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 {mathrm{mathcal{M}}}_3times {mathrm{S}}^3times {mathbbm{T}}^4 (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold SymN ( {mathbbm{T}}^4 ). We strengthen the analysis of [1] 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 ℳ3. We also emphasize the fact that string perturbation theory computes the grand canonical partition function of the family of theories ⊕N SymN ( {mathbbm{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 {mathbbm{T}}^4 and of deforming away from the orbifold by the marginal deformation.
Highlights
The AdS/CFT correspondence has provided us with a unique glimpse into the properties of quantum gravity and consistency of the correspondence is still surprising from a variety of angles.There are essentially two classes of proposals that seem to have qualitatively different properties
We examine the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions
We strengthen the analysis of [1] 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
Summary
The AdS/CFT correspondence has provided us with a unique glimpse into the properties of quantum gravity and consistency of the correspondence is still surprising from a variety of angles. We will essentially replace the notion of summing over geometries by sums over different worldsheet theories While questions such as ‘which manifolds should we sum over to obtain the boundary partition function?’ have a clear answer in this framework, the result is somewhat difficult to interpret from a semiclassical gravity point of view. We use this formalism to demonstrate that the string path integral does localize in the moduli space of Riemann surface. They are mostly not necessary to understand the main text
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