arXiv subjects

Eric Zaslow

Publications and source records attributed to Eric Zaslow.

At least 19 recordsLinked to original sources

arXiv2026

Mirror Counts of Spectral Curves

Given a convex lattice polygon $\Delta\subset \mathbb R^2$, let $N_\Delta$ be the count of rational, nodal curves in the linear system of the ample line bundle $L_\Delta$ on the toric variety $\mathbb P_\Delta$ defined by $\Delta$, having fixed intersection with the toric boundary. We show that the relative Jacobian of the linear system defines an integrable system that has a mirror dual in the language of constructible sheaves on a two-torus microsupported on a Legendrian link. In this setting, we define the dual counting problem using an analogue of rulings of Legendrian links in three-space, then prove equivalence with $N_\Delta$. We perform calculations of $N_\Delta$ in several examples using constructible methods, tropical curve counting, and localization in logarithmic Gromov-Witten theory to demonstrate the equality of the different approaches. Moreover, the rulings give rise to a stratification of the moduli of constructible sheaves. We conjecture that this ruling decomposition recovers the refined tropical invariants of Block-G\"ottsche, and hence encodes higher-genus logarithmic Gromov--Witten invariants, by a theorem of Bousseau.

math.SG↗

arXiv2025

Evasive Random Walks and the Clairvoyant Demon

A pair of random walks $(R,S)$ on the vertices of a graph $G$ is {\it successful} if two tokens can be scheduled (moving only one token at a time) to travel along $R$ and $S$ without colliding. We consider questions related to P. Winkler's {\it clairvoyant demon problem}, which asks whether for random walks $R$ and $S$ on $G$, $Pr[\ (R,S) \mbox{ is successful }] >0$. We introduce the notion of an {\it evasive} walk on $G$: a walk $S$ so that for a random walk $R$ on $G$, $Pr[\ (R,S) \mbox{ is successful }]>0$. We characterize graphs $G$ having evasive walks, giving explicit constructions on such $G$. On a cycle, we show that with high probability the tokens must collide quickly. Finally we consider two variants of the problem for which, under certain assumptions on the graph $G$, we provide algorithms that schedule $(R,S)$ successfully with positive probability.

math.CO↗

arXiv2025

An iterated random function with Lipschitz number one

Consider the set of functions $f_{\theta}(x)=|\theta -x|$ on $\mathbb{R}$. Define a Markov process that starts with a point $x_0 \in \mathbb{R}$ and continues with $x_{k+1}=f_{\theta_{k+1}}(x_{k})$ with each $\theta _{k+1}$ picked from a fixed bounded distribution $\mu$ on $\mathbb{R}^+$. We prove the conjecture of G. Letac that if $\mu$ is not supported on a lattice, then this process has a unique stationary distribution $\pi_{\mu}$ and any distribution converges under iteration to $\pi_{\mu}$ (in the weak-$^*$ topology). We also give a bound on the rate of convergence in the special case that $\mu$ is supported on a two-point set. We hope that the techniques will be useful for the study of other Markov processes where the transition functions have Lipschitz number one.

math.PR↗

arXiv2025

Optimal estimators for threshold-based quality measures

We consider a problem in parametric estimation: given $n$ samples from an unknown distribution, we want to estimate which distribution, from a given one-parameter family, produced the data. Following Schulman and Vazirani, we evaluate an estimator in terms of the chance of being within a specified tolerance of the correct answer, in the worst case. We provide optimal estimators for several families of distributions on $\mathbb{R}$. We prove that for distributions on a compact space, there is always an optimal estimator that is translation-invariant, and we conjecture that this conclusion also holds for any distribution on $\mathbb{R}$. By contrast, we give an example showing it does not hold for a certain distribution on an infinite tree.

math.ST↗

arXiv2025

Enumerative Geometry of Quantum Periods

We interpret the $q$-refined theta function $\vartheta_1$ of a log Calabi-Yau surface $(\mathbb{P},E)$ as a natural $q$-refinement of the open mirror map, defined by quantum periods of mirror curves for outer Aganagic-Vafa branes on the local Calabi-Yau $K_{\mathbb{P}}$. The series coefficients are all-genus logarithmic two-point invariants, directly extending the relation found in [GRZ]. Yet we find an explicit discrepancy at higher genus in the relation to open Gromov-Witten invariants of the Aganagic-Vafa brane. Using a degeneration argument, we express the difference in terms of relative invariants of an elliptic curve. With $\pi: \widehat{\mathbb{P}} \rightarrow \mathbb{P}$ the toric blow up of a point, we use the Topological Vertex [AKMV] to show a correspondence between open invariants of $K_{\mathbb{P}}$ and closed invariants of $K_{\widehat{\mathbb{P}}}$ generalizing a variant of [CLLT][LLW] to arbitrary genus and winding. We also equate winding-1, open-BPS invariants with closed Gopakumar-Vafa invariants.

