# Sum over backgrounds in String Theory

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Summary:
Are there any attempts to see what kind of spacetimes need to be integrated in the "path-integral" of string theory?
Usually, I saw that string theory (perturbative, or matrix models) are made in a fixed background. Even if you consider that the metric is quantized and etc. there is an apparent physically motivated need for making a sum over topologies (manifolds, conifolds, orbifolds, and etc), for example, to take into account the possibility of microscopic formation of wormholes or other kind of spacetime defects, like there is a sum over topologies on perturbative string theory. I know that manifolds in more than 3+0 dimensions aren't even non-classifiable, but maybe there is a restricted category of this generalized manifolds that is manageable (or not). What is string theory depends in part on what kind of "manifolds" are part of the sum (integration).

Have there been any attempts to see what kind of spacetimes need to be integrated in the "path-integral" of string theory?

Can be justified making string theory in a fixed background?

At least, when the background isn't fully well defined (for example, there is a superposition of metrics or Kalb-Ramond or dilaton background), have there been made a calculation on that situation (maybe it's not physical, but it can be interesting)?

pd: Part of the justification can be made by superselection arguments, I guess. $E_8\times E_8$ and $IIA$ appear to be part of different superselection sectors (because of the branes in M-theory construction), I think.

Demystifier
Gold Member
In the path integral quantization of perturbative string theory, one integrates over all world-sheet geometries, but spacetime geometry is held fixed.

Yes, you do with a spacetime geometry held fixed but (ignoring non-Borel-summability and non-convergence of the series, and all the etcetera that can be made about this), inserting sum over product of vertex of gravitons , kalb-ramonds and dilatons you can get "any background" with the same space-time topology, and even superpositions of backgrounds. Making a calculation of it can be really be painful, but maybe there is a method to get that kind of thing without having so much trouble.

Part of the question is because, while metric and torsion can be put by gravitons and Kalb-Ramonds ($$H_{\mu\nu\rho}$$ can be taken as part of the torsion, as is widely known I suppose), I don't see if there is a topology-changing quantum or similar (at least, in the string modes doesn't seem to be one), or some good criteria to know what kind of "manifolds" are allowed.

Also, orbifold to toroidal compactification transitions are allowed? I know that dimension-changing is allowed (usually when you have Tachyon modes in your spectrum).

pd: Sorry for saying "...manifolds in more than 3+0 dimensions aren't even non-classifiable..." when what I would have had said is "...manifolds in more than 3+0 dimensions aren't even classifiable...".

king vitamin
Gold Member
In the path integral quantization of perturbative string theory, one integrates over all world-sheet geometries, but spacetime geometry is held fixed.

This is true in the textbook perturbative string theory, but isn't in believed that the full non-perturbative theory should somehow involve a sum over spacetime topologies? I think of the first paragraph of page 8 of this Witten article: https://arxiv.org/abs/1710.01791.

Demystifier
Yes, it's there (sorry for replying so late, busy weeks). Also, these last months there was a lot of talking about sum over backgrounds and black hole information paradox resolution in the context of JK-gravity (a 2+1 dimensional theory), and some theories (like Colored Group Field Theories and causal triangulations) naturally incorpore this feature to their approach.

String Theory MUST incorpore this to their approach, maybe by some kind of stringy restrictions. Also, theories like ASQG (asymptotically safe quantum gravity) must be capable of doing this, and taking different actions terms on it's relationship with topology can, in principle, make very different theories, to the point that many alternatives of quantum gravity can in principle be ruled out by daily life for their predictions of geometric-topological nature (non-abundance of dog-sized wormholes is easy to verify in experiments at earth, non-abundance of bubbles of nothing, etc).

king vitamin