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Register a new userDynamical generalization of Yetters model based on a crossed module of discrete groups
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We construct a dynamical lattice model based on a crossed module of possibly non-abelian finite groups. Its degrees of freedom are defined on links and plaquettes, while gauge transformations are based on vertices and links of the underlying lattice. We specify the Hilbert space, define basic observables (including the Hamiltonian) and initiate a~discussion on the models phase diagram. The constructed model generalizes, and in appropriate limits reduces to, topological theories with symmetries described by groups and crossed modules, lattice Yang-Mills theory and $2$-form electrodynamics. We conclude by reviewing classifying spaces of crossed modules, with an emphasis on the direct relation between their geometry and properties of gauge theories under consideration.
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Prolongations of a group extension can be studied in a more general situation that we call group extensions of the co-type of a crossed module. Cohomology classification of such extensions is obtained by applying the obstruction theory of monoidal functors.
In this paper we state some applications of Gr-category theory on the classification of crossed modules and on the classification of extensions of groups of the type of a crossed module.
We show there exist UV-complete field-theoretic models in general dimension, including $2+1$, with the spontaneous breaking of a global symmetry, which persists to the arbitrarily high temperatures. Our example is a conformal vector model with the $O(N)times mathbb{Z}_2$ symmetry at zero temperature. Using conformal perturbation theory we establish $mathbb{Z}_2$ symmetry is broken at finite temperature for $N>10$. Similar to recent constructions, in the infinite $N$ limit our model has a non-trivial conformal manifold, a moduli space of vacua, which gets deformed at finite temperature. Furthermore, in this regime the model admits a persistent breaking of $O(N)$ in $2+1$ dimensions, therefore providing another example where the Coleman-Hohenberg-Mermin-Wagner theorem can be bypassed.
We construct lattice parafermions for the $Z(N)$ chiral Potts model in terms of quasi-local currents of the underlying quantum group. We show that the conservation of the quantum group currents leads to twisted discrete-holomorphicity (DH) conditions for the parafermions. At the critical Fateev-Zamolodchikov point the parafermions are the usual ones, and the DH conditions coincide with those found previously by Rajabpour and Cardy. Away from the critical point, we show that our twisted DH conditions can be understood as deformed lattice current conservation conditions for an underlying perturbed conformal field theory in both the general $Ngeq 3$ and $N=2$ Ising cases.
Based on the dAlembert-Lagrange-Poincar{e} variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincar{e}-Hamilton equations, and study a version of corresponding Poincar{e}-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincar{e} variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain line integral under the motion of a mechanical system of the type considered characterizes the Poincar{e}-Hamilton equations as underlying equations of the motion. As a special case, an invariant analogous to Poincar{e} linear integral invariant is obtained.
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