Algebraic Bundles Research Articles

On Holonomy Groupoid of Vertex Operator Algebra Bundles on Foliations

On Holonomy Groupoid of Vertex Operator Algebra Bundles on Foliations

Bundles of Holomorphic Function Algebras on Subvarieties of the Noncommutative Ball

Bundles of Holomorphic Function Algebras on Subvarieties of the Noncommutative Ball

Connes fusion of spinors on loop space

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

Relativity of Reductive Chain Complexes of Non-abelian Simplexes

Chain total double complexes with reductive differentials for non-abelian simplexes with associated spaces are considered. It is conjectured that corresponding relative cohomology is equivalent to the coset space of vanishing functionals over non-vanishing functionals related to differentials of complexes. The conjecture is supported by the theorem for the case of spaces of correlation functions and generalized connections on vertex operator algebra bundles.

The Clifford Algebra Bundle on Loop Space

We construct a Clifford algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that this bundle is in general non-trivial, more precisely that its triviality is obstructed by the transgressions of the second Stiefel-Whitney class and the first (fractional) Pontrjagin class of the manifold.

Algebraic Lie Algebra Bundles and Derivations of Lie Algebra Bundles

In this paper, we define algebraic Lie algebra bundles, discuss some results on algebraic Lie algebra bundles and derivations of Lie algebra bundles. Some results involving inner derivations and central derivations of Lie algebra bundles are obtained.

LIE ALGEBRA BUNDLES OF FINITE TYPE AND NUMERABLE BUNDLES

In this paper, we study the relation between numerable Lie algebra bundle and Lie algebra bundle of finite type. The effect of finite type on shrinkable maps and homotopy equivalence are examined. Further, we obtain some results on Section Extension Property, Covering Homotopy Property and Weak Covering Homotopy Property for Lie algebra bundles of finite type.

K-theory of flag Bott manifolds

Abstract The aim of this paper is to describe the topological K-ring in terms of generators and relations of a flag Bott manifold. We apply our results to give a presentation for the topological K-ring, and hence the Grothendieck ring of algebraic vector bundles over flag Bott–Samelson varieties.

Structure theorem for Jordan algebra bundles

PurposeThe aims of this paper is to prove that every semisimple Jordan algebra bundle is locally trivial and establish the decomposition theorem for locally trivial Jordan algebra bundles using the decomposition theorem of Lie algebra bundles.Design/methodology/approachUsing the decomposition theorem of Lie algebra bundles, this paper proves the decomposition theorem for locally trivial Jordan algebra bundles.FindingsFindings of this paper establish the decomposition theorem for locally trivial Jordan algebra bundles.Originality/valueTo the best of the author’s knowledge, all the results are new and interesting to the field of Mathematics and Theoretical Physics community.

Gauge theory on fiber bundle of hypercomplex algebras

I introduce a way of constructing a fiber bundle whose fibers are given by hypercomplex algebras and woven by appropriate structure group, and present that a novel gauge theory can be built on the hypercomplex fiber bundle. In this work, I aim to answer a question about how nature selects one preferred vacuum among degenerate physical vacua, called vacuum selection problem. In the end, I found presence of the impenetrable domain wall that prohibits phase transition between the two vacua. To be specific, I found that in this theory, one particular vacuum between two degenerate physical vacua for Higgs-like scalar potential can be dynamically chosen with priority due to intrinsic even parity of both a scalar field and its vacuum under a Z2 symmetry, even though its scalar potential is given to be Z2-symmetric under both odd- and even-parity transformations of the scalar field. This means that the vacuum selection problem can be resolved in this gauge theory. I suggest that this work may be a gateway to addressing the theoretical origin of the true physical vacuum that nature takes.

