Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 020, 29 pages      nlin.SI/0605027      https://doi.org/10.3842/SIGMA.2007.020
Contribution to the Vadim Kuznetsov Memorial Issue

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

Henrik Aratyn a and Johan van de Leur b
a) Department of Physics, University of Illinois at Chicago, 845 W. Taylor St., Chicago, IL 60607-7059, USA
b) Mathematical Institute, University of Utrecht, P.O. Box 80010, 3508 TA Utrecht, The Netherlands

Received October 11, 2006, in final form January 09, 2007; Published online February 06, 2007

Abstract
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 × 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.

Key words: Clifford algebra; tau-functions; Kac-Moody algebras; loop groups; Camassa-Holm equation; Cecotti-Vafa equations; AKNS hierarchy.

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