A mathematical introduction to conformal field theory by Martin Schottenloher

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 decided and the appearence of the Virasoro algebra within the context of the quantization of two-dimensional conformal symmetry is defined through the type of principal extensions of Lie algebras and teams. half II surveys extra complicated themes of conformal box concept similar to the illustration concept of the Virasoro algebra, conformal symmetry inside string conception, an axiomatic method of Euclidean conformally covariant quantum box thought 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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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. 22 2. 3 The restriction of 7 to the product of spheres p n+l } C Rn+2~ x sq "= {~ ~ R "+2" j=o j=p+ l gives a differentiable 2-to-1 covering 71" "--" 7[$PxSq " SP X ~q ~ N p'q. P r o o f Obviously 7 ( ~ x S q) c N p'q. For ~, ~' E S p x S q it follows from 7(~) = 7(~') that ~ = A~' with A E R \ {0}.

1 ) . _ 7(~) with respect to the so-called homogeneous coordinates. Obviously, we have (~0..... ~n+l) _. (/~0 . . . /~n+l) for all A e R \ {0}. We are looking for a suitable compactification of RP,q. n+I(R). R e m a r k 2 . 2 z(Rv,q) = N p,q, where Np,q := { ( ~ o . . . e. ~(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-(~.

40 Central Extensions of Groups All these groups are topological groups in a natural way. 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).

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