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The torus is special because it's so simple, and because it provides the most tractable example of Mirror Symmetry https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory), a generalization of T-duality (which relates Type IIB with Type IIA with one another). Toric compactifications are rather special, they're a special case of an incredibly large ...


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Let us suppress (world-sheet) time $\tau$ in what follows, i.e. consider a fixed time $\tau$. Let there be given a continuous map $\phi:\Sigma\to M$, where the world-space $\Sigma$ and the target space $M$ are both 1D manifolds. We will assume that such a 1D manifold is either a real line $\mathbb{R}$ or a circle $S^1\cong\mathbb{R}/\mathbb{Z}$. That gives ...


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Notice first that even before restricting the domain of $\phi$, we are considering the theory on the cylinder and identifying the boundary condition $\phi(x + L,t) = \phi(x,t)$. Now to explain the restriction, let's take this example. Consider a field configuration at some fixed time $\phi(x,0)$, we only have to study this in the domain $[0,L]$. Now pick ...


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There are models where the extra dimensions don't need to be curled up. The main issue with extra dimensions is, 'why don't the particles/fields we interact with travel in those directions?' We have extremely good limits on standard model particles (electrons, photons) travelling in extra dimensions. However, it is possible to imagine a string inspired ...


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I don't think its a weakness in any sense. Because in all string theories, $10+1$ dimensional Lorentz transformations ARE a symmetry of the action itself. However not only in order to agree with phenomenology, but also as an attempt (not completely successful so far) to reproduce the entire structure of the standard model interactions, string theory ...



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