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Since the worldsheet theory is conformal, you are allowed to "shrink the boundaries to a point". So the usual viewpoint is that the worldsheet are boundary-less with certain points on them corresponding to the former boundaries. The cylinder, for instance, becomes a twice-punctured sphere - the punctures are the places where one inserts the vertex operators ...

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Naively, the collection of data which describes a model of anyons is not its fusion rules, but rather its modular data - that is, a pair of matrices $S$ and $T$ which generate a representation of the (modular) group $SL(2,\mathbb Z)$. The (diagonal) matrix $T$ encodes the mutual statistics of quasi-particles - that is what happens when you exchange two ...

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I) It seems the resolution to OP's question lies in the difference between the Levi-Civita symbol, which is not a tensor and whose values are only $0$ and $\pm 1$; and the Levi-Civita tensor, whose definition differs from the Levi-Civita symbol by a factor of $\sqrt{|\det(g_{\mu\nu})|}$. II) The 2D Euler-density is $$E_2~=~ \frac{1}{8\pi} ... 1 This may not be exactly what you want, but it does go over some classical field theory and good chuck of differential geometry. http://www.gravity-and-light.org/lectures 3 We consider the metric$$\mathrm{d}s^2=-\mathrm{d}t^2+a^2(t)\mathrm{d}\vec x^2 $$where a(t):= a_0e^{Ht}. To show that these coordinates do not cover the entire spacetime manifold, we consider the trajectory of a freely falling observer, which of course extremizes the proper time$$\tau=\int\mathrm{d}t\sqrt{1-a^2\dot{\vec x^2}}  Performing the ...

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The geometry and topology of the relevant phase space is identical for both classical and quantum problems: it is the very same phase space. The scale of the former is the small $\hbar$ limit of the latter. Extended WFs appear like δ-fctn spikes ("points") in the small $\hbar$ limit, once the phase space-variables are suitably rescaled by $\sqrt{\hbar}$. ...

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