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

By Martin Schottenloher (auth.)

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Extra 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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These neighborhoods form a subbasis of the strong topology. The strong topology on U (H) is not metrizable. , ;. ~-1 ( B ) C U(][-]I) open. 2 Quantization of Symmetries 41 The strong topology can be defined on any subset M C B(]HI)"= { B" ]HI ---, IHIIB is R-linear and bounded} of JR-linear continuous operators, hence in particular on M~ = { U" ][-]I---, IHI[U unitary or anti-unitary}. ) is induced by ~" M~ Aut(IP) (cf. 2). This topology on Aut(IP) (or U(IP)) is called the strong topology as well.

3. e. it is already a continuous homomorphism T : G --~ U(P). This is the reason why one assumes a continuous homomorphism T : G --, U(P) instead of T : G --, Aut(P) in the quantization of symmetries. 4. ~ " U(]E) ---, U(P) is a continuous homomorphism and has local continuous sections (cf. 8). 8 A n exact sequence of group homomorphisms 1 A ~E ~G ~1 splits if there is a homomorphism a • G ~ E such that r o a = idG. leading to the trivial extension 1 ~ A---~A × G - - * G ~ 1, which is equivalent to the original sequence in the following sense: the diagram 1 1 :A ~AxG id ¢ ~A ,E ~G ,1 id 'G ~1 commutes.

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. 6 Let n = p + q > 2. Every conformal transformation on a connected open subset M C R p'q has a unique conformal continuation to N p'q. The group of all conformal transformations N p'q N p,q is isomorphic to O(p+ 1, q+ 1)/{:kl}. 1 the conformal group Conf (R p'q) - is isomorphic to SO(p+ 1, q+ 1).

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