Yau 3-folds Research Articles

Counting Invariant of Perverse Coherent Sheaves and its Wall-crossing

We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi–Yau 3-folds and consider their Donaldson–Thomas-type counting invariants. The stability depends on the choice of a component ( = a chamber) in the complement of finitely many lines ( = walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of chambers when the Calabi–Yau is the resolved conifold. For suitable choices of chambers, our invariants are specialized to Donaldson–Thomas, Pandharipande–Thomas and Szendroi invariants.

Geometry of minimal energy Yang-Mills connections

We prove that energy minimizing Yang–Mills connections on compact homogeneous 4-manifolds are either instantons or split into a sum of instantons on passage to the adjoint bundle. We prove related results for Calabi–Yau 3-folds and for 3-dimensional manifolds of nonnegative Ricci curvature.

Donaldson–Thomas invariants of certain Calabi–Yau 3-folds

We compute the Donaldson-Thomas invariants for two types of Calabi-Yau 3-folds. These invariants are associated to the moduli spaces of rank-2 Gieseker semistable sheaves. None of the sheaves are locally free, and their double duals are locally free stable sheaves investigated earlier in (DT, Tho, LQ2). We show that these Gieseker moduli spaces are isomorphic to some Quot-schemes. We prove a formula for Behrend's �-functions when torus ac- tions present with positive dimensional fixed point sets, and use it to obtain the generating series of the relevant Donaldson-Thomas invariants in terms of the McMahon function. Our results might shed some light on the wall-crossing phenomena of Donaldson-Thomas invariants.

Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions

We construct a surprisingly large class of new Calabi–Yau 3-folds $X$ with small Picard numbers and propose a construction of their mirrors $X^∗$ using smoothings of toric hypersurfaces with conifold singularities. These new examples are related to the previously known ones via conifold transitions. Our results generalize the mirror construction for Calabi– Yau complete intersections in Grassmannians and flag manifolds via toric degenerations. There exist exactly 198849 reflexive four-polytopes whose two-faces are only triangles or parallelograms of minimal volume. Every such polytope gives rise to a family of Calabi–Yau hypersurfaces with at worst conifold singularities. Using a criterion of Namikawa we found 30241 reflexive four-polytopes such that the corresponding Calabi–Yau hypersurfaces are smoothable by a flat deformation. In particular, we found 210 reflexive four-polytopes defining 68 topologically different Calabi–Yau 3-folds with $h_11 = 1$. We explain the mirror construction and compute several new Picard–Fuchs operators for the respective oneparameter families of mirror Calabi–Yau 3-folds.

Examples of Calabi–Yau 3-Folds of ℙ7 with ρ = 1

Abstract. We give some examples of Calabi–Yau 3-folds with $\rho =1$ and $\rho =2$, defined over $\mathbb{Q}$ and constructed as 4-codimensional subvarieties of ${{\mathbb{P}}^{7}}$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top Chern classes from computer based computations over some finite field ${{\mathbb{F}}_{p}}$. Three of our examples (of degree 17 and 20) are new. The two others (degree 15 and 18) are known, and we recover their well-known invariants with ourmethod. These examples are built out of Gulliksen–Neg°ard and Kustin–Miller complexes of locally free sheaves.Finally, we give two new examples of Calabi–Yau 3-folds of ${{\mathbb{P}}^{6}}$ of degree 14 and 15 (defined over $\mathbb{Q}$). We show that they are not deformation equivalent to Tonoli's examples of the same degree, despite the fact that they have the same invariants $({{H}^{3}},{{c}_{2}}.H,{{c}_{3}})$ and $\rho =1$.

On the Crepant Resolution Conjecture in the Local Case

In this paper we analyze four examples of birational transformations between local Calabi–Yau 3-folds: two crepant resolutions, a crepant partial resolution, and a flop. We study the effect of these transformations on genus-zero Gromov–Witten invariants, proving the Coates–Iritani–Tseng/Ruan form of the Crepant Resolution Conjecture in each case. Our results suggest that this form of the Crepant Resolution Conjecture may also hold for more general crepant birational transformations. They also suggest that Ruan’s original Crepant Resolution Conjecture should be modified, by including appropriate “quantum corrections”, and that there is no straightforward generalization of either Ruan’s original Conjecture or the Cohomological Crepant Resolution Conjecture to the case of crepant partial resolutions. Our methods are based on mirror symmetry for toric orbifolds.

