Four-dimensional Yang-Mills Theory Research Articles

ASPECTS OF INTEGRABILITY IN $\mathcal{N} = 4$ SYM

Various recently developed connections between supersymmetric Yang–Mills theories in four dimensions and two-dimensional integrable systems serve as crucial ingredients in improving our understanding of the AdS/CFT correspondence. In this review, we highlight some connections between superconformal four-dimensional Yang–Mills theory and various integrable systems. In particular, we focus on the role of Yangian symmetries in studying the gauge theory dual of closed string excitations. We also briefly review how the gauge theory connects to Calogero models and open quantum spin chains through the study of the gauge theory duals of D3 branes and open strings ending on them. This invited review is based on a seminar given at the Institute of Advanced Study, Princeton.

Scaling phenomena in gravity from QCD

We present holographic arguments to predict properties of strongly coupled gravitational systems in terms of weakly coupled gauge theories. In particular we relate the latest computed value for the Choptuik critical exponent in black hole formation in five dimensions, γ5D=0.412±1%, to the saturation exponent of four-dimensional Yang–Mills theory in the Regge limit, γBFKL≃0.410.

Renormalizable parameters of the sine-Gordon model

The well-known phase structure of the two-dimensional sine-Gordon model is reconstructed by means of its renormalization group flow, the study of the sensitivity of the dynamics on microscopic parameters. Such an analysis resolves the apparent contradiction between the phase structure and the triviality of the effective potential in either phases, provides a case where usual classification of operators based on the linearization of the scaling relation around a fixed point is not available and shows that the Maxwell-cut generates an unusually strong universality at long distances. Possible analogies with four-dimensional Yang–Mills theories are mentioned, too.

Baryons from Instantons in Holographic QCD

We consider aspects of dynamical baryons in a holographic dual of QCD that is proposed on the basis of a D4/D8-brane configuration. We construct a soliton solution carrying a unit baryon number and show that it is given by an instanton solution of four-dimensional Yang-Mills theory with fixed size. The Chern-Simons term on the flavor D8-branes plays a crucial role of protecting the instanton from collapsing to zero size. By quantizing the collective coordinates of the soliton, we work out the baryon spectra. Negative-parity baryons as well as baryons with higher spins and isospins can be obtained in a simple manner.

Non-Abelian strings and axions

We address two distinct but related issues: (i) the impact of (two-dimensional) axions in a two-dimensional theory known to model confinement, the $CP(N\ensuremath{-}1)$ model; (ii) bulk axions in four-dimensional Yang-Mills theory supporting non-Abelian strings. In the first case $n$, $\overline{n}$ kinks play the role of ``quarks.'' They are known to be confined. We show that introduction of axions leads to deconfinement (at very large distances). This is akin to the phenomenon of wall liberation in four-dimensional Yang-Mills theory. In the second case we demonstrate that the bulk axion does not liberate confined (anti)monopoles, in contradistinction with the two-dimensional model. A novel physical effect which we observe is the axion radiation caused by monopole-antimonopole pairs attached to the non-Abelian strings.

Baryon Vertices in AdS Black Hole Backgrounds

We numerically computed the energy of QCD string junctions in non-supersymmetric SU(N) gauge theories as the energy of brane configurations in the background of AdS black hole solutions, and extracted the contribution of baryon vertices. We obtain a negative vertex energy for the three- and four-dimensional Yang-Mills theories and a vanishing vertex energy for the five-dimensional theory.

Asymptotically free improvement of Yang–Mills classical mechanics

We construct a classical mechanics Hamiltonian which exhibits spontaneous symmetry breaking akin the Coleman–Weinberg mechanism, dimensional transmutation, and asymptotically free self-similarity congruent with the beta-function of four-dimensional Yang–Mills theory. Its classical equations of motion support stable periodic orbits and in a three-dimensional projection these orbits are self-linked into topologically non-trivial knots.

Adrian Nicolae Patrascioiu

Adrian Nicolae Patrascioiu

ADE-quiver theories and mirror symmetry

We show that the Higgs branch of a four-dimensional Yang-Mills theory, with gauge and matter content summarised by an ADE quiver diagram, is identical to the generalised Coulomb branch of a four-dimensional superconformal strongly coupled gauge theory with ADE global symmetry. This equivalence suggests the existence of a mirror symmetry between the quiver theories and the strongly coupled theories.

