We study Chern-Simons theory on 3-manifolds M that are circle-bundles over 2dimensional surfaces Σ and show that the method of Abelianisation, previously employed for trivial bundles Σ× S, can be adapted to this case. This reduces the non-Abelian theory onM to a 2-dimensional Abelian theory on Σ which we identify with q-deformed Yang-Mills theory, as anticipated by Vafa et al. We compare and co...

The formal extension of the T-duality rules for open strings from Abelian to non-Abelian gauge field background leads in a well known manner to the notion of matrix valued D-brane position. The application of this concept to the non-Abelian gauge field RG β-function of the corresponding σ-model yields a mass term in the gauge field dynamics on the matrix D-brane. The direct calculation in a cor...

The topology of a normal surface singularity does not determine the analytical invariants of its equisingularity class, but recent partial results indicated that this should be true under two restrictions, a topological one, that the link of the singularity is a rational homology sphere, and an analytical one, that the singularity is Q-Gorenstein. Neumann and Wahl conjectured that the singulari...

Sunada’s work on topological crystallography emphasizes the role of the ‘maximal abelian cover’ of a graph X. This is a covering space of X for which the group of deck transformations is the first homology group H1(X,Z). An embedding of the maximal abelian cover in a vector space can serve as the pattern for a crystal: atoms are located at the vertices, while bonds lie on the edges. We prove th...

In this paper we decompose the rational homology of ordered configuration space n open unit-diameter disks in infinite strip width 2 as a direct sum induced $$S_{n}$$ -representations. (Alpert Generalized representation stability for and no-k-equal spaces, arXiv:2006.01240 , 2020), Alpert proves that $$k^{\text {th}}$$ -integral is an FI $$_{k+1}$$ -module by studying certain operations on call...

The size function is defined for points in projective space over any field K, finitely generated field over Q, generalizing the height function for number fields. We prove that the size function on the Jf-rational points of an abelian variety is bounded by a quadratic function. Introduction. In his book, Introduction to transcendental numbers, Lang showed how one can extend some of the theorems...

We give a formulation of the abelian case of Stark’s Main Conjecture in terms of determinants of projective modules and briefly show how this formulation leads naturally to its Equivariant Tamagawa Number Conjecture (ETNC) – type integral refinements. We discuss the Rubin-Stark integral refinement of an idempotent p1 iece of Stark’s Abelian Main Conjecture. In the process, we give a new formula...

Suppose we are given a differentiable path η̂. Is it possible to derive sub-minimal subsets? We show that every integral monodromy is rightJacobi and countably stochastic. This leaves open the question of uncountability. Unfortunately, we cannot assume that there exists an abelian unique topos.

A reduction of Benney's equations is constructed corresponding to Schwartz-Christoffel maps associated with a family of genus six cyclic tetragonal curves. The mapping function, a second kind Abelian integral on the associated Riemann surface, is constructed explicitly as a rational expression in derivatives of the Kleinian σ-function of the curve.