Orbifold Singularities Research Articles

Flux-induced SUSY-breaking soft terms on D7–D3 brane systems

We study the effect of RR and NSNS 3-form fluxes on the effective action of the world-volume fields of Type IIB D7/D3 brane configurations. The D7-branes wrap 4-cycles Σ 4 on a local Calabi–Yau geometry. This is an extension of previous work on hep-th/0311241, where a similar analysis was applied to the case of D3-branes. Our present analysis is based on the D7- and D3-brane Dirac–Born–Infeld and Chern–Simons actions, and makes full use of the R-symmetries of the system, which allow us to compute explicitly results for the fields lying at the D3–D7 intersections. A number of interesting new properties appear as compared to the simpler case of configurations with only D3-branes. As a general result one finds that fluxes stabilize some or all of the D7-brane moduli. We argue that this is important for the problem of stabilizing Kähler moduli through non-perturbative effects in KKLT-like vacua. We also show that ( 0 , 3 ) imaginary self-dual fluxes, which lead to compactifications with zero vacuum energy, give rise to SUSY-breaking soft terms including gaugino and scalar masses, and trilinear terms. Particular examples of chiral MSSM-like models of this class of vacua, based on D3–D7 brane systems at orbifold singularities are presented.

Localized Tachyons and the Quantum McKay Correspondence

The condensation of closed string tachyons localized at the fixed point of a d/Γ orbifold can be studied in the framework of renormalization group flow in a gauged linear sigma model. The evolution of the Higgs branch along the flow describes a resolution of singularities via the process of tachyon condensation. The study of the fate of D-branes in this process has lead to a notion of a ``quantum McKay correspondence.'' This is a hypothetical correspondence between fractional branes in an orbifold singularity in the ultraviolet with the Coulomb and Higgs branch branes in the infrared. In this paper we present some nontrivial evidence for this correspondence in the case 2/n by relating the intersection form of fractional branes to that of ``Higgs branch branes,'' the latter being branes which wrap nontrivial cycles in the resolved space.

Witten's formulas for intersection pairings on moduli spaces of flat G-bundles

In a 1992 paper (J. Geom. Phys. 9 (1992) 303), Witten gave a formula for the intersection pairings of the moduli space of flat G-bundles over an oriented surface, possibly with markings. In this paper, we give a general proof of Witten's formula, for arbitrary compact, simple groups, and any markings for which the moduli space has at most orbifold singularities.

Localized tachyons inC3/ZN

We study the condensation of localized closed string tachyons in C^3/Z_N nonsupersymmetric noncompact orbifold singularities via renormalization group flows that preserve supersymmetry in the worldsheet conformal field theory and their interrelations with the toric geometry of these orbifolds. We show that for worldsheet supersymmetric tachyons, the endpoint of tachyon condensation generically includes ``geometric'' terminal singularities (orbifolds that do not have any marginal or relevant Kahler blowup modes) as well as singularities in codimension two. Some of the various possible distinct geometric resolutions are related by flip transitions. For Type II theories, we show that the residual singularities that arise under tachyon condensation in various classes of Type II theories also admit a Type II GSO projection. We further show that Type II orbifolds entirely devoid of marginal or relevant blowup modes (Kahler or otherwise) cannot exist, which thus implies that the endpoints of tachyon condensation in Type II theories are always smooth spaces.

Type IIA on a compact Calabi‐Yau and D = 11 supergravity uplift of its orientifold

