Orbifold Singularities Research Articles

Instanton Counting and Wall-Crossing for Orbifold Quivers

Noncommutative Donaldson–Thomas invariants for abelian orbifold singularities can be studied via the enumeration of instanton solutions in a six-dimensional noncommutative \({{\mathcal N}=2}\) gauge theory; this construction is based on the generalized McKay correspondence and identifies the instanton counting with the counting of framed representations of a quiver which is naturally associated with the geometry of the singularity. We extend these constructions to compute BPS partition functions for higher-rank refined and motivic noncommutative Donaldson–Thomas invariants in the Coulomb branch in terms of gauge theory variables and orbifold data. We introduce the notion of virtual instanton quiver associated with the natural symplectic charge lattice which governs the quantum wall-crossing behaviour of BPS states in this context. The McKay correspondence naturally connects our formalism with other approaches to wall-crossing based on quantum monodromy operators and cluster algebras.

Non-perturbative gauge/gravity correspondence in $ \mathcal{N} $ = 2 theories

We derive the exact supergravity profile for the twisted scalar field emitted by a system of fractional D3 branes at a $ {\mathbb{Z}_2} $ orbifold singularity supporting $ \mathcal{N} $ = 2 quiver gauge theories with unitary groups and bifundamental matter. At the perturbative level this twisted field is “dual” to the gauge coupling but it is corrected non-perturbatively by an infinite tower of fractional D-instantons. The explicit microscopic description allows to derive the gravity profile from disk amplitudes computing the emission rate of the twisted scalar field in terms of chiral correlators in the dual gauge theory. We compute these quantum correlators using multi-instanton localization techniques and/or Seiberg-Witten analysis. Finally, we discuss a non-perturbative relation between the twisted scalar and the effective coupling of the gauge theory for some simple choices of the brane set ups.

5d quivers and their AdS 6 duals

We consider an infinite class of 5d supersymmetric gauge theories involving products of symplectic and unitary groups that arise from D4-branes at orbifold singularities in Type I' string theory. The theories are argued to be dual to warped AdS(6)x S4/Zn backgrounds in massive Type IIA supergravity. In particular, this demonstrates the existence of supersymmetric 5d fixed points of quiver type. We analyze the spectrum of gauge fields and charged states in the supergravity dual, and find a precise agreement with the symmetries and charged operators in the quiver theories. We also comment on other brane objects in the supergravity dual and their interpretation in the field theories.

Hirzebruch-Riemann-Roch-type formula for DG algebras

For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ringel's formula for quivers with relations. The second example is related to orbifold singularities of the form V/G where V is a complex vector space and G is a finite subgroup of SL(V). Furthermore, we prove that the above pairing on the Hochschild homology is non-degenerate when the DG algebra is smooth. We also formulate the conjecture that for a Calabi-Yau DG algebra A the pairing coincides with the one coming from the Topological Field Theory associated with A and verify it in the case of Frobenius algebras.

Unoriented D‐brane instantons

AbstractWe give a pedagogical introduction to D‐brane instanton effects in vacuum configurations with open and unoriented strings. We focus on quiver gauge theories for unoriented D‐branes at orbifold singularities and describe in some detail the ℤ3 case, where both ‘gauge’ and ‘exotic’ instantons can generate non‐perturbative super potentials, and the ℤ5 case, where supersymmetry breaking may arise from the combined effect of ‘gauge’ instantons and a FI D‐term.

Gauged Linear Sigma Models for toroidal orbifold resolutions

Toroidal orbifolds and their resolutions are described within the framework of (2,2) Gauged Linear Sigma Models (GLSMs). Our procedure describes two-tori as hypersurfaces in (weighted) projective spaces. The description is chosen such that the orbifold singularities correspond to the zeros of their homogeneous coordinates. The individual orbifold singularities are resolved using a GLSM guise of non-compact toric resolutions, i.e. replacing discrete orbifold actions by Abelian worldsheet gaugings. Given that we employ the same global coordinates for both the toroidal orbifold and its resolutions, our GLSM formalism confirms the gluing procedure on the level of divisors discussed by Lust et al. Using our global GLSM description we can study the moduli space of such toroidal orbifolds as a whole. In particular, changes in topology can be described as phase transitions of the underlying GLSM. Finally, we argue that certain partially resolvable GLSMs, in which a certain number of fixed points can never be resolved, might be useful for the study of mini-landscape orbifold MSSMs.

Black holes and singularity resolution in higher spin gravity

We investigate higher spin theories of gravity in three dimensions based on the gauge group SL(N,R)*SL(N,R). In these theories the usual diffeomorphism symmetry is enhanced to include higher spin gauge transformations under which traditional geometric notions of curvature and causality are no longer invariant. This implies, for example, that apparently singular geometries can be rendered smooth by a gauge transformation, much like the resolution of orbifold singularities in string theory. The classical solutions, including the recently constructed higher spin black hole, are characterized by their holonomies around the non-contractible cycles of space-time. The black hole solutions are shown to be gauge equivalent to a BTZ black hole which is charged under a set of U(1) Chern-Simons fields. Nevertheless, depending on the choice of embedding of the gravitational gauge group, the space-time geometry may be non-trivial. We study in detail the N=3 example, where this observation allows us to find a gauge where the black hole geometry takes a simple form and the thermodynamic properties can be studied.

