We also discuss the relation of our work to the earlier work on the decomposition of the ζ-determinant by Burghelea, Friedlander and Kappeler (from this point on referred to as BFK). The present work is companion to the paper [10] and in several places we refer to [10] for the proof of a given statement and a more detailed discussion. Let D : C(M ;S) → C(M ;S) be a compatible Dirac operator act...

Dirac cohomology is a new tool to study unitary and admissible representations of semisimple Lie groups. It was introduced by Vogan and further studied by Kostant and ourselves [V2], [HP1], [K4]. The aim of this paper is to study the Dirac cohomology for the Kostant cubic Dirac operator and its relation to Lie algebra cohomology. We show that the Dirac cohomology coincides with the correspondin...

For a continuous curve of families of Dirac type operators we define a higher spectral flow as a K-group element. We show that this higher spectral flow can be computed analytically by '̂-forms and is related to the family index in the same way as the spectral flow is related to the index. We introduce a notion of Toeplitz family and relate its index to the higher spectral flow. Applications to ...

We provide a method of inserting and removing any finite number of prescribed eigenvalues into spectral gaps of a given one-dimensional Dirac operator. This is done in such a way that the original and deformed operator are unitarily equivalent when restricted to the complement of the subspace spanned by the newly inserted eigenvalue. Moreover, the unitary transformation operator which links the...

For a continuous curve of families of Dirac type operators we define a higher spectral flow as a K-group element. We show that this higher spectral flow can be computed analytically by η̂-forms, and is related to the family index in the same way as the spectral flow is related to the index. We introduce a notion of Toeplitz family and relate its index to the higher spectral flow. Applications to...

We extend classical results of Kostant and al. on multiplets of representations of finite-dimensional Lie algebras and on the cubic Dirac operator to the setting of affine Lie algebras and twisted affine cubic Dirac operator. We prove in this setting an analogue of Vogan’s conjecture on infinitesimal characters of Harish–Chandra modules in terms of Dirac cohomology. For our calculations we use ...

With respect to the Dirac operator and the conformally invariant Laplacian, an explicit description of the inverse Penrose transform on Riemannian twistor spaces is given. A Dolbeault representative of cohomology on the twistor space is constructed from a solution of the field equation on the base manifold.

We study the asymptotic of the Bergman kernel of the spin Dirac operator on high tensor powers of a line bundle.

O. Obregón∗ Instituto de F́ısica de la Universidad de Guanajuato, P.O. Box E-143, 37150 León Gto., México (Dated: June 4, 2005) Abstract A non-Abelian Born-Infeld theory is presented. The square root structure that characterizes the Dirac-Born-Infeld (DBI) action does not appear. The procedure is based on an Abelian theory proposed by Erwin Schrödinger that, as he showed, is equivalent to Born-I...

We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on R. We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small diierences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operat...