Discrete and continuous Yang-Mills measure for non-trivial bundles over compact surfaces

In this paper, we construct one Yang-Mills measure on an orientable compact surface for each isomorphism class of principal bundles with compact connected structure group over this surface. For this, we refine the discretization procedure used in a previous construction [9] and define a discrete theory on a new configuration space which is essentially a covering of the usual one. We prove that the measures corresponding to different isomorphism classes of bundles or to different total areas of the base space are mutually singular. We give also a combinatorial computation of the partition functions which relies on the formalism of fat graphs.     

Cohomology rings of multiplicative fibrations

Let p be an odd prime. Let be the fibre space induced from an H-map where K is a generalized Eilenberg MacLane space and X is a simply connected H-space. Such spaces occur frequently in Postnikov towers and connective covers. In this paper, we compute the mod p cohomology of as a ring. The ring depends on the structure of imf^* and the structure of subkerf^* as modules over the Steenrod algebra.     

Rational homotopy of the (homotopy) fixed point sets of circle actions

In this paper, when G is the circle S¹ and M is a G-space, we study the rational homotopy type of the fixed point set MG, the homotopy fixed point set MhG, and the natural injection MG→MhG.     

Maps between classifying spaces of Kac-Moody groups

We classify up to homotopy the self-maps of the classifying space of any non-affine Kac-Moody group of rank two.     

On string topology of classifying spaces

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.     

Radford's biproduct for quasi-Hopf algebras and bosonization

Let B be a braided Hopf algebra (with bijective antipode) in the category of left Yetter-Drinfeld modules over a quasi-Hopf algebra H. As in the case of Hopf algebras (J. Algebra 92 (1985) 322), the smash product B#H defined in (Comm. Algebra 28(2) (2000) 631) and a kind of smash coproduct afford a quasi-Hopf algebra structure on B⊗H. Using this, we obtain the structure of quasi-Hopf algebras with a projection. Further we will use this biproduct to describe the Majid bosonization (J. Algebra 163 (1994) 165) for quasi-Hopf algebras.     

Localization and duality in topology and modular representation theory

We develop a duality theory for localizations in the context of ring spectra in algebraic topology. We apply this to prove a theorem in the modular representation theory of finite groups.Let G be a finite group and k be an algebraically closed field of characteristic p. If p is a homogeneous nonmaximal prime ideal in H^∗(G,k), then there is an idempotent module κ_p which picks out the layer of the stable module category corresponding to p, and which was used by Benson, Carlson and Rickard [D.J. Benson, J.F. Carlson, J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997) 59-80] in their development of varieties for infinitely generated kG-modules. Our main theorem states that the Tate cohomology is a shift of the injective hull of H^∗(G,k)/p as a graded H^∗(G,k)-module. Since κ_p can be constructed using a version of the stable Koszul complex, this can be viewed as a statement of localized Gorenstein duality in modular representation theory. Various consequences of this theorem are given, including the statement that the stable endomorphism ring of the module κ_p is the p-completion of cohomology , and the statement that κ_p is a pure injective kG-module.In the course of proving the theorem, we further develop the framework introduced by Dwyer, Greenlees and Iyengar [W.G. Dwyer, J.P.C. Greenlees, S. Iyengar, Duality in algebra and topology, Adv. Math. 200 (2006) 357-402] for translating between the unbounded derived categories and . We also construct a functor to the full stable module category, which extends the usual functor and which preserves Tate cohomology. The main theorem is formulated and proved in , and then translated to and finally to .The main theorem in can be viewed as stating that a version of Gorenstein duality holds after localizing at a prime ideal in H^∗(BG;k). This version of the theorem holds more generally for a compact Lie group satisfying a mild orientation condition. This duality lies behind the local cohomology spectral sequence of Greenlees and Lyubeznik for localizations of H^∗(BG;k).In a companion paper [D.J. Benson, Idempotent kG-modules with injective cohomology, J. Pure Appl. Algebra 212 (7) (2008) 1744-1746], a more recent and shorter proof of the main theorem is given. The more recent proof seems less natural, and does not say anything about localization of the Gorenstein condition for compact Lie groups.     

Frobenius rational loop algebra

Recently R. Cohen and V. Godin have proved that the loop homology of a closed oriented m dimensional manifold M with coefficients in a field k has the structure of a unital Frobenius algebra without counit. When the characteristic of k is zero we describe explicitly the dual of the coproduct and prove that it respects the Hodge decomposition of .     

