A Mathematical Introduction to Conformal Field Theory by Martin Schottenloher

By Martin Schottenloher

The first a part of this e-book supplies a self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in dimensions. the second one half surveys a few extra complicated themes of conformal box theory.

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E. so-called projective representations. As we have explained in Sect. 2, the conformal group of R ~'~ is isomorphic to Diff+ (S) x Diff+ (S) (here and in the following S "= S 1 is the unit circle). Hence, given a theory with this conformal group as symmetry group, one studies the group Diff+ (S) and its Lie algebra first. After quantization one is interested in the unitary representations of the central extensions of Diff+ (S) or Lie (Diff+ (S)) in order to get representations in the Hilbert space as we have explained in the preceding two sections.

O. A, g . 2 The Conformal Group of IRp,v for p + q > 2 25 for ( { 0 . . . {,+1) e N p'q (cf. 4). For x 6 ]Rp,q we have ~(~(~)) = = ( 1-(x> A'x2 " . l+(x})2 (1 - 1 + (A'x)) ~ • A'~. ~ , , since A' e O(p, q) implies (x} = (A'x}. z(~o(x)) for all x e IRp'q. Hence, ~ ( z ( x ) ) = 2. Translations. For a translation ~o(x) = x + c, c E IR", one has the continuation ~ ( d . . . ~ -+~) := (~o _ (~,, c) - ~+ (c} • ~' + 2~+c • ~+~ + <~',~)+ ~+ <~)) for ( ~ 0 . . . ~,+1) E N p'q. Here, = ~ +d) and ~' = ( ~ 1 , .

1) E N p'q. Here, = ~ +d) and ~' = ( ~ 1 , . . , ~ ) . e. a conformal continuation of ~o. The matrix we look for can be found directly from the definition of ~. , i, - I , . . , - I ) 1 26 2. The Conformal Group is the (n x n) diagonal matrix representing gP'q. The proof of Ac E O(p + 1, q + 1) requires some elementary calculation. A~ E SO(p + 1, q + 1) can be shown by looking at the curve t ~ At~ connecting E~+2 and A~. 3. Dilatations. , 0 0 E~ l-r2 2r 0 l--r2 / 2r 0 l+r2 2r (A~ E O(p + 1, q + 1) requires a short calculation again).

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