Compactifying N=(1,0) theories on a torus, with additional fluxes for global symmetries, we obtain N=1 supersymmetric theories in four dimensions. It is shown that for many choices of flux these models are toric quiver gauge theories with singlet fields. In particular we compare the anomalies deduced from the description of the six-dimensional theory and the anomalies of the quiver gauge theories. We also give predictions for anomalies of four-dimensional theories corresponding to general compactifications of M5-branes probing C_2/Z_k singularities.
We discuss the four dimensional models obtained by compactifying a single M5 brane probing $D_{N}$ singularity (minimal D-type $(1,0)$ conformal matter in six dimensions) on a torus with flux for abelian subgroups of the $SO(4N)$ flavor symmetry. We derive the resulting quiver field theories in four dimensions by first compactifying on a circle and relating the flux to duality domain walls in five dimensions. This leads to novel ${\cal N}=1$ dualities in 4 dimensions which arise from distinct five dimensional realizations of the circle compactifications of the D-type conformal matter.
We calculate the most general causal N=1 three-dimensional, gauge invariant action coupled to matter in superspace and derive its component form using Ectoplasmic integration theory. One example of such an action can be obtained by compactifying M-theory on a Spin(7) holonomy manifold taking non-vanishing fluxes into account. We show that the resulting three-dimensional theory is in agreement with the more general construction. The scalar potential resulting from Kaluza-Klein compactification stabilizes all the moduli fields describing deformations of the metric except for the radial modulus. …
We consider the torus compactifications with flux of a class of $6d$ $(1,0)$ SCFTs that can be engineered as the low-energy theories on M$5$-branes near an M$9$-plane on a $C^2/Z_2$ singularity. Specifically, we concentrate on the two SCFTs where the $Z_2$ orbifold action acts non-trivially on the $E_8$ global symmetry. We analyze this problem by compactifying to $5d$, where we can exploit the relation to $5d$ duality domain walls. By a suitable guess of the domain wall theories, the resulting $4d$ theories can be conjectured. These can then be tested by comparing their properties, notably ano…
We study (0,2) supersymmetric two-dimensional theories obtained by compactifying four-dimensional N=1 supersymmetric theories on a two-torus, with a magnetic field for a global U(1) symmetry, and present evidence that Seiberg duality in four dimensions leads to an identification of different models of this type.
In this paper we study T-duality for principal torus bundles with H-flux. We identify a subset of fluxes which are T-dualizable, and compute both the dual torus bundle as well as the dual H-flux. We briefly discuss the generalized Gysin sequence behind this construction and provide examples both of non T-dualizable and of T-dualizable H-fluxes.
We show that a four-parameter class of 3+1 dimensional NCOS theories can be obtained by dimensional reduction on a general 2-torus from OM theory. Compactifying two spatial directions of NCOS theory on a 2-torus, we study the transformation properties under the $SO(2,2;Z)$ T-duality group. We then discuss non-perturbative configurations of non-commutative super Yang-Mills theory. In particular, we calculate the tension for magnetic monopoles and (p,q) dyons and exhibit their six-dimensional origin, and construct a supergravity solution representing an instanton in the gauge theory. We also com…
In this paper, we describe non-abelian gauge bundles with magnetic and electric fluxes on higher dimensional noncommutative tori. We give an explicit construction of a large class of bundles with nonzero magnetic 't Hooft fluxes. We discuss Morita equivalence between these bundles. The action of the duality is worked out in detail for the four-torus. As an application, we discuss Born-Infeld on this torus, as a description of compactified string theory. We show that the resulting theory, including the fluctuations, is manifestly invariant under the T-duality group SO(4,4;Z). The U-duality inva…
We study discrete fluxes in four dimensional SU(N) gauge theories with a mass gap by using brane compactifications which give ${\cal{N}} = 1$ or ${\cal{N}} = 0$ supersymmetry. We show that when such theories are compactified further on a torus, the t'Hooft magnetic flux $m$ is related to the NS two-form modulus $B$ by $B = 2π{m\over N}$. These values of $B$ label degenerate brane vacua, giving a simple demonstration of magnetic screening. Furthermore, for these values of $B$ one has a conventional gauge theory on a commutative torus, without having to perform any T-dualities. Because of the ma…
We show that the $D=11$ Supermembrane theory (M2-brane) compactified on a $M_9 \times T^2$ target space, with constant fluxes $C_{\pm}$ naturally incorporates the geometrical structure of a twisted torus. We extend the M2-brane theory to a formulation on a twisted torus bundle. It is consistently fibered over the world volume of the M2-brane. It can also be interpreted as a torus bundle with a nontrivial $U(1)$ connection associated to the fluxes. The structure group $G$ is the area preserving diffeomorphisms. The torus bundle is defined in terms of the monodromy associated to the isotopy clas…
We propose a universal geometric formulation of gauged supergravity in terms of a twisted doubled torus. We focus on string theory (M-theory) reductions with generalized Scherk-Schwarz twists residing in the O(n,n) (E_{7(7)}) duality group. The set of doubled geometric fluxes, associated with the duality twists and identified naturally with the embedding tensor of gauged supergravity, captures all known fluxes, i.e. physical form fluxes, ordinary geometric fluxes, as well as their non-geometric counterparts. Furthermore, we propose a prescription for obtaining the effective geometry embedded i…
We present the supersymmetric completion of the M-theory free differential algebra resulting from a compactification to four dimensions on a twisted seven-torus with 4-form and 7-form fluxes turned on. The super--curvatures are given and the local supersymmetry transformations derived. Dual formulations of the theory are discussed in connection with classes of gaugings corresponding to diverse choices of vacua. This also includes seven dimensional compactifications on more general spaces not described by group manifolds.
