The Lagrangian Amplituhedron
The article constructs the Lagrangian analog of the amplituhedron, studies its geometric properties and BCFW decompositions.
hep-th↗
arXiv subjects
Ran J. Tessler
Publications and source records attributed to Ran J. Tessler.
The Structure of Cycles in Projective Geometry over $\mathbb{F}_q$
A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of $\mathbb{F}_q^n$, determining the minimum support size of a nonzero $d$-cycle over a field $K$ of characteristic $p \mid q+1$. Surprisingly, the boundary of a $(d+1)$-space is not always optimal. Shorter cycles occur for $d=1$, and for $d=2$ when $n\ge4$, and otherwise, the boundary of a $(d+1)$-space is shortest. For $d\ge4$, we prove a gap-stability result: every cycle with support less than $(2-10/q)$ times the minimum is a multiple of the boundary of a $d+1$ space. We also construct support-controlled cones, yielding a direct geometric analysis of the dimensions in which the homology groups of the subspace incidence complex vanish and explicit lower bounds on its coboundary expansion. The degree-$1$ expansion estimate is an ingredient in the stability theorem.
math.CO↗
Open FJRW Theory and Mirror Symmetry
We construct an open enumerative theory for the Landau-Ginzburg (LG) model $(\mathbb{C}^2, μ_r\times μ_s, x^r+y^s)$. The invariants are defined as integrals of multisections of a Witten bundle with descendents over a moduli space that is a real orbifold with corners. In turn, a generating function for these open invariants yields the mirror LG model and a versal deformation of it with flat coordinates. After establishing an open topological recursion result, we prove an LG/LG open mirror symmetry theorem in dimension two with all descendents. The open invariants we define are not unique but depend on boundary conditions that, when altered, exhibit wall-crossing phenomena for the invariants. We describe an LG wall-crossing group classifying the wall-crossing transformations that can occur.
math.AG↗
Spectral Analysis of Word Statistics
Given a random text over a finite alphabet, we study the frequencies at which fixed-length words occur as subsequences. As the data size grows, the joint distribution of word counts exhibits a rich asymptotic structure. We investigate all linear combinations of subword statistics, and fully characterize their different orders of magnitude using diverse algebraic tools. Moreover, we establish the spectral decomposition of the space of word statistics of each order. We provide explicit formulas for the eigenvectors and eigenvalues of the covariance matrix of the multivariate distribution of these statistics. Our techniques include and elaborate on a set of algebraic word operators, recently studied and employed by Dieker and Saliola (Adv Math, 2018). Subword counts find applications in Combinatorics, Statistics, and Computer Science. We revisit special cases from the combinatorial literature, such as intransitive dice, random core partitions, and questions on random walk. Our structural approach describes in a unified framework several classical statistical tests. We propose further potential applications to data analysis and machine learning.
math.PR↗
Open enumerative geometries for Landau-Ginzburg models
We survey the recent progress in defining open enumerative theories for Landau-Ginzburg models. We illustrate the ideas required to develop these new foundations. In particular, we describe how to define the open enumerative invariants as integrals of multisections of certain vector bundles over a moduli space that is a real orbifold with corners, after prescribing boundary conditions for the multisections. We then explain the known situations where the open invariants satisfy certain forms of topological recursion relations, integrable hierarchies, or mirror symmetry. We end with a list of open questions and problems.
math.AG↗
The point insertion technique and open $r$-spin theories I: moduli and orientation
The papers [3,1,4,10] constructed an intersection theory on the moduli space of $r$-spin disks, and proved it satisfies mirror symmetry and relations with integrable hierarchies. That theory considered only disks with a single boundary state. In this work, we initiate the study of more general $r$-spin surfaces. We define graded $r$-spin surfaces with multiple internal and boundary states, together with their moduli spaces. In genus zero, the disk case, we define the associated open Witten bundle and prove that it is canonically oriented relative to the moduli space. We also describe a gluing construction for moduli spaces along boundaries, show that it lifts to the Witten bundle and relative cotangent line bundles, and that the result remains canonically relatively oriented. We then study the genus-one cylinder case. Here foundational difficulties arise because the Witten "bundle" is no longer an orbifold vector bundle. We resolve this by removing strata with incorrect fibre dimension, obtaining an orbibundle on the complement. The gluing method extends to genus one, and we prove that the Witten bundle again admits a canonical relative orientation. In the sequel [20], we construct a family of $\lfloor r/2\rfloor$ intersection theories in genus-zero indexed by $\mathfrak h\in\{0,\ldots,\lfloor r/2\rfloor-1\}$, where the $\mathfrak h$-th theory has $\mathfrak h+1$ boundary states, and compute their intersection numbers. The case $\mathfrak h=0$ recovers the theory of [3,1]. In the sequel [21], restricting to the $\mathfrak h=0$ case, we construct an intersection theory on the moduli space of $r$-spin cylinders and show that its potential yields, after a change of variables, the genus-one part of the $r$th Gelfand-Dikii wave function, proving the genus-one case of the main conjecture of [4].
