Chern-Simons Numbers Research Articles

Heavy holographic exotics: Tetraquarks as Efimov states

We provide a holographic description of nonstrange multiquark exotics as compact topological molecules by binding heavy-light mesons to a tunneling configuration in $\mathrm{D}8\text{\ensuremath{-}}\mathrm{D}\overline{8}$ that is homotopic to the vacuum state with a fixed Chern-Simons number. In the tunneling process, the heavy-light mesons transmute to fermions. Their binding is generic and arises from a trade-off between the dipole attraction induced by the Chern-Simons term and the U(1) fermionic repulsion. In the heavy quark limit, the open-flavor tetraquark exotics $QQ\overline{q}\overline{q}$ and $\overline{Q}\overline{Q}qq$ emerge as bound Efimov states in a degenerate multiplet $I{J}^{\ensuremath{\pi}}=(0{0}^{+},0{1}^{+})$ with opposite intrinsic Chern-Simons numbers $\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$. The hidden-flavor tetraquark exotics such as $Q\overline{Q}q\overline{q}$, $QQ\overline{Q}\overline{q}$, and $QQ\overline{Q}\overline{Q}$ as compact topological molecules are unbound. Other exotics are also discussed.

Generating luminous and dark matter during inflation

We propose a new mechanism for generating both luminous and dark matter during cosmic inflation. According to this mechanism, ordinary and dark matter carry common charge which is associated with an anomalous U(1)[Formula: see text] group. Anomaly terms source [Formula: see text] and U(1)[Formula: see text] charge violating processes during inflation, producing corresponding nonzero Chern–Simons numbers which are subsequently reprocessed into baryon and dark matter densities. The general framework developed is then applied to two possible extensions of the Standard Model with anomalous gauged [Formula: see text] and [Formula: see text], each with an additional dark matter candidate. In each scenario, we consider the parameter choices that predict the correct dark matter to baryonic matter density ratio and baryon asymmetry. Interestingly, under these conditions, for the U(1)[Formula: see text] extension we obtain a prediction for the mass of the dark matter candidate which is independent of the other choice of parameters, when assuming an [Formula: see text] and [Formula: see text].

Topological charge carried by nexuses and center vortices

In this paper we further explore some questions of topological charge arising from transverse intersections of center vortices and nexuses. Topological charge can also be carried by twist or writhe; we introduce a new nexus carrying twist (or equivalently writhe) which carries Chern-Simons number and in some cases topological charge without reference to an intersection with a separate center vortex. This new nexus can also carry topological charge by linking to vortices. In general, no topological charge in $d=4$ arises from static new nexuses, since the charge is the difference of two (equal) Chern-Simons numbers, but it can arise through dynamic reconnection processes, as the author had suggested earlier. Generally, charge carried by transverse intersections is locally fractionalized in units of $1/N$ for gauge group $\mathrm{SU}(N),$ but globally quantized in integral units. We show explicitly that in $d=4$ such global topological charge is a linkage number of the closed two-surface of a center vortex with a nexus world line, and relate this linkage to the Hopf fibration, with homotopy ${\ensuremath{\Pi}}_{3}{(S}^{2})\ensuremath{\simeq}Z;$ this homotopy ensures integrality of the global topological charge. We show that a standard nexus form used earlier, when linked to a center vortex, gives rise naturally to a homotopy ${\ensuremath{\Pi}}_{2}{(S}^{2})\ensuremath{\simeq}Z,$ a homotopy usually associated with 't Hooft--Polyakov monopoles and similar objects which exist by virtue of the presence of an adjoint scalar field which gives rise to spontaneous symmetry breaking. We show that certain integrals related to monopole or topological charge in gauge theories with adjoint scalars also appear in the center vortex-nexus picture, but with a different physical interpretation. We complete earlier work on vortex-nexus intersections to show explicitly how to express globally integral topological charge as composed of essentially independent units of charge $1/N.$

Classical Yang-Mills vacua onT3:Explicit constructions

Flat connections for unitary gauge groups on a 3-torus with twisted boundary conditions as well as recently discovered periodic nontrivial flat connections with ''nondiagonalizable'' triples of holonomies for higher orthogonal and exceptional groups are constructed explicitly in terms of Jacobi theta functions with rational characteristics. The (fractional) Chern-Simons numbers of these vacuum gauge field configurations are verified by direct computation.

Calorons in the Weyl gauge

We demonstrate by explicit construction that while the untwisted Harrington-Shepard caloron ${A}_{\ensuremath{\mu}}$ is manifestly periodic in Euclidean time, with a period $\ensuremath{\beta}=1/T,$ when transformed to the Weyl ${(A}_{0}=0)$ gauge, the caloron gauge field ${A}_{i}$ is periodic only up to a large gauge transformation, with the winding number equal to the caloron's topological charge. This helps clarify the tunneling interpretation of these solutions, and their relation to Chern-Simons numbers and winding numbers.

