Orbifold Singularities Research Articles

Linear sigma models for the [formula omitted] orbifold

We construct N=4 gauged linear sigma models in two dimensions whose Higgs branch has a R8/Zk orbifold singularity or its generalization. Our linear sigma models have either ALF or ALE type hyper-Kähler 8-manifolds as their Higgs branch. For the ALE case, the matter content of our model is specified by a quiver diagram which is a union of two A-type extended Dynkin diagrams overlapping at one link.

Superpotentials from stringy instantons without orientifolds

In this paper we show that it is possible to derive non-perturbative superpotential terms from a stringy instanton without introducing orientifold planes. The instanton is realized by a Euclidean D brane wrapping a non-trivial cycle upon which we also wrap a single space-filling D brane. The standard problem of unwanted neutral fermionic zero modes is evaded by the appearance of couplings to charged bosonic zero modes in the instanton moduli action. Since the Euclidean D brane wraps a cycle which is not associated to any low energy gauge dynamics, it can not be interpreted as an ordinary gauge instanton, but rather as a stringy one. By considering such a brane configuration at an orbifold singularity, we can explicitly evaluate the instanton moduli space integral and find a holomorphic superpotential term with the structure of a baryonic mass term.

Compact heterotic orbifolds in blow-up

We compare heterotic string models on orbifolds with supergravity models on smooth compact spaces, obtained by resolving the orbifold singularities. Our main focus is on heterotic E8 × E'8 models on the resolution of the compact T6/3 orbifold with Wilson lines. We explain how different gauge fluxes at various resolved fixed points can be interpreted in blow down as Wilson lines. Even when such Wilson lines are trivial from the orbifold perspective, they can still lead to additional symmetry breaking in blow-up. Full agreement is achieved between orbifold and resolved models, at the level of gauge interactions, massless spectrum and anomaly cancellation. In this matching the blow-up modes are of crucial importance: they play the role of model-dependent axions involved in the cancellation of multiple anomalous U(1)'s on the resolution. We illustrate various aspects by investigating blow-ups of a 3 MSSM model with two Wilson lines: if all its fixed points are resolved simultaneously, the SM gauge group is necessarily broken. Finally, we explore in detail the anomaly cancellation on the complex two dimensional resolution of 2/2.

Symplectic resolutions, Lefschetz property and formality

We introduce a method to resolve a symplectic orbifold ( M , ω ) into a smooth symplectic manifold ( M ˜ , ω ˜ ) . Then we study how the formality and the Lefschetz property of ( M ˜ , ω ˜ ) are compared with that of ( M , ω ) . We also study the formality of the symplectic blow-up of ( M , ω ) along symplectic submanifolds disjoint from the orbifold singularities. This allows us to construct the first example of a simply connected compact symplectic manifold of dimension 8 which satisfies the Lefschetz property but is not formal, therefore giving a counter-example to a conjecture of Babenko and Taimanov.

Instanton induced open string superpotentials and branes at singularities

We study different aspects of the non-perturbative superpotentials induced by Euclidean E3-branes on systems of D3/D7-branes located at Abelian orbifold singularities. We discuss in detail how the induced couplings are consistent with the U(1) symmetries carried by the D3/D7 branes. We construct different compact and non-compact examples, and show phenomenologically relevant couplings like μ-terms or certain Yukawa couplings generated by these E3 instantons. Some other novel effects are described. We show an example where E3 instantons combine with standard gauge instantons to yield new multi-instanton effects contributing to superpotential, along the lines of ref. [28]. In the case of non-SUSY ZN tachyon-free singularities it is shown how E3-instantons give rise to non perturbative scalar couplings including exponentially suppressed scalar bilinears.

Branes in the [formula omitted] WZNW model

We initiate a systematic study of boundary conditions in conformal field theories with target space supersymmetry. The WZNW model on GL ( 1 | 1 ) is used as a prototypical example for which we find the complete set of maximally symmetric branes. This includes a unique brane of maximal super-dimension 2 | 2 , a 2-parameter family of branes with super-dimension 0 | 2 and an infinite set of fully localized branes possessing a single modulus. Members of the latter family can only exist along certain lines on the bosonic base, much like fractional branes at orbifold singularities. Our results establish that all essential algebraic features of Cardy-type boundary theories carry over to the non-rational logarithmic WZNW model on GL ( 1 | 1 ) .

Stringy instantons at orbifold singularities

We study the effects produced by D-brane instantons on the holomorphic quantities of a D-brane gauge theory at an orbifold singularity. These effects are not limited to reproducing the well known contributions of the gauge theory instantons but also generate extra terms in the superpotential or the prepotential. On these brane instantons there are some neutral fermionic zero-modes in addition to the ones expected from broken supertranslations. They are crucial in correctly reproducing effects which are dual to gauge theory instantons, but they may make some other interesting contributions vanish. We analyze how orientifold projections can remove these zero-modes and thus allow for new superpotential terms. These terms contribute to the dynamics of the effective gauge theory, for instance in the stabilization of runaway directions.

