Narrow Results

Rigidity theorems for orbifold elliptic genus of complete simplicial multi-fans are formulated and proved. Applications to n-dimesional Q-factorial toric varieties whose canonical divisors are divisible by n are given.

Given a Morse–Smale function on an effective orientable orbifold, we construct its Morse–Smale–Witten complex. We show that critical points of a certain type have to be discarded to build a complex properly, and that gradient flows should be counted with suitable weights. Its homology is proven to be isomorphic to the singular homology of the quotient space under the self-indexing assumption. For a global quotient orbifold [M/G], such a complex can be understood as the G-invariant part of the Morse complex of M, where the G-action on generators of the Morse complex has to be defined including orientation spaces of unstable manifolds at the critical points. Alternatively in the case of global quotients, we introduce the notion of weak group actions on Morse–Smale–Witten complexes for non-invariant Morse–Smale functions on M, which give rise to genuine group actions on the level of homology.

We discuss the behavior of the bottom of the spectrum of scalar Schrödinger operators under Riemannian coverings of orbifolds. We apply our results to geometrically finite and to conformally compact orbifolds.

The theory of twistors on foliated manifolds is developed. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings.

We consider N = 2 superconformal field theories on a two-dimensional torus with central charge c = 3. In particular, we present the partition function for this theory. Furthermore, to generate new theories, we recall general orbifold prescription. Finally, we construct the modular invariant ℤ₃ orbifold-prime model.

The instability of protons is a crucial prediction of supersymmetric GUTs. We review the decay in minimal supersymmetric SU(5), which is dominated by dimension-five operators, and discuss the implications of the failure of Yukawa unification for the decay rate. In a consistent SU(5) model, where SU(5) relations among Yukawa couplings hold, the proton decay rate can be several orders of magnitude smaller than the present experimental bound. Finally, we discuss orbifold GUTs, where proton decay via dimension-five operators is absent. The branching ratios of dimension-six decay can significantly differ from those in four dimensions.

We review the gauge symmetry breaking mechanism due to orbifold projections in orbifold model building. We explicitly show the existence of a scale of breaking if such a symmetry breaking is due to freely-acting orbifold operators only, i.e. in case the breaking is realized nonlocally in the internal space. We show that such a scale is related to the compactification moduli only, and that there are no extra continuous parameters, at least in semirealistic models with SUSY in four dimensions. In this sense, the mechanism is peculiarly different from the standard Higgs (or Hosotani) symmetry breaking mechanism. We show that the mechanism also differs from that present in standard orbifold models where, even in the presence of discrete Wilson lines, a scale of breaking is generically missing, since the breaking is localized in specific points in the internal space.

We review a set of background geometries where the described nonlocal breaking is realized, both in the case of two and six extra dimensions. In the latter case, relevant in string model building, we consider both heterotic and open string compactifications.

One of the challenges in connecting higher dimensional theories to cosmology is stabilization of the moduli fields. We investigate the role of a Lorentz violating vector field in the context of stabilization. Specifically, we compute the one-loop Casimir energy in Randall–Sundrum five-dimensional (non-supersymmetric) S¹/ Z₂ orbifolds resulting from the interaction of a real scalar field with periodic boundary conditions with a Lorentz violating vector field. We find that the result is an enhanced attractive Casimir force. Hence, for stability, positive contributions to the Casimir force from branes and additional fields would be required to counter the destabilizing, attractive effect of Lorentz violating fields.

We study conformal field theory on two-dimensional orbifolds and show this to be an effective way to analyze physical effects of geometric singularities with angular deficits. They are closely related to boundaries and cross caps. Representative classes of singularities can be described exactly using generalizations of boundary states. From this we compute correlation functions and derive the spectra of excitations localized at the singularities.

Recently an action formulation, called the general WZW orbifold action, was given for each sector of every WZW orbifold. In this paper we gauge this action by general twisted gauge groups to find the action formulation of each sector of every coset orbifold. Connection with the known current-algebraic formulation of coset orbifolds is discussed as needed, and some large examples are worked out in further detail.