AbstractUsing the prescription of [1] for defining period integrals in the Landau‐Ginsburg theory for compact Calabi‐Yau's, we obtain the Picard‐Fuchs equation and the Meijer basis of solutions for the compact Calabi‐Yau CY3(3,243) expressed as a degree‐24 Fermat hypersurface after resolution of the orbifold singularities. This is similar in spirit to the method of obtaining Meijer basis of solutions in [2] for the case wherein one is away from the orbifold singularities, and one is considering the large‐base limit of the Calabi‐Yau. The importance of the method lies in the ease with which one can consider the large and small complex structure limits, as well as the ability to get the “ln”‐terms in the periods without having to parametrically differentiate infinite series. We consider in detail the evaluation of the monodromy matrix in the large and small complex structure limits. We also consider the action of the freely acting antiholomorphic involution of [2,3] on D = 11 supergravity compactified on CY3(3,243) × S1 [4] and obtain the Kähler potential for the same in the limit of large volume of the Calabi‐Yau. As a by‐product, we also give a conjecture for the action of the orientation‐reversing antiholomorphic involution on the periods, given its action on the cohomology, using a canonical (co)homology basis. Finally, we also consider showing a null superpotential on the orientifold of type IIA on CY3(3,243), having taken care of the orbifold singularities, thereby completing the argument initiated in [2].

Inflation on Fractional Branes: D–Brane Inflation as D–Term Inflation

We describe a D--brane inflation model which consists of two fractional D3 branes separated on a transverse $T^2 \times K3$. Inflation arises due to the resolved orbifold singularity of $K3$ which corresponds to an anomalous D--term on the brane. We show that D--brane inflation in the bulk corresponds to D--term inflation on the brane. The inflaton and the trigger field parametrize the interbrane distances on $T^2$ an $K3$ respectively. After inflation the branes reach a supersymmetric configuration in which they are at the origin of $T^2$ but separated along the $K3$ directions.

Non-minimal Higgs content in standard-like models from D-branes at a [formula omitted] singularity

We show that attempts to construct the Standard Model, or the MSSM, by placing D3-branes and D7-branes at a ZN orbifold or orientifold singularity all require that the electroweak Higgs content is non-minimal. For the orbifold the lower bound on the number n(H)+n(H¯) of electroweak Higgs doublets is the number n(qLc)=6 of quark singlets, and for the orientifold the lower bound can be one less. As a consequence there is a generic flavour changing neutral current problem in such models.

ANOMALIES IN FIELD THEORIES WITH EXTRA DIMENSIONS

We give an overview of the issue of anomalies in field theories with extra dimensions. We start by reviewing in a pedagogical way the computation of the standard perturbative gauge and gravitational anomalies on noncompact spaces, using Fujikawa's approach and functional integral methods, and discuss the available mechanisms for their cancellation. We then generalize these analyses to the case of orbifold field theories with compact internal dimensions, emphasizing the new aspects related to the presence of orbifold singularities and discrete Wilson lines, and the new cancellation mechanisms that are becoming available. We conclude with a very brief discussion on global and parity anomalies.

Witten-Veneziano from Green-Schwarz

We consider the U(1) problem within the AdS/CFT framework. We explain how the Witten-Veneziano formula for the eta' mass is related to a generalized Green-Schwarz mechanism. The closed string mode, that cancels the anomaly of the gauged U(1) axial symmetry, is identified with the eta' meson. In a particular set-up of D3-branes on a C3/(Z3xZ3) orbifold singularity, the eta' meson is a twisted-sector R-R field.

Realistic D-Brane Models on Warped Throats: Fluxes, Hierarchies and Moduli Stabilization

We describe the construction of string theory models with semirealistic spectrum in a sector of (anti) D3-branes located at an orbifold singularity at the bottom of a highly warped throat geometry, which is a generalisation of the Klebanov-Strassler deformed conifold. These models realise the Randall-Sundrum proposal to naturally generate the Planck/electroweak hierarchy in a concrete string theory embedding, and yielding interesting chiral open string spectra. We describe examples with Standard Model gauge group (or left-right symmetric extensions) and three families of SM fermions, with correct quantum numbers including hypercharge. The dilaton and complex structure moduli of the geometry are stabilised by the 3-form fluxes required to build the throat. We describe diverse issues concerning the stabilisation of geometric Kahler moduli, like blow-up modes of the orbifold singularities, via D term potentials and gauge theory non-perturbative effects, like gaugino condensation. This local geometry, once embedded in a full compactification, could give rise to models with all moduli stabilised, and with the potential to lead to de Sitter vacua. Issues of gauge unification, proton stability, supersymmetry breaking and Yukawa couplings are also discussed.