math.AG↗
Open $r$-spin theory in genus one, and the Gelfand-Dikii wave function
We construct the $g=1$ sector of the open $r$-spin theory, that is, an open $r$-spin theory on the moduli space of cylinders. This is the second construction of a $g>0$ open intersection theory, which includes descendents (the first is the all genus construction of the intersection theory on moduli of open Riemann surfaces with boundaries [23,30], whose $g=1$ case equals to the $r=2$ case of our construction). Unlike the construction of [30], in order to construct the $r$-spin cylinder theory we had to overcome the foundational problem of dimension jump loci, which in analogous closed theories has been treated using virtual fundamental class techniques, that are currently absent in the open setting. For this reason our construction is much more involved, and relies on the point insertion technique developed in [31,32]. We prove that the open $g=1$ potential equals, after a coordinate change, to the $g=1$ part of the Gelfand-Dikii wave function, thus confirming a conjecture of [7]. We also prove that our $g=1$ intersection numbers satisfy a $g=1$ recursion, also predicted in [7,15]. This recursion is the $g=1$ analogue of Solomon's famous $g=0$ Open WDVV equation [25], with descendents, and is also the universal $g=1$ recursion for $F$-Cohomological field theories [1]. Again, this is first geometric construction which is not the $g=1$ sector of [23,30], proven to satisfy this universal recursion.
math.AG↗
Error Resilient Space Partitioning
A major research area in discrete geometry is to consider the best way to partition the $d$-dimensional Euclidean space $\mathbb{R}^d$ under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space $\mathbb{R}^d$ to a discrete subset of representative values. Specifically, we study partitions of $\mathbb{R}^d$ into bounded-size tiles colored by one of $k$ colors, such that tiles of the same color have a distance of at least $t$ from each other. Such tilings allow for \emph{error-resilient} rounding, as two points of the same color and distance less than $t$ from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors $k$ and the distance $t$, for various dimensions $d$. On the qualitative side, we show that in $\mathbb{R}^d$, using $k=d+1$ colors is both sufficient and necessary to achieve $t>0$. On the quantitative side, we achieve numerous upper and lower bounds on $t$ as a function of $k$. In particular, for $d=3,4,8,24$, we obtain sharp asymptotic bounds on $t$, as $k \to \infty$. We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat's connector-free lemma, and Čech cohomology.
cs.CG↗
Open Hurwitz numbers and the mKP hierarchy
We give a natural definition of open Hurwitz numbers, where the weight of each ramified covering includes an integer parameter $N$ taken to the power that is equal to the number of boundary components of a Riemann surface with boundary mapping to $\mathbb{CP}^1$. We prove that the resulting sequence of partition functions, depending on $N\in\mathbb{Z}$, is a tau-sequence of the mKP hierarchy, or in other words it is a sequence of tau-functions of the KP hierarchy where each tau-function is obtained from the previous one by a Bäcklund-Darboux transformation. Our result is motivated by a previous observation of Alexandrov and the first two authors that the refined intersection numbers on the moduli spaces of Riemann surfaces with boundary give a tau-sequence of the mKP hierarchy.
math-ph↗
Notes on the one-loop amplituhedron and its BCFW tiling
These notes are based on talks I gave in the seminar "Mathematical structures in scattering amplitudes in quantum field theories" I organized in Weizmann Institute on Fall 24'. They study amplituhedra, and extend the proof of \cite{even2021amplituhedron} of the BCFW conjecture for tree amplituhedra to one loop.
