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Anthony M. Licata

Publications and source records attributed to Anthony M. Licata.

17 recordsLinked to original sources
arXiv2025

A sphere of spherical objects

Given a Bridgeland stability condition on a 2-Calabi--Yau category, we define a simplicial complex that encodes the Harder--Narasimhan filtrations of spherical objects. For 2-Calabi--Yau categories of type A, we relate this complex to the complex of pointed pseudo-triangulations on configurations of points on the plane. Using this connection, we prove that the complex undergoes piecewise-linear wall-crossings as we vary the stability condition, and is piecewise-linearly homeomorphic to a sphere. Additionally, we prove that for a generic stability condition on a 2-Calabi--Yau category, a spherical object is determined by the ordered list of its Harder--Narasimhan factors.

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arXiv2025

Periodic elements in finite type Artin-Tits groups and stability conditions

Periodic elements in finite type Artin--Tits groups are elements some positive power of which is central. We give a dynamical characterisation of periodic elements via their action on the corresponding 2-Calabi--Yau category and on its space of (fusion equivariant) Bridgeland stability conditions. The main theorem is that an element $β$ is periodic if and only if $β$ has a fixed point in the stability manifold.

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arXiv2025

Fusion-equivariant stability conditions and Morita duality

Given a triangulated category $D$ with an action of a fusion category $C$, we study the moduli space $Stab_{C}(D)$ of fusion-equivariant Bridgeland stability conditions on $D$. The main theorem is that the fusion-equivariant stability conditions form a closed, complex submanifold of the moduli space of stability conditions on $D$. As an application of this framework to finite group actions on categories, we generalise a result of Macrì--Mehrotra--Stellari by establishing a biholomorphism between the space of $G$-invariant stability conditions on $D$ and the space of $rep(G)$-equivariant stability conditions on the equivariant category $D^G$. We also describe applications to the study of stability conditions associated to McKay quivers and to geometric stability conditions on free quotients of smooth projective varieties.

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arXiv2024

Stability conditions and Artin--Tits groups

We describe spaces of Bridgeland stability conditions on certain triangulated categories associated to Coxeter systems. These categories are defined algebraically using the category of modules for zigzag algebras associated to Coxeter systems, which we construct as distinguished (quadratic, graded) algebra objects in fusion categories. The resulting stability spaces are closely related to conjectural K($π$,1) spaces for Artin--Tits groups.

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arXiv2023

A Thurston compactification of the space of stability conditions

We propose compactifications of the moduli space of Bridgeland stability conditions of a triangulated category. Our construction arises from a viewing a stability condition as a metric on the underlying category and is inspired by the Thurston compactification of the Teichmüller space of hyperbolic metrics on a surface. The key ingredient in the construction are maps from the stability manifold to an infinite projective space. We prove that, under suitable hypotheses, these maps are injective and their image has a compact closure. We identify a family of points in the boundary that are categorical analogous to the intersection functionals in Teichmüller theory. We study in detail the geometry of the resulting compactification for the 2-Calabi--Yau categories of quivers, and fully work out the cases of the \(A_2\) and \(\widehat{A_1}\) quivers. To do so, we carefully examine the dynamics of Harder--Narasimhan multiplicities under auto-equivalences of the category. We introduce a finite automaton to study this dynamics and employ it in our analysis of the \(A_{2}\) and \(\widehat{A_1}\) categories.

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arXiv2023

Strands algebras and the affine highest weight property for equivariant hypertoric categories

We show that the equivariant hypertoric convolution algebras introduced by Braden-Licata-Proudfoot-Webster are affine quasi hereditary in the sense of Kleshchev and compute the Ext groups between standard modules. Together with the main result of arXiv:2009.03981, this implies a number of new homological results about the bordered Floer algebras of Ozsvath-Szabo, including the existence of standard modules over these algebras. We prove that the Ext groups between standard modules are isomorphic to the homology of a variant of the Lipshitz-Ozsvath-Thurston bordered strands dg algebras.

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arXiv2022

$q$-deformed rational numbers and the 2-Calabi--Yau category of type $A_2$

We describe a family of compactifications of the space of Bridgeland stability conditions of any triangulated category following earlier work by Bapat, Deopurkar, and Licata. We particularly consider the case of the 2-Calabi--Yau category of the $A_2$ quiver. The compactification is the closure of an embedding (depending on $q$) of the stability space into an infinite-dimensional projective space. In the $A_2$ case, the three-strand braid group $B_3$ acts on this closure. We describe two distinguished braid group orbits in the boundary, points of which can be identified with certain rational functions in $q$. Points in one of the orbits are exactly the $q$-deformed rational numbers recently introduced by Morier-Genoud and Ovsienko, while the other orbit gives a new $q$-deformation of the rational numbers. Specialising $q$ to a positive real number, we obtain a complete description of the boundary of the compactification.

