Topological Toric Manifolds

In this article, we provide an exposition about symplectic toric manifolds, which are symplectic manifolds (M2n, ω) equipped with an effective Hamiltonian \(\mathbb {T}^n\cong (S^1)^n\)-action. We summarize the construction of M as a symplectic quotient of \(\mathbb {C}^d\), the \(\mathbb {T}^n\)-actions on M and their moment maps, and Guillemin’s Kähler potential on M. While the theories presented in this paper are for compact toric manifolds, they do carry over for some noncompact examples as well, such as the canonical line bundle KM, which is one of our main running examples, along with the complex projective space \(\mathbb {P}^n\) and its canonical bundle \(K_{\mathbb {P}^n}\). One main topic explored in this article is how to write the moment map in terms of the complex homogeneous coordinates \(z\in \mathbb {C}^d\), or equivalently, the relationship between the action-angle coordinates and the complex toric coordinates. We end with a brief review of homological mirror symmetry for toric geometries, where the main connection with the rest of the paper is that KM provides a prototypical class of examples of a Calabi-Yau toric manifold Y which serves as the total space of a symplectic fibration \(W: Y \to \mathbb {C}\) with a singular fiber above 0, known as a Landau-Ginzburg model in mirror symmetry. Here we write W in terms of the action-angle coordinates, which will prove to be useful in understanding the geometry of the fibration in our forthcoming work (Azam et al., Global homological mirror symmetry for genus two curves, in preparation).

In this paper, we study the uniqueness of the direct decomposition of a toric manifold. We first observe that the direct decomposition of a toric manifold as algebraic varieties is unique up to order of the factors. An algebraically indecomposable toric manifold happens to decompose as smooth manifold and no criterion is known for two toric manifolds to be diffeomorphic, so the unique decomposition problem for toric manifolds as smooth manifolds is highly nontrivial and nothing seems known for the problem so far. We prove that this problem is affirmative if the complex dimension of each factor in the decomposition is less than or equal to two. A similar argument shows that the direct decomposition of a smooth manifold into copies of $\mathbb{C}P^{1}$ and simply connected closed smooth 4-manifolds with smooth actions of $(S^{1})^{2}$ is unique up to order of the factors.

We provide a general set of rules for extracting the data defining a quiver gauge theory from a given toric Calabi-Yau singularity. Our method combines information from the geometry and topology of Sasaki-Einstein manifolds, AdS/CFT, dimers, and brane tilings. We explain how the field content, quantum numbers, and superpotential of a superconformal gauge theory on D3-branes probing a toric Calabi-Yau singularity can be deduced. The infinite family of toric singularities with known horizon Sasaki-Einstein manifolds La,b,c is used to illustrate these ideas. We construct the corresponding quiver gauge theories, which may be fully specified by giving a tiling of the plane by hexagons with certain gluing rules. As checks of this construction, we perform a-maximisation as well as Z-minimisation to compute the exact R-charges of an arbitrary such quiver. We also examine a number of examples in detail, including the infinite subfamily La,b,a, whose smallest member is the Suspended Pinch Point.

We describe a technique which enables one to quickly compute an infinite number of toric geometries and their dual quiver gauge theories. The central object in this construction is a ``brane tiling,'' which is a collection of D5-branes ending on an NS5-brane wrapping a holomorphic curve that can be represented as a periodic tiling of the plane. This construction solves the longstanding problem of computing superpotentials for D-branes probing a singular non-compact toric Calabi-Yau manifold, and overcomes many difficulties which were encountered in previous work. The brane tilings give the largest class of N=1 quiver gauge theories yet studied. A central feature of this work is the relation of these tilings to dimer constructions previously studied in a variety of contexts. We do many examples of computations with dimers, which give new results as well as confirm previous computations. Using our methods we explicitly derive the moduli space of the entire Y^{p,q} family of quiver theories, verifying that they correspond to the appropriate geometries. Our results may be interpreted as a generalization of the McKay correspondence to non-compact 3-dimensional toric Calabi-Yau manifolds.

