Semisimple Gauge Group Research Articles

Three-loop OPE Wilson coefficients of dimension-four operators for (axial-)vector and (pseudo-)scalar current correlators

We calculate the three-loop Wilson coefficients of all physically relevant dimension-four operators, i.e. GμνaGa,μν, miq¯jqj and mimjmk2, in the short-distance expansion of the time-ordered product of a pair of gauge-singlet vector, axial-vector, scalar and pseudo-scalar currents. The results are given for a general non-Abelian gauge theory with arbitrary (compact semi-simple) gauge group and nf light fermion flavors (quarks) in a common arbitrary representation of the gauge group, which includes QCD as a special case. In particular, we allow for arbitrary flavor contents of each of the currents. For the axial-vector current the included contributions from so-called singlet diagrams are consistent with the one-loop axial anomaly.

Global exterior stability of the Minkowski space: Coupled Einstein–Yang–Mills perturbations

Here we prove a global gauge-invariant radiation estimates for the perturbations of the 3 + 1 dimensional Minkowski spacetime in the presence of Yang–Mills sources. In particular, we obtain a novel gauge invariant estimate for the Yang–Mills fields coupled to gravity in a double null framework in the Causal complement of a compact set of a Cauchy slice. A consequence of our result is the global exterior stability of the Minkowski space under coupled Yang–Mills perturbations. A special structure present both in the null Bianchi equations and the null Yang–Mills equations is utilized crucially to obtain the dispersive estimates necessary to conclude the global existence property. Direct use of Bel-Robinson and Yang–Mills stress-energy tensor to obtain the energy estimates is avoided in favor of weighted integration by parts taking advantage of the manifestly symmetric hyperbolic characteristics of null Bianchi and null Yang–Mills equations. Our result holds for any compact semi-simple gauge group. This is the first stability result of Minkowski space including a non-linear source.

Towards a complete classification of 6D supergravities

The constraints arising from anomaly cancellation are particular strong for chiral theories in six dimensions. We make progress towards a complete classification of 6D supergravities with minimal supersymmetry and non-abelian gauge group. First, we generalize a previously known infinite class of anomaly-free theories which has \U0001d447 ≫ 9 to essentially any semi-simple gauge group and infinitely many choices for hypermultiplets. The construction relies on having many decoupled sectors all selected from a list of four simple theories which we identify. Second, we use ideas from graph theory to rephrase the task of finding anomaly-free theories as constructing cliques in a certain multigraph. A branch-and-bound type algorithm is described which can be used to explicitly construct, in a \U0001d447-independent way, anomaly-free theories with an arbitrary number of simple factors in the gauge group. We implement these ideas to generate an ensemble of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}(107) irreducible cliques from which anomaly-free theories may be easily built, and as a special case obtain a complete list of 19,847 consistent theories for \U0001d447 = 0, for which the maximal gauge group rank is 24. Modulo U(1), SU(2) and SU(3) simple factors and the new infinite families, we give a complete characterization of anomaly-free theories and show that the bound \U0001d447 ≤ 273 is sharp.

Yang\u2013Mills black holes in quasitopological gravity

In this paper, we formulate two new classes of black hole solutions in higher curvature quartic quasitopological gravity with nonabelian Yang–Mills theory. At first step, we consider the SO(n) and SO(n-1,1) semisimple gauge groups. We obtain the analytic quartic quasitopological Yang–Mills black hole solutions. Real solutions are only accessible for the positive value of the redefined quartic quasitopological gravity coefficient, mu _{4}. These solutions have a finite value and an essential singularity at the origin, r=0 for space dimension higher than 8. We also probe the thermodynamic and critical behavior of the quasitopological Yang–Mills black hole. The obtained solutions may be thermally stable only in the canonical ensemble. They may also show a first order phase transition from a small to a large black hole. In the second step, we obtain the pure quasitopological Yang–Mills black hole solutions. For the positive cosmological constant and the space dimensions greater than eight, the pure quasitopological Yang–Mills solutions have the ability to produce both the asymptotically AdS and dS black holes for respectively the negative and positive constant curvatures, k=-1 and k=+1. This is unlike the quasitopological Yang–Mills theory which can lead to just the asymptotically dS solutions for Lambda >0. The pure quasitopological Yang–Mills black hole is not thermally stable.

Mathematical Quantum Yang—Mills Theory Revisited II: Mass without Mass

Massless Dirac equation for spinor multiplets is minimally coupled with a unitary representation of an arbitrary compact semisimple gauge group. The spectrum of the second quantized interaction Hamiltonian has a positive mass gap running along the classical energy scale.