Stringy profiles of gauge field tadpoles near orbifold singularities: II. field theoretical implications

Recent work has provided a direct string calculation of the internal coordinate dependence of gauge field tadpoles on the orbifold C^3/Z_3. We investigate the structure of these profiles in momentum and coordinate space representations using both analytical and numerical methods. The twisted states are determined to be localized within a few string lengths of the singularity. This provides an example of how delta-like singularities, which are typical for field theories on orbifolds, are avoided in string theory. A systematic expansion of the full string result allows us to speculate on how the leading stringy effects can be incorporated in a field theoretical description. In one heterotic model, we find that even though tachyons are not present in the physical spectrum, they totally dominate the tadpole profile in the vicinity of the fixed point.

On (orientifold of) type IIA on a compact Calabi‐Yau

AbstractWe study the gauged sigma model and its mirror Landau‐Ginsburg model corresponding to type IIA on the Fermat degree‐24 hypersurface in WCP4[1,1,2,8,12] (whose blow‐up gives the smooth CY3(3,243)) away from the orbifold singularities, and its orientifold by a freely‐acting antiholomorphic involution. We derive the Picard‐Fuchs equation obeyed by the period integral as defined in [1, 2], of the parent 𝒩 = 2 type IIA theory of [3]. We obtain the Meijer's basis of solutions to the equation in the large and small complex structure limits (on the mirror Landau‐Ginsburg side) of the abovementioned Calabi‐Yau, and make some remarks about the monodromy properties associated based on [4], at the same and another MATHEMATICAlly interesting point. Based on a recently shown 𝒩 = 1 four‐dimensional triality [6] between Heterotic on the self‐mirror Calabi‐Yau CY3(11,11), M theory on ${CY_3(3,243)\times S^1\over{\bf Z}_2}$ and F‐theory on an elliptically fibered CY4 with the base given by CP1 × Enriques surface, we first give a heuristic argument that there can be no superpotential generated in the orientifold of of CY3(3,243), and then explicitly verify the same using mirror symmetry formulation of [2] for the abovementioned hypersurface away from its orbifold singularities. We then discuss briefly the sigma model and the mirror Landau‐Ginsburg model corresponding to the resolved Calabi‐Yau as well.

Stringy profiles of gauge field tadpoles near orbifold singularities: I. heterotic string calculations

Closed string theories on orbifolds contain both untwisted and twisted states. The latter are normally assumed to live exactly at the orbifold fixed points. We perform a calculation of a gauge field tadpole amplitude and show that off-shell both the twisted and untwisted states give rise to non-trivial momentum profiles over the orbifold C^3/Z_3. These profiles take the form of Gaussian distributions integrated over the fundamental domain of the modular parameter of the torus. The propagators of the internal coordinate fields on the torus world sheet determine the width of the Gaussian profiles. These propagators are determined up to a single normal ordering constant which must be bounded below to allow the existence of the coordinate space representation of these Gaussians. Apart from the expected massless states, massive and even tachyonic string excitations contribute to the profiles in some anomalous U(1) models. However, when a tadpole is integrated over the internal dimensions, these tachyonic contributions cancel in a non-trivial manner.

Einstein metrics and complex singularities

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kaehler metric (which is hyperkaehler if and only if c_1(X)=0), and that if c_1(X)<0 then X also admits a complete (non-Kaehler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number. Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable. All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.

