Topological Responses of the Standard Model Gauge Group

The gauge group of strong and electroweak interactions in Nature could be any of the four that share the same Lie algebra, $SU(3)_c\times SU(2)_L\times U(1)_Y/Z_p\equiv G_p$ with $Z_p=\left\{Z_6,Z_3,Z_2,Z_1\right\}$. Each of these cases allows in its spectrum for the matter fields of the SM but also for new distinctive representations, e.g. under the assumption that $q_L$ possesses the minimum possible hypercharge in Nature, $G_p$ allows for particles with a multiple of $p\,e...

We study the topological structure of the symmetry group of the standard model, $G_{SM}=U(1)\times SU(2)\times SU(3)$. Locally, $G_{SM}\cong S^1\times (S^3)^2\times S^5$. For SU(3), which is an $S^3$ bundle over $S^5$ (and therefore a local product of these spheres) we give a canonical gauge i.e. a canonical set of local trivializations. These formulae give the matrices of SU(3) in terms of points of spheres. Globally, we prove that the characteristic function of SU(3) is the...

We analyze $2+1d$ and $3+1d$ Bosonic Symmetry Protected Topological (SPT) phases of matter protected by onsite symmetry group $G$ by using dual bulk and boundary approaches. In the bulk we study an effective field theory which upon coupling to a background flat $G$ gauge field furnishes a purely topological response theory. The response action evaluated on certain manifolds, with appropriate choice of background gauge field, defines a set of SPT topological invariants. Furthe...

Clean isotropic quantum Hall fluids in the continuum possess a host of symmetry-protected quantized invariants, such as the Hall conductivity, shift and Hall viscosity. Here we develop a theory of symmetry-protected quantized invariants for topological phases defined on a lattice, where quantized invariants with no continuum analog can arise. We develop topological field theories using discrete crystalline gauge fields to fully characterize quantized invariants of (2+1)D Abel...

In the Standard Model, some combination of the baryon $\bf B$ and lepton $\bf L$ number symmetry is free of mixed anomalies with strong and electroweak $su(3) \times su(2) \times u(1)_{\tilde Y}$ gauge forces. However, it can still suffer from a mixed gravitational anomaly, hypothetically pertinent to leptogenesis in the very early universe. This happens when the total "sterile right-handed" neutrino number $n_{\nu_R}$ is not equal to the family number $N_f$. Thus the inverti...

We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,\nu_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $\nu_d$ an element in group cohomology $\cH^d(G,\R/\Z)$. When $d=3$ and/or when $G_g$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriche...

We examine the interplay of symmetry and topological order in $2+1$ dimensional topological phases of matter. We present a definition of the \it topological symmetry \rm group, which characterizes the symmetry of the emergent topological quantum numbers of a topological phase, and we describe its relation with the microscopic symmetry of the underlying physical system. We derive a general framework to characterize and classify symmetry fractionalization in topological phases,...

In this work, we uncover a collection of non invertible topological operators linked to the 0-, 2-, 4- and 6-form symmetries arising from the type IIB superstring effective pseudo action. By pinpointing the $\text{SL}(2,\mathbb{R})$-covariant conserved currents corresponding to these symmetries, we first derive a set of $\text{SL}(2,\mathbb{Z})$ invariant invertible topological operators that encapsulate the integer BPS charges inherent to the theory. Moving forward, by incor...

We use the intrinsic one-form and two-form global symmetries of (3+1)$d$ bosonic field theories to classify quantum phases enriched by ordinary ($0$-form) global symmetry. Different symmetry-enriched phases correspond to different ways of coupling the theory to the background gauge field of the ordinary symmetry. The input of the classification is the higher-form symmetries and a permutation action of the $0$-form symmetry on the lines and surfaces of the theory. From these d...

In the standard lore, the baryon asymmetry of the present universe is attributed to the leptogenesis from the sterile right-handed neutrino with heavy Majorana mass decaying into the Standard Model's leptons at the very early universe -- called the Majorana fermion's leptogenesis; while the electroweak sphaleron causes baryogenesis at a later time. In this work, we propose a new mechanism, named topological leptogenesis, to explain the lepton asymmetry. Topological leptogenes...