Some Global Aspects of Gauge Anomalies of Semisimple Gauge Groups and Fermion Generations in GUT and Superstring Theories

We report on a detailed calculation of the anomaly coefficients for the odd and even parts of the $Z_2$-graded representation $\theta$ of the Lie algebra Lie$ G$ on the exterior algebra of dimension $2^n$ assuming that $G\subset U(n)$. The coefficients vanish provided $G\subset SU(n)$ and $n\ne3$. The singular role of the gauge group SU(3) is emphasized. The Standard Model is covered by this result.

The mechanism by which gauge and gravitational anomalies cancel in certain string theories is reviewed. The presentation is aimed at theorists who do not necessarily specialize in string theory.

We discuss anomaly cancellation in $U(2)$ gauge theories in four dimensions. For a $U(2)$ gauge theory defined with a spin structure, the vanishing of the bordism group $\Omega_5^{\text{Spin}}(BU(2))$ implies that there can be no global anomalies, in contrast to the related case of an $SU(2)$ gauge theory. We show explicitly that the familiar $SU(2)$ global anomaly is replaced by a local anomaly when $SU(2)$ is embedded in $U(2)$. There must be an even number of fermions with...

We discuss a global anomaly associated with the coupling of chiral Weyl fermions to gravity. The Standard Model based upon $SU(3){\times}SU(2){\times}{U(1)}$ which has 15 fermions per generation is shown to be inconsistent if all background spin manifolds with signature invariant $\tau=8k$ are allowed. Similarly, GUTs based on odd number of Weyl fermions are inconsistent. Consistency can be achieved by adding an extra Weyl fermion which needs to couple only to gravity. For ar...

The axial anomaly arising from the fermion sector of $\U(N)$ or $\SU(N)$ reduced model is studied under a certain restriction of gauge field configurations (the ``$\U(1)$ embedding'' with $N=L^d$). We use the overlap-Dirac operator and consider how the anomaly changes as a function of a gauge-group representation of the fermion. A simple argument shows that the anomaly vanishes for an irreducible representation expressed by a Young tableau whose number of boxes is a multiple ...

In this talk, we briefly review the basic concepts of anomalous gauge theories. It has been known for some time how theories with local anomalies can be handled. Recently it has been pointed out that global anomalies, which obstruct the quantization of certain gauge theories in the temporal gauge, get bypassed in canonical quantization.

A recent work [2006.16996] suggests that a 4d nonperturbative global anomaly of mod 16 class hinting a possible new hidden gapped topological sector beyond the Standard Model (SM) and Georgi-Glashow $su(5)$ Grand Unified Theory (GUT) with 15n chiral Weyl fermions and a discrete $\mathbb{Z}_{4,X}$ symmetry of $X=5({\bf B- L})-4Y$. This $\mathbb{Z}_{16}$ class global anomaly is a mixed gauge-gravitational anomaly between the discrete $X$ and spacetime backgrounds. The new topol...

We give a pedagogical introduction to quantum anomalies, how they are calculated using various methods, and why they are important in condensed matter theory. We discuss axial, chiral, and gravitational anomalies as well as global anomalies. We illustrate the theory with examples such as quantum Hall liquids, Fermi liquids, Weyl semi-metals, topological insulators and topological superconductors. The required background is basic knowledge of quantum field theory, including fe...

These lecture notes review the structure of anomalies and present some of their applications in field theory, string theory and M theory. They expand on material presented at the TASI 2003 summer school and the 2005 International Spring School on String Theory in Hangzhou, China.

I review results from recent investigations of anomalies in fermion--Yang Mills systems in which basic notions from noncommutative geometry (NCG) where found to naturally appear. The general theme is that derivations of anomalies from quantum field theory lead to objects which have a natural interpretation as generalization of de Rham forms to NCG, and that this allows a geometric interpretation of anomaly derivations which is useful e.g. for making these calculations efficie...