We investigate Yang-Mills theory on a spatial torus at finite temperature in the presence of discrete electric and magnetic fluxes using the AdS/CFT correspondence. We calculate the leading dependence of the partition function on the fluxes using the dual supergravity theory and comment upon the interpretation of these fluxes as discrete quantum hair for black holes in AdS spacetime.
We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "non-classical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted $K$-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the non-classical case by an isomorphism of twisted cyclic homology. An important part…
Gauge theories in four dimensions can exhibit interesting low energy phenomena, such as infrared enhancements of global symmetry. We explore a class of 4d N=1 gauge theories arising from a construction that is motivated by duality walls in 5d gauge theories. Their quiver descriptions bear a resemblance to 4d theories obtained by compactifying 6d N=(1,0) superconformal field theories on a torus with fluxes, but with lower number of flavours and different number of gauge singlets and superpotentials. One of the main features of these theories is that they exhibit a flavour symmetry enhancement, …
As the effective field theory of the superstring theory, ten-dimensional ${\cal N}=1$ supersymmetric Yang-Mills theory is induced. We consider the ten-dimensional space-time ${\cal M}_{10}$ as direct products of our four-dimensional space-time ${\cal M}_{4}$ and three two-dimensional compact spaces $X_i\ (i=1,2,3)$, i.e. ${\cal M}_{10} = {\cal M}_4 \times X_2 \times X_2 \times X_3$. In particular, we consider $X_i$ as a torus, the torus orbifold, and also the blow-up manifold of the torus orbifold with background magnetic fluxes. We discuss the modular symmetry in the magnetized torus and its …
Compactification of type IIB theory on torus, in the presence of fluxes, is considered. The reduced effective action is expressed in manifestly S-duality invariant form. Cosmological solutions of the model are discussed in several cases in the Pre-Big Bang scenario.
We compute flux-dependent corrections in the four-dimensional F-theory effective action using the M-theory dual description. In M-theory the 7-brane fluxes are encoded by four-form flux and modify the background geometry and Kaluza-Klein reduction ansatz. In particular, the flux sources a warp factor which also depends on the torus directions of the compactification fourfold. This dependence is crucial in the derivation of the four-dimensional action, although the torus fiber is auxiliary in F-theory. In M-theory the 7-branes are described by an infinite array of Taub-NUT spaces. We use the ex…
Classical configurations of a M2-brane, a D2-brane and D0-branes are investigated in the background of an infinite array of M5-branes or NS5-branes. On the M2-brane, we discuss three kinds of configurations, such as a sphere, a cylinder and a torus-like one. These are stabilized by virtue of the background fluxes of M5-branes. The torus-like M2-brane configuration has winding and momentum numbers of 11th direction, and in terms of the type IIA superstring theory, this corresponds to a torus-like D2-brane with electric and magnetic fluxes on it. We also reproduce the same configuration from a n…
String compactifications with non-abelian gauge fields localized on D-branes, with background NSNS and RR 3-form fluxes, and with non-trivial warp factors, can naturally exist within T-dual versions of type I string theory. We develop a systematic procedure to construct the effective bosonic Lagrangian of type I T-dualized along a six-torus, including the coupling to gauge multiplets on D3-branes and the modifications due to 3-form fluxes. Looking for solutions to the ten-dimensional equations of motion, we find warped products of Minkowski space and Ricci-flat internal manifolds. Once the war…
In this paper we consider heterotic compactifications on K3 x T2 as well as type II compactifications on K3-fibred Calabi-Yau spaces with certain fluxes for the gauge and RR field strengths F and H turned on. By providing an identification of corresponding fluxes we show that the well-known N=2 heterotic/type II string-string duality still holds for a subset of all possible fluxes, namely those which arise from six-dimensional gauge fields with internal magnetic flux on the common two-sphere P1, which is the base space of the type II K3-fibration. On the other hand, F- and H-fluxes without P1-…
