A Mathematical Introduction to Conformal Field Theory: Based by Martin Schottenloher (auth.)

By Martin Schottenloher (auth.)

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Additional resources for A Mathematical Introduction to Conformal Field Theory: Based on a Series of Lectures given at the Mathematisches Institut der Universität Hamburg

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8 ~ • U(IHI) ---. U(]P) has local continuous sections and therefore can be regarded as a principal fibre bundle with structure group U(1). P r o o f (cf. [Sim68, p. 10]). For f e E let Y$ "= {U e U(IHI)" (U f, f ) ¢ 0}. Then VI is open in U(IHI), since U ~ U f is continuous in the strong topology. Hence, U ~ (U f, f ) is continuous as well. ) since ~ - l ( W f ) -- Yf is open. ). Let j~f " Vf --"+ U(1), U I(Uf, f)l

In this way, N p'q obtains a conformal structure. e. an injective conformal transformation. As a consequence, N p'q is a conformal compactification of R v'q. 4 CA" N p'q ~ For every m a t r i x A E O ( p + 1, q + 1) the m a p ¢ = N p'q defined by ~3(~0..... = 7(A~), ( ~ o . . . 1 Con[ormal Compactification o [ R p'q 23 is a conformal transformation and a diffeomorphism. The inverse transformation ¢-1 = CA-~ is also conformal. The map A ~-, CA is not injective. However, CA = CA' implies A = A ~ or A = - A ' .

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. 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 .

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