In Segal’s axiomatisation, conformal field theories (CFT) can be constructed from (projective) representations of the semigroup of annuli (the set of annuli with parametrised boundaries endowed with the gluing operation). In the case of the Liouville CFT, we realise this idea by defining a certain family of Markov processes on the space of distributions on the unit circle. Their generators exist as unbounded operators on the Hilbert space and represent the Virasoro algebra (they are equivalent to the Sugawara construction). This representation is instrumental in the study of the conformal blocks of the theory. We can define them using pants decomposition of the underlying surface, and associate an operator in the Hilbert space to each pair of pants. The gluing of pairs of pants is represented by the composition of operators, and the conformal blocks are essentially the spectral decomposition of these operators. Their variation with respect to the complex structure of the surface is governed by the semigroup, which is a version of the statement that conformal blocks are horizontal with respect to the connection defined by the stress-energy tensor. This allows us to show that the blocks do not depend on the choice of curves representing the pants decomposition. Based on joint and ongoing works with Guillarmou, Kupiainen, Rhodes & Vargas. [The talk will be streamed online, please ask the organizers for the link.]

Humboldt-Universität zu Berlin