By Martin Schottenloher
Half I offers a close, 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 category of primary extensions of Lie algebras and teams. half II surveys extra complex issues of conformal box conception resembling the illustration idea of the Virasoro algebra, conformal symmetry inside string idea, an axiomatic method of Euclidean conformally covariant quantum box concept 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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Additional resources for A mathematical introduction to conformal field theory
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. Conversely, if such a commutative diagram with a group isomorphism ¢ exists, the sequence 1 ~A ~E ,G splits with splitting map a ( g ) : = ¢(1A, g). ,1 3. 9 Let 1 ~A--~E "~G ~1 be a central extension and let T • G ---* E be a map (not necessarily a homomorphism) with r o T = ida and T(1) = 1. We set T~ "= T (X) for x 6 G and define a map G ×G (~,y) 02 " , ) A~-~(A) C E ~ ~~ This map is well-defined since TzTyT~1 6 ker r , and it satisfies w(1, 1) = 1 and w (x, y)w (xy, z) = w (x, yz) w (y, z) (4) for x, y, z 6 G.
This is the reason why in the context of quantization of classical field theories with conformal symmetry Diff+ (S) x Diff+ (S) one is interested in the cohomology group g 2 (Diff+ (S), V(1)). 4 Central Extensions of Lie Algebras and Bargmann's Theorem In this section some basic results on Lie groups and Lie algebras are assumed to be known, as presented, for instance, in [HN91] or [BR77]. For example, every finite-dimensional Lie group G has a corresponding Lie algebra Lie G determined up to isomorphism, and every differentiable homomorphism R" G --, H of Lie groups induces a Lie-algebra homomorphism Lie R = / ~ " Lie G --, Lie H.
Hence, on the infinitesimal level the cases (p, q) = (2, 0) and (p, q) = (1, 1) seem to be quite similar. However, in the interpretation and within the representation theory there are differences, which we will not discuss here. We shall just mention that the Lie algebra ~[(2, C) belongs to the Witt algebra in the Euclidean case (as the Lie algebra of Mb generated by L_I, L0, L1), while in the Minkowski case it is only generated by complexification of ~[(2, R) . 4 The Conformal Group of RI,1 33 2 For M = N 1,1 one u~ = -vy, uy = -v~, and, in addition, u~2 > v~.
A mathematical introduction to conformal field theory by Martin Schottenloher