By Leonardo Et. Al. Castellani
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Additional resources for Supergravity and Superstrings: A Geometric Perspective, Volume 3: Superstrings
Example text
A surface integral, the only thing that matters is its topology, which is a sophisticated word for boundary conditions. Specifically, using a path integral quantization scheme, the functional integral oveT the 2~dimensional 'metries becomes, after division 1393 by the group of Diffeomorphism (Diff), a discrete sum over the topologies, labelled by a positive integer number g (the genus of the surface), and, at fixed topology, a multiple integral over a finitedimensional complex parameter space M (the moduli space) whose g coordinates label the conformal classes of the world-sheet.
It follows that elliptic elements cannot be used to represent the generators of the homotopy group. For a similar reason the parabolic elements are also excluded. C} is a discrete subgroup of the automorphism group containing parabolic elements, then one can show that 1: is not compact since it has as many punctures as there are parabolic elements in G. A surface L with punctures is obtained from a compact surface 1: by removing a finite number of points (the punctures) in such a way that the region around each puncture is conformally equivalent to the pointed open disk {z, 0 < Izi < l}.
2. 75) ii) T is said to be parabolic if it is similar to a translation (VI. 2. 76) iii) T is said to be hyperbolic if it is similar to a dilatation (VI. 2. 77) 1415 One can establish a simple criterion to decide the elliptic. parabolic or hyperbolic character of any group element T by looking at the trace of T which is a similarity invariant. 78) Clearly elliptic elements have fixed points on H (Im ~ > 0). 75) via a matrix A then it has a fixed point in the A-image of z '" i. It follows that elliptic elements cannot be used to represent the generators of the homotopy group.