Exceptional Gauge Groups Research Articles

On S-duality for non-simply-laced gauge groups

We point out that for N=4 gauge theories with exceptional gauge groups G_2 and F_4 the S-duality transformation acts on the moduli space by a nontrivial involution. We note that the duality groups of these theories are the Hecke groups with elliptic elements of order six and four, respectively. These groups extend certain subgroups of SL(2,Z) by elements with a non-trivial action on the moduli space. We show that under an embedding of these gauge theories into string theory, the Hecke duality groups are represented by T-duality transformations.

Monopoles, affine algebras and the gluino condensate

We examine the low-energy dynamics of four-dimensional supersymmetric gauge theories and calculate the values of the gluino condensate for all simple gauge groups. By initially compactifying the theory on a cylinder we are able to perform calculations in a controlled weakly coupled way for a small radius. The dominant contributions to the path integral on the cylinder arise from magnetic monopoles which play the role of instanton constituents. We find that the semi-classically generated superpotential of the theory is the affine Toda potential for an associated twisted affine algebra. We determine the supersymmetric vacua and calculate the values of the gluino condensate. The number of supersymmetric vacua is equal to c2, the dual Coxeter number, and in each vacuum the monopoles carry a fraction 1/c2 of topological charge. As the results are independent of the radius of the circle, they are also valid in the strong coupling regime where the theory becomes decompactified. For gauge groups SU(N), SO(N) and USp(2N) our results for the gluino condensate are in precise agreement with the “weak coupling instanton” expressions (and not with the “strong coupling instanton” calculations). For the exceptional gauge groups we calculate the values of the gluino condensate for the first time.

Exceptional confinement in G(2) gauge theory

We study theories with the exceptional gauge group G(2). The 14 adjoint “gluons” of a G(2) gauge theory transform as {3}, { 3 ̄ } and {8} under the subgroup SU(3), and hence have the color quantum numbers of ordinary quarks, anti-quarks and gluons in QCD. Since G(2) has a trivial center, a “quark” in the {7} representation of G(2) can be screened by “gluons”. As a result, in G(2) Yang–Mills theory the string between a pair of static “quarks” can break. In G(2) QCD there is a hybrid consisting of one “quark” and three “gluons”. In supersymmetric G(2) Yang–Mills theory with a {14} Majorana “gluino” the chiral symmetry is Z(4) χ . Chiral symmetry breaking gives rise to distinct confined phases separated by confined–confined domain walls. A scalar Higgs field in the {7} representation breaks G(2) to SU(3) and allows us to interpolate between theories with exceptional and ordinary confinement. We also present strong coupling lattice calculations that reveal basic features of G(2) confinement. Just as in QCD, where dynamical quarks break the Z(3) symmetry explicitly, G(2) gauge theories confine even without a center. However, there is not necessarily a deconfinement phase transition at finite temperature.

Chiral Rings, Superpotentials and the Vacuum Structure of <i>N</i> = 1 Supersymmetric Gauge Theories

We use the Konishi anomaly equations to construct the exact effective superpotential of the glueball superfields in various N = 1 supersymmetric gauge theories. We use the superpotentials to study in detail the structure of the spaces of vacua of these theories. We consider chiral and non-chiral SU(N) models, the exceptional gauge group G_2 and models that break supersymmetry dynamically.

Direct derivation of scaling relation of prepotential in N=2 supersymmetric G2 Yang–Mills theory

In contrast to the classical gauge group cases, any method to exactly prove the scaling relation which relates moduli and prepotential is not known in the case of exceptional gauge groups. This paper provides a direct method to establish this relation by using Picard–Fuchs equations. In particular, it is shown that the scaling relation found by Ito in N=2 supersymmetric G2 Yang–Mills theory actually holds exactly.

Flat connections with non-diagonalizable holonomies

Recently the long-standing puzzle about counting the Witten index in N=1 supersymmetric gauge theories was resolved. The resolution was based on existence (for higher orthogonal SO( N), N≥7 and exceptional gauge groups) of flat connections on T 3 which have commuting holonomies but cannot be gauged to a Cartan torus. A number of papers has been published which studied moduli spaces and some topological characteristics of those flat connections. In the present letter an explicit description of such flat connection for the basic case of Spin(7) is given.

Calabi-Yau spaces and five-dimensional field theories with exceptional gauge symmetry

Five-dimensional field theories with exceptional gauge groups are engineered from degenerations of Calabi-Yau threefolds. The structure of the Coulomb branch is analyzed in terms of relative Kähler cones. For low number of flavors, the geometric construction leads to new five-dimensional fixed points.

Duality and superconvergence relation in supersymmetric gauge theories

We investigate the phase structures of various N=1 supersymmetric gauge theories including even the exceptional gauge group from the viewpoint of superconvergence of the gauge field propagator. Especially we analyze in detail whether a new type of duality recently discovered by Oehme in $SU(N_c)$ gauge theory coupled to fundamental matter fields can be found in more general gauge theories with more general matter representations or not. The result is that in the cases of theories including matter fields in only the fundamental representation, Oehme's duality holds but otherwise it does not. In the former case, superconvergence relation might give good criterion to describe the interacting non-Abelian Coulomb phase without using some information from dual magnetic theory.

ADE confining phase superpotentials

We obtain a low-energy effective superpotential for a phase with a single confined photon in N = 1 gauge theory with adjoint matter with ADE gauge groups. The expectation values of gauge invariants constructed out of the adjoint field parametrize the singularities of moduli space of the Coulomb phase. The result can be used to derive the N = 2 curve in the form of a foliation over CP 1. Our N = 1 theory exhibits non-trivial fixed points which naturally inherit the properties of the ADE classification of N = 2 superconformal field theories in four dimensions. We also discuss how to include matter hypermultiplets in deriving the Riemann surface which describes N = 2 QCD with exceptional gauge groups.

