scholarly journals Gauge groups and bialgebroids

2021 ◽  
Vol 111 (6) ◽  
Author(s):  
Xiao Han ◽  
Giovanni Landi
Keyword(s):  
Gauge Group ◽  
Galois Extension ◽  
Principal Bundle ◽  
Module Structure ◽  
Crossed Module ◽  
Gauge Groups ◽  
Quantum Sphere ◽  
Hopf Algebroid ◽  
Commutative Algebras

AbstractWe study the Ehresmann–Schauenburg bialgebroid of a noncommutative principal bundle as a quantization of the gauge groupoid of a classical principal bundle. We show that the gauge group of the noncommutative bundle is isomorphic to the group of bisections of the bialgebroid, and we give a crossed module structure for the bisections and the automorphisms of the bialgebroid. Examples include: Galois objects of Taft algebras, a monopole bundle over a quantum sphere and a not faithfully flat Hopf–Galois extension of commutative algebras. For each of the latter two examples, there is in fact a suitable invertible antipode for the bialgebroid making it a Hopf algebroid.

scholarly journals CONTINUOUS TRACE C*-ALGEBRAS, GAUGE GROUPS AND RATIONALIZATION

2009 ◽  
Vol 01 (03) ◽  
pp. 261-288 ◽  
Author(s):  
JOHN R. KLEIN ◽  
CLAUDE L. SCHOCHET ◽  
SAMUEL B. SMITH
Keyword(s):  
Gauge Group ◽  
Metric Space ◽  
Principal Bundle ◽  
Rational Homotopy ◽  
Compact Metric Space ◽  
Gauge Groups ◽  
C Algebra ◽  
C Algebras ◽  
Rational Cohomology ◽  
Rational Homotopy Type

Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζdenote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

On p-local homotopy types of gauge groups

2014 ◽  
Vol 144 (1) ◽  
pp. 149-160 ◽  
Author(s):  
Daisuke Kishimoto ◽  
Akira Kono ◽  
Mitsunobu Tsutaya
Keyword(s):  
Gauge Group ◽  
Homotopy Type ◽  
Principal Bundle ◽  
Dimensional Sphere ◽  
Gauge Groups ◽  
Homotopy Types

The aim of this paper is to show that the p-local homotopy type of the gauge group of a principal bundle over an even-dimensional sphere is completely determined by the divisibility of the classifying map by p. In particular, for gauge groups of principal SU(n)-bundles over S2d for 2 ≤ d ≤ p − 1 and n ≤ 2p − 1, we give a concrete classification of their p-local homotopy types.

ENERGY REPRESENTATIONS OF INFINITE DIMENSIONAL GAUGE GROUPS IN NONCOMMUTATIVE GEOMETRY

1994 ◽  
Vol 05 (03) ◽  
pp. 329-348
Author(s):  
JEAN MARION
Keyword(s):  
Gauge Group ◽  
Noncommutative Geometry ◽  
Von Neumann Algebra ◽  
Linear Operators ◽  
Consistent Theory ◽  
Bounded Linear ◽  
Von Neumann ◽  
Gauge Groups ◽  
Infinite Dimensional ◽  
Involutive Algebra

Let M be a compact smooth manifold, let [Formula: see text] be a unital involutive subalgebra of the von Neumann algebra £ (H) of bounded linear operators of some Hilbert space H, let [Formula: see text] be the unital involutive algebra [Formula: see text], let [Formula: see text] be an hermitian projective right [Formula: see text]-module of finite type, and let [Formula: see text] be the gauge group of unitary elements of the unital involutive algebra [Formula: see text] of right [Formula: see text]-linear endomorphisms of [Formula: see text]. We first prove that noncommutative geometry provides the suitable setting upon which a consistent theory of energy representations [Formula: see text] can be built. Three series of energy representations are constructed. The first consists of energy representations of the gauge group [Formula: see text], [Formula: see text] being the group of unitary elements of [Formula: see text], associated with integrable Riemannian structures of M, and the second series consists of energy representations associated with (d, ∞)-summable K-cycles over [Formula: see text]. In the case where [Formula: see text] is a von Neumann algebra of type II 1 a third series is given: we introduce the notion of regular quasi K-cycle, we prove that regular quasi K-cycles over [Formula: see text] always exist, and that each of them induces an energy representation.