Homotopy monomorphisms and H-splitting in loop space fibrations

Let be a fibration, the holonomy action of this fibration and the connecting map. It is shown that if the fibre F admits an H-structure ν such that ρ?ν○(1×∂) (principal fibrations of all kinds satisfy such a condition), then i is a monomorphism if and only if it is weak monomorphism, the latter is equivalent to that Ωp has a homotopy right inverse Γ. If in addition Γ is an H-map, then ΩE has the same H-type as ΩB×ΩF.     

Renormalization as a functor on bialgebras

The Hopf algebra of renormalization in quantum field theory is described at a general level. The products of fields at a point are assumed to form a bialgebra B and renormalization endows T(T(B)⁺), the double tensor algebra of B, with the structure of a noncommutative bialgebra. When the bialgebra B is commutative, renormalization turns S(S(B)⁺), the double symmetric algebra of B, into a commutative bialgebra. The usual Hopf algebra of renormalization is recovered when the elements of S¹(B) are not renormalized, i.e., when Feynman diagrams containing one single vertex are not renormalized. When B is the Hopf algebra of a commutative group, a homomorphism is established between the bialgebra S(S(B)⁺) and the Faà di Bruno bialgebra of composition of series. The relation with the Connes-Moscovici Hopf algebra is given. Finally, the bialgebra S(S(B)⁺) is shown to give the same results as the standard renormalization procedure for the scalar field.     

Force distribution for double-walled carbon nanotubes and gigahertz oscillators

Advances in nanotechnology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices that might be instrumental in the micro-computer industry which are predominantly based on electron transport phenomena. There are many experiments and molecular dynamical simulations which show that a wave is generated on the outer cylinder as a result of the oscillation of the inner carbon nanotube and that the frequency of this wave is also in the gigahertz range. As a preliminary to analyze and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here we determine some new analytical expressions for the van der Waals force using the Lennard–Jones potential for general lengths of the inner and outer tubes. These expressions are utilized together with Newton’s second law to determine the motion of an oscillating inner tube, assuming that any frictional effects may be neglected. An idealized and much simplified representation of the Lennard–Jones force is used to determine a simple formula for the oscillation frequency resulting from an initial extrusion of the inner tube. This simple formula is entirely consistent with the existing known behavior of the frequency and predicts a maximum oscillation frequency occurring when the extrusion length is (L ₂ – L ₁)/2 where L ₁ and L ₂ are the respective half-lengths of the inner and outer tubes (L ₁ < L ₂).     

Interpolation categories for homology theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions, we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As an application, we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.     

Blow-up results for a class of first-order nonlinear evolution inequalities

We investigate the non-existence of solutions to a class of evolution inequalities; in this case, as it happens in a relatively small number of blow-up studies, nonlinearities depend also on time-variable t and spatial derivatives of the unknown. The present results, which in great part do not require any assumption on the regularity of data, are completely new and shown with various applications. Some of these results referring to the problem u_t=Δu+a(x)|u|^p+λf(x) in R^N, t>0 include the non-existence results of positive global solutions obtained by Fujita and others when a≡1 and f≡0, Bandle-Levine and Levine-Meier when a≡|x|^m and f≡0, Pinsky when either f≡0 or f?0 and λ>0, Zhang and Bandle-Levine-Zhang when a≡1 and λ=1.     

An algebraic model of fibration with the fiberK(π,n)-space

For a fibration with the fiberK(,n)-space, the algebraic model as a twisted tensor product of chains of the base with standard chains ofK(,n)-complex is given which preserves multiplicative structure as well. In terms of this model the action of then-cohomology of the base with coefficients in on the homology of fibration is described.     

Nonself-dual Chern-Simons and Maxwell-Chern-Simons vortices on bounded domains

We first show the existence of the multivortex solutions of Maxwell-Chern-Simons-Higgs (MCSH) model on bounded domains and prove that the solutions of the Euler-Lagrange equations of the MCSH energy functional converge to those of the Abelian-Higgs (AH) model and the Chern-Simons-Higgs (CSH) model in suitable limits, respectively. We also show the existence of the multivortex solutions of the nonself-dual CSH model on bounded domains. Besides, we study asymptotics for the minimizers of MCSH energy functional when the gauge field vanishes.     

Correspondences of ribbon categories

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally, we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.