Non-compact Gauge Groups Research Articles

Non-compact gauge groups, tensor fields and Yang-Mills-Einstein amplitudes

Scattering amplitudes in Yang-Mills-Einstein theories have been investigated mostly for compact gauge groups. While non-compact gauge groups are not physically viable in Yang-Mills theory, non-compact gaugings feature prominently in the supergravity literature, where any choice of perturbative vacuum spontaneously breaks the gauge group to a compact subgroup. In this paper, we formulate double-copy constructions for several five-dimensional 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 supergravities with non-compact gauge groups. On one side of the double copy, we employ amplitudes from a super-Yang-Mills theory with a massive hypermultiplet. On the other, we use amplitudes from particular non-supersymmetric Yang-Mills-scalar theories with massive fermions, chosen to obey constraints coming from color/kinematics duality. Supergravities with massive self-dual tensors in five dimensions are also considered, showing that tensors are straightforwardly realized as double copies of gauge-theory fermions with suitable choices of signs in the corresponding solutions of the Dirac equation. We present several examples of these constructions, noting in particular the appearance of Heisenberg groups in the supergravity gauge symmetry and, in some cases, the possibility of exotic tensor-vector matter couplings.

On Yang–Mills fields from anti-de Sitter spaces

Motivated by some recent progress involving a non-compact gauge group, we obtain classical gauge fields using non-compact foliations of anti-de Sitter space in 4 dimensions (required dimensionality for conformal invariance of Yang–Mills theory) and transfer these to Minkowski spacetime using a series of conformal maps. This construction also builds upon some previous works involving SU(2) gauge group in that we now use its non-compact counterpart SU(1, 1) here. We note down gauge fields in both Abelian as well as non-Abelian settings and find them to be divergent at some hyperboloid, which is a hypersurface of co-dimension 1 inside the conformal boundary of AdS4. In spite of this hurdle we find a physically relevant field configuration in the Abelian case, reproducing a known result.

3 definitions of BF theory on homology 3-spheres

3-dimensional BF theory with gauge group G (= Chern-Simons theory with non-compact gauge group TG) is a deceptively simple yet subtle topological gauge theory. Formally, its partition function is a sum/integral over the moduli space mathcal{M} of flat connections, weighted by the Ray-Singer torsion. In practice, however, this formal expression is almost invariably singular and ill-defined.In order to improve upon this, we perform a direct evaluation of the path integral for certain classes of 3-manifolds (namely integral and rational Seifert homology spheres). By a suitable choice of gauge, we sidestep the issue of having to integrate over mathcal{M} and reduce the partition function to a finite-dimensional Abelian matrix integral which, however, itself requires a definition. We offer 3 definitions of this integral, firstly via residues, and then via a large k limit of the corresponding G × G or Gℂ Chern-Simons matrix integrals (obtained previously). We then check and discuss to which extent the results capture the expected sum/integral over all flat connections.

Optimal regularity and Uhlenbeck compactness for general relativity and Yang–Mills theory

We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of general relativity (GR). Compactness is the essential tool of mathematical analysis for establishing the validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with L p curvature. Solutions of the RT-equations furnish coordinate and gauge transformations which give a non-optimal connection a gain of one derivative over its Riemann curvature (i.e. to optimal regularity). The RT-equations are elliptic regardless of metric signature, and regularize singularities in solutions of the hyperbolic Einstein equations. As an application, singularities at GR shock waves are removable, implying geodesic curves, locally inertial coordinates and the Newtonian limit all exist. By the extra derivative, we extend Uhlenbeck compactness from Uhlenbeck’s setting of vector bundles with compact gauge groups over Riemannian manifolds, to the case of compact and non-compact gauge groups over non-Riemannian manifolds. Our version of Uhlenbeck compactness can also be viewed as a ‘geometric’ improvement of the Div-Curl Lemma, improving weak continuity of wedge products to strong convergence.

