Torus Fiber Research Articles

Dimensions of Level-1 Group-Based Phylogenetic Networks

Phylogenetic networks represent evolutionary histories of sets of taxa where horizontal evolution or hybridization has occurred. Placing a Markov model of evolution on a phylogenetic network gives a model that is particularly amenable to algebraic study by representing it as an algebraic variety. In this paper, we give a formula for the dimension of the variety corresponding to a triangle-free level-1 phylogenetic network under a group-based evolutionary model. On our way to this, we give a dimension formula for codimension zero toric fiber products. We conclude by illustrating applications to identifiability.

Enumerative geometry of del Pezzo surfaces

We prove an equivalence between the superpotential defined via tropical geometry and Lagrangian Floer theory for special Lagrangian torus fibres in del Pezzo surfaces constructed by Collins-Jacob-Lin [Duke Math. J. 170 (2021), pp. 1291–1375]. We also include some explicit calculations for the projective plane, which confirm some folklore conjectures in this case.

Virasoro Constraints for Toric Bundles

Abstract We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.

Chekanov torus and Gelfand–Zeitlin torus in S2 × S2

The Chekanov torus is the first known exotic torus, a monotone Lagrangian torus that is not Hamiltonian isotopic to the standard monotone Lagrangian torus. We explore the relationship between the Chekanov torus in S2×S2 and a monotone Lagrangian torus that had been constructed before Chekanov's construction [6]. We prove that the monotone Lagrangian torus fiber in a certain Gelfand–Zeitlin system is related to the Chekanov torus in S2×S2 by a symplectomorphism.

Torus fibers and the weight filtration

We show that if ( X , Y ) (X,Y) is a simple normal crossings log Calabi–Yau pair, then there is a real torus of dimension equal to the codimension of the smallest stratum of Y Y which can be used to construct W 2 k − 1 H k ( X ∖ Y ; Q ) W_{2k-1}\mathrm {H}^k(X \setminus Y;\mathbb {Q}) for all k k . We use this to show that a P=W type result holds for pairs ( X , Y ) (X,Y) consisting of a rational surface X X and a nodal anticanonical divisor Y Y .

Classical iterative proportional scaling of log-linear models with rational maximum likelihood estimator

In this work we investigate multipartition models, the subset of log-linear models for which one can perform the classical iterative proportional scaling (IPS) algorithm to numerically compute the maximum likelihood estimate (MLE). Multipartition models include families of models such as hierarchical models and balanced, stratified staged trees. We define a sufficient condition, called the Generalized Running Intersection Property (GRIP), on the matrix representation of a multipartition model under which the classical IPS algorithm produces the exact MLE in one cycle. In this case, the MLE is a rational function of the data. Additionally we connect the GRIP to the toric fiber product and to previous results for hierarchical models and balanced, stratified staged trees. This leads to a characterization of balanced, stratified staged trees in terms of the GRIP.

Disk potential functions for quadrics

We compute the disk potential of Gelfand–Zeitlin monotone torus fiber in a quadric hypersurface by exploiting toric degenerations, Lie theoretical mirror symmetry, and the structural result of the monotone Fukaya category.

Fibering flat manifolds of diagonal type and their fundamental groups

An [Formula: see text]-dimensional closed flat manifold is said to be of diagonal type if the standard representation of its holonomy group [Formula: see text] is diagonal. An [Formula: see text]-dimensional Bieberbach group of diagonal type is the fundamental group of such a manifold. We introduce the diagonal Vasquez invariant of [Formula: see text] as the least integer [Formula: see text] such that every flat manifold of diagonal type with holonomy [Formula: see text] fibers over a flat manifold of dimension at most [Formula: see text] with flat torus fibers. Using a combinatorial description of Bieberbach groups of diagonal type, we give both upper and lower bounds for this invariant. We show that the lower bounds are exact when [Formula: see text] has low rank. We apply this to analyze diffuseness properties of Bieberbach groups of diagonal type. This leads to a complete classification of Bieberbach groups of diagonal type with Klein four-group holonomy and to an application to Kaplansky’s Unit Conjecture.

