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

Orbifolds, fuzzy spheres and chiral fermions

Starting with a N=4 supersymmetric Yang-Mills theory in four dimensions with gauge group SU(3N) we perform an orbifold projection leading to a N=1 supersymmetric SU(N)^3 Yang-Mills theory with matter supermultiplets in bifundamental representations of the gauge group, which is chiral and anomaly free. Subsequently, we search for vacua of the projected theory which can be interpreted as spontaneously generated twisted fuzzy spheres. We show that by adding the appropriate soft supersymmetry breaking terms we can indeed reveal such vacua. Three cases are studied, where the gauge group is spontaneously broken further to the low-energy gauge groups SU(4)xSU(2)xSU(2), SU(4)^3 and SU(3)^3. Such models behave in intermediate scales as higher-dimensional theories with a finite Kaluza-Klein tower, while their low-energy physics is governed by the corresponding zero-modes and exhibit chirality in the fermionic sector. The most interesting case from the phenomenological point of view turns out to be the SU(3)^3 unified theory, which has several interesting features such as (i) it can be promoted to a finite theory, (ii) it breaks further spontaneously first to the MSSM and then to SU(3)xU(1)_{em} due to its own scalar sector, i.e. without the need of additional superfields and (iii) the corresponding vacua lead to spontaneously generated fuzzy spheres.Comment: 24 pages, minor changes, references added, matching with the published versio

Dirac structures on nilmanifolds and coexistence of fluxes

We study some aspects of the generalized geometry of nilmanifolds and examine to which extent different types of fluxes can coexist on them. Nilmanifolds constitute a class of homogeneous spaces which are interesting in string compactifications with fluxes since they carry geometric flux by construction. They are generalized Calabi-Yau spaces and therefore simple examples of generalized geometry at work. We identify and classify Dirac structures on nilmanifolds, which are maximally isotropic subbundles closed under the Courant bracket. In the presence of non-vanishing fluxes, these structures are twisted and closed under appropriate extensions of the Courant bracket. Twisted Dirac structures on a nilmanifold may carry multiple coexistent fluxes of any type. We also show how dual Dirac structures combine to Courant algebroids and work out an explicit example where all types of generalized fluxes coexist. These results may be useful in the context of general flux compactifications in string theory.Comment: 1+25 pages; v2: clarifying comments and 6 references added, published versio

Sigma models for genuinely non-geometric backgrounds

The existence of genuinely non-geometric backgrounds, i.e. ones without geometric dual, is an important question in string theory. In this paper we examine this question from a sigma model perspective. First we construct a particular class of Courant algebroids as protobialgebroids with all types of geometric and non-geometric fluxes. For such structures we apply the mathematical result that any Courant algebroid gives rise to a 3D topological sigma model of the AKSZ type and we discuss the corresponding 2D field theories. It is found that these models are always geometric, even when both 2-form and 2-vector fields are neither vanishing nor inverse of one another. Taking a further step, we suggest an extended class of 3D sigma models, whose world volume is embedded in phase space, which allow for genuinely non-geometric backgrounds. Adopting the doubled formalism such models can be related to double field theory, albeit from a world sheet perspective.Comment: 1+34 pages, v2. added references and additional comments; published versio

T-duality without isometry via extended gauge symmetries of 2D sigma models

Target space duality is one of the most profound properties of string theory. However it customarily requires that the background fields satisfy certain invariance conditions in order to perform it consistently; for instance the vector fields along the directions that T-duality is performed have to generate isometries. In the present paper we examine in detail the possibility to perform T-duality along non-isometric directions. In particular, based on a recent work of Kotov and Strobl, we study gauged 2D sigma models where gauge invariance for an extended set of gauge transformations imposes weaker constraints than in the standard case, notably the corresponding vector fields are not Killing. This formulation enables us to follow a procedure analogous to the derivation of the Buscher rules and obtain two dual models, by integrating out once the Lagrange multipliers and once the gauge fields. We show that this construction indeed works in non-trivial cases by examining an explicit class of examples based on step 2 nilmanifolds.Comment: 1+18 pages; version 2: corrections and improvements, more complete version than the published on

Topological Field Theories induced by twisted R-Poisson structure in any dimension

We construct a class of topological field theories with Wess-Zumino term in spacetime dimensions $\ge 2$ whose target space has a geometrical structure that suitably generalizes Poisson or twisted Poisson manifolds. Assuming a field content comprising a set of scalar fields accompanied by gauge fields of degree $(1,p-1,p)$ we determine a generic Wess-Zumino topological field theory in $p+1$ dimensions with background data consisting of a Poisson 2-vector, a $(p+1)$-vector $R$ and a $(p+2)$-form $H$ satisfying a specific geometrical condition that defines a $H$-twisted $R$-Poisson structure of order $p+1$. For this class of theories we demonstrate how a target space covariant formulation can be found by means of an auxiliary connection without torsion. Furthermore, we study admissible deformations of the generic class in special spacetime dimensions and find that they exist in dimensions 2, 3 and 4. The two-dimensional deformed field theory includes the twisted Poisson sigma model, whereas in three dimensions we find a more general structure that we call bi-twisted $R$-Poisson. This extends the twisted $R$-Poisson structure of order 3 by a non-closed 3-form and gives rise to a topological field theory whose covariant formulation requires a connection with torsion and includes a twisted Poisson sigma model in three dimensions as a special case. The relation of the corresponding structures to differential graded Q-manifolds based on the degree shifted cotangent bundle $T^{\ast}[p]T^{\ast}[1]M$ is discussed, as well as the obstruction to them being QP-manifolds due to the Wess-Zumino term.Comment: 40 page

Matrix theory compactifications on twisted tori

We study compactifications of Matrix theory on twisted tori and non-commutative versions of them. As a first step, we review the construction of multidimensional twisted tori realized as nilmanifolds based on certain nilpotent Lie algebras. Subsequently, matrix compactifications on tori are revisited and the previously known results are supplemented with a background of a non-commutative torus with non-constant non-commutativity and an underlying non-associative structure on its phase space. Next we turn our attention to 3- and 6-dimensional twisted tori and we describe consistent backgrounds of Matrix theory on them by stating and solving the conditions which describe the corresponding compactification. Both commutative and non-commutative solutions are found in all cases. Finally, we comment on the correspondence among the obtained solutions and flux compactifications of 11-dimensional supergravity, as well as on relations among themselves, such as Seiberg-Witten maps and T-duality.Comment: 1+31 pages, v2: some comments and clarifications added, accepted for publication in Physical Review