A Z2 structure in the configuration space of Yang–Mills theories

We argue for the presence of a Z2 topological structure in the space of static gauge–Higgs field configurations of SU(2n) and SO(2n) Yang–Mills theories. We rigorously prove the existence of a Z2 homotopy group of mappings from the two-dimensional projective sphere RP2 into SU(2n)/Z2 and SO(2n)/Z2 Lie groups, respectively. Consequently the symmetric phase of these theories admits infinite surfaces of odd-parity static and unstable gauge field configurations which divide into two disconnected sectors with integer Chern–Simons numbers n and n+1/2, respectively. Such a Z2 structure persists in the Higgs phase of the above theories and accounts for the existence of CS=1/2 odd-parity saddle point solutions to the field equations which correspond to spontaneous symmetry breaking mass scales.

Thermal fluctuations of Chern-Simons numbers in the lattice SU(2) Higgs model.

We study the temperature dependence of the Chern-Simons number fluctuations in the SU(2) Higgs model on Euclidean lattices with spatial sizes up to ${20}^{3}$. Temperatures well above the Higgs phase transition ${\mathit{T}}_{\mathit{H}}$ are achieved on anisotropic lattices. Numerical results are compared to perturbative results on finite lattices as well as in continuum perturbation theory. We find qualitative agreement with perturbative estimates and see at high temperatures a tendency towards static configurations. Up to temperatures T\ensuremath{\simeq}2${\mathit{T}}_{\mathit{H}}$ we find an indication that tunneling between vacuums with different Chern-Simons numbers is still exponentially suppressed.

Vacuum tunneling and periodic structure in lattice Higgs models

Using a geometruc definition for the lattice Chern-Simons term in even dimensions, we have studied the distribution of Chern-Simons numbers for the 2d U(1) and the 4d SU(2) lattice Higgs models. The periodic structure of the distributions is preserved in our lattice formulation and has been examined in detail. In both cases the finite-size effects visible in the distribution of Chern-Simons numbers are well accounted for by the Haar measure. Moreover, we find that 〈 N CS 2〉 grows with the spatial volume. We also find numerical evidence that tunneling in 4d is increased at high temperature.

Self-gravitating Yang–Mills solitons and their Chern–Simons numbers

A classification of the possible regular, spherically symmetric solutions of the Einstein–Yang–Mills system which is based on a bundle theoretical analysis for arbitrary gauge groups is presented herein. It is shown that such solitons must be of magnetic type, at least if the magnetic Yang–Mills charge vanishes. Explicit expressions for the Chern–Simons numbers of these self-gravitating Yang–Mills solitons are derived, which involve only properties of irreducible root systems and some information about the asymptotics of the solutions. It turns out, as an example, that the Chern–Simons numbers are always half integers or integers for the gauge groups SU(n). Possible physical implications of these results, which are based on analogies with the unstable sphaleron solution of the electroweak theory, are briefly indicated.

Sphaleron topography

By numerical simulations in real time we provide evidence in favour of sphaleron like transitions in the hot, symmetric phase of the electroweak theory. Earlier performed observations of a change in the Chern-Simons number are supplemented with a measurement of the lowest eigenvalues of the three-dimensional staggered fermion Dirac operator and observations of the spatial extension of energy lumps associated with the transition. The observations corroborate on the interpretation of the change in Chern-Simons numbers as representing continuum physics, not lattice artifacts. By combining the various observations it is possible to follow in considerable detail the time-history of thermal fluctuations of the classical gauge-field configurations responsible for the change in the Chern-Simons number.

DOES THE STRONG CP-PROBLEM REALLY EXIST?

It is argued that the principle of absolute gauge invariance (which implies that the functional manifold of the gauge fields is a “circle,” where the integer-valued Chern-Simons numbers present the same point at the “circle”) provides a natural solution of the CP-problem. It is shown in simple mechanical models that the θ-term being a time derivative of some function does not change the quantum dynamics just as it does not change the classical one. In this context the two-dimensional electrodynamics is also considered.

Topography of the hot sphaleron transitions

By numerical simulations in real time we provide evidence in favour of sphaleron-like transitions in the hot, symmetric phase of the electroweak theory. Earlier performed observations of a change in the Chern-Simons number are supplemented with a measurement of the lowest eigenvalues of the three-dimensional staggered fermion Dirac operator and observations of the spatial extension of energy lumps associated with the transition. The observations corroborate on the interpretation of the change in Chern-Simons numbers as representing continuum physics, not lattice artifacts. By combining the various observations it is possible to follow in considerable detail the time-history of thermal fluctuations of the classical gauge-field configurations responsible for the change in the Chern-Simons number.

Study of sphaleron transitions by means of the real-time Langevin equation.

Microscopic Langevin evolution is introduced in Abelian gauge field theory. It is studied numerically in the (1 + 1)-dimensional Abelian Higgs model at finite temperature. The Brownian behavior of the Chern-Simons number is established. Sharp transitions between the classical vacua with different Chern-Simons numbers are observed. The rate of these transitions depends exponentially on the temperature, indicating the relevance of the thermal activation theory involving the sphaleron saddle point.

Eta Invariants of Homogeneous Spaces

We derive a formula for the eta invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extension. As an example, we give some computations for spheres.