math.AG↗

arXiv2023

Skeins, clusters and wavefunctions

In previous work of the second- and third-named authors with Linhui Shen, cluster theory was used to construct wavefunctions for branes in threespace and conjecturally relate them to open Gromov-Witten invariants. This was done by defining a quantum Lagrangian subvariety of a quantum cluster variety, and mutating a simple solution to the defining equations in a distinguished seed. In this paper, we extend the construction to incorporate the skein-theoretic approach to open Gromov-Witten theory of Ekholm-Shende. In particular, we define a skein-theoretic version of cluster theory, including the groupoid of seeds and mutations and a skein-theoretic version of the quantum dilogarithm. We prove a pentagon relation in the skein of the closed torus in this context, and give strong evidence that its analogue holds for arbitrary surfaces. We propose face relations satisfied by the skein-theoretic wavefunction, prove their invariance under mutations, and show their solution is unique. We define a skein version of framings in the story, and use the novel cluster structure to compute wavefunctions in several examples. The skein approach incorporates moduli spaces of sheaves of higher microlocal rank and their quantizations.

math.SG↗

arXiv2023

The Chromatic Lagrangian: Wavefunctions and Open Gromov-Witten Conjectures

Inside a symplectic leaf of the cluster Poisson variety of Borel-decorated $PGL_2$ local systems on a punctured surface is an isotropic subvariety we will call the chromatic Lagrangian. Local charts for the quantized cluster variety are quantum tori defined by cubic planar graphs, and can be put in standard form after some additional markings giving the notion of a framed seed. The mutation structure is encoded as a groupoid. The local description of the chromatic Lagrangian defines a wavefunction which, we conjecture, encodes open Gromov-Witten invariants of a Lagrangian threefold in threespace defined by the cubic graph and the other data of the framed seed. We also find a relationship we call framing duality: for a family of "canoe" graphs, wavefunctions for different framings encode DT invariants of symmetric quivers.

math.RT↗

arXiv2022

The proper Landau-Ginzburg potential is the open mirror map

The mirror dual of a smooth toric Fano surface $X$ equipped with an anticanonical divisor $E$ is a Landau-Ginzburg model with superpotential, W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair $(X,E)$. When $E$ is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle $K_X$, in framing zero. As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.

math.AG↗

arXiv2020

Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications

We study a class of Legendrian surfaces in contact five-folds by encoding their wavefronts via planar combinatorial structures. We refer to these surfaces as Legendrian weaves, and to the combinatorial objects as N-graphs. First, we develop a diagrammatic calculus which encodes contact geometric operations on Legendrian surfaces as multi-colored planar combinatorics. Second, we present an algebraic-geometric characterization for the moduli space of microlocal constructible sheaves associated to these Legendrian surfaces. Then we use these N-graphs and the flag moduli description of these Legendrian invariants for several new applications to contact and symplectic topology. Applications include showing that any finite group can be realized as a subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in the 1-jet space of the two-sphere, a new construction of infinitely many exact Lagrangian fillings for Legendrian links in the standard contact three-sphere, and performing rational point counts over finite fields that distinguish Legendrian surfaces in the standard five-dimensional Darboux chart. In addition, the manuscript develops the notion of Legendrian mutation, studying microlocal monodromies and their transformations. The appendix illustrates the connection between our N-graph calculus for Lagrangian cobordisms and Elias-Khovanov-Williamson's Soergel Calculus.

math.SG↗

arXiv2018

Kasteleyn operators from mirror symmetry

Given a consistent bipartite graph $Γ$ in $T^2$ with a complex-valued edge weighting $\mathcal{E}$ we show the following two constructions are the same. The first is to form the Kasteleyn operator of $(Γ, \mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\mathbb{C}^\times)^2$. The second is to form the conjugate Lagrangian $L \subset T^* T^2$ of $Γ$, equip it with a brane structure prescribed by $\mathcal{E}$, and pass to its mirror coherent sheaf. This lives on a stacky toric compactification of $(\mathbb{C}^\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $Γ$ (and to which the noncompact Lagrangian $L$ is asymptotic). We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry. We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the asymptotic boundary of $L$.

math.AG↗

arXiv2018

Cluster varieties from Legendrian knots

Many interesting spaces --- including all positroid strata and wild character varieties --- are moduli of constructible sheaves on a surface with microsupport in a Legendrian link. We show that the existence of cluster structures on these spaces may be deduced in a uniform, systematic fashion by constructing and taking the sheaf quantizations of a set of exact Lagrangian fillings in correspondence with isotopy representatives whose front projections have crossings with alternating orientations. It follows in turn that results in cluster algebra may be used to construct and distinguish exact Lagrangian fillings of Legendrian links in the standard contact three space.

