T-duality of gauge theories on a noncommutative T d can be extended to include fields with twisted boundary conditions. The resulting T-dual theories contain novel nonlocal fields. These fields represent dipoles of constant magnitude. Several unique properties of field theories on noncommutative spaces have simpler counterparts in the dipole-theories.

The gravity duals of nonlocal field theories in the large N limit exhibit a novel behavior near the boundary. To explore this, we present and study the duals of dipole theories -a particular class of nonlocal theories with fundamental dipole fields. The nonlocal interactions are manifest in the metric of the gravity dual and type-0 string theories make a surprising appearance. We compare the situation to that in noncommutative SYM.

We calculate the volumes of a large class of Einstein manifolds, namely Sasaki-Einstein manifolds which are the bases of Ricci-flat affine cones described by polynomial embedding relations in C n . These volumes are important because they allow us to extend and test the AdS/CFT correspondence. We use these volumes to extend the central charge calculation of Gubser (1998) to the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999). These volumes also allow one to quantize precisely the D-brane flux of the AdS supergravity solution. We end by demonstrating a relationship between the volumes of these Einstein spaces and the number of holomorphic polynomials (which correspond to chiral primary operators in the field theory dual) on the corresponding affine cone.

We show that the data of a principal G bundle over a principal circle bundle is equivalent to that of a LG def = U(1) ⋉ LG bundle over the base of the circle bundle. We apply this to the Kaluza-Klein reduction of M-theory to IIA and show that certain generalized characteristic classes of the loop group bundle encode the Bianchi identities of the antisymmetric tensor fields of IIA supergravity. We further show that the low dimensional characteristic classes of the central extension of the loop group encode the Bianchi identities of massive IIA, thereby adding support to the conjectures of hep-th/0203218. K(Z, 4) in low dimensions is LE k=1 8 bundles, where LE k=1 8gauge bundles in M-theory were, in this sense, compatible with the K-theoretic quantization of fluxes IIA. However, their calculation indicated that there did not seem to exist a 1-1 correspondence between quantized flux configurations in M-theory and IIA. Rather, it was only after performing a sum over configurations in the partition functions that agreement was observed. It was therefore suggested that the relationship between the antisymmetric tensor fields in M-theory and IIA must be understood as a quantum equivalence.Here, we take a very modest step towards better understanding this relationship by describing how loop groups of E 8 can allow one to organize and generalize the Kaluza-Klein reduction of the quantization data in M-theory to include all nontrivial RR and NS fluxes, even G 0 = 0. To do this, we will consider in detail the Kaluza-Klein reduction of the E 8bundle data and certain generalizations suggested by this process. It was conjectured [7] that the resulting data might be understood in terms of bundles of a loop group of E 8 and that such a description might be related to the K-theory description in a more transparent way, perhaps along the lines suggested in [8]. This idea was first explored in [9] and later in [10,11], where it was suggested that for trivial M-theory circle bundles, the dimensional reduction of the E 8 -bundle in M-theory would be a LE 8 -bundle over the ten dimensional base (related ideas were further developed in [12,13,14,15,16]). Here, LE 8 is the free loop group of E 8 defined as maps from S 1 into E 8 with pointwise multiplication. However, we will find that this conjecture must be modified in order to include the case of a nontrivial M-theory principal circle bundle. We will establish a correspondence between E 8 -bundles in M-theory and LE 8 -bundles in IIA, where LE 8 is a slightly modified version of the loop group given by LE 8 def = U(1) ⋉ LE 8 . It was further conjectured in [9] that massive IIA supergravity [17] would be related to the centrally extended loop group LE k 8 . If this could be verified and an appropriate loop group generalization of the η invariants used in the work of [6,18] defined, one might be able to extend their results to G 0 = 0 by adding to the M-theory partition function sectors corresponding to the LE k 8 quantization of massive IIA for all k = G 0 . 2 While such a task is far...

Given a sufficiently nice collection of sheaves on an algebraic variety V , Bondal explained how to build a quiver Q along with an ideal of relations in the path algebra of Q such that the derived category of representations of Q subject to these relations is equivalent to the derived category of coherent sheaves on V . We consider the case in which these sheaves are all locally free and study the moduli spaces of semistable representations of our quiver with relations for various stability conditions. We show that V can often be recovered as a connected component of such a moduli space and we describe the line bundle induced by a GIT construction of the moduli space in terms of the input data. In certain special cases, we interpret our results in the language of topological string theory.