Four-dimensional Gauge Research Articles

Gauge-Higgs unification in spontaneously created fuzzy extra dimensions

We propose gauge-Higgs unification in fuzzy extra dimensions as a possible solution to the Higgs naturalness problem. In our approach, the fuzzy extra dimensions are created spontaneously as a vacuum solution of certain four-dimensional gauge theory. As an example, we construct a model which has a fuzzy torus as its vacuum. The Higgs field in our model is associated with the Wilson loop wrapped on the fuzzy torus. We show that the quadratic divergence in the mass of the Higgs field in the one-loop effective potential is absent. We then argue based on symmetries that the quantum corrections to the Higgs mass is suppressed including all loop contributions. We also consider a realization on the worldvolume theory of D3-branes probing $C^3/(Z_N \times Z_N)$ orbifold with discrete torsion.

SUPERGRAVITY DUAL OF THE SUPERCONFORMAL ANOMALY

The supergravity dual of the superconformal anomaly multiplet in a four-dimensional supersymmetric gauge theory is investigated. We consider a well-established dual correspondence between an [Formula: see text]SU(N+M) × SU(N) supersymmetric gauge theory and type IIB superstring in a space–time background described by the Klebanov–Strassler solution. Based on the fact that fractional D3-branes lead to superconformal anomaly on the field theory side and in the meantime deform AdS 5 × T1, 1 space–time background on the gravity side, we observe the five-dimensional gauged supergravity yielded from the spontaneous compactification on the deformed T1, 1, and find that the spontaneous breaking of local symmetries and the consequent super-Higgs effect in the gauged AdS5 supergravity should be the dual of the superconformal anomaly of the four-dimensional supersymmetric gauge theory.

Review of AdS/CFT Integrability, Chapter IV.4: Integrability in QCD and $${\fancyscript{N} <4 }$$ SYM

There is a growing amount of evidence that QCD (and four-dimensional gauge theories in general) possesses a hidden symmetry which does not exhibit itself as a symmetry of classical Lagrangians but is only revealed on the quantum level. In this review we consider the scale dependence of local gauge invariant operators and high-energy (Regge) behavior of scattering amplitudes to explain that the effective QCD dynamics in both cases is described by completely integrable systems that prove to be related to the celebrated Heisenberg spin chain and its generalizations.

Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries

We review the role of integrability in the planar spectral problem of four-dimensional superconformal gauge theories besides N=4 SYM. The cases considered include the Leigh-Strassler marginal deformations of N=4 SYM, quiver theories which arise as orbifolds of AdS5xS5 on the dual gravity side, as well as various theories involving open spin chains.

Surface operator, bubbling Calabi-Yau and AGT relation

Surface operators in N=2 four-dimensional gauge theories are interesting half-BPS objects. These operators inherit the connection of gauge theory with the Liouville conformal field theory, which was discovered by Alday, Gaiotto and Tachikawa. Moreover it has been proposed that toric branes in the A-model topological strings lead to surface operators via the geometric engineering. We analyze the surface operators by making good use of topological string theory. Starting from this point of view, we propose that the wave-function behavior of the topological open string amplitudes geometrically engineers the surface operator partition functions and the Gaiotto curves of corresponding gauge theories. We then study a peculiar feature that the surface operator corresponds to the insertion of the degenerate fields in the conformal field theory side. We show that this aspect can be realized as the geometric transition in topological string theory, and the insertion of a surface operator leads to the bubbling of the toric Calabi-Yau geometry.

On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture

In their recent paper \cite{Alday:2009aq} Alday, Gaiotto and Tachikawa proposed a relation between $\mathcal{N}=2$ four-dimensional supersymmetric gauge theories and two-dimensional conformal field theories. As part of their conjecture they gave an explicit combinatorial formula for the expansion of the conformal blocks inspired from the exact form of instanton part of the Nekrasov partition function. In this paper we study the origin of such an expansion from a CFT point of view. We consider the algebra $\mathcal{A}=\text{\sf Vir}\otimes\mathcal{H}$ which is the tensor product of mutually commuting Virasoro and Heisenberg algebras and discover the special orthogonal basis of states in the highest weight representations of $\mathcal{A}$. The matrix elements of primary fields in this basis have a very simple factorized form and coincide with the function called $Z_{\text{\sf{bif}}}$ appearing in the instanton counting literature. Having such a simple basis, the problem of computation of the conformal blocks simplifies drastically and can be shown to lead to the expansion proposed in \cite{Alday:2009aq}. We found that this basis diagonalizes an infinite system of commuting Integrals of Motion related to Benjamin-Ono integrable hierarchy.

Microscopic structure of magnetic bions

Magnetic bions — stable bound states of monopoles and twisted (“Kaluza-Klein”) anti-monopoles, carrying two units of magnetic charge — have been shown to be the leading cause of confinement and mass gap in four-dimensional gauge theories with massless adjoint fermions compactified on \( {\mathbb{R}^{1,2}} \times {\mathbb{S}^1} \), at least at small \( {\mathbb{S}^1} \). In this paper, we study in detail the bion mechanism and the scales involved for an SU(2) gauge group, using traditional QCD instanton methods. We represent the vacuum functional as the partition function of a bion-anti-bion plasma and obtain the next-to-leading dependence of the mass gap on the \( {\mathbb{S}^1} \) size L at fixed strong-coupling scale Λ. We find that, at small ΛL, the mass gap is an increasing function of L for theories with four massless adjoint Weyl fermions, a case left undetermined by the previous leading-order analysis, and comment on the approach to \( {\mathbb{R}^4} \).

