Subset SU Research Articles

Anomaly and global inconsistency matching: θ angles, SU(3)/U(1)2 nonlinear sigma model, SU(3) chains, and generalizations

We discuss the $SU(3)/[U(1)\times U(1)]$ nonlinear sigma model in 1+1D and, more broadly, its linearized counterparts. Such theories can be expressed as $U(1)\times U(1)$ gauge theories and therefore allow for two topological $\theta$-angles. These models provide a field theoretic description of the $SU(3)$ chains. We show that, for particular values of $\theta$-angles, a global symmetry group of such systems has a 't Hooft anomaly, which manifests itself as an inability to gauge the global symmetry group. By applying anomaly matching, the ground-state properties can be severely constrained. The anomaly matching is an avatar of the Lieb-Schultz-Mattis (LSM) theorem for the spin chain from which the field theory descends, and it forbids a trivially gapped ground state for particular $\theta$-angles. We generalize the statement of the LSM theorem and show that 't Hooft anomalies persist even under perturbations which break the spin-symmetry down to the discrete subgroup $\mathbb Z_3\times\mathbb Z_3\subset SU(3)/\mathbb Z_3$. In addition the model can further be constrained by applying global inconsistency matching, which indicates the presence of a phase transition between different regions of $\theta$-angles. We use these constraints to give possible scenarios of the phase diagram. We also argue that at the special points of the phase diagram the anomalies are matched by the $SU(3)$ Wess-Zumino-Witten model. We generalize the discussion to the $SU(N)/U(1)^{N-1}$ nonlinear sigma models as well as the 't Hooft anomaly of the $SU(N)$ Wess-Zumino-Witten model, and show that they match. Finally the $(2+1)$-dimensional extension is considered briefly, and we show that it has various 't Hooft anomalies leading to nontrivial consequences.

Complex Hantzsche-Wendt manifolds

Complex Hantzsche-Wendt manifolds are flat Kähler manifolds with holonomy group {{mathrm{mathbb {Z}}}}_2^{n-1}subset SU(n). They are important examples of Calabi-Yau manifolds of abelian type. In this paper we describe them as quotients of a product of elliptic curves by a finite group tilde{G}. This will allow us to classify all possible integral holonomy representations and give an algorithm classifying their diffeomorphism types.

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.

Quantum critical behavior of semisimple gauge theories

We study the perturbative phase diagram of semi-simple fermionic gauge theories resembling the Standard Model. We investigate an $SU(N)$ gauge theory with $M$ Dirac flavors where we gauge first an $SU(M)_L$ and then an $SU(2)_L \subset SU(M)_L$ of the original global symmetry $SU(M)_L\times SU(M)_R \times U(1) $ of the theory. To avoid gauge anomalies we add lepton-like particles. At the two-loops level an intriguing phase diagram appears. We uncover phases in which one, two or three fixed points exist and discuss the associated flows of the coupling constants. We discover a phase featuring complete asymptotic freedom and simultaneously an interacting infrared fixed point in both couplings. The analysis further reveals special renormalisation group trajectories along which one coupling displays asymptotic freedom and the other asymptotic safety, while both flowing in the infrared to an interacting fixed point. These are \emph{safety free} trajectories. We briefly sketch out possible phenomenological implications, among which an independent way to generate near-conformal dynamics a l\'a walking is investigated.

Simultaneous selection of treatments better and worse than the best and worst controls

In this paper, we consider p(p≥2) and q(q≥2) independent treatment and control populations respectively, such that an appropriate probability model for the data from ith(jth) treatment (control) population is a member of absolutely continuous location and scale family of distributions which have common scale parameter and possibly differ in location parameters. For example, there may be p newly invented drugs/varieties of seeds/components which have to compete with their existing q standard competitors in terms of their average responses. A newly invented drug/variety of seed/component is said to be good (bad) if the distance of its average response from the largest (smallest) average response of q control populations is more (less) than δ1(δ2) units, where δ1 and δ2 are positive constants to be specified by the experimenter. In this setting a selection procedure is proposed to select simultaneously two subsets SU and SL of the p treatment populations such that the subset SU contains all the good treatments and the subset SL contains all the bad treatments with probability at least P∗, where P∗ is a pre-assigned value. The proposed procedure was applied to normal and two parameters exponential probability models separately and the relevant selection constants have been tabulated. The implementation of the proposed methodology is demonstrated through a numerical example based on real life data. The authenticity of numerically computed critical constants have been verified through simulation. Further, if we define the ith treatment population as bad (good) if the distance of its average response from the largest (smallest) average response of q control populations is less (more) than δ3(δ4) units, where δ3 and δ4 are to be specified by the experimenter such that δ4>δ3>0, then we have proposed a simultaneous selection procedure to select SU and SL and a sample size is determined so that the probability of omitting a good (bad) treatment population from SU(SL) or selecting a bad (good) treatment population in SU(SL) is at most 1−P∗.

