By Martin Schottenloher

Half I supplies a close, self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in dimensions. The conformal teams are made up our minds and the appearence of the Virasoro algebra within the context of the quantization of two-dimensional conformal symmetry is defined through the category of critical extensions of Lie algebras and teams. half II surveys extra complicated subject matters of conformal box conception resembling the illustration conception of the Virasoro algebra, conformal symmetry inside string idea, an axiomatic method of Euclidean conformally covariant quantum box idea and a mathematical interpretation of the Verlinde formulation within the context of moduli areas of holomorphic vector bundles on a Riemann floor.

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Diff+ (S) is equipped with the topology of uniform convergence of the differentiable mappings ~ : S ~ S and all their derivatives. This topology is metrizable. Similarly, Vect (S) carries the topology of uniform convergence of the differentiable vector fields X :S ~ TS and all their derivatives. With this topology, Vect (S) is a Fr@chet space. In fact, Vect (S) is isomorphic to C°°(S, R), as we will see shortly. The proof that Diff+ (S) in this way actually becomes a differentiable manifold modeled on Vect (S) and that the group operation and the inversion are differentiable, is elementary and can be carried out for arbitrary-oriented, compact (finite-dimensional) manifolds M instead of S (cf.

Obviously, CA = ¢-A and ¢~1 = Ch-~. In the case CA = CA' for A,A' e O ( p + 1, q+ 1) we have ~/(h~) = ~(A'~) for all ~ e R ~+2 with (~) = 0. Hence, h = r h ' with r e R\{0}. Now A, A' e O(p+l, q+l) implies r - 1 or r - - 1 . 5 Let ~ : M --, ~P,q be a conformal transformation on a connected open subset M C R p'q. Then ~ " N p'q --, N p'q is called a conforrnal continuation of ~, if ~ is a conformal diffeomorphism (with conformal inverse) a n d / f z ( ~ ( x ) ) = ~(z(x)) for all x E M . In other words, the following diagram is commutative: M • R p'q A Np,q ~ Np,q 2.

I? of the Riemann sphere I?. ) using the compactification C ~ I? 6. 8 says even more: Mb is also isomorphic to the proper Lorentz group SO(3, 1). An interpretation of the 2. The Conformal Group 30 isomorphism Aut(IP) =~ S0(3, 1) from a physical viewpoint was given by Penrose, cf. g. [Sch95, p. 210]. g. [BPZ84, p. 335] "The situation is somewhat better in two dimensions. " [FQS84, p. 4 2 0 ] "Two dimensions is an especially promising place to apply notions of conformal field invariance, because there the group of conformal transformations is infinite dimensional.