On Type IIA geometries dual to [formula omitted] SCFTs

We provide explicit solutions of Type IIA supergravity which are believed to be dual to N = 2 superconformal four-dimensional gauge theories. These explicit solutions are based on the general ansatz for such a type of backgrounds introduced by Gaiotto and Maldacena.

Non-Abelian tensor gauge fields

Recently proposed extension of Yang-Mills theory contains non-Abelian tensor gauge fields. The Lagrangian has quadratic kinetic terms, as well as cubic and quartic terms describing non-linear interaction of tensor gauge fields with the dimensionless coupling constant. We analyze particle content of non-Abelian tensor gauge fields. In four-dimensional space-time the rank-2 gauge field describes propagating modes of helicity 2 and 0. We introduce interaction of the non-Abelian tensor gauge field with fermions and demonstrate that the free equation of motion for the spin-vector field correctly describes the propagation of massless modes of helicity 3/2. We have found a new metric-independent gauge invariant density which is a four-dimensional analog of the Chern-Simons density. The Lagrangian augmented by this Chern-Simons-like invariant describes massive Yang-Mills boson, providing a gauge-invariant mass gap for a four-dimensional gauge field theory.

The U(1) lattice gauge theory universally connects all classical models with continuous variables,including background gravity

We show that the partition function of many classical models with continuous degrees of freedom,e.g. Abelian lattice gauge theories and statistical mechanical models, can be written as thepartition function of an (enlarged) four-dimensional lattice gauge theory (LGT) with gauge groupU(1). This result is very general in that it includes models in differentdimensions with different symmetries. In particular, we show that aU(1) LGT defined in a curved spacetime can be mapped to aU(1) LGT with a flat background metric. The result is achieved by expressing theU(1) LGT partition function as an inner product between two quantum states.

Light scalars from a compact fifth dimension

We consider a general five-dimensional sigma-model coupled to gravity, with any number of scalars and general sigma-model metric and potential. We discuss in detail the problem of the boundary conditions for the scalar fluctuations, in the case where the fifth dimension is compact, and provide a simple (and very general) algorithmic procedure for computing the spectrum of physical scalar fluctuations of the fully back-reacted system. Focusing in particular on the conditions under which the spectrum of scalar excitations (glueballs) contains parametrically light states, we apply the formalism to some especially simple toy models, which can be thought of as the gauge/gravity duals of strongly-coupled, non-conformal four-dimensional gauge theories. Our examples are chosen both within the context of phenomenological effective field theory constructions (bottom-up approach), and within the context of consistent truncations of ten-dimensional string theories in the supergravity limit (top-down approach). In one of the examples, a light dilaton is present in the spectrum in spite of the presence of a bad naked singularity in the deep IR, near which the RG flow of the dual theory is certainly very far away from any fixed point. If this feature were to persist in a complete model in which the singularity is resolved, this would prove that a light dilaton is to be expected in at least certain walking technicolor theories. We provide here all the technical details for testing this statement, once such a complete model is identified.

Complete four-loop four-point amplitude inN=4super-Yang-Mills theory

We present the complete four-loop four-point amplitude in N=4 super-Yang-Mills theory, for a general gauge group and general D-dimensional covariant kinematics, and including all non-planar contributions. We use the method of maximal cuts --- an efficient application of the unitarity method --- to construct the result in terms of 50 four-loop integrals. We give graphical rules, valid in D-dimensions, for obtaining various non-planar contributions from previously-determined terms. We examine the ultraviolet behavior of the amplitude near D=11/2. The non-planar terms are as well-behaved in the ultraviolet as the planar terms. However, in the color decomposition of the three- and four-loop amplitude for an SU(N_c) gauge group, the coefficients of the double-trace terms are better behaved in the ultraviolet than are the single-trace terms. The results from this paper were an important step toward obtaining the corresponding amplitude in N=8 supergravity, which confirmed the existence of cancellations beyond those needed for ultraviolet finiteness at four loops in four dimensions. Evaluation of the loop integrals near D=4 would permit tests of recent conjectures and results concerning the infrared behavior of four-dimensional massless gauge theory.

MagneticSParameter

We propose a direct test of the existence of gauge duals for nonsupersymmetric asymptotically free gauge theories developing an infrared fixed point by computing the S parameter in the electric and dual magnetic description. In particular we show that at the lower bound of the conformal window the magnetic S parameter, i.e., the one determined via the dual magnetic gauge theory, assumes a simple expression in terms of the elementary magnetic degrees of freedom. The results further support our recent conjecture of the existence of a universal lower bound on the S parameter and indicates that it is an ideal operator for counting the active physical degrees of freedom within the conformal window. Our results can be directly used to unveil possible four-dimensional gauge duals and constitute the first explicit computation of a nonperturbative quantity, in the electric variables, via nonsupersymmetric gauge duality.

