ar X iv : 0 70 5 . 16 99 v 2 [ m at h . A P ] 1 4 M ay 2 00 7 Subelliptic Spin C Dirac operators , II Basic Estimates

ar X iv : 0 70 5 . 07 12 v 1 [ m at h - ph ] 4 M ay 2 00 7 REFLECTION POSITIVITY AND MONOTONICITY

We prove the most general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality C D ≤ C N between Dirichlet and Neumann covariance operators on a manifold with a reflection.

ar X iv : 0 70 5 . 07 12 v 2 [ m at h - ph ] 9 M ay 2 00 7 REFLECTION POSITIVITY AND MONOTONICITY

We prove general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality C D ≤ C N between Dirichlet and Neumann covariance operators on a manifold with a reflection.

ar X iv : h ep - p h / 97 05 45 5 v 1 2 8 M ay 1 99 7 LECTURE 1 Spectral Fluctuations of the QCD Dirac Operator

ar X iv : 0 70 5 . 08 66 v 2 [ he p - th ] 1 0 M ay 2 00 7 Infinite loop superalgebras of the Dirac theory on the Euclidean Taub - NUT space

The Dirac theory in the Euclidean Taub-NUT space gives rise to a large collection of conserved operators associated to genuine or hidden symmetries. They are involved in interesting algebraic structures as dynamical algebras or even infinite-dimensional algebras or superalgebras. One presents here the infinite-dimensional superalgebra specific to the Dirac theory in manifolds carrying the Gross...

ar X iv : h ep - l at / 0 00 30 13 v 1 1 7 M ar 2 00 0 BUTP / 2000 - 6 The construction of generalized Dirac operators on the lattice

We discuss the steps to construct Dirac operators which have arbitrary fermion offsets, gauge paths, a general structure in Dirac space and satisfy the basic symmetries (gauge symmetry, hermiticity condition , charge conjugation, hypercubic rotations and reflections) on the lattice. We give an extensive set of examples and offer help to add further structures.