May 11, 2005
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 Z_2 global gauge anomaly given in ref.1 for a gauge theory with the semisimple gauge group SU(2)\times SU(2)\times SU(2) in D=4 dimensions and Weyl fermions in the irreducible representation (IR) \omega=(2,2,2) with 2 denoting the corresponding dimensions. This Z_2 anomaly was used in the discussions related to generic SO(10) and supersymmetric SO(10) unification theories^1 for the total generation numbers of fermions and mirror fermions. Our result^1 that the global anomaly coefficient formula is given by A(\omega)=exp[i{\pi}Q_2(\Box)]=-1 in this case with Q_2(\Box) being the Dynkin index for SU(8) in the fundamental IR (\Box)=(8) is also discussed, and as shown by our results^1 that 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 Z_2 global gauge anomaly of gauge group SU(2)\times SU(2) is also discussed. We also give a complete proof for some relevent topological results. 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.14. We also remark the connection of our results and discussions to the total generation numbers in relevant theories.
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December 29, 1994
The possibilities of global (non-perturbative) gauge anomalies for a class of gauge groups are investigated. Intimately connected to branching rules and topological aspect of gauge groups, the results are applied to the study of unification gauge groups such as SO(10), SU(5), $E_6$, $E_8$ etc. Especially, we discuss extensively about the selection rule for generation numbers $N_f+N_{mf}=even\ge 4$ in SO(10) and supersymmetric SO(10) unification theories as originally proposed...
July 13, 1994
One of the interesting features in unification models and supersymmetric unification models is that the chiral states of quarks and leptons in a family including a right-handed neutrino can be fitted neatly into a fundamental spinor representation (f.s) of dimension 16 for the SO(10) gauge group. However, it is shown in this paper that such a fundamental spinor representation of SO(10) for Weyl fermions will generate global (non-perturbative) gauge anomalies (of new type) whe...
October 11, 2017
String theory provides us with 8d supersymmetric gauge theory with gauge algebras $\mathfrak{su}(N)$, $\mathfrak{so}(2N)$, $\mathfrak{sp}(N)$, $\mathfrak{e}_{6}$, $\mathfrak{e}_{7}$ and $\mathfrak{e}_{8}$, but no construction for $\mathfrak{so}(2N{+}1)$, $\mathfrak{f}_4$ and $\mathfrak{g}_2$ is known. In this paper, we show that the theories for $\mathfrak{f}_4$ and $\mathfrak{so}(2N{+}1)$ have a global gauge anomaly associated to $\pi_{d=8}$, while $\mathfrak{g}_2$ does not ...
August 7, 2017
We discuss a grand unified theory (GUT) based on an $SO(32)$ GUT gauge group broken to its subgroups including a special subgroup. In the $SO(32)$ GUT on six-dimensional (6D) orbifold space $M^4\times T^2/\mathbb{Z}_2$, one generation of the SM fermions can be embedded into a 6D bulk Weyl fermion in the $SO(32)$ vector representation. We show that for a three generation model, all the 6D and 4D gauge anomalies in the bulk and on the fixed points are canceled out without exoti...
February 5, 2008
These lectures on anomalies are relatively self-contained and intended for graduate students who are familiar with the basics of quantum field theory. We begin with several derivations of the abelian anomaly: anomalous transformation of the measure, explicit computation of the triangle Feynman diagram, relation to the index of the Euclidean Dirac operator. The chiral (non-abelian) gauge anomaly is derived by evaluating the anomalous triangle diagram with three non-abelian gau...
January 21, 2000
We discuss the issue of global anomalies in chiral gauge theories on the lattice. In Luscher's approach, these obstructions make it impossible to define consistently a fermionic measure for the path integral. We show that an SU(2) theory has such a global anomaly if the Weyl fermion is in the fundamental representation. The anomaly in higher representations is also discussed. We finally show that this obstruction is the lattice analogue of the SU(2) anomaly first discovered b...
June 28, 2023
We present a Mathematica package that takes any reductive gauge algebra and fully-reducible fermion representation, and outputs all semisimple gauge extensions under the condition that they have no additional fermions, and are free of local anomalies. These include all simple completions, also known as grand unified theories (GUT). We additionally provide a list of all semisimple completions for 5835 fermionic extensions of the one-generation Standard Model.
September 26, 1995
We investigate more generally the possible unification Yang-Mills groups $G_{YM}$ and representations with CP as a gauge symmetry. Besides the possible Yang-Mills groups $E_8$, $E_7$, $SO(2n+1)$, $SO(4n)$, $SP(2n)$, $G_2$ or $F_4$ (or a product of them) which only allow self-contragredient representations, we present other unification groups $G_{YM}$ and representations which may allow CP as a gauge symmetry. These include especially $SU(N)$ containing Weyl fermions and their...
October 24, 2019
We analyse global anomalies and related constraints in the Standard Model (SM) and various Beyond the Standard Model (BSM) theories. We begin by considering four distinct, but equally valid, versions of the SM, in which the gauge group is taken to be $G=G_{\text{SM}}/\Gamma_n$, with $G_{\text{SM}}=SU(3)\times SU(2) \times U(1)$ and $\Gamma_n$ isomorphic to $\mathbb{Z}/n$ where $n\in\left\{1,2,3,6\right\}$. In addition to deriving constraints on the hypercharges of fields tran...
June 30, 2020
Standard lore uses local anomalies to check the kinematic consistency of gauge theories coupled to chiral fermions, e.g. Standard Models (SM). Based on a systematic cobordism classification, we examine constraints from invertible quantum anomalies (including all perturbative local and nonperturbative global anomalies) for gauge theories. We also clarify the different uses of these anomalies: including (1) anomaly cancellations of dynamical gauge fields, (2) 't Hooft anomaly m...