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arXiv2021

Spherical objects and stability conditions on 2-Calabi--Yau quiver categories

Consider a 2-Calabi--Yau triangulated category with a Bridgeland stability condition. We devise an effective procedure to reduce the phase spread of an object by applying spherical twists. Using this, we give new proofs of the following theorems for 2-Calabi--Yau categories associated to ADE quivers: (1) all spherical objects lie in a single orbit of the braid group, and (2) the space of Bridgeland stability conditions is connected.

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arXiv2020

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in arXiv:0905.1335 from the equivariant cohomology of toric varieties. We prove this conjecture for cyclic arrangements by showing that these algebras are isomorphic to algebras appearing in work of Ozsvath-Szabo arXiv:1603.06559 in bordered Heegaard Floer homology arXiv:0810.0687. The proof of our conjecture in the cyclic case extends work of Karp-Williams arXiv:1608.08288 on sign variation and the combinatorics of the m=1 amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of gl(1|1).

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arXiv2019

Koszul Algebras and Flow Lattices

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph $Γ$ with a spanning tree $T$, we associate a finite dimensional Koszul algebra $A_{Γ,T}$. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated $A_{Γ,T}$ modules is isomorphic to the Euclidean lattice $\mathbb Z^{E(Γ)}$, and we describe the sublattices of integer cuts and integer flows on $Γ$ in terms of the representation theory of $A_{Γ,T}$. The grading on $A_{Γ,T}$ gives rise to $q$-analogs of the lattices of integer cuts and flows; these $q$-lattices depend non-trivially on the choice of spanning tree. We give a $q$-analog of the matrix-tree theorem, and prove that the $q$-flow lattice of $(Γ_1,T_1)$ is isomorphic to the $q$-flow lattice of $(Γ_2,T_2)$ if and only if there is a cycle preserving bijection from the edges of $Γ_1$ to the edges of $Γ_2$ taking the spanning tree $T_1$ to the spanning tree $T_2$. This gives a $q$-analog of a classical theorem of Caporaso-Viviani and Su-Wagner.

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arXiv2017

Braid groups of type ADE, Garside monoids, and the categorified root lattice

We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$, a baric structure on $\mathcal{K}$. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid $\mathbb{B}_{W.c}^\vee$. As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of $\mathbb{B}_W$ on $\mathcal{K}$. Finally, we use the compatibility of the baric and $t$-structures on $\mathcal{K}$ to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.

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arXiv2016

Annular Khovanov-Lee homology, braids, and cobordisms

We prove that the Khovanov-Lee complex of an oriented link, L, in a thickened annulus, A x I, has the structure of a bifiltered complex whose filtered chain homotopy type is an invariant of the isotopy class of L in A x I. Using ideas of Ozsvath-Stipsicz-Szabo as reinterpreted by Livingston, we use this structure to define a family of annular Rasmussen invariants that yield information about annular and non-annular cobordisms. Focusing on the special case of annular links obtained as braid closures, we use the behavior of the annular Rasmussen invariants to obtain a necessary condition for braid quasipositivity and a sufficient condition for right-veeringness.

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arXiv2016

Khovanov's Heisenberg category, moments in free probability, and shifted symmetric functions

We establish an isomorphism between the center of the Heisenberg category defined by Khovanov and the algebra $Λ^*$ of shifted symmetric functions defined by Okounkov-Olshanski. We give a graphical description of the shifted power and Schur bases of $Λ^*$ as elements of the center, and describe the curl generators of the center in the language of shifted symmetric functions. This latter description makes use of the transition and co-transition measures of Kerov and the noncommutative probability spaces of Biane.

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arXiv2016

On the 2-linearity of the free group

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag modules and explore the connection between gradings on the zigzag algebra and monoids in $F_n$. We use this connection to give homological constructions of the standard and Bessis dual word length metrics on the free group.

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arXiv2015

Annular Khovanov homology and knotted Schur-Weyl representations

Let L be a link in a thickened annulus. We show that its sutured annular Khovanov homology carries an action of the exterior current algebra of the Lie algebra sl_2. When L is an m-framed n-cable of a knot K in the three-sphere, its sutured annular Khovanov homology carries a commuting action of the symmetric group S_n. One therefore obtains a "knotted" Schur-Weyl representation that agrees with classical sl_2 Schur-Weyl duality when K is the Seifert-framed unknot.

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arXiv2015

W-algebras from Heisenberg categories

The trace (or zeroth Hochschild homology) of Khovanov's Heisenberg category is identified with a quotient of the algebra W_{1+\infty}. This induces an action of W_{1+\infty} on symmetric functions.

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