Non‐commutative amoebas

The group of isometries of the hyperbolic space H 3 $ \mathbb {H}^3$ is the 3-dimensional group PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ , which is one of the simplest non-commutative complex Lie groups. Its quotient by the subgroup SO ( 3 ) ⊂ PSL 2 ( C ) $\operatorname{SO}(3)\subset \operatorname{PSL}_2(\mathbb {C})$ naturally maps it back to H 3 $ \mathbb {H}^3$ . Each fiber of this map is diffeomorphic to the real projective 3-space R P 3 ${\mathbb {R}}{\mathbb {P}}^3$ . The resulting map PSL 2 ( C ) → H 3 $\operatorname{PSL}_2(\mathbb {C})\rightarrow \mathbb {H}^3$ can be viewed as the simplest non-commutative counterpart of the map Log : ( C × ) n → R n $\operatorname{Log}:(\mathbb {C}^\times )^n\rightarrow \mathbb {R}^n$ from the commutative complex Lie group ( C × ) n $(\mathbb {C}^\times )^n$ with the Lagrangian torus fibers that can be considered as a Liouville–Arnold type integrable system. Gelfand, Kapranov and Zelevinsky have introduced amoebas of algebraic varieties V ⊂ ( C × ) n $V\subset (\mathbb {C}^\times )^n$ as images Log ( V ) ⊂ R n $\operatorname{Log}(V)\subset \mathbb {R}^n$ . We define the amoeba of an algebraic subvariety of PSL 2 ( C ) $\operatorname{PSL}_2(\mathbb {C})$ as its image in H 3 $ \mathbb {H}^3$ . The paper surveys basic properties of the resulting hyperbolic amoebas and compares them against the commutative amoebas R n $\mathbb {R}^n$ .

Numerical characterization of some toric fiber bundles

Given a complex projective manifold X and a divisor D with normal crossings, we say that the logarithmic tangent bundle $$T_X(-\hbox {log}\, D)$$ is R-flat if its pull-back to the normalization of any rational curve contained in X is the trivial vector bundle. If moreover $$-(K_X+D)$$ is nef, then the log canonical divisor $$K_X+D$$ is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber F being a toric variety with boundary divisor $$D_{|F}$$ .

Gröbner bases for staged trees

In this article we consider the problem of finding generators of the toric ideal associated to a combinatorial object called a staged tree. Our main theorem states that toric ideals of staged trees that are balanced and stratified are generated by a quadratic Gr\"obner basis whose intial terms are squarefree. The proof of this result is based on Sullivant's toric fiber product construction.

Lagrangian cobordism and tropical curves

Abstract We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.

Toric geometry of the Cavender-Farris-Neyman model with a molecular clock

We give a combinatorial description of the toric ideal of invariants of the Cavender-Farris-Neyman model with a molecular clock (CFN-MC) on a rooted binary phylogenetic tree and prove results about the polytope associated to this toric ideal. Key results about the polyhedral structure include that the number of vertices of this polytope is a Fibonacci number, the facets of the polytope can be described using the combinatorial “cluster” structure of the underlying rooted tree, and the volume is equal to an Euler zig-zag number. The toric ideal of invariants of the CFN-MC model has a quadratic Gröbner basis with squarefree initial terms. Finally, we show that the Ehrhart polynomial of these polytopes, and therefore the Hilbert series of the ideals, depends only on the number of leaves of the underlying binary tree, and not on the topology of the tree itself. These results are analogous to classic results for the Cavender-Farris-Neyman model without a molecular clock. However, new techniques are required because the molecular clock assumption destroys the toric fiber product structure that governs group-based models without the molecular clock.

Maximum likelihood estimation of toric Fano varieties

We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points and provide explicit expressions that allow one to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using $A$-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees.

Fukaya category of Grassmannians: Rectangles

We show that the monotone Lagrangian torus fiber of the Gelfand-Cetlin integrable system on the complex Grassmannian Gr(k,n) supports generators for all maximum modulus summands in the spectral decomposition of the Fukaya category over C, generalizing the example of the Clifford torus in projective space. We introduce an action of the dihedral group Dn on the Landau-Ginzburg mirror proposed by Marsh-Rietsch [27] that makes it equivariant and use it to show that, given a lower modulus, the torus supports nonzero objects in none or many summands of the Fukaya category with that modulus. The alternative is controlled by the vanishing of rectangular Schur polynomials at the n-th roots of unity, and for n=p prime this suffices to give a complete set of generators and prove homological mirror symmetry for Gr(k,p).

