Unified Gauge Theories and Reduction of Couplings: from Finiteness to Fuzzy Extra Dimensions
Myriam Mondragon, George Zoupanos
备注：
This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
Finite Unified Theories (FUTs) are N=1 supersymmetric Grand Unified Theories, which can be made all-loop finite, both in the dimensionless (gauge and Yukawa couplings) and dimensionful (soft supersymmetry breaking terms) sectors. This remarkable property, based on the reduction of couplings at the quantum level, provides a drastic reduction in the number of free parameters, which in turn leads to an accurate prediction of the top quark mass in the dimensionless sector, and predictions for the Higgs boson mass and the supersymmetric spectrum in the dimensionful sector. Here we examine the predictions of two such FUTs. Next we consider gauge theories defined in higher dimensions, where the extra dimensions form a fuzzy space (a finite matrix manifold). We reinterpret these gauge theories as four-dimensional theories with Kaluza-Klein modes. We then perform a generalized a la Forgacs-Manton dimensional reduction. We emphasize some striking features emerging such as (i)the appearance of non-Abelian gauge theories in four dimensions starting from an Abelian gauge theory in higher dimensions, (ii) the fact that the spontaneous symmetry breaking of the theory takes place entirely in the extra dimensions and (iii) the renormalizability of the theory both in higher as well as in four dimensions. Then reversing the above approach we present a renormalizable four dimensional SU(N) gauge theory with a suitable multiplet of scalar fields, which via spontaneous symmetry breaking dynamically develops extra dimensions in the form of a fuzzy sphere $S^2_N$.
We propose gauge-Higgs unification in fuzzy extra dimensions as a possible solution to the Higgs naturalness problem. In our approach, the fuzzy extra dimensions are created spontaneously as a vacuum solution of certain four-dimensional gauge theory. As an example, we construct a model which has a fuzzy torus as its vacuum. The Higgs field in our model is associated with the Wilson loop wrapped on the fuzzy torus. We show that the quadratic divergence in the mass of the Higgs field in the one-loop effective potential is absent. We then argue based on symmetries that the quantum corrections to …
Theories defined in higher than four dimensions have been used in various frameworks and have a long and interesting history. Here we review certain attempts, developed over the last years, towards the construction of unified particle physics models in the context of higher-dimensional gauge theories with non-commutative extra dimensions. These ideas have been developed in two complementary ways, namely (i) starting with a higher-dimensional gauge theory and dimensionally reducing it to four dimensions over fuzzy internal spaces and (ii) starting with a four-dimensional, renormalizable gauge t…
In the present contribution the construction of particle physics models in theories with fuzzy extra dimensions is discussed. We focus on a bottom-up approach where the structure of a higher-dimensional theory arises within ordinary four-dimensional field theory via an appropriate mechanism based on spontaneous symmetry breaking. Subsequently, possible ways to obtain particle physics models with realistic spectrum in this framework are exhibited.
We briefly review the Coset Space Dimensional Reduction (CSDR) programme and the best model constructed so far and then we present some details of the corresponding programme in the case that the extra dimensions are considered to be fuzzy. In particular, we present a four-dimensional $\mathcal{N} = 4$ Super Yang Mills Theory, orbifolded by $\mathbb{Z}_3$, which mimics the behaviour of a dimensionally reduced $\mathcal{N} = 1$, 10-dimensional gauge theory over a set of fuzzy spheres at intermediate high scales and leads to the trinification GUT $SU(3)^3$ at slightly lower, which in turn can be…
We first briefly review the Coset Space Dimensional Reduction (CSDR) programme and present the results of the best model so far, based on the $\mathcal{N} = 1$, $d = 10$, $E_8$ gauge theory reduced over the nearly-Kähler manifold $SU(3)/U(1)\times U(1)$. Then, we present the adjustment of the CSDR programme in the case that the extra dimensions are considered to be fuzzy coset spaces and then, the best model constructed in this framework, too, which is the trinification GUT, $SU(3)^3$.
We first review the Coset Space Dimensional Reduction (CSDR) programme and present the best model constructed so far based on the $\mathcal{N} = 1$, $10$-dimensional $E_8$ gauge theory reduced over the nearly-Kähler manifold $SU(3)/U(1)\times U(1)$ with the additional use of the Wilson flux mechanism. Then we present the corresponding programme in the case that the extra dimensions are considered to be fuzzy coset spaces and the best model that has been constructed in this framework too. In both cases the best model appears to be the trinification GUT $SU(3)^3$.
We revisit an orbifold matrix model obtained as a restriction of the type IIB matrix model on a Z_3-invariant sector. An investigation of its moduli space of vacua is performed and issues related to chiral gauge theory and gravity are discussed. Modifications of the orbifolded model triggered by Chern-Simons or mass deformations are also analyzed. Certain vacua of the modified models exhibit higher-dimensional behaviour with internal geometries related to fuzzy spheres.
In this article, we explore the low energy structure of a $U(3)$ gauge theory over spaces with fuzzy sphere(s) as extra dimensions. In particular, we determine the equivariant parametrization of the gauge fields, which transform either invariantly or as vectors under the combined action of $SU(2)$ rotations of the fuzzy spheres and those $U(3)$ gauge transformations generated by $SU(2) \subset U(3)$ carrying the spin $1$ irreducible representation of $SU(2)$. The cases of a single fuzzy sphere $S_F^2$ and a particular direct sum of concentric fuzzy spheres, $S_F^{2 \, Int}$, covering the monop…
We consider certain vacua of four-dimensional SU(N) gauge theory with the same field content as the N=4 supersymmetric Yang-Mills theory, resulting from potentials which break the N=4 supersymmetry as well as its global SO(6) symmetry down to SO(3) x SO(3). We show that the theory behaves at intermediate scales as Yang-Mills theory on M^4 x S^2 x S^2, where the extra dimensions are fuzzy spheres with magnetic fluxes. We determine in particular the structure of the zero modes due to the fluxes, which leads to low-energy mirror models.
