Twisted Cocycle Research Articles

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.

Pointed Hopf algebras over nonabelian groups with nonsimple standard braidings

AbstractWe construct finite‐dimensional Hopf algebras whose coradical is the group algebra of a central extension of an abelian group. They fall into families associated to a semisimple Lie algebra together with a Dynkin diagram automorphism. We show conversely that every finite‐dimensional pointed Hopf algebra over a nonabelian group with nonsimple infinitesimal braiding of rank at least 4 is of this form. We follow the steps of the Lifting Method by Andruskiewitsch–Schneider. Our starting point is the classification of finite‐dimensional Nichols algebras over nonabelian groups by Heckenberger–Vendramin, which consist of low‐rank exceptions and large‐rank families. We prove that the large‐rank families are cocycle twists of Nichols algebras constructed by the second author as foldings of Nichols algebras of Cartan type over abelian groups by outer automorphisms. This enables us to give uniform Lie‐theoretic descriptions of the large‐rank families, prove generation in degree 1, and construct liftings. We also show that every lifting is a cocycle deformation of the corresponding coradically graded Hopf algebra using an explicit presentation by generators and relations of the Nichols algebra. On the level of tensor categories, we construct families of graded extensions of the representation category of a quantum group by a group of diagram automorphism.

An $$\mathfrak {sl}_2$$-type tensor category for the Virasoro algebra at central charge 25 and applications

Let $\mathcal{O}_{25}$ be the vertex algebraic braided tensor category of finite-length modules for the Virasoro Lie algebra at central charge $25$ whose composition factors are the irreducible quotients of reducible Verma modules. We show that $\mathcal{O}_{25}$ is rigid and that its simple objects generate a semisimple tensor subcategory that is braided tensor equivalent to an abelian $3$-cocycle twist of the category of finite-dimensional $\mathfrak{sl}_2$-modules. We also show that this $\mathfrak{sl}_2$-type subcategory is braid-reversed tensor equivalent to a similar category for the Virasoro algebra at central charge $1$. As an application, we construct a simple conformal vertex algebra which contains the Virasoro vertex operator algebra of central charge $25$ as a $PSL_2(\mathbb{C})$-orbifold. We also use our results to study Arakawa's chiral universal centralizer algebra of $SL_2$ at level $-1$, showing that it has a symmetric tensor category of representations equivalent to $\mathrm{Rep}\,PSL_2(\mathbb{C})$. This algebra is an extension of the tensor product of Virasoro vertex operator algebras of central charges $1$ and $25$, analogous to the modified regular representations of the Virasoro algebra constructed earlier for generic central charges by I. Frenkel-Styrkas and I. Frenkel-M. Zhu.

Twists of rational Cherednik algebras

Abstract We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one.

Twisted translation flows and effective weak mixing

We introduce a twisted cohomology cocycle over the Teichmüller flow and prove a “spectral gap” for its Lyapunov spectrum with respect to the Masur–Veech measures. We then derive Hölder estimates on spectral measures and bounds on the speed of weak mixing for almost all translation flows in every stratum of Abelian differentials on Riemann surfaces, as well as bounds on the deviation of ergodic averages for product translation flows on the product of a translation surface with a circle.

Quantum coordinate ring in WZW model and affine vertex algebra extensions

In this paper, we construct various simple vertex superalgebras which are extensions of affine vertex algebras, by using abelian cocycle twists of representation categories of quantum groups. This solves the Creutzig and Gaiotto conjectures (Creutzig and Gaiotto in Comm Math Phys 379:785–845, 2020, Conjecture 1.1 and 1.4) in the case of type ABC. If the twist is trivial, the resulting algebras correspond to chiral differential operators in the chiral case, and to WZW models in the non-chiral case.

On the computation of intersection numbers for twisted cocycles

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example, square roots) although the final result may be expressed without algebraic extensions. In this article, I present an improvement of this algorithm, which avoids algebraic extensions.

(Re)constructing Code Loops

The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops’ twisted cocycles satisfy. As a byproduct we demonstrate that one can reconstruct the multiplication in Parker’s loop from a mere fragment of its twisted cocycle. We also give relatively large subspaces of the Golay code over which Parker’s loop splits as a direct product.

