ADE Singularities Research Articles

Matter from geometry without resolution

We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of Z^N, where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal's recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E_6. We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D_4 to G_2 in an example. A companion mathematical paper will follow.

Modular forms and special cubic fourfolds

We study the degrees of special cubic divisors on moduli space of cubic fourfolds with at worst ADE singularities. In this paper, we show that the generating series of the degrees of such divisors is a level three modular form.

ADE Bundles over Surfaces with ADE Singularities

Given a complex projective surface with an ADE singularity and p_{g}=0, we construct ADE bundles over it and its minimal resolution. Furthermore, we descibe their minuscule representation bundles in terms of configurations of (reducible) (-1)-curves.

Mordell–Weil groups of elliptic threefolds and the Alexander module of plane curves

Abstract We establish a correspondence between the rank of Mordell–Weil group of the complex elliptic threefold associated with a plane curve 𝒞 ⊂ ℙ2(𝔻) with equation F = 0, certain roots of the Alexander polynomial associated with the fundamental group π1(ℙ2(𝔻)∖𝒞) and the polynomial solutions for the functional equation of type h 1 p F 1 + h 2 q F 2 + h 3 r F 3 = 0 where F = F 1 F 2 F 3. This correspondence is obtained for curves in a certain class which includes the curves having introduced here δ-essential singularities and in particular for all curves with ADE singularities. As a consequence we find a linear bound for the degree of the Alexander polynomial in terms of the degree of 𝒞 for curves with δ-essential singularities and in particular arbitrary ADE singularities.

Simultaneous semi-stable reduction for curves with ADE singularities

A key tool in the study of algebraic surfaces and their moduli is Brieskorn's simultaneous resolution for families of algebraic surfaces with simple (du Val or ADE) singularities. In this paper we show that a similar statement holds for families of curves with at worst simple (ADE) singularities. For a family X ! B of ADE curves, we give an explicit and natural resolution of the rational map B 99K Mg. Moreover, we discuss a lifting of this map to the moduli stack Mg, i.e. a simultaneous semi-stable reduction for the family X /B. In particular, we note that in contrast to what might be expected from the case of surfaces, the natural Weyl cover of B is not a sufficient base change for a lifting of the map B 99K Mg to Mg.

Deformation of a singularity of type E8 and Mordell-Weil lattices in characteristic 2

In this paper, we study the Mordell-Weil lattices of the family of elliptic surfaces which is arising from the E84 singularity, one of the ADE singularities in characteristic 2. And we construct a subfamily of the universal family of supersingular K 3 surfaces in characteristic 2 as an application (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Matrix model superpotentials and ADE singularities

We use F. Ferrari’s methods relating matrix models to Calabi-Yau spaces in order to explain much of Intriligator and Wecht’s ADE classification of N = 1 superconformal theories which arise as RG fixed points of N = 1 SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator–Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yaus with corresponding ADE singularities. Moreover, in the additional b O, b A, b D and b E cases we find new singular geometries. These ‘hat’ geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two coordinate charts. To obtain these results we develop an algorithm for blowing down exceptional P 1 s, described in the appendix.

CHEN–RUAN COHOMOLOGY OF ADE SINGULARITIES

We study Ruan's cohomological crepant resolution conjecture [41] for orbifolds with transversal ADE singularities. In the An-case, we compute both the Chen–Ruan cohomology ring [Formula: see text] and the quantum corrected cohomology ring H*(Z)(q1,…,qn). The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between [Formula: see text] and H*(Z)(-1) in the A1-case, verifying Ruan's conjecture. In the An-case, the family H*(Z)(q1,…,qn) is not defined for q1 = ⋯ = qn = -1. This implies that the conjecture should be slightly modified. We propose a new conjecture in the An-case (Conjecture 1.9). Finally, we prove Conjecture 1.9 in the A2-case by constructing an explicit isomorphism.

Black holes in type IIA string on Calabi–Yau threefolds with affine ADE geometries and q-deformed 2d quiver gauge theories

