Givental formula in terms of Virasoro operators

TL;DR

The paper conjectures a deep connection between differential operators and Virasoro algebra in the context of Givental's formula, providing a new decomposition approach and verifying constraints up to a certain order.

Contribution

It introduces a conjectural decomposition of differential operators into Virasoro generators and applies this to analyze Givental's formula, with explicit checks and a proposed factorization.

Findings

Decomposition of differential operators into Virasoro operators is possible at the origin.

Constraints from Givental's formula are verified up to order q^4.

A conjecture on factorization modulo Hirota equation is proposed and tested.

Abstract

We present a conjecture that the universal enveloping algebra of differential operators $\frac{\p}{\p t_k}$ over $\mathbb{C}$ 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 math.AG/0008067 as a constant part of the differential operator we introduce. In the case of $\mathbb{CP}^1$ studied in hep-th/0103254, 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.