Holographic RG flows in $$N=4$$ N = 4 SCFTs from half-maximal gauged supergravity

We study four-dimensional N=4 gauged supergravity coupled to six vector multiplets with semisimple gauge groups SO(4)times SO(4), SO(3,1)times SO(3,1) and SO(4)times SO(3,1). All of these gauge groups are dyonically embedded in the global symmetry group SO(6, 6) via its maximal subgroup SO(3,3)times SO(3,3). For SO(4)times SO(4) gauge group, there are four N=4 supersymmetric AdS_4 vacua with SO(4)times SO(4), SO(4)times SO(3), SO(3)times SO(4) and SO(3)times SO(3) symmetries, respectively. These AdS_4 vacua correspond to N=4 SCFTs in three dimensions with SO(4) R-symmetry and different flavor symmetries. We explicitly compute the full scalar mass spectra at all these vacua. Holographic RG flows interpolating between these conformal fixed points are also given. The solutions describe supersymmetric deformations of N=4 SCFTs by relevant operators of dimensions Delta =1,2. A number of these solutions can be found analytically although some of them can only be obtained numerically. These results provide a rich and novel class of N=4 fixed points in three-dimensional Chern–Simons-Matter theories and possible RG flows between them in the framework of N=4 gauged supergravity in four dimensions. Similar studies are carried out for non-compact gauge groups, but the SO(4)times SO(4) gauge group exhibits a much richer structure.

On the existence of topological dyons and dyonic black holes in anti-de Sitter Einstein-Yang-Mills theories with compact semisimple gauge groups

Here we study the global existence of “hairy” dyonic black hole and dyon solutions to four-dimensional, anti-de Sitter Einstein-Yang-Mills theories for a general simply connected and semisimple gauge group G, for the so-called topologically symmetric systems, concentrating here on the regular case. We generalise here cases in the literature which considered purely magnetic spherically symmetric solutions for a general gauge group and topological dyonic solutions for su(N). We are able to establish the global existence of non-trivial solutions to all such systems, both near existing embedded solutions and as |Λ| → ∞. In particular, we can identify non-trivial solutions where the gauge field functions have no zeroes, which in the su(N) case proved important to stability. We believe that these are the most general analytically proven solutions in 4D anti-de Sitter Einstein-Yang-Mills systems to date.

On the global existence of hairy black holes and solitons in anti-de Sitter Einstein–Yang–Mills theories with compact semisimple gauge groups

We investigate the existence of black hole and soliton solutions to four dimensional, anti-de Sitter (adS), Einstein-Yang-Mills theories with general semisimple connected and simply connected gauge groups, concentrating on the so-called "regular" case. We here generalise results for the asymptotically flat case, and compare our system with similar results from the well-researched adS $\mathfrak{su}(N)$ system. We find the analysis differs from the asymptotically flat case in some important ways: the biggest difference is that for $\Lambda<0$, solutions are much less constrained as $r\rightarrow\infty$, making it possible to prove the existence of global solutions to the field equations in some neighbourhood of existing trivial solutions, and in the limit of $|\Lambda|\rightarrow\infty$. In particular, we can identify non-trivial solutions where the gauge field functions have no zeroes, which in the $\mathfrak{su}(N)$ case proved important to stability.

Gaugings of four-dimensionalN=3supergravity andAdS4/CFT3holography

We study matter-coupled $N=3$ gauged supergravity in four dimensions with various semisimple gauge groups. When coupled to $n$ vector multiplets, the gauged supergravity contains $3+n$ vector fields and $3n$ complex scalars parametrized by $SU(3,n)/SU(3)\times SU(n)\times U(1)$ coset manifold. Semisimple gauge groups take the form of $G_0\times H\subset SO(3,n)\subset SU(3,n)$ with $H$ being a compact subgroup of $SO(n+3-\textrm{dim}(G_0))$. The $G_0$ groups considered in this paper are of the form $SO(3)$, $SO(3,1)$, $SO(2,2)$, $SL(3,\mathbb{R})$ and $SO(2,1)\times SO(2,2)$. We find that $SO(3)\times SO(3)$, $SO(3,1)$ and $SL(3,\mathbb{R})$ gauge groups admit a maximally supersymmetric $AdS_4$ critical point. The $SO(2,1)\times SO(2,2)$ gauge group admits a supersymmetric Minkowski vacuum while the remaining gauge groups admit both half-supersymmetric domain wall vacua and $AdS_4$ vacua with completely broken supersymmetry. For the $SO(3)\times SO(3)$ gauge group, there exists another supersymmetric $N=3$ $AdS_4$ critical point with $SO(3)_{\textrm{diag}}$ symmetry. We explicitly give a detailed study of various holographic RG flows between $AdS_4$ critical points, flows to non-conformal theories and supersymmetric domain walls in each gauge group. The results provide gravity duals of $N=3$ Chern-Simons-Matter theories in three dimensions.

