Exceptional Gauge Groups Research Articles

5d Conformal matter

Six-dimensional superconformal field theories (SCFTs) have an atomic classification in terms of elementary building blocks, conformal systems that generalize matter and can be fused together to form all known 6d SCFTs in terms of generalized 6d quivers. It is therefore natural to ask whether 5d SCFTs can be organized in a similar manner, as the outcome of fusions of certain elementary building blocks, which we call 5d conformal matter theories. In this project we begin exploring this idea and we give a systematic construction of 5d generalized “bifundamental” SCFTs, building from geometric engineering techniques in M-theory. In particular, we find several examples of (e6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_6 $$\\end{document}, e6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_6 $$\\end{document}), (e7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_7 $$\\end{document}, e7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_7 $$\\end{document}) and (e8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_8 $$\\end{document}, e8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathfrak{e}}_8 $$\\end{document}) 5d bifundamental SCFTs beyond the ones arising from (elementary) KK reductions of the 6d conformal matter theories. We show that these can be fused together giving rise to 5d SCFTs captured by 5d generalized linear quivers with exceptional gauge groups as nodes, and links given by 5d conformal matter. As a first application of these models we uncover a large class of novel 5d dualites, that generalize the well-known fiber/base dualities outside the toric realm.

The Chiral Ring of D=4$D=4$, N=1${\mathcal {N}}=1$ SYM with Exceptional Gauge Groups

AbstractThe Cachazo–Douglas–Seiberg–Witten conjecture, concerning the algebraic structure of the chiral ring in , supersymmetric Yang–Mills theory, is proven for exceptional gauge groups. This completes the proof of the conjecture.

3d exceptional gauge theories and boundary confinement

We find boundary confining dualities of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric gauge theories with exceptional gauge groups. The half-indices which enumerate the boundary BPS local operators in the presence of Neumann and Dirichlet boundary conditions for gauge fields are identified with the Askey-Wilson type q-beta integrals and Macdonald type sums respectively. New conjectural identities of E6 and E7 integrals and sums are derived from the boundary confining dualities. We also consider theories with a vector multiplet and adjoint chiral, which correspond to an N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 vector multiplet, with appropriate boundary conditions. We argue for the boundary confinement of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 vector multiplet and comment on such theories also with classical gauge groups.

Exceptionally simple integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory

Supersymmetric localisation has led to several modern developments in the study of integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills (SYM) theory. In particular, exact results have been derived for certain integrated four-point functions of superconformal primary operators in the stress tensor multiplet which are valid for all classical gauge groups, SU(N), SO(N), and USp(2N), and for all values of the complex coupling, τ = θ/(2π) + 4πi/gYM2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {g}_{YM}^2 $$\\end{document}. In this work we extend this analysis and provide a unified two-dimensional lattice sum representation valid for all simple gauge groups, in particular for the exceptional series Er (with r = 6, 7, 8), F4 and G2. These expressions are manifestly covariant under Goddard-Nuyts-Olive duality which for the cases of F4 and G2 is given by particular Fuchsian groups. We show that the perturbation expansion of these integrated correlators is universal in the sense that it can be written as a single function of three parameters, called Vogel parameters, and a suitable ’t Hooft-like coupling. To obtain the perturbative expansion for the integrated correlator with a given gauge group we simply need substituting in this single universal expression specific values for the Vogel parameters. At the non-perturbative level we conjecture a formula for the one-instanton Nekrasov partition function valid for all simple gauge groups and for general Ω-deformation background. We check that our expression reduces in various limits to known results and that it produces, via supersymmetric localisation, the same one-instanton contribution to the integrated correlator as the one derived from the lattice sum representation. Finally, we consider the action of the hyperbolic Laplace operator with respect to τ on the integrated correlators with exceptional gauge groups and derive inhomogeneous Laplace equations very similar to the ones previously obtained for classical gauge groups.

