Cohomology Groups Research Articles

One-loop double copy relation from twisted (co)homology

We propose a geometric relation between closed and open string amplitudes at one-loop. After imposing a homological splitting on the world-sheet torus, twisted intersection theory is used to establish a one-loop double copy relation. The latter expresses a closed string amplitude by a pair of open string amplitudes and twisted intersection numbers. These inner products on the vector space of twisted differential forms are related to the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral.

Algebraic algorithms for vector bundles over curves

In this paper, we represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and images of global homomorphisms, isomorphisms between vector bundles, cohomology groups, extensions, and splitting into a direct sum of indecomposables. Most algorithms are deterministic except for computing isomorphisms when the base field is infinite. Some algorithms are only polynomial time if we may compute Hermite forms of pseudo-matrices in polynomial time. All algorithms rely exclusively on algebraic operations in function fields. For applications, we give an algorithm enumerating isomorphism classes of vector bundles on an elliptic curve, and to construct algebraic geometry codes over vector bundles. We implement all our algorithms into a SageMath package.

Deformations and abelian extensions of compatible pre-Lie algebras

In this paper, first, we give the notion of a compatible pre-Lie algebra and its representation. We study the relation between compatible Lie algebras and compatible pre-Lie algebras. We also construct a new bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible pre-Lie structures. We give the bidifferential graded Lie algebra which controls deformations of a compatible pre-Lie algebra. Then, we introduce a cohomology of a compatible pre-Lie algebra with coefficients in itself. We study infinitesimal deformations of compatible pre-Lie algebras and show that equivalent infinitesimal deformations are in the same second cohomology group. We further give the notion of a Nijenhuis operator on a compatible pre-Lie algebra. We study formal deformations of compatible pre-Lie algebras. If the second cohomology group H2(g;g) is trivial, then the compatible pre-Lie algebra is rigid. Finally, we give a cohomology of a compatible pre-Lie algebra with coefficients in arbitrary representation and study abelian extensions of compatible pre-Lie algebras using this cohomology. We show that abelian extensions are classified by the second cohomology group.

Volume and Euler classes in bounded cohomology of transformation groups

Abstract Let $M$ be an oriented smooth manifold and $\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of $M$ , where $\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0\!(M,\omega )$ and $\operatorname{Homeo}_+\!(M,\omega )$ , respectively, and in several cases prove their non-triviality. More precisely, we define: • Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$ , where $M$ is a hyperbolic manifold of dimension $n$ . • Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$ , where $S$ is an oriented closed hyperbolic surface. We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$ -manifolds; hence, they are non-trivial.

Crossed Modules and Non-Abelian Extensions of Differential Leibniz Conformal Algebras

In this paper, we introduce two-term differential Leib∞-conformal algebras and give characterizations of some particular classes of such two-term differential Leib∞-conformal algebras. Furthermore, we discuss the classification of the non-Abelian extensions in terms of non-Abelian cohomology groups. Finally, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of differential Leibniz conformal algebras.

Curves are algebraic K(π,1)$K(\pi,1)$: Theoretical and practical aspects

AbstractWe prove that any geometrically connected curve over a field is an algebraic , as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible étale sheaf of ‐modules, with invertible in , is canonically isomorphic to the cohomology of its corresponding ‐module. To this end, we explicitly construct some Galois coverings of corresponding to Galois coverings of the normalisation of its irreducible components. When is finite or algebraically closed, we precisely describe finite quotients of that allow to compute the cohomology groups of the sheaf, and give explicit descriptions of the cup products and in étale cohomology in terms of finite group cohomology.

