Virasoro vacuum block at next-to-leading order in the heavy-light limit

We consider the semiclassical limit of the vacuum virasoro block describing the diagonal 4-point correlation functions on the sphere. at large central charge c, after exponentiation, it depends on two fixed ratios h h /c and h l /c, where h h,l are the conformal dimensions of the 4-point function operators. the semiclassical block be expanded in powers of the light ratio h l /c and the leading non-trivial (linear) order is known in closed form as a function of h h /c. recently, this contribution has been matched against ads3 gravity calculations where heavy operators build up a classical geometry corresponding to a btz black hole, while the light operators are described by a geodesic in this background. here, we compute for the first time the next-to-leading quadratic correction $$ mathcal{o} $$ ((h l /c)2), again in closed form for generic heavy operator ratio h h /c. the result is a highly non-trivial extension of the leading order and be relevant for further refined ads3/cft2 tests. applications to the two-interval rényi entropy are also presented.