Abstract.
We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick’s theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half-plane. This is then applied to stochastic Loewner evolutions (SLE). We give a rigorous derivation of the equations, obtained previously by the authors, that connect the stochastic Loewner equation to the representation theory of the Virasoro algebra. We give a new proof that this construction enumerates all polynomial SLE martingales. When one of the hulls removed from the upper half-plane is the SLE hull, we show that the partition function reduces to a useful local martingale known to probabilists, thereby unraveling its CFT origin.
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Communicated by Vincent Rivasseau
Submitted 13/06/03, accepted 21/10/03
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Bauer, M., Bernard, D. Conformal Transformations and the SLE Partition Function Martingale . Ann. Henri Poincaré 5, 289–326 (2004). https://doi.org/10.1007/s00023-004-0170-z
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DOI: https://doi.org/10.1007/s00023-004-0170-z