Toric Manifolds Research Articles

Toric origami structures on quasitoric manifolds

We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given, and the statement is reformulated in discrete geometrical terms. The problem reduces to the existence of planar triangulations with certain coloring and metric properties.

Gross-Siebert’s slab functions and open GW invariants for toric Calabi-Yau manifolds

This paper derives an equality between the slab functions in Gross-Siebert program and generating functions of open Gromov-Witten invariants for toric Calabi-Yau manifolds, and thereby confirms a conjecture of Gross-Siebert on symplectic enumerative meaning of slab functions. The proof is based on the open mirror theorem of Chan-Cho-Lau-Tseng [CCLT13]. It shows an instance of correspondence between tropical and symplectic geometry in the open sector.

Erratum to "SYZ mirror symmetry for toric Calabi-Yau manifolds"

Erratum to "SYZ mirror symmetry for toric Calabi-Yau manifolds"

Symplectic Toric Geometry and the Regular Dodecahedron

The regular dodecahedron is the only simple polytope among the platonic solids which is not rational. Therefore, it corresponds neither to a symplectic toric manifold nor to a symplectic toric orbifold. In this paper, we associate to the regular dodecahedron a highly singular space called symplectic toric quasifold.

Killing forms and toric Sasaki-Einstein spaces

The construction of the special Killing forms on toric Sasaki-Einstein manifolds is presented. This goal is achieved using the interplay between complex coordinates of the Calabi-Yau metric cone and the special Killing forms on the toric Sasaki-Einstein space. As a concrete example, we present the complete set of special Killing forms on the five-dimensional Einstein-Sasaki Yp,q spaces. It is pointed out the existence of two additional special Killing forms associated with the complex holomorphic volume form of Calabi-Yau cone manifold.

A class of torus manifolds with nonconvex orbit space

We study a class of smooth torus manifolds whose orbit space has the struc- ture of a simple polytope with holes. We prove that these manifolds have stable almost complex structure and give combinatorial formula for some of their Hirzebruch genera. They have (invariant) almost complex structure if they admit positive omniorientation. In dimension four, we calculate the homology groups, construct symplectic structure on a large class of these manifolds, and give a family which is symplectic but not complex.

Examples of smooth compact toric varieties that are not quasitoric manifolds

We construct toric manifolds of complex dimension $\geq 4$, whose orbit spaces by the action of the compact torus are not homeomorphic to simple polytopes (as manifolds with corners). These provide the first known examples of toric manifolds which are not quasitoric manifolds.

5D Super Yang–Mills on Y p,q Sasaki–Einstein Manifolds

On any simply connected Sasaki-Einstein five dimensional manifold one can construct a super Yang-Mills theory which preserves at least two supersymmetries. We study the special case of toric Sasaki-Einstein manifolds known as $Y^{p,q}$ manifolds. We use the localisation technique to compute the full perturbative part of the partition function. The full equivariant result is expressed in terms of certain special function which appears to be a curious generalisation of the triple sine function. As an application of our general result we study the large $N$ behaviour for the case of single hypermultiplet in adjoint representation and we derive the $N^3$-behaviour in this case.

On a Morelli type expression of cohomology classes of torus orbifolds

Let X be a complete toric variety of dimension n and \del the fan in a lattice N associated to X. For each cone \sigma of \del there corresponds an orbit closure V(\sigma) of the action of complex torus on X. The homology classes {[V(\sigma)]| \dim \sigma=k} form a set of specified generators of H_{n-k}(X,Q). Then any x\in H_{n-k}(X,Q) can be written in the form \[ x=\sum_{\sigma\in\del_X, \dim\sigma=k}\mu(x,\sigma)[V(\sigma)]. \] A question occurs whether there is some canonical way to express \mu(x,\sigma). Morelli gave an answer when X is non-singular and at least for x= \T_{n-k}(X) the Todd class of X. However his answer takes coefficients in the field of rational functions of degree 0 on the Grassmann manifold G_{n-k+1}(N_Q) of (n-k+1)-planes in N_Q. His proof uses Baum-Bott's residue formula for holomorphic foliations applied to the action of complex torus on X. On the other hand there appeared several attempts for generalizing non-singular toric varieties in topological contexts. Such generalized manifolds of dimension 2n acted on by a compact n dimensional torus T are called by the names quasi-toric manifolds, torus manifolds, toric manifolds, toric origami manifolds, topological toric manifolds and so on. Similarly torus orbifold can be considered. To a torus orbifold $X$ a simplicial set \del_X called multi-fan of X is associated. A question occurs whether a similar expression to Morelli's formula holds for torus orbifolds. It will be shown the answer is yes in this case too at least when the rational cohomology ring H^*(X)_Q is generated by H^2(X)_Q. Under this assumption the equivariant cohomology ring with rational coefficients H^*_T(X,Q) is isomorphic to H^*_T(\del_X,Q), the face ring of the multi-fan \del_X, and the proof is carried out on H^*_T(\del_X,Q) by using completely combinatorial terms.

Toric origami manifolds and multi-fans

The notion of a toric origami manifold, which weakens the notion of a symplectic toric manifold, was introduced by Cannas da Silva-Guillemin-Pires \cite{ca-gu-pi11} and they show that toric origami manifolds bijectively correspond to origami templates via moment maps, where an origami template is a collection of Delzant polytopes with some folding data. Like a fan is associated to a Delzant polytope, a multi-fan introduced in \cite{ha-ma03} and \cite{masu99} can be associated to an oriented origami template. In this paper, we discuss their relationship and show that any simply connected compact smooth 4-manifold with a smooth action of $T^2$ can be a toric origami manifold. We also characterize products of even dimensional spheres which can be toric origami manifolds.