A note on the homotopy type of certain gauge groups

1991 ◽  
Vol 117 (3-4) ◽  
pp. 295-297 ◽  
Author(s):  
Akira Kono
Keyword(s):  
Gauge Group ◽  
Homotopy Type ◽  
Gauge Groups

SynopsisLet Gk be the gauge group of Pk, the principal SU(2) bundle over S4 with c2(Pk) = k. In this paper we show that Gk ≃ Gk. if and only if (12, k) = (12, k′) where (12, k) is the GCD of 12 and k.

scholarly journals SOME GLOBAL ASPECTS OF GAUGE ANOMALIES OF SEMISIMPLE GAUGE GROUPS AND FERMION GENERATIONS IN GUT AND SUPERSTRING THEORIES

2006 ◽  
Vol 21 (05) ◽  
pp. 1033-1052
Author(s):  
HUAZHONG ZHANG
Keyword(s):  
Gauge Group ◽  
Complete Proof ◽  
Gauge Groups ◽  
Gauge Transformations ◽  
Global Gauge ◽  
Global Anomaly ◽  
Gauge Anomalies ◽  
Physical Consequences ◽  
Gauge Anomaly ◽  
Weyl Fermions

We study more extensively and completely for global gauge anomalies with some semisimple gauge groups as initiated in Ref. 1. A detailed and complete proof or derivation is provided for the Z2 global (nonperturbative) gauge anomaly given in Ref. 1 for a gauge theory with the semisimple gauge group SU (2) × SU (2) × SU (2) in D = 4 dimensions and Weyl fermions in the irreducible representation (IR) ω = (2, 2, 2) with 2 denoting the corresponding dimensions. This Z2 anomaly was used in the discussions related to all the generic SO (10) and supersymmetric SO (10) unification theories1 for the total generation numbers of fermions and mirror fermions. Our result1 shows that the global anomaly coefficient formula is given by A(ω) = exp [iπQ2(□)] = -1 in this case with Q2(□) being the Dynkin index for SU (8) in the fundamental IR (□) = (8) and that the corresponding gauge transformations need to be topologically nontrivial simultaneously in all the three SU (2) factors for the homotopy group Π4( SU (2) × SU (2) × SU (2))is also discussed, and as shown by the results1 the semisimple gauge transformations collectively may have physical consequences which do not correspond to successive simple gauge transformations. The similar result given in Ref. 1 for the Z2 global gauge anomaly of gauge group SU (2) × SU (2) with Weyl fermions in the IR ω = (2, 2) with 2 denoting the corresponding dimensions is also discussed with proof similar to the case of SU (2) × SU (2) × SU (2). We also give a complete proof for some relevant topological results. We expect that our results and discussions may also be useful in more general studies related to global aspects of gauge theories. Gauge anomalies for the relevant semisimple gauge groups are also briefly discussed in higher dimensions, especially for self-contragredient representations, with discussions involving trace identities relating to Ref. 15. We also relate the discussions to our results and propositions in our previous studies of global gauge anomalies. We also remark the connection of our results and discussions to the total generation numbers in relevant theories.

scholarly journals A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in $d>2$ dimensions

2021 ◽  
Vol 11 (6) ◽  
Author(s):  
Yin-Chen He ◽  
Junchen Rong ◽  
Ning Su
Keyword(s):  
Gauge Group ◽  
Gauge Theories ◽  
Higher Dimensions ◽  
Equation Of Motion ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Gauge Groups ◽  
Conformal Field ◽  
Conformal Bootstrap ◽  
Null Operator

We propose a roadmap for bootstrapping conformal field theories (CFTs) described by gauge theories in dimensions d>2d>2. In particular, we provide a simple and workable answer to the question of how to detect the gauge group in the bootstrap calculation. Our recipe is based on the notion of decoupling operator, which has a simple (gauge) group theoretical origin, and is reminiscent of the null operator of 2d2d Wess-Zumino-Witten CFTs in higher dimensions. Using the decoupling operator we can efficiently detect the rank (i.e. color number) of gauge groups, e.g., by imposing gap conditions in the CFT spectrum. We also discuss the physics of the equation of motion, which has interesting consequences in the CFT spectrum as well. As an application of our recipes, we study a prototypical critical gauge theory, namely the scalar QED which has a U(1)U(1) gauge field interacting with critical bosons. We show that the scalar QED can be solved by conformal bootstrap, namely we have obtained its kinks and islands in both d=3d=3 and d=2+\epsilond=2+ϵ dimensions.