Cox ring of the generic fiber

Given a surjective morphism π:X→Y of normal varieties satisfying some regularity hypotheses we prove how to recover a Cox ring of the generic fiber of π from the Cox ring of X. As a corollary we show that in some cases it is also possible to recover the Cox ring of a very general fiber, and finally we give an application in the case of the blowing-up of a toric fiber space.

Potential functions on Grassmannians of planes and cluster transformations

With a triangulation of a planar polygon with $n$ sides, one can associate an integrable system on the Grassmannian of 2-planes in an $n$-space. In this paper, we show that the potential functions of Lagrangian torus fibers of the integrable systems associated with different triangulations glue together by cluster transformations. We also prove that the cluster transformations coincide with the wall-crossing formula in Lagrangian intersection Floer theory.

Noncommutative gauge theories on D-branes in non-geometric backgrounds

We investigate the noncommutative gauge theories arising on the worldvolumes of D-branes in non-geometric backgrounds obtained by T-duality from twisted tori. We revisit the low-energy effective description of D-branes on three-dimensional T-folds, examining both cases of parabolic and elliptic twists in detail. We give a detailed description of the decoupling limits and explore various physical consequences of the open string non-geometry. The T-duality monodromies of the non-geometric backgrounds lead to Morita duality monodromies of the noncommutative Yang-Mills theories induced on the D-branes. While the parabolic twists recover the well-known examples of noncommutative principal torus bundles from topological T-duality, the elliptic twists give new examples of noncommutative fibrations with non-geometric torus fibres. We extend these considerations to D-branes in backgrounds with R-flux, using the doubled geometry formulation, finding that both the non-geometric background and the D-brane gauge theory necessarily have explicit dependence on the dual coordinates, and so have no conventional formulation in spacetime.

Spectral Invariants with Bulk, Quasi-Morphisms and Lagrangian Floer Theory

In this paper we first develop various enhancements of the theory of spectral invariants of Hamiltonian Floer homology and of Entovi-Polterovich theory of spectral symplectic quasi-states and quasimorphisms by incorporating \emph{bulk deformations}, i.e., deformations by ambient cycles of symplectic manifolds, of the Floer homology and quantum cohomology. Essentially the same kind of construction is independently carried out by Usher [Us4] in a slightly less general context. Then we explore various applications of these enhancements to the symplectic topology, especially new construction of symplectic quasi-states, quasimorphisms and new Lagrangian intersection results on toric manifolds. The most novel part of this paper is to use open-closed Gromov-Witten theory (operator $\frak q$ in [FOOO1] and its variant involving closed orbits of periodic Hamiltonian system) to connect spectral invariants (with bulk deformation), symplectic quasi-states, quasimorphism to the Lagrangian Floer theory (with bulk deformation). We use this open-closed Gromov-Witten theory to produce new examples. Especially using the calculation of Lagrangian Floer homology with bulk deformation in [FOOO3,FOOO4], we produce examples of compact toric manifolds $(M,\omega)$ which admits uncountably many independent quasimorphisms $\widetilde{\operatorname{Ham}}(M,\omega) \to \mathbb R$. We also obtain a new intersection result of Lagrangian submanifolds on $S^2 \times S^2$ discovered in [FOOO6]. Many of these applications were announced in [FOOO3,FOOO4,FOOO6].

On the prevalence of elliptic and genus one fibrations among toric hypersurface Calabi-Yau threefolds

We systematically analyze the fibration structure of toric hypersurface Calabi-Yau threefolds with large and small Hodge numbers. We show that there are only four such Calabi-Yau threefolds with h1,1 ≥ 140 or h2,1 ≥ 140 that do not have manifest elliptic or genus one fibers arising from a fibration of the associated 4D polytope. There is a genus one fibration whenever either Hodge number is 150 or greater, and an elliptic fibration when either Hodge number is 228 or greater. We find that for small h1,1 the fraction of polytopes in the KS database that do not have a genus one or elliptic fibration drops exponentially as h1,1 increases. We also consider the different toric fiber types that arise in the polytopes of elliptic Calabi-Yau threefolds.