Langevin Dynamics of Lattice Yang–Mills–Higgs and Applications

We develop a new stochastic analysis approach to the lattice Yang--Mills model at strong coupling in any dimension $d>1$, with t' Hooft scaling $\beta N$ for the inverse coupling strength. We study their Langevin dynamics, ergodicity, functional inequalities, large $N$ limits, and mass gap. Assuming $|\beta| < \frac{N-2}{32(d-1)N}$ for the structure group $SO(N)$, or $|\beta| < \frac{1}{16(d-1)}$ for $SU(N)$, we prove the following results. The invariant measure for the corresponding Langevin dynamic is unique on the entire lattice, and the dynamic is exponentially ergodic under a Wasserstein distance. The finite volume Yang--Mills measures converge to this unique invariant measure in the infinite volume limit, for which Log-Sobolev and Poincar\'e inequalities hold. These functional inequalities imply that the suitably rescaled Wilson loops for the infinite volume measure has factorized correlations and converges in probability to deterministic limits in the large $N$ limit, and correlations of a large class of observables decay exponentially, namely the infinite volume measure has a strictly positive mass gap. Our method improves earlier results or simplifies the proofs, and provides some new perspectives to the study of lattice Yang--Mills model.

Smooth gauge strings and lattice Yang–Mills theories

An effective action for monopoles and knot solitons in Yang–Mills theory

We report on the results of numerical simulations of $SU(N)$ lattice Yang Mills with two flavors of (light) Wilson fermion in the adjoint representation. We analytically and numerically address the question of center symmetry realization on lattices with $\Gamma$ sites in each direction in the large-$N$ limit. We show, by a weak coupling calculation that, for massless fermions, center symmetry realization is independent of $\Gamma$, and is unbroken. Then, we extend our result by conducting simulations at non zero mass and finite gauge coupling. Our results indicate that center symmetry is intact for a range of fermion mass in the vicinity of the critical line on lattices of volume $2^4$. This observation makes it possible to compute infinite volume physical observables using small volume simulations in the limit $N\to\infty$, with possible applications to the determination of the conformal window in gauge theories with adjoint fermions.

Center vortex model for the infrared sector of Yang–Mills theory—quenched Dirac spectrum and chiral condensate

We give new descriptions of lattice SU(N) Yang–Mills theory in terms of new lattice variables. The validity of such descriptions has already been demonstrated in the SU(2) Yang–Mills theory by our previous works from the viewpoint of defining and extracting topological degrees of freedom such as gauge-invariant magnetic monopoles and vortices which play the dominant role in quark confinement. In particular, we have found that the SU(3) lattice Yang–Mills theory has two possible options, maximal and minimal: The existence of the minimal option has been overlooked so far, while the maximal option reproduces the conventional SU(3) Cho–Faddeev–Niemi–Shabanov decomposition in the naive continuum limit. The new description gives an important framework for understanding the mechanism of quark confinement based on the dual superconductivity.

The phase diagram ofSU(3) four-dimensional space-time lattice Yang-Mills field theory at finite temperature is analyzed by the extended mean-field technique. With this, finite-temperature effects are present already at the level of the saddle point approximation. A deconfining second-order phase transition is obtained which is in agreement with a recent Monte Carlo numerical simulation.

Renormalized field strength approach as continuum limit of lattice Yang-Mills theory

Center vortex model for the infrared sector of SU(3) Yang–Mills theory—confinement and deconfinement

Lattice Yang–Mills theory at finite densities of heavy quarks

We study a discretization of = 2 super Yang–Mills theory which possesses a single exact supersymmetry at non-zero lattice spacing. This supersymmetry arises after a reformulation of the theory in terms of twisted fields. In this paper we derive the action of the other twisted supersymmetries on the component fields and study, using Monte Carlo simulation, a series of corresponding Ward identities. Our results for SU(2) and SU(3) support a restoration of these additional supersymmetries without fine tuning in the infinite volume continuum limit. Additionally we present evidence supporting a restoration of (twisted) rotational invariance in the same limit. Finally we have examined the distribution of scalar field eigenvalues and find evidence for power law tails extending out to large eigenvalue. We argue that these tails indicate that the classical moduli space does not survive in the quantum theory.

The construction of a consistent measure for Yang–Mills is a precondition for an accurate formulation of nonperturbative approaches to QCD, both analytical and numerical. Using projective limits as subsets of Cartesian products of homomorphisms from a lattice to the structure group, a consistent interaction measure and an infinite-dimensional calculus have been constructed for a theory of non-Abelian generalized connections on a hypercubic lattice. Here, after reviewing and clarifying past work, new results are obtained for the mass gap when the structure group is compact.

In this Letter, we consider lattice versions of the decomposition of the Yang–Mills field a la Cho–Faddeev–Niemi, which was extended by Kondo, Shinohara and Murakami in the continuum formulation. For the SU(N) gauge group, we propose a set of defining equations for specifying the decomposition of the gauge link variable and solve them exactly without using the ansatz adopted in the previous studies for SU(2) and SU(3). As a result, we obtain the general form of the decomposition for SU(N) gauge link variables and confirm the previous results obtained for SU(2) and SU(3).

It is shown that the physical phase space of the γ-deformed Hamiltonian lattice in the Yang-Mills theory coincides as a Poisson manifold with the moduli space of flat connections on a Riemann surface with L−V+1 handles and, therefore, with the physical phase space of the corresponding (2+1)-dimensional Chern-Simons model. Here, L and V are, respectively, the total number of links and vertices of the lattice. The deformation parameter γ is identified with 2π/k, where k is an integer appearing in the Chern-Simons action.

Center vortex model for the infrared sector of Yang–Mills theory — topological susceptibility