scholarly journals Exploring the landscape of heterotic strings on Td

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Anamaría Font ◽  
Bernardo Fraiman ◽  
Mariana Graña ◽  
Carmen A. Núñez ◽  
Héctor Parra De Freitas
Keyword(s):  
Gauge Group ◽  
Moduli Space ◽  
Relevant Information ◽  
Duality Group ◽  
Simple Method ◽  
Gauge Groups ◽  
Heterotic Strings ◽  
Singular Fibers ◽  
Special Points ◽  
Extended Dynkin Diagram

Abstract Compactifications of the heterotic string on Td are the simplest, yet rich enough playgrounds to uncover swampland ideas: the U(1)d+16 left-moving gauge symmetry gets enhanced at special points in moduli space only to certain groups. We state criteria, based on lattice embedding techniques, to establish whether a gauge group is realized or not. For generic d, we further show how to obtain the moduli that lead to a given gauge group by modifying the method of deleting nodes in the extended Dynkin diagram of the Narain lattice II1,17. More general algorithms to explore the moduli space are also developed. For d = 1 and 2 we list all the maximally enhanced gauge groups, moduli, and other relevant information about the embedding in IId,d+16. In agreement with the duality between heterotic on T2 and F-theory on K3, all possible gauge groups on T2 match all possible ADE types of singular fibers of elliptic K3 surfaces. We also present a simple method to transform the moduli under the duality group, and we build the map that relates the charge lattices and moduli of the compactification of the E8 × E8 and Spin(32)/ℤ2 heterotic theories.

Homotopy commutativity in p-localized gauge groups

2013 ◽  
Vol 143 (4) ◽  
pp. 851-870 ◽  
Author(s):  
D. Kishimoto ◽  
A. Kono ◽  
S. Theriault
Keyword(s):  
Gauge Group ◽  
Lie Group ◽  
Chern Class ◽  
Compact Lie Group ◽  
Gauge Groups

Let G be a simple, compact Lie group and let $\mathcal{G}_k(G)$ be the gauge group of the principal G-bundle over S4 with second Chern class k. McGibbon classified the groups G that are homotopy commutative when localized at a prime p. We show that in many cases the homotopy commutativity of G, or its failure, determines that of $\mathcal{G}_k(G)$.

scholarly journals RADIATIVE PRODUCTION OF LIGHTEST NEUTRALINOS IN e+e-COLLISIONS IN SUPERSYMMETRIC GRAND UNIFIED MODELS

2012 ◽  
Vol 27 (29) ◽  
pp. 1250172 ◽  
Author(s):  
P. N. PANDITA ◽  
MONALISA PATRA
Keyword(s):  
Gauge Group ◽  
Linear Collider ◽  
International Linear Collider ◽  
Signal Process ◽  
Gauge Groups ◽  
Neutrino Production ◽  
The Standard Model ◽  
Compare And Contrast ◽  
Signal Cross Section ◽  
Comprehensive Study

We study the production of the lightest neutralinos in the radiative process [Formula: see text] in supersymmetric models with grand unification. We consider models wherein the standard model (SM) gauge group SU(3)c×SU(2)L×U(1)Yis unified into the grand unified gauge groups SU(5) or SO(10). We study this process at energies that may be accessible at a future International Linear Collider (ILC). We compare and contrast the dependence of the signal cross-section on the grand unified gauge group, and different representations of the grand unified gauge group, into which the SM gauge group is unified. We carry out a comprehensive study of the radiative production process which includes higher-order QED corrections in our calculations. In addition we carry out a detailed study of the background to the signal process coming from the SM radiative neutrino production [Formula: see text], as well as from the radiative production of the scalar partners of the neutrinos (sneutrinos) [Formula: see text]. The latter can be a major supersymmetric background to the radiative production of neutralinos when the sneutrinos decay invisibly. It is likely that the radiative production of the lightest neutralinos may be a viable channel to study supersymmetric partners of the SM particles at the first stage of an ILC, where heavier sparticles may be too heavy to be produced in pairs.

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