Fuzzy Extra Dimensions Research Articles

Quantum [formula omitted]-Yang-Mills from the IKKT matrix model

We study the one-loop effective action of the higher-spin gauge theory induced by the IKKT matrix model on a M1,3×K background, where M1,3 is an FLRW cosmological spacetime brane and K are compact fuzzy extra dimensions. In particular, we show that all non-abelian (hs-valued) gauge fields in this model acquire mass via quantum effects, thus avoiding no-go theorems. This leads to a massive non-abelian quantum hs-Yang-Mills theory, whose detailed structure depends on K. The stabilization of K at one loop is understood as a result of the coupling between K and the U(1)-flux bundle on space-time. This flux stabilization induces the KK scale into the N=4 SYM sector of the model, which break superconformal symmetry.

One-loop effective action and emergent gravity on quantum spaces in the IKKT matrix model

A detailed derivation of 3 + 1 dimensional induced or emergent gravity in the IKKT matrix model at one loop is given, as announced in [1]. The mechanism requires a brane configuration with structure {mathcal{M}}^{3,1}times mathcal{K}subset {mathbb{R}}^{9,1} , where {mathcal{M}}^{3,1} is the noncommutative space-time brane, and mathcal{K} are compact fuzzy extra dimensions embedded in target space. The 3+1-dimensional Einstein-Hilbert action then arises in the one loop effective action of the maximally supersymmetric IIB or IKKT matrix model, with effective Newton constant determined by the Kaluza-Klein scale of mathcal{K} . At weak coupling, all physical modes are confined to the brane, leading to 3 + 1-dimensional low-energy physics. The Einstein-Hilbert action can be interpreted as interaction of mathcal{K} with the space-time brane via IIB supergravity. The vacuum energy does not act as cosmological constant, but stabilizes the brane structure at one loop.

Gravity as a quantum effect on quantum space-time

The 3+1-dimensional Einstein-Hilbert action is obtained from the 1-loop effective action on noncommutative branes in the IIB or IKKT matrix model. The presence of compact fuzzy extra dimensions K as well as maximal supersymmetry of the model is essential. The E-H action can be interpreted as interaction of K with the space-time brane via IIB supergravity, and the effective Newton constant is determined by the Kaluza-Klein scale of K. The classical matrix model defines a pre-gravity action with 2 derivatives less than the induced E-H action, governing the cosmological regime. The perturbative physics is confined to the space-time brane, which for covariant quantum space-times includes all dof of gravity, as well as a tower of higher-spin modes. The vacuum energy of the background is given in terms of the symplectic volume form, and hence does not act as cosmological constant.

Gauge Theories: From Kaluza–Klein to noncommutative gravity theories

First, the Coset Space Dimensional Reduction scheme and the best particle physics model so far resulting from it are reviewed. Then, a higher-dimensional theory in which the extra dimensions are fuzzy coset spaces is described and a dimensional reduction to four-dimensional theory is performed. Afterwards, another scheme including fuzzy extra dimensions is presented, but this time the starting theory is four-dimensional while the fuzzy extra dimensions are generated dynamically. The resulting theory and its particle content is discussed. Besides the particle physics models discussed above, gravity theories as gauge theories are reviewed and then, the whole methodology is modified in the case that the background spacetimes are noncommutative. For this reason, specific covariant fuzzy spaces are introduced and, eventually, the program is written for both the 3-d and 4-d cases.

Covariant 4-dimensional fuzzy spheres, matrix models and higher spin

We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as -equivariant projections of quantized coadjoint orbits of . We show that they arise as solutions in Yang–Mills matrix models, which naturally leads to higher-spin gauge theories on S4. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions , which is expected to entail 2 + 1 generations.

Equivariant fields in anSU(N)gauge theory with new spontaneously generated fuzzy extra dimensions

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 around this vacuum have the structure of gauge fields over $S_F^{2\, Int}\times S_F^{2\, Int}$, and this enables us to conjecture the spontaneous broken model as an effective $U(n)$ $(n < {\cal N})$ gauge theory on the product manifold $M^4 \times S_F^{2\, Int} \times S_F^{2\, Int}$. We support this interpretation by examining the $U(4)$ theory and determining all of the $SU(2)\times SU(2)$ equivariant fields in the model, characterizing its low energy degrees of freedom. Monopole sectors with winding numbers $(\pm 1,0),\,(0,\pm1),\,(\pm1,\pm 1)$ are accessed from $S_F^{2\, Int}\times S_F^{2\, Int}$ after suitable projections and subsequently equivariant fields in these sectors are obtained. We indicate how Abelian Higgs type models with vortex solutions emerge after dimensionally reducing over the fuzzy monopole sectors as well. A family of fuzzy vacua is determined by giving a systematic treatment for the splitting of the scalar fields and it is made manifest that suitable projections of these vacuum solutions yield all higher winding number fuzzy monopole sectors. We observe that the vacuum configuration $S_F^{2\, Int}\times S_F^{2\, Int}$ identifies with the bosonic part of the product of two fuzzy superspheres with $OSP(2,2)\times OSP(2,2)$ supersymmetry and elaborate on this feature.