Gauge symmetry breaking with fluxes and natural Standard Model structure from exceptional GUTs in F-theory

We give a general description of gauge symmetry breaking using vertical and remainder fluxes in 4D F-theory models. The fluxes can break a geometric gauge group to a smaller group and induce chiral matter, even when the larger group admits no chiral matter representations. We focus specifically on applications to realizations of the Standard Model gauge group and chiral matter spectrum through breaking of rigid exceptional gauge groups E7, E6, which are ubiquitous in the 4D F-theory landscape. Supplemented by an intermediate SU(5) group, these large classes of models give natural constructions of Standard Model-like theories with small numbers of generations of matter in F-theory.

6D strings and exceptional instantons

We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5d and 6d supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study $G_2$ theories with $n_{\bf 7}\leq 3$ matters in ${\bf 7}$, and $SO(7)$ theories with $n_{\bf 8}\leq 4$ matters in the spinor representation ${\bf 8}$. We also study the elliptic genera of self-dual instanton strings of 6d SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank $2$ or $3$ tensor branches. Some of them are tested with topological vertex calculus. We also explore a D-brane-based method to study instanton particles of 5d $SO(7)$ and $SO(8)$ gauge theories with matters in spinor representations, which further tests our ADHM-like proposals.

Chern-Simons theory with the exceptional gauge group as a refined topological string

We present the partition function of Chern-Simons theory with the exceptional gauge group on three-sphere in the form of a partition function of the refined closed topological string with relation 2τ=gs(1−b) between single Kähler parameter τ, string coupling constant gs and refinement parameter b, where b=53,52,3,4,6 for G2,F4,E6,E7,E8, respectively. The non-zero BPS invariants NJL,JRd (d - degree) are N0,122=1,N0,111=1. Besides these terms, partition function of Chern-Simons theory contains term corresponding to the refined constant maps of string theory.Derivation is based on the universal (in Vogel's sense) form of a Chern-Simons partition function on three-sphere, restricted to exceptional line Exc with Vogel's parameters satisfying γ=2(α+β). This line contains points, corresponding to the all exceptional groups. The same results are obtained for F line γ=α+β (containing SU(4),SO(10) and E6 groups), with the non-zero N0,122=1,N0,17=1.In both cases refinement parameter b (=−ϵ2/ϵ1 in terms of Nekrasov's parameters) is given in terms of universal parameters, restricted to the line, by b=−β/α.

4D gauge theories with conformal matter

One of the hallmarks of 6D superconformal field theories (SCFTs) is that on a partial tensor branch, all known theories resemble quiver gauge theories with links comprised of 6D conformal matter, a generalization of weakly coupled hypermultiplets. In this paper we construct 4D quiverlike gauge theories in which the links are obtained from compactifications of 6D conformal matter on Riemann surfaces with flavor symmetry fluxes. This includes generalizations of super QCD with exceptional gauge groups and quarks replaced by 4D conformal matter. Just as in super QCD, we find evidence for a conformal window as well as confining gauge group factors depending on the total amount of matter. We also present F-theory realizations of these field theories via elliptically fibered Calabi-Yau fourfolds. Gauge groups (and flavor symmetries) come from 7-branes wrapped on surfaces, conformal matter localizes at the intersection of pairs of 7-branes, and Yukawas between 4D conformal matter localize at points coming from triple intersections of 7-branes. Quantum corrections can also modify the classical moduli space of the F-theory model, matching expectations from effective field theory.