We find that the IIA matrix models defined on the non-compact C³/Z₆, C²/Z₂ and C²/Z₄ orbifolds preserve supersymmetry where the fermions are on-mass-shell Majorana–Weyl fermions. In these examples supersymmetry is preserved both in the orbifolded space and in the non-orbifolded space at the same time. The matrix model on C³/Z₆ orbifold has the same supersymmetry as the case of C³/Z₃ orbifold which was pointed out previously. On the other hand the matrix models on C²/Z₂ and C²/Z₄ orbifold have a half of the supersymmetry. We further find that the matrix model on C²/Z₂ orbifold with a parity-like identification preserves supersymmetry.

We consider orbifolds of two-dimensional unitary toroidal superconformal field theories with target spaces of arbitrary dimensions, where the orbifold group involves the space–time fermion number operator. We construct all so-called superaffine, orbifold prime and super-M-orbifold models by generalizing the constructions of Dixon, Ginsparg and Harvey. We also correct claims made by Dixon, Ginsparg and Harvey about multicritical points among those models with central charge .

We summarize calculation of one-loop beta functions in orbifolds of the SYM theory by a discrete subgroup Γ of the SU(4) R-symmetry, which are dual to string theory on AdS₅ × S⁵/Γ, and the general strategy for determining whether there is a fixed line passing through the origin of the coupling constant space. We emphasize the importance of decoupling the U(1) factors. We describe a class of non-supersymmetric field theories corresponding to orbifolds acting freely on the S⁵ which preserve some of its global symmetries. In this class we do not find any large N non-supersymmetric theories with fixed lines passing through the origin. Connection of these results with closed string tachyon condensation in AdS₅ × S⁵/Γ is discussed.

We discuss tachyon-free examples of (Type IIB on) noncompact nonsupersymmetric orbifolds. Tachyons are projected out by discrete torsion between orbifold twists, while supersymmetry is broken by a Scherk–Schwarz phase (+1/-1 when acting on space–time bosons/fermions) accompanying some even order twists. The absence of tachyons is encouraging for constructing nonsupersymmetric D3-brane gauge theories with stable infrared fixed points. The D3-brane gauge theories in our orbifold backgrounds have chiral supersymmetric spectra, but nonsupersymmetric interactions.

We propose a mechanism that the soft supersymmetry breaking masses and μ parameter can be induced from the dynamical rearrangement of local U(1) symmetries in a five-dimensional model. It offers to a solution to μ problem if there is a large hierarchy among the relevant U(1) charge of Higgsinos and that of hidden fields which stabilize the extra-dimensional component of U(1) gauge boson.

The symmetry breaking of five-dimensional SU(6) GUT is realized by Scherk–Schwarz mechanisms through trivial and pseudo nontrivial orbifold S¹/Z₂ breakings to produce dimensional deconstruction 5D SU(6) → 4DSU(6). The latter also induces near-brane weakly-coupled SU(6) Baby Higgs to further break the symmetry into SU(3)_C ⊗ SU(3)_H ⊗ U(1)_C. The model successfully provides a scenario of the origin of (Little) Higgs from GUT scale, produces the (intermediate and light) Higgs boson with the most preferred range and establishes coupling unification and compactification scale correctly.

The moduli space of the maximally supersymmetric heterotic string in $d$-dimensional Minkowski space contains various components characterized by the rank of the gauge symmetries of the vacua they parametrize. We develop an approach for describing in a unified way continuous Wilson lines which parametrize a component of the moduli space, together with discrete deformations responsible for the switch from one component to the other. Applied to a component that contains vacua with $SO(2n+1)$ gauge-symmetry factors, our approach yields a description of all backgrounds of the component in terms of free-orbifold models. The orbifold generators turn out to act symmetrically or asymmetrically on the internal space, with or without discrete torsion. Our derivations use extensively affine characters of $SO(2n+1)$. As a by-product, we find a peculiar orbifold description of the heterotic string in 10 dimensions, where all gauge degrees of freedom arise as twisted states, while the untwisted sector reduces to the gravitational degrees of freedom.

In this paper, we exhibit three new commensurability classes of arithmetic link complements, bringing the known total to seventeen.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

We determine the maximum order of cyclic group actions on the pair $(T^3,{\mathrm{\Sigma }}_g)$ among all embeddings of closed surface ${\mathrm{\Sigma }}_g(g>1)$ into the 3-dimensional torus $T^3$ in the orientable category.