Motivated by studies on 4d black holes and q-deformed 2d Yang–Mills theory, and borrowing ideas from compact geometry of the blowing up of affine ADE singularities, we build a class of local Calabi–Yau threefolds (CY 3) extending the local 2-torus model O ( m ) ⊕ O ( − m ) → T 2 considered in [C. Gomez, S. Montanez, A comment on quantum distribution functions and the OSV conjecture, hep-th/0608162] to test OSV conjecture. We first study toric realizations of T 2 and then build a toric representation of X 3 using intersections of local Calabi–Yau threefolds O ( m ) ⊕ O ( − m − 2 ) → P 1 . We develop the 2d N = 2 linear σ-model for this class of toric CY 3s. Then we use these local backgrounds to study partition function of 4d black holes in type IIA string theory and the underlying q-deformed 2d quiver gauge theories. We also make comments on 4d black holes obtained from D-branes wrapping cycles in O ( m ) ⊕ O ( − m − 2 ) → B k with m = ( m 1 , … , m k ) a k-dim integer vector and B k a compact complex one dimension base consisting of the intersection of k 2-spheres S i 2 with generic intersection matrix I i j . We give as well the explicit expression of the q-deformed path integral measure of the partition function of the 2d quiver gauge theory in terms of I i j . A comment on the link between our analysis and the construction of [N. Caporaso, M. Cirafici, L. Griguolo, S. Pasquetti, D. Seminara, R.J. Szabo, Topological strings, two-dimensional Yang–Mills theory and Chern–Simons theory on torus bundles, hep-th/0609129] is also given.

Theta-characteristics on singular curves

On a smooth curve a theta–characteristic is a line bundle L with square that is the canonical line bundle ω. The equivalent conditionHom(L, ω) ∼= L generalizes well to singular curves, as applications show. More precisely, a theta–characteristic is a torsion–free sheaf F of rank 1 with Hom(F , ω) ∼= F . If the curve has non ADE–singularities then there are infinitely many theta–characteristics. Therefore, theta–characteristics are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta–characteristics (i.e. F with h(C,F) ≡ 0 resp. h(C,F) ≡ 1 modulo 2) in terms of the geometric genus of the curve and certain discrete invariants of a fixed local type. Mathematics Subject Classification (2000). 14H60, 14H42, 14H40. A theta–characteristic on a smooth projective curve C over C is a line bundle L whose square is the canonical bundle K. Let Θ be the set of all theta–characteristics. If g is the genus of the curve, then there are 2 theta– characteristics on C. Naturally, one would like to know the dimension of the linear systems |L|, L ∈ Θ. However, these dimensions depend not only on the genus alone, but also on the complex structure of the curve. Nevertheless, using the his theory of theta–functions Riemann proved that the dimension modulo 2 depends only on the genus. If Θ = {L ∈ Θ | h(L) is even.} Θ− = {L ∈ Θ | h(L) is odd.} are the even resp. odd theta–characteristics, then there are 2g−1(2g + 1) even and 2g−1(2g − 1) odd theta–characteristics. Atiyah gave another analytic proof of this, and Mumford the first algebraic one [A, M]. Mumford’s ideas were refined and extended by Harris to include singular curves into the theory — at least Gorenstein curves, for example plane curves [H]. On a singular curve a line bundle is defined to be a locally free sheaf of rank 1. Harris showed that the number of even and odd theta–characteristics can be computed in terms of the genus of the curve and certain discrete invariants of the singularaties. He also remarked that it would be desirable to be able to treat torsion–free sheaves of rank 1 alongside with the line bundles. This is what we want to do in this article.

Extended Seiberg-Witten theory and integrable hierarchy

The prepotential of the effective N=2 super-Yang-Mills theory perturbed in the ultraviolet by the descendents of the single-trace chiral operators is shown to be a particular tau-function of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory with 2N-2 fundamental hypermultiplets at the appropriate locus of the moduli space of vacua) or a theory on a single fractional D3 brane at the ADE singularity the hierarchy is the dispersionless Toda chain. We present its explicit solutions. Our results generalize the limit shape analysis of Logan-Schepp and Vershik-Kerov, support the prior work hep-th/0302191 which established the equivalence of these N=2 theories with the topological A string on CP^1 and clarify the origin of the Eguchi-Yang matrix integral. In the higher rank case we find an appropriate variant of the quasiclassical tau-function, show how the Seiberg-Witten curve is deformed by Toda flows, and fix the contact term ambiguity.

Distribution of flux vacua around singular points in Calabi-Yau moduli space

We study the distribution of type IIB flux vacua in the moduli space near various singular loci, e.g. conifolds, ADE singularities on P1, Argyres-Douglas point etc, using the Ashok- Douglas density det(R + omega). We find that the vacuum density is integrable around each of them, irrespective of the type of the singularities. We study in detail an explicit example of an Argyres-Douglas point embedded in a compact Calabi-Yau manifold.

