Principal G-bundle Research Articles

Boundedness of Differential of Symplectic Vortices in Open Cylinder Model

Let G be a compact connected Lie group, (X,ω,μ) a Hamiltonian G-manifold with moment map μ, and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices (A,u) near Z, where A is a connection 1-form of a principal G-bundle P over a punctured Riemann surface Σ˚, and u is a G-equivariant map from P to an open cylinder model near Z. We show that if the total energy of a family of symplectic vortices on Σ˚≅[0,+∞)×S1 is finite, then the A-twisted differential dAu(r,θ) is uniformly bounded for all (r,θ)∈[0,+∞)×S1.

Torelli theorem for moduli stacks of vector bundles and principal G-bundles

Given any irreducible smooth complex projective curve X, of genus at least 2, consider the moduli stack of vector bundles on X of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve X and the rank of the vector bundles. The case of trivial determinant, rank 2 and genus 2 is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is P3 thus independently of the curve).We also prove a Torelli theorem for moduli stacks of principal G-bundles on a curve of genus at least 3, where G is any non-abelian reductive group.

Counting Parabolic Principal G-Bundles with Nilpotent Sections Over $$\mathbb {P}^{1}$$

Counting Parabolic Principal G-Bundles with Nilpotent Sections Over $$\mathbb {P}^{1}$$

Holonomy of the Planar Brownian Motion in a Poisson Punctured Plane

We define a family of diffeomorphism-invariant models of random connections on principal G-bundles over the plane, whose curvatures are concentrated on singular points. We study the limit when the number of points grows to infinity whilst the singular curvature on each point diminishes, and prove that the holonomy along a Brownian trajectory converges towards an explicit limit.

Torsors on moduli spaces of principal G-bundles on curves

Let [Formula: see text] be a semisimple complex algebraic group with a simple Lie algebra [Formula: see text], and let [Formula: see text] denote the moduli stack of topologically trivial stable [Formula: see text]-bundles on a smooth projective curve [Formula: see text]. Fix a theta characteristic [Formula: see text] on [Formula: see text] which is even in case [Formula: see text] is odd. We show that there is a nonempty Zariski open substack [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text], for all [Formula: see text]. It is shown that any such [Formula: see text] has a canonical connection. It is also shown that the tangent bundle [Formula: see text] has a natural splitting, where [Formula: see text] is the restriction of [Formula: see text] to the semi-stable locus. We also produce an isomorphism between two naturally occurring [Formula: see text]-torsors on the moduli space of regularly stable [Formula: see text].

Noncommutative numerable principal bundles from group actions on C⁎-algebras

We introduce the notion of a locally trivial G-C⁎-algebra, which is a noncommutative counterpart of the total space of a locally compact Hausdorff numerable principal G-bundle. To obtain this generalization, we have to go beyond the Gelfand–Naimark duality and use the multipliers of the Pedersen ideal. Our new concept enables us to investigate local triviality of noncommutative principal bundles coming from group actions on non-unital C⁎-algebras, which we illustrate through examples coming from C0(Y)-algebras and graph C⁎-algebras. In the case of an action of a compact Hausdorff group on a unital C⁎-algebra, local triviality in our sense is implied by the finiteness of the local-triviality dimension of the action. Furthermore, we prove that if A is a locally trivial G-C⁎-algebra, then the G-action on A is free in a certain sense, which in many cases coincides with the known notions of freeness due to Rieffel and Ellwood.

Meromorphic parahoric Higgs torsors and filtered Stokes G-local systems on curves

In this paper, we consider the wild nonabelian Hodge correspondence for principal G-bundles on curves, where G is a connected complex reductive group. We establish the correspondence under a “very good” condition on the irregular type of the meromorphic G-connections introduced by Boalch, and thus confirm a conjecture in [9, §1.5]. We first give a version of Kobayashi–Hitchin correspondence, which induces a one-to-one correspondence between stable meromorphic parahoric Higgs torsors of degree zero (Dolbeault side) and stable meromorphic parahoric connections of degree zero (de Rham side). Then, by introducing a notion of stability condition on filtered Stokes G-local systems, we prove a one-to-one correspondence between stable meromorphic parahoric connections of degree zero (de Rham side) and stable filtered Stokes G-local systems of degree zero (Betti side). When G=GLn(C), the main result in this paper reduces to that in [4].

Homotopy types of {{textrm{Spin}}}^{{textrm{c}}}(n)-gauge groups over S^4

The gauge group of a principal G-bundle P over a space X is the group of G-equivariant homeomorphisms of P that cover the identity on X. We consider the gauge groups of bundles over S^4 with {{textrm{Spin}}}^{{textrm{c}}}(n), the complex spin group, as structure group and show how the study of their homotopy types reduces to that of {{textrm{Spin}}}(n)-gauge groups over S^4. We then advance on what is known by providing a partial classification for {{textrm{Spin}}}(7)- and {{textrm{Spin}}}(8)-gauge groups over S^4.

