Toric Divisors Research Articles

On the intermediate Jacobian of M5-branes

We study Euclidean M5-branes wrapping vertical divisors in elliptic Calabi-Yau fourfold compactifications of M/F-theory that admit a Sen limit. We construct these Calabi-Yau fourfolds as elliptic fibrations over coordinate flip O3/O7 orientifolds of toric hypersurface Calabi-Yau threefolds. We devise a method to analyze the Hodge structure (and hence the dimension of the intermediate Jacobian) of vertical divisors in these fourfolds, using only the data available from a type IIB compactification on the O3/O7 Calabi-Yau orientifold. Our method utilizes simple combinatorial formulae (that we prove) for the equivariant Hodge numbers of the Calabi-Yau orientifolds and their prime toric divisors, along with a formula for the Euler characteristic of vertical divisors in the corresponding elliptic Calabi-Yau fourfold. Our formula for the Euler characteristic includes a conjectured correction term that accounts for the contributions of pointlike terminal ℤ2 singularities corresponding to perturbative O3-planes. We check our conjecture in a number of explicit examples and find perfect agreement with the results of direct computations.

Examples of Weinstein Domains in the Complement of Smoothed Total Toric Divisors

ABSTRACT In [ACG+1] and [ACG+2], it is shown that a complement of a neighborhood of a partially smoothed total toric divisor of a closed toric symplectic 4-manifold is a Weinstein domain. In this article, we extend the family of Weinstein domains that can be realized as such complements by producing an infinite family of new examples.

Open/closed correspondence via relative/local correspondence

We establish a correspondence between the disk invariants of a smooth toric Calabi-Yau 3-fold X with boundary condition specified by a framed Aganagic-Vafa outer brane (L,f) and the genus-zero closed Gromov-Witten invariants of a smooth toric Calabi-Yau 4-fold X˜, proving the open/closed correspondence proposed by Mayr and developed by Lerche-Mayr. Our correspondence is the composition of two intermediate steps:•First, a correspondence between the disk invariants of (X,L,f) and the genus-zero maximally-tangent relative Gromov-Witten invariants of a relative Calabi-Yau 3-fold (Y,D), where Y is a toric partial compactification of X by adding a smooth toric divisor D. This correspondence can be obtained as a consequence of the topological vertex (Li-Liu-Liu-Zhou) and Fang-Liu where the all-genus open Gromov-Witten invariants of (X,L,f) are identified with the formal relative Gromov-Witten invariants of the formal completion of (Y,D) along the toric 1-skeleton. Here, we present a proof without resorting to formal geometry.•Second, a correspondence in genus zero between the maximally-tangent relative Gromov-Witten invariants of (Y,D) and the closed Gromov-Witten invariants of the toric Calabi-Yau 4-fold X˜=OY(−D). This can be viewed as an instantiation of the log-local principle of van Garrel-Graber-Ruddat in the non-compact setting.

An imaginary refined count for some real rational curves

In 2015, Mikhalkin introduced a refined count for the real rational curves in a toric surface which pass through a set consisting of real points and pairs of complex conjugated points chosen generically on the toric boundary of the surface. He then proved that the result of this refined count depends only on the number of pairs of complex conjugated points on each toric divisor. Using the tropical geometry approach and the correspondence theorem, we address the computation of the refined count when the pairs of complex conjugated points are chosen purely imaginary and belonging to the same component of the toric boundary. Despite the non-genericity, we relate this refined count for purely imaginary values to the refined invariant of Mikhalkin for generic values. That allows us to extend the relation between these classical refined invariants and the tropical refined invariants from Block–Göttsche.

Faking gauge coupling unification in string theory

Gauge coupling unification misleads infrared observers if new gauge bosons do not simultaneously come into the spectrum. Though easy to engineer in gauge theory, the situation in string theory is nuanced due to moduli dependence. We study the possibility of faking gauge coupling unification in the context of 4D F-theory compactifications. Specifically, we formulate a sufficient condition that we call a strong calibration, under which seven-brane gauge couplings on homologically distinct divisors become equal at codimension one in K\"ahler moduli space. We prove that a strong calibration is preserved under appropriate topological transitions and that a pair of nonintersecting divisors each admitting a contraction can always be strongly calibrated. Within the tree ensemble [J. Halverson et al., Phys. Rev. D 96, 126006 (2017)], we find that $\ensuremath{\approx}77.12%$ of pairs of intersecting toric divisors can be strongly calibrated and $\ensuremath{\approx}3.22%$ can never be calibrated. Physically, this means that gauge coupling unification can be faked in most cases that we study, but in others it can not, which is surprising from a gauge theoretic perspective.

Toric Sasaki–Einstein metrics with conical singularities

We show that any toric Kähler cone with smooth compact cross-section admits a family of Calabi–Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi–Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki–Einstein metrics.