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities. II. Examples

This article is a sequel to Chan (Ann Glob Anal Geom, to appear) on simultaneous desingularizations of Calabi–Yau and special Lagrangian (SL) 3-folds with conical singularities. In Chan (Ann Glob Anal Geom, to appear) we treated the question of starting with a conically singular Calabi–Yau 3-fold and an SL 3-fold with conical singularities at the same points and deforming both together to get a smooth situation. In this article, we survey the major result from Chan (Ann Glob Anal Geom, to appear) and describe some examples from our earlier articles (Chan Q J Math 57:151–181, 2006, Q J Math, to appear) on Calabi–Yau desingularizations. We then provide many explicit examples of Asymptotically Conical (AC) SL submanifolds in two specific AC Calabi–Yau manifolds. Using the result in Chan (Ann Glob Anal Geom, to appear), we construct smooth examples of compact SL 3-folds in compact Calabi–Yau 3-folds by gluing those AC SL 3-folds into some conically singular SL 3-folds at the singular points.

Simultaneous desingularizations of Calabi–Yau and special Lagrangian 3-folds with conical singularities: I. The gluing construction

In this paper we extend our previous results on resolving conically singular Calabi–Yau 3-folds (Chan, Quart. J. Math. 57:151–181, 2006; Quart. J. Math., to appear) to include the desingularizations of special Lagrangian (SL) 3-folds with conical singularities that occur at the same points of the ambient Calabi–Yau. The gluing construction of the SL 3-folds is achieved by applying Joyce’s analytic result (Joyce, Ann. Global. Anal. Geom. 26: 1–58, 2004, Thm. 5.3) on deforming Lagrangian submanifolds to nearby special Lagrangian submanifolds. Our result will in principle be able to construct more examples of compact SL submanifolds in compact Calabi–Yau manifolds. Various explicit examples and applications illustrating the result in this paper can be found in the sequel (Chan, Ann. Global. Anal. Geom., to appear).

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

This is the second of two papers studying Calabi–Yau 3-folds with conical singularities and their desingularizations. In our first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181] we constructed the desingularization of the conically singular manifold M0 by gluing an asymptotically conical (AC) Calabi–Yau 3-fold Y into M0 at the singular point, thus obtaining a 1-parameter family of compact, non-singular Calabi–Yau 3-folds Mt for small t > 0. During the gluing process one may encounter a kind of cohomological obstruction to defining a 3-form Ωt on Mt which interpolates between the 3-form Ω0 on M0 and the scaled 3-form t3 ΩY on Y if the rate λ at which the AC Calabi–Yau 3-fold Y converges to the Calabi–Yau cone is equal to − 3. The first paper [3] studied the simpler case λ < −3 where there is no obstruction. This paper extends the result in the first one by considering a more complicated situtation when λ = −3. Assuming the existence of singular Calabi–Yau metrics on compact complex 3-folds with ordinary double points, our result in this paper can be applied to repairing such kinds of singularities, which is an analytic version of Friedman's result giving necessary and sufficient conditions for smoothing ordinary double points.

Calabi–Yau coverings over some singular varieties and new Calabi–Yau 3-folds with Picard number one

This note is a report on the observation that some singular varieties admit Calabi–Yau coverings. As an application, we construct 18 new Calabi–Yau 3-folds with Picard number one that have some interesting properties.

Calabi–Yau 3-folds from 2-folds

We consider type IIA string theory on a Calabi–Yau 2-fold with D6-branes wrapping 2-cycles in the 2-fold. We find a complete set of conditions on the supergravity solution for any given wrapped brane configuration in terms of SU(2) structures. We reduce the problem of finding a supergravity solution for the wrapped branes to finding a harmonic function on R × CY2. We then lift this solution to eleven dimensions as a product of R(4,1) and a Calabi–Yau 3-fold. We show how the metric on the 3-fold is determined in terms of the wrapped brane solution. We write down the distinguished (3, 0) form and the Kähler form of the 3-fold in terms of structures defined on the base 2D complex manifold. We discuss the topology of the 3-fold in terms of the D6-branes and the underlying 2-fold. We show that in addition to the non-trivial cycles inherited from the underlying 2-fold there are N − 1 new 2-cycles. We construct closed (1, 1) forms corresponding to these new cycles. We also display some explicit examples. One of our examples is that of D6-branes wrapping the 2-cycle in an A1 ALE space, the resulting 3-fold has h(1,1) = N, where N is the number of D6-branes.