Commuting conformal and dual conformal symmetries in the Regge limit

In this paper we continue our study of the dual SL ( 2 , C ) symmetry of the BFKL equation, analogous to the dual conformal symmetry of N = 4 super-Yang–Mills. We find that the ordinary and dual SL ( 2 , C ) symmetries do not generate a Yangian, in contrast to the ordinary and dual conformal symmetries in the four-dimensional gauge theory. The algebraic structure is still reminiscent of that of N = 4 SYM, however, and one can extract a generator from the dual SL ( 2 , C ) close to the bi-local form associated with Yangian algebras. We also discuss the issue of whether the dual SL ( 2 , C ) symmetry, which in its original form is broken by IR effects, is broken in a controlled way, similar to the way the dual conformal symmetry of N = 4 satisfies an anomalous Ward identity. At least for the lowest orders it seems possible to recover the dual SL ( 2 , C ) by deforming its representation, keeping open the possibility that it is an exact symmetry of BFKL. Independently of a possible relation to N = 4 scattering amplitudes, this opens an avenue for explaining the integrability of BFKL in terms of two finite-dimensional subalgebras.

Lectures on gauge theory

In these two lectures I will review classic results on four-dimensional supersymmetric gauge theory, which are a useful prerequisite to learn more recent advances in the field. The lectures were given at the CERN Winter School on Supergravity, Strings and Gauge Theory (CERN, 25–29 January 2010).

Magnetic field-induced quantum criticality via new asymptotically AdS5 solutions

Using analytical methods, we derive and extend previously obtained numerical results on the low temperature properties of holographic duals to four-dimensional gauge theories at finite density in a nonzero magnetic field. We find a new asymptotically AdS5 solution representing the system at zero temperature. This solution has vanishing entropy density, and the charge density in the bulk is carried entirely by fluxes. The dimensionless magnetic field to charge density ratio for these solutions is bounded from below, with a quantum critical point appearing at the lower bound. Using matched asymptotic expansions, we extract the low temperature thermodynamics of the system. Above the critical magnetic field, the low temperature entropy density takes a simple form, linear in the temperature, and with a specific heat coefficient diverging at the critical point. At the critical magnetic field, we derive the scaling law s ∼ T1/3 inferred previously from the numerical analysis. We also compute the full scaling function describing the region near the critical point and identify the dynamical critical exponent: z = 3. These solutions are expected to holographically represent boundary theories in which strongly interacting fermions are filling up a Fermi sea. They are fully top–down constructions in which both the bulk and boundary theories have well-known embeddings in the string theory.Communicated by S F Ross

Topological mass generation four-dimensional gauge theory

The Lagrangian of non-Abelian tensor gauge fields describes the interaction of the Yang–Mills and massless tensor bosons of increasing helicities. We have found a metric-independent gauge invariant density which is a four-dimensional analog of the Chern–Simons density. The Lagrangian augmented by this Chern–Simons-like invariant describes massive Yang–Mills boson, providing a gauge-invariant mass gap for a four-dimensional gauge field theory. We present invariant densities which can provide masses to the high-rank tensor bosons.

The omega deformation, branes, integrability and Liouville theory

We reformulate the Ω-deformation of four-dimensional gauge theory in a way that is valid away from fixed points of the associated group action. We use this reformulation together with the theory of coisotropic A-branes to explain recent results linking the Ω-deformation to integrable Hamiltonian systems in one direction and Liouville theory of two-dimensional conformal field theory in another direction.

Universal extra dimensions on real projective plane

We propose a six-dimensional Universal Extra Dimensions (UED) model compactified on a real projective plane RP2, a two-sphere with its antipodal points being identified. We utilize the Randjbar-Daemi–Salam–Strathdee spontaneous sphere compactification with a monopole configuration of an extra U(1)X gauge field that leads to a spontaneous radius stabilization. Unlike the sphere and the S2/Z2 orbifold compactifications, the massless U(1)X gauge boson is safely projected out. We show how a compactification on a non-orientable manifold results in a chiral four-dimensional gauge theory by utilizing 6D chiral gauge and Yukawa interactions. The resultant Kaluza–Klein mass spectra are distinct from the ordinary UED models compactified on torus. We briefly comment on the anomaly cancellation and also on a possible dark matter candidate in our model.

Non-Abelian Gauge Field Localized on Walls with Four-Dimensional World Volume

A mechanism using the position-dependent gauge coupling is proposed to localize non-Abelian gauge fields on domain walls in five-dimensional space-time. Low-energy effective theory posseses a massless vector field, and a mass gap. The four-dimensional gauge invariance is maintained intact. We obtain perturbatively the four-dimensional Coulomb law for static sources on the domain wall. BPS domain wall solutions with the localization mechanism are explicitly constructed in the U(1)xU(1) supersymmetric gauge theory coupling to the non-Abelian gauge fields only through the cubic prepotential, which is consistent with the general principle of supersymmetry in five-dimensional space-time.