Twisted deformations vs. cocycle deformations for quantum groups

In this paper, we study two deformation procedures for quantum groups: deformations by twists, that we call “comultiplication twisting”, as they modify the coalgebra structure, while keeping the algebra one — and deformations by [Formula: see text]-cocycle, that we call “multiplication twisting”, as they deform the algebra structure, but save the coalgebra one. We deal with quantized universal enveloping algebras (in short QUEAs), for which we accordingly consider those arising from twisted deformations (in short TwQUEAs) and those arising from [Formula: see text]-cocycle deformations, usually called multiparameter QUEAs (in short MpQUEAs). Up to technicalities, we show that the two deformation methods are equivalent, in that they eventually provide isomorphic outputs, which are deformations (of either kinds) of the “canonical”, well-known one-parameter QUEA by Jimbo and Lusztig. It follows that the two notions of TwQUEAs and of MpQUEAs — which, in Hopf algebra theoretical terms are naturally dual to each other — actually coincide; thus, that there exists in fact only one type of “pluriparametric deformation” for QUEAs. In particular, the link between the realization of any such QUEA as a MpQUEA and that as a TwQUEA is just a (very simple, and rather explicit) change of presentation.

Cohomology of fiber-bunched twisted cocycles over hyperbolic systems

AbstractA twisted cocycle taking values on a Lie group G is a cocycle that is twisted by an automorphism of G in each step. In the case where G = GL(d, ℝ), we prove that if two Hölder continuous twisted cocycles satisfying the so-called fiber-bunching condition have the same periodic data then they are cohomologous.

Correlation functions on the lattice and twisted cocycles

We study linear relations among correlation functions on a lattice obtained from integration-by-parts identities. We use the framework of twisted cocycles and determine for a scalar theory a basis of correlation functions, in which all other correlation functions may be expressed.

Variational principles for spectral radius of weighted endomorphisms of 𝐶(𝑋,𝐷)

We give formulas for the spectral radius of weighted endomorphisms a α : C ( X , D ) → C ( X , D ) a\alpha : C(X,D)\to C(X,D) , a ∈ C ( X , D ) a\in C(X,D) , where X X is a compact Hausdorff space and D D is a unital Banach algebra. Under the assumption that α \alpha generates a partial dynamical system ( X , φ ) (X,\varphi ) , we establish two kinds of variational principles for r ( a α ) r(a\alpha ) : using linear extensions of ( X , φ ) (X,\varphi ) and using Lyapunov exponents associated with ergodic measures for ( X , φ ) (X,\varphi ) . This requires considering (twisted) cocycles over ( X , φ ) (X,\varphi ) with values in an arbitrary Banach algebra D D , and thus our analysis cannot be reduced to any of the multiplicative ergodic theorems known so far. The established variational principles apply not only to weighted endomorphisms but also to a vast class of operators acting on Banach spaces that we call abstract weighted shifts associated with α : C ( X , D ) → C ( X , D ) \alpha : C(X,D)\to C(X,D) . In particular, they are far-reaching generalizations of formulas obtained by Kitover, Lebedev, Latushkin, Stepin, and others. They are most efficient when D = B ( F ) D=\mathcal {B}(F) , for a Banach space F F , and endomorphisms of B ( F ) \mathcal {B}(F) induced by α \alpha are inner isometric. As a by-product we obtain a dynamical variational principle for an arbitrary operator b ∈ B ( F ) b\in \mathcal {B}(F) and that its spectral radius is always a Lyapunov exponent in some direction v ∈ F v\in F when F F is reflexive.

Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf\u2013Witten Theory

Given a finite mathbb {Z}_2-graded group hat{mathsf {G}} with ungraded subgroup mathsf {G} and a twisted cocycle hat{lambda } in Z^n(B hat{mathsf {G}}; mathsf {U}(1)_{pi }) which restricts to lambda in Z^n(B mathsf {G}; mathsf {U}(1)), we construct a lift of lambda -twisted mathsf {G}-Dijkgraaf–Witten theory to an unoriented topological quantum field theory. Our construction uses a new class of homotopy field theories, which we call orientation twisted. We also introduce an orientation twisted variant of the orbifold procedure, which produces an unoriented topological field theory from an orientation twisted mathsf {G}-equivariant topological field theory.

Graded-simple algebras and cocycle twisted loop algebras

The loop algebra construction by Allison, Berman, Faulkner, and Pianzola, describes graded-central-simple algebras with split centroid in terms of central simple algebras graded by a quotient of the original grading group. Here the restriction on the centroid is removed, at the expense of allowing some deformations (cocycle twists) of the loop algebras.

