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Is the Lorentzian path integral in string theory well defined, as opposed to the usual Euclidian path integral that is commonly used for simplicity? The path integral is roughly

$$\sum_{\mathbf{\mathcal M \in Top}}\int \mathscr DX \mathscr Dg\ e^{i\lambda_{\chi}[\mathcal M ]}\exp{\lbrack i \int d^2 \sigma \sqrt{-g} g^{ab} \partial_a X^\mu \partial_b X_\mu \rbrack}$$

with a sum over all worldsheet topologies and states for a 2D spacetime with $n$ conformal fields on it.

But most topologies involved in string theory for such a thing involve interpolating topologies between two boundaries. In a Lorentzian context, this would be a Lorentz cobordism, which, for two non-diffeomorphic boundaries, always involve either a loss of time orientability or chronology (I think singularities are also an option but I'm not sure), by a theorem of Geroch.

The only two terms of the sums that would not involve any cobordism will be the open string to open string transition (path integral on the topology $\Bbb R^2$) and the closed string to closed string transition (path integral on the topology $\Bbb R \times S$), as well as any disjoint union of those. All other topological terms will involve some causal pathologies.

Scalar field theories in such contexts are pretty badly divergent (cf Kay's paper for instance), which makes me wonder if a Lorentzian integral in such a context would converge at all. Is this a problem?

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  • $\begingroup$ Usually the integral is defined as a complex continuation of the Euclidean integral. The question of causality is investigated in the results of the theory, not in first principles. $\endgroup$ – Prof. Legolasov Jun 7 '17 at 2:14

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