/ 93 09 01 7 v 1 2 S ep 1 99 3 Tau functions for the Dirac operator on the Poincaré disk

Table of Contents Introduction—1 §1 The Dirac operator on the Hyperbolic disk A covering of the frame bundle—4 The Dirac operator—7 Covariance of the Dirac operator—10 §2 Local Expansions Eigenfunctions for infinitesimal rotations—14 Differentiating local expansions with respect to the branch points—21 Estimates at infinity—24 §3 L 2 existence results A model for the simply connected covering D R (a)—28 Multivalued solutions with specified branching—30 Existence for a canonical L 2 basis—35 The response functions—38 §4 The Green function G a,λ for the Dirac operator Green functions in the absence of branch points—40 The Green function for the Helmholtz operator with branch points—44 The Green function for the Dirac operator with branch points—52 The derivative of the Green function G a,λ —54 §5 Deformation Equations A holonomic system—58 The holonomic extension—63 The deformation equations—64 Further identities—66 The two point deformation theory in a special case—71 §6 The tau function The Grassmannian formalism—80 The log derivative of τ in the two point case—88 Bibliography—91 Introduction