Royal Institution of Great Britain

Family puzzle asks why the Standard Model (SM) features exactly 3 families of quarks and leptons. Motivated by topological constraints, we study 4-dimensional fermionic anomalies with discrete Z_n symmetry, classified by the 5d spin bordism group. We show that only the group-cohomology subclass H^5(Z_n,U(1))≅ Z_n can be canceled by an anomalous Z_n-symmetric 4d Z_n-gauge topological quantum field theory (TQFT), while beyond-group-cohomology A_Z_n p_1 involving the Pontryagin class p_1 cannot (except n=2,3). More generally, we prove that any cocycle α_d∈H^d(Z_n,U(1)) in odd spacetime dimension d≥3 is trivialised by the symmetry extension 1→ Z_n→ Z_n^2→ Z_n→ 1, and we construct the corresponding symmetric anomalous boundary TQFT. For d=5 and n=3, this yields a Spin× Z_3-symmetric 4d Z_3-gauge TQFT that cancels the mixed discrete ( B+L)-gauge-gravitational anomaly of the SM in the absence of 3 "sterile" right-handed neutrinos ν_R. We further analyze a generalized SM with N_c colors and N_f families and argue that missing N_f copies of the ν_R can be naturally replaced by that 4d anomalous Spin×_Z_2^F Z_2 N_f, B + L symmetric Z_N_c-gauge TQFT under the anomaly cancellation, via an appropriate Z_N_c-color symmetry extension construction 1→ Z_N_c→ Spin× Z_N_cN_f→ Spin×_Z_2^F Z_2N_f→1 of anomalous topological order. For minimal nonzero positive integers N_c and N_f, we find the minimal color extensions: N_c=3, N_f ≥ 3; N_c=4, N_f ≥ 2; and N_c=12, N_f ≥ 6. If we further require that an SM baryon is a fermion so N_c is odd, then Z_N_c=Z(SU(N_c)) color center, we prove 3 families and 3 colors, N_c=N_f=3, is the unique case that stands out. We also prove that A_Z_3p_1= 03 for the mod 3 cohomology class in an appropriate context.

CIPro is a Mathematica package for constructing and analyzing complete intersections in products of projective spaces and line bundles over such varieties. It computes properties of complete intersections, such as Chern classes, Hilbert series, GV invariants and symmetries, as well as properties of line bundles on complete intersections, including their cohomology groups. The package also consolidates a number of data sets available in the literature into a single system, including the lists of complete intersection Calabi-Yau three- and four-folds. This short tutorial introduces the package, provides a brief discussion of some of the mathematical background underlying its computations, and gives a series of examples to illustrate its use. These tools are of utility for many computations in string compactifications, especially for Calabi-Yau geometries appearing in Heterotic and Type II constructions. Many tools apply beyond the Calabi-Yau context, including for example, almost Fano bases in F-theory.

We investigate the full counting statistics of a one-dimensional multicomponent impenetrable gas released from a bipartite state in the presence of a local defect. Spin–charge separation reduces the multivariate generating function to an exact Fredholm determinant whose kernel is built from time-evolved single-particle orbitals. Expressing these orbitals in terms of scattering data gives access to both transient dynamics and the nonequilibrium steady state. We derive the leading long-time asymptotics and identify persistent oscillations in densities and currents caused by multiple bound states at the junction. We argue that interactions between components lead to a linear low-temperature correction to the current, in contrast to the quadratic correction in the single-component case. At equilibrium, we obtain the M-Wright distribution of spin transfer with t^1/4 scaling.

Spacetime metamaterials are engineered media whose constitutive parameters such as permittivity, permeability, stiffness, or mass density are modulated simultaneously in both space and time. These additional degrees of freedom, absent in conventional static metamaterials, unlock a cabinet of wave phenomena that cannot be achieved in time-invariant structures, e.g. compact nonreciprocal devices, topological insulators, and devices for efficient frequency conversion and mixing and pulse shaping. The vast majority of these studies, however, operate in the stable regime, where modulation parameters are chosen to yield linear wave propagation. Here, we push spacetime metamaterials into the regime of parametric instability, and discover a novel type of “dynamically frustrated” oscillating states, where nonlinear non-reciprocal, topologically protected phase dislocations emerge. We control these dislocations and make them stop, split, and recombine. We harness this control to create devices for cascading logic in branched networks, and synchronization in 2D metamaterials. Our findings are broadly applicable anywhere where spacetime modulation can be pushed beyond linear stability, from cold atoms and superconducting circuits to acoustics and RF circuits.

We obtain an exact analytic expression for the dynamical structure factor of one-dimensional quantum gas of hard rods. Our result is valid for an arbitrary many-body state of the system, with finite temperature states and the ground state being important special cases that we analyze in detail. We demonstrate that the expression obeys fundamental relations such like the f-sum rule and the detailed balance. We also reveal the hidden fermionic structure behind the correlator. In the static limit we show that it can be written in terms of universal functions, which, at zero temperature, coincide with the level spacing distribution function of the Gaussian unitary ensemble. Our Letter provides a full and exact characterization of a dynamic correlation function in a strongly correlated interacting quantum many-body system.