Geometric Confinement in Gauge Theories

In 1978, Friedberg and Lee introduced the phenomenological soliton bag model of hadrons, generalizing the MIT bag model developed in 1974 shortly after the formulation of QCD. In this model, quarks and gluons are confined due to coupling with a real scalar field ρ, which tends to zero outside some compact region S⊂R3 determined dynamically from the equations of motion. The gauge coupling in the soliton bag model runs as the inverse power of ρ, already at the semiclassical level. We show that this model arises naturally as a consequence of introducing the warped product metric dsM2+ρ2dsG2 on the principal G-bundle P(M,G)≅M×G with a non-Abelian group G over Minkowski space M=R3,1. Confinement of quarks and gluons in a compact domain S⊂R3 is a consequence of the collapse of the bundle manifold M×G to M outside S due to shrinking of the group manifold G to a point. We describe the formation of such regions S as a dynamical process controlled by the order parameter field ρ.

Generalized parabolic structures over smooth curves with many components and principal bundles over reducible nodal curves

Let Y be a possibly non-connected smooth projective curve, $$y_{1}^{1},y_{2}^{1},\ldots ,y_{1}^{\nu },y_{2}^{\nu }$$ $$2\nu $$ different points of Y, $$r\in \mathbb {N}$$ , $$d\in \mathbb {Z}$$ , $$\delta \in \mathbb {Q}_{>0}$$ , $${\underline{\kappa }}=(\kappa _1,\ldots ,\kappa _{\nu })\in \mathbb {Q}^{\nu }_{\ge 0}$$ and $${\underline{e}}=(e_1,\ldots ,e_{\nu })\in \mathbb {Z}_{\ge 0}^{\nu }$$ with $$e_i \le r$$ . We construct a projective moduli space of $$({\underline{\kappa }},\delta )$$ -(semi)stable singular principal G-bundles of rank r, degree d, with generalized parabolic structure of type $${\underline{e}}$$ supported on the divisors $$D_{1}=y_{1}^{1}+y_{2}^{1},\ldots ,D_{\nu }=y_{1}^{\nu }+y_{2}^{\nu }$$ . In case Y is the normalization of a connected and reducible projective nodal curve X, there exists a closed subscheme coarsely representing the subfunctor corresponding to those bundles that descend to X. We prove that the descent operation gives a bijection between the set of isomorphism classes of singular principal G-bundles of type $${\underline{e}}$$ on X and the set of isomorphism classes of descending singular principal G-bundles with generalized parabolic structures of type $${\underline{e}}$$ satisfying certain condition on Y. If the stable locus is dense inside the moduli space of descending singular principal G-bundles, the descent operation induces a birational, surjective and proper morphism onto the schematic closure of the space of $$\delta $$ -stable singular principal G-bundles of type $${\underline{e}}$$ . This generalizes the known results over irreducible curves.

Principal bundles on metric graphs: The GLn case

Using the notion of a root datum of a reductive group G we propose a tropical analogue of a principal G-bundle on a metric graph. We focus on the case G=GLn, i.e. the case of vector bundles. Here we give a characterization of vector bundles in terms of multidivisors and use this description to prove analogues of the Weil–Riemann–Roch theorem and the Narasimhan–Seshadri correspondence. We proceed by studying the process of tropicalization. In particular, we show that the non-Archimedean skeleton of the moduli space of semistable vector bundles on a Tate curve is isomorphic to a certain component of the moduli space of semistable tropical vector bundles on its dual metric graph.

The Bestvina–Edwards theorem and the Hilbert–Smith conjecture

We prove a number of results surrounding the Borsuk–Ulam-type conjecture of Baum, Dabrowski, and Hajac (BDH, for short), which states that given a free action of a compact group G on a compact space X, there are no G-equivariant maps X∗G→X (with ∗ denoting the topological join). Mainly, we prove the BDH conjecture for locally trivial principal G-bundles. The proof relies on the nonexistence of G-equivariant maps G∗(n+1)→G∗n, which in turn is a strengthening of an unpublished result of Bestvina and Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev which implies the weak version of the Hilbert–Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold so that the orbit space is finite-dimensional.

Toric principal bundles, piecewise linear maps and Tits buildings

We define the notion of a piecewise linear map from a fan $\Sigma$ to $\tilde{\mathfrak{B}}(G)$, the cone over the Tits building of a linear algebraic group $G$. Let $X_\Sigma$ be a toric variety with fan $\Sigma$. We show that when $G$ is reductive the set of integral piecewise linear maps from $\Sigma$ to $\tilde{\mathfrak{B}}(G)$ classifies the isomorphism classes of (framed) toric principal $G$-bundles on $X_\Sigma$. This in particular recovers Klyachko's classification of toric vector bundles, and gives new classification results for the orthogonal and symplectic toric principal bundles.

