Precise constraints on a $\tau$ function in 2D quantum gravity

TL;DR

This paper introduces a new method to precisely determine unknown constants in constraints on a tau function in 2D quantum gravity, revealing a unique algebra structure that includes the Virasoro algebra.

Contribution

It proposes a computable approach to find all non-zero constants in tau function constraints, expanding understanding of the algebraic structure in 2D quantum gravity.

Findings

All constants in the constraints are non-zero.

The resulting algebra includes the Virasoro algebra as a subalgebra.

The method applies to arbitrary p in the hierarchy.

Abstract

For an arbitrary $p$, propose a new and computable method which can determine the values of unknown constants in constraints on a tau function which satisfies both the p-reduced KP hierarchy and the sting equation. All the constants do not equal 0, unlike what people usually think of. With these values, obtain the precise algebra that the constraints compose. This algebra includes none of $\{t_{mp}\}$ and also includes the Virasoro algebra as a subalgebra.