Algebraic Bundles Research Articles

Representations of elementary abelian p-groups and bundles on Grassmannians

We initiate the study of representations of elementary abelian p-groups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements, k[t1,…,tr]/(t1p,…,trp). We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples.

Fermion Flavors in Quaternion Basis and Infrared QCD

I analyze the lattice simulation data of the Domain Wall Fermion in quaternion basis. As pointed out by Atiyah and Ward, the minimum action solution for SU(2) Yang–Mills fields in Euclidean 4-space correspond, via Penrose twistor transform, to algebraic bundles on the complex projective 3-space. Assuming dominance of correlation between the fermions on the domain walls via exchange of instantons, I extract parameters necessary for defining gauge fields of Atiyah–Ward ansatz. The QCD effective coupling in the infrared and the relation between the number of flavors and the infrared fixed point is investigated. Consequence of this lepton flavor assignment to phenomenology of baryons is also discussed.

Universal vector bundle over the reals

Let X_R be a geometrically irreducible smooth projective curve, defined over R, such that X_R does not have any real points. Let X= X_R\times_R C be the complex curve. We show that there is a universal real algebraic line bundle over X_R x Pic^d(X_R)$ if and only if $\chi(L)$ is odd for L in Pic^d(X_R)$. There is a universal quaternionic algebraic line bundle over X x Pic^d(X) if and only if the degree d is odd. Take integers r and d such that r > 1, and d is coprime to r. Let M_{X_R}(r,d) (respectively, M_X(r,d)$) be the moduli space of stable vector bundles over X_R (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over X_R x M_{X_R}(r,d) if and only if \chi(E) is odd for E in M_{X_R}(r,d). There is a universal quaternionic vector bundle over X x M_X(r,d) if and only if the degree d is odd. The cases where X_R is geometrically reducible or X_R has real points are also investigated.

Independent parameters for special instanton bundles on [formula omitted

Motivated by Yang–Mills theory in 4 n dimensions, and generalizing the notion due to Atiyah, Drinfeld, Hitchin and Manin for n = 1 , Okonek, Spindler and Trautmann introduced instanton bundles and special instanton bundles as certain algebraic vector bundles of rank 2 n on the complex projective space P 2 n + 1 . The moduli space of special instanton bundles is shown to be rational.

Constructions for infinitesimal group schemes

Let G be an infinitesimal group scheme over a field k of char- acteristic p > 0. We introduce the universal p-nilpotent operatorG 2 Homk(k(G),k(V (G))), where V (G) is the scheme which represents1-parameter subgroups of G. This operatorG applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain kG-modules M (including those of constant Jordan type), we employG to associate various algebraic vector bundles on P(G), the pro- jectivization of V (G). These vector bundles not only distinguish certain rep- resentations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on P(G).

On Utiyama’s Theme Through “ $${\mathcal {A}}$$ -Invariance”

The term “\({\mathcal {A}}\) -invariance” refers to the invariance of our results, with respect to the “arithmetic” employed, viz. to an appropriate algebra sheaf\({\mathcal {A}}\) . This, combined with the categorical notion of “adjunction”, particularized here with the homological, in nature Hom-⊗ adjunction, affords the classical perspective of Utiyama, pertaining to the characteristic type of field interactions. Yet, all this, without any “space-time” support, in the classical sense of the term (: smooth manifolds), but, just based on the “functorial” character of ADG (acronym of “Abstract Differential Geometry”) and the aforementioned two fundamental principles (:“\({\mathcal {A}}\) -invariance” and “adjunction”).

ON THE EQUIVALENCE BETWEEN THE CATEGORIES OF CERTAIN BUNDLES AND CERTAIN SHEAVES

In this paper the notion of universal algebra bundle is introduced. We show that several properties of local sections of a certain sheaf hold also for local sections of module bundles and universal algebra bundles. As a main result of this paper, we construct an equivalence between the category of module bundles (or universal algebra bundles) and a certain subcategory of the category of sheaves or a certain subcategory of the category of complete presheaves.

Gaudin subalgebras and stable rational curves

AbstractGaudin subalgebras are abelian Lie subalgebras of maximal dimension spanned by generators of the Kohno–Drinfeld Lie algebra $\Xmathfrak {t}_{\hspace *{.3pt}n}$. We show that Gaudin subalgebras form a variety isomorphic to the moduli space $\bar M_{0,n+1}$ of stable curves of genus zero with n+1 marked points. In particular, this gives an embedding of $\bar M_{0,n+1}$ in a Grassmannian of (n−1)-planes in an n(n−1)/2-dimensional space. We show that the sheaf of Gaudin subalgebras over $\bar M_{0,n+1}$ is isomorphic to a sheaf of twisted first-order differential operators. For each representation of the Kohno–Drinfeld Lie algebra with fixed central character, we obtain a sheaf of commutative algebras whose spectrum is a coisotropic subscheme of a twisted version of the logarithmic cotangent bundle of $\bar M_{0,n+1}$.

