By locality I mean something like the Atiyah-Segal axioms for Riemannian cobordi

Question

By locality I mean something like the Atiyah-Segal axioms for Riemannian cobordisms (see e.g. http://ncatlab.org/nlab/show/FQFT). I.e. to any (spacelike) hypersurface in the target we associate a Hilbert space and to any cobordism an S-matrix.

I am familiar with the S-matrix prescription for the target being Rn and the hypersurfaces being the asymptotic time infinities. Can one extend that to any cobordism?

Does locality appear only when we integrate over the worldsheet conformal structures and sum over all genera, or can we see it even for a fixed conformal structure?

I believe this is what string field theory is about, but why would one expect locality from the (perturbative) string theory point of view?

Explanation / Answer

String theory as we know it admits only S-matrix as an observable. By its definition an S-matrix is a non-local object, it tells you about transition amplitudes between asymptotic states in past and future infinity. You cannot even ask local questions in spacetime, unless you somehow extend the formalism (which is the goal of string field theory, more about this below).

This is not (in my opinion) a quirk of the formalism. String theory is a quantum theory of gravity, and at long distances it coincides with General Relativity. GR also does not allow for local observables. Mathematically it is because there are no local diffeomorphism invariant quantities. Physically it is because there is no systematic way to locally probe the system without disturbing it

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