scholarly journals Dynamics of a lattice 2-group gauge theory model

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
A. Bochniak ◽  
L. Hadasz ◽  
P. Korcyl ◽  
B. Ruba
Keyword(s):  
Degrees Of Freedom ◽  
Local Symmetry ◽  
Theory Model ◽  
Factorization Theorem ◽  
Quantum Field Theories ◽  
Crossed Module ◽  
A Value ◽  
Topological Quantum ◽  
Local Observables ◽  
Dynamical Degrees

Abstract We study a simple lattice model with local symmetry, whose construction is based on a crossed module of finite groups. Its dynamical degrees of freedom are associated both to links and faces of a four-dimensional lattice. In special limits the discussed model reduces to certain known topological quantum field theories. In this work we focus on its dynamics, which we study both analytically and using Monte Carlo simulations. We prove a factorization theorem which reduces computation of correlation functions of local observables to known, simpler models. This, combined with standard Krammers-Wannier type dualities, allows us to propose a detailed phase diagram, which form is then confirmed in numerical simulations. We describe also topological charges present in the model, its symmetries and symmetry breaking patterns. The corresponding order parameters are the Polyakov loop and its generalization, which we call a Polyakov surface. The latter is particularly interesting, as it is beyond the scope of the factorization theorem. As shown by the numerical results, expectation value of Polyakov surface may serve to detects all phase transitions and is sensitive to a value of the topological charge.

scholarly journals Dynamical generalization of Yetter’s model based on a crossed module of discrete groups

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Arkadiusz Bochniak ◽  
Leszek Hadasz ◽  
Błażej Ruba
Keyword(s):  
Degrees Of Freedom ◽  
Discrete Groups ◽  
Quantum Field Theories ◽  
Crossed Module ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Model Based ◽  
Topological Quantum ◽  
Crossed Modules ◽  
Topological Theories

Abstract We construct a lattice model based on a crossed module of possibly non-abelian finite groups. It generalizes known topological quantum field theories, but in contrast to these models admits local physical excitations. 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 model’s phase diagram. The constructed model reduces in appropriate limits 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.

Dynamical degrees of freedom of constrained quantum field theories (?-model)

1985 ◽  
Vol 28 (4) ◽  
pp. 587-589
Author(s):  
F. Alonso ◽  
J. Julve ◽  
A. Tiemblo ◽  
R. Tresguerres
Keyword(s):  
Degrees Of Freedom ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Dynamical Degrees

scholarly journals Domain walls in 4d $$ \mathcal{N} $$ = 1 SYM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Diego Delmastro ◽  
Jaume Gomis
Keyword(s):  
Domain Wall ◽  
Domain Walls ◽  
Simons Theory ◽  
Partition Functions ◽  
Quantum Field Theories ◽  
Yang Mills ◽  
Stable Domain ◽  
Coxeter Number ◽  
Topological Quantum ◽  
Simply Connected

Abstract 4d$$ \mathcal{N} $$ N = 1 super Yang-Mills (SYM) with simply connected gauge group G has h gapped vacua arising from the spontaneously broken discrete R-symmetry, where h is the dual Coxeter number of G. Therefore, the theory admits stable domain walls interpolating between any two vacua, but it is a nonperturbative problem to determine the low energy theory on the domain wall. We put forward an explicit answer to this question for all the domain walls for G = SU(N), Sp(N), Spin(N) and G2, and for the minimal domain wall connecting neighboring vacua for arbitrary G. We propose that the domain wall theories support specific nontrivial topological quantum field theories (TQFTs), which include the Chern-Simons theory proposed long ago by Acharya-Vafa for SU(N). We provide nontrivial evidence for our proposals by exactly matching renormalization group invariant partition functions twisted by global symmetries of SYM computed in the ultraviolet with those computed in our proposed infrared TQFTs. A crucial element in this matching is constructing the Hilbert space of spin TQFTs, that is, theories that depend on the spin structure of spacetime and admit fermionic states — a subject we delve into in some detail.

GENERAL COVARIANCE, TOPOLOGICAL QUANTUM FIELD THEORIES AND FRACTIONAL STATISTICS

1992 ◽  
Vol 07 (02) ◽  
pp. 209-234 ◽  
Author(s):  
J. GAMBOA
Keyword(s):  
Hamiltonian Formalism ◽  
Field Theories ◽  
General Covariance ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Fractional Statistics ◽  
Multiply Connected ◽  
Topological Quantum ◽  
Gauge Fixing ◽  
Topological Quantum Field Theories