New fuzzy extra dimensions fromSU(N)gauge theories

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 broken model as a $U(n)$ gauge theory over ${\cal M} \times S_F^{2 \, Int}$. Focusing on a $U(2)$ gauge theory we present complete parameterizations of the $SU(2)$-equivariant, scalar, spinor and vector fields characterizing the effective low energy features of this model. Next, we direct our attention to the monopole bundles $S_F^{2 \, \pm} := S_F^2 (\ell) \oplus S_F^2 \left ( \ell \pm \frac{1}{2} \right )$ over $S_F^2 (\ell)$ with winding numbers $\pm 1$, which naturally come forth through certain projections of $S_F^{2 \, Int}$, and discuss the low energy behaviour of the $U(2)$ gauge theory over ${\cal M} \times S_F^{2 \, \pm}$. We study models with $k$-component multiplet of the global $SU(2)$, give their vacuum solutions and obtain a class of winding number $\pm (k-1)$ monopole bundles $S_F^{2 \,, \pm (k-1)}$ as certain projections of these vacuum solutions. We make the observation that $S_F^{2 \, Int}$ is indeed the bosonic part of the $N=2$ fuzzy supersphere with $OSP(2,2)$ supersymmetry and construct the generators of the $osp(2,2)$ Lie superalgebra in two of its irreducible representations using the matrix content of the vacuum solution $S_F^{2 \, Int}$. Finally, we show that our vacuum solutions are stable by demonstrating that they form mixed states with non-zero von Neumann entropy.

Self-intersecting fuzzy extra dimensions from squashed coadjoint orbits in N = 4 $$ \mathcal{N}=4 $$ SYM and matrix models

We find new vacuum solutions of $$ \mathcal{N}=4 $$ super-Yang-Mills with totally antisymmetric cubic soft SUSY breaking terms, or equivalently solutions of the IKKT matrix model of type $$ {\mathbb{R}}_{\theta}^4\times {\mathcal{K}}_N $$ with flux terms. The solutions can be understood in terms of 4- and 6-dimensional fuzzy branes $$ {\mathcal{K}}_N $$ 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-intersection spanning all 6 internal directions. The solutions have lower energy than the trivial vacuum, and we prove that there are no negative modes. The massless modes are identified explicitly. In particular there are chiral fermionic zero modes, linking the coincident sheets with opposite flux at the origin. They have a $$ {\mathbb{Z}}_3 $$ family symmetry, originating from the Weyl group rotations.

Equivariant Reduction of Gauge Theories over Fuzzy Extra Dimensions

In SU(N) Yang-Mills theories on a manifold M, which are suitably coupled to a set of scalars, fuzzy spheres may be generated as extra dimensions by spontaneous symmetry breaking. This process results in gauge theories over the product space of the manifold M and the fuzzy spheres with smaller gauge groups. Here we present the SU(2)– and SU(2) × SU(2)-equivariant parametrization of U(2) and U(4) gauge fields on S2F and S2F × S2F respectively and outline the dimensional reduction of these theories over the fuzzy extra dimensions. The emerging dimensionally reduced theories are Higgs type models. Some vortex type solutions of these theories are briefly discussed.

Gauge-Higgs unification in spontaneously created fuzzy extra dimensions

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 the Higgs mass is suppressed including all loop contributions. We also consider a realization on the worldvolume theory of D3-branes probing $C^3/(Z_N \times Z_N)$ orbifold with discrete torsion.

Gauge Theories with Fuzzy Extra Dimensions and Noncommutative Vortices and Fluxons

A U (2) Yang-Mills theory on the space ℳ × S2F is considered, where it is assumed that ℳ is an arbitrary noncommutative space and S2F is a fuzzy sphere spontaneously generated from a noncommutative U(\U0001d4a9) Yang-Mills theory on ℳ coupled to a triplet of scalars in the adjoint of U(\U0001d4a9). SU(2)-equivariant reduction of this theory leads to a noncommutative U(1) gauge theory coupled adjointly to a set of scalar fields. The emergent model is studied on the Groenewald-Moyal plane ℛ2θ and it is found that, in certain limits, it admits noncommutative, non-BPS vortex as well as fluxon solutions.

