Universal Bundle Research Articles

Appearance of universal bundle structure in four dimensional topological gravity

We calculate the connection and curvature of the universal fibration of the Riemannian manifolds and compare them with the BSRT algebra of four dimensional topological gravity. We also comment on the dimension of the moduli space, conformal gauge fixing and the construction of non-trivial observables.

The algebraic structure of cohomological field theory

The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occuring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.

The distribution of bidegrees of smooth surfaces in Gr(1, P3)

The study of algebraic surfaces in Gr (1, p3), the Grassmann variety parametrising lines in projective three-space, was a popular one algebraic geometers of the late nineteenth and early twentieth centuries. Calling them line congruences, researchers such as Kummer, Fano, Roth and many others published many papers on the topic, classifying congruences and studying their invariants. Since that time, the field has lain dormant until very recently. The classical geometers identified two numbers associated with a given congruence: the order and the class. Thinking of a congruence as a two dimensional family of lines, the order is the number of lines in the family passing through a general point in p3, and the class the number of lines in the family contained in a general plane. Together, these two numbers make up the bidegree of the congruence. In modern terms, the bidegree gives the class of the congruence in the Chow ring of Gr (1, p3). In this paper, we consider the question: for what values of a and b does there exist (or not exist) a smooth congruence of bidegree (a, b)? In particular, we try to find restrictions on the bidegree, using an approach suggested by Dolgachev and Reider in [8]. This approach is to study the restriction of the universal bundle g of Gr (1, p3), which appears in the exact sequence

A field theoretic realization of a universal bundle for gravity

Based upon a local vector supersymmetry algebra, we discuss the general structure of the quantum action for topological gravity theories in arbitrary dimensions. The precise form of the action depends on the particular dimension, and also on the moduli space of interest. We describe the general features by examining a theory of topological gravity in two dimensions, with a moduli space specified by vanishing curvature two-form. It is shown that these topological gravity models together with their observables provide a field theoretic realization of a universal bundle for gravity.

Vector supersymmetry in the universal bundle

We present a vector supersymmetry for Witten-type topological gauge theories, and examine its algebra in terms of a superconnection formalism. When covariant constraints on the supercurvature are chosen, a correspondence is established with the universal bundle construction of Atiyah and Singer. The vector supersymmetry represents a certain shift operator in the curvature of the universal bundle, and can be used to generate the hierarchy of observables in these theories. This formalism should lead to the construction of vector supergravity theories, and perhaps to the gravitational analogue of the universal bundle.

On the universal bundle for gravity

We construct a supergravity type theory based on a superspace whose odd directions consist of a vector, together with a scalar representing a topological BRST shift symmetry. As such, the resulting theory is a theory of topological gravity. The gravitino is interpreted as a ghost field for this shift symmetry and plays the usual role of gauge field for local supersymmetry. Our construction is within the bundle of frames approach to superspace where covariant torsion constraints are analyzed, and we find that the resulting theory contains additional fields which are not present in existing theories of topological gravity. In particular, a minimal solution exists which contains a BRST invariant scalar field. Vector supergravity presents a field theoretic model of a universal bundle for gravity, and we show how the vector supersymmetry operator can be used to generate a hierachy of observables.

String structures and the path fibration of a group

We use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class, that is the obstruction to a loop group bundle lifting to a Kac-Moody group bundle. In the case that the loop group bundle is constructed by taking loops into a principal bundle we show that the classifying map is the holonomy around loops and give an explicit formula for the string class relating it to the Pontrjangin class of the principal bunble.

GEOMETRY AND QUANTIZATION OF TOPOLOGICAL GAUGE THEORIES

A general method for constructing interesting topological gauge theories in arbitrary dimensions is presented. The basic framework upon which these models are built is given by the geometrical data of the "universal bundle with connection" of Atiyah and Singer. The models considered include theories which represent the moduli spaces of flat connections and solutions to the Yang-Mills equations. The former theories correspond to supersymmetric versions of the recently introduced BF systems. In all cases we show explicitly that the quantization can be carried out through the construction of an off-shell nilpotent BRST operator, thus guaranteeing the metric independence of these models.

