Study of a lattice 2-group gauge model Bochniak, Arkadiusz; Hadasz, Leszek; Korcyl, Piotr ...
arXiv (Cornell University),
09/2021
Paper, Journal Article
Open access
Gauge theories admit a generalisation in which the gauge group is replaced by a finer algebraic structure, known as a 2-group. The first model of this type is a Topological Quantum Field Theory ...introduced by Yetter. We discuss a common generalisation of both the Yetter's model and Yang-Mills theory and in particular we focus on the lattice formulation of such model for finite 2-groups. In the second part we present a particular realization based on a 2-group constructed from \(\mathbb Z_4\) groups. In the selected model, independent degrees of freedom are associated to both links and faces of a four-dimensional lattice and are subject to a certain constraint. We present the details of this construction, discuss the expected dynamics in different regions of phase space and show numerical results from Monte Carlo simulations corroborating these expectations.
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.
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 model's 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.
Many research programs aiming to deal with the sign problem were proposed since the advent of lattice field theory. Several of these try to achieve this by exploiting properties of analytic ...functions. This is also the case for our study. There auxiliary complex variables are introduced and desired weight is obtained after integrating them out. In this note we clarify conceptual difficulties with this procedure encountered in previous works. In the process we observe an exciting connection with thimbles and discover an interesting hidden symmetry present in the problem. Problem of negative eigenvalues of the action will be revisited and considered from a different perspective. As a byproduct we perform simulations of simple quantum systems directly in Minkowski time.
The problem of finding a positive distribution, which corresponds to a given complex density, is studied. By the requirement that the moments of the positive distribution and of the complex density ...are equal, one can reduce the problem to solving the matching conditions. These conditions are a set of quadratic equations, thus Groebner basis method was used to find its solutions when it is restricted to a few lowest-order moments. For a Gaussian complex density, these approximate solutions are compared with the exact solution, that is known in this special case.
We compute amplitudes for the process $g^* g^* \to q \overline q V^*$ (two
virtual gluons into a quark, antiquark and a boson) at the tree level using the
spinor-helicity formalism. The resulting ...analytic expressions are much shorter
than squared amplitudes obtained using trace methods. Our results can be used
to expedite numerical calculations in phenomenological studies of the Drell-Yan
process in high energy factorization framework.
We construct highest weight vectors of ${\widehat{\mathfrak{sl}_2}}_{,k+1}
\oplus \mathsf{Vir}$ in tensor products of highest weight modules of
${\widehat{\mathfrak{sl}_2}}_{,k}$ and ...${\widehat{\mathfrak{sl}_2}}_{,1}$, and
thus for generic weights we find the decomposition of the tensor product into
irreducibles of ${\widehat{\mathfrak{sl}_2}}_{,k+1} \oplus \mathsf{Vir}$. The
construction uses Wakimoto representations of
${\widehat{\mathfrak{sl}_2}}_{,k}$, but the obtained vectors can be mapped back
to Verma modules. Singularities of this mapping are cancelled by a
renormalization. A detailed study of ``degenerations'' of Wakimoto modules
allowed to find the renormalization factor explicitly. The obtained result is a
significant step forward in a proof of equivalence of certain two-dimesnional
CFT models.
We study translationally invariant Pauli stabilizer codes with qudits of arbitrary, not necessarily uniform, dimensions. Using homological methods, we define a series of invariants called charge ...modules. We describe their properties and physical meaning. The most complete results are obtained for codes whose charge modules have Krull dimension zero. This condition is interpreted as mobility of excitations. We show that it is always satisfied for translation invariant 2D codes with unique ground state in infinite volume, which was previously known only in the case of uniform, prime qudit dimension. For codes all of whose excitations are mobile we construct a \(p\)-dimensional excitation and a \((D-p-1)\)-form symmetry for every element of the \(p\)-th charge module. Moreover, we define a braiding pairing between charge modules in complementary degrees. We discuss examples which illustrate how charge modules and braiding can be computed in practice.
We compute amplitudes for the process \(g^* g^* \to q \overline q V^*\) (two virtual gluons into a quark, antiquark and a boson) at the tree level using the spinor-helicity formalism. The resulting ...analytic expressions are much shorter than squared amplitudes obtained using trace methods. Our results can be used to expedite numerical calculations in phenomenological studies of the Drell-Yan process in high energy factorization framework.
We construct highest weight vectors of \({\widehat{\mathfrak{sl}_2}}_{,k+1} \oplus \mathsf{Vir}\) in tensor products of highest weight modules of \({\widehat{\mathfrak{sl}_2}}_{,k}\) and ...\({\widehat{\mathfrak{sl}_2}}_{,1}\), and thus for generic weights we find the decomposition of the tensor product into irreducibles of \({\widehat{\mathfrak{sl}_2}}_{,k+1} \oplus \mathsf{Vir}\). The construction uses Wakimoto representations of \({\widehat{\mathfrak{sl}_2}}_{,k}\), but the obtained vectors can be mapped back to Verma modules. Singularities of this mapping are cancelled by a renormalization. A detailed study of ``degenerations'' of Wakimoto modules allowed to find the renormalization factor explicitly. The obtained result is a significant step forward in a proof of equivalence of certain two-dimesnional CFT models.