Aspects of functoriality in homological mirror symmetry for toric varieties

We study homological mirror symmetry for toric varieties, exploring the relationship between various Fukaya-Seidel categories which have been employed for constructing the mirror to a toric variety. In particular, we realize tropical Lagrangian sections as objects of a partially wrapped category and construct a Lagrangian correspondence mirror to the inclusion of a toric divisor. As a corollary, we prove that tropical sections generate the Fukaya-Seidel category, completing a Floer-theoretic proof of homological mirror symmetry for projective toric varieties. In the course of the proof, we develop techniques for constructing Lagrangian cobordisms and Lagrangian correspondences in Liouville domains, which may be of independent interest.

A note on h2,1 of divisors in CY fourfolds. Part I

In this note, we prove combinatorial formulas for the Hodge number h2,1 of prime toric divisors in an arbitrary toric hypersurface Calabi-Yau fourfold Y4. We show that it is possible to find a toric hypersurface Calabi-Yau in which there are more than h1,1(Y4) non-perturbative superpotential terms with trivial intermediate Jacobian. Hodge numbers of divisors in toric complete intersection Calabi-Yaus are the subjects of the sequel.

Homological mirror symmetry of CPn and their products via Morse homotopy

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many examples, the derived category Db(coh(X)) of coherent sheaves on a toric manifold X is compared with the Fukaya–Seidel category of the Milnor fiber of the corresponding Landau–Ginzburg potential. We instead consider the dual torus fibration π: M → B of the complement of the toric divisors in X, where B̄ is the dual polytope of the toric manifold X. A natural formulation of homological mirror symmetry in this setup is to define Fuk(M̄) a variant of the Fukaya category and show the equivalence Db(coh(X))≃Db(Fuk(M̄)). As an intermediate step, we construct the category Mo(P) of weighted Morse homotopy on P≔B̄ as a natural generalization of the weighted Fukaya–Oh category proposed by Kontsevich and Soibelman [Symplectic geometry and mirror symmetry (Seoul, 2000) (World Science Publishing, 2001), p. 203]. We then show that a full subcategory MoE(P) of Mo(P) generates Db(coh(X)) for the cases X is a complex projective space and their products.

Intersection theory of toric 𝑏-divisors in toric varieties

We introduce toric b b -divisors on complete smooth toric varieties and a notion of integrability of such divisors. We show that under some positivity assumptions toric b b -divisors are integrable and that their degree is given as the volume of a convex set. Moreover, we show that the dimension of the space of global sections of a nef toric b b -divisor is equal to the number of lattice points in this convex set and we give a Hilbert–Samuel-type formula for its asymptotic growth. This generalizes classical results for classical toric divisors on toric varieties. Finally, we relate convex bodies associated to b b -divisors with Newton–Okounkov bodies. The main motivation for studying toric b b -divisors is that they locally encode the singularities of the invariant metric on an automorphic line bundle over a toroidal compactification of a mixed Shimura variety of non-compact type.

Heights of hypersurfaces in toric varieties

For a cycle of codimension 1 in a toric variety, its degree with respect to a nef toric divisor can be understood in terms of the mixed volume of the polytopes associated to the divisor and to the cycle. We prove here that an analogous combinatorial formula holds in the arithmetic setting: the global height of a 1-codimensional cycle with respect to a toric divisor equipped with a semipositive toric metric can be expressed in terms of mixed integrals of the $v$-adic roof functions associated to the metric and the Legendre-Fenchel dual of the $v$-adic Ronkin function of the Laurent polynomial of the cycle.

Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds

We study Hilbert schemes of points on a smooth projective Calabi–Yau 4-fold X. We define DT4 invariants by integrating the Euler class of a tautological vector bundle L[n] against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when L corresponds to a smooth divisor on X. A parallel equivariant conjecture for toric Calabi–Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples.Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by exp⁡(M(q)−1), where M(q) denotes the MacMahon function.

New large volume Calabi-Yau threefolds

In previous work, we have commenced the task of unpacking the $473,800,776$ reflexive polyhedra by Kreuzer and Skarke into a database of Calabi-Yau threefolds (see http://www.rossealtman.com). In this paper, following a pedagogical introduction, we present a new algorithm to isolate Swiss cheese solutions characterized by "holes," or small 4-cycles, descending from the toric divisors inherent to the original four dimensional reflexive polyhedra. Implementing these methods, we find $2,268$ explicit Swiss cheese manifolds, over half of which have $h^{1,1}=6$. Many of our solutions have multiple large cycles. Such Swiss cheese geometries facilitate moduli stabilization in string compactifications and provide flat directions for cosmological inflation.

On homological mirror symmetry of toric Calabi–Yau threefolds

We use Lagrangian torus fibrations on the mirror $X$ of a toric Calabi-Yau threefold $\check X$ to construct Lagrangian sections and various Lagrangian spheres on $X$. We then propose an explicit correspondence between the sections and line bundles on $\check X$ and between spheres and sheaves supported on the toric divisors of $\check X$. We conjecture that these correspondences induce an embedding of the relevant derived Fukaya category of $X$ inside the derived category of coherent sheaves on $\check X$.