ON THE PICARD–FUCHS EQUATIONS OF N= 2 SUPERSYMMETRIC E6 YANG–MILLS THEORY

We obtain the Picard–Fuchs equations of N=2 supersymmetric Yang–Mills theory with the exceptional gauge group E6. Such equations are based on E6 spectral curve.

Discrete anomaly matching

We extend the well-known 't Hooft anomaly matching conditions for continuous global symmetries to discrete groups. We state explicitly the matching conditions for all possible anomalies which involve discrete symmetries. There are two types of discrete anomalies. For Type I anomalies, the matching conditions always have to be satisfied regardless of the details of the massive bound state spectrum. The Type II anomalies have to be also matched except if there are fractionally charged massive bound states in the theory. We check discrete anomaly matching in recent solutions of certain N = 1 supersymmetric gauge theories, most of which satisfy these constraints. The excluded examples include the chirally symmetric phase of N = 1 pure symmetric Yang-Mills theories described by the Veneziano-Yankielowicz Lagrangian and certain non-supersymmetric confining theories. The conjectured self-dual theories based in exceptional gauge groups do not satisfy discrete anomaly matching nor mapping of operators, and are viable only if the discrete symmetry in the electric theory appears as an accidental symmetry in the magnetic theory and vice versa.

Periods and prepotential in N=2 supersymmetric E6 Yang-Mills theory

We obtain the periods and one-instanton coefficient of the N=2 superrsymmetric Yang-Mills theory with the exceptional gauge group E_6. These calculations are based on the E_6 spectral curve and the obtained one-instanton coefficient is in agreement with the microscopic results.

More on N = 1 self-dualities and exceptional gauge groups

Starting from a generalization of a recent result on self-duality we systematically analyze self-dual models. We find a criterion to judge whether a given model is self-dual or not. With this tool we construct some new self-dual pairs, focussing on examples with exceptional gauge groups.

Picard-Fuchs equations and prepotential in N = 2 supersymmetric G2 Yang-Mills theory

We study the low-energy effective theory of N=2 supersymmetric Yang-Mills theory with the exceptional gauge group $G_{2}$. We obtain the Picard-Fuchs equations for the $G_{2}$ spectral curve and compute multi-instanton contribution to the prepotential. We find that the spectral curve is consistent with the microscopic supersymmetric instanton calculus. It is also shown that $G_{2}$ hyperelliptic curve does not reproduce the microscopic result.

Moduli space ofN=2supersymmetricG2gauge theory

We apply the method of confining phase superpotentials to N = 2 supersymmetric Yang-Mills theory with the exceptional gauge group G_2. Our findings are consistent with the spectral curve of the periodic Toda lattice, but do not agree with the hyperelliptic curve suggested previously in the literature. We also apply the method to theories with fundamental matter, treating both the example of SO(5) and G_2.

N = 1 Dualities for Exceptional Gauge Groups and Quantum Global Symmetries

We discuss our attempts to generalize the known examples of dualities in N=1 supersymmetric gauge theories to exceptional gauge groups. We derive some dual pairs from known examples connected to exceptional groups and find an interesting phenomenon: sometimes the full global symmetry is ``hidden'' on the magnetic side. It is not realized as a symmetry on the fundamental fields in the Lagrangian. Rather, it emerges as a symmetry of the quantum theory. We then focus on an approach based on self-dual models. We construct duals for some very special matter content of $E_ 7$, $E_6$ and $F_4$. Again we find that the full global symmetry is not realized on the fundamental fields.

The moduli space and monodromies of the N = 2 supersymmetric Yang-Mills theory with any Lie gauge groups

We propose a unified scheme for finding the hyperelliptic curve of N = 2 SUSY YM theory with any Lie gauge groups. Our general scheme gives the well-known results for the classical gauge groups and the exceptional G 2 group. In particular, we present the curve for the exceptional gauge groups F 4, E 6,7,8 and check the consistency condition for them. The exact monodromies and the dyon spectrum of these theories are determined. We note that for any Lie gauge groups, the exact monodromies could be obtained only from the Cartan matrix.

Exceptional equivalences in N = 2 supersymmetric Yang-Mills theory

We find low-energy equivalences between N = 2 supersymmetric gauge theories with different simple gauge groups with and without matter. We give a construction of equivalences based on subgroups and find all examples with maximal simple subgroups. This is used to solve some theories with exceptional gauge groups G 2 and F 4. We are also able to solve an E 6 theory on a codimension one submanifold of its moduli space.

Chiral duals of non-chiral SUSY gauge theories

We study N = 1 SUSY gauge theories in four dimensions with gauge group Spin(7) and N f flavors of quarks in the spinorial representation. We find that in the range 6 < N f < 15, this theory has a long distance description in terms of an SU( N f - 4) gauge theory with a symmetric tensor and N f antifundamentals. As a spin-off, we obtain by deforming along a flat direction a dual description of the theories based on the exceptional gauge group G 2 with N f fundamental flavors of quarks.

Seven-dimensional octonionic Yang-Mills instanton and its extension to an heterotic string soliton

We construct an octonionic instanton solution to the seven-dimensional Yang-Mills theory based on the exceptional gauge group G 2 which is the automorphism group of the division algebra of octonions. This octonionic has an extension to a solitonic solution of the low energy effective theory of the heterotic string that preserves two of the sixteen supersymmetries and hence corresponds to N = 1 space-time supersymmetry in (2+1) dimensions transverse to the seven dimensions where the Yang-Mills instanton is defined.