Clifford Algebra Derivations of Tau-Functions for Two-Dimensional Integrable Models with Positive and Negative Flows

We use a Grassmannian framework to define multi-component tau functions as expectation values of certain multi-component Fermi operators satisfying simple bilinear commutation relations on Clifford algebra. The tau functions contain both positive and negative flows and are shown to satisfy the $2n$-component KP hierarchy. The hierarchy equations can be formulated in terms of pseudo-differential equations for $n\times n$ matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato–Wilson relations. A reduction process leads to the AKNS, two-component Camassa–Holm and Cecotti–Vafa models and the formalism provides simple formulas for their solutions.