Five-branes wrapped on topological disks from 7D N=2 gauged supergravity

We study supersymmetric $AdS_5\times \Sigma$ solutions, for $\Sigma$ being a topological disk with non-trivial $U(1)$ holonomy at the boundary or "half-spindle", in seven-dimensional $N=2$ gauged supergravity coupled to three vector multiplets. We consider compact and non-compact gauge groups, $SO(4-p,p)$, for $p=0,1,2$, and find a number of $AdS_5\times \Sigma$ solutions with $SO(2)\times SO(2)$ and $SO(2)_{\text{diag}}$ residual symmetry from $SO(4)$ and $SO(2,2)$ gauge groups. We also find an $SO(2)_R\subset SO(3)_R\sim SU(2)_R$ symmetric solution which can be regarded as a solution of pure $N=2$ gauged supergravity with $SO(3)_R$ gauge group and all the fields from vector multiplets vanishing. The solutions preserve $\frac{1}{2}$ of the original $N=2$ supersymmetry and could be interpreted as supergravity duals of $N=1$ superconformal field theories in four dimensions. In particular, some of these solutions can be embedded in ten or eleven dimensions in which a description in terms of five-branes wrapped on a topological disk can be given.

Non-invertible global symmetries and completeness of the spectrum

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

Supersymmetric AdS_5 black holes and strings from 5D N=4 gauged supergravity

We study supersymmetric AdS_3times Sigma _2 and AdS_2times Sigma _3 solutions, with Sigma _2=S^2,H^2 and Sigma _3=S^3,H^3, in five-dimensional N=4 gauged supergravity coupled to five vector multiplets. The gauge groups considered here are U(1)times SU(2)times SU(2), U(1)times SO(3,1) and U(1)times SL(3,mathbb {R}). For U(1)times SU(2)times SU(2) gauge group admitting two supersymmetric N=4AdS_5 vacua, we identify a new class of AdS_3times Sigma _2 and AdS_2times H^3 solutions preserving four supercharges. Holographic RG flows describing twisted compactifications of N=2 four-dimensional SCFTs dual to the AdS_5 vacua to the SCFTs in two and one dimensions dual to these geometries are numerically given. The solutions can also be interpreted as supersymmetric black strings and black holes in asymptotically AdS_5 spaces with near horizon geometries given by AdS_3times Sigma _2 and AdS_2times H^3, respectively. These solutions broaden previously known black brane solutions including half-supersymmetric AdS_5 black strings recently found in N=4 gauged supergravity. Similar solutions are also studied in non-compact gauge groups U(1)times SO(3,1) and U(1)times SL(3,mathbb {R}).

Holographic RG flows and AdS_5 black strings from 5D half-maximal gauged supergravity

We study five-dimensional N=4 gauged supergravity coupled to five vector multiplets with compact and non-compact gauge groups U(1)times SU(2)times SU(2) and U(1)times SO(3,1). For U(1)times SU(2)times SU(2) gauge group, we identify N=4AdS_5 vacua with U(1)times SU(2)times SU(2) and U(1)times SU(2)_{text {diag}} symmetries and analytically construct the corresponding holographic RG flow interpolating between these critical points. The flow describes a deformation of the dual N=2 SCFT driven by vacuum expectation values of dimension-two operators. In addition, we study AdS_3times Sigma _2 geometries, for Sigma _2 being a two-sphere S^2 or a two-dimensional hyperbolic space H^2, dual to twisted compactifications of N=2 SCFTs with flavor symmetry SU(2). We find a number of AdS_3times H^2 solutions preserving eight supercharges for different twists from U(1)times U(1)times U(1) and U(1)times U(1)_{text {diag}} gauge fields. We numerically construct various RG flow solutions interpolating between N=4AdS_5 critical points and these AdS_3times H^2 geometries in the IR. The solutions can also be interpreted as supersymmetric black strings in asymptotically AdS_5 space. These types of holographic solutions are also studied in non-compact U(1)times SO(3,1) gauge group. In this case, only one N=4AdS_5 vacuum exists, and we give an RG flow solution from this AdS_5 to a singular geometry in the IR corresponding to an N=2 non-conformal field theory. An AdS_3times H^2 solution together with an RG flow between this vacuum and the N=4AdS_5 are also given.

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.

Manton’s five vortex equations from self-duality

We demonstrate that the five vortex equations recently introduced by Manton arise as symmetry reductions of the anti-self-dual Yang–Mills equations in four dimensions. In particular the Jackiw–Pi vortex and the Ambjørn–Olesen vortex correspond to the gauge group , and respectively the Euclidean or the symmetry groups acting with two-dimensional orbits. We show how to obtain vortices with higher vortex numbers, by superposing vortex equations of different types. Finally we use the kinetic energy of the Yang–Mills theory in 4 + 1 dimensions to construct a metric on vortex moduli spaces. This metric is not positive-definite in cases of non-compact gauge groups.