We consider M-theory compactified on a twisted 7-torus with fluxes when all the seven antisymmetric tensor fields in four dimensions have been dualized into scalars and thus the E_{7(7)} symmetry is recovered. We find that the Scherk--Schwarz and flux gaugings define a ``dual'' gauge algebra, subalgbra of E_{7(7)}, where some of the generators are associated with vector fields which are dual to part of the original vector fields (deriving from the 3-form). In particular they are dual to those vector fields which have been ``eaten'' by the antisymmetric tensors in the original theory by the (an…
We study $SO(32)$ heterotic string theory on torus with magnetic fluxes. Non-vanishing fluxes can lead to non-universal gauge kinetic functions for $SU(3) \times SU(2) \times U(1)_Y$ which is the important features of $SO(32)$ heterotic string theory in contrast to the $E_8\times E_8$ theory. It is found that the experimental values of gauge couplings are realized with ${\cal O}(1)$ values of moduli fields based on the realistic models with the $SU(3) \times SU(2) \times U(1)_Y$ gauge symmetry and three chiral generations of quarks and leptons without chiral exotics.
We consider minimally supersymmetric Yang-Mills theory with a Chern-Simons term on a flat spatial two-torus. The Witten index may be computed in the weak coupling limit, where the ground state wave-functions localize on the moduli space of flat gauge connections. We perform such computations by considering this moduli space as an orbifold of a certain flat complex torus. Our results agree with those obtained previously by instead considering the moduli space as a complex projective space. An advantage of the present method is that it allows for a more straightforward determination of the discr…
We shall review a novel formulation of four dimensional gauged supergravity which is manifestly covariant with respect to the non-perturbative electric-magnetic duality symmetry transformations of the ungauged theory, at the level of the equations of motion and Bianchi identities. We shall also discuss the application of this formalism to the description of M-theory compactified on a twisted torus in the presence of fluxes and to the interpretation from a M/Type IIA theory perspective of the D=5 --> D=4 generalized Scherk-Schwarz reduction. This latter analysis will bring up the issue of non-g…
We consider the field theories on multiple stacks of D5 branes wrapped on four cycles of resolved/deformed conifold geometries fibered over a two torus. The central charges of the D5 branes are slightly misaligned when the branes are wrapped on various rigid holomorphic two cycles or when they have different charges with respect to a magnetic flux turned on the two torus. The wrapped D5 branes preserve (0,2) supersymmetry in two dimensions if the Kahler moduli and the magnetic flux are related. Our geometries are T-dual to the brane configurations considered by Kutasov-Lin and we provide a geo…
We analyse different N=4 supergravities coupled to six vector multiplets corresponding to low-energy descriptions of the bulk sector of T6/Z2 orientifolds with p-brane in IIB (p odd) and in IIA (p even) superstrings. When fluxes are turned on, a gauging emerges corresponding to some non-semisimple Lie algebra related to nilpotent algebras N_p inside so(6,6), with dimension 15 + (p-3)(9-p). The non-metric axions have Stueckelberg couplings that induce a spontaneous breaking of gauge symmetries. In four cases the gauge algebra is non-abelian with a non-commutative structure of the compactificati…
We study 10D super Yang-Mills theory with the gauge groups $E_6$, $E_7$ and $E_8$. We consider the torus/orbifold compacfitication with magnetic fluxes and Wilson lines. They lead to 4D interesting models with three families of quarks and leptons, whose profiles in extra dimensions are quasi-localized because of magnetic fluxes.
We discuss compactifications of rank $Q$ E-string theory on a torus with fluxes for abelian subgroups of the $E_8$ global symmetry of the $6d$ SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as $E[USp(2Q)]$. This model has a variety of non trivial properties. In particular the global symmetry is $USp(2Q)\times USp(2Q)\times U(1)^2$ with one of the two $USp(2Q)$ symmetries emerging in the IR as an enhancement of an $SU(2)^Q$ symmetry of the UV Lagrangian. The $E[USp(2Q)]$ model after dimensional reduction to $3d$ and a subsequent Coulomb branc…
We report on some recent investigations of the structure of the four dimensional gauged supergravity Lagrangian which emerges from flux and Scherk-Schwarz compactifications in higher dimensions. Special attention is given to the gauge structure of M-theory compactified on a seven torus with 4-form and geometrical (spin connection) fluxes turned on. A class of vacua, with flat space-time and described by ``no-scale'' supergravity models, is analyzed.
