Abstract:
In this paper we make two observations related to discrete torsion. First, we observe that an old obscure degree of freedom (momentum/translation shifts) in (symmetric) string orbifolds is related to discrete torsion. We point out how our previous derivation of discrete torsion from orbifold group actions on B fields includes these momentum lattice shift phases, and discuss how they are realized in terms of orbifold group actions on D-branes. Second, we describe the M theory dual of IIA discrete torsion, a duality relation to our knowledge not previously understood. We show that IIA discrete torsion is encoded in analogues of the shift orbifolds above for the M theory C field.

Abstract:
This is the writeup of a lecture given at the May Wisconsin workshop on mathematical aspects of orbifold string theory. In the first part of this lecture, we review recent work on discrete torsion, and outline how it is currently understood in terms of the B field. In the second part of this lecture, we discuss the relationship between quotient stacks and string orbifolds.

Abstract:
D-branes on orbifolds with and without discrete torsion are analysed in a unified way using the boundary state formalism. For the example of the Z2 x Z2 orbifold it is found that both the theory with and without discrete torsion possess D-branes whose world-volume carries conventional and projective representations of the orbifold group. The resulting D-brane spectrum is shown to be consistent with T-duality.

Abstract:
The consistency of the orbifold action on open strings between D-branes in orbifold theories with and without discrete torsion is analysed carefully. For the example of the C^3/Z_2 x Z_2 theory, it is found that the consistency of the orbifold action requires that the D-brane spectrum contains branes that give rise to a conventional representation of the orbifold group as well as branes for which the representation is projective. It is also shown how the results generalise to the orbifolds C^3/Z_N x Z_N for which a number of novel features arise. In particular, the N>2 theories with minimal discrete torsion have non-BPS branes charged under twisted R-R potentials that couple to none of the (known) BPS branes.

Abstract:
Armed with the explicit computation of Schur Multipliers, we offer a classification of SU(n) orbifolds for n = 2,3,4 which permit the turning on of discrete torsion. This is in response to the host of activity lately in vogue on the application of discrete torsion to D-brane orbifold theories. As a by-product, we find a hitherto unknown class of N = 1 orbifolds with non-cyclic discrete torsion group. Furthermore, we supplement the status quo ante by investigating a first example of a non-Abelian orbifold admitting discrete torsion, namely the ordinary dihedral group as a subgroup of SU(3). A comparison of the quiver theory thereof with that of its covering group, the binary dihedral group, without discrete torsion, is also performed.

Abstract:
Using the algebraic geometric approach of Berenstein et {\it al} (hep-th/005087 and hep-th/009209) and methods of toric geometry, we study non commutative (NC) orbifolds of Calabi-Yau hypersurfaces in toric varieties with discrete torsion. We first develop a new way of getting complex $d$ mirror Calabi-Yau hypersurfaces $H_{\Delta}^{\ast d}$ in toric manifolds $M_{\Delta }^{\ast (d+1)}$ with a $C^{\ast r}$ action and analyze the general group of the discrete isometries of $H_{\Delta}^{\ast d}$. Then we build a general class of $d$ complex dimension NC mirror Calabi-Yau orbifolds where the non commutativity parameters $\theta_{\mu \nu}$ are solved in terms of discrete torsion and toric geometry data of $M_{\Delta}^{(d+1)}$ in which the original Calabi-Yau hypersurfaces is embedded. Next we work out a generalization of the NC algebra for generic $d$ dimensions NC Calabi-Yau manifolds and give various representations depending on different choices of the Calabi-Yau toric geometry data. We also study fractional D-branes at orbifold points. We refine and extend the result for NC $% (T^{2}\times T^{2}\times T^{2})/(\mathbf{{Z_{2}}\times {Z_{2})}}$ to higher dimensional torii orbifolds in terms of Clifford algebra.

Abstract:
We discuss some Z_N^L x Z_N^R orbifold compactifications of the type IIB superstring to D= 4,6 dimensions and their type I descendants. Although the Z_N^L x Z_N^R generators act asymmetrically on the chiral string modes, they result into left-right symmetric models that admit sensible unorientable reductions. We carefully work out the phases that appear in the modular transformations of the chiral amplitudes and identify the possibility of introducing discrete torsion. We propose a simplifying ansatz for the construction of the open-string descendants in which the transverse-channel Klein-bottle, annulus and Moebius-strip amplitudes are numerically identical in the proper parametrization of the world-sheet. A simple variant of the ansatz for the Z_2^L x Z_2^R orbifold gives rise to models with supersymmetry breaking in the open-string sector.

Abstract:
In this article we explain discrete torsion. Put simply, discrete torsion is the choice of orbifold group action on the B field. We derive the classification H^2(G, U(1)), we derive the twisted sector phases appearing in string loop partition functions, we derive M. Douglas's description of discrete torsion for D-branes in terms of a projective representation of the orbifold group, and we outline how the results of Vafa-Witten fit into this framework. In addition, we observe that additional degrees of freedom (known as shift orbifolds) appear in describing orbifold group actions on B fields, in addition to those classified by H^2(G, U(1)), and explain how these new degrees of freedom appear in terms of twisted sector contributions to partition functions and in terms of orbifold group actions on D-brane worldvolumes. This paper represents a technically simplified version of prior papers by the author on discrete torsion. We repeat here technically simplified versions of results from those papers, and have included some new material.

Abstract:
We analyse D-branes on orbifolds with discrete torsion, extending earlier results. We analyze certain Abelian orbifolds of the type C^3/ \Gamma, where \Gamma is given by Z_m x Z_n, for the most general choice of discrete torsion parameter. By comparing with the AdS/CFT correspondence, we can consider different geometries which give rise to the same physics. This identifies new mirror pairs and suggests new dualities at large N. As a by-product we also get a more geometric picture of discrete torsion.

Abstract:
The theory of permutation orbifolds is reviewed and applied to the study of symmetric product orbifolds and the congruence subgroup problem. The issue of discrete torsion, the combinatorics of symmetric products, the Galois action and questions related to the classification of RCFTs are also discussed.