Picard groups of certain compact complex parallelizable manifolds and related spaces

Let G be a complex simply connected semisimple Lie group and let Γ be a torsionless uniform irreducible lattice in G. Then Γ﹨G is a compact complex non-Kähler manifold whose tangent bundle is holomorphically trivial. In this note we compute the Picard group of Γ﹨G when rank(G)≥3. When rank(G)<3, we determine the group Pic0(Γ﹨G)⊂Pic(Γ﹨G) of topologically trivial holomorphic line bundles. When rank(G)≥2, we also show that Pic0(PΓ) is isomorphic to Pic0(Y) where PΓ is a Γ﹨G-bundle associated to a principal G-bundle over a compact connected complex manifold Y, and, when rank(G)≥3, we show that Pic(Y)→Pic(PΓ) is injective with finite cokernel.

On birational Torelli theorems

Let G be a simple simply-connected connected linear algebraic group over \({\mathbb {C}}\). We prove a 2-birational Torelli theorem for the moduli space of semistable principal G-bundles over a smooth curve of genus \(\ge 3\), which says that if two such moduli spaces are 2-birational, then the curves are isomorphic. We also prove a 3-birational Torelli theorem for the moduli space of stable symplectic parabolic bundles over a smooth curve of genus \(\ge 4\).

Core Fundamental Groupoid Bundle, Its Sections and Relatedness of Homeomorphisms and Sections

In this paper, we introduce bundle, fibre bundle and principal G-bundle structures on the Core fundamental groupoid keeping its standard projections and quotient topology intact. We give an explicit description of Core fundamental groupoids as such bundles including for the uniquely geodesic spaces and formulate some results on bundle maps. Further, we introduce sections on the Core fundamental groupoid bundle, and also, present some basic properties including composition and inverse with the help of the induced groupoid homomorphisms on the Core fundamental groupoids. With a group structure on the set of all continuous sections of the Core fundamental groupoid bundle, a group action has been built on the Core fundamental groupoid. A notion of relatedness of homeomorphism and section are defined and discussed the pushforward, pullback of sections and their properties. Finally, we investigate more about relatedness notions relΓ0(¯π1M)(f), relHomeo(M)(X) and same on the subsets based on both section related homeomorphisms and homeomorphism related sections. Further, some consequences based on an algebraic structure on the new class of relΓ0(¯π1M)(f), relHomeo(M)(X), etc. have been placed. We present an interrelationship between subsets of Homeo(M) and Γ 0 (¯π1M), which have nice applications in the leftinvariant sections and topological group

A Compactification of the Universal Moduli Space of Principal G-Bundles

Let G be a semisimple linear algebraic group, rho :Ghookrightarrow text {SL}(V) a finite-dimensional faithful representation, gge 2 a natural number, and delta a positive rational number. We prove the existence of a compactification of the universal moduli space of semistable principal G-bundles over overline{text {M}}_{g}, provided that delta is sufficiently large, having the following property: the fibers over singular curves are the moduli spaces of delta -semistable singular principal G-bundles.

Cardinal-indexed classifying spaces for families of subgroups of any topological group

For G a topological group, existence theorems by Milnor (1956), Gelfand–Fuks (1968), and Segal (1975) of classifying spaces for principal G-bundles are generalized to G-spaces with torsion. Namely, any G-space approximately covered by tubes (a generalization of local trivialization) is the pullback of a universal space indexed by the orbit types of tubes and cardinality of the cover. For G a Lie group, via a metric model we generalize the corresponding uniqueness theorem by Palais (1960) and Bredon (1972) for compact G. Namely, the G-homeomorphism types of proper G-spaces over a metric space correspond to stratified-homotopy classes of orbit classifying maps.The former existence result is enabled by Segal's clever but esoteric use of non-Hausdorff spaces. The latter uniqueness result is enabled by our own development of equivariant ANR theory for noncompact Lie G. Applications include the existence part of classification for unstructured fiber bundles with locally compact Hausdorff fiber and with locally connected base or fiber, as well as for equivariant principal bundles which in certain cases via other models is due to Lashof–May (1986) and to Lück–Uribe (2014). From a categorical perspective, our general model EFκG is a final object inspired by the formulation of the Baum–Connes conjecture (1994).

Principal bundles on two-dimensional CW-complexes with disconnected structure group

AbstractGiven any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

Bigerbes

The bigerbes introduced here give a refinement of the notion of 2-gerbes, representing degree four integral cohomology classes of a space. Defined in terms of bisimplicial line bundles, bigerbes have a symmetry with respect to which they form 'bundle 2-gerbes' in two ways; this structure replaces higher associativity conditions. We provide natural examples, including a Brylinski-McLaughlin bigerbe associated to a principal G-bundle for a simply connected simple Lie group. This represents the first Pontryagin class of the bundle, and is the obstruction to the lifting problem on the associated principal bundle over the loop space to the structure group consisting of a central extension of the loop group; in particular, trivializations of this bigerbe for a spin manifold are in bijection with string structures on the original manifold. Other natural examples represent 'decomposable' 4-classes arising as cup products, a universal bigerbe on K(Z,4) involving its based double loop space, and the representation of any 4-class on a space by a bigerbe involving its free double loop space. The generalization to 'multigerbes' of arbitrary degree is also described.