The idea that one could prove by analytic methods the topological invariance of characteristic classes was suggested in an influential lecture by Singer. The concrete realization of this, in the case of rational Pontrjagin classes, was the result of works by Sullivan, who showed that topological manifolds have essentially unique Lipschitz structures, and Teleman, who showed that Lipschitz manifolds are the type of object for which one can analyze the signature operator. (A refinement of the same method proves the KO[l/2] orientability of topological bundles.) Much of this can be done for manifolds with group actions. The main implication is that for certain sorts of topological group actions (including all smooth and PL actions) one can construct an signature operator whose class in equivariant if-homology is a topological invariant. This class satisfies the G-signature theorem. The topological method has implications for various precise classification problems (e.g. semifree topological actions on the sphere), but here we shall present only the consequences of these Lipschitzian ideas: (A) Topologically locally linear G-manifolds for odd-order G have KOG[1/2] orientations (Madsen-Rothenberg). For all finite G there are Lipschitz structures, but the signature class is a zero divisor for even-order groups. The same method produces, upon application of the localization theorem, Atiyah-Singer classes for some actions with nonmanifold fixed set (Weinberger, extending earlier joint work with Cappell). (B) For odd-order groups, topological and linear conjugacy of linear representations are equivalent (Hsiang-Pardon, Madsen-Rothenberg). This is a consequence of (A). Alternatively one can return to the original argument of Atiyah-Bott-Milnor for semifree representations. A stronger result is true: the Atiyah-Bott local numbers associated to representations agree for elements that have discrete fixed sets in topologically equivalent representations. This distinguishes many pairs of representations of even-order groups. (C) R Top(G) = R Lip(G) for all finite G, where R Cat is the Grothendieck group of representations up to Cat conjugacy (Weinberger, after inverting 2). This implies that there is no analytic definition of Reidemeister torsion that works well in the Lipschitz setting. The reason is that, following

Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus Tk. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus Φ, which was introduced independently by Krichever and Löffler in 1974, albeit from radically different viewpoints. In fact, Φ is a version of tom Dieck’s bundling transformation of 1970, defined on Tk-equivariant complex cobordism classes and taking values in the complex cobordism algebra of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera and refer to the index-theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework and to introduce parametrized versions that apply to bundles equipped with a stably complex structure on the tangents along their fibers. In the presence of isolated fixed points, we obtain universal localization formulae, whose applications include the identification of Krichever’s generalized elliptic genus as universal among the genera that are rigid on SU-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites, we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s.

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Algebraic topology of manifolds defined by simple polytopes V M Bukhshtaber and T E Panov Torus actions, combinatorial topology, and homological algebra V M Buchstaber and T E Panov Hirzebruch genera of manifolds with torus action T E Panov The index of an equivariant vector field and addition theorems for Pontryagin classes V M Bukhshtaber and K E Feldman Embedding of compacta, stable homotopy groups of spheres, and singularity theory P M Akhmet'ev Complex cobordism and formal groups Viktor M Buchstaber Massey Products and the Bukhshtaber spectral sequence I V Artel'nykh Toric manifolds and complex cobordisms

In view of various results and the nature of their constructions for real toric objects, it is an interesting and also natural problem to know how much torsion can be contained in their cohomology groups. The aim of this paper is to answer this question by explicitly constructing examples of real toric objects to show the richness of torsion which can appear in the cohomology groups with coefficients in a locally constant presheaf. That is, we show that a real quasitoric manifold (or small cover) which plays an important role in the category of real toric objects can have an arbitrary amount of torsion in its cohomology groups with coefficients in a locally constant presheaf. This will be achieved by crucially using the Torsion Theorem for links of Bosio and Meersseman.

The classification problem for quasitoric manifolds over a given polytope Pn is considered. The known construction of weights, which was used in the study of the similar problem for toric manifolds, is modified. The construction thus obtained is applied to the solution of the above problem in small dimensions. Classification results are also obtained for polytopes that are products of finitely many simplices of arbitrary dimensions.

