By Martin Schottenloher

Half I provides 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 class of relevant extensions of Lie algebras and teams. half II surveys extra complex subject matters of conformal box concept equivalent to the illustration concept of the Virasoro algebra, conformal symmetry inside string concept, 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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**Extra info for A mathematical introduction to conformal field theory**

**Example text**

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. N p'q is a regular quadric in IP,+I(R). Hence it is an n-dimensional compact submanifold of IP~+I(R). N p'q contains ,(R v,q) as a dense subset. We get another description of N p,q using the quotient map "y.

E. e. A ~ ~ A~. Altogether, ¢ : N p'q --, N p'q is well-defined. Because of the fact that the metric on R p÷l'q÷l is invariant with respect to A, CA turns out to be conformal. if P is . represented by E Sp x S q. ) 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.

A topological group is a group G equipped with a topology, such that the group operation G × G --, G, (g, h) ~ gh, and the inversion map G --, G, g ~ g-i, are continuous. The first three examples are finite-dimensional Lie groups, while the last two examples are infinite-dimensional Lie groups (modeled on Fr~chet spaces). The topology of Diff+(S) will be discussed shortly at the beginning of Sect. 5. ) remains, which is continuous for the strong topology on U(P) (see below for the definition of the strong topology).