Searcharxiv⌕ Search

arXiv subjects

Fabian Haiden

Publications and source records attributed to Fabian Haiden.

18 recordsLinked to original sources
arXiv2026

Towards Categorical K\"ahler Geometry

We outline the contours of an emerging theory of K\"ahler metrics in derived noncommutative geometry. This is a refinement of the theory of Bridgeland stability conditions encoding underlying differential-geometric structures. We propose elements of such a structure in both Archimedean and non-Archimedean settings, including metrized objects, mass measures satisfying a BPS inequality, harmonic metrics, minimizing flows, and complexified K\"ahler potentials. We develop the framework through examples and constructions involving Fukaya categories, quiver representations and associated C$^*$-algebras, spectral networks, and comonadic adjunctions of stable $\infty$-categories.

math.AG↗
arXiv2026

Gluing abelian categories and stability conditions

We study the support property for stability conditions obtained by gluing along semiorthogonal decompositions. Our main tool is a new gluing construction for abelian categories along a bimodule: under natural exactness assumptions the glued category is abelian, and it recovers the heart obtained by gluing $t$-structures. This gives a concrete description of objects in the glued heart which we use to prove the support property given a phase gap for glued semistable objects. In particular, we prove the support property assuming Collins--Polishchuk phase bounds. As applications, we construct stability conditions on dg-comma categories, including examples related to augmented curves in the sense of Alexeev and Kuznetsov, and analyze the behavior of the construction under mutations of semiorthogonal decompositions.

math.AG↗
arXiv2025

A counterexample to the Jordan-Hölder property for polarizable semiorthogonal decompositions

We show that the Jordan-Hölder property fails for polarizable semiorthogonal decompositions -- those where every factor admits a Bridgeland stability condition. Counterexamples exist among Fukaya categories of surfaces and bounded derived categories of smooth projective varieties. Furthermore, we give an example of a smooth and proper pre-triangulated dg category with positive rank Grothendieck group which does not admit a stability condition.

math.RT↗
arXiv2024

Counting in Calabi--Yau categories, with applications to Hall algebras and knot polynomials

We show that homotopy cardinality -- a priori ill-defined for many dg-categories, including all periodic ones -- has a reasonable definition for even-dimensional Calabi--Yau (evenCY) categories and their relative generalizations (under appropriate finiteness conditions). As a first application we solve the problem of defining an intrinsic Hall algebra for degreewise finite pre-triangulated dg-categories in the case of oddCY categories. We compare this definition with Toën's derived Hall algebras (in case they are well-defined) and with other approaches based on extended Hall algebras and central reduction, including a construction of Hall algebras associated with Calabi--Yau triples of triangulated categories. For a category equivalent to the root category of a 1CY abelian category $\mathcal A$, the algebra is shown to be isomorphic to the Drinfeld double of the twisted Ringel--Hall algebra of $\mathcal A$, thus resolving in the Calabi--Yau case the long-standing problem of realizing the latter as a Hall algebra intrinsically defined for such a triangulated category. Our second application is the proof of a conjecture of Ng--Rutherford--Shende--Sivek, which provides an intrinsic formula for the ruling polynomial of a Legendrian knot $L$, and its generalization to Legendrian tangles, in terms of the augmentation category of $L$.

math.QA↗
arXiv2024

The stability manifold of $E{\times} E{\times} E$

We determine a full component of the space of stability conditions on $D^b(E^3)$ where $E$ is an elliptic curve without complex multiplication. The component has complex dimension 14 and a very concrete description in terms of alternating trilinear forms. This confirms a conjecture of Kontsevich, motivated by homological mirror symmetry, in the case of dimension $3$.

math.AG↗
arXiv2024

Perverse schobers, stability conditions and quadratic differentials II: relative graded Brauer graph algebras

We introduce a class of dg-algebras which generalize the classical Brauer graph algebras. They are constructed from mixed-angulations of surfaces and often admit a (relative) Calabi--Yau structure. We discovered these algebras through two very distinct routes, one involving perverse schobers whose stalks are cyclic quotients of the derived categories of relative Ginzburg algebras, and another involving deformations of partially wrapped Fukaya categories of surfaces. Applying the results of our previous work arXiv:2303.18249, we describe the spaces of stability conditions on the derived categories of these algebras in terms of spaces of quadratic differentials.

math.RT↗
arXiv2023

On pseudo-Anosov autoequivalences

Motivated by results of Thurston, we prove that any autoequivalence of a triangulated category induces a filtration by triangulated subcategories, provided the existence of Bridgeland stability conditions. The filtration is given by the exponential growth rate of masses under iterates of the autoequivalence, and only depends on the choice of a connected component of the stability manifold. We then propose a new definition of pseudo-Anosov autoequivalences, and prove that our definition is more general than the one previously proposed by Dimitrov, Haiden, Katzarkov, and Kontsevich. We construct new examples of pseudo-Anosov autoequivalences on the derived categories of quintic Calabi-Yau threefolds and quiver Calabi-Yau categories. Finally, we prove that certain pseudo-Anosov autoequivalences on quiver 3-Calabi-Yau categories act hyperbolically on the space of Bridgeland stability conditions.

