Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

G_2 invariant 7D Euclidean super Yang-Mills theory as a higher-dimensional analogue of the 3D super-BF theory

A formulation of the N_T=1, D=8 Euclidean super Yang-Mills theory with generalized self-duality and reduced Spin(7)-invariance is given which avoids the peculiar extra constraints of Nishino and Rajpoot, hep-th/0210132. Its reduction to 7 dimensions leads to the G_2-invariant N_T=2, D=7 super Yang-Mills theory which may be regarded as a higher-dimensional analogue of the N=2, D=3 super-BF theory. When reducing further that G_2-invariant theory to 3 dimensions one gets the N_T=2 super-BF theory coupled to a spinorial hypermultiplet.Comment: 9 pages, Late

A Comparative Study of Laplacians and Schroedinger-Lichnerowicz-Weitzenboeck Identities in Riemannian and Antisymplectic Geometry

We introduce an antisymplectic Dirac operator and antisymplectic gamma matrices. We explore similarities between, on one hand, the Schroedinger-Lichnerowicz formula for spinor bundles in Riemannian spin geometry, which contains a zeroth-order term proportional to the Levi-Civita scalar curvature, and, on the other hand, the nilpotent, Grassmann-odd, second-order \Delta operator in antisymplectic geometry, which in general has a zeroth-order term proportional to the odd scalar curvature of an arbitrary antisymplectic and torsionfree connection that is compatible with the measure density. Finally, we discuss the close relationship with the two-loop scalar curvature term in the quantum Hamiltonian for a particle in a curved Riemannian space.Comment: 55 pages, LaTeX. v2: Subsection 3.10 expanded. v3: Reference added. v4: Published versio

Generalizations of Eulerian partially ordered sets, flag numbers, and the Mobius function

A partially ordered set is r-thick if every nonempty open interval contains at least r elements. This paper studies the flag vectors of graded, r-thick posets and shows the smallest convex cone containing them is isomorphic to the cone of flag vectors of all graded posets. It also defines a k-analogue of the Mobius function and k-Eulerian posets, which are 2k-thick. Several characterizations of k-Eulerian posets are given. The generalized Dehn-Sommerville equations are proved for flag vectors of k-Eulerian posets. A new inequality is proved to be valid and sharp for rank 8 Eulerian posets