A remark on the asymptotic form of BPS multi-dyon solutions and their conserved charges

We evaluate the gauge invariant, dynamically conserved charges, recently obtained from the integral form of the Yang-Mills equations, for the BPS multi-dyon solutions of a Yang-Mills-Higgs theory associated to any compact semi-simple gauge group G. Those charges are shown to correspond to the eigenvalues of the next-to-leading term of the asymptotic form of the Higgs field at spatial infinity, and so coinciding with the usual topological charges of those solutions. Such results show that many of the topological charges considered in the literature are in fact dynamical charges, which conservation follows from the global properties of classical Yang-Mills theories encoded into their integral dynamical equations. The conservation of those charges can not be obtained from the differential form of Yang-Mills equations.

Classification of 6d N = 1 0 $$ \mathcal{N}=\left(1,0\right) $$ gauge theories

We delineate a procedure to classify 6d $$ \mathcal{N}=\left(1,0\right) $$ gauge theories composed, in part, of a semi-simple gauge group and hypermultiplets. We classify these theories by requiring that satisfy some consistency conditions. The primary consistency condition is that the gauge anomaly can be cancelled by adding tensor multiplets which couple to the gauge fields by acting as sources of instanton strings. Based on the number of tensor multiplets required to cancel the anomaly, we conjecture that the UV completion of these consistent gauge theories (if it exists) should be either a 6d $$ \mathcal{N}=\left(1,0\right) $$ SCFT or a 6d $$ \mathcal{N}=\left(1,0\right) $$ little string theory.

Stable Asymptotically Free Extensions (SAFEs) of the Standard Model

We consider possible extensions of the standard model that are not only completely asymptotically free, but are such that the UV fixed point is completely UV attractive. All couplings flow towards a set of fixed ratios in the UV. Motivated by low scale unification, semi-simple gauge groups with elementary scalars in various representations are explored. The simplest model is a version of the Pati-Salam model. The Higgs boson is truly elementary but dynamical symmetry breaking from strong interactions may be needed at the unification scale. A hierarchy problem, much reduced from grand unified theories, is still in need of a solution.

Loop quantization of a model for D = 1 + 2 (anti)de Sitter gravity coupled to topological matter

We present a complete quantization of Lorentzian gravity with cosmological constant, coupled to a set of topological matter fields. The approach of loop quantum gravity is used thanks to a partial gauge fixing leaving a residual gauge invariance under a compact semi-simple gauge group, namely Spin(4) = SU(2) × SU(2). A pair of quantum observables is constructed, which are non-trivial despite being gauge-equivalent to zero at the classical level. A semi-classical approximation based on appropriately defined coherent states shows non-vanishing expectation values for them, thus not reproducing their classical behaviour.

New N = 5, 6, 3D gauged supergravities and holography

We study $N=5$ gauged supergravity in three dimensions with compact, non-compact and non-semisimple gauge groups. The theory under consideration is of Chern-Simons type with $USp(4,k)/USp(4)\times USp(k)$ scalar manifold. We classify possible semisimple gauge groups of the $k=2,4$ cases and identify some of their critical points. A number of supersymmetric $AdS_3$ critical points are found, and holographic RG flows interpolating between these critical points are also investigated. As one of our main results, we consider a non-semisimple gauge group $SO(5)\ltimes \mathbf{T}^{10}$ for the theory with $USp(4,4)/USp(4)\times USp(4)$ scalar manifold. The resulting theory describes $N=5$ gauged supergravity in four dimensions reduced on $S^1/\mathbb{Z}_2$ and admits a maximally supersymmetric $AdS_3$ critical point with $Osp(5|2,\mathbb{R})\times Sp(2,\mathbb{R})$ superconformal symmetry. We end the paper with the construction of $SO(6)\ltimes \mathbf{T}^{15}$ gauged supergravity with $N=6$ supersymmetry. The theory admits a half-supersymmetric domain wall as a vacuum solution and may be obtained from an $S^1/\mathbb{Z}_2$ reduction of $N=6$ gauged supergravity in four dimensions.

Classification of 4d $ \mathcal{N} $ =2 gauge theories

We classify all possible four-dimensional N=2 supersymmetric UV-complete gauge theories composed of semi-simple gauge groups and hypermultiplets. We also give appropriate references for all theories with known Seiberg-Witten solutions.