Supergravity description of spacetime instantons

We present and discuss BPS instanton solutions that appear in type II string theory compactifications on Calabi–Yau 3-folds. From an effective action point of view, these arise as finite action solutions of the Euclidean equations of motion in four-dimensional N = 2 supergravity coupled to tensor multiplets. As a solution generating technique we make use of the c-map, which produces instanton solutions from either Euclidean black holes or from Taub–NUT like geometries.

Gromov–Witten theory and Donaldson–Thomas theory, I

We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the change of variables is the Euler characteristic parameter of Donaldson–Thomas theory. The conjecture is proven for local Calabi–Yau toric surfaces.

String loop corrections to the universal hypermultiplet

We study loop corrections to the universal dilaton supermultiplet for type IIA strings compactified on Calabi–Yau 3-folds. We show that the corresponding quaternionic kinetic terms receive non-trivial one-loop contributions proportional to the Euler number of the Calabi–Yau manifold, while the higher-loop corrections can be absorbed by field redefinitions. The corrected metric is no longer Kähler. Our analysis implies in particular that the Calabi–Yau volume is renormalized by loop effects which are present even in higher orders, while there are also one-loop corrections to the Bianchi identities for the NS and RR field strengths.

Singularities of special Lagrangian fibrations and the SYZ Conjecture

The SYZ Conjecture explains Mirror Symmetry between Calabi–Yau 3-folds M, ˆ M in terms of special Lagrangian fibrations f : M ! B and ˆ f : ˆ M ! B over the same base B, whose fibres are dual 3-tori, except for singular fibres. This paper studies the singularities of special Lagrangian fibrations. We construct many examples of special Lagrangian fibrations on open subsets of C 3 . The simplest are given explicitly, and the rest use analytic existence results for U(1)-invariant special Lagrangian 3-folds in C 3 . We then argue that some features of our examples should also hold for generic special Lagrangian fibrations of (almost) Calabi–Yau 3-folds, and draw some conclusions on the SYZ Conjecture.

Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in Cm. The previous paper (Math. Ann.320 (2001), 757–797), defined the idea of evolution data, which includes an (m − 1)-submanifold P in Rn, and constructed a family of special Lagrangian m-folds N in Cm, which are swept out by the image of P under a 1-parameter family of affine maps φt: Rn→ Cm, satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C3. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C3.Our main results are a number of new families of special Lagrangian 3-foldsin C3, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R3 or Open image in new window1× R2. Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.

RR flux on Calabi–Yau and partial supersymmetry breaking

We show how turning on Flux for RR (and NS-NS) field strengths on non-compact Calabi–Yau 3-folds can serve as a way to partially break supersymmetry from N=2 to N=1 by mass deformation. The freezing of the moduli of Calabi–Yau in the presence of the flux is the familiar phenomenon of freezing of fields in supersymmetric theories upon mass deformations.

On supermembrane actions on Calabi–Yau 3-folds

In this note we examine the supermembrane action on Calabi–Yau 3-folds. We write down the Dirac–Born–Infeld part of the action, and show that it is invariant under the rigid spacetime supersymmetry.

Sigma models for bundles on Calabi–Yau: A proposal for Matrix string compactifications

We describe a class of supersymmetric gauged linear sigma-model, whose target space is the infinite-dimensional space of bundles on a Calabi–Yau 3- or 2-fold. This target space can be considered the configuration space of D-branes wrapped around the Calabi–Yau. We propose that this model can be used to define matrix string theory compactifications. In the infrared limit the model flows to a superconformal non-linear sigma-model whose target space is the moduli space of BPS configurations of branes on the compact space, containing the moduli space of semi-stable bundles. We argue that the bulk degrees of freedom decouple in the infrared limit if semi-stability implies stability. We study topological versions of the model on Calabi–Yau 3-folds. The resulting B-model is argued to be equivalent to the holomorphic Chern–Simons theory proposed by Witten. The A-model and half-twisted model define the quantum cohomology ring and the elliptic genus, respectively, of the moduli space of stable bundles on a Calabi–Yau 3-fold.

Supersymmetry and gauge theory on Calabi–Yau 3-folds

We consider the dimensional reduction of supersymmetric Yang-Mills on a Calabi-Yau 3-fold. We show by construction how the resulting cohomological theory is related to the balanced field theory of the Kaehler Yang-Mills equations introduced by Donaldson and Uhlenbeck-Yau.