Non-Abelian topological mass generation in four dimensions

Topological mass generation in non-Abelian gauge theory—geometric BRST quantization approach

The Becchi–Rouet–Stora–Tyutin (BRST) quantization of a gauge theory in noncommutative geometry is carried out in the ‘‘matrix derivative’’ approach. BRST/anti-BRST transformation rules are obtained by applying the horizontality condition, in the superconnection formalism. A BRST/anti-BRST invariant quantum action is then constructed, using an adaptation of the method devised by Baulieu and Thierry-Mieg for the Yang–Mills case. The resulting quantum action turns out to be the same as that of a gauge theory in the ’t Hooft gauge with spontaneously broken symmetry. Our result shows that only the even part of the supergroup acts as a gauge symmetry, while the odd part effectively provides a global symmetry. We treat the general formalism first, then work out the SU(2/1) and SU(2/2) cases explicitly.

A dynamical non-Abelian two-form potential gives masses to vector bosons via a topological coupling.1 Unlike in the Abelian case, the two-form cannot be dualized to Goldstone bosons. Duality is restored by coupling a flat connection to the theory in a particular way, and the new action is then dualized to a nonlinear sigma model. The presence of the flat connection is crucial, which saves the original mechanism of Higgs-free topological mass generation from being dualized to a sigma model.

Providing a new perspective on quantum field theory, this book gives a pedagogical and up-to-date exposition of non-perturbative methods in relativistic quantum field theory and introduces the reader to modern research work in theoretical physics. It describes in detail non-perturbative methods in quantum field theory, and explores two- dimensional and four- dimensional gauge dynamics using those methods. The book concludes with a summary emphasizing the interplay between two- and four- dimensional gauge theories. Aimed at graduate students and researchers, this book covers topics from two-dimensional conformal symmetry, affine Lie algebras, solitons, integrable models, bosonization, and 't Hooft model, to four-dimensional conformal invariance, integrability, large N expansion, Skyrme model, monopoles and instantons. Applications, first to simple field theories and gauge dynamics in two dimensions, and then to gauge theories in four dimensions and quantum chromodynamics (QCD) in particular, are thoroughly described.

The 4-dimensional model with topological mass generation that has recently been presented by Dvali, Jackiw and Pi [G. Dvali, R. Jackiw, and S.-Y. Pi, Phys. Rev. Lett. 96, 081602 (2006), hep-th/0610228] is generalized to any even number of dimensions. As in the 4-dimensional model, the 2n-dimensional model describes a mass-generation phenomenon due to the presence of the chiral anomaly. In addition to this model, new 2n-dimensional models with topological mass generation are proposed, in which a Stueckelberg-type mass term plays a crucial role in the mass generation. The mass generation of a pseudoscalar field such as the eta-prime meson is discussed within this framework.

A massive B∧F theory in (3+1) dimensions with a Kalb-Ramond field isdimensionally reduced, giving rise to two different topological terms, oneinvolving a scalar and a Kalb-Ramond field and another one equivalent tothe four-dimensional B∧F term. We constructed two actions with thesetopological terms and show that a topological mass generation mechanism canbe implemented. Using the non–Chern-Simons topological term, an action isproposed leading to a classical duality relation between Klein-Gordon andMaxwell actions. We also show that an action in (2+1) dimensions with theKalb-Ramond field is related by Buscher's duality transformation to amassive gauge-invariant Stükelberg-type theory.

Toroidal compactification in string theory from Chern–Simons theory

We present a gauge and Lorentz invariant effective field theory model for the interaction of a charged scalar matter field with a magnetic monopole source, described by an external magnetic current. The quantum fluctuations of the monopole field are described effectively by a strongly coupled “dual” Ud(1) gauge field, which is independent of the electromagnetic Uem(1) gauge field. The effective interactions of the charged matter with the monopole source are described by a gauge invariant mixed Chern-Simons-like (Pontryagin-density) term between the two U(1) gauge fields. The latter interaction coupling is left free, and a lattice study of the system is performed with the aim of determining the phase structure of this effective theory. Our study shows that, in the spontaneously broken-symmetry phase, the monopole source triggers, via the mixed Chern-Simons term, which is nontrivial in its presence, the generation of a dynamical singular configuration (magnetic-monopolelike) for the respective gauge fields. The scalar field also behaves in the broken phase in a way similar to that of the scalar sector of the ’t Hooft-Polyakov monopole. Moreover, we show that the modest size of the lattices involved does not have significant effects on the main conclusions of our analysis. Published by the American Physical Society 2024

We consider SU(2) gauge potentials over a space with a compactified\ndimension. A non-Abelian Fourier transform of the gauge potential in the\ncompactified dimension is defined in such a way that the Fourier coefficients\nare (almost) gauge invariant. The functional measure and the gauge field\nstrengths are expressed in terms of these Fourier coefficients. The emerging\nformulation of the non-Abelian gauge theory turns out to be an Abelian gauge\ntheory of a set of fields defined over the initial space with the compactified\ndimension excluded. The Abelian theory contains an Abelian gauge field, a\nscalar field, and an infinite tower of vector matter fields, some of which\ncarry Abelian charges. Possible applications of this formalism are discussed\nbriefly.\n

The exact results for the topological susceptibility and the mean value of the topological charge density in (1+1)-dimensional lattice gauge theory are obtained by means of the Hamiltonian method.