F4 , E6 and G2 exceptional gauge groups in the vacuum domain structure model

Using vacuum domain structure model, trivial static potentials in various representations of $F_4$, $E_6$ and $G_2$ exceptional groups are calculated by means of the unit center element. Due to the absence of the non-trivial center elements, the potential of every representation is screened at far distances. However, the linear part is observed at intermediate quark separations which is investigated by the decomposition of the exceptional group to its maximal subgroups. Comparing the group factor of the super-group with the corresponding one obtained from the non-trivial center elements of $SU(3)$ subgroup, shows that $SU(3)$ is not the direct cause of temporary confinement in any of the exceptional groups. However, the trivial potential obtained from the group decomposition to the $SU(3)$ subgroup is the same as the potential of the super-group itself. In addition, any regular or singular decomposition to the $SU(2)$ subgroup which produces the Cartan generator with the same elements as $h_1$, in any exceptional group, leads to the linear intermediate potential of the exceptional gauge groups. The other $SU(2)$ decompositions with the Cartan generator different from $h_1$, are still able to describe the linear potential if the number of $SU(2)$ non-trivial center element which emerge in the decompositions is the same. As a result, it is the center vortices quantized in terms of non-trivial center element of the $SU(2)$ subgroup which give rise to the intermediate confinement in the static potentials.

Confinement inF4Exceptional Gauge Group Using Domain Structures

We calculate the potential between static quarks in the fundamental representation of the F 4 exceptional gauge group using domain structures of the thick center vortex model. As non-trivial center elements are absent, the asymptotic string tension is lost while an intermediate linear potential is observed. SU (2) is a subgroup of F 4 . Investigating the decomposition of the 26 dimensional representation of F 4 to the SU (2) representations, might explain what accounts for the intermediate linear potential, in the exceptional groups with no center element.

Gauge-covariant decomposition and magnetic monopole forG(2)Yang-Mills field

We give a gauge-covariant decomposition of the Yang-Mills field with an exceptional gauge group $G(2)$, which extends the field decomposition invented by Cho, Duan-Ge, and Faddeev-Niemi for the $SU(N)$ Yang-Mills field. As an application of the decomposition, we derive a new expression of the non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation of $G(2)$. The resulting new form is used to define gauge-invariant magnetic monopoles in the $G(2)$ Yang-Mills theory. Moreover, we obtain the quantization condition to be satisfied by the resulting magnetic charge. The method given in this paper is general enough to be applicable to any semi-simple Lie group other than $SU(N)$ and $G(2)$.

Higher‐dimensional gauge theories from string theory

We review some recent developments regarding supersymmetric field theories in six and five dimensions. In particular, we will describe the classification of supersymmetric six‐dimensional theories with a holographic IIA dual; they are “linear quivers” consisting of chains of many SU (or SO/Sp) gauge groups connected by hypermultiplets and tensor multiplets. We will also describe the wider classification of supersymmetric six‐dimensional theories that can be engineered in F‐theory; these are also chains, but they include exceptional gauge groups and copies of a more exotic “E‐string” theory with a single tensor and E8 flavor symmetry. Finally we discuss some properties of these theories under compactification to lower dimensions.

Gauge groups and matter fields on some models of F-theory without section

We investigate F-theory on an elliptic Calabi-Yau 4-fold without a section to the fibration. To construct an elliptic Calabi-Yau 4-fold without a section, we introduce families of elliptic K3 surfaces which do not admit a section. A product K3 $\times$ K3, with one of the K3's chosen from these families of elliptic K3 surfaces without a section, realises an elliptic Calabi-Yau 4-fold without a section. We then compactify F-theory on such K3 $\times$ K3's. We determine the gauge groups and matter fields which arise on 7-branes for these models of F-theory compactifications without a section. Since each K3 $\times$ K3 constructed does not have a section, gauge groups arising on 7-branes for F-theory models on constructed K3 $\times$ K3's do not have $U(1)$-part. Interestingly, exceptional gauge group $E_6$ appears for some cases.

G2monopoles

We investigate some aspects of Bogomolny-Prasad-Sommerfield monopole solutions in the Yang-Mills-Higgs theory with exceptional gauge group $G_2$ spontaneously broken to $U(1)\times U(1)$. Corresponding homotopy group is $\pi_2\left(G_2/U(1)\times U(1)\right)$ and similar to the $SU(3)$ theory, the $G_2$ monopoles are classified by two topological charges $(n_1,n_2)$. In fundamental representation these yield a subset of $SO(7)$ monopole configurations. Through inspection of the structure of $Alg(G_2)$, we propose an extension of the Nahm construction to the $(n,1)_{G_2}$ monopoles. For $(1,1)_{G_2}$ monopole the Nahm data are written explicitly.