Universals: an opinionated introduction

* The Problem * Primitive Natural Classes * Resemblance Nominalism * Particulars as Bundles of Universals * Universals as Attributes * Tropes * Summing Up

Quantum adiabatic theorem and universal holonomy

We give a simplified proof of the quantum adiabatic theorem for a system of possibly degenerate Hamiltonians by taking Berry's phase into account. We also relate the adiabatic transformation to the parallel transport induced by the holonomy in the universal bundle over a Grassman manifold. The special case of a nondegenerate Hamiltonian is precisely the cyclic quantum evolution studied by Aharanov and Anandan

Spinor bundles on quadrics

We define some stable vector bundles on the complex quadric hypersurface Q n {Q_n} of dimension n n as the natural generalization of the universal bundle and the dual of the quotient bundle on Q 4 ≃ Gr ⁡ ( 1 , 3 ) {Q_4} \simeq \operatorname {Gr} (1,\,3) . We call them spinor bundles. When n = 2 k − 1 n = 2k - 1 there is one spinor bundle of rank 2 k − 1 {2^{k - 1}} . When n = 2 k n = 2k there are two spinor bundles of rank 2 k − 1 {2^{k - 1}} . Their behavior is slightly different according as n ≡ 0 ( mod 4 ) n \equiv 0\;(\bmod 4) or n ≡ 2 ( mod 4 ) n \equiv 2\;(\bmod 4) . As an application, we describe some moduli spaces of rank 3 3 vector bundles on Q 5 {Q_5} and Q 6 {Q_6} .

Characteristic classes in borel cohomology

Let G be a topological group and let EC be a free contractible G-space. The Bore1 construction on a G-space X is the orbit space Xo = EC x G X. When asked what equivariant cohomology is, most people would answer Bore1 cohomology, namely H:(X) = H*(X,). This theory has the claim of priority and the merit of ready computability, and many very beautiful results have been proven with it. However, it suffers from the defects of its virtues. Precisely, it is ‘invariant’, in the sense that a G-map f: X + Y which is a nonequivariant homotopy equivalence induces an isomorphism on Bore1 cohomology. A quick way to see this is to observe that 1 x f: EC xX+ EC x Y is a map of principal G-bundles with base map lxof=fo:Xo-‘Y,, so that fc is a weak homotopy equivalence. As explained in [l], this invariance property is the crudest of a hierarchy of such properties that a theory might have. We shall show how to compute all characteristic classes in any invariant equivariant cohomology theory, the conclusion being that no such theory is powerful enough to support a very useful theory of characteristic classes. As explained in [2], a less crude invariance property can sometimes be exploited to obtain a calculation of equivariant characteristic classes in more powerful theories, such as equivariant K-theory. To establish context, consider a closed normal subgroup Z7 of a topological group r with quotient group G. If Y is a n-free r-space, we think of the orbit projection q : Y-* Y/Z7 as a particular kind of equivariant bundle. It is a principal U-bundle in the usual sense, its base space is a G-space and thus a Z7-trivial r-space, and its projection is a r-map. The classical case is r= G x Z7. Here q is called a principal (G, Ill)-bundle, and there is a theory of associated (G, I;T)-bundles exactly as in the nonequivariant case. For example, if G is a compact Lie group acting smoothly on a differentiable manifold M”, then the tangent n-plane bundle of M is a (G, O(n))bundle and is determined by its associated principal (G, O(n))-bundle. See e.g. [4; 5; 7, V $11 for background. There is a universal bundle E(Z7;r) -+ B(I7;I’) = E(Zl7; r)/Ii’ of the sort just specified. Its total space E(L7;T) is a Z7-free T-CW complex such that the /l-fixed point space E(Z7;r)” is contractible for any closed subgroup A of r such that

A grand superspace for unified field theories

Agrand superspace is proposed as the phase space for gauge field theories with a fixed structure groupG over a fixed space-time manifoldM. This superspace incorporatesall principal fiber bundles with these data. This phase space is the space of isomorphism classes ofall connections onall G-principal fiber bundles overM (fixedG andM). The justification for choosing this grand superspace for the phase space is that the space-time and the structure group are determinants of the physical theory, but the principal fiber bundle with the givenG andM is not. Grand superspace is studied in terms of a natural universal principal fiber bundle overM, canonically associated withM alone, and with a natural universal connection on this bundle. This bundle and its connection are universal in the sense that all connections on allG-principal fiber bundles (anyG) overM can be recovered from this universal bundle and its universal connection by a canonical construction. WhenG is Abelian, grand superspace is shown to be an Abelian group. Various subspaces of grand superspace consisting of the isomorphism classes of flat connections and of Yang-Mills connections are also discussed.