Gaugings of N = 4 three dimensional gauged supergravity with exceptional coset manifolds

Some admissible gauge groups of N=4 Chern-Simons gauged supergravity in three dimensions with exceptional scalar manifolds $G_{2(2)}/SO(4)$, $F_{4(4)}/USp(6)\times SU(2)$, $E_{6(2)}/SU(6)\times SU(2)$, $E_{7(-5)}/SO(12)\times SU(2)$ and $E_{8(-24)}/E_7\times SU(2)$ are identified. In particular, a complete list of all possible gauge groups is given for the theory with $G_{2(2)}/SO(4)$ coset space. We also study scalar potentials for all of these gauge groups and find some critical points. In the case of $F_{4(4)}/USp(6)\times SU(2)$ target space, we give some semisimple gauge groups which are maximal subgroups of $F_{4(4)}$. Most importantly, we construct the $SO(4)\ltimes \mathbf{T}^6$ gauged supergravity which is equivalent to N=4 SO(4) Yang-Mills gauged supergravity. The latter is proposed to be obtained from an $S^3$ reduction of $(1,0)$ six dimensional supergravity coupled to two vector and two tensor multiplets. The scalar potential of this theory on the scalar fields which are invariant under SO(4) is explicitly computed. Depending on the value of the coupling constants, the theory admits both dS and AdS vacua when all of the 28 scalars vanish. The maximal N=4 supersymmetric $AdS_3$ should correspond to the $AdS_3\times S^3$ solution of the $(1,0)$ six dimensional theory. Finally, some gauge groups of the theories with $E_{6(2)}/SU(6)\times SU(2)$, $E_{7(-5)}/SO(12)\times SU(2)$ and $E_{8(-24)}/E_7\times SU(2)$ scalar manifolds are identified.

Effective Action and Phase Transitions in Yang-Mills Theory on Spheres

We study the covariantly constant Savvidy-type chromomagnetic vacuum in finite-temperature Yang-Mills theory on the four-dimensional curved spacetime. Motivated by the fact that a positive spatial curvature acts as an effective gluon mass we consider the compact Euclidean spacetime $S^1\times S^1\times S^2$, with the radius of the first circle determined by the temperature $a_1=(2\pi T)^{-1}$. We show that covariantly constant Yang-Mills fields on $S^2$ cannot be arbitrary but are rather a collection of monopole-antimonopole pairs. We compute the heat kernels of all relevant operators exactly and show that the gluon operator on such a background has negative modes for any compact semi-simple gauge group. We compute the infrared regularized effective action and apply the result for the computation of the entropy and the heat capacity of the quark-gluon gas. We compute the heat capacity for the gauge group SU(2N) for a field configuration of $N$ monopole-antimonopole pairs. We show that in the high-temperature limit the heat capacity is well defined in the infrared limit and exhibits a typical behavior of second-order phase transition $\sim (T-T_c)^{-3/2}$ with the critical temperature $T_c=(2\pi a)^{-1}$, where $a$ is the radius of the 2-sphere $S^2$.

On quasi-local charges and Newman–Penrose type quantities in Yang–Mills theories

We generalize the notion of quasi-local charges, introduced by P Tod for Yang–Mills fields with unitary gauge groups, to non-Abelian gauge theories with arbitrary gauge groups, and calculate its small sphere and large sphere limits both at spatial and null infinity. We show that for semisimple gauge groups no reasonable definition yield conserved total charges and Newman–Penrose (NP) type quantities at null infinity in generic, radiative configurations. The conditions of their conservation, both in terms of the field configurations and the structure of the gauge group, are clarified. We also calculate the NP quantities for stationary, asymptotic solutions of the field equations with vanishing magnetic charges, and illustrate these by explicit solutions with various gauge groups.

A bound on 6D 𝒩 = 1 supergravities

We prove that there are only finitely many distinct semi-simple gauge groups and matter representations possible in consistent 6D chiral (1,0) supergravity theories with one tensor multiplet. The proof relies only on features of the low-energy theory; the consistency conditions we impose are that anomalies should be cancelled by the Green-Schwarz mechanism, and that the kinetic terms for all fields should be positive in some region of moduli space. This result does not apply to the case of the non-chiral (1,1) supergravities, which are not constrained by anomaly cancellation.

Renormalisability of the matter determinants in noncommutative gauge theory in the enveloping-algebra formalism

We consider noncommutative gauge theory defined by means of Seiberg–Witten maps for an arbitrary semisimple gauge group. We compute the one-loop UV divergent matter contributions to the gauge field effective action to all orders in the noncommutative parameters θ. We do this for Dirac fermions and complex scalars carrying arbitrary representations of the gauge group. We use path-integral methods in the framework of dimensional regularisation and consider arbitrary invertible Seiberg–Witten maps that are linear in the matter fields. Surprisingly, it turns out that the UV divergent parts of the matter contributions are proportional to the noncommutative Yang–Mills action where traces are taken over the representation of the matter fields; this result supports the need to include such traces in the classical action of the gauge sector of the noncommutative theory.