The Virasoro symmetries of the bigraded modified Toda hierarchy
Yi Yang
TL;DR
The paper develops a comprehensive symmetry framework for the modified Toda hierarchy by constructing its additional symmetries via Orlov–Schulman operators and vertex-operator formalisms, and proves the Adler–Shiota–van Moerbeke (ASvM) formula linking wave-function and tau-function dynamics. It extends these results to the $(N,M)$-bigraded modified Toda hierarchy (BMTH), showing that Virasoro-type symmetries arise from linear combinations of the additional flows and form a centerless Virasoro subalgebra under the reduction $L_1^N=L_2^M$ with $\,\mathcal{L}(1)=0$. The analysis also reveals that the MToda flows generate a centerless $W_{\\infty}\times W_{\\infty}$ algebra, providing a robust algebraic underpinning for string-equation constraints and related matrix-model applications. Together, these results deepen the connection between integrable hierarchies, Virasoro symmetries, and bigraded reductions, with potential impact on constrained Toda systems and their applications in mathematical physics.
Abstract
Modified Toda hierarchy is a two-component generalization of the 1st modified KP hierarchy, which has been widely applied to analyze constraints of the Toda hierarchy, including the B--Toda and C--Toda hierarchies. In this paper, we construct additional symmetries for the modified Toda hierarchy and derive the corresponding Adler-Shiota-van Moerbeke formula. In addition, we also show that $(N,M)$--bigraded modified Toda hierarchy(BMTH), which is a special reduction for modified Toda, possess additional symmetries that form a subalgebra of the Virasoro algebra.