Noncommutative Gauge Theories on Fuzzy Sphere and Fuzzy Torus from Matrix Model

We consider a reduced model of four-dimensional Yang-Mills theory with a mass term. This matrix model has two classical solutions, a two-dimensional fuzzy sphere and a two-dimensional fuzzy torus. These classical solutions are constructed by embedding them into three or four dimensional flat space. They exist for finite size matrices. then, the number of quanta on these manifolds is finite. Noncommutative gauge theories on these noncommutative manifolds are derived by expanding the model around these classical solutions and studied by taking two large N limits, the commutative limit and the large radius limit. The behaviors of gauge invariant operators are also discussed.

E8 quiver gauge theory and mirror symmetry

We show that the Higgs branch of a four-dimensional Yang-Mills theory, with gauge and matter content summarised by an E_8 quiver diagram, is identical to the generalised Coulomb branch of a four-dimensional superconformal strongly coupled gauge theory with E_8 global symmetry. This is the final step in showing that there is a Higgs-Coulomb identity of this kind for each of the cases {0}, A_1, A_2, D_4, E_6, E_7 and E_8. This series of equivalences suggests the existence of a mirror symmetry between the quiver theories and the strongly coupled theories. We also discuss how to interpret the parameters of the quiver gauge theory in terms of the Hanany-Witten picture.

Zero curvature formalism of the four-dimensional Yang–Mills theory in superspace

The supersymmetric descent equations in N=1 superspace are discussed by means of the introduction of two operators ζα,ζ̄α̇, which allow to decompose the supersymmetric covariant derivatives Dα,D̄α̇ as BRS commutators.

Scale invariance of Dirac condition gegm=1 in type 0 string approach to gauge theory

In this letter we shall discuss a description of non-supersymmetric four-dimensional Yang–Mills theory based on Type 0 strings recently proposed by Klebanov and Tseytlin. The three brane near-horizon geometry allows one to study the UV behaviour of the gauge theory. Following Minahan and Klebanov and Tseytlin we shall discuss how the gravity solution reproduces logarithmic renormalization of coupling constant g e extracted from quark-antiquark potential and then show that effective coupling constant g m describing monopole-antimonopole interactions is of zero-charge type and Dirac condition g e g m =1 is scale invariant in logarithmic approximation.

Yang-Mills theory as a deformation of topological field theory, dimensional reduction, and quark confinement

We propose a reformulation of Yang-Mills theory as a perturbative deformation of a novel topological (quantum) field theory. We prove that this reformulation of four-dimensional QCD leads to quark confinement in the sense of an area law of the Wilson loop. First, Yang-Mills theory with a non-Abelian gauge group G is reformulated as a deformation of a novel topological field theory. Next, a special class of topological field theories is defined by both Becchi-Rouet-Stora-Tyupin (BRST) and anti-BRST exact actions corresponding to the maximal Abelian gauge leaving the maximal torus group Hof G invariant. Then we find topological field theory $(D>2)$ has a hidden supersymmetry for a choice of maximal Abelian gauge. As a result, the D-dimensional topological field theory is equivalent to the $(D\ensuremath{-}2)$-dimensional coset $G/H$ nonlinear sigma model in the sense of the Parisi-Sourlas dimensional reduction. After maximal Abelian gauge fixing, the topological property of the magnetic monopole and antimonopole of four-dimensional Yang-Mills theory is translated into that of an instanton and anti-instanton in a two-dimensional equivalent model. It is shown that the linear static potential in four dimensions follows from the instanton--anti-instanton gas in the equivalent two-dimensional nonlinear sigma model obtained from the four-dimensional topological field theory by dimensional reduction, while the remaining Coulomb potential comes from the perturbative part in four-dimensional Yang-Mills theory. The dimensional reduction opens a path for applying various exact methods developed in two-dimensional quantum field theory to study the nonperturbative problem in low-energy physics of four-dimensional quantum field theories.

