# Ward and Belavin-Polyakov-Zamolodchikov (BPZ) identities for Liouville quantum field theory on the Riemann sphere

# Ward and Belavin-Polyakov-Zamolodchikov (BPZ) identities for Liouville quantum field theory on the Riemann sphere

The foundations of modern conformal field theory (CFT) were introduced in a 1984 seminal paper by Belavin, Polyakov and Zamolodchikov (BPZ). Though the CFT formalism is widespread in the physics literature, it remains a challenge for mathematicians to make sense out of it. Liouville CFT (or quantum field theory), introduced by Polyakov in his 1981 theory of Liouville quantum gravity, is a class of CFTs which can be seen as a random version of the theory of Riemann surfaces. In a recent work, we constructed the correlation functions (and the random measures) of Liouville CFT in the Feynman path formalism using probabilistic techniques. In this talk, I will present a rigorous derivation of the so-called Ward and BPZ identities for Liouville CFT. These identities are the building blocks of the CFT formalism. Based on joint works with F. David, A. Kupiainen and R. Rhodes.