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![Figure 1: 3-dimensional section of squashed C P 2 , taken from [9].](profile/Harold-Steinacker/publication/275364157/figure/fig1/AS:310161244737543@1450959611688/3-dimensional-section-of-squashed-C-P-2-taken-from-9_Q320.jpg)
![Figure 4: C [(1 , 0)] + C [(0 , 1)] with Higgs φ .](profile/Harold-Steinacker/publication/275364157/figure/fig4/AS:310161244737547@1450959611948/C-1-0-C-0-1-with-Higgs-ph_Q320.jpg)
We discuss the low-energy physics which arises on stacks of squashed brane
solutions of $SU(N)$ ${\cal N}=4$ SYM, deformed by a cubic soft SUSY breaking
potential. A brane configuration is found which leads to a low-energy physics
similar to the standard model in the broken phase, assuming suitable VEV's of
the scalar zero modes. Due to the triple...
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Citations
... In particular, we observe an interesting triple self-intersection (or covering) structure in the extra dimensions of S 4 Λ , which is best understood in terms of the CP 2 fiber. This is very similar to the squashed CP 2 found as extra dimensions in [21,22], and it strongly suggests 3 families of fermionic (near-)zero modes. In fact, one of these families would be distinct from the other two, which suggests a "2+1" family structure. ...
... as extra embedding coordinates. Then (t 1 , t 2 , t 3 , t 4 , s 3 , s 4 ) describes precisely the squashed CP 2 as discussed in [21,22], which has a triple self-intersection at the origin as in Figure 2a. Projecting out the s 3,4 leads to a further projection along 2 of the 6 directions. ...
... It suggests that the low-energy physics on S 4 will have 3 generations, which arise from fermionic strings connecting these 3 sheets at the origin t = 0. This is very close to the situation studied in [21,22], where a similar squashed fuzzy CP 2 led to 3 generations with low-energy physics not far from the Standard Model. Interestingly, here we would expect 2+1 generations, since the string connecting the two degenerate sheets is different from the 2 strings connecting one degenerate sheet with the regular sheet. ...
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.
... Models involving matrix valued fields in the adjoint of SU (N ) have been proposed for inflation models in [35, 36]. Recently, new 4-and 6-dimensional fuzzy vacuum configurations in SSB deformed N = 4 SYM are reported in [38, 39, 40]. The outlined developments call for further investigations on the low energy structure around such fuzzy vacua in a diverse class of models with larger gauge groups in order to better assess the potential value of these models from a phenomenological point of view. ...
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 monopole bundle sectors with windings $\pm 1$ are treated in full and the low energy degrees of freedom for the gauge fields are obtained. Employing the parametrizations of the fields in the former case, we determine a low energy action by tracing over the fuzzy sphere and show that the emerging model is abelian Higgs type with $U(1) \times U(1)$ gauge symmetry and possess vortex solutions on ${\mathbb R}^2$, which we discuss in some detail. Generalization of our formulation to the equivariant parametrization of gauge fields in $U(n)$ theories is also briefly addressed.
... If it is possible to obtain also a (near-) realistic low-energy particle physics in this framework (e.g. along the lines of [47][48][49]), this would offer an extremely simple and attractive approach to a quantum theory of fundamental interactions. ...
We study perturbations of the 4-dimensional fuzzy sphere as a background in the IKKT or IIB matrix model. The linearized 4-dimensional Einstein equations are shown to arise from the classical matrix model action, without adding an Einstein-Hilbert term. The excitation modes with lowest spin are identified as gauge fields, metric and connection fields. In addition to the usual gravitational waves, there are also physical "torsion" wave excitations. The quantum structure of the geometry encodes a twisted bundle of self-dual 2-forms, which leads to a covariant 4-dimensional noncommutative geometry. The formalism of string states is used to compute one-loop corrections to the effective action. This leads to a mass term for the gravitons which is significant for $S^4$, but argued to be small in the Minkowski case.
... The general ideas in this paper are tested and elaborated in detail for the standard examples such as the fuzzy sphere S 2 N , fuzzy torus T 2 N , fuzzy CP 2 N , and squashed fuzzy CP 2 N . The latter is a very interesting and non-trivial example which arises in N = 4 SYM with a cubic potential [13, 14], and which does not correspond to a Kähler (nor an almost-Kähler) manifold. In particular, we find that the (numerically obtained) quasi-coherent states have smaller dispersion than the Perelomov-type states, and we find strong evidence that the semi-classical geometry is again recovered from exact zero modes of / D x . ...
... However, there are many interesting examples with degenerate embedding. For example, it turns out that squashed CP 2 has a triple self-intersection at the origin in target space, which leads to interesting physics [13, 14] . A more drastic example is the fuzzy foursphere S 4 N [25], which can be interpreted as a twisted N -fold degenerate embedding of fuzzy CP 3 N in R 5 [26, 27]. ...
We develop a systematic approach to determine and measure numerically the geometry of generic quantum or "fuzzy" geometries realized by a set of finite-dimensional hermitian matrices. The method is designed to recover the semi-classical limit of quantized symplectic spaces embedded in $\mathbb{R}^d$ including the well-known examples of fuzzy spaces, but it applies much more generally. The central tool is provided by quasi-coherent states, which are defined as ground states of Laplace- or Dirac operators corresponding to localized point branes in target space. The displacement energy of these quasi-coherent states is used to extract the local dimension and tangent space of the semi-classical geometry, and provides a measure for the quality and self-consistency of the semi-classical approximation. The method is discussed and tested with various examples, and implemented in an open-source Mathematica package.
