Multi-component Pfaff-Toda hierarchy within bilinear formalism

TL;DR

This work extends the Pfaff-Toda hierarchy to a multi-component setting using free fermions and Clifford-group operators, defining the tau-function as a vacuum expectation value $\tau({\bf n}, \bar{\bf n}, {\bf t}, \bar{\bf t})=\langle {\bf n}| e^{J({\bf t})} g e^{-\bar{J}(\bar{\bf t})}|-\bar{\bf n}\rangle$ and deriving a generating bilinear equation that encodes the entire hierarchy. The authors derive an operator bilinear identity and, via non-abelian bosonization, obtain an integral generating relation for the tau-function, from which Hirota-Miwa type functional equations follow as corollaries. They classify and present explicit non-degenerate 4-point relations and show how to obtain numerous 2-point reductions through Miwa-type degenerations, illustrating the rich structure of inter-component couplings. The paper also outlines connections to the DKP hierarchy and discusses future directions, including dispersionless limits and the potential equivalence with Lax-Sato formulations, with a second part focusing on dispersionless aspects promised. Overall, the work provides a robust bilinear framework for multi-component Pfaff-Toda hierarchies and lays groundwork for both analytical and geometric explorations of these integrable systems.

Abstract

Using the free fermions technique and non-abelian bosonization rules we introduce the multi-component Pfaff-Toda hierarchy. The tau-function is defined as vacuum expectation value of a Clifford group element of the algebra of Fermi-operators. A generating bilinear integral equation for the tau-function is obtained. A number of bilinear functional relations for the tau-function of the Hirota-Miwa type are derived as corollaries of the generating bilinear equation.