Noncompact gauging of N=2 7D supergravity and AdS/CFT holography

Half-maximal gauged supergravity in seven dimensions coupled to $n$ vector multiplets contains $n+3$ vectors and $3n+1$ scalars parametrized by $\mathbb{R}^+\times SO(3,n)/SO(3)\times SO(n)$ coset manifold. The two-form field in the gravity multiplet can be dualized to a three-form field which admits a topological mass term. Possible non-compact gauge groups take the form of $G_0\times H\subset SO(3,n)$ with a compact group $H$. $G_0$ is one of the five possibilities; $SO(3,1)$, $SL(3,\mathbb{R})$, $SO(2,2)$, $SO(2,1)$ and $SO(2,2)\times SO(2,1)$. We investigate all of these possible non-compact gauge groups and classify their vacua. Unlike the gauged supergravity without a topological mass term, there are new supersymmetric $AdS_7$ vacua in the $SO(3,1)$ and $SL(3,\mathbb{R})$ gaugings. These correspond to new $N=(1,0)$ superconformal field theories (SCFT) in six dimensions. Additionally, we find a class of $AdS_5\times S^2$ and $AdS_5\times H^2$ backgrounds with $SO(2)$ and $SO(2)\times SO(2)$ symmetries. These should correspond to $N=1$ SCFTs in four dimensions obtained from twisted compactifications of six-dimensional field theories on $S^2$ or $H^2$. We also study RG flows from six-dimensional $N=(1,0)$ SCFT to $N=1$ SCFT in four dimensions and RG flows from a four-dimensional $N=1$ SCFT to a six-dimensional SYM in the IR. The former are driven by a vacuum expectation value of a dimension-four operator dual to the supergravity dilaton while the latter are driven by vacuum expectation values of marginal operators.

Consistent Kaluza-Klein truncations via exceptional field theory

We present the generalized Scherk-Schwarz reduction ansatz for the full supersymmetric exceptional field theory in terms of group valued twist matrices subject to consistency equations. With this ansatz the field equations precisely reduce to those of lower-dimensional gauged supergravity parametrized by an embedding tensor. We explicitly construct a family of twist matrices as solutions of the consistency equations. They induce gauged supergravities with gauge groups SO(p,q) and CSO(p,q,r). Geometrically, they describe compactifications on internal spaces given by spheres and (warped) hyperboloides $H^{p,q}$, thus extending the applicability of generalized Scherk-Schwarz reductions beyond homogeneous spaces. Together with the dictionary that relates exceptional field theory to D=11 and IIB supergravity, respectively, the construction defines an entire new family of consistent truncations of the original theories. These include not only compactifications on spheres of different dimensions (such as AdS$_5\times S^5$), but also various hyperboloid compactifications giving rise to a higher-dimensional embedding of supergravities with non-compact and non-semisimple gauge groups.

Disk entanglement entropy for a Maxwell field

In three dimensions, the pure Maxwell theory with compact U(1) gauge group is dual to a free compact scalar, and flows from the Maxwell theory with non-compact gauge group in the ultraviolet to a non-compact free massless scalar theory in the infrared. We compute the vacuum disk entanglement entropy all along this flow, and show that the renormalized entropy F(r) decreases monotonically with the radius r as predicted by the F-theorem, interpolating between a logarithmic growth for small r (matching the behavior of the S^3 free energy) and a constant at large r (equal to the free energy of the conformal scalar). The calculation is carried out by the replica trick, employing the scalar formulation of the theory. The Renyi entropies for n>1 are given by a sum over winding sectors, leading to a Riemann-Siegel theta function. The extrapolation to n=1, to obtain the von Neumann entropy, is done by analytic continuation in the large- and small-r limits and by a numerical extrapolation method at intermediate values. We also compute the leading contribution to the renormalized entanglement entropy of the compact free scalar in higher dimensions. Finally, we point out some interesting features of the reduced density matrix for the compact scalar, and its relation to that for the non-compact theory.

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.