We study some phenomenological models in a matrix model corresponding to the IIB matrix model compactified on a six-dimensional torus with magnetic fluxes. Extending our previous works, we examine a wider class of models: a Pati-Salam-like model with a gauge group U(4)*U_L(2)*U_R(2), and models where the gauge group U(4) is broken down to U_c(3)*U(1) and/or U_R(2) is broken down to U(1)^2. We find all the matrix configurations that yield matter content of all the phenomenological models whose gauge group is a subgroup of U(8). We then estimate semiclassically a probability distribution for the…
In this work we construct an effective four-dimensional model by compactifying a ten-dimensional theory of gravity coupled with a real scalar dilaton field on a time-dependent torus. The corresponding action in four dimensions is similar to the action of K-essence theories. This approach is applied to anisotropic cosmological Bianchi type $VI_{(h=-1)}$ model for which we study the classical coupling of the anisotropic scale factors with the two real scalar moduli produced by the compactification process. The classical Einstein field equations give us a hidden symmetry, corresponding to equal r…
We study macroscopically two dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product $\mathbb{T}^{d} \times \mathbb{R}^{2}_ε$ of a $d$-dimensional torus and a two dimensional cigar with $Ω$-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on $\mathbb{R}^{2-d} \times \mathbb{T}^{2+d}$ and singular monopoles on $\mathbb{R}^{2-d} \times \mathbb{T}^…
We continue our studies of the low-energy spectrum of N=4 super-Yang-Mills theory on a spatial three-torus. In two previous papers, we computed the spectrum of normalizable zero-energy states for all choices of gauge group and all values of the electric and magnetic 't Hooft fluxes, and checked its invariance under the SL_2(Z) S-duality group. In this paper, we refine the analysis by also decomposing the space of bound states into irreducible unitary representations of the SL_3(Z) mapping class group of the three-torus. We perform a detailed study of the S-dual pairs of theories with gauge gro…
We study 10D super Yang-Mills E8 theory on the 6D torus compactification with magnetic fluxes. We study systematically the possibilities for realizing 4D supersymmetric standard models with three generations of quarks and leptons. We also study quark mass matrices.
We study the flavor structure in $SO(32)$ heterotic string theory on six-dimensional torus with magnetic fluxes. In particular, we focus on models with the flavor symmetries $SU(3)_f$ and $Δ(27)$. In both models, we can realize the realistic quark masses and mixing angles.
The $q$-deformed loop gravity framework was introduced as a canonical formalism for the Turaev-Viro model (with $Λ< 0$), allowing to quantize 3D Euclidean gravity with a (negative) cosmological constant using a quantum deformation of the gauge group. We describe its application to the 2-torus, explicitly writing the $q$-deformed gauge symmetries and deriving the reduced physical phase space of Dirac observables, which leads back to the Goldman brackets for the moduli space of flat connections. Furthermore it turns out that the $q$-deformed loop gravity can be derived through a gauge fixing fro…
We consider an exactly solvable model in 3+1 dimensions, based on a finite group, which is a natural generalization of Kitaev's quantum double model. The corresponding lattice Hamiltonian yields excitations located at torus-boundaries. By cutting open the three-torus, we obtain a manifold bounded by two tori which supports states satisfying a higher-dimensional version of Ocneanu's tube algebra. This defines an algebraic structure extending the Drinfel'd double. Its irreducible representations, labeled by two fluxes and one charge, characterize the torus-excitations. The tensor product of such…
We present exact solutions of four-dimensional Einstein's equations related to Minkoswki vacuum constructed from Type IIB string theory with non-trivial fluxes. Following recent works, we study a non-trivial flux compactification on a fibered product by a four-dimensional torus and a two-dimensional sphere punctured by 5- and 7-branes. By considering only 3-form fluxes and the dilaton, as functions on the internal sphere coordinates, we show that these solutions correspond to a family of supersymmetric solutions constructed by the use of G-theory. Meromorphicity on functions constructed in ter…
We study a systematic derivation of four dimensional $\mathcal{N}=1$ supersymmetric effective theory from ten dimensional non-Abelian Dirac-Born-Infeld action compactified on a six dimensional torus with magnetic fluxes on the D-branes. We find a new type of matter Kähler metric while gauge kinetic function and superpotential are consistent with previous studies. For the ten dimensional action, we use a symmetrized trace prescription and focus on the bosonic part up to $\mathcal{O}(F^4)$. In the presence of the supersymmetry, four dimensional chiral fermions can be obtained via index theorem. …
