Klein Bottles and Simple Currents

The standard Klein bottle coefficient in the construction of open descendants is shown to equal the Frobenius-Schur indicator of a conformal field theory. Other consistent Klein bottle projections are shown to correspond to simple currents. These observations enable us to generalize the standard open string construction from C-diagonal parent theories to include non-standard Klein bottles. Using (generalizations of) the Frobenius-Schur indicator we prove positivity and integrality of the resulting open and closed string state multiplicities for standard as well as non-standard Klein bottles.Comment: 11 pages, LaTeX. References added, minor error correcte

Crosscaps, Boundaries and T-Duality

Open descendants with boundaries and crosscaps of non-trivial automorphism type are studied. We focus on the case where the bulk symmetry is broken to a Z_2 orbifold subalgebra. By requiring positivity and integrality for the open sector, we derive a unique crosscap of automorphism type g \in Z_2 and a corresponding g-twisted Klein bottle for a charge conjugation invariant. As a specific example, we use T-duality to construct the descendants of the true diagonal invariant with symmetry preserving crosscaps and boundaries.Comment: Latex, 14 page

Open Descendants of Non-Diagonal Invariants

The open descendants of simple current automorphism invariants are constructed. We consider the case where the order of the current is two or odd. We prove that our solutions satisfy the completeness conditions, positivity and integrality of the open and closed sectors and the Klein bottle constraint (apart from an interesting exception). In order to do this, we derive some new relations between the tensor Y and the fixed point conformal field theory. Some non-standard Klein bottle projections are considered as well.Comment: 21 pages, LaTe

WZW orientifolds and finite group cohomology

The simplest orientifolds of the WZW models are obtained by gauging a Z_2 symmetry group generated by a combined involution of the target Lie group G and of the worldsheet. The action of the involution on the target is by a twisted inversion g \mapsto (\zeta g)^{-1}, where \zeta is an element of the center of G. It reverses the sign of the Kalb-Ramond torsion field H given by a bi-invariant closed 3-form on G. The action on the worldsheet reverses its orientation. An unambiguous definition of Feynman amplitudes of the orientifold theory requires a choice of a gerbe with curvature H on the target group G, together with a so-called Jandl structure introduced in hep-th/0512283. More generally, one may gauge orientifold symmetry groups \Gamma = Z_2 \ltimes Z that combine the Z_2-action described above with the target symmetry induced by a subgroup Z of the center of G. To define the orientifold theory in such a situation, one needs a gerbe on G with a Z-equivariant Jandl structure. We reduce the study of the existence of such structures and of their inequivalent choices to a problem in group-\Gamma cohomology that we solve for all simple simply-connected compact Lie groups G and all orientifold groups \Gamma = Z_2 \ltimes Z.Comment: 48+1 pages, 11 figure

Boundaries, crosscaps and simple currents

Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory.Comment: 11 pages, LaTeX2e, reference added, typos correcte

Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models

We construct d=4,N=1 orientifolds of Gepner models with just the chiral spectrum of the standard model. We consider all simple current modular invariants of c=9 tensor products of N=2 minimal models. For some very specific tensor combinations, and very specific modular invariants and orientifold projections, we find a large number of such spectra. We allow for standard model singlet (dark) matter and non-chiral exotics. The Chan-Paton gauge group is either U(3) x Sp(2) x U(1) x U(1) or U(3) x U(2) x U(1) x U(1). In many cases the standard model hypercharge U(1) has no coupling to RR 2-forms and hence remains massless; in some of those models the B-L gauge boson does acquire a mass.Comment: 16 pages, LaTeX, minor corrections, references added Link added to updated and almost complete result

Chiral Supersymmetric Gepner Model Orientifolds

We explicitly construct A-type orientifolds of supersymmetric Gepner models. In order to reduce the tadpole cancellation conditions to a treatable number we explicitly work out the generic form of the one-loop Klein bottle, annulus and Moebius strip amplitudes for simple current extensions of Gepner models. Equipped with these formulas, we discuss two examples in detail to provide evidence that in this setting certain features of the MSSM like unitary gauge groups with large enough rank, chirality and family replication can be achieved.Comment: 37 pages, TeX (harvmac), minor changes, typos corrected, to appear in JHE

Geometry of WZW Orientifolds

We analyze unoriented Wess-Zumino-Witten models from a geometrical point of view. We show that the geometric interpretation of simple current crosscap states is as centre orientifold planes localized on conjugacy classes of the group manifold. We determine the locations and dimensions of these planes for arbitrary simply-connected groups and orbifolds thereof. The dimensions of the O-planes turn out to be given by the dimensions of symmetric coset manifolds based on regular embeddings. Furthermore, we give a geometrical interpretation of boundary conjugation in open unoriented WZW models; it yields D-branes together with their images under the orientifold projection. To find the agreement between O-planes and crosscap states, we find explicit answers for lattice extensions of Gaussian sums. These results allow us to express the modular P-matrix, which is directly related to the crosscap coefficient, in terms of characters of the horizontal subgroup of the affine Lie algebra. A corollary of this relation is that there exists a formal linear relation between the modular P- and the modular S-matrix.Comment: 35 pages LaTeX, 2 tables; Proof added for symmetric space relation; minor improvements; references adde

Supersymmetric Orientifolds of Gepner Models

Supersymmetric orientifolds of four dimensional Gepner Models are constructed in a systematic way. For all levels of the Gepner model being odd the generic expression for both the A-type and the B-type Klein bottle amplitude is derived. The appearing massless tadpoles are canceled by introducing appropriate boundary states of Recknagel/Schomerus(RS). After determining the Moebius strip amplitude we extract general expressions for the tadpole cancellation conditions. We discuss the issue of chirality for such supersymmetric orientifold models and finally present a couple of examples in detail.Comment: 38 pages, TeX harvmac, ref. adde

Liouville Field Theory on an Unoriented Surface

Liouville field theory on an unoriented surface is investigated, in particular, the one point function on a RP^2 is calculated. The constraint of the one point function is obtained by using the crossing symmetry of the two point function. There are many solutions of the constraint and we can choose one of them by considering the modular bootstrap.Comment: 13 pages, no figures, LaTeX, minor changes, equations in section 4 are correcte