math.SG↗

arXiv2018

Wavefunctions for a Class of Branes in Three-space

Wavefunctions are proposed for a class of Lagrangian branes in three complex-dimensional space. The branes are asymptotic to Legendrian surfaces of genus g. The expansion of these wavefunctions in appropriate coordinates conjecturally encodes all-genus open Gromov-Witten invariants, i.e. the free energy of the topological open string. This paper is written in physics language, but tries to welcome mathematicians. Most results stem from joint mathematical works with Linhui Shen and David Treumann.

hep-th↗

arXiv2017

Cubic Planar Graphs and Legendrian Surface Theory

We study Legendrian surfaces determined by cubic planar graphs. Graphs with distinct chromatic polynomials determine surfaces that are not Legendrian isotopic, thus giving many examples of non-isotopic Legendrian surfaces with the same classical invariants. The Legendrians have no exact Lagrangian fillings, but have many interesting non-exact fillings. We obtain these results by studying sheaves on a three-ball with microsupport in the surface. The moduli of such sheaves has a concrete description in terms of the graph and a beautiful embedding as a holomorphic Lagrangian submanifold of a symplectic period domain, a Lagrangian that has appeared in the work of Dimofte-Gabella-Goncharov [DGGo]. We exploit this structure to find conjectural open Gromov-Witten invariants for the non-exact filling, following Aganagic-Vafa [AV, AV2].

math.SG↗

arXiv2016

Legendrian knots and constructible sheaves

We study the unwrapped Fukaya category of Lagrangian branes ending on a Legendrian knot. Our knots live at contact infinity in the cotangent bundle of a surface, the Fukaya category of which is equivalent to the category of constructible sheaves on the surface itself. Consequently, our category can be described as constructible sheaves with singular support controlled by the front projection of the knot. We use a theorem of Guillermou-Kashiwara-Schapira to show that the resulting category is invariant under Legendrian isotopies, and conjecture it is equivalent to the representation category of the Chekanov-Eliashberg differential graded algebra. We also find two connections to topological knot theory. First, drawing a positive braid closure on the annulus, the moduli space of rank-n objects maps to the space of local systems on a circle. The second page of the spectral sequence associated to the weight filtration on the pushforward of the constant sheaf is the (colored-by-n) triply-graded Khovanov-Rozansky homology. Second, drawing a positive braid closure in the plane, the number of points of our moduli spaces over a finite field with q elements recovers the lowest coefficient in 'a' of the HOMFLY polynomial of the braid closure.

math.SG↗

arXiv2015

Augmentations are Sheaves

We show that the set of augmentations of the Chekanov-Eliashberg algebra of a Legendrian link underlies the structure of a unital A-infinity category. This differs from the non-unital category constructed in [BC], but is related to it in the same way that cohomology is related to compactly supported cohomology. The existence of such a category was predicted by [STZ], who moreover conjectured its equivalence to a category of sheaves on the front plane with singular support meeting infinity in the knot. After showing that the augmentation category forms a sheaf over the x-line, we are able to prove this conjecture by calculating both categories on thin slices of the front plane. In particular, we conclude that every augmentation comes from geometry.

math.SG↗

arXiv2013

Skeleta of Affine Hypersurfaces

A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S.

math.AG↗

arXiv2011

Polytopes and Skeleta

To each simplicial reflexive polytope in Z^(n+1), we attach an n-dimensional space, Lambda^\infty. It is the Legendrian boundary of a conic Lagrangian considered in work of the authors, Fang and Liu, and because of this it carries a sheaf of dg categories called the "Kashiwara-Schapira sheaf." We discuss some conjectures and results about the role that Lambda^\infty and the Kashiwara-Schapira sheaf should play in homological mirror symmetry.

math.SG↗

arXiv2011

Ribbon Graphs and Mirror Symmetry I

Given a ribbon graph $Γ$ with some extra structure, we define, using constructible sheaves, a dg category $CPM(Γ)$ meant to model the Fukaya category of a Riemann surface in the cell of Teichmüller space described by $Γ.$ When $Γ$ is appropriately decorated and admits a combinatorial "torus fibration with section," we construct from $Γ$ a one-dimensional algebraic stack $\widetilde{X}_Γ$ with toric components. We prove that our model is equivalent to $Perf(\widetilde{X}_Γ)$, the dg category of perfect complexes on $\widetilde{X}_Γ$.

math.AT↗