Black holes, instanton counting on toric singularities and q-deformed two-dimensional Yang–Mills theory

We study the relationship between instanton counting in N = 4 Yang–Mills theory on a generic four-dimensional toric orbifold and the semi-classical expansion of q-deformed Yang–Mills theory on the blowups of the minimal resolution of the orbifold singularity, with an eye to clarifying the recent proposal of using two-dimensional gauge theories to count microstates of black holes in four dimensions. We describe explicitly the instanton contributions to the counting of D-brane bound states which are captured by the two-dimensional gauge theory. We derive an intimate relationship between the two-dimensional Yang–Mills theory and Chern–Simons theory on generic Lens spaces, and use it to show that the known instanton counting is only reproduced when the Chern–Simons contributions are treated as non-dynamical boundary conditions in the D4-brane gauge theory. We also use this correspondence to discuss the counting of instantons on higher genus ruled Riemann surfaces.

On tachyons in generic orbifolds of Bbb Crand gauged linear sigma models

We study some aspects of Gauged Linear Sigma Models corresponding to orbifold singularities of the form $\BC^r/\Gamma$, for $r=2,3$ and $\Gamma = \BZ_n$ and $\BZ_n\times \BZ_m$. These orbifolds might be tachyonic in general. We compute expressions for the multi parameter sigma model Lagrangians for these orbifolds, in terms of their toric geometry data. Using this, we analyze some aspects of the phases of generic orbifolds of $\BC^r$.

Propagation of a string across the cosmological singularity

Our results concern the transition of a quantum string through the singularity of the compactified Milne (CM) space. We restrict our analysis to the string winding around the compact dimension (CD) of spacetime. The CD undergoes contraction to a point followed by re-expansion. We demonstrate that both the classical and quantum dynamics of the considered string are well defined. Most of the presently available calculations strongly suggest that the singularity of a time-dependent orbifold is useless as a model of the cosmological singularity. We believe that our results bring, to some extent, this claim into question.

Non-Abelian vortices of higher winding numbers

We make a detailed study of the moduli space of winding number two ($k=2$) axially symmetric vortices (or equivalently, of coaxial composite of two fundamental vortices), occurring in $U(2)$ gauge theory with two flavors in the Higgs phase, recently discussed by Hashimoto and Tong and by Auzzi, Shifman, and Yung. We find that it is a weighted projective space $W\mathbf{C}{P}_{(2,1,1)}^{2}\ensuremath{\simeq}\mathbf{C}{P}^{2}/{\mathbf{Z}}_{2}$. This manifold contains an ${A}_{1}$-type (${\mathbf{Z}}_{2}$) orbifold singularity even though the full moduli space including the relative position moduli is smooth. The $SU(2)$ transformation properties of such vortices are studied. Our results are then generalized to $U(N)$ gauge theory with $N$ flavors, where the internal moduli space of $k=2$ axially symmetric vortices is found to be a weighted Grassmannian manifold. It contains singularities along a submanifold.

Radiative symmetry breaking on D-branes at non-supersymmetric singularities

The possibility of radiative gauge symmetry breaking on D3-branes at non-supersymmetric orbifold singularities is examined. As an example, a simple model of D3-branes at non-supersymmetric C 3 / Z 6 singularity with some D7-branes for the cancellations of RR tadpoles in twisted sectors is analyzed in detail. We find that there are no tachyon modes in twisted sectors, and NS–NS tadpoles in twisted sectors are canceled out, though uncanceled tadpoles and tachyon modes exist in untwisted sectors. This means that this singularity background is a stable solution of string theory at tree level, though some specific compactification of six-dimensional space should be considered for a consistent untwisted sector. On D3-brane three massless “Higgs doublet fields” and three family “up-type quarks” are realized at tree level. Other fermion fields, “down-type quarks” and “leptons”, can be realized as massless modes of the open strings stretching between D3-branes and D7-branes. The Higgs doublet fields have Yukawa couplings with up-type quarks, and they also have self-couplings which give a scalar potential without flat directions. Since there is no supersymmetry, the radiative corrections may naturally develop negative Higgs mass squared and “electroweak symmetry breaking”. We explicitly calculate the open string one-loop correction to the Higgs mass squared from twisted sectors, and find that the negative value is indeed realized in this specific model.

Fractional branes and the gravity dual of partial supersymmetry breaking

In type IIB string theory, we consider fractional D3-branes in the orbifold background dual to four-dimensional N = 2 supersymmetric Yang–Mills theory. We find the gravitational dual description of the generation of a non-trivial field theory potential on the Coulomb branch. When the orbifold singularity is softened to the smooth Eguchi–Hanson space, the resulting potential induces spontaneous partial supersymmetry breaking. We study the N = 1 theory arising in those vacua and we see how the resolved conifold geometry emerges in this way. It will be natural to identify the size of the blown-up two-cycle of that geometry with the inverse mass of the adjoint chiral scalar. We finally discuss the issue of geometric transition in this context.

