbb{T}^4)$. The authors show the perturbative string partition function around a fixed background already captures a sum over semi-classical geometries and that large stringy corrections can be interpreted as different semi-classical geometries; remarkably, the Euclidean wormhole partition function factorizes because the worldsheet path integral localizes on covering maps of the boundary, effectively removing cross-boundary correlations. They express the boundary theory as a grand canonical partition function over symmetric orbifolds $\bigoplus_N \mathrm{Sym}^N(\mathbb{T}^4)$ and demonstrate how averaging over Narain moduli and marginal deformations can produce bulk geometries as condensates of stringy geometries, yielding an emergent classical bulk in suitable limits. The paper also develops a detailed formalism for higher-genus partition functions, localization to covering maps, and the role of discrete torsion and multiple chemical potentials, and it discusses how a bulk emerges from a boundary perspective via the condensation of stringy geometries, with implications for ensemble interpretations and potential generalizations to higher-dimensional AdS/CFT. Overall, the results provide a concrete string-theoretic mechanism for bulk emergence in a tensionless regime, linking worldsheet localization to a semi-classical bulk geometry and clarifying the interplay between ensemble averaging, marginal deformations, and wormhole physics.

Abstract

We examine the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions. We discuss this problem with the help of the tensionless string on $\mathcal{M}_3 \times \mathrm{S}^3 \times \mathbb{T}^4$ (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold $\text{Sym}^N(\mathbb{T}^4)$. We strengthen the analysis of arXiv:2008.07533 and show that the perturbative string partition function around a fixed bulk background already includes a sum over semi-classical geometries and large stringy corrections can be interpreted as various semi-classical geometries. We argue in particular that the string partition function on a Euclidean wormhole geometry factorizes completely into factors associated to the two boundaries of spacetime. Central to this is the remarkable property of the moduli space integral of string theory to localize on covering spaces of the conformal boundary of $\mathcal{M}_3$. We also emphasize the fact that string perturbation theory computes the grand canonical partition function of the family of theories $\bigoplus_N\text{Sym}^N(\mathbb{T}^4)$. The boundary partition function is naturally expressed as a sum over winding worldsheets, each of which we interpret as a `stringy geometry'. We argue that the semi-classical bulk geometry can be understood as a condensate of such stringy geometries. We also briefly discuss the effect of ensemble averaging over the Narain moduli space of $\mathbb{T}^4$ and of deforming away from the orbifold by the marginal deformation.

Figures (12)

• Figure 1: The gravitational computation of the partition function with two boundary circles in a two-dimensional theory of gravity. The presence of the wormhole geometry generically destroys factorization of the answer and leads to an ensemble dual on the blue boundary.
• Figure 2: The wormholes with perturbative string excitations on top of them. These pictures are cartoons, since the actual model that we will consider is three-dimensional and instead of the green worldline curves, we will have worldsheets wrapping the geometry. It turns out that the model realizes only the possibility in the right picture: there are no perturbative string excitations that connect the two asymptotic regions, but only separate string excitations that stay close to the two boundaries.
• Figure 3: A typical string configuration for string theory on the wormhole geometry. As in Figure \ref{['fig:2d bulk']}, this picture is drawn in one dimension lower. The small strings in the middle are spheres, whereas the winding strings close to the boundary are genus $g \ge 1$ surfaces.
• Figure 4: The basic organization of the paper. Arrows indicate strong logical dependencies and we recommend to read the relevant sections first. Dashed lines indicate weaker dependencies.
• Figure 5: The $S_N$-bundles in the toy model for $N=2$ and $N=3$.