Global anomalies on orbifolds

Communicated by A. Jaffe     

Global gravitational anomalies

A general formula for global gauge and gravitational anomalies is derived. It is used to show that the anomaly free supergravity and superstring theories in ten dimensions are all free of global anomalies that might have ruined their consistency. However, it is shown that global anomalies lead to some restrictions on allowed compactifications of these theories. For example, in the case of O(32) superstring theory, it is shown that a global anomaly related to ₇(O(32)) leads to a Dirac-like quantization condition for the field strength of the antisymmetric tensor field.Related to global anomalies is the question of the number of fermion zero modes in an instanton field. It is argued that the relevant gravitational instantons are exotic spheres. It is shown that the number of fermion zero modes in an instanton field is always even in ten dimensional supergravity.Supported in part by NSF Grant No. PHY80-19754     

Global gravitational anomalies from perturbative chiral anomalies

The relation between (4k ? 2)-dimensional global gravitational anomalies and perturbative chiral anomalies in 4k dimensions is clarified using an open space generalization of the Atiyah-Patodi-Singer index theorem. These anomalies are then reduced to a chiral anomaly in a two-dimensional Schwinger model. It is argued that “all” anomalies can be similarly reduced.     

Sphalerons on orbifolds

In this work, we study the electroweak sphalerons in a 5D background, where the fifth dimension lies on an interval. We consider two specific cases: flat space-time and the anti-de Sitter space-time compactified on S ¹/Z ₂. In our work, we take the SU(2) gauge–Higgs model, where the gauge fields reside in the 5D bulk; but the Higgs doublet is confined in one brane. We find that the results in this model are close to those of the 4D Standard Model (SM). The existence of the warp effect, as well as the heaviness of the gauge Kaluza–Klein modes make the results extremely close to the SM ones.     

Constraints on asymmetric orbifolds

It is claimed that besides the left-right matching condition, one has to check the relative phases dictated by modular invariance for an asymmetric orbifold to give rise to an acceptable model.     

Open strings on orbifolds

We consider propagation of type I SO(32) superstrings on orbifolds. It is shown that anomaly cancellation requires the existence of “twisted” open strings and we determine the Chan-Paton factors for these strings in a simple example.     

Permutation orbifolds

A general theory of permutation orbifolds is developed for arbitrary twist groups. Explicit expressions for the number of primaries, the partition function, the genus one characters, the matrix elements of modular transformations and for fusion rule coefficients are presented, together with the relevant mathematical concepts, such as Λ-matrices and twisted dimensions. The arithmetic restrictions implied by the theory for the allowed modular representations in CFT are discussed. The simplest nonabelian example with twist group S₃ is described to illustrate the general theory.     

Degenerate orbifolds

We present evidence for the existence of new four-dimensional string theories, obtained from a smooth variation of background fields in the twisted sectors of symmetric and asymmetric orbifolds. Flat directions only in the untwisted sector are shown to reproduce previously constructed models in terms of Wilson lines, exhibiting a Three-Higgs-Rule (THR). The new models provide a mechanism to lower the rank of the gauge group, lead to more flexible Yukawa couplings and give a strict separation of hidden and observable sectors, which are usually mixed in (2, 0)-models. Even though Fayet-Iliopoulos terms are induced in some of the models due to the presence of anomalous U(1)'s supersymmetry remains, in general, unbroken. Particular examples of the new models correspond to “blown up” versions of (2, 0)-orbifolds.     

Dirac operators on non-compact orbifolds

In this paper we prove that Dirac operators on non-compact almost complex, complete orbifolds which are sufficiently regular at infinity, admit a unique extension. Additonally, we prove a generalized orbifold Stokes’/Divergence theorem.     

Generalized symmetry breaking on orbifolds

We reconsider the phenomenon of mass generation via coordinate-dependent compactifications of higher-dimensional theories on orbifolds. For definiteness, we study a generic five-dimensional theory compactified on S(1)/Z(2). We show that the presence of fixed points, where the fields or their derivatives may be discontinuous, permits new realizations of the Scherk-Schwarz mechanism where, for example, the mass terms are localized at the orbifold fixed points. Our technique can be used to describe the explicit breaking of global flavor symmetries and supersymmetries by brane-localized mass terms. It can also be applied to the spontaneous breaking of local symmetries, such as gauge symmetries or supergravities.     

Global anomalies and homotopy group cocycles

A consistency condition for global anomalies is expressed in terms of cocycles of homotopy groups and is used to characterise all possible global anomalies.     

Duality and orbifolds

We construct several examples where duality transformation commutes with the orbifolding procedure even when the orbifolding group does not act freely, and there are massless states from the twisted sector at a generic point in the moduli space. Often the matching of spectrum in the dual theories is a result of nontrivial identities satisfied by the coefficients on one loop tadpoles in the heterotic, type II and type I string theories.     

Strings on orbifolds (II)

A general framework for analyzing string propagation on the quotient of a manifold by a discrete group is presented.The conditions necessary for modular invariance and absence of tachyons are discussed. Models with four or fewer generations which correspond to the standard embedding of the spin connection in the gauge group are analyzed. We all discuss some generalizations which may lead to more phenomenologically promising models.     

Non-renormalization of Yukawa couplings on orbifolds

We discuss the non-renormalization of Yukawa couplings in the orbifold compactification of the heterotic string, at the one-loop level. Special attention is paid to on-shell infrared divergences. The non-vanishing part of the one-loop quark-quark-quark scattering amplitude is identified with the one-loop correction of the external legs.     

Classical and quantum ergodicity on orbifolds

We extend classical results on quantum ergodicity due to Shnirelman, Colin de Verdière, and Zelditch to orbifolds, proving that, for any positive, first-order self-adjoint elliptic pseudodifferential operator P on a compact orbifold X with positive principal symbol p, the ergodicity of the Hamiltonian flow of p implies the quantum ergodicity for the operator P. We also prove the ergodicity of the geodesic flow on a compact Riemannian orbifold of negative sectional curvature.     

Cosmic strings as orbifolds

We construct string theory versions of cosmic strings by considering orbifold compactifications of spacetime down to two dimensions.     

Equivariant quantization of orbifolds

Equivariant quantization is a new theory that highlights the role of symmetries in the relationship between classical and quantum dynamical systems. These symmetries are also one of the reasons for the recent interest in quantization of singular spaces, orbifolds, stratified spaces, etc. In this work, we prove the existence of an equivariant quantization for orbifolds. Our construction combines an appropriate desingularization of any Riemannian orbifold by a foliated smooth manifold, with the foliated equivariant quantization that we built in Poncin et al. (2009) [19]. Further, we suggest definitions of the common geometric objects on orbifolds, which capture the nature of these spaces and guarantee, together with the properties of the mentioned foliated resolution, the needed correspondences between singular objects of the orbifold and the respective foliated objects of its desingularization.