It has been conjectured that a pair ofD5−D5 branes wrapped on some non-trivial two cycle of a Calabi-Yau manifold becomes a stable BPS D3 brane in the presence of a very large B field and magnetic fluxes on their worldvolumes. We discuss this by considering the non-commutative field theory on the worldvolume of the pair of branes whose field multiplication is made with respect to two different ...

We consider the standard model in the formulationo of non-commutative geometry, for a Euclidean space-time consisting of two copies. The electroweak scale is set by the vacuum expectation value of a scalar field and is undetermined at the classical level. By adding the Coleman-Weinberg effective potential, that scale turns out to be fixed. Provided that the renormalized form of the Lagrangian m...

This paper consists of two parts. In the first part we show that any Poisson algebraic group over a field of characteristic zero and any Poisson Lie group admits a local quantization. This answers positively a question of Drinfeld and generalizes the results of [BFGP] and [BP]. In the second part we apply our techniques of quan-tization to obtain some nontrivial examples of quantization of Pois...

Let 3⁄4 be an SO.n/-bundle over a simply connected manifold M with a spin structure Q ! M . The string class is an obstruction to lift the structure group LSpin.n/ of the loop group bundle L Q ! L M to the universal central extension of LSpin.n/ by the circle. We prove that the string class vanishes if and only if 1=2 the first Pontrjagin class of 3⁄4 vanishes when M is a compact simply connect...

Let Mn be a closed, connected n-manifold. Let M−τ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that M−τ is homotopy equivalent to the Spanier-Whitehead dual of M with a disjoint basepoint, M+. This dual can be viewed as the function spectrum, F (M,S), where S is the sphere spectrum. F (M,S) has the structure of a commutative, symmetric ring spectru...

It is shown that a (curved) projective structure on a smooth manifold determines on the Poisson algebra of smooth functions on the cotangent bundle, fiberwise-polynomial of bounded degree, a one-parameter family of graded star products. For a particular value of the parameter (corresponding to half-densities) the star product is symmetric, and specializes in the projectively flat case to the on...

Given a commutative semigroup S with 0, where 0 is the unique singleton ideal, we associate a simple graph Γ(S), whose vertices are labeled with the nonzero elements in S. Two vertices in Γ(S) are adjacent if and only if the corresponding elements multiply to 0. The inverse problem, i.e., given an arbitrary simple graph, whether or not it can be associated to some commutative semigroup, has pro...

This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter −A. There is an application to Penrose’s binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this ten...

We continue the investigation of Hopf-Galois extensions with central invariants started in [29]. Our objective is not to imitate algebraic geometry using Hopf-Galois extension but to understand their geometric properties. Let H be a finite-dimensional Hopf algebra over a ground field k. Our main object of study is an H-Galois extension U ⊇ O such that O is a central subalgebra of U . Let us bri...

We present Lie group integrators for nonlinear stochastic differential equations with non-commutative vector fields whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action, and subsequently pull back the flow to the corresponding Lie alge...