The noncommutative geometry of Yang–Mills fields

We generalize to topologically non-trivial gauge configurations the description of the Einstein–Yang–Mills system in terms of a noncommutative manifold, as was done previously by Chamseddine and Connes. Starting with an algebra bundle and a connection thereon, we obtain a spectral triple, a construction that can be related to the internal Kasparov product in unbounded KK-theory. In the case that the algebra bundle has typical fiber M N ( C ) , we construct a P S U ( N ) -principal bundle for which it is an associated bundle. The so-called internal fluctuations of the spectral triple are parametrized by connections on this principal bundle and the spectral action gives the Yang–Mills action for these gauge fields, minimally coupled to gravity. Finally, we formulate a definition for a topological spectral action.

Suppleness of the sheaf of algebras of generalized functions on manifolds

We show that the sheaves of algebras of generalized functions Ω → G ( Ω ) and Ω → G ∞ ( Ω ) , Ω are open sets in a manifold X, are supple, contrary to the non-suppleness of the sheaf of distributions.

An application of ample vector bundles in real algebraic geometry

Let E be an algebraic vector bundle on a compact nonsingular real algebraic set X, and let Z be the zero locus of a generic algebraic section of E. We investigate how certain cohomological invariants of X and Z are related. A crucial role in the proof is played by ample vector bundles on a suitable complexification of X.

HODGE STRUCTURES OF THE MODULI SPACES OF PAIRS

Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let [Formula: see text] be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of [Formula: see text] are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

Quaternionic structures

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spin c manifolds in Čadek et al. (2008) [5] and may be of some interest, also, in quaternionic and algebraic geometry.

A CHARACTERIZATION OF SATURATED C*-ALGEBRAIC BUNDLES OVER FINITE GROUPS

AbstractLet A be a unital C*-algebra. Let (B,E) be a pair consisting of a unital C*-algebra B containing A as a C*-subalgebra with a unit that is also the unit of B, and a conditional expectation E from B onto A that is of index-finite type and of depth 2. Let B1 be the C*-basic construction induced by (B,E). In this paper, we shall show that any such pair (B,E) satisfying the conditions that A′∩B=ℂ1 and that A′∩B1 is commutative is constructed by a saturated C*-algebraic bundle over a finite group. Furthermore, we shall give a necessary and sufficient condition for B to be described as a twisted crossed product of A by its twisted action of a finite group under the condition that A′∩B1 is commutative.

Affine varieties with stably trivial algebraic vector bundles

Let $k$ be an algebraically closed field. For every affine variety $X$ with dim $X\ge 7$ we construct a smooth affine variety $Y$ which is birationally equivalent to $X$ and which possesses a stably trivial but not trivial algebraic vector bundle. We give some application of this fact to the cancellation problem.

The Beilinson equivalence for differential operators

Let D be the ring of differential operators on a smooth irreducible affine variety X over C , or, more generally, the enveloping algebra of any locally free Lie algebroid on X . The category of finitely generated graded modules of the Rees algebra D ˜ has a natural quotient category P D which imitates the category of modules on P r o j of a graded commutative ring. We show that the derived category D b ( P D ) is equivalent to the derived category of finitely generated modules of a sheaf of algebras E on X which is coherent over X . This generalizes the usual Beilinson equivalence for projective space, and also the Beilinson equivalence for differential operators on a smooth curve used by Ben-Zvi and Nevins in [6] to describe the moduli space of left ideals in D .

Lie Algebra bundles on s-Kähler manifolds, with applications to Abelian varieties

Nous prouvons que on peut obtenir fibrés naturels des algèbres de Lie so(s+1,s+1), su(s+1,s+1) et sl(2s+2,ℝ) sur variétés s-Kähler de rang 2. Ces fibrés ont connexions naturelles dont les sections globales généralisent les opérateurs de Lefschetz de la géométrie de Kähler et agissent d’une façon naturelle sur la cohomologie. Pour première application nous construisons une représentation irréductible d’une forme rationnelle de su(s+1,s+1) sur les classes de Hodge (rationelles) de variétés abéliennes dont la matrice des periodes est rationelle.

Fibrations with noncommutative fibers

We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C*-algebra bundles. We then derive an analogue of the Leray–Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature.

Multisymplectic structures and the variational bicomplex

AbstractMultisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

Moduli Spaces of Coherent Systems of Small Slope on Algebraic Curves

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E, V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for 0 < d ≤ 2n. We show that these spaces are irreducible whenever they are nonempty and obtain necessary and sufficient conditions for nonemptiness.