Twists of quantum Borel algebras

We classify Drinfeld twists for the quantum Borel subalgebra uq(b) in the Frobenius–Lusztig kernel uq(g), where g is a simple Lie algebra over C and q an odd root of unity. More specifically, we show that alternating forms on the character group of the group of grouplikes for uq(b) generate all twists for uq(b), under a certain algebraic group action. This implies a simple classification of finite-dimensional Hopf algebras whose categories of representations are tensor equivalent to that of uq(b). We also show that cocycle twists for the corresponding De Concini–Kac algebra are in bijection with alternating forms on the aforementioned character group.

Twisted sigma-model solitons on the quantum projective line

On the configuration space of projections in a noncommutative algebra, and for an automorphism of the algebra, we use a twisted Hochschild cocycle for an action functional and a twisted cyclic cocycle for a topological term. The latter is Hochschild-cohomologous to the former and positivity in twisted Hochschild cohomology results into a lower bound for the action functional. While the equations for the critical points are rather involved, the use of the positivity and the bound by the topological term lead to self-duality equations (thus yielding twisted noncommutative sigma-model solitons, or instantons). We present explicit nontrivial solutions on the quantum projective line.

Scattering Amplitudes from Intersection Theory.

We use Picard-Lefschetz theory to prove a new formula for intersection numbers of twisted cocycles associated with a given arrangement of hyperplanes. In a special case when this arrangement produces the moduli space of punctured Riemann spheres, intersection numbers become tree-level scattering amplitudes of quantum field theories in the Cachazo-He-Yuan formulation.

Cocycle rigidity of abelian partially hyperbolic actions

Suppose G is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let G = G or G = G ⋉ V, where V is a finite-dimensional vector space V. For any unitary representation (π,H) of G, we study the twisted cohomological equation π(a)f − λf = g for partially hyperbolic element a ∈ G and λ ∈ U(1), as well as the twisted cocycle equation π(a1)f − λ1f = π(a2)g − λ2g for commuting partially hyperbolic elements a1, a2 ∈ G. We characterize the obstructions to solving these equations, construct smooth solutions and obtain tame Sobolev estimates for the solutions. These results can be extended to partially hyperbolic flows in parallel. As an application, we prove cocycle rigidity for any abelian higher-rank partially hyperbolic algebraic actions. This is the first paper exploring rigidity properties of partially hyperbolic that the hyperbolic directions don’t generate the whole tangent space. The result can be viewed as a first step toward the application of KAM method in obtaining differential rigidity for these actions in future works.

Combinatorics and topology of Kawai-Lewellen-Tye relations

We revisit the relations between open and closed string scattering amplitudes discovered by Kawai, Lewellen, and Tye (KLT). We show that they emerge from the un-derlying algebro-topological identities known as the twisted period relations. In order to do so, we formulate tree-level string theory amplitudes in the language of twisted de Rham theory. There, open string amplitudes are understood as pairings between twisted cycles and cocycles. Similarly, closed string amplitudes are given as a pairing between two twisted cocycles. Finally, objects relating the two types of string amplitudes are the α′-corrected bi-adjoint scalar amplitudes recently defined by the author [1]. We show that they naturally arise as intersection numbers of twisted cycles. In this work we focus on the combinatorial and topological description of twisted cycles relevant for string theory amplitudes. In this setting, each twisted cycle is a polytope, known in combinatorics as the associahedron, together with an additional structure encoding monodromy properties of string integrals. In fact, this additional structure is given by higher-dimensional generalizations of the Pochhammer contour. An open string amplitude is then computed as an integral of a logarithmic form over an associahedron. We show that the inverse of the KLT kernel can be calculated from the knowledge of how pairs of associahedra intersect one another in the moduli space. In the field theory limit, contributions from these intersections localize to vertices of the associahedra, giving rise to the bi-adjoint scalar partial amplitudes.

Twisted bialgebroids versus bialgebroids from a Drinfeld twist

Bialgebroids (respectively Hopf algebroids) are bialgebras (Hopf algebras) over noncommutative rings. Drinfeld twist techniques are particularly useful in the (deformation) quantization of Lie algebras as well as the underlying module algebras (=quantum spaces). A smash product construction combines both of them into the new algebra which, in fact, does not depend on the twist. However, we can turn it into a bialgebroid in a twist-dependent way. Alternatively, one can use Drinfeld twist techniques in a category of bialgebroids. We show that both the techniques indicated in the title—the twisting of a bialgebroid or constructing a bialgebroid from the twisted bialgebra—give rise to the same result in the case of a normalized cocycle twist. This can be useful for the better description of a quantum deformed phase space. We argue that within this bialgebroid framework one can justify the use of deformed coordinates (i.e. spacetime noncommutativity), which are frequently postulated in order to explain quantum gravity effects.