math-ph↗
The Amplituhedron BCFW Triangulation
The amplituhedron Ank4 is a geometric object, introduced by Arkani-Hamed and Trnka (2013) in the study of scattering amplitudes in quantum field theories. They conjecture that Ank4 admits a decomposition into images of BCFW positroid cells, arising from the Britto--Cachazo--Feng--Witten recurrence (2005). We prove that this conjecture is true.
math-ph↗
The Tropical Amplituhedron
The Amplituhedron is a subspace of the Grassmannian that was recently defined by Arkani-Hamed and Trnka in their study of scattering amplitudes in planar $\mathcal{N}=4$ super Yang Mills theory (arXiv:1312.2007), and was the subject of many papers in the last decade. In this work we define a tropical analog of the amplituhedron, and develop techniques to address it. We prove that many of the key properties of the amplituhedron hold also in this simpler, piecewise linear, model.
hep-th↗
The point insertion technique and open $r$-spin theories II: intersection theories in genus-zero
The papers [5, 3, 6, 19, 20] initiated the study of open $r$-spin and open FJRW intersection theories, and related them to integrable hierarchies and mirror symmetry. This paper uses a new technique, the point insertion technique, developed in the prequel [36], to define new open r-spin and open FJRW intersection theories. These new constructions provide potential candidates for theories whose existence was conjectured before: $\bullet$ K. Hori [23] predicted the existence of open $r$-spin theory with $\lfloor\frac{r}{2}\rfloor$ types of boundary states. The one constructed in [5, 3] has only one type of boundary state. In this work we describe $\lfloor\frac{r}{2}\rfloor$ open $r$-spin theories, labelled by $\mathfrak{h}\in\{0,\ldots,\lfloor\frac{r}{2}\rfloor-1\},$ where the $\mathfrak{h}$-th one has $\mathfrak{h}+1$ boundary states. We prove that the $\mathfrak{h}=0$ theory is equivalent to the [5, 3] construction, and calculate all intersection numbers for all these theories. $\bullet$ In [1] K. Aleshkin and C.C.M. Liu conjectured the existence of a quintic Fermat FJRW theory. We construct such an FJRW theory, and provide evidence that this is the conjectured theory. We also explain how the point insertion technique can be used for constructing other open enumerative theories, satisfying the same universal recursions.
math.AG↗
Open $r$-spin theory II: The analogue of Witten's conjecture for $r$-spin disks
We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand-Dickey integrable hierarchy. This provides an analogue of Witten's $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan-Jarvis-Ruan-Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Witten's closed theory.
math.SG↗
Open $r$-spin theory III: a prediction for higher genus
In our previous two papers, we constructed an $r$-spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of Witten's $r$-spin conjecture in genus zero in the open setting. In particular, we proved that the generating series of open $r$-spin intersection numbers is determined by the genus-zero part of a special solution of a certain extension of the Gelfand-Dickey hierarchy, and we conjectured that the whole solution controls the open $r$-spin intersection numbers in all genera, which do not yet have a geometric definition. In this paper, we provide geometric and algebraic evidence for the correctness of this conjecture.
math-ph↗
Garland's Technique for Posets and High Dimensional Grassmannian Expanders
Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [G] to our days. In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high dimensional random walks generalizing [KO,AL], an equivalence with a global random walk definition, generalizing [DDFH] and a trickling down theorem, generalizing [O]. In particular, we show that some posets, such as the Grassmannian poset, exhibit qualitatively stronger trickling down effect than simplicial complexes. Using these methods, and the novel idea of Posetification, to Ramanujan complexes [LSV1,LSV2], we construct a constant degree expanding Grassmannian poset, and analyze its expansion. This it the first construction of such object, whose existence was conjectured in [DDFH].
math.CO↗
Open $\mathbb{CP}^1$ descendent theory I: The stationary sector
We define stationary descendent integrals on the moduli space of stable maps from disks to $(\mathbb{CP}^1,\mathbb{RP}^1)$. We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus $0$ disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.
math.SG↗
Mirror Symmetry for open r-spin invariants
We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau-Ginzburg model $(\mathbb{C},x^r)$ via Saito-Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in a sequel paper.
math.AG↗