arXiv subjects

Severin Barmeier

Publications and source records attributed to Severin Barmeier.

15 recordsLinked to original sources

arXiv2026

Fukaya categories of orbifold surfaces in representation theory

We give an introduction to partially wrapped Fukaya categories of surfaces with orbifold singularities. Dissecting an orbifold surface $\mathbf S$ into polygons, certain dissections give rise to formal generators, inducing a triangulated equivalence between the derived Fukaya category of $\mathbf S$ and the perfect derived category of a graded associative algebra. This provides a geometric means for obtaining associative algebras -- conjecturally all -- which are derived equivalent to skew-gentle algebras. We include a new perspective on the partially wrapped Fukaya category of an orbifold disk which serves as a local model for the Fukaya categories of general orbifold surfaces. This perspective yields an equivalence between the perfect derived category of a quiver of type $\mathrm D_{n+1}$ and the perfect derived category of a graded quiver of type $\widetilde{\mathrm A}_{n-1}$, the latter being equipped with quadratic zero relations and a nontrivial A$_\infty$ structure. This equivalence elucidates the relationship between skew-gentle algebras and orbifold surfaces, and the role of deformation theory in this relationship.

math.RT↗

arXiv2025

Deformations of partially wrapped Fukaya categories of surfaces

We give a complete description of the A$_\infty$ deformation theory of partially wrapped Fukaya categories of graded surfaces. We show that any abstract A$_\infty$ deformation is "geometric", namely it is equivalent to the partially wrapped Fukaya category of an orbifold surface obtained as a partial compactification of the original surface. For certain genus 0 surfaces, these deformations are generically Fukaya categories of compact pillowcases. We introduce the notion of a weak dual and use unbounded twisted complexes to overcome the curvature problem that naturally arises when some of the boundary components are fully wrapped. Our results provide a first account of the relationship between A$_\infty$ deformations of Fukaya categories and partial compactifications, as advocated in P. Seidel's ICM 2002 address, in the presence of stop data. All of our results also hold when the original surface has finitely many order 2 orbifold points.

math.SG↗

arXiv2024

Partially wrapped Fukaya categories of orbifold surfaces

We give a complete description of partially wrapped Fukaya categories of graded orbifold surfaces with stops. We show that a construction via global sections of a natural cosheaf of A$_\infty$ categories on a Lagrangian core of the surface is equivalent to a global construction via the (equivariant) orbit category of a smooth cover. We therefore establish the local-to-global properties of partially wrapped Fukaya categories of orbifold surfaces closely paralleling a proposal by Kontsevich for Fukaya categories of smooth Weinstein manifolds. From the viewpoint of Weinstein sectorial descent in the sense of Ganatra, Pardon and Shende, our results show that orbifold surfaces also have Weinstein sectors of type $\mathrm D$ besides the type $\mathrm A$ or type $\widetilde{\mathrm A}$ sectors on smooth surfaces. We describe the global sections of the cosheaf explicitly for any generator given by an admissible dissection of the orