Bilinear equations on Painleve tau functions from CFT

Understand

In 2012 Gamayun, Iorgov, Lisovyy conjectured an explicit expression for the Painlev\'e VI $\tau$~function in terms of the Liouville conformal blocks with central charge $c=1$.

• We prove that proposed expression satisfies Painlev\'e VI $\tau$~function bilinear equations (and therefore prove the conjecture).
• The proof reduces to the proof of bilinear relations on conformal blocks.
• These relations were studied using the embedding of a direct sum of two Virasoro algebras into a sum of Majorana fermion and Super Virasoro algebra.