Gauge Theories: From Kaluza–Klein to noncommutative gravity theories

Abstract

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.

Intertwining noncommutativity with gravity and particle physics

“…The correct dimensions regarding the integration are restored by the presence of the Tr which is effectively the integration operator referring to the coordinates of the extra dimensions, since they now are represented by matrices. Moreover, k is a parameter related to the volume of the fuzzy coset space (see [106] for the case of the fuzzy sphere) and F MN is the field strength tensor of the gauge theory with M, N = 0, . .…”

Matrix-Formulated Noncommutative Gauge Theories of Gravity

“…In [62] (see also [63,64]), we started a programme realizing gravity as noncommutative gauge theory in three dimensions. In our next contributions in the subject, we worked on the more realistic four-dimensional case [65][66][67][68][69][70][71]. Both the constructions and the details involved will be presented in the following.…”

Gauge theoretic approach to (ordinary) gravity and its fuzzy extensions in three and four dimensions

“…Among the various gauge fields, the vielbein and the spin connection are introduced and their transformation rules are determined. Consequently, noncommutative deformations of field theories have been constructed [14,[96][97][98][99] (see also [79][80][81][82][83]).…”