(Pseudo-)Synthetic BRST quantisation of the bosonic string and the higher quantum origin of dualities

In this article I am arguing in favour of the hypothesis that the origin of gauge and string dualities in general can be found in a higher-categorical interpretation of basic quantum mechanics. It is interesting to observe that the Galilei group has a non-trivial cohomology, while the Lorentz/Poincare group has trivial cohomology. When we constructed quantum mechanics, we noticed the non-trivial cohomology structure of the Galilei group and hence, we required for a proper quantisation procedure that would be compatible with the symmetry group of our theory, to go to a central extension of the Galilei group universal covering by co-cycle. This would be the Bargmann group. However, Nature didn’t choose this path. Instead in nature, the Galilei group is not realised, while the Lorentz group is. The fact that the Galilei group has topological obstructions leads to a central charge, the mass, and a superselection rule, required to implement the Galilei symmetry, that forbids transitions between states of different mass. The topological structure of the Lorentz group however lacks such an obstruction, and hence allows for transitions between states of different mass. The connectivity structure of the Lorentz group as opposed to that of the Galilei group can be interpreted in the sense of an ER=EPR duality for the topological space associated to group cohomology. In string theory we started with the Witt algebra, and due to similar quantisation issues, we employed the central extension by co-cycle to obtain the Virasoro algebra. This is a unique extension for orientation preserving diffeomorphisms on a circle, but there is no reason to believe that, at the high energy domain in physics where this would apply, we do not have a totally different structure altogether and the degrees of freedom present there would require something vastly more general and global.

Strong surjections from two-complexes with trivial top-cohomology onto nonorientable surfaces

For every nonorientable closed surface $\U$, we present a strong surjection $f\colon X\to\U$, where $X$ is a finite two-dimensional {\sc cw}-complex with trivial second integer cohomology group. This provides an answer, for all nonorientable closed surfaces, to a problem in topological root theory for which we have hitherto known solutions only for the sphere, the torus, the projective plane and the Klein bottle.

Rationality of extended unipotent characters

We determine the rationality properties of unipotent characters of finite reductive groups arising as fixed points of disconnected reductive groups under a Frobenius map. In the proof, we use realisations of characters in ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-adic cohomology groups of Deligne–Lusztig varieties as well as block theoretic considerations.

Intersection numbers, polynomial division and relative cohomology

We present a simplification of the recursive algorithm for the evaluation of intersection numbers for differential n-forms, by combining the advantages emerging from the choice of delta-forms as generators of relative twisted cohomology groups and the polynomial division technique, recently proposed in the literature. We show that delta-forms capture the leading behaviour of the intersection numbers in presence of evanescent analytic regulators, whose use is, therefore, bypassed. This simplified algorithm is applied to derive the complete decomposition of two-loop planar and non-planar Feynman integrals in terms of a master integral basis. More generally, it can be applied to derive relations among twisted period integrals, relevant for physics and mathematical studies.

Equivariant 𝒟-modules on 2×2×𝓃 hypermatrices

We study D \mathcal {D} -modules and related invariants on the space of 2 × 2 × n 2\times 2\times n hypermatrices for n ≥ 3 n\geq 3 , which has finitely many orbits under the action of G = G L 2 ( C ) × G L 2 ( C ) × G L n ( C ) G=GL_2(\mathbb {C}) \times GL_2(\mathbb {C}) \times GL_n(\mathbb {C}) . We describe the category of coherent G G -equivariant D \mathcal {D} -modules as the category of representations of a quiver with relations. We classify the simple equivariant D \mathcal {D} -modules, determine their characteristic cycles and find special representations that appear in their G G -structures. We determine the explicit D \mathcal {D} -module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen–Macaulay. While our results display special behavior in the cases n = 3 n=3 and n = 4 n=4 , they are completely uniform for n ≥ 5 n\geq 5 .

Formal deformations, cohomology theory and L∞[1]-structures for differential Lie algebras of arbitrary weight

We introduced a cohomology theory for differential Lie algebras of arbitrary weight which generalised a previous work of Jiang and Sheng. The underlying L∞[1]-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between L∞[1]-structures for absolute and relative differential Lie algebras is established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras.

