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

Courant sigma model and L∞L_\infty-algebras

The Courant sigma model is a 3-dimensional topological sigma model of AKSZ type which has been used for the systematic description of closed strings in non-geometric flux backgrounds. In particular, the expression for the fluxes and their Bianchi identities coincide with the local form of the axioms of a Courant algebroid. On the other hand, the axioms of a Courant algebroid also coincide with the conditions for gauge invariance of the Courant sigma model. In this paper we embed this interplay between background fluxes of closed strings, gauge (or more precisely BRST) symmetries of the Courant sigma model and axioms of a Courant algebroid into an $L_\infty$-algebra structure. We show how the complete BV-BRST formulation of the Courant sigma model is described in terms of $L_\infty$-algebras. Moreover, the morphism between the $L_\infty$-algebra for a Courant algebroid and the one for the corresponding sigma model is constructed.Comment: 34 pages. v2: typos corrected, published versio

Instantons and Chern-Simons flows in 6, 7 and 8 dimensions

The existence of K-instantons on a cylinder M^7 = R_tau x K/H over a homogeneous nearly K"ahler 6-manifold K/H requires a conformally parallel or a cocalibrated G_2-structure on M^7. The generalized anti-self-duality on M^7 implies a Chern-Simons flow on K/H which runs between instantons on the coset. For K-equivariant connections, the torsionful Yang-Mills equation reduces to a particular quartic dynamics for a Newtonian particle on C. When the torsion corresponds to one of the G_2-structures, this dynamics follows from a gradient or hamiltonian flow equation, respectively. We present the analytic (kink-type) solutions and plot numerical non-BPS solutions for general torsion values interpolating between the instantonic ones.Comment: 1+8 pages, 14 figures; talk presented at SQS-11 during 18-23 July, 2011, at JINR, Dubna, Russia; v2: missing * in eq.(1) adde

Nearly K\"ahler heterotic compactifications with fermion condensates

We revisit AdS_4 heterotic compactifications on nearly K\"ahler manifolds in the presence of H-flux and certain fermion condensates. Unlike previous studies, we do not assume the vanishing of the supersymmetry variations. Instead we determine the full equations of motion originating from the ten-dimensional action, and subsequently we provide explicit solutions to them on nearly K\"ahler manifolds at first order in alpha'. The Bianchi identity is also taken into account in order to guarantee the absence of all anomalies. In the presence of H-flux, which is identified with the torsion of the internal space, as well as of fermion condensates in the gaugino and dilatino sectors, new solutions are determined. These solutions provide a full classification of consistent backgrounds of heterotic supergravity under our assumptions. All the new solutions are non-supersymmetric, while previously known supersymmetric ones are recovered too. Our results indicate that fully consistent (supersymmetric or not) heterotic vacua on nearly K\"ahler manifolds are scarce, even on AdS_4, and they can be completely classified.Comment: 1+17 pages, 1 figure; v2: remark and two references added, published versio

Bundles over Nearly-Kahler Homogeneous Spaces in Heterotic String Theory

We construct heterotic vacua based on six-dimensional nearly-Kahler homogeneous manifolds and non-trivial vector bundles thereon. Our examples are based on three specific group coset spaces. It is shown how to construct line bundles over these spaces, compute their properties and build up vector bundles consistent with supersymmetry and anomaly cancelation. It turns out that the most interesting coset is $SU(3)/U(1)^2$. This space supports a large number of vector bundles which lead to consistent heterotic vacua, some of them with three chiral families.Comment: 32 pages, reference adde

Heat kernel expansion and induced action for matrix models

In this proceeding note, I review some recent results concerning the quantum effective action of certain matrix models, i.e. the supersymmetric IKKT model, in the context of emergent gravity. The absence of pathological UV/IR mixing is discussed, as well as dynamical SUSY breaking and some relations with string theory and supergravity.Comment: 11 pages, 1 figure; talk given at the 7th International Conference on Quantum Theory and Symmetries, August 7-13, 2011, Prague/Czech Republi

On Lie-algebraic solutions of the type IIB matrix model

A systematic search for Lie algebra solutions of the type IIB matrix model is performed. Our survey is based on the classification of all Lie algebras for dimensions up to five and of all nilpotent Lie algebras of dimension six. It is shown that Lie-type solutions of the equations of motion of the type IIB matrix model exist and they correspond to certain nilpotent and solvable Lie algebras. Their representation in terms of Hermitian matrices is discussed in detail. These algebras give rise to certain non-commutative spaces for which the corresponding star-products are provided. Finally the issue of constructing quantized compact nilmanifolds and solvmanifolds based on the above algebras is addressed.Comment: 22 page

Matrix theory origins of non-geometric fluxes

We explore the origins of non-geometric fluxes within the context of M theory described as a matrix model. Building upon compactifications of Matrix theory on non-commutative tori and twisted tori, we formulate the conditions which describe compactifications with non-geometric fluxes. These turn out to be related to certain deformations of tori with non-commutative and non-associative structures on their phase space. Quantization of flux appears as a natural consequence of the framework and leads to the resolution of non-associativity at the level of the unitary operators. The quantum-mechanical nature of the model bestows an important role on the phase space. In particular, the geometric and non-geometric fluxes exchange their properties when going from position space to momentum space thus providing a duality among the two. Moreover, the operations which connect solutions with different fluxes are described and their relation to T-duality is discussed. Finally, we provide some insights on the effective gauge theories obtained from these matrix compactifications.Comment: 1+31 pages, reference list update

Yang-Mills instantons and dyons on homogeneous G_2-manifolds

We consider Lie G-valued Yang-Mills fields on the space R x G/H, where G/H is a compact nearly K"ahler six-dimensional homogeneous space, and the manifold R x G/H carries a G_2-structure. After imposing a general G-invariance condition, Yang-Mills theory with torsion on R x G/H is reduced to Newtonian mechanics of a particle moving in R^6, R^4 or R^2 under the influence of an inverted double-well-type potential for the cases G/H = SU(3)/U(1)xU(1), Sp(2)/Sp(1)xU(1) or G_2/SU(3), respectively. We analyze all critical points and present analytical and numerical kink- and bounce-type solutions, which yield G-invariant instanton configurations on those cosets. Periodic solutions on S^1 x G/H and dyons on iR x G/H are also given.Comment: 1+26 pages, 14 figures, 6 miniplot

Gravity and compactified branes in matrix models

A mechanism for emergent gravity on brane solutions in Yang-Mills matrix models is exhibited. Newtonian gravity and a partial relation between the Einstein tensor and the energy-momentum tensor can arise from the basic matrix model action, without invoking an Einstein-Hilbert-type term. The key requirements are compactified extra dimensions with extrinsic curvature M^4 x K \subset R^D and split noncommutativity, with a Poisson tensor \theta^{ab} linking the compact with the noncompact directions. The moduli of the compactification provide the dominant degrees of freedom for gravity, which are transmitted to the 4 noncompact directions via the Poisson tensor. The effective Newton constant is determined by the scale of noncommutativity and the compactification. This gravity theory is well suited for quantization, and argued to be perturbatively finite for the IKKT model. Since no compactification of the target space is needed, it might provide a way to avoid the landscape problem in string theory.Comment: 35 pages. V2: substantially revised and improved, conclusion weakened. V3: some clarifications, published version. V4: minor correctio