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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

Example text

G. for the case (p, q ) = (2, 1). 3. 1. Orthogonal transformations. The easiest case is the continuation of an orthogonal transformation ~(x) = A~x represented by a matrix A~ E O(p, q). , 1 , - 1 , . . , - 1 ) is the matrix representing gp+l'q+l. Furthermore, h E S O ( p + 1 , q + 1) ~ h' E SO(p,q). e. (~(%~0..... ~(,~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'~.

F:,- g. ) g . ) v~ = ~1 ( f ; + g . ) it follows immediately that u~ = vy, uy = v~. Conversely, let (~,v) ~ c°°(R2, R 2) 2. 34 The Conformal Group w i t h u , = vy, u~ = v,. T h e n u , , = vw = u~y. Since a solution of the one-dimensional wave equation u has the form u(x, y) = 51 (f+ (x, y) + g_ (X, y)) with f, g e C °° (R). Because 1 g,_ 1 g,_ , ofv~=uy=5(f~)andvy=u~=~(f~+ ) we have 1 v = ~ ( f + - g_) where f and g possibly have to be changed by a constant. 2. For (u, v) = ~ ( f , g) one has u~2 - v~2 = f +~ g/.

For p = q = 1, the global conformal transformations need no continuation at all, hence a conformal compactification is not necessary. 4 The Conformal Group of ]R1,1 35 conformal group of ~1,1 simply as the connected component containing the identity of the group of conformal transformations R 1'1 ---, R 1,1. 10 isomorphic to Diff+ (]~) x Diff+ (R). However, for various reasons one wants to work with a group of transformations on a compact manifold with a conformal structure. Therefore, one replaces R 1'1 with S 1'1 in the sense of the conformal compactification of the Minkowski plane which we described at the beginning (cf.