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.

GLOBAL ANOMALIES IN SIX DIMENSIONS

1986 ◽  
Vol 01 (04) ◽  
pp. 267-276 ◽  
Author(s):  
E. BERGSHOEFF ◽  
T.W. KEPHART ◽  
ABDUS SALAM ◽  
E. SEZGIN
Keyword(s):  
Betti Number ◽  
Arbitrary Number ◽  
Space Time ◽  
Gauge Groups ◽  
Yang Mills ◽  
Global Gauge ◽  
Six Dimensions ◽  
Global Anomaly ◽  
Gravitational Anomalies

Applying Witten’s formula for global gauge and gravitational anomalies to 6-dimensional supergravities, we find: (a) The perturbatively anomaly free N=4 chiral supergravity coupled to 21 tensor multiplets is global anomaly free for any choice of space-time manifold with vanishing third Betti number (b3); (b) The perturbatively anomaly free matter coupled N=2 chiral super-gravities with arbitrary number of tensor multiplets, whose Yang-Mills gauge groups do not include G2, SU(2), or SU(3) are free of global anomalies if the theory is formulated on S6. In the case of 9 tensor multiplets coupled to supergravity, this result holds for any space-time with vanishing b3. (c) The N=6 chiral supergravity has perturbative gravitational anomalies and therefore the global anomalies need not be considered in this case.

scholarly journals A note on gauge anomaly cancellation in effective field theories

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Ferruccio Feruglio
Keyword(s):  
Effective Field Theories ◽  
Effective Field ◽  
Field Theories ◽  
Anomaly Cancellation ◽  
Charge Assignment ◽  
The Standard Model ◽  
Renormalizable Theory ◽  
Complete Set ◽  
Gauge Anomalies ◽  
Gauge Anomaly

Abstract The conditions for the absence of gauge anomalies in effective field theories (EFT) are rivisited. General results from the cohomology of the BRST operator do not prevent potential anomalies arising from the non-renormalizable sector, when the gauge group is not semi-simple, like in the Standard Model EFT (SMEFT). By considering a simple explicit model that mimics the SMEFT properties, we compute the anomaly in the regularized theory, including a complete set of dimension six operators. We show that the dependence of the anomaly on the non-renormalizable part can be removed by adding a local counterterm to the theory. As a result the condition for gauge anomaly cancellation is completely controlled by the charge assignment of the fermion sector, as in the renormalizable theory.

scholarly journals Omega vs. pi, and 6d anomaly cancellation

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Joe Davighi ◽  
Nakarin Lohitsiri
Keyword(s):  
Gauge Theories ◽  
Necessary Conditions ◽  
Homotopy Groups ◽  
Mapping Torus ◽  
Anomaly Cancellation ◽  
Global Anomaly ◽  
Gauge Anomalies ◽  
Local Anomalies ◽  
Dimensional Mapping

Abstract In this note we review the role of homotopy groups in determining non-perturbative (henceforth ‘global’) gauge anomalies, in light of recent progress understanding global anomalies using bordism. We explain why non-vanishing of πd(G) is neither a necessary nor a sufficient condition for there being a possible global anomaly in a d-dimensional chiral gauge theory with gauge group G. To showcase the failure of sufficiency, we revisit ‘global anomalies’ that have been previously studied in 6d gauge theories with G = SU(2), SU(3), or G2. Even though π6(G) ≠ 0, the bordism groups $$ {\Omega}_7^{\mathrm{Spin}}(BG) $$ Ω 7 Spin BG vanish in all three cases, implying there are no global anomalies. In the case of G = SU(2) we carefully scrutinize the role of homotopy, and explain why any 7-dimensional mapping torus must be trivial from the bordism perspective. In all these 6d examples, the conditions previously thought to be necessary for global anomaly cancellation are in fact necessary conditions for the local anomalies to vanish.

scholarly journals Some comments on 6d global gauge anomalies

2021 ◽  
Author(s):  
Yasunori Lee ◽  
Yuji Tachikawa
Keyword(s):  
Point Of View ◽  
Homotopy Groups ◽  
Proper Treatment ◽  
Anomaly Cancellation ◽  
The Past ◽  
Global Gauge ◽  
Gauge Anomalies

Abstract Global gauge anomalies in 6d associated with non-trivial homotopy groups π6(G) for G = SU(2), SU(3), and G2 were computed and utilized in the past. In the modern bordism point of view of anomalies, however, they come from the bordism groups Ω7spin (BG), which are in fact trivial and therefore preclude their existence. Instead, it was noticed that a proper treatment of the 6d Green-Schwarz mechanism reproduces the same anomaly cancellation conditions derived from π6(G). In this paper, we revisit and clarify the relation between these two different approaches.

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.

ON THE QUANTIZATION OF A THEORY WITH GLOBAL GAUGE ANOMALIES

1987 ◽  
Vol 02 (12) ◽  
pp. 977-982 ◽  
Author(s):  
ROBERTO PERCACCI
Keyword(s):  
Gauge Theories ◽  
Global Gauge ◽  
Gauge Anomalies

It is shown that the Faddeev-Shatashvili approach to the quantization of anomalous gauge theories can be extended to the case of global gauge anomalies.

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.

Global gauge anomaly and James numbers of Stiefel manifolds

1988 ◽  
Vol 38 (6) ◽  
pp. 1880-1887 ◽  
Author(s):  
Huazhong Zhang ◽  
Susumu Okubo
Keyword(s):  
Stiefel Manifolds ◽  
Global Gauge ◽  
Gauge Anomaly
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