Options for gauge groups in five-dimensional supergravity

Quantum mechanics of Klein–Gordon fields I: Hilbert Space, localized states, and chiral symmetry

This paper describes CosmoLattice, a modern package for lattice simulations of the dynamics of interacting scalar and gauge fields in an expanding universe. CosmoLattice incorporates a series of features that makes it very versatile and powerful: i) it is written in C++ fully exploiting the object oriented programming paradigm, with a modular structure and a clear separation between the physics and the technical details, ii) it is MPI-based and uses a discrete Fourier transform parallelized in multiple spatial dimensions, which makes it specially appropriate for probing scenarios with well-separated scales, running very high resolution simulations, or simply very long ones, iii) it introduces its own symbolic language, defining field variables and operations over them, so that one can introduce differential equations and operators in a manner as close as possible to the continuum, iv) it includes a library of numerical algorithms, ranging from O(δt2) to O(δt10) methods, suitable for simulating global and gauge theories in an expanding grid, including the case of ‘self-consistent’ expansion sourced by the fields themselves. Relevant observables are provided for each algorithm (e.g. energy densities, field spectra, lattice snapshots) and we note that, remarkably, all our algorithms for gauge theories (Abelian or non-Abelian) always respect the Gauss constraint to machine precision. Program summaryProgram Title::CosmoLatticeCPC Library link to program files:https://doi.org/10.17632/44vr5xssc6.1Developer's repository link:http://github.com/cosmolattice/cosmolatticeLicensing provisions: MITProgramming language: C++, MPINature of problem: The phenomenology of high energy physics in the early universe is typically characterized by non-linear dynamics, which cannot be captured accurately with analytical techniques. In order to fully understand the non-linearities developed in a given scenario, one needs to carry out lattice simulations. A number of public packages for lattice simulations have appeared over the years, but most of them are only capable of simulating scalar fields. However, realistic models of particle physics do contain other kind of field species, such as (Abelian or non-Abelian) gauge fields, whose non-linear dynamics can also play a relevant role in the early universe. Tensor modes representing gravitational waves are also naturally expected in many scenarios.Solution method:CosmoLattice represents a modern code for lattice simulations of scalar-gauge field theories in an expanding universe. It allows for the simulation of the evolution of interacting (singlet) scalar fields, charged scalar fields under U(1) and/or SU(2) gauge groups, and the corresponding associated Abelian and/or non-Abelian gauge fields. From version 1.1 onward, CosmoLattice also allows to simulate the production of gravitational waves. Simulations can be done either in a flat space-time background, or in a homogeneous and isotropic (spatially flat) expanding FLRW background. CosmoLattice provides symplectic integrators, with accuracy ranging from O(δt2) up to O(δt10), to simulate the non-linear dynamics of the appropriate fields in comoving three-dimensional lattices. The code is parallelized with MPI, and uses a discrete Fourier Transform parallelized in multiple spatial dimensions, which makes it a very powerful code for probing physical problems with well-separated scales. Moreover, the code has been designed as a ‘platform’ to implement any system of dynamical equations suitable for discretization on a lattice.

One of the fundamental problems of gravity to be an infinite range interaction is whether it has a gauge group via the internal structure, so that Einstein type theories couldn’t explain it yet. Gauge groups in Yang Mill type theories, i.e. weak [11] and strong [3, 8] interactions, are generally non Abelian Lie group via the Hermitian structure induced by complex valued smooth vector fields. A group labeled by U(1)⊗Ũ(d) obtained as the reduction of unitary Lie groups is presented as candidate a gauge group for each behavior at large scale and at small scale of a gauge field, in this context, integrated Einstein Yang Mills formalism on a unique geometrical frame is given. Also as seen that the (anti) self dual gauge theory of the group G = U(1) ⊗ Ũ(4) is indeed a G = U(1) theory in real 4 dimension, i.e the Maxwell theory for the electromagnetism to be an infinite range interaction. Mathematics Subject Classification: 22E15, 53C07, 53C80, 70S05, 70S15, 83C05