On the equality case in Ehrhart’s volume conjecture

Abstract Ehrhart’s conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has the maximal anticanonical degree among all toric Kähler- Einstein Fano manifolds. In this note, we prove that projective space is the only such toric manifold with maximal degree by proving the corresponding convex-geometric statement. We also discuss a generalized version of Ehrhart’s conjecture involving an invariant corresponding to the so-called greatest lower bound on the Ricci curvature.

Holomorphic orbi-discs and Lagrangian Floer cohomology of symplectic toric orbifolds

We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential.We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta, and Ono.

Geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds

This paper is devoted to a systematic study of the geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where these actions are related to a lot of other geometric objects, including reflection groups, singular affine structures, toric and quasi-toric manifolds, monodromy phenomena, topological invariants, etc. We construct a geometric theory of these actions, and obtain a series of results, including: local and semi-local normal forms, automorphism and twisting groups, the reflection principle, the toric degree, the monodromy, complete fans associated to hyperbolic domains, quotient spaces, elbolic actions and toric manifolds, existence and classification theorems.

Curvature of scalar-flat Kähler metrics on non-compact symplectic toric 4-manifolds

In this paper, we show that the complete scalar-flat Kähler metrics constructed in [4] on strictly unbounded toric 4-dimensional orbifolds have finite L2 norm of the full Riemannian tensor. In particular, this answers a question of Donaldson's from [12] on the corresponding Generalized Taub-NUT metric on R4. This norm is explicitly determined when the underlying toric manifold is the minimal resolution of a cyclic singularity of the form C2/Zk. In the Ricci-flat case corresponding to gravitational instantons, this recovers a recent result in [5].

Line bundle twisted chiral de Rham complex, chiral Riemann-Roch formula and D-branes on toric manifolds

I present the results of the elliptic genus calculations in various examples of twisted chiral de Rham complex on one- and two-dimensional toric compact manifolds. The explicit calculations are made for line bundle twisted chiral de Rham complex on ℙ1, ℙ2 and Hirzebruch surface. Based on these results I propose the elliptic genus expression of the bundle twisted chiral de Rham complex for general smooth compact two dimensional toric manifold. The expression resembles Riemann-Roch formula and coincides with the later in certain limit. I interpret the result in terms of infinite tower of open string oscillator contributions and identify directly the open string boundary conditions of the corresponding bound state of D-branes.

New compact forms of the trigonometric Ruijsenaars–Schneider system

The reduction of the quasi-Hamiltonian double of SU(n) that has been shown to underlie Ruijsenaars' compactified trigonometric n-body system is studied in its natural generality. The constraints contain a parameter y, restricted in previous works to 0<y<π/n because Ruijsenaars' original compactification relies on an equivalent condition. It is found that allowing generic 0<y<π/2 results in the appearance of new self-dual compact forms of two qualitatively different types depending on the value of y. The type (i) cases are similar to the standard case in that the reduced phase space comes equipped with globally smooth action and position variables, and turns out to be symplectomorphic to CPn−1 as a Hamiltonian toric manifold. In the type (ii) cases both the position variables and the action variables develop singularities on a nowhere dense subset. A full classification is derived for the parameter y according to the type (i) versus type (ii) dichotomy. The simplest new type (i) systems, for which π/n<y<π/(n−1), are described in some detail as an illustration.

Line bundle twisted chiral de Rham complex and bound states of D-branes on toric manifolds

In this note we calculate elliptic genus in various examples of twisted chiral de Rham complex on two-dimensional toric compact manifolds and Calabi–Yau hypersurfaces in toric manifolds. At first the elliptic genus is calculated for the line bundle twisted chiral de Rham complex on a compact smooth toric manifold and K3 hypersurface in P3. Then we twist chiral de Rham complex by sheaves localized on positive codimension submanifolds in P2 and calculate in each case the elliptic genus. In the last example the elliptic genus of chiral de Rham complex on P2 twisted by SL(N) vector bundle with instanton number k is calculated. In all the cases considered we find the infinite tower of open string oscillator contributions and identify directly the open string boundary conditions of the corresponding bound state of D-branes.

Logarithmic Frobenius manifolds, hypergeometric systems and quantum 𝒟-modules

We describe mirror symmetry for weak Fano toric manifolds as an equivalence of filtered D \mathcal {D} -modules. We discuss in particular the logarithmic degeneration behavior at the large radius limit point and express the mirror correspondence as an isomorphism of Frobenius manifolds with logarithmic poles. The main tool is an identification of the Gauß-Manin system of the mirror Landau-Ginzburg model with a hypergeometric D \mathcal {D} -module, and a detailed study of a natural filtration defined on this differential system. We obtain a solution of the Birkhoff problem for lattices defined by this filtration and show the existence of a primitive form, which yields the construction of Frobenius structures with logarithmic poles associated to the mirror Laurent polynomial. As a final application, we show the existence of a pure polarized non-commutative Hodge structure on a Zariski open subset of the complexified Kähler moduli space of the variety.

Displacing Lagrangian toric fibers by extended probes

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff's original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space P(1,3,5) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

Extremal metrics on toric manifolds&amp;mdash;&amp;mdash;existence and K-stability

We study the existence extremal metrics and K-stability on toric manifolds. We extend Donaldsons notion of stability related to constant scalar metrics on toric manifolds to the case of general extremal metrics. Based on this reduction, we show that the existence of extremal metrics on toric manifolds implies the relative K-stability for toric degenerations. A sufficient condition for K-stability is established whether or not there exists extremal metrics. On toric surfaces, we prove that every toric surfaces admits Kahler classes which contain extremal metrics by Arrezo-Pacard-Singers work. In the meanwhile, many counter-examples of unstable Kahler classes are given. Finally, we discuss the existence of weak solutions in general dimensions under the assumption that the modified K-energy is proper.