Morphisms Cohomology and Deformations of Hom-algebras

The purpose of this paper is to define cohomology complexes and study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.     

On the obstructions to non-Cliffordian pin structures

We derive the topological obstructions to the existence of non-Cliffordian pin structures on four-dimensional spacetimes. We apply these obstructions to the study of non-Cliffordian pin-Lorentz cobordism. We note that our method of derivation applies equally well in any dimension and in any signature, and we present a general format for calculating obstructions in these situations. Finally, we interpret the breakdown of pin structure and discuss the relevance of this to aspects of physics.m     

Non-linear Schrödinger Equations,Separation and Symmetry

Abstract

We investigate the symmetry properties of hierarchies of non-linear Schrödinger equations, introduced in [2], which describe non-interacting systems in which tensor product wave-functions evolve by independent evolution of the factors (the separation property). We show that there are obstructions to lifting symmetries existing at a certain number of particles to higher numbers. Such obstructions vanish for particles without internal degrees of freedom and the usual space-time symmetries. For particles with internal degrees of freedom, such as spin, these obstructions are present and their circumvention requires a choice of a new term in the equation for each particle number. A Lie-algebra approach for non-linear theories is developed.     

Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.     

Hardy Uncertainty Principle,Convexity and Parabolic Evolutions

A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of projective geometry combines with structures known as Yang–Mills detour complexes to produce a general tool for generating invariant pseudo-Riemannian gauge theories. This produces (detour) complexes of differential operators corresponding to gauge invariances and dynamics. We show, as an application, that curved versions of these sequences give geometric characterizations of the obstructions to propagation of higher spins in Einstein spaces. Further, we show that projective BGG detour complexes generate both gauge invariances and gauge invariant constraint systems for partially massless models: the input for this machinery is a projectively invariant gauge operator corresponding to the first operator of a certain BGG sequence. We also connect this technology to the log-radial reduction method and extend the latter to Einstein backgrounds.     

Obstructions to the Existence of Sasaki–Einstein Metrics

We describe two simple obstructions to the existence of Ricci-flat Kähler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kähler–Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R–charge of a gauge invariant chiral primary operator violates the unitarity bound.     

Generalized Deformations and Hochschild Cohomology

We present a generalization of Gerstenhaber's theory of deformations. We no longer assume that the deformation parameter t acts in its usual free and symmetric way on the elements of the original algebra A, but in the following manner: t · a = (a)t and a · t = (a)t, where and are endomorphisms of A. We develop the cohomological framework adapted to these deformations.     

Cyclic Cohomology and Hopf Algebras

We associate canonically a cyclic module to any Hopf algebra endowed with a modular pair in involution, consisting of a group-like element and a character. This provides the key construction for allowing the extension of cyclic cohomology to Hopf algebras in the nonunimodular case and, further, to developing a theory of characteristic classes for actions of Hopf algebras compatible not only with traces but also with the modular theory of weights. This applies to both ribbon and coribbon algebras as well as to quantum groups and their duals.     

The field copy problem: to what extent do curvature (gauge field) and its covariant derivatives determine connection (gauge potential)?

We show that a connection of a principal bundle is determined up to (global) gauge equivalence by the curvature and its covariant derivatives provided that the infinitesimal holonomy group is of constant dimension and the base space is simply connected. If the dimension of the infinitesimal holonomy group varies, there may be obstructions of a topological nature to the existence of a global or even local gauge equivalence between two connections whose curvatures and covariant derivatives of curvature agree everywhere. These obstructions are analyzed and illustrated by examples.     

Gauge fixing, families index theory, and topological features of the space of lattice gauge fields

The families index theory for the overlap lattice Dirac operator is applied to derive topological features of the space of SU(N) lattice gauge fields on the 4-torus: the topological sectors, specified by the fermionic topological charge, are shown to contain noncontractible even-dimensional spheres when N3, and noncontractible circles in the N=2 case. We describe how certain obstructions to the existence of gauge fixings without the Gribov problem in the continuum setting correspond on the lattice to obstructions to the contractibility of these spheres and circles. We also point out a canonical connection on the space of lattice gauge fields with monopole-like singularities associated with the spheres.     

Orbifold construction in subfactors

We use the lattice models to determine the obstructions to the flatness of the orbifold connections in some finite depth subsfactors.     

Nested T-Duality

We identify the obstructions for Kiritsis–Obers T-duality of boundary WZW models. The open string duality pattern is much richer than in the closed strings case since it depends substantially on the geometry of branes. In particular, the duality obstructions disappear for certain brane configurations associated to non-regular elements of the Cartan torus. It is shown in this case that the boundary WZW model is “nested” in the twisted boundary WZW model as the dynamical subsystem of the latter     

Brownian surfaces with boundary and Deligne cohomology

We define differentiable random surfaces, which realize a kind of generalized parallel transport for line bundles over the loop space. This gives a realization of one of the axioms of Segal of conformal field theory.     

Differential cohomology of C (a)

Let be a finite dimensional real Lie algebra and ^* its dual. ^* is a Poisson manifold. Thus the space C^∞( ^*) of C^∞ functions on ^* has an associative and a Lie algebra structure. The problem of formal deformations of such a structure needs the determination of some cohomology groups of C^∞( ^*), considered as a module on itself for left multiplication or adjoint representation. We determine here these groups. The result is very similar to the case of C^∞(W), where W is a symplectic manifold except for the Lie algebras h_r × ^m, direct products of Heisenberg and abelian Lie algebras.     

Dual gravity and matter

We consider the problem of finding a dual formulation of gravity in the presence of non-trivial matter couplings. In the absence of matter a dual graviton can be introduced only for linearised gravitational interactions. We show that the coupling of linearised gravity to matter poses obstructions to the usual construction and comment on possible resolutions of this difficulty.     

Pontrjagin forms, Chern Simons classes, Codazzi transformations, and affine hypersurfaces

We show that the primary and secondary characteristic classes vanish in the context of affine differential geometry. This gives rise to obstructions to realizing a conformal class of metrics on a manifold either as the first or as the second fundamental form of an affine immersion.     

The d = [δ, b] Decomposition of Topological Yang–Mills Theory

We review the construction of topological Yang–Mills theory (TYMT) from the point of view of an operator satisfying d = [, b] with b the BRST operator. We focus our attention on a situation in which [, d] 0 and show how this leads us to consider a more general derivative ~d = b + d + _{k 2 k} ^1–k .     

Formal Poisson Cohomology of Quadratic Poisson Structures

We compute the formal Poisson cohomology of quadratic Poisson structures. We first recall that, generically, quadratic Poisson structures are diagonalizable. Then we compute the formal cohomology of diagonal Poisson structures.     

A Note on Symplectic Algorithm

We present the symplectic algorithm in the Lagrangian formalism for the Hamiltonian systems by virtue of the noncommutative differential calculus with respect to the discrete time and the Euler-Lagrange cohomological concepts. We also show that the trapezoidal integrator is symplectic in certain sense.``