We explore the consequences of the hypothesis that $B\ensuremath{-}L$ symmetry, which is exact in the standard model based on the $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(3)}_{c}$ gauge group and its extension to the minimal SU(5) theory, is a spontaneously broken global symmetry. This results in possibly observable rates for $\ensuremath{\Delta}(B\ensuremath{-}L)=0$, $\ensuremath{\Delta}B=2$ hydrogen-antihydrogen transitions and double proton decay $\mathrm{pp}\ensuremath{\rightarrow}{e}^{+}{e}^{+}({\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{+})$ and $\ensuremath{\Delta}(B\ensuremath{-}L)\ensuremath{\ne}0$, $\ensuremath{\Delta}B=2$ neutron-antineutron mixing and $p+n\ensuremath{\rightarrow}\ensuremath{\pi}'\mathrm{s}$; furthermore, a relation between the transition times ${\ensuremath{\tau}}_{\mathrm{H}\ensuremath{-}\overline{\mathrm{H}}}$, ${\ensuremath{\tau}}_{n\ensuremath{-}\overline{n}}$, Majorana neutrino mass, and the mass of a doubly charged Higgs particle ${\ensuremath{\Delta}}^{++}$ emerges. At the level of $\mathrm{SU}{(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(3)}_{c}$, for ${\ensuremath{\tau}}_{n\ensuremath{-}\overline{n}}\ensuremath{\simeq}{10}^{7}$ sec and ${m}_{{\ensuremath{\Delta}}^{++}}\ensuremath{\simeq}20$ GeV, we predict ${\ensuremath{\tau}}_{\mathrm{H}\ensuremath{-}\overline{\mathrm{H}}}\ensuremath{\simeq}{10}^{3}$ yr and ${\ensuremath{\tau}}_{\mathrm{pp}}\ensuremath{\simeq}{10}^{31}$ yr, both of which are experimentally testable numbers. The embedding of our model in SU(5) is discussed in detail, with the analysis of ${\ensuremath{\tau}}_{p}$ and ${{sin}^{2}\ensuremath{\theta}}_{W}$ besides matter oscillations.

We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,\nu_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $\nu_d$ an element in group cohomology $\cH^d(G,\R/\Z)$. When $d=3$ and/or when $G_g$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e. gapped long-range entangled phases with symmetry). Thus, those SET phases are classified by $\cH^d(G,\R/\Z)$, where $G/G_g=G_s$. We also apply our theory to a simple case $G_s=G_g=Z_2$, which leads to 12 different SET phases in 2+1D, where quasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry $G_s$, which may lead to different fractionalizations of $G_s$ quantum numbers and different fractional statistics (if in 2+1D).

We determine the phase structure of an $SU(2)$ gauge theory with an adjoint scalar on ${R}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ using semiclassical methods. There are two global symmetries: a $Z(2{)}_{H}$ symmetry associated with the Higgs field and a $Z(2{)}_{C}$ center symmetry associated with the Polyakov loop in the compact direction. The order of the deconfining phase transition can be either second-order or first-order for $SU(2)$, depending on the deformation used. After finding order parameters for the global symmetries, we show that there are four distinct phases: a deconfined phase, a confined phase, a Higgs phase, and a mixed confined phase. The mixed confined phase occurs where one might expect a phase in which there is both confinement and the Higgs mechanism, but the behavior of the order parameters distinguishes the two phases. In the mixed confined phase, the $Z(2{)}_{C}\ifmmode\times\else\texttimes\fi{}Z(2{)}_{H}$ global symmetry breaks spontaneously to a $Z(2)$ subgroup that acts nontrivially on both the scalar field and the Polyakov loop. We find explicitly the BPS and KK monopole solutions of the Euclidean field equations in the BPS limit; these monopoles are extensions of similar pure gauge theory solutions, where they are constituents of instantons. In the mixed phase, a linear combination of the Higgs field $\ensuremath{\phi}$ and ${A}_{4}$, the component of the gauge field in the compact direction, enters into the monopole solutions. In all four phases, Wilson loops orthogonal to the compact direction are expected to show area-law behavior. We show that this confining behavior can be attributed to a dilute monopole gas in a broad region that includes portions of all four phases. The dilute monopole gas picture breaks down when the action of a BPS monopole is zero. A duality argument similar to that applied recently [40] to the Seiberg-Witten model on ${R}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ shows that the monopole gas picture, arrived at using Euclidean instanton methods, can be interpreted as a gas of finite-energy dyons.