We consider a U(2) Yang-Mills theory on M x S_F^2 where M is an arbitrary noncommutative manifold and S_F^2 is a fuzzy sphere spontaneously generated from a noncommutative U(N) Yang-Mills theory on M, coupled to a triplet of scalars in the adjoint of U(N). Employing the SU(2)-equivariant gauge field constructed in arXiv:0905.2338, we perform the dimensional reduction of the theory over the fuzzy sphere. The emergent model is a noncommutative U(1) gauge theory coupled adjointly to a set of scalar fields. We study this model on the Groenewald-Moyal plane and find that, in certain limits, it admi…
我们结合并利用陪集空间降维（CSDR）方法和非交换几何的思想。本文研究了定义在高维中的规范理论的降维问题，其中紧致方向是一个模糊空间（矩阵流形）。在CSDR中，我们假设时空的形式是M^D=M^4 x S/R，其中S/R是齐次空间。然后，一个具有定义在M^D上的规范群G的规范理论可以利用S/R的对称性以一种优雅的方式在维数上约化为M ^4，特别地，所得到的四维规范是G的一个子群。在目前的工作中，我们表明，可以应用CSDR的想法的情况下，紧凑的部分的时空是一个有限的近似的齐次空间S/R，即模糊陪集。特别是我们研究的模糊球的情况下。
We find new spontaneously generated fuzzy extra dimensions emerging from a certain deformation of $N=4$ supersymmetric Yang-Mills (SYM) theory with cubic soft supersymmetry breaking and mass deformation terms. First, we determine a particular four dimensional fuzzy vacuum that may be expressed in terms of a direct sum of product of two fuzzy spheres, and denote it in short as $S_F^{2\, Int}\times S_F^{2\, Int}$. The direct sum structure of the vacuum is revealed by a suitable splitting of the scalar fields in the model in a manner that generalizes our approach in \cite{Seckinson}. Fluctuations…
We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as $SO(5)$-equivariant projections of quantized coadjoint orbits of $SO(6)$. We show that they arise as solutions in Yang-Mills matrix models, which naturally leads to higher-spin gauge theories on $S^4$. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions $S^4 \times \mathcal{K}$, which is expected to entail 2+1 generations.
Renormalizable Theories from Fuzzy Higher Dimensions
我们考虑定义在高维中的规范理论，其中额外的维度形成一个模糊空间（有限矩阵流形）。我们将这些规范理论重新解释为具有卡鲁扎-克莱因模的四维理论。然后，我们执行一个广义的`a la Forgacs-Manton降维。我们强调一些引人注目的特点出现，如（i）外观的非阿贝尔规范理论在四维开始从阿贝尔规范理论在高维，（ii）的事实，即自发对称性破缺的理论完全发生在额外的维度和（iii）重正化的理论都在更高以及在四维。
We find new vacuum solutions of ${\cal N}=4$ super-Yang-Mills with totally anti-symmetric cubic soft SUSY breaking terms, or equivalently solutions of the IKKT matrix model of type $\mathbb{R}^4_θ\times {\cal K}_N$ with flux terms. The solutions can be understood in terms of 4- and 6- dimensional fuzzy branes ${\cal K}_N$ in extra dimensions, describing self-intersecting projections of compact flag manifolds of $SU(3)$. The 6-dimensional solutions provide a 6-fold covering of the internal space near the origin, while the 4-dimensional branes have a triple self-intersections spanning all 6 inte…
In M(atrix) theory, there exist membranes and longitudinal 5-branes (L5-branes) as extended objects. Transverse components of these brane solutions are known to be described by fuzzy CP^k (k=1,2), where k=1 and k=2 correspond to spherical membranes and L5-branes of CP^2 \times S^1 world-volume geometry, respectively. In addition to these solutions, we here show the existence of L7-branes of CP^3 \times S^1 geometry, introducing extra potentials to the M(atrix) theory Lagrangian. As in the cases of k=1,2, the L7-branes (corresponding to k=3) also break the supersymmetries of M(atrix) theory. Th…
We start with an $SU(\cal {N})$ Yang-Mills theory on a manifold ${\cal M}$, suitably coupled to two distinct set of scalar fields in the adjoint representation of $SU({\cal N})$, which are forming a doublet and a triplet, respectively under a global $SU(2)$ symmetry. We show that a direct sum of fuzzy spheres $S_F^{2 \, Int} := S_F^2(\ell) \oplus S_F^2 (\ell) \oplus S_F^2 \left ( \ell + \frac{1}{2} \right ) \oplus S_F^2 \left ( \ell - \frac{1}{2} \right )$ emerges as the vacuum solution after the spontaneous breaking of the gauge symmetry and lay the way for us to interpret the spontaneously b…
A general framework is described which associates geometrical structures to any set of $D$ finite-dimensional hermitian matrices $X^a, \ a=1,...,D$. This framework generalizes and systematizes the well-known examples of fuzzy spaces, and allows to extract the underlying classical space without requiring the limit of large matrices or representation theory. The approach is based on the previously introduced concept of quasi-coherent states. In particular, a concept of quantum Kähler geometry arises naturally, which includes the well-known quantized coadjoint orbits such as the fuzzy sphere $S^2…
Emergence of 3D Superconformal Ising Criticality on the Fuzzy Sphere
Westlake Institute of Advanced Study, Westlake University（西湖高等研究院，西湖大学）；Institute for Advanced Study, Kyushu University（九州大学高级研究院）；School of Physics and Astronomy, University of Leeds（利兹大学物理与天文学院）