Evidence for F(uzz) theory

We show that in the decoupling limit of an F-theory compactification, the internal directions of the seven-branes must wrap a non-commutative four-cycle \( \mathcal{S} \). We introduce a general method for obtaining fuzzy geometric spaces via toric geometry, and develop tools for engineering four-dimensional GUT models from this non-commutative setup. We obtain the chiral matter content and Yukawa couplings, and show that the theory has a finite Kaluza-Klein spectrum. The value of 1/α GUT is predicted to be equal to the number of fuzzy points on the internal four-cycle \( \mathcal{S} \). This relation puts a non-trivial restriction on the space of gauge theories that can arise as a limit of F-theory. By viewing the seven-brane as tiled by D3-branes sitting at the N fuzzy points of the geometry, we argue that this theory admits a holographic dual description in the large N limit. We also entertain the possibility of constructing string models with large fuzzy extra dimensions, but with a high scale for quantum gravity.

Noncommutative vortices and flux tubes from Yang-Mills theories with spontaneously generated fuzzy extra dimensions

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 admits noncommutative, non-BPS vortex as well as flux-tube (fluxon) solutions and discuss some of their properties.

Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions

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 theory and dynamically generating fuzzy extra dimensions. We describe the above approaches and moreover we discuss the inclusion of fermions and the construction of realistic chiral theories in this context.

On the fermion spectrum of spontaneously generated fuzzy extra dimensions with fluxes

AbstractWe 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) × SO(3). We show that the theory behaves at intermediate scales as Yang‐Mills theory on M4 × SL2 × SR2, 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.

Unified Gauge Theories and Reduction of Couplings: from Finiteness to Fuzzy Extra Dimensions

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 reduc- tion 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 theo- ries as four-dimensional theories with Kaluza-Klein modes. We then perform a generalized ` 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 renormaliza- bility 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 explicitly find the tower of massive Kaluza-Klein modes consistent with an interpretation as gauge theory on M 4 ◊ S 2 , the scalars being interpreted as gauge fields on S 2 . Depending on the parameters of the model the low-energy gauge group can be SU(n), or broken further to SU(n1) ◊ SU(n2) ◊ U(1). Therefore the second picture justifies the first one in a renormalizable framework but in addition has the potential to reveal new aspects of the theory.

Zero point energy on extra dimensions: Noncommutative torus

In this paper we calculate the zero point energy density experienced by observers on ${\mathcal{M}}^{4}$ due to a massless scalar field defined throughout ${\mathcal{M}}^{4}\ifmmode\times\else\texttimes\fi{}{\mathbf{T}}_{F}^{2}$, where ${\mathbf{T}}_{F}^{2}$ are fuzzy extra dimensions. Using the Green's function approach we calculate the energy density for the commutative torus and the fuzzy torus. We also calculate the energy density for the fuzzy torus using the Hamiltonian approach. Agreement is shown between the Green's function and Hamiltonian approaches.

Fermions on spontaneously generated spherical extra dimensions

We include fermions to the model proposed in hep-th/0606021, and obtain a renormalizable 4-dimensional SU(N) gauge theory which spontaneously generates fuzzy extra dimensions and behaves like Yang-Mills theory on M^4 \times S^2. We find a truncated tower of fermionic Kaluza-Klein states transforming under the low-energy gauge group, which is found to be either SU(n), or SU(n_1) x SU(n_2) x U(1). The latter case implies a nontrivial U(1) flux on S^2, leading to would-be zero modes for the bifundamental fermions. In the non-chiral case they may pair up to acquire a mass, and the emerging picture is that of mirror fermions. We discuss the possible implementation of a chirality constraint in 6 dimensions, which is nontrivial at the quantum level due to the fuzzy nature of the extra dimensions.

Unified Theories: From Finiteness to Fuzzy Extra Dimensions

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 FUTs taking into account all theoretical andexperimental constraints. Unified theories defined in more than four dimensions provide after dimensional reduction in four dimensions a further unification of the gauge, Yukawa and Higgs sectors as well as of the soft supersymmetry breaking sector in the case of supersymmetric theories. Furthermore, if the extra dimensions are fuzzy instead of continuous, the theories become renormalizable. Due to lack of space we cannot review the last two subjects here and we suggest consultation of the original papers.

Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking

We present a renormalizable 4-dimensional SU(N) gauge theory with a suitable multiplet of scalar fields, which dynamically develops extra dimensions in the form of a fuzzy sphere S^2. We explicitly find the tower of massive Kaluza-Klein modes consistent with an interpretation as gauge theory on M^4 x S^2, the scalars being interpreted as gauge fields on S^2. The gauge group is broken dynamically, and the low-energy content of the model is determined. Depending on the parameters of the model the low-energy gauge group can be SU(n), or broken further to SU(n_1) x SU(n_2) x U(1), with mass scale determined by the size of the extra dimension.