Four-Dimensional Yang-Mills Theory as a Deformation of Topological BF Theory

The classical action for pure Yang–Mills gauge theory can be formulated as a deformation of the topological BF theory where, beside the two-form field B, one has to add one extra-field η given by a one-form which transforms as the difference of two connections. The ensuing action functional gives a theory that is both classically and quantistically equivalent to the original Yang–Mills theory. In order to prove such an equivalence, it is shown that the dependency on the field η can be gauged away completely. This gives rise to a field theory that, for this reason, can be considered as semi-topological or topological in some but not all the fields of the theory. The symmetry group involved in this theory is an affine extension of the tangent gauge group acting on the tangent bundle of the space of connections. A mathematical analysis of this group action and of the relevant BRST complex is discussed in detail.

Renormalization of the Topological Charge in Yang–Mills Theory

The conditions leading to a nontrivial renormalization of the topological charge in four-dimensional Yang–Mills theory are discussed. It is shown that if the topological term is regarded as the limit of a certain nontopological interaction, quantum effects due to the gauge bosons lead to a finite multiplicative renormalization of the θ-parameter while fermions give rise to an additional shift of θ. A truncated form of an exact renormalization group equation is used to study the scale dependence of the θ-parameter. Possible implications for the strong CP-problem of QCD are discussed.

SUPER-LAX OPERATOR EMBEDDED IN SELF-DUAL SUPERSYMMETRIC YANG-MILLS THEORY

We show that the super-Lax operator for N=1 supersymmetric Kadomtsev-Petviashvili (SKP) equation of Manin and Radul in three dimensions can be embedded into recently developed self-dual supersymmetric Yang-Mills theory in 2+2 dimensions, based on general features of its underlying super-Lax equation. The whole hierarchy of the SKP equations of Manin and Radul is generated by geometrical superfield equations of self-dual supersymmetric Yang-Mills theory. The differential geometrical relationship in superspace between the embedding principle of the super-Lax operator and its associated super-Sato equation is clarified. This result provides a good guiding principle for the embedding of other integrable subsystems in the super-Lax equation into the four-dimensional self-dual supersymmetric Yang-Mills theory, which is the consistent background for N=2 superstring theory, and potentially generates other unknown supersymmetric integrable models in lower dimensions.

Infinite conformal algebras in supersymmetric theories on four-manifolds

We study a supersymmetric theory twisted on a Kähler four-manifold M = Σ 1 × Σ 2, where Σ 1,2 are two-dimensional Riemann surfaces. We demonstrate that it possesses a “left-moving” conformal stress tensor on Σ 1 ( Σ 2) in a BRST cohomology, which generates the Virasoro algebra with the conventional commutation relations. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic χ of the Σ 2 ( Σ 1) surface. It is shown that this construction can be extended to include a realization of a Kac-Moody algebra in BRST cohomology with a level proportional to the Euler characteristic χ. This structure is shown to be invariant under renormalization group. A representation of the algebra W 1+∞ in terms of a free chiral supermultiplet is also given. We discuss the role of instantons and a possible relation between the dynamics of four-dimensional Yang-Mills theories and those of two-dimensional sigma models.

Continuum strong-coupling expansion of Yang-Mills theory: quark confinement and infra-red slavery

We solve Schrödinger's equation for the ground-state of four-dimensional Yang-Mills theory as an expansion in inverse powers of the coupling. Expectation values computed with the leading-order approximation are reduced to a calculation in two-dimensional Yang-Mills theory which is known to confine. Consequently the Wilson loop in the four-dimensional theory obeys an area law to leading order and the coupling becomes infinite as the mass scale goes to zero.

Affine Yang-Mills theory and affine self-dual Chern-Simons Solitons

A four-dimensional affine Yang-Mills theory, i.e. Yang—Mills gauge theory with values in an affine Kac-Moody algebra, is constructed which can give rise to a spontaneous breakdown of the affine symmetry. The affine self-dual Yang-Mills equation (which is a special kind of affine YM theory) in four dimensions is dimensionally reduced to the affine self-dual Chem-Simons equation in two dimensions. The latter is shown to have soliton solutions which satisfy the conformal affine Toda equations.