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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.

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.