Quantum dilaton equation for the quantum tau-function F^(q)

Establish that the quantum tau-function F^(q), defined by the commutator relations in equation (1.1), the quantum string equation (1.2), and the specified constant term F^(q)|_{t*=0} = -i ε, satisfies the quantum dilaton equation (2.1) for general ε (i.e., without the substitution ε = 0).

Background

The paper considers the quantum tau-function F^q associated to the quantization of the KdV hierarchy via commuting quantum Hamiltonians H_d. F^q is characterized by the multiple commutator relations (1.1), the quantum string equation (1.2), and a fixed constant term, providing a quantum analog of the classical Witten–Kontsevich generating series.

The authors highlight a quantum analog of the classical dilaton equation, presented as equation (2.1). While compatibility with the classical dilaton equation is noted and the validity of the quantum dilaton equation after setting ε = 0 has been shown, the full proof for general ε remains unresolved.