Reduction cohomology of Riemann surfaces

We study the algebraic conditions leading to the chain property of complexes for vertex operator algebra n-point functions (with their convergence assumed) with differential being defined through reduction formulas. The notion of the reduction cohomology of Riemann surfaces is introduced. Algebraic, geometrical, and cohomological meanings of reduction formulas are clarified. A counterpart of the Bott–Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven. It is shown that the reduction cohomology is given by the cohomology of n-point connections over the vertex operator algebra bundle defined on a genus g Riemann surface [Formula: see text]. The reduction cohomology for a vertex operator algebra with formal parameters identified with local coordinates around marked points on [Formula: see text] is found in terms of the space of analytical continuations of solutions to Knizhnik–Zamolodchikov equations. For the reduction cohomology, the Euler–Poincaré formula is derived. Examples for various genera and vertex operator cluster algebras are provided.

Cosimplicial meromorphic functions cohomology on complex manifolds

Developing ideas of [B. L. Feigin, Conformal field theory and cohomologies of the Lie algebra of holomorphic vector fields on a complex curve, in Proc. Int. Congress of Mathematicians (Kyoto, 1990), Vols. 1 and 2 (Mathematical Society of Japan, Tokyo, 1991), pp. 71–85], we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold [Formula: see text]. Graded differential cohomology of a sheaf of Lie algebras [Formula: see text] via the cosimplicial cohomology of [Formula: see text]-formal series for any covering by Stein spaces on [Formula: see text] is computed. A relation between cosimplicial cohomology (on a special set of open domains of [Formula: see text]) of formal series of an infinite-dimensional Lie algebra [Formula: see text] and singular cohomology of auxiliary manifold associated to a [Formula: see text]-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.

New Quiver-Like Varieties and Lie Superalgebras

In order to extend the geometrization of Yangian R-matrices from Lie algebras $$ \mathfrak {gl}(n) $$ to superalgebras $$\mathfrak {gl}(M|N)$$ , we introduce new quiver-related varieties which are associated with representations of $$ \mathfrak {gl}(M|N) $$ . In order to define them similarly to the Nakajima-Cherkis varieties, we reformulate the construction of the latter by replacing the Hamiltonian reduction with the intersection of generalized Lagrangian subvarieties in the cotangent bundles of Lie algebras sitting at the vertices of the quiver. The new varieties come from replacing some Lagrangian subvarieties with their Legendre transforms. We present superalgebra versions of stable envelopes for the new quiver-like varieties that generalize the cotangent bundle of a Grassmannian. We define superalgebra generalizations of the Tarasov–Varchenko weight functions, and show that they represent the super stable envelopes. Both super stable envelopes and super weight functions transform according to Yangian $$\check{R}$$ -matrices of $$ \mathfrak {gl}(M|N) $$ with $$M+N=2$$ .

On Lie algebroid over algebraic spaces

We consider Lie algebroids over algebraic spaces (in short we call it as a-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid of a Lie algebroid over an a-space . This is done by sheafification of the presheaf of universal enveloping algebras for Lie-Rinehart algebras. We review the extent to which structure of the universal enveloping algebroid of Lie algebroids (over special a-spaces) resembles a sheaf of bialgebras. In the sequel we present a version of Poincare-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for the Lie algebraoid.

Cohomology of the moduli stack of algebraic vector bundles

Let Vectn be the moduli stack of vector bundles of rank n on derived schemes. We prove that, if E is a Zariski sheaf of ring spectra which is equipped with finite quasi-smooth transfers and satisfies projective bundle formula, then E⁎(Vectn,S) is freely generated by Chern classes c1,…,cn over E⁎(S) for any qcqs derived scheme S. Examples include all multiplicative localizing invariants.

Logarithmic Riemann–Hilbert correspondences for rigid varieties

On any smooth algebraic variety over a p p -adic local field, we construct a tensor functor from the category of de Rham p p -adic étale local systems to the category of filtered algebraic vector bundles with integrable connections satisfying the Griffiths transversality, which we view as a p p -adic analogue of Deligne’s classical Riemann–Hilbert correspondence. A crucial step is to construct canonical extensions of the desired connections to suitable compactifications of the algebraic variety with logarithmic poles along the boundary, in a precise sense characterized by the eigenvalues of residues; hence the title of the paper. As an application, we show that this p p -adic Riemann–Hilbert functor is compatible with the classical one over all Shimura varieties, for local systems attached to representations of the associated reductive algebraic groups.