Seven-dimensional Einstein manifolds from Tod–Hitchin geometry

We construct infinitely many seven-dimensional Einstein metrics of weak holonomy G 2 . These metrics are defined on principal SO(3) bundles over four-dimensional Bianchi IX orbifolds with the Tod–Hitchin metrics. The Tod–Hitchin metric has an orbifold singularity parametrized by an integer, and is shown to be similar near the singularity to the Taub-NUT de Sitter metric with a special charge. We show, however, that the seven-dimensional metrics on the total space are actually smooth. The geodesics on the weak G 2 manifolds are discussed. It is shown that the geodesic equation is equivalent to the Hamiltonian equation of an interacting rigid body system. We also discuss M-theory on the product space of AdS 4 and the seven-dimensional manifolds, and the dual gauge theories in three dimensions.

DESINGULARIZATIONS OF CALABI–YAU 3-FOLDS WITH A CONICAL SINGULARITY

We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on the same cone V, and then glue into M_0 at x after applying a homothety to Y. We then get a 1-parameter family of nearly Calabi-Yau 3-folds M_t depending on a small real variable t. For sufficiently small t, we show that the nearly Calabi-Yau structures on M_t can be deformed to genuine Calabi-Yau structures, and therefore obtaining the desingularizations of M_0. Our result can be applied to resolving orbifold singularities and hence provides a quantitative description of the Calabi-Yau metrics on the crepant resolutions.

Predictions for orientifold field theories from type [formula omitted] string theory

Two predictions about finite-N non-supersymmetric “orientifold field theories” are made by using the dual type 0′ string theory on C3/Z2×Z2 orbifold singularity. First, the mass ratio between the lowest pseudoscalar and scalar color-singlets is estimated to be equal to the ratio between the axial anomaly and the scale anomaly at strong coupling, M−/M+∼C−/C+. Second, the ratio between the domain wall tension and the value of the quark condensate is computed.

D-brane construction for non-Abelian walls

Supersymmetric $U({N}_{\mathrm{C}})$ gauge theory with ${N}_{\mathrm{F}}$ massive hypermultiplets in the fundamental representation is given by the brane configuration made of ${N}_{\mathrm{C}}$ fractional $\mathrm{D}p$-branes stuck at the ${\mathbf{Z}}_{2}$ orbifold singularity on ${N}_{\mathrm{F}}$ separated $\mathrm{D}(p+4)$-branes. We show that non-Abelian walls in this theory are realized as kinky fractional $\mathrm{D}p$-branes interpolating between $\mathrm{D}(p+4)$-branes. Wall solutions and their duality between ${N}_{\mathrm{C}}$ and ${N}_{\mathrm{F}}\ensuremath{-}{N}_{\mathrm{C}}$ imply extensions of the s-rule and the Hanany-Witten effect in brane dynamics. We also find that the reconnection of fractional D-branes occurs in this system. Diverse phenomena in non-Abelian walls found in field theory can be understood very easily by this brane configuration.

The Coulomb branch in gauged linear sigma models

We investigate toric GLSMs as models for tachyon condensation in type II strings on space-time non-supersymmetric orbifold singularities. The A-model correlators in these theories satisfy a set of relations related to the topology of the resolved orbifold. Using these relations we compute the correlators and find a non-trivial chiral ring in the IR, which we interpret as supported on isolated Coulomb vacua of the theory.

On tachyons, gauged linear sigma models, and flip transitions

We study systems of multiple localized closed string tachyons and the phenomena associated with their condensation, in C3/ZN nonsupersymmetric noncompact orbifold singularities using gauged linear sigma model constructions, following hep-th/0406039. Our study reveals close connections between the combinatorics of nonsupersymmetric flip transitions (between topologically distinct resolutions of the original singularity), the physics of tachyons of different degrees of relevance and the singularity structure of the corresponding residual endpoint geometries. This in turn can be used to study the stability of the phases of gauged linear sigma models and gain qualitative insight into the closed string tachyon potential.

N=1field theories and fluxes in IIB string theory

Deformation of $\mathcal{N}=2$ quiver gauge theories by adjoint masses leads to fixed manifolds of $\mathcal{N}=1$ superconformal field theories. We elaborate on the role of the complex three-form flux in the IIB duals to these fixed point theories, primarily using field theory techniques. We study the moduli space at a fixed point and find that it is either the two (complex) dimensional asymptotically locally Euclidean (ALE) space or three-dimensional generalized conifold, depending on the type of three-form flux that is present. We describe the exactly marginal operators that parameterize the fixed manifolds and find the operators which preserve the dimension of the moduli space. We also study deformations by arbitrary superpotentials $W({\ensuremath{\Phi}}_{i})$ for the adjoints. We invoke the $a$-theorem to show that there are no dangerously irrelevant operators like $\mathrm{T}\mathrm{r}{\ensuremath{\Phi}}_{i}^{k+1}$, $k>2$ in the $\mathcal{N}=2$ quiver gauge theories. The moduli space of the IR fixed point theory generally contains orbifold singularities if $W({\ensuremath{\Phi}}_{i})$ does not give a mass to the adjoints. Finally we examine some nonconformal $\mathcal{N}=1$ quiver theories. We find evidence that the moduli space at the endpoint of a Seiberg duality cascade is always a three-dimensional generalized conifold. In general, the low-energy theory receives quantum corrections. In several noncascading theories we find that the moduli space is a generalized conifold realized as a monodromic fibration.