3D N = 6 gauged supergravity: admissible gauge groups, vacua and RG flows

We study N=6 gauged supergravity in three dimensions with scalar manifolds $\frac{SU(4,k)}{S(U(4)\times U(k))}$ for $k=1,2,3,4$ in great details. We classify some admissible non-compact gauge groups which can be consistently gauged and preserve all supersymmetries. We give the explicit form of the embedding tensors for these gauge groups as well as study their scalar potentials on the full scalar manifold for each value of $k=1,2,3,4$ along with the corresponding vacua. Furthermore, the potentials for the compact gauge groups, $SO(p)\times SO(6-p)\times SU(k)\times U(1)$ for $p=3,4,5,6$, identified previously in the literature are partially studied on a submanifold of the full scalar manifold. This submanifold is invariant under a certain subgroup of the corresponding gauge group. We find a number of supersymmetric AdS vacua in the case of compact gauge groups. We then consider holographic RG flow solutions in the compact gauge groups $SO(6)\times SU(4)\times U(1)$ and $SO(4)\times SO(2)\times SU(4)\times U(1)$ for the k=4 case. The solutions involving one active scalar can be found analytically and describe operator flows driven by a relevant operator of dimension 3/2. For non-compact gauge groups, we find all types of vacua namely AdS, Minkowski and dS, but there is no possibility of RG flows in the AdS/CFT sense for all gauge groups considered here.

Vacua of N = 10 three-dimensional gauged supergravity

We study scalar potentials and the corresponding vacua of N = 10 three-dimensional gauged supergravity. The theory contains 32 scalar fields parametrizing the exceptional coset space . The admissible gauge groups considered in this work involve both compact and non-compact gauge groups which are maximal subgroups of SO(10) × U(1) and E6( − 14), respectively. These gauge groups are given by SO(p) × SO(10 − p) × U(1) for p = 6, …, 10, SO(5) × SO(5), SU(4, 2) × SU(2), G2( − 14) × SU(2, 1) and F4( − 20). We find many AdS3 critical points with various unbroken gauge symmetries. The relevant background isometries associated with the maximally supersymmetric critical points at which all scalars vanish are also given. These correspond to the superconformal symmetries of the dual conformal field theories in two dimensions.

INTEGRATION OVER CONNECTIONS IN THE DISCRETIZED GRAVITATIONAL FUNCTIONAL INTEGRALS

The result of performing integrations over connection type variables in the path integral for the discrete field theory may be poorly defined in the case of non-compact gauge group with the Haar measure exponentially growing in some directions. This point is studied in the case of the discrete form of the first-order formulation of the Einstein gravity theory. Here the result of interest can be defined as generalized function (of the rest of variables of the type of tetrad or elementary areas), i.e. a functional on a set of probe functions. To define this functional, we calculate its values on the products of components of the area tensors, the so-called moments. The resulting distribution (in fact, probability distribution) has singular (δ-function-like) part with support in the nonphysical region of the complex plane of area tensors and regular part (usual function) which decays exponentially at large areas. As we discuss, this also provides suppression of large edge lengths which is important for internal consistency, if one asks whether gravity on short distances can be discrete. Some other features of the obtained probability distribution including occurrence of the local maxima at a number of the approximately equidistant values of area are also considered.

Noncompact quantum knot invariants

We describe one avenue to the explicit calculation of partition functions of knot complements in Chern-Simons theory with noncompact gauge group S L ( 2 , C ) , following [T.D. Dimofte, S. Gukov, J. Lenells, and D. Zagier, to appear]. Our techniques involve geometric quantization of the moduli space of flat connections on the torus, combined with quantization of the classical A-polynomial of a knot complement. We also compare these methods to known results for compact gauge group SU (2).

Nonsingular solutions of Hitchin's equations for noncompact gauge groups

We consider a general ansatz for solving the 2-dimensional Hitchin's equations, which arise as dimensional reduction of the 4-dimensional self-dual Yang–Mills equations, with remarkable integrability properties. We focus on the case when the gauge group G is given by a real form of . For G = SO(2,1), the resulting field equations are shown to reduce to either the Liouville, elliptic sinh-Gordon or elliptic sine-Gordon equations. As opposed to the compact case, given by G = SU(2), the field equations associated with the noncompact group SO(2,1) are shown to have smooth real solutions with nonsingular action densities, which are furthermore localized in some sense. We conclude by discussing some particular solutions, defined on , S2 and T2, that come out of this ansatz.

De Sitter solutions inN= 4 matter coupled supergravity

We investigate the scalar potential of gauged N=4 supergravity with matter. The extremum in the SU(1,1)/U(1) scalars is obtained for an arbitrary number of matter multiplets. The constraints on the matter scalars are solved in terms of an explicit parametrisation of an SO(6,6+n) element. For the case of six matter multiplets we discuss both compact and noncompact gauge groups. In an example involving noncompact groups and four scalars we find a potential with an absolute minimum and a positive cosmological constant.