We construct the Green-Schwarz terms of six-dimensional supergravity theories on spacetimes with non-trivial topology and gauge bundle. We prove the cancellation of all global gauge and gravitational anomalies for theories with gauge groups given by products of $U(n)$, $SU(n)$ and $Sp(n)$ factors, as well as for $E_8$. For other gauge groups, anomaly cancellation is equivalent to the triviality of a certain 7-dimensional spin topological field theory. We show in the case of a finite Abelian gauge group that there are residual global anomalies imposing constraints on the 6d supergravity. These constraints are compatible with the known F-theory models. Interestingly, our construction requires that the gravitational anomaly coefficient of the 6d supergravity theory is a characteristic element of the lattice of string charges, a fact true in six-dimensional F-theory compactifications but that until now was lacking a low-energy explanation. We also discover a new anomaly coefficient associated with a torsion characteristic class in theories with a disconnected gauge group.

Surface quasi-geostrophic (SQG) theory describes the two-dimensional active transport of a scalar field, such as temperature, which – when properly rescaled – shares the same physical dimension of length/time as the advecting velocity field. This duality has motivated analogies with fully developed three-dimensional turbulence. In particular, the Kraichnan – Leith – Batchelor similarity theory predicts a Kolmogorov-type inertial range scaling for both scalar and velocity fields, and the presence of intermittency through multifractal scaling was pointed out by Sukhatme & Pierrehumbert (2002 Chaos 12 , 439–450), in unforced settings. In this work, we refine the discussion of these statistical analogies, using numerical simulations with up to $16\,384^2$ collocation points in a steady-state regime dominated by the direct cascade of scalar variance. We show that mixed structure functions, coupling velocity increments with scalar differences, develop well-defined scaling ranges, highlighting the role of anomalous fluxes of all the scalar moments. However, the clean multiscaling properties of SQG transport are blurred when considering velocity and scalar fields separately. In particular, the usual (unmixed) structure functions do no follow any power-law scaling in any range of scales, neither for the velocity nor for the scalar increments. This specific form of the intermittency phenomenon reflects the specific kinematic properties of SQG turbulence, involving the interplay between long-range interactions, structures and geometry. Revealing the multiscaling in single-field statistics requires us to resort to generalised notions of scale invariance, such as extended self-similarity and a specific form of refined self-similarity. Our findings emphasise the fundamental entanglement of scalar and velocity fields in SQG turbulence: they evolve hand in hand and any attempt to isolate them destroys scaling in its usual sense. This perspective sheds new lights on the discrepancies in spectra and structure functions that have been repeatedly observed in SQG numerics for the past 20 years.

M5 branes on a D-type ALE singularity display various phenomena that introduce additional massless degrees of freedom. The M5 branes are known to fractionate on a D-type singularity. Whenever two fractional M5 branes coincide, tensionless strings arise. Therefore, these systems do not admit a low-energy Lagrangian description. Focusing on the 6-dimensional mathcal{N} = (1, 0) world-volume theories on the M5 branes, the vacuum moduli space has two branches were either the scalar fields in the tensor multiplet or the scalars in the hypermultiplets acquire a non-trivial vacuum expectation value. As suggested in previous work, the Higgs branch may change drastically whenever a BPS-string becomes tensionless. Recently, magnetic quivers have been introduced with the aim to capture all Higgs branches over any point of the tensor branch. In this paper, the formalism is extended to Type IIA brane configurations involving O6 planes. Since the 6d mathcal{N} = (1, 0) theories are composed of orthosymplectic gauge groups, the derivation rules for the magnetic quiver in the presence of O6 planes have to be conjectured. This is achieved by analysing the 6d theories for a single M5 brane on a D-type singularity and deriving the magnetic quivers for the finite and infinite gauge coupling Higgs branch from a brane configuration. The validity of the proposed derivation rules is underpinned by deriving the associated Hasse diagram. For multiple M5 branes, the approach of this paper provides magnetic quivers for all Higgs branches over any point of the tensor branch. In particular, an interesting infinite gauge coupling transition is found that is related to the SO(8) non-Higgsable cluster.