Topological quantum field theories and fractional statistics are both defined in multiply connected manifolds. We study the relationship between both theories in 2 + 1 dimensions and we show that, due to the multiply-connected character of the manifold, the propagator for any quantum (field) theory always contains a first order pole that can be identified with a physical excitation with fractional spin. The article starts by reviewing the definition of general covariance in the Hamiltonian formalism, the gauge-fixing problem and the quantization following the lines of Batalin, Fradkin and Vilkovisky. The BRST–BFV quantization is reviewed in order to understand the topological approach proposed here.

scholarly journals ON ALGEBRAIC STRUCTURES IMPLICIT IN TOPOLOGICAL QUANTUM FIELD THEORIES

1999 ◽  
Vol 08 (02) ◽  
pp. 125-163 ◽  
Author(s):  
Louis Crane ◽  
David Yetter
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Hopf Algebra ◽  
Tensor Category ◽  
Topological Quantum Field Theory ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Topological Quantum ◽  
Natural Way

We show that any 3D topological quantum field theory satisfying physically reasonable factorizability conditions has associated to it in a natural way a Hopf algebra object in a suitable tensor category. We also show that all objects in the tensor category have the structure of left-left crossed bimodules over the Hopf algebra object. For 4D factorizable topological quantum filed theories, we provide by analogous methods a construction of a Hopf algebra category.

scholarly journals Towards an Explicit Construction of Local Observables in Integrable Quantum Field Theories

2019 ◽  
Vol 20 (12) ◽  
pp. 3889-3926
Author(s):  
Henning Bostelmann ◽  
Daniela Cadamuro
Keyword(s):  
Bound States ◽  
Von Neumann Algebras ◽  
Form Factors ◽  
Explicit Construction ◽  
Local Fields ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Von Neumann ◽  
Local Observables ◽  
Wightman Axioms

Abstract We present a new viewpoint on the construction of pointlike local fields in integrable models of quantum field theory. As usual, we define these local observables by their form factors; but rather than exhibiting their n-point functions and verifying the Wightman axioms, we aim to establish them as closed operators affiliated with a net of local von Neumann algebras, which is defined indirectly via wedge-local quantities. We also investigate whether these fields have the Reeh–Schlieder property, and in which sense they generate the net of algebras. Our investigation focuses on scalar models without bound states. We establish sufficient criteria for the existence of averaged fields as closable operators, and complete the construction in the specific case of the massive Ising model.

TWO-DIMENSIONAL TOPOLOGICAL QUANTUM FIELD THEORIES AND FROBENIUS ALGEBRAS

1996 ◽  
Vol 05 (05) ◽  
pp. 569-587 ◽  
Author(s):  
LOWELL ABRAMS
Keyword(s):  
Field Theories ◽  
Algebra Structure ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Two Dimensional ◽  
Direct Sums ◽  
Frobenius Algebras ◽  
Topological Quantum ◽  
Simple Demonstration ◽  
Topological Quantum Field Theories

We characterize Frobenius algebras A as algebras having a comultiplication which is a map of A-modules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either “annihilator algebras” — algebras whose socle is a principal ideal — or field extensions. The relationship between two-dimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable two-dimensional topological quantum field theories.

Towards a Virtual Laboratory for Grain Boundaries and Dislocations

2009 ◽  
Vol 1224 ◽  
Author(s):  
Sebastián Echeverri Restrepo ◽  
Barend J. Thijsse
Keyword(s):  
Molecular Dynamics ◽  
Grain Boundaries ◽  
Systematic Study ◽  
Degrees Of Freedom ◽  
Minimum Energy ◽  
Local Symmetry ◽  
Virtual Laboratory ◽  
Face Centered Cubic ◽  
Face Centered ◽  
Edge Dislocations

AbstractIn order to perform a systematic study of the interaction between grain boundaries (GBs) and dislocations using molecular dynamics (MD), several tools need to be available. A combination of computational geometry and MD was used to build the foundations of what we call a virtual laboratory. First, an algorithm to generate GBs on face-centered cubic bicrystals was developed. Two crystals with different orientations are placed together. Then, by applying “microscopic” rigid body translations along the GB plane to one of the crystals and removing overlapping atoms, a set of initial configurations is sampled and a minimum energy configuration is found. Second, to classify the geometry of the GBs a local symmetry type (LST) describing the angular environment of each atom is calculated. It is found that for a given relaxed GB the number of atoms with different LSTs is not very large and that it is possible to find unique geometrical patterns in each GB. For instance, the LSTs of two GBs having the same “macroscopic” configuration but different “microscopic” degrees of freedom can be dissimilar: the configurations with higher GB energy tend to have a higher number of atoms with different LSTs. Third, edge dislocations are introduced into the bicrystals. We see that full edge dislocations split into Shockley partials. Finally, by loading the bicrystals with tensile stresses the edge dislocations are put into motion. Various examples of dislocation-GB interactions in Cu are presented.

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