Non-Abelian monopole-vortex complex

In the context of softly broken N=2 supersymmetric quantum chromodynamics (SQCD), with a hierarchical gauge symmetry breaking SU(N+1) -> U(N) -> 1, at scales v1 and v2, respectively, where v1 >> v2, we construct monopole-vortex complex soliton-like solutions and examine their properties. They represent the minimum of the static energy under the constraint that the monopole and antimonopole positions sitting at the extremes of the vortex are kept fixed. They interpolate the 't Hooft-Polyakov-like regular monopole solution near the monopole centers and a vortex solution far from them and in between. The main result, obtained in the theory with Nf=N equal-mass flavors, is concerned with the existence of exact orientational CP(N-1) zero modes, arising from the exact color-flavor diagonal SU(N)_{C+F} global symmetry. The "unbroken" subgroup SU(N) \subset SU(N+1) with which the na\"ive notion of non-Abelian monopoles and the related difficulties were associated, is explicitly broken at low energies. The monopole transforms nevertheless according to the fundamental representation of a new exact, unbroken SU(N) symmetry group, as does the vortex attached to it. We argue that this explains the origin of the dual non-Abelian gauge symmetry.

Flexibility of surface groups in classical simple Lie groups

We show that a surface group of high genus contained in a classical simple Lie group can be deformed to become Zariski dense, unless the Lie group is SU(p,q) (resp. SO^* (2n) , n odd) and the surface group is maximal in some S(U(p,p) \times U(q-p)) \subset SU(p,q) (resp. SO^* (2n-2) \times SO(2) \subset SO^* (2n) ). This is a converse, for classical groups, to a rigidity result of S. Bradlow, O. García-Prada and P. Gothen.

Closing theSU(3)L⊗U(1)Xsymmetry at the electroweak scale

We show that some models with $SU(3)_C\otimes SU(3)_L\otimes U(1)_X$ gauge symmetry can be realized at the electroweak scale and that this is a consequence of an approximate global $SU(2)_{L+R}$ symmetry. This symmetry implies a condition among the vacuum expectation value of one of the neutral Higgs scalars, the $U(1)_X$'s coupling constant, $g_X$, the sine of the weak mixing angle $\sin\theta_W$, and the mass of the $W$ boson, $M_W$. In the limit in which this symmetry is valid it avoids the tree level mixing of the $Z$ boson of the Standard Model with the extra $Z^\prime$ boson. We have verified that the oblique $T$ parameter is within the allowed range indicating that the radiative corrections that induce such a mixing at the 1-loop level are small. We also show that a $SU(3)_{L+R}$ custodial symmetry implies that in some of the models we have to include sterile (singlets of the 3-3-1 symmetry) right-handed neutrinos with Majorana masses, being the see-saw mechanism mandatory to obtain light active neutrinos. Moreover, the approximate $SU(2)_{L+R}\subset SU(3)_{L+R}$ symmetry implies that the extra non-standard particles of these 3-3-1 models can be considerably lighter than it had been thought before so that new physics can be really just around the corner.

Running couplings in equivariantly gauge-fixedSU(N)Yang-Mills theories

In equivariantly gauge-fixed SU(N) Yang--Mills theories, the gauge symmetry is only partially fixed, leaving a subgroup $H\subset SU(N)$ unfixed. Such theories avoid Neuberger's nogo theorem if the subgroup $H$ contains at least the Cartan subgroup $U(1)^{N-1}$, and they are thus non-perturbatively well defined if regulated on a finite lattice. We calculate the one-loop beta function for the coupling $\tilde{g}^2=\xi g^2$, where $g$ is the gauge coupling and $\xi$ is the gauge parameter, for a class of subgroups including the cases that $H=U(1)^{N-1}$ or $H=SU(M)\times SU(N-M)\times U(1)$. The coupling $\tilde{g}$ represents the strength of the interaction of the gauge degrees of freedom associated with the coset $SU(N)/H$. We find that $\tilde{g}$, like $g$, is asymptotically free. We solve the renormalization-group equations for the running of the couplings $g$ and $\tilde{g}$, and find that dimensional transmutation takes place also for the coupling $\tilde{g}$, generating a scale $\tilde{\Lambda}$ which can be larger than or equal to the scale $\Lambda$ associated with the gauge coupling $g$, but not smaller. We speculate on the possible implications of these results.

