Orientifold Projection Research Articles

Noncommutative and gauge theories

We study the generalization of noncommutative gauge theories to the case of orthogonal and symplectic groups. We find out that this is possible, since we are allowed to define orthogonal and symplectic subgroups of noncommutative unitary gauge transformations even though the gauge potentials and gauge transformations are not valued in the orthogonal and symplectic subalgebras of the Lie algebra of antihermitean matrices. Our construction relies on an antiautomorphism of the basic noncommutative algebra of functions which generalizes the charge conjugation operator of ordinary field theory. We show that the corresponding noncommutative picture from low energy string theory is obtained via orientifold projection in the presence of a non-trivial NSNS B-field.

GEOMETRY OF ORIENTIFOLDS WITH NS–NS B FLUX

We discuss geometry underlying orientifolds with nontrivial NS–NS B flux. If D-branes wrap a torus with B flux the rank of the gauge group is reduced due to noncommuting Wilson lines whose presence is implied by the B flux. In the case of c-branes transverse to a torus with B flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation conditions in the presence of B flux as some of the orientifold planes now have the opposite orientifold projection. We point out that T duality in the presence of B flux is more subtle than in the case with trivial B flux, and it is precisely consistent with the qualitative difference between the aforementioned two setups. In the case where both types of branes are present, the states in the mixed (e.g. 59) open string sectors come with a nontrivial multiplicity, which we relate to a discrete gauge symmetry due to nonzero B flux, and construct vertex operators for the mixed sector states. Using these results we revisit K3 orientifolds with B flux (where K3 is a T4/ZM orbifold) and point out various subtleties arising in some of these models. For instance, in the Z2 case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds with B flux. This is related to the fact that nonzero B flux requires the presence of both O5-- as well as O5+-planes at various Z2 orbifold fixed points, which appears to be inconsistent with the presence of the twistedB flux in the conformal field theory orbifold. We also consider four-dimensional [Formula: see text] and [Formula: see text] supersymmetric orientifolds. We construct consistent four-dimensional models with B flux which do not suffer from difficulties encountered in the K3 cases.

A new orientifold of C 2/Z N and six-dimensional RG fixed points

We discuss the consistency conditions of a novel orientifold projection of type IIB string theory on C 2/ Z N singularities, in which one mods out by the combined action of world-sheet parity and a geometric operation which exchanges the two complex planes. The field theory on the world-volume of D5-brane probes defines a family of six-dimensional RG fixed points, which had been previously constructed using type IIA configurations of NS-branes and D6-branes in the presence of O6-planes. Both constructions are related by a T-duality transforming the set of NS-branes into the C 2/ Z N singularity. We also construct additional models, where both the standard and the novel orientifold projections are imposed. They have an interesting relation with orientifolds of D K singularities, and provide the T-duals of certain type IIA configurations containing both O6- and O8-planes.

Geometry of Orientifolds with NS-NS B-flux

We discuss geometry underlying orientifolds with non-trivial NS-NS B-flux. If D-branes wrap a torus with B-flux the rank of the gauge group is reduced due to non-commuting Wilson lines whose presence is implied by the B-flux. In the case of D-branes transverse to a torus with B-flux the rank reduction is due to a smaller number of D-branes required by tadpole cancellation conditions in the presence of B-flux as some of the orientifold planes now have the opposite orientifold projection. We point out that T-duality in the presence of B-flux is more subtle than in the case with trivial B-flux, and it is precisely consistent with the qualitative difference between the aforementioned two setups. In the case where both types of branes are present, the states in the mixed (e.g., 59) open string sectors come with a non-trivial multiplicity, which we relate to a discrete gauge symmetry due to non-zero B-flux, and construct vertex operators for the the mixed sector states. Using these results we revisit K3 orientifolds with B-flux (where K3 is a T^4/Z_M orbifold) and point out various subtleties arising in some of these models. For instance, in the Z_2 case the conformal field theory orbifold does not appear to be the consistent background for the corresponding orientifolds with B-flux. This is related to the fact that non-zero B-flux requires the presence of both O5^- as well as O5^+ planes at various Z_2 orbifold fixed points, which appears to be inconsistent with the presence of the twisted B-flux in the conformal field theory orbifold. We also consider four dimensional N=2 and N=1 supersymmetric orientifolds. We construct consistent four dimensional models with B-flux which do not suffer from difficulties encountered in the K3 cases.

On second-quantized open superstring theory

The SO(32) theory, in the limit where it is an open superstring theory, is completely specified in the light-cone gauge as a second-quantized string theory in terms of a “matrix string” model. The theory is defined by the neighborhood of a 1 + 1-dimensional fixed point theory, characterized by an Abelian gauge theory with type IB Green-Schwarz form. Non-orientability and SO(32) gauge symmetry arise naturally, and the theory effectively constructs an orientifold projection of the (weakly coupled) matrix type IIB theory (also discussed herein). The fixed point theory is a conformal field theory with boundary, defining the free string theory. Interactions involving the interior of open and closed strings are governed by a twist operator in the bulk, while string endpoints are created and destroyed by a boundary twist operator.

