In Segal's definition of conformal field theory (CFT), one key ingredient is to construct a representation of the semigroup of complex annuli with parametrised boundaries. In the first part of the talk, I will explain how this statement can be understood as a generalisation of the Hille-Yosida theorem, using only an analytic language. In particular, the (infinite dimensional) family of generators form a representation of the Virasoro algebra and encode the conformal symmetry of the theory. In the second part, I will give an example of this construction using the Gaussian free field, and show how it can be extended to treat the Liouville CFT. Based on joint works with Guillarmou, Kupiainen, Rhodes, Vargas.