Octonions,G2Symmetry, Generalized Self-Duality and Supersymmetries in Dimensions D 8

We establish N=(1/8,1) supersymmetric Yang-Mills vector multiplet with generalized self-duality in Euclidian eight-dimensions with the original full SO(8) Lorentz covariance reduced to SO(7). The key ingredient is the usage of octonion structure constants made compatible with SO(7) covariance and chirality in 8D. By a simple dimensional reduction together with extra constraints, we derive N=1/8+7/8 supersymmetric self-dual vector multiplet in 7D with the full SO(7) Lorentz covariance reduced to G_2. We find that extra constraints needed on fields and supersymmetry parameter are not obtained from a simple dimensional reduction from 8D. We conjecture that other self-dual supersymmetric theories in lower dimensions D =6 and 4 with respective reduced global Lorentz covariances such as SU(3) \subset SO(6) and SU(2) \subset SO(4) can be obtained in a similar fashion.

Investigation of no-scale supersymmetry breaking models with a gaugedU(1)B−Lsymmetry

Noscale supersymmetry (SUSY) breaking model is investigated in the minimal extension of the minimal supersymmetric standard model (MSSM) with a gauged U(1)_{B-L} symmetry. We specifically consider a unification-inspired model with the gauge groups SU(3)_{C} \times SU(2)_{L} \times U(1)_Y \times U(1)_{B-L} \subset SU(5) \times U(1)_{5} for illustration. While the noscale boundary condition at the grand unification scale (M_{G}\simeq 2\times 10^{16} GeV) in the MSSM is not consistent with phenomenological constraints, we show that it is if the gaugino of the U(1)_{5} multiplet is several times heavier than the gauginos of the MSSM. However, if SU(5) \times U(1)_{5} is further embedded in a larger simple group, e.g. SO(10), the noscale boundary condition at M_{G} is inconsistent with phenomenological constraints. If we relax the noscale boundary condition and allow non-zero soft scalar masses for the Higgs fields which spontaneously break the U(1)_{5} symmetry, the resultant spectrum of SUSY particles becomes consistent with all the phenomenological constraints, even if we impose the GUT relation on the gauge coupling and the gaugino mass of the U(1)_{5}. In this case, the SUSY CP problem is also solved, since the condition B\mu =0 at the boundary can be imposed consistently with the electroweak symmetry breaking.

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

In this paper we further develop the theory of $\alpha$-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two ``chiral'' induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU(n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices i.e. on the ``physical spectrum'' of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU(n) and many conformal inclusions, covering in particular all type I modular invariants of SU(2) and SU(3), and we conjecture that it holds also for any other conformal inclusion of SU(n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion $SU(3)_5 \subset SU(6)_1$ is computed for the first time.

Branes at Bbb C4/Γ singularity from toric geometry

We study toric singularities of the form of $\C^4/\Ga$ for finite abelian groups $\Ga \subset SU(4)$. In particular, we consider the simplest case $\Ga=\Z_2 \times \Z_2 \times \Z_2$ and find explicitly charge matrices for partial resolutions of this orbifold by extending the method by Morrison and Plesser. We obtain three kinds of algebraic equations, $z_1 z_2 z_3 z_4=z_5^2, z_1 z_2 z_3=z_4^2 z_5 $ and $z_1 z_2 z_5 = z_3 z_4$ where $z_i$'s parametrize $\C^5$. When we put $N$ D1 branes at this singularity, it is known that the field theory on the worldvolume of $N$ D1 branes is T-dual to $2 \times 2 \times 2 $ brane cub model. We analyze geometric interpretation for field theory parameters and moduli space.

Applications of braided endomorphisms from conformal inclusions

We give three applications of general theory about braided endomorphisms from conformal inclusions developed previously by us. The first is an example of subfactors associated with conformal inclusion whose dual fusion ring is non-commutative. In the second application we show that the Kac-Wakimoto hypothesis about certain relations between branching rules and S-matrices, which has existed for almost a decade, is not true in at least three examples. Finally we show that the fusion rings of subfactors associated with the conformal inclusions $SU(n)_{(n+2)}\subset SU(n(n+1)/2)$ and $SU(n+2)_n\subset SU((n+1)(n+2)/2)$ are canonically isomorphic using a version of level-rank duality.

New Braided Endomorphisms from Conformal Inclusions

We give three applications of general theory about braided endomorphisms from conformal inclusions developed previously by us. The first is an example of subfactors associated with conformal inclusion whose dual fusion ring is non-commutative. In the second application we show that the Kac-Wakimoto hypothesis about certain relations between branching rules and S-matrices, which has existed for almost a decade, is not true in at least three examples. Finally we show that the fusion rings of subfactors associated with the conformal inclusions $SU(n)_{(n+2)}\subset SU(n(n+1)/2)$ and $SU(n+2)_n\subset SU((n+1)(n+2)/2)$ are canonically isomorphic using a version of level-rank duality.