4D non-Abelian Gauge Theory Research Articles

Nilpotent (anti-)BRST and (anti-)co-BRST symmetries in 2D non-Abelian gauge theory: Some novel observations

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-) co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two [Formula: see text]-dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel.

Categorical Webs and S-Duality in 4d $${\mathcal {N}}$$ = 2 QFT

We review the categorical approach to the BPS sector of a 4d $$\mathcal {N}=2$$ QFT, clarifying many tricky issues and presenting a few novel results. To a given $$\mathcal {N}=2$$ QFT one associates several triangulated categories: they describe various kinds of BPS objects from different physical viewpoints (e.g. IR versus UV). These diverse categories are related by a web of exact functors expressing physical relations between the various objects/pictures. A basic theme of this review is the emphasis on the full web of categories, rather than on what we can learn from a single description. A second general theme is viewing the cluster category as a sort of ‘categorification’ of ’t Hooft’s theory of quantum phases for a 4d non-Abelian gauge theory. The S-duality group is best described as the auto-equivalences of the full web of categories. This viewpoint leads to a combinatorial algorithm to search for S-dualities of the given $$\mathcal {N}=2$$ theory. If the ranks of the gauge and flavor groups are not too big, the algorithm may be effectively run on a laptop. This viewpoint also leads to a clearer view of 3d mirror symmetry. For class $$\mathcal {S}$$ theories, all the relevant triangulated categories may also be constructed in terms of geometric objects on the Gaiotto curve, and we present the dictionary between triangulated categories and the WKB approach of GMN. We also review how the VEV’s of UV line operators are related to cluster characters.

An alternative to the horizontality condition in the superfield approach to BRST symmetries

We provide an alternative to the gauge covariant horizontality condition, which is responsible for the derivation of the nilpotent (anti-) BRST symmetry transformations for the gauge and (anti-) ghost fields of a (3+1)-dimensional (4D) interacting 1-form non-Abelian gauge theory in the framework of the usual superfield approach to the Becchi–Rouet–Stora–Tyutin (BRST) formalism. The above covariant horizontality condition is replaced by a gauge invariant restriction on the (4,2)-dimensional supermanifold, parameterised by a set of four spacetime coordinates, xμ(μ=0,1,2,3), and a pair of Grassmannian variables, θ and θ. The latter condition enables us to derive the nilpotent (anti-) BRST symmetry transformations for all the fields of an interacting 1-form 4D non-Abelian gauge theory in which there is an explicit coupling between the gauge field and the Dirac fields. The key differences and the striking similarities between the above two conditions are pointed out clearly.

Systematics of coupling flows in AdS backgrounds

We give an effective field theory derivation, based on the running of Planck brane gauge correlators, of the large logarithms that arise in the predictions for low energy gauge couplings in compactified ${\mathrm{AdS}}_{5}$ backgrounds, including the one-loop effects of bulk scalars, fermions, and gauge bosons. In contrast with the case of charged scalars coupled to Abelian gauge fields that has been considered previously in the literature, the one-loop corrections are not dominated by a single 4D Kaluza-Klein mode. Nevertheless, in the case of gauge field loops, the amplitudes can be reorganized into a leading logarithmic contribution that is identical to the running in 4D non-Abelian gauge theory, and a term which is not logarithmically enhanced and is analogous to a two-loop effect in 4D. In a warped grand unified theory (GUT) model broken by the Higgs mechanism in the bulk, we show that the matching scale that appears in the large logarithms induced by the non-Abelian gauge fields is ${m}_{\mathrm{XY}}^{2}/k$ where ${m}_{\mathrm{XY}}$ is the bulk mass of the XY bosons and k is the AdS curvature. This is in contrast with the UV scale in the logarithmic contributions of scalars, which is simply the bulk mass m. Our results are summarized in a set of simple rules that can be applied to compute the leading logarithmic predictions for coupling constant relations within a given warped GUT model. We present results for both boundary breaking of the GUT gauge group and bulk breaking through the Higgs mechanism.

Vortex condensation and mass gap generation in two-dimensional principal chiral models

We consider the two-dimensional $\mathrm{SU}(N)$ principal chiral model and discuss a vortex condensation mechanism which could explain the existence of a non-zero mass gap at arbitrarily small values of the coupling constant. The mechanism is an analogue of the vortex condensation mechanism of confinement in 4D non-Abelian gauge theories. We formulate a sufficient condition for the mass gap to be non-vanishing in terms of the behavior of the vortex free energy. The $\mathrm{SU}(2)$ model is studied in detail. In one dimension we calculate the vortex free energy exactly. An effective model for the center variables of the spin configurations of the 2D $\mathrm{SU}(2)$ model is proposed and the $Z(2)$ correlation function is derived in this model. We define a $Z(2)$ mass gap in both the full and effective model and argue that they should coincide whenever the genuine mass gap is non-zero. We show via Monte Carlo simulations of the $\mathrm{SU}(2)$ model that the $Z(2)$ mass gap reproduces the full mass gap with perfect accuracy. We also test this mechanism in the positive link model which is an analogue of the positive plaquette model in gauge theories and find excellent agreement between the full and the $Z(2)$ mass gap.

Renormalization and topological susceptibility on the lattice

We study the renormalization of the topological susceptibility, χ, when the density of topological charge is defined as a local operator on the lattice. In the 2D O(3) δ model we determine both multiplicative and additive renormalizations by heating configurations with a definite number of instantons. We find results that are consistent with perturbation theory. The method is also applicable to the physically interesting case of 4D non-abelian gauge theories.