A number‐theoretic classification of toroidal solenoids

AbstractWe classify toroidal solenoids defined by nonsingular ‐matrices with integer coefficients by studying associated first Ĉech cohomology groups. In a previous work, we classified the groups in the case using generalized ideal classes in the splitting field of the characteristic polynomial of . In this paper, we explore the classification problem for an arbitrary .

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

AbstractWe compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the rational homology of the unit tangent bundles of the surfaces. These coefficients define a covariant functor over the classical category associated to mapping class groups, rather than a contravariant one, and are thus out of the scope of the traditional framework to study twisted cohomological stability. A remarkable property is that the computed stable twisted cohomology is not free as a module over the stable cohomology algebra with constant rational coefficients. For comparison, we also compute the stable cohomology group with coefficients in the first rational cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

On the top-dimensional cohomology of arithmetic Chevalley groups

Let K \mathbb {K} be a number field with ring of integers O \mathfrak {O} and let G \mathcal {G} be a Chevalley group scheme not of type E 8 \mathtt {E}_8 , F 4 \mathtt {F}_4 or G 2 \mathtt {G}_2 . We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that H v c d ( G ( O ) ; Q ) = 0 H^{{vcd}}(\mathcal {G}(\mathfrak {O}); \mathbb {Q}) = 0 if O \mathfrak {O} is Euclidean.

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

Central extensions, derivations, and automorphisms of the super Heisenberg-Virasoro algebra

In this paper, we use the Balinsky-Novikov construction to naturally realize the super Heisenberg-Virasoro algebra g . We then proceed to compute the second cohomology of g with trivial coefficients and determine its universal central extension. This yields two odd 2-cocycles in addition to the Virasoro 2-cocycle. We also calculate the first cohomology group of g in the adjoint module to classify all derivations of g . Finally, we classify all automorphisms of g and describe its automorphism group.

$L^{2}$-cohomology and quasi-isometries on the ends of unbounded geometry

The paper explores the minimal and maximal L^{2} -cohomology of oriented Riemannian manifolds, focusing on both the reduced and the unreduced versions. The main result is the proof of the invariance of the L^{2} -cohomology groups under uniform homotopy equivalences that are quasi-isometric on the unbounded ends. A uniform map is a uniformly continuous map such that the diameter of the preimage of a subset is bounded in terms of the diameter of the subset itself. Moreover, a map f between two Riemannian manifolds (X,g) and (Y,h) is quasi-isometric on the unbounded ends if X = M \cup E_{X} , where M is the interior of a manifold of bounded geometry with boundary, E_{X} is an open subset of X and the restriction of f to E_{X} is a quasi-isometry. Finally, some consequences are shown: the main ones are the definition of a mapping cone for L^{2} -cohomology and the invariance of the L^{2} -signature.

On unstable twisted rational cohomology groups of the automorphism groups of free groups

In this paper, we consider low-dimensional twisted rational cohomology groups of the automorphism groups of free groups. In particular, we study behaviors of Kawazumi’s cocycles by observing their restrictions to the IA-automorphism groups. We show the nontrivialities of certain unstable twisted cocycles constructed from Kawazumi’s cocycles, and compute the second and the third cohomology groups of the automorphism group of the free group of rank three with coefficients in the exterior powers of the abelianization of the free group.

Mini-Workshop: Homological aspects of TDLC-groups

This mini-workshop aimed at bringing together experts and early career researchers on finiteness conditions for discrete groups, and experts on varying aspects of locally compact groups to find a common framework to develop a systematic theory of homological finiteness conditions for totally disconnected locally compact groups. Whereas the homological theory of finiteness conditions of discrete groups is well developed and the structure theory of totally disconnected locally compact groups has seen some important breakthroughs in the last decade, the homological theory for (non-compact) totally disconnected locally compact groups is an emerging research area. Specific topics include finiteness conditions for locally compact groups, Mackey functors and Bredon cohomology for topological groups, connections to condensed mathematics, connections to \ell^{2} -invariants and \Sigma -invariants.