Givental formula in terms of Virasoro operators

Abstract

We present a conjecture that the universal enveloping algebra of differential operators ∂ ∂t k over C coincides in the origin with the universal enveloping algebra of the (Borel subalgebra of) Virasoro generators from the Kontsevich model. Thus, we can decompose any (pseudo)differential operator to a combination of the Virasoro operators. Using this decomposition we present the r.h.s. of the Givental formula [1] as a constant part of the differential operator we introduce. In the case of CP 1 studied in [2], the l.h.s. of the Givental formula is a unit, which imposes certain constraints on this differential operator. We explicitly check that these constraints are correct up to O(q 4 ). We also propose a conjecture of factorization modulo Hirota equation of the differential operator introduced and check this conjecture with the same accuracy.