We study $D$-dimensional gauge theory with an extra dimension of a circle at finite temperature. We mainly focus on the expectation value of the gauge field for the direction of the extra dimension, which is the order parameter of the gauge symmetry breaking. We evaluate the effective potential in the one-loop approximation at finite temperature. We show that the vacuum configuration of the theory at finite temperature is determined by a ($D\ensuremath{-}1$)-dimensional gauge theory defined by removing the Euclidean time coordinate as well as all of the fermions from the original $D$-dimensional gauge theory on the circle. It is pointed out that gauge symmetry nonrestoration and/or inverse symmetry breaking can occur at high temperature in a class of gauge theories on circles and that phase transitions (if they occur) are, in general, expected to be first order.

We discuss reductions of general N=1 four dimensional gauge theories on S^2. The effective two dimensional theory one obtains depends on the details of the coupling of the theory to background fields, which can be translated to a choice of R-symmetry. We argue that, for special choices of R-symmetry, the resulting two dimensional theory has a natural interpretation as an N=(0,2) gauge theory. As an application of our general observations, we discuss reductions of N=1 and N=2 dualities and argue that they imply certain two dimensional dualities.

We review the non-anticommutative Q-deformations of $$ \mathcal{N} $$ = (1, 1) supersymmetric theories in four-dimensional Euclidean harmonic superspace. These deformations preserve chirality and harmonic Grassmann analyticity. The associated field theories arise as a low-energy limit of string theory in specific backgrounds and generalize the Moyal-deformed supersymmetric field theories. A characteristic feature of the Q-deformed theories is the half-breaking of supersymmetry in the chiral sector of the Euclidean superspace. Our main focus is on the chiral singlet Q-deformation, which is distinguished by preserving the SO(4) ∼ Spin(4) “Lorentz” symmetry and the SU(2) R-symmetry. We present the superfield and component structures of the deformed $$ \mathcal{N} $$ = (1, 0) supersymmetric gauge theory as well as of hypermultiplets coupled to a gauge superfield: invariant actions, deformed transformation rules, and so on. We discuss quantum aspects of these models and prove their renormalizability in the Abelian case. For the charged hypermultiplet in an Abelian gauge superfield background we construct the deformed holomorphic effective action.

A toy model is proposed for four dimensional non-abelian gauge theories coupled to a large number of fermionic degrees of freedom. As the number of flavors is varied the gauge theory may be confining, walking or conformal. The toy model mimicking this feature is the two dimensional O(3) sigma model with a theta-term. For all theta the model is asymptotically free. For small theta the model is confining in the infra red, for theta = pi the model has a non-trivial infra red fixed point and consequently for theta slightly below pi the coupling walks. The first step in investigating the notoriously difficult systematic effects of the gauge theory in the toy model is to establish non-perturbatively that the theta parameter is actually a relevant coupling. This is done by showing that there exist quantities that are entirely given by the total topological charge and are well defined in the continuum limit and are non-zero, despite the fact that the topological susceptibility is divergent. More precisely it is established that the differences of connected correlation functions of the topological charge (the cumulants) are finite and non-zero and consequently there is only a single divergent parameter in Z(theta) but otherwise it is finite. This divergent constant can be removed by an appropriate counter term rendering the theory completely finite even at theta > 0.

Hartle's generalized quantum mechanics in the sum-over-histories formalism is used to describe a non-Abelian gauge theory. Predictions are made for certain alternatives, with particular attention given to coarse grainings involving the constraint. In this way, the theory is compared to other quantum-mechanical descriptions of gauge theories in which the constraints are imposed by hand. The vanishing of the momentum space constraint is seen to hold, both through a simple formal argument and via a more careful description of the Lorentzian path integral as defined on a spacetime lattice. The configuration space realization of the constraint is shown to behave in a more complicated fashion. For some coarse grainings, we recover the known result from an Abelian theory, that coarse grainings by values of the constraint either predict its vanishing or fail to decohere. However, sets of alternatives defined in terms of a more complicated quantity in the Abelian case are exhibited where definite predictions can be made which disagree with the assumption that the constraints vanish. Finally, the configuration space sum-over-histories theory is exhibited in a manifestly Lorentz-invariant formulation.