A mathematical introduction to conformal field theory by Martin Schottenloher

By Martin Schottenloher

Half I offers an in depth, 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 themes of conformal box conception corresponding to the illustration idea of the Virasoro algebra, conformal symmetry inside string concept, 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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The extension is called central if the image ~ (A) of A is in the center of E, i. e. a e A,b e E =~ ~ (a)b = b~(a). Examples: • A trivial extension has the form 1 , A i ~AxG pr2> G >1, where A x G denotes the product group and where i • A --. G is given by a ~ (a, 1). This extension is central. • An example sequence for a non-trivial central extension is the exact 1 ~Zk ~E=U(1) ~U(1) ;1 with ~ (z) "= z k for k E N, k >_ 2. This extension cannot be trivial, since E = U(1) and Zk x U(1) are not isomorphic.

For every continuous homomorphism T : G ---, U(IP) there is a continuous homomorphism S" G ---, U(H) with T = ~ o S. U(1) u(~) T ^ , u(~) ,1 4. Central Extensions of Lie Algebras 52 A Here, E = { (U, g) e U(]HI) x G[ ~(U) = Tg}, r = pr 2 and T = pr:. E is a topological group as a subgroup of the topological group U(]E) x G (cf. 7) and T is a continuous homomorphism. 8: For every A E U(P) there is an open neighborhood W C U(IP) and a continuous map u : W ---, U(H) with o u = idw. Let now V "= T-i(W).

Proof By definition of w we have -- TxTyT~I T x y T z T x y -Iz "-- TxTyTzTxy = = ~~z%z~z~-~z ~ (y, z)~z~-~l~ = ~~;~lz = ~ (~, y z ) ~ (y, z). 10 Let w • G × G ~ A be a map having the property (~). , G denotes the product A × G endowed with the multiplication defined by (a, ~) (b, y) "= (~ (~, y) ab, ~y) for (a, x), (b, y) 6 A x G. 3 Equivalence of Central Extensions 45 It has to be shown that this multiplication defines a group structure on A ×~ G for which z and pr2 are homomorphisms. The crucial property is the associativity of the multiplication, which is guaranteed by the condition (4)" ((a, x)(b, y))(c, z) = (w(x, y)ab, xy)(c, z) = = = = y)abc, yz) (a, (a, (y, z) z, Vz) y), z)).

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A mathematical introduction to conformal field theory by Martin Schottenloher
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