Scattering and sequestering of blow-up moduli in local string models

We study the scattering and sequestering of blow-up fields - either local to or distant from a visible matter sector - through a CFT computation of the dependence of physical Yukawa couplings on the blow-up moduli. For a visible sector of D3-branes on orbifold singularities we compute the disk correlator < \tau_s^{(1)} \tau_s^{(2)} ... \tau_s^{(n)} \psi \psi \phi > between orbifold blow-up moduli and matter Yukawa couplings. For n = 1 we determine the full quantum and classical correlator. This result has the correct factorisation onto lower 3-point functions and also passes numerous other consistency checks. For n > 1 we show that the structure of picture-changing applied to the twist operators establishes the sequestering of distant blow-up moduli at disk level to all orders in \alpha'. We explain how these results are relevant to suppressing soft terms to scales parametrically below the gravitino mass. By giving vevs to the blow-up fields we can move into the smooth limit and thereby derive CFT results for the smooth Swiss-cheese Calabi-Yaus that appear in the Large Volume Scenario.

Instantons, quivers and noncommutative Donaldson–Thomas theory

We construct noncommutative Donaldson–Thomas invariants associated with abelian orbifold singularities by analyzing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson–Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.

= 2 Instanton Effective action in Ω-background and D3/D(−1)-brane System in R-R background

We study the relation between the ADHM construction of instantons in the Ω-background and the fractional D3/D(−1)-branes at the orbifold singularity of ℂ × ℂ2/ℤ2 in Ramond-Ramond (R-R) 3-form field strength background. We calculate disk amplitudes of open strings connecting the D3/D(−1)-branes in certain R-R background to obtain the D(−1)-brane effective action deformed by the R-R background. We show that the deformed D(−1)-brane effective action agrees with the instanton effective action in the Ω-background.

Convergence of Calabi–Yau manifolds

In this paper, we study the convergence of Calabi–Yau manifolds under Kähler degeneration to orbifold singularities and complex degeneration to canonical singularities (including the conifold singularities), and the collapsing of a family of Calabi–Yau manifolds.

Parameter spaces of massive IIA solutions

We find a new class of N=2 massive IIA solutions whose internal spaces are S^2 fibrations over S^2 x S^2. These solutions appear naturally as massive deformations of the type IIA reduction of Sasaki-Einstein manifolds in M-theory, including Q^{1,1,1} and Y^{p,k}, and play a role in the AdS4/CFT3 correspondence. We use this example to initiate a systematic study of the parameter space of massive solutions with fluxes. We define and study the natural parameter space of the solutions, which is a certain dense subset of R^3, whose boundaries correspond to orbifold or conifold singularities. On a codimension-one subset of the parameter space, where the Romans mass vanishes, it is possible to perform a lift to M-theory; extending earlier work, we produce a family A^{p,q,r} of Sasaki-Einstein manifolds with cohomogeneity one and SU(2) x SU(2) x U(1) isometry. We also propose a Chern-Simons theory describing the duals of the massless and massive solutions.

Orbifold singularities, Lie algebras of the third kind (LATKes), and pure Yang–Mills with matter

We discover the unique, simple Lie algebra of the third kind, or LATKe, that stems from codimension 6 orbifold singularities and gives rise to a new kind of Yang–Mills theory which simultaneously is pure and contains matter. The root space of the LATKe is one-dimensional and its Dynkin diagram consists of one point. The uniqueness of the LATKe is a vacuum selection mechanism. \documentclass[12pt]{minimal}\begin{document}\[\begin{array}{c}\hbox{The World in a Point?}\\[12pt]\hbox{Blow-up of} C^3/Z_3|\,\, \hbox{Dynkin diagram of the LATKe}\\[12pt]\bullet \\[12pt]\hbox{Pure Yang-Mills with matter}\end{array}\]\end{document}TheWorldinaPoint?Blow-upofC3/Z3|DynkindiagramoftheLATKe●PureYang-Millswithmatter

Kinetic mixing of U(1)’s for local string models

We study kinetic mixing between massless U(1)s in toroidal orbifolds with D3-branes at orbifold singularities. We focus in particular on C^3/Z_4 singularities but also study C^3/Z_6 and C^3/Z'_6 singularities. We find kinetic mixing can be present and describe the conditions for it to occur. Kinetic mixing comes from winding modes in the N=2 sector of the orbifold. If kinetic mixing is present its size depends only on the complex structure modulus of the torus and is independent of the K\"ahler moduli. We also study gauge threshold corrections for local Z_M x Z_N orbifold models finding that, consistent with previous studies, gauge couplings run from the bulk winding scale rather than the string scale.