math.AG↗
arXiv2023

Perverse schobers, stability conditions and quadratic differentials I

We develop a unified approach for identifying spaces of stability conditions of triangulated categories arising from weighted marked surfaces with moduli spaces of quadratic differentials. This identification is based on the use of perverse schobers (perverse sheaves of triangulated categories) and a notion of positive arc system kit on a perverse schober $\mathcal F$, which provides a systematic way of assigning to a graded curve on the surface a global section of $\mathcal F$. This assignment allows us to identify mixed-angulations and their flips with finite-length hearts and their tilts. As an application we obtain a generalization of the results of Bridgeland--Smith to quadratic differentials with arbitrary singularity type (zero/pole/exponential).

math.RT↗
arXiv2023

3-d Calabi--Yau categories for Teichmüller theory

For $g,n\geq 0$ a 3-dimensional Calabi-Yau $A_\infty$-category $\mathcal C_{g,n}$ is constructed such that a component of the space of Bridgeland stability conditions, $\mathrm{Stab}(\mathcal C_{g,n})$, is a moduli space of quadratic differentials on a genus $g$ surface with simple zeros and $n$ simple poles. For a generic point in the moduli space the corresponding quantum/refined Donaldson--Thomas invariants are computed in terms of counts of finite-length geodesics on the flat surface determined by the quadratic differential. As a consequence, these counts satisfy wall-crossing formulas.

math.AG↗
arXiv2023

Braid group actions on branched coverings and full exceptional sequences

We relate full exceptional sequences in Fukaya categories of surfaces or equivalently in derived categories of graded gentle algebras to branched coverings over the disk, building on a previous classification result of the first and third author. This allows us to apply tools from the theory of branched coverings such as Birman--Hilden theory and Hurwitz systems to study the natural braid group action on exceptional sequences. As an application, counterexamples are given to a conjecture of Bondal--Polishchuk on the transitivity of the braid group action on full exceptional sequences in a triangulated category.

math.RT↗
arXiv2021

Spectral networks and stability conditions for Fukaya categories with coefficients

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto--Moore--Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an $A_2$ quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

math.AG↗
arXiv2021

Legendrian skein algebras and Hall algebras

We compare two associative algebras which encode the "quantum topology" of Legendrian curves in contact threefolds of product type $S\times\mathbb R$. The first is the skein algebra of graded Legendrian links and the second is the Hall algebra of the Fukaya category of $S$. We construct a natural homomorphism from the former to the latter, which we show is an isomorphism if $S$ is a disk with marked points and injective if $S$ is the annulus.

math.SG↗
arXiv2020

Semistability, modular lattices, and iterated logarithms

We provide a complete description of the asymptotics of the gradient flow on the space of metrics on any semistable quiver representation. This involves a recursive construction of approximate solutions and the appearance of iterated logarithms and a limiting filtration of the representation. The filtration turns out to have an algebraic definition which makes sense in any finite length modular lattice. This is part of a larger project by the authors to study iterated logarithms in the asymptotics of gradient flows, both in finite and infinite dimensional settings.

math.RT↗
arXiv2020

Flags and tangles

We show that two constructions yield equivalent braided monoidal categories. The first is topological, based on Legendrian tangles and skein relations, while the second is algebraic, in terms of chain complexes with complete flag and convolution-type products. The category contains Iwahori--Hecke algebras of type $A_n$ as endomorphism algebras of certain objects.

math.QA↗
arXiv2018

Iterated logarithms and gradient flows

We consider applications of the theory of balanced weight filtrations and iterated logarithms, initiated in arXiv:1706.01073, to PDEs. The main result is a complete description of the asymptotics of the Yang--Mills flow on the space of metrics on a holomorphic bundle over a Riemann surface. A key ingredient in the argument is a monotonicity property of the flow which holds in arbitrary dimension. The A-side analog is a modified curve shortening flow for which we provide a heuristic calculation in support of a detailed conjectural picture.

math.RT↗
arXiv2015

Flat surfaces and stability structures

We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is achieved by new methods involving the complete classification of objects in these categories, which are defined in an elementary way. We also introduce a number of tools to deal with surfaces of infinite area, where structures similar to those in cluster algebra appear.

math.AG↗
arXiv2013

Dynamical systems and categories

We study questions motivated by results in the classical theory of dynamical systems in the context of triangulated and A-infinity categories. First, entropy is defined for exact endofunctors and computed in a variety of examples. In particular, the classical entropy of a pseudo-Anosov map is recovered from the induced functor on the Fukaya category. Second, the density of the set of phases of a Bridgeland stability condition is studied and a complete answer is given in the case of bounded derived categories of quivers. Certain exceptional pairs in triangulated categories, which we call Kronecker pairs, are used to construct stability conditions with density of phases. Some open questions and further directions are outlined as well.

math.CT↗