RG fixed points and flows in SQCD with adjoints

We map out and explore the zoo of possible 4d N=1 superconformal theories which are obtained as RG fixed points of N=1 SQCD with N f fundamental and N a adjoint matter representations. Using “a-maximization”, we obtain exact operator dimensions at all RG fixed points and classify all relevant, Landau–Ginzburg type, adjoint superpotential deformations. Such deformations can be used to RG flow to new SCFTs, which are then similarly analyzed. Remarkably, the resulting 4d SCFT classification coincides with Arnold's ADE singularity classification. The exact superconformal R-charge and the central charge a are computed for all of these theories. RG flows between the different fixed points are analyzed, and all flows are verified to be compatible with the conjectured a-theorem.

Anomalies in M-theory on singular G2-manifolds

When M-theory is compactified on G 2-holonomy manifolds with conical singularities, charged chiral fermions are present and the low-energy four-dimensional theory is potentially anomalous. We reconsider the issue of anomaly cancellation, first studied by Witten. We propose a mechanism that provides local cancellation of all gauge and mixed gauge-gravitational anomalies, i.e., separately for each conical singularity. It is similar in spirit to the one used to cancel the normal bundle anomaly in the presence of five-branes. It involves smoothly cutting off all fields close to the conical singularities, resulting in an anomalous variation of the 3-form C and of the non-abelian gauge fields present if there are also ADE singularities.

Comments on quiver gauge theories and matrix models

Dijkgraaf and Vafa have conjectured the correspondences between topological string theories, N=1 gauge theories and matrix models. By the use of this conjecture, we calculate the quantum deformations of Calabi–Yau threefolds with ADE singularities from ADE multi-matrix models. We analyse the effective superpotentials of the dual quiver gauge theories in terms of the geometric engineering for the deformed geometries. We find a part of the Veneziano–Yankielowicz terms in the effective superpotentials.

Metrics building of pp-wave orbifold geometries

We study strings on orbifolds of AdS×S5 by SU(2) discrete groups in the Penrose limit. We derive the degenerate metrics of pp wave of AdS×S5/Γ using ordinary ADE and affine ADE singularities of complex surfaces and results on N=4 CFT4's. We also give explicit metric moduli dependences for both Abelian and non-Abelian orbifolds.

Mixed characteristic hypersurfaces of finite Cohen–Macaulay type

We define the mixed ADE singularities, which are generalizations of the ADE plane curve singularities to the case of mixed characteristic. The ADE plane curve singularities are precisely the equicharacteristic plane curve singularities of finite Cohen–Macaulay type; we show that the mixed ADE singularities also have finite Cohen–Macaulay type.

ADE singularities and coset models

We consider the compactification of the IIA string to (1+1) dimensions on non-compact 4-folds that are ALE fibrations. Supersymmetry requires that the compactification include 4-form fluxes, and a particular class of these models has been argued by Gukov, Vafa and Witten to give rise to a set of perturbed superconformal coset models that also have a Landau-Ginzburg description. We examine all these ADE models in detail, including the exceptional cosets. We identify which perturbations are induced by the deformation of the singularity, and compute the Landau-Ginzburg potentials exactly. We also show how the the Landau-Ginzburg fields and their superpotentials arise from the geometric data of the singularity, and we find that this is most naturally described in terms of non-compact, holomorphic 4-cycles.

Orientifold dual for stuck NS5 branes

We establish T-duality between NS5 branes stuck on an orientifold 8-plane in type I' and an orientifold construction in type IIB with D7 branes intersecting at angles. Two applications are discussed. For one we obtain new brane constructions, realizing field theories with gauge group a product of symplectic factors, giving rise to a large new class of conformal N=1 theories embedded in string theory. Second, by studying a D2 brane probe in the type I' background, we get some information on the still elusive (0,4) linear sigma model describing a perturbative heterotic string on an ADE singularity.

Singular Calabi–Yau manifolds and ADE classification of CFTs

We study superstring propagations on the Calabi-Yau manifold which develops an isolated ADE singularity. This theory has been conjectured to have a holographic dual description in terms of N=2 Landau-Ginzburg theory and Liouville theory. If the Landau-Ginzburg description precisely reflects the information of ADE singularity, the Landau-Ginzburg model of $D_4,E_6,E_8$ and Gepner model of $A_2\otimes A_2, A_2\otimes A_3, A_2\otimes A_4$ should give the same result. We compute the elements of $D_4,E_6,E_8$ modular invariants for the singular Calabi-Yau compactification in terms of the spectral flow invariant orbits of the tensor product theories with the theta function which encodes the momentum mode of the Liouville theory. Furthermore we find the interesting identity among characters in minimal models at different levels. We give the complete proof for the identity.