This paper establishes a connection between equivariant bordisms and universal complexes, providing a new approach to the study of equivariant bordisms. We show that Z n ( Z 2 n ) \mathcal {Z}_n(\mathbb {Z}_2^n) and Z 2 n U ( T n ) \mathcal {Z}_{2n}^U(T^n) are isomorphic to the reduced homology groups H ~ n − 1 ( X ( Z 2 n ) ; Z 2 ) \widetilde {H}_{n-1}(X(\mathbb {Z}_2^n);\mathbb {Z}_2) and H ~ n − 1 ( X ( Z n ) ; Z ) \widetilde {H}_{n-1}(X(\mathbb {Z}^n);\mathbb {Z}) , respectively. Here, Z n ( Z 2 n ) \mathcal {Z}_n(\mathbb {Z}_2^n) (resp. Z 2 n U ( T n ) \mathcal {Z}_{2n}^U(T^n) ) denotes the group formed by the equivariant unoriented bordism classes of all n n -dimensional 2-torus manifolds (resp. the equivariant unitary bordism classes of all 2 n 2n -dimensional unitary torus manifolds), and X ( Z 2 n ) X(\mathbb {Z}_2^n) (resp. X ( Z n ) X(\mathbb {Z}^n) ) is the universal complex determined by Z 2 n \mathbb {Z}_2^n (resp. Z n \mathbb {Z}^n ). Furthermore, by using known theorems on the homotopy type of X ( Z 2 n ) X(\mathbb {Z}_2^n) , we determine the dimension of Z n ( Z 2 n ) \mathcal {Z}_n(\mathbb {Z}_2^n) as a Z 2 \mathbb {Z}_2 -vector space. As an additional application of this connection, utilizing matroid theory, we show that the geometric generators of the equivariant bordism groups Z n ( Z 2 n ) \mathcal {Z}_n(\mathbb {Z}_2^n) and Z 2 n U ( T n ) \mathcal {Z}_{2n}^U(T^n) can be chosen from small covers and quasitoric manifolds of Davis–Januszkiewicz theory, respectively.

In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov–Hausdorff distance.

Toric varieties play an important role both in symplectic and complex geometry. In symplectic geometry, the construction of a symplectic toric manifold from a smooth polytope is due to Delzant. In algebraic geometry, there is a more general construction using fans rather than polytopes. However, in case the fan is induced by a smooth polytope Audin showed both constructions to give isomorphic projective varieties. For rational but not necessarily smooth polytopes the Delzant construction was refined by Lerman and Tolman, leading to symplectic toric orbifolds or more generally, symplectic toric DM stacks (Lerman and Malkin). We show that the stacks resulting from the Lerman-Tolman construction are isomorphic to the stacks obtained by Borisov et al. in case the stacky fan is induced by a polytope. No originality is claimed (cf. also an article by Sakai). Rather we hope that this text serves as an example driven introduction to symplectic toric geometry for the algebraically minded reader.

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.

Dark solitons are some of the most interesting nonlinear excitations and are considered to be the one-dimensional topological analogs of vortices. However, in contrast to their two-dimensional vortex counterparts, the topological characteristics of a dark soliton are far from fully understood because the topological charge defined according to the phase jump cannot reflect its essential property. Here, similar to the complex extension used in the exploration of the partition-function zeros to depict thermodynamic states, we extend the complex coordinate space to explore the density zeros of dark solitons. Surprisingly we find that these zeros constitute some pointlike magnetic fields, each of which has a quantized magnetic flux of elementary π. The corresponding vector potential fields demonstrate the topology of the Wess-Zumino term and can depict the essential characteristics of dark solitons. Then we classify the dark solitons according to the Euler characteristic of the topological manifold of the vector potential fields. Our study not only reveals the topological features of dark solitons but can also be applied to explore and identify new dark solitons with high topological complexity.