Systematics of axion inflation in Calabi-Yau hypersurfaces

We initiate a comprehensive survey of axion inflation in compactifications of type IIB string theory on Calabi-Yau hypersurfaces in toric varieties. For every threefold with h1,1 ≤ 4 in the Kreuzer-Skarke database, we compute the metric on Kähler moduli space, as well as the matrix of four-form axion charges of Euclidean D3-branes on rigid divisors. These charges encode the possibility of enlarging the field range via alignment. We then determine an upper bound on the inflationary field range Δϕ that results from the leading instanton potential, in the absence of monodromy. The bound on the field range in this ensemble is Δϕ ≲ 0.3Mpl, in a compactification where the smallest curve volume is (2π)2α′, and we argue that the sigma model expansion is adequately controlled. The largest increase resulting from alignment is a factor ≈ 2.6. We also examine a set of threefolds with h1,1 up to 100 and characterize their axion charge matrices. While we find modest alignment in this ensemble, the maximum field range is ultimately suppressed by the volume of the internal space, which typically grows quickly with h1,1. Furthermore, we find that many toric divisors are rigid — and the corresponding charge matrices are relatively trivial — at large h1,1. It is therefore challenging to realize alignment via superpotentials generated only by Euclidean D3-branes, without taking into account the effects of flux, D7-branes, and orientifolding.

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).

Growth of balls of holomorphic sections on projective toric varieties

Let [Formula: see text] be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety [Formula: see text]. Given a continuous toric metric [Formula: see text] on [Formula: see text], we define the energy at equilibrium of [Formula: see text] where [Formula: see text] is the weight of the metrized toric divisor [Formula: see text]. We show that this energy describes the asymptotic behavior as [Formula: see text] of the volume of the [Formula: see text]-norm unit ball induced by [Formula: see text] on the space of global holomorphic sections [Formula: see text].

De Sitter vacua from a D-term generated racetrack potential in hypersurface Calabi-Yau compactifications

In arXiv:1407.7580 a mechanism to fix the closed string moduli in a de Sitter minimum was proposed: a D-term potential generates a linear relation between the volumes of two rigid divisors which in turn produces at lower energies a race-track potential with de Sitter minima at exponentially large volume. In this paper, we systematically search for implementations of this mechanism among all toric Calabi-Yau hypersurfaces with $h^{1,1}\leq 4$ from the Kreuzer-Skarke list. For these, topological data can be computed explicitly allowing us to find the subset of three-folds which have two rigid toric divisors that do not intersect each other and that are orthogonal to $h^{1,1}-2$ independent four-cycles. These manifolds allow to find D7-brane configurations compatible with the de Sitter uplift mechanism and we find an abundance of consistent choices of D7-brane fluxes inducing D-terms leading to a de Sitter minimum. Finally, we work out a couple of models in detail, checking the global consistency conditions and computing the value of the potential at the minimum.

Complex symplectomorphisms and pseudo-Kähler islands in the quantization of toric manifolds

Let $$P$$ be a Delzant polytope. We show that the quantization of the corresponding toric manifold $$X_{P}$$ in toric Kähler polarizations and in the toric real polarization are related by analytic continuation of Hamiltonian flows evaluated at time $$t = {\sqrt{-1}}s$$ . We relate the quantization of $$X_{P}$$ in two different toric Kähler polarizations by taking the time- $${\sqrt{-1}}s$$ Hamiltonian “flow” of strongly convex functions on the moment polytope $$P$$ . By taking $$s$$ to infinity, we obtain the quantization of $$X_{P}$$ in the (singular) real toric polarization. Recall that $$X_{P}$$ has an open dense subset which is biholomorphic to $$({\mathbb {C}}^{*})^{n}$$ . The quantization of $$X_{P}$$ in a toric Kähler polarization can also be described by applying the complexified Hamiltonian flow of the Abreu–Guillemin symplectic potential $$g$$ , at time $$t={\sqrt{-1}}$$ , to an appropriate finite-dimensional subspace of quantum states in the quantization of $$T^{*}{\mathbb {T}}^{n}$$ in the vertical polarization. By taking other imaginary times, $$t= k {\sqrt{-1}}, k\in {\mathbb {R}}$$ , we describe toric Kähler metrics with cone singularities along the toric divisors in $$X_{P}$$ . For convex Hamiltonian functions and sufficiently negative imaginary part of the complex time, we obtain degenerate Kähler structures which are negative definite in some regions of $$X_{P}$$ . We show that the pointwise and $$L^2$$ -norms of quantum states are asymptotically vanishing on negative-definite regions.

Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities

We construct a two-dimensional crystal melting model which reproduces the BPS index of D2-D0 states bound to a non-compact D4-brane on an arbitrary toric Calabi-Yau singularity. The crystalline structure depends on the toric divisor wrapped by the D4-brane. The molten crystals are in one-to-one correspondence with the torus fixed points of the moduli space of the quiver gauge theory on D-branes. The F- and D-term constraints of the gauge theory are regarded as a generalization of the ADHM constraints on instantons. We also show in several examples that our model is consistent with the wall-crossing formula for the BPS index.