Instanton Expansion of Noncommutative Gauge Theory in Two Dimensions

We show that noncommutative gauge theory in two dimensions is an exactly solvable model. A cohomological formulation of gauge theory defined on the noncommutative torus is used to show that its quantum partition function can be written as a sum over contributions from classical solutions. We derive an explicit formula for the partition function of Yang-Mills theory defined on a projective module for arbitrary noncommutativity parameter \theta which is manifestly invariant under gauge Morita equivalence. The energy observables are shown to be smooth functions of \theta. The construction of noncommutative instanton contributions to the path integral is described in some detail. In general, there are infinitely many gauge inequivalent contributions of fixed topological charge, along with a finite number of quantum fluctuations about each instanton. The associated moduli spaces are combinations of symmetric products of an ordinary two-torus whose orbifold singularities are not resolved by noncommutativity. In particular, the weak coupling limit of the gauge theory is independent of \theta and computes the symplectic volume of the moduli space of constant curvature connections on the noncommutative torus.

Rotationally Symmetric Shrinking and Expanding Gradient Kähler-Ricci Solitons

We construct new families of Kähler-Ricci solitons on complex line bundles over ℂℙn−1, n ≥ 2. Among these are examples whose initial or final condition is equal to a metric cone ℂn/ℤk. We exhibit a noncompact Ricci flow that shrinks smoothly and self-similarly for t < 0, becomes a cone at t = 0, and then expands smoothly and self-similarly for t > 0; this evolution is smooth in space-time except at a single point, at which there is a blowdown of a ℂℙn−1. We also construct certain shrinking solitons with orbifold point singularities.

NC geometry and fractional branes

Considering complex n-dimension Calabi–Yau homogeneous hypersurfaces ℋn with discrete torsion and using the Berenstein and Leigh algebraic geometry method, we study fractional D-branes that result from stringy resolution of singularities. We first develop the method introduced by Berenstein and Leigh (Preprint hep-th/0105229) and then build the non-commutative (NC) geometries for orbifolds \U0001d4aa = ℋn/Zn+2n with a discrete torsion matrix tab = exp[i2π/n+2(ηab − ηba)], ηab ∊ SL(n, Z). We show that the NC manifolds \U0001d4aa(nc) are given by the algebra of functions on the real (2n + 4) fuzzy torus \U0001d4afβij2(n+2) with deformation parameters βij = exp i2π/n+2[(ηab−1 − ηba−1)qai qbj] with qai being charges of Znn+2. We also give graphic rules to represent \U0001d4aa(nc) by quiver diagrams which become completely reducible at orbifold singularities. It is also shown that regular points in these NC geometries are represented by polygons with (n + 2) vertices linked by (n + 2) edges while singular ones are given by (n + 2) non-connected loops. We study the various singular spaces of quintic orbifolds and analyse the varieties of fractional D-branes at singularities as well as the spectrum of massless fields. Explicit solutions for the NC quintic \U0001d4ac(nc) are derived with details and general results for complex n-dimension orbifolds with discrete torsion are presented.

Cohomology pairings on singular quotients in geometric invariant theory

In this paper we shall give formulas for the pairings of intersection cohomology classes of complementary dimensions in the intersection cohomology of geometric invariant-theoretic quotients for which semistability is not necessarily the same as stability (although we make some weaker assumptions on the action). We also give formulas for intersection pairings on resolutions of singularities (or more precisely partial resolutions, since orbifold singularities are allowed) of the quotients.

On the classical stability of orientifold cosmologies

We analyse the classical stability of string cosmologies driven by the dynamics of orientifold planes. These models are related to time-dependent orbifolds, and resolve the orbifold singularities which are otherwise problematic by introducing orientifold planes. In particular, we show that the instability discussed by Horowitz and Polchinski for pure orbifold models is resolved by the presence of the orientifolds. Moreover, we discuss the issue of stability of the cosmological Cauchy horizon, and we show that it is stable to small perturbations due to in-falling matter.

Compact Einstein spaces based on quaternionic Kähler manifolds

We investigate the Einstein equation with a positive cosmological constant for (4n + 4)-dimensional metrics on bundles over quaternionic Kähler base manifolds whose fibres are four-dimensional Bianchi IX manifolds. The Einstein equations are reduced to a set of nonlinear ordinary differential equations. We numerically find inhomogeneous compact Einstein spaces with orbifold singularity.