D = 4, N = 1, type IIB orientifolds

We study different aspects of the construction of D = 4, N = 1 type IIB orientifolds based on toroidal Z N and Z M × Z N, D = 4 ortifolds . We find that tadpole cancellation conditions are in general more constraining than in six dimensions and that the standard Gimon-Polchinski orientifold projection leads to the impossibility of tadpole cancellations in a number of Z N orientifolds with even N including Z 4, Z 8, Z 8′ and Z 12′ . We construct D = 4, Z N and Z N × Z M orientifolds with different configurations of 9-branes, 5-branes and 7-branes, most of them chiral. Models including the analogue of discrete torsion are constructed and shown to have features previously conjectured on the basis of F-theory compactified on four-folds. Different properties of the D = 4, N = 1 models obtained are discussed including their possible heterotic duals and effective low-energy action. These models have in general more than one anomalous U(1) and the anomalies are cancelled by a D = 4 generalized Green-Schwarz mechanism involving dilaton and moduli fields.

Brane dynamics and chiral non-chiral transitions

We study brane realizations of chiral matter in N = 1 supersymmetric gauge theories in four dimensions. A “cross” configuration which leads to “flavor doubling” is found to have a superpotential. The main example is realized using a special “fork” configuration. Many of the results are found by studying a SU × SU product gauge group first. The chiral theory is then an orientifold projection of the product gauge group. An interesting observation in the brane picture is that there are transitions between chiral and non-chiral models. These transitions are closely related to small instanton transitions in six dimensions.

Gauge theories with tensors from branes and orientifolds

We present brane constructions in type-IIA string theory for $\mathcal{N}=1$ supersymmetric SO and Sp gauge theories with tensor representations using an orientifold six-plane. One limit of these setups corresponds to $\mathcal{N}=2$ theories previously constructed by Landsteiner and Lopez, while a different limit yields $\mathcal{N}=1$ SO or Sp theories with a massless tensor and no superpotential. For the Sp-type orientifold projection, a comparison with the field-theory moduli space leads us to postulate two new rules governing the stability of configurations of $D$-branes intersecting the orientifold. Lifting one of our configurations to $M$ theory by finding the corresponding curves, we rederive the $\mathcal{N}=1$ dualities for SO and Sp groups using semi-infinite $D4$ branes.

Tensor constructions of open string theories II: Vector bundles, D-branes and orientifold groups

A generalized Chan-Paton construction is presented which is analogous to the tensor product of vector bundles. To this end open string theories are considered where the space of states decomposes into sectors whose product is described by a semigroup. The cyclicity properties of the open string theory are used to prove that the relevant semigroups are direct unions of Brandt semigroups. The known classification of Brandt semigroups then implies that all such theories have the structure of a theory with Dirichlet-branes. We also describe the structure of an arbitrary orientifold group, and show that the truncation to the invariant subspace defines a consistent open string theory. Finally, we analyze the possible orientifold projections of a theory with several kinds of branes.

Tensors fromK3orientifolds

Recently Gimon and Johnson (hep-th/9604129) and Dabholkar and Park (hep-th/9604178) have constructed Type I theories on K3 orbifolds. The spectra differ from that of Type I on a smooth K3, having extra tensors. We show that the orbifold theories cannot be blown up to smooth K3's, but rather $Z_2$ orbifold singularities always remain. Douglas's recent proposal to use D-branes as probes is useful in understanding the geometry. The $Z_2$ singularities are of a new type, with a different orientifold projection from those previously considered. We also find a new world-sheet consistency condition that must be satisfied by orientifold models.

Orbifold and orientifold compactifications of F-theory and M-theory to six and four dimensions

We study orbifold compactifications of F-theory which lead to N = 1 supersymmetry in six and four space-time dimensions. These are dual to specific orientifolds of M-theory, and in many cases to orientifolds of type IIB string theory. The equivalences are demonstrated by mapping the orbifolding transformations in the F-, M- and string theories to each other using dualities. We observe that M- and F-theory appear to possess a property similar to discrete torsion in string theory. This is related to an ambiguity recently noted by Polchinski in the orientifold projection for six-dimensional models. The four-dimensional compactifications exhibit similar features, from which we predict the existence of certain new orientifolds of type IIB. Some orbifolds with higher supersymmetry are also examined.

An orbifold and an orientifold of type IIB theory on K3 × K3

We consider the compactification of type IIB superstring theory of K3 × K3. We obtain the massless spectrum of the resulting two dimensional theory and show that the model is free of gravitational anomaly. We then consider an orbifold and an orientifold projection of the above model and find that their spectrum match identically and are anomaly-free as well. This gives a dual pair of type IIB theory in two dimensions and can be understood as a consequence of SL(2, Z) symmetry of the ten dimensional theory. We also point out the M-theory duals of the type IIB compactifications considered here.

Consistency conditions for orientifolds and D-manifolds.

We study superstrings with orientifold projections and with generalized open string boundary conditions (D-branes). We find two types of consistency condition, one related to the algebra of Chan-Paton factors and the other to cancellation of divergences. One consequence is that the Dirichlet 5-branes of the Type I theory carry a symplectic gauge group, as required by string duality. As another application we study the Type I theory on a $K3$ $Z_2$ orbifold, finding a family of consistent theories with various unitary and symplectic subgroups of $U(16) \times U(16)$. We argue that the $K3$ orbifold with spin connection embedded in gauge connection corresponds to an interacting conformal field theory in the Type I theory.