Coulomb branch and the moduli space of instantons

The moduli space of instantons on C^2 for any simple gauge group is studied using the Coulomb branch of N=4 gauge theories in three dimensions. For a given simple group G, the Hilbert series of such an instanton moduli space is computed from the Coulomb branch of the quiver given by the over-extended Dynkin diagram of G. The computation includes the cases of non-simply-laced gauge groups G, complementing the ADHM constructions which are not available for exceptional gauge groups. Even though the Lagrangian description for non-simply laced Dynkin diagrams is not currently known, the prescription for computing the Coulomb branch Hilbert series of such diagrams is very simple. For instanton numbers one and two, the results are in agreement with previous works. New results and general features for the moduli spaces of three and higher instanton numbers are reported and discussed in detail.

Counting exceptional instantons

We show how to obtain the instanton partition function of N=2 SYM with exceptional gauge group EFG using blow-up recursion relations derived by Nakajima and Yoshioka. We compute the two instanton contribution and match it with the recent proposal for the superconformal index of rank 2 SCFTs with E6, E7 global symmetry.

Distorted mass edges at LHC from supersymmetric leptoquarks

Supersymmetric (SUSY) grand unified theories based on exceptional gauge groups such as E6 have recently triggered a lot of interest. Aside from top-down motivations, they contain phenomenologically interesting states with leptoquark quantum numbers. Their SUSY partners, leptoquarkinos, will appear similar to all R-odd particles in decay cascades, but mass edges in kinematic distributions- originating from the same semiexclusive final states-will however have major differences to the corresponding edges of ordinary squarks. This distortion of standard observables bears the opportunity to detect them at the LHC, but may also pose significant confusion of underlying model assumptions, which should be handled with care and, if interpreted falsely, might even prevent a possible discovery.

On instanton effects in F-theory

We revisit the issue of M5-brane instanton corrections to the superpotential in F-theory compactifications on elliptically fibered Calabi-Yau fourfolds. Elaborating on concrete geometries, we compare the instanton zero modes for non-perturbative F-theory models with the zero modes in their perturbative Sen limit. The fermionic matter zero modes localized on the intersection of the instanton with the space-time filling D7-branes show up in a geometric way in F-theory. Methods for their computation are developed and, not surprisingly, exceptional gauge group structures do appear. Finally, quite intriguing geometrical aspects of the one-loop determinant are discussed.

The Hilbert series of the one instanton moduli space

The moduli space of k G-instantons on R^4 for a classical gauge group G is known to be given by the Higgs branch of a supersymmetric gauge theory that lives on Dp branes probing D(p + 4) branes in Type II theories. For p = 3, these (3 + 1) dimensional gauge theories have N = 2 supersymmetry and can be represented by quiver diagrams. The F and D term equations coincide with the ADHM construction. The Hilbert series of the moduli spaces of one instanton for classical gauge groups is easy to compute and turns out to take a particularly simple form which is previously unknown. This allows for a G invariant character expansion and hence easily generalisable for exceptional gauge groups, where an ADHM construction is not known. The conjectures for exceptional groups are further checked using some new techniques like sewing relations in Hilbert Series. This is applied to Argyres-Seiberg dualities.

Octonions, [formula omitted] and generalized Lie 3-algebras

We construct an explicit example of a generalized Lie 3-algebra from the octonions. In combination with the result of [S. Cherkis, C. Saemann, arXiv: 0807.0808 [hep-th]], this gives rise to a three-dimensional N=2 Chern–Simons-matter theory with exceptional gauge group G2 and with global symmetry SU(4)×U(1). This gives a possible candidate for the theory on multiple M2-branes with G2 gauge symmetry.