On spinor bundles

A. A. Beĭlinson's spectral sequence [Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69;] gives a way of reconstructing any coherent sheaf on P n from the cohomology on P n of this sheaf and its twists. In this article the author shows that a similar construction is possible on the Grassmannian of 2-planes in C 4 , where twisting with the hyperplane section bundle on P n is replaced by tensoring with various spinor bundles'', namely, associated bundles of the universal bundle. Indeed it is clear that the author's generalization applies to an arbitrary Grassmannian. The author illustrates his theorem with an example (Proposition 5), and in fact most of this paper is concerned with calculations of cohomology for this example. These calculations may be effected alternatively by means of the Bott-Borel-Weil theorem [ R. Bott , Ann. of Math. (2) 66 (1957), 203–248;]. The paper is plagued with misprints: e.g., in Proposition 5 replace E(1) by E(−1) , ⊗ by ⊕ , and (n+2 2) by (n+2 3) .

SU(n) bundles over the configuration space of three identical particles moving on R3

The authors study the systems of three identical spinless particles moving on R3 and possessing an SU(n) gauge symmetry. Three such systems are possible, corresponding to the three non-isomorphic SU(n) bundles over C3(R3), the configuration space. The authors retract C3(R3) to a subcomplex which shows clearly how its homology arises. The three bundles can be realised as pull-backs of the universal bundle S7 to S4 using three non-homotopic maps C3(R3) to S4. The two non-trivial bundles admit no flat connection; they do not correspond to Bose, Fermi or parastatistics.

On the tangent bundle of a flag variety

In this paper we give a description of the tangent bundle of a flag variety in terms of the universal bundles on such a variety (Theorem 2.1).

Universal bundles for deformations of asymmetric structures

Riemannian parallel transport of frames and Cartan connections can be considered as special cases of a type of structure in which the connection form on the principal bundle can transform by an arbitrary extension of the adjoint representation of the fiber group; there always exist universal deformations for asymmetric objects in any of these geometries.

Killing rational characteristic classes by surgery

Let fr:Xr → BO(r) be a sequence of fibrations with maps gr:Xr → Xr+1 such that the usual diagram commutes. For such a situation R. Lashof defines the concept of an X-structure on manifolds (see [3]), and proves a Thom-isomorphism for the cobordism groups of such manifolds. Let n, m be positive integers which are fixed throughout this paper. If r is very big in comparison with n + m then Xr is a simply connected CW-complex and the map (gr)*:H*(Xr; Q)→ H*(Xr+l; Q) is an isomorphism up to dimension n. Let γ be the pull-back over Xr of the universal r-linear bundle (which is, of course, a bundle over BO(r)). If r is very big in comparison with n + m, then we put X = Xr, and we assume that γ is orientable and oriented. The elements of H*(X; Q) of dimension not greater than n, will be called rational universal X-characteristic classes. It is well-known that many of the usual classes of manifolds may be described in terms of X-structures, (e.g. SO, SU, Spin-manifolds etc.).

On the Height of the First Stiefel-Whitney Class

Let ${G_k}({{\mathbf {R}}^{n + k}})$ denote the grassmann manifold of k-planes in real $(n + k)$-space and ${w_1} \in {H^1}({G_k}({{\mathbf {R}}^{n + k}});{{\mathbf {Z}}_2})$ the first Stiefel-Whitney class of the universal bundle. We determine, for many (k, n), the exact height of ${w_1}$ in the cohomology ring. We also indicate the combinatorial significance of the complex analogue of these computations.

On the cohomology of real Grassmanians

LetGk(Rn+k){G_k}({\textbf {R}^{n + k}})denote the grassman manifold ofk-planes in real(n+k)(n\, +\, k)-space andw1∈H1(Gk(Rn+k);Z2){w_1}\, \in \, {H^1}({G_k}({\textbf {R}^{n + k}});\,{\textbf {Z}_2})the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height ofw1{w_1}in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.