Absolute Hodge and ℓ-adic monodromy

Let $\mathbb {V}$ be a motivic variation of Hodge structure on a $K$-variety $S$, let $\mathcal {H}$ be the associated $K$-algebraic Hodge bundle, and let $\sigma \in \mathrm {Aut}(\mathbb {C}/K)$ be an automorphism. The absolute Hodge conjecture predicts that given a Hodge vector $v \in \mathcal {H}_{\mathbb {C}, s}$ above $s \in S(\mathbb {C})$ which lies inside $\mathbb {V}_{s}$, the conjugate vector $v_{\sigma } \in \mathcal {H}_{\mathbb {C}, s_{\sigma }}$ is Hodge and lies inside $\mathbb {V}_{s_{\sigma }}$. We study this problem in the situation where we have an algebraic subvariety $Z \subset S_{\mathbb {C}}$ containing $s$ whose algebraic monodromy group $\textbf {H}_{Z}$ fixes $v$. Using relationships between $\textbf {H}_{Z}$ and $\textbf {H}_{Z_{\sigma }}$ coming from the theories of complex and $\ell$-adic local systems, we establish a criterion that implies the absolute Hodge conjecture for $v$ subject to a group-theoretic condition on $\textbf {H}_{Z}$. We then use our criterion to establish new cases of the absolute Hodge conjecture.

Trace Densities and Algebraic Index Theorems for Sheaves of Formal Cherednik Algebras

Abstract We show how a novel construction of the sheaf of Cherednik algebras $\mathscr {H}_{1, c, X, G}$ on a quotient orbifold $Y:=X/G$ in author’s prior work leads to results for $\mathscr {H}_{1, c, X, G}$, which until recently were viewed as intractable. First, for every orbit type stratum in $X$, we define a trace density map for the Hochschild chain complex of $\mathscr {H}_{1, c, X, G}$, which generalizes the standard Engeli–Felder’s trace density construction for the sheaf of differential operators $\mathscr {D}_X$. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of $\mathbb {C}_{Y}\llbracket \hbar \rrbracket $-modules, which computes the hypercohomology of the Hochschild chain complex of the sheaf of formal Cherednik algebras $\mathscr {H}_{1, \hbar , X, G}$. We show that this hypercohomology is isomorphic to the Chen–Ruan cohomology of the orbifold $Y$ with values in the ring of formal power series $\mathbb {C}\llbracket \hbar \rrbracket $. We infer that the Hochschild chain complex of the sheaf of skew group algebras $\mathscr {H}_{1, 0, X, G}$ has a well-defined Euler characteristic that is equal to the orbifold Euler characteristic of $Y$. Finally, we prove an algebraic index theorem.

Group Bundles and Group Connections

We consider smooth families of Lie groups (group bundles) and connections that are compatible with the group operation. We characterize the space of group connections on a group bundle as an affine space modeled over the vector space of $$1$$ -forms with values cocycles in the Lie algebra bundle of the aforementioned group bundle. We show that group connections satisfy the Ambrose–Singer theorem and that group bundles can be seen as a particular case of associated bundles realizing group connections as associated connections. We give a construction of the Moduli space of group connections with fixed base and fiber, as a space of representations of the fundamental group of the base.

Comparison of Poisson structures on moduli spaces

Let X be a complex irreducible smooth projective curve, and let {{mathbb {L}}} be an algebraic line bundle on X with a nonzero section sigma _0. Let {mathcal {M}} denote the moduli space of stable Hitchin pairs (E,, theta ), where E is an algebraic vector bundle on X of fixed rank r and degree delta , and theta , in , H^0(X,, {mathcal {E}nd}(E)otimes K_Xotimes {{mathbb {L}}}). Associating to every stable Hitchin pair its spectral data, an isomorphism of {mathcal {M}} with a moduli space {mathcal {P}} of stable sheaves of pure dimension one on the total space of K_Xotimes {{mathbb {L}}} is obtained. Both the moduli spaces {mathcal {P}} and {mathcal {M}} are equipped with algebraic Poisson structures, which are constructed using sigma _0. Here we prove that the above isomorphism between {mathcal {P}} and {mathcal {M}} preserves the Poisson structures.