Topological Responses of the Standard Model Gauge Group

We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit sy...

Fractional quantum Hall (FQH) states are examples of symmetry-enriched topological states (SETs): in addition to the intrinsic topological order, which is robust to symmetry breaking, they possess symmetry-protected topological invariants, such as fractional charge of anyons and fractional Hall conductivity. In this paper we develop a comprehensive theory of symmetry-protected topological invariants for FQH states with spatial symmetries, which applies to Abelian and non-Abel...

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...

The boundary of symmetry-protected topological states (SPTs) can harbor new quantum anomaly phenomena. In this work, we characterize the bosonic anomalies introduced by the 1+1D non-onsite-symmetric gapless edge modes of 2+1D bulk bosonic SPTs with a generic finite Abelian group symmetry (isomorphic to $G=\prod_i Z_{N_i}=Z_{N_1} \times Z_{N_2} \times Z_{N_3} \times ...$). We demonstrate that some classes of SPTs (termed "Type II") trap fractional quantum numbers (such as frac...

Classifying Higgs phases within the landscape of gapped and symmetry preserving states of matter presents a conceptual challenge. We argue that $U(1)$ Higgs phases are symmetry-protected topological (SPT) phases and we derive their topological response theory and boundary anomaly -- applicable to superconductors treated with dynamical electromagnetic field. This generalizes the discussion of discrete gauge theories by Verresen et al., arXiv:2211.01376. We show that a Higgs ph...

Study of the Weyl and Dirac topological materials (topological semimetals, insulators, superfluids and superconductors) opens the route for the investigation of the topological quantum vacua of relativistic fields. The symmetric phase of the Standard Model (SM), where both electroweak and chiral symmetry are not broken, represents the topological semimetal. The vacua of the SM (and its extensions) in the phases with broken Electroweak symmetry represent the topological insula...

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...

We study the Standard Model with gauged $\mathbb Z_{N=2,3,6}^{(1)}$ subgroups of its $\mathbb Z_6^{(1)}$ 1-form global symmetry, making the gauge group $SU(3) \times SU(2)\times U(1) \over \mathbb Z_N$. We show that, on a finite $\mathbb T^3$, there are self-dual instantons of fractional topological charge. They mediate baryon- and lepton-number violating processes. We compare their amplitudes to the ones due to the usual BPST-instantons. We find that the small hypercharge co...

Standard Model (SM) with 15 Weyl fermions per family (lacking the 16th, the sterile right-handed neutrino $\nu_R$) suffers from mixed gauge-gravitational anomalies tied to baryon number plus or minus lepton number ${\bf B} \pm {\bf L}$ symmetry. Including $\nu_R$ per family can cancel these anomalies, but when ${\bf B} \pm {\bf L}$ symmetry is preserved as discrete finite subgroups rather than a continuous U(1), the perturbative local anomalies become nonperturbative global a...

The challenge of identifying symmetry-protected topological states (SPTs) is due to their lack of symmetry-breaking order parameters and intrinsic topological orders. For this reason, it is impossible to formulate SPTs under Ginzburg-Landau theory or probe SPTs via fractionalized bulk excitations and topology-dependent ground state degeneracy. However, the partition functions from path integrals with various symmetry twists are the universal SPT invariants defining topologica...