We study critical points of $F(4)$ gauged supergravity in six dimensions coupled to three vector multiplets. Scalar fields are described by $\mathbb{R}^+\times \frac{SO(4,3)}{SO(4)\times SO(3)}$ coset space, and the gauge group is given by $SO(3)_R\times SO(3)$ with $SO(3)_R$ being the R-symmetry. The maximally supersymmetric critical point with all scalars vanishing preserves the full $SO(3)_R\times SO(3)$ symmetry. This is dual to a superconformal field theories (SCFT$_5$) arising from a near horizon geometry of the D4-D8 brane system in type I$'$ theory with an enhanced global symmetry $E_1\sim SU(2)$. Apart from this trivial critical point, we identify a new supersymmetric critical point preserving the full supersymmetry with the $SO(3)_R\times SO(3)$ symmetry broken to its diagonal subgroup. This critical point should correspond to a new SCFT in five dimensions. We study an RG flow solution interpolating between the SCFT with $E_1$ symmetry and the new supersymmetric critical point. The flow describes a supersymmetric deformation driven by a vacuum expectation value of relevant operators of dimension $3$. We identify the dual operators with the mass terms for hypermultiplet scalars in the dual field theory. The solution provides an example of analytic supersymmetric RG flows in AdS$_6$/CFT$_5$ correspondence.

We study critical points of F (4) gauged supergravity in six dimensions coupled to three vector multiplets. Scalar fields are described by $ {{\mathbb{R}}^{+}}\times \frac{{\mathrm{SO}\left( {4,3} \right)}}{{\mathrm{SO}(4)\times \mathrm{SO}(3)}} $ coset space, and the gauge group is given by SO(3)R × SO(3) with SO(3)R being the R-symmetry. We identify new non-supersymmetric critical points of the scalar potential. One of these new critical points is shown to be stable with all scalar masses above the BF bound and should correspond to a new non-supersymmetric CFT in five dimensions. On the other hand, the maximally supersymmetric critical point with all scalars vanishing is dual to an SCFT5 arising from a near horizon geometry of the D4-D8 brane system in type I′ theory with an enhanced global symmetry E 1 ~ SU(2). We give a numerical RG flow solution interpolating between this SCFT and the new stable critical point. The flow describes a non-supersymmetric deformation driven by relevant operators of dimension 3. We identify the dual operators with the mass terms for hypermultiplet scalars in the dual field theory. The solution provides another example of holographic RG flows in AdS6/CFT5 correspondence.

We consider an extension of the Standard Model within the framework of Noncommutative Geometry. The model is based on an older model [C. A. Stephan, Phys. Rev. D79, 065013 (2009)] which extends the Standard Model by new fermions, a new U(1)-gauge group and, crucially, a new scalar field which couples to the Higgs field. This new scalar field allows to lower the mass of the Higgs mass from ~170 GeV, as predicted by the Spectral Action for the Standard Model, to a value of 120–130 GeV. The shortcoming of the previous model lay in its inability to meet all the constraints on the gauge couplings implied by the Spectral Action. These shortcomings are cured in the present model which also features a "dark sector" containing fermions and scalar particles.

We study a gauge theory in a 5D warped space via the dimensional deconstruction that a higher dimensional gauge theory is constructed from a moose of 4D gauge groups. By the AdS/CFT correspondence, a 5D warped gauge theory is dual to a 4D conformal field theory (CFT) with a global symmetry. As far as physics of the gauge theory, we obtain the one-to-one correspondence between each component of a moose of gauge groups and that of a CFT. We formulate a supersymmetric extension of deconstruction and explore the framework of natural supersymmetry in a 5D warped space — the super-symmetric Randall-Sundrum model with the IR-brane localized Higgs and bulk fermions — via the gauge moose. In this model, a supersymmetry breaking source is located at the end of the moose corresponding to the UV brane and the first two generations of squarks are decoupled. With left-right gauge symmetries in the bulk of the moose, we demonstrate realization of accidental or emergent supersymmetry of the Higgs sector in comparison with the proposed “Moose/CFT correspondence.”

We show that there are two supersymmetric completions of the three-dimensional Chern-Simons theory of level k with gauge group U(N)xU(N) coupled to four sets of massless scalars and spinors in the bi-fundamental representation, if we require Sp(2) global symmetry with the matter fields in the fundamental representation of SU(4). One is the N=6 superconformal theory recently studied in arXiv:0806.1218 [hep-th] and another is a new theory with N=1 superconformal symmetry. We conjecture that the N=1 theory is dual to M theory on AdS_4 x Squashed S^7/Z_k.

Is CP a gauge symmetry?