$ \mathcal{N} = 2 $ instanton effective action in Ω-background and D3/D(−1)-brane system in R-R background

We study the relation between the ADHM construction of instantons in the Ω-background and the fractional D3/D(−1)-branes at the orbifold singularity of $ \mathbb{C} \times {{{{\mathbb{C}^2}}} \left/ {{{\mathbb{Z}_2}}} \right.} $ in Ramond-Ramond (R-R) 3-form field strength background. We calculate disk amplitudes of open strings connecting the D3/D(−1)-branes in certain R-R background to obtain the D(−1)-brane effective action deformed by the R-R background. We show that the deformed D(−1)-brane effective action agrees with the instanton effective action in the Ω-background.

Anomaly mediation in superstring theory

AbstractWe study anomaly mediated supersymmetry breaking in type IIB string theory and use our results to test the supergravity formula for anomaly mediated gaugino masses. We compute 1‐loop gaugino masses for models of D3‐branes on orbifold singularities with 3‐form fluxes by calculating the annulus correlator of 3‐form flux and two gauginos in the zero momentum limit. Consistent with supergravity expectations we find both anomalous and running contributions to 1‐loop gaugino masses. For background Neveu‐Schwarz H‐flux we find an exact match with the supergravity formula. For Ramond‐Ramond flux there is an off‐shell ambiguity that precludes a full matching. The anomaly mediated gaugino masses, while determined by the infrared spectrum, arise from an explicit sum over UV open string winding modes. We also calculate brane‐to‐brane tree‐level gravity mediated gaugino masses and show that there are two contributions coming from the dilaton and from the twisted modes, which are suppressed by the full T6 volume and the untwisted T2 volume respectively.

Dirac quantization of membrane winding uniformly in time dependent orbifold

We present quantum theory of a membrane propagating in the vicinity of a time dependent orbifold singularity. The dynamics of a membrane, with the parameters space topology of a torus, winding uniformly around compact dimension of the embedding spacetime is mathematically equivalent to the dynamics of a closed string in a flat FRW spacetime. The construction of the physical Hilbert space of a membrane makes use of the kernel space of self-adjoint constraint operators. It is a subspace of the representation space of the constraints algebra. There exist non-trivial quantum states of a membrane evolving across the singularity.

Constructing Calabi–Yau metrics from hyperkähler spaces

Recently, a metric construction for Calabi–Yau threefolds from a four-dimensional hyperkähler space by adding a complex line bundle was proposed. We extend the construction by adding a U(1) factor to the holomorphic (3, 0)-form and obtain the explicit formalism for a generic hyperkähler base. We find that a discrete choice arises: the U(1) factor can either depend solely on the fiber coordinates or vanish. In each case, the metric is determined by a differential equation for the modified Kähler potential. As explicit examples, we obtain the generalized resolutions (up to orbifold singularity) of the cone of the Einstein–Sasaki spaces Yp, q. We also obtain a large class of new singular CY3 metrics with SU(2) × U(1) or SU(2) × U(1)2 isometries.

Rod-structure classification of gravitational instantons with [formula omitted] isometry

The rod-structure formalism has played an important role in the study of black holes in D = 4 and 5 dimensions with R × U ( 1 ) D − 3 isometry. In this paper, we apply this formalism to the study of four-dimensional gravitational instantons with U ( 1 ) × U ( 1 ) isometry, which could serve as spatial backgrounds for five-dimensional black holes. We first introduce a stronger version of the rod structure with the rod directions appropriately normalised, and show how the regularity conditions can be read off from it. Requiring the absence of conical and orbifold singularities will in general impose periodicity conditions on the coordinates, and we illustrate this by considering known gravitational instantons in this class. Some previous results regarding certain gravitational instantons are clarified in the process. Finally, we show how the rod-structure formalism is able to provide a classification of gravitational instantons, and speculate on the existence of possible new gravitational instantons.

DYNAMICAL ELECTROWEAK SYMMETRY BREAKING IN STRING MODELS WITH D-BRANES

The possibility of dynamical electroweak symmetry breaking by strong coupling gauge interaction in models with D-branes in String Theory is examined. Instead of elementary scalar Higgs doublet fields, the gauge symmetry with strong coupling (technicolor) is introduced. As the first step of this direction, a toy model, which is not fully realistic, is concretely analyzed in some detail. The model consists of D-branes and anti-D-branes at orbifold singularities in (T2 × T2 × T2)/Z3 which preserves supersymmetry. Supersymmetry is broken through the brane supersymmetry breaking. It is pointed out that the problem of large S parameter in dynamical electroweak symmetry breaking scenario may be solved by natural existence of kinetic term mixings between hypercharge U(1) gauge boson and massive anomalous U(1) gauge bosons. The problems to be solved toward constructing more realistic models are clarified in the analysis.