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By Martin Schottenloher

Half I offers an in depth, 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 type of crucial extensions of Lie algebras and teams. half II surveys extra complex issues of conformal box thought resembling the illustration conception of the Virasoro algebra, conformal symmetry inside string idea, an axiomatic method of Euclidean conformally covariant quantum box conception 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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RP,q) C N p'q. First of all, ( ~ o . . . ,~n)) __ ( ~ 0 . . . ~n+l)E ~(R p'q) for )~ "-- ~o_[_ ~n+l ¢ 0. G i v e n (~o . . . ~n+l) E N p'q t h e r e always exist sequences ek --* 0, 5k --* 0 with ek ¢ 0 ¢ 5k and 2~1ck q-c 2 = 2~n+1(~k q-(~. For p _ 1 we have Pk "= (~0 . ~1 + £k " ~2 . . . ~n . 4_ ~k) C= N ~''q Moreover, ~o + ~n+l _[_ (~k -- ¢~k # 0 implies Pk E z(RP'q). Finally, since Pk --* (~o. ~+1) for k --, co it follows that (~o . e. N p'q C ~(Rv,q). m We therefore choose N p'q as the underlying manifold of the conformal compactification.

Is a conformal continuation of ~, since ~ = CA with A = E S O ( p + 1, q + 1). En -½ (b) 1+ ½ (b) To sum up, for all conformal transformations ~ on Rp,q we have constructed continuations to conformal transformations ~" NP,q --. N v'q of the type ~ ( ~ 0 . . . ~+1) = 7(A~) with A 6 S O ( p + 1, q+ 1) having a conformal inverse ~-1 = ¢^_~. e. M = Rv,q or M = M1, cf. 9). e. ~ has a conformal continuation ~, which must be equal to ¢. Furthermore, the group of conformal transformations N p'q ~ N v'q is isomorphic to O(p + 1, q + 1)/{4-1}, since ~ can be described by the uniquely determined set {A,-A} of matrices in O(p + 1, q + 1).

3 The Conformal Group of lR2'° 29 In view of this result, it is justified to call the connected component containing the identity the conformal group Conf(R 2,°) of R 2'°. Another reason for this comes from the impossibility of enlarging this group by additional conformal transformations discussed below. 8 shows the following exceptional situation of the case p + q > 2: every conformal transformation, which is defined on a connected open subset M C R p'q, is injective and has a unique continuation to a global conformal transformation.

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