Toric Manifolds Research Articles

Mathcal{N}=2 supersymmetric gauge theory on connected sums of S2 \xd7 S2

We construct 4D mathcal{N}=2 theories on an infinite family of 4D toric manifolds with the topology of connected sums of S2× S2. These theories are constructed through the dimensional reduction along a non-trivial U(1)-fiber of 5D theories on toric Sasaki-Einstein manifolds. We discuss the conditions under which such reductions can be carried out and give a partial classification result of the resulting 4D manifolds. We calculate the partition functions of these 4D theories and they involve both instanton and anti-instanton contributions, thus generalizing Pestun’s famous result on S4.

Shift operators and toric mirror theorem

We give a new proof of Givental's mirror theorem for toric manifolds using shift operators of equivariant parameters. The proof is almost tautological: it gives an A-model construction of the I-function and the mirror map. It also works for non-compact or non-semipositive toric manifolds.

On toric locally conformally Kähler manifolds

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $$-\infty $$ , and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.

A note on torus actions and the Witten genus

We show that the Witten genus of a string manifold $M$ vanishes, if there is an effective action of a torus $T$ on $M$ such that $\dim T>b_2(M)$. We apply this result to study group actions on $M\times G/T$, where $G$ is a compact connected Lie group and $T$ a maximal torus of $G$. Moreover, we use the methods which are needed to prove these results to the study of torus manifolds. We show that up to diffeomorphism there are only finitely many quasitoric manifolds $M$ with the same cohomology ring as $#_{i=1}^k \pm\mathbb{C} P^n$ with $k<n$.

Cohomology of toric origami manifolds with acyclic proper faces

Toric origami manifolds are generalizations of symplectic toric manifolds, where the origami symplectic form, in contrast to the usual sym- plectic form, is allowed to degenerate in a good controllable way. It is widely known that symplectic toric manifolds are encoded by Delzant polytopes. The cohomology and equivariant cohomology rings of a symplectic toric manifold can be described in terms of the corresponding polytope. One can obtain a similar description for the cohomology of a toric origami manifold M in terms of the orbit space M=T when M is orientable and the orbit space M=T is con- tractible; this was done by Holm and Pires in (9). Generally, the orbit space of a toric origami manifold need not be contractible. In this paper we study the topology of orientable toric origami manifolds for the wider class of examples: we require that any proper face of the orbit space is acyclic, while the orbit space itself may be arbitrary. Furthermore, we give a general description of the equivariant cohomology ring of torus manifolds with locally standard torus action in the case when proper faces of the orbit space are acyclic and the free part of the action is a trivial torus bundle.

Contact blow up and cylindrical contact homology of toric contact manifolds of Reeb type

Let (V,?) be a toric contact manifold of Reeb type that is a prequantization of a toric symplectic manifold (M,?). A contact blow up of (V,?) is the prequantization of a symplectic blow up of (M,?). Thus, a contact blow up of (V,?) is a new toric contact manifold of Reeb type. In some special cases we are able to compute the cylindrical contact homology for the contact blowup using only the cylindrical contact homology of the contact manifold we started with.

Toric constructions of monotone Lagrangian submanifolds in $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$

We extract from a toric model of the Chekanov-Schlenk exotic torus in $\mathbb{CP}^2$ methods of construction of Lagrangian submanifolds in toric symplectic manifolds. These constructions allow for some control of the monotonicity. We recover this way some known monotone Lagrangians in the toric symplectic manifolds $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$ as well as new examples.

Non-affine Landau-Ginzburg models and intersection cohomology

We construct Landau-Ginzburg models for numerically effective complete intersections in toric manifolds as partial compactifications of families of Laurent polynomials. We show a mirror statement saying that the quantum D-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology D-module defined from this partial compactification and we deduce Hodge properties of these differential systems.

Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds

We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus in contrast to the case of parallel spinors the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.

Intersection Spaces, Equivariant Moore Approximation and the Signature

We generalize the first author's construction of intersection spaces to the case of stratified pseudomanifolds of stratification depth 1 with twisted link bundles, assuming that each link possesses an equivariant Moore approximation for a suitable choice of structure group. As a by-product, we find new characteristic classes for fiber bundles admitting such approximations. For trivial bundles and flat bundles whose base has finite fundamental group these classes vanish. For oriented closed pseudomanifolds, we prove that the reduced rational cohomology of the intersection spaces satisfies global Poincare duality across complementary perversities if the characteristic classes vanish. The signature of the intersection spaces agrees with the Novikov signature of the top stratum. As an application, these methods yield new results about the Goresky-MacPherson intersection homology signature of pseudomanifolds. We discuss several nontrivial examples, such as the case of flat bundles and symplectic toric manifolds.

On displaceability of pre-Lagrangian toric fibers in contact toric manifolds

In this paper, we analyze displaceability of pre-Lagrangian toric fibers in contact toric manifolds. While every symplectic toric manifold contains at least one non-displaceable Lagrangian toric fiber and infinitely many displaceable ones, we show that this is not the case for contact toric manifolds. More precisely, we prove that for the contact toric manifolds [Formula: see text] and [Formula: see text] all pre-Lagrangian toric fibers are displaceable, and that for all contact toric manifolds for which the toric action is free, except possibly non-trivial [Formula: see text]-bundles over [Formula: see text], all pre-Lagrangian toric fibers are non-displaceable. Moreover, we also prove that if for a compact connected contact toric manifold all but finitely many pre-Lagrangian toric fibers are non-displaceable then the action is necessarily free. On the other hand, as we will discuss, displaceability of all pre-Lagrangian toric fibers seems to be related to the non-orderability of the underlying contact manifolds.

Toric 2-Fano manifolds and extremal contractions

We show that for a projective toric manifold with the ample second Chern character, if there exists a Fano contraction, then it is isomorphic to the projective space. For the case that the second Chern character is nef, the Fano contraction gives either a projective line bundle structure or a direct product structure. We also show that, for a toric weakly 2-Fano manifold, there does not exist a divisorial contraction to a point.

On toric generators in the unitary and special unitary bordism rings

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions >8. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.

Global embedding of fibre inflation models

We present concrete embeddings of fibre inflation models in globally consistent type IIB Calabi-Yau orientifolds with closed string moduli stabilisation. After performing a systematic search through the existing list of toric Calabi-Yau manifolds, we find several examples that reproduce the minimal setup to embed fibre inflation models. This involves Calabi-Yau manifolds with h 1,1 = 3 which are K3 fibrations over a ℙ1 base with an additional shrinkable rigid divisor. We then provide different consistent choices of the underlying brane set-up which generate a non-perturbative superpotential suitable for moduli stabilisation and string loop corrections with the correct form to drive inflation. For each Calabi-Yau orientifold setting, we also compute the effect of higher derivative contributions and study their influence on the inflationary dynamics.

Circle actions, quantum cohomology, and the Fukaya category of Fano toric varieties

We define a class of non-compact Fano toric manifolds, called admissible toric manifolds, for which Floer theory and quantum cohomology are defined. The class includes Fano toric negative line bundles, and it allows blow-ups along fixed point sets. We prove closed-string mirror symmetry for this class of manifolds: the Jacobian ring of the superpotential is the symplectic cohomology (not the quantum cohomology). Moreover, SH(M) is obtained from QH(M) by localizing at the toric divisors. We give explicit presentations of SH(M) and QH(M), using ideas of Batyrev, McDuff and Tolman. Assuming that the superpotential is Morse (or a milder semisimplicity assumption), we prove that the wrapped Fukaya category for this class of manifolds satisfies the toric generation criterion, i.e. is split-generated by the natural Lagrangian torus fibres of the moment map with suitable holonomies. In particular, the wrapped category is compactly generated and cohomologically finite. The proof uses a deformation argument, via a generic generation theorem and an argument about continuity of eigenspaces. We also prove that for any closed Fano toric manifold, if the superpotential is Morse (or a milder semisimplicity assumption) then the Fukaya category satisfies the toric generation criterion. The key ingredients are non-vanishing results for the open-closed string map, using tools from the paper by Ritter-Smith (we also prove a conjecture from that paper that any monotone toric negative line bundle contains a non-displaceable monotone Lagrangian torus). We also need to extend the class of Hamiltonians for which the maximum principle holds for symplectic manifolds conical at infinity, thus extending the class of Hamiltonian circle actions for which invertible elements can be constructed in SH(M).

On K-stability and the volume functions of ℚ-Fano varieties: Table 1.

We introduce a new effective stability named "divisorial stability" for Fano manifolds which is weaker than K-stability and is stronger than slope stability along divisors. We show that we can test divisorial stability via the volume function. As a corollary, we prove that the first coordinate of the barycenter of the Okounkov body of the anticanonical divisor is not bigger than one for any K\"ahler-Einstein Fano manifold. In particular, for toric Fano manifolds, the existence of K\"ahler-Einstein metrics is equivalent to divisorial semistability. Moreover, we find many non-K\"ahler-Einstein Fano manifolds of dimension three.

On the cohomology and their torsion of real toric objects

Abstract In this paper, we do the following two things: (i) We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for an arbitrary coefficient ring G in which 2 is a unit. (ii) We construct infinitely many real toric manifolds and small covers whose integral cohomology rings have a q-torsion for any positive odd integer q.

Non-perturbative quantum geometry III

The Nekrasov-Shatashvili limit of the refined topological string on toric Calabi-Yau manifolds and the resulting quantum geometry is studied from a non-perturbative perspective. The quantum differential and thus the quantum periods exhibit Stokes phenomena over the combined string coupling and quantized Kaehler moduli space. We outline that the underlying formalism of exact quantization is generally applicable to points in moduli space featuring massless hypermultiplets, leading to non-perturbative band splitting. Our prime example is local P1xP1 near a conifold point in moduli space. In particular, we will present numerical evidence that in a Stokes chamber of interest the string based quantum geometry reproduces the non-perturbative corrections for the Nekrasov-Shatashvili limit of 4d supersymmetric SU(2) gauge theory at strong coupling found in the previous part of this series. A preliminary discussion of local P2 near the conifold point in moduli space is also provided.

Volume Polynomials and Duality Algebras of Multi-Fans

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(\Delta)$. We study the structure and properties of $V_\Delta$ and $\mathcal{A}^*(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik--Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.

On the density function on moduli spaces of toric 4-manifolds

AbstractThe optimal density function assigns to each symplectic toric manifoldMa positive numberd ≤1 obtained by considering the ratio between the maximum volume ofMwhich can be filled by symplectically embedded disjoint balls and the total symplectic volume ofM. In the toric version,Mis toric and the balls need to be embedded respecting the toric action onM. We give a brief survey of toric symplectic manifolds and the recent constructions of moduli space structure on them, and then we recall how to define a natural density function on this moduli space. We review previous work which explains how the study of the density function can be reduced to a problem in convex geometry, and use this correspondence to to give a simple description of the regions of continuity of the maximal density function when the dimension is 4.