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Recently I've came to some references on mathematical aspects on string theory that deal with the Polyakov euclidean path integral. An example is the book "Quantum Fields and Strings: A Course for Mathematicians. Volume 2", where it is stated roughly that the path integral is
$$A = \sum_{\text{topologies}} \int_{\text{Met}(\Sigma)} \frac{1}{\mathcal{N}(g)} \int_{\text{Map}(\Sigma, M)} Dg Dx e^{-S[x,g,G]},$$
where ##(\Sigma, g), \, (M, G)## are Riemannian manifolds and ##x: \Sigma \to M## is assumed to be only continuous (?), ##Dg, Dx## being "measures". The main problem to me is that one of the spaces over which the integral is taken is the space of ALL maps ##x: \Sigma \to M##. I'm my understanding, this should be a space of embeddings, just as it is assumed in Chapter 3 in Polchinski's Vol.1, when he constructs the Polyakov path integral. It is physically relevant to just integrante over all maps from the worldsheet to the manifold ##M##?
$$A = \sum_{\text{topologies}} \int_{\text{Met}(\Sigma)} \frac{1}{\mathcal{N}(g)} \int_{\text{Map}(\Sigma, M)} Dg Dx e^{-S[x,g,G]},$$
where ##(\Sigma, g), \, (M, G)## are Riemannian manifolds and ##x: \Sigma \to M## is assumed to be only continuous (?), ##Dg, Dx## being "measures". The main problem to me is that one of the spaces over which the integral is taken is the space of ALL maps ##x: \Sigma \to M##. I'm my understanding, this should be a space of embeddings, just as it is assumed in Chapter 3 in Polchinski's Vol.1, when he constructs the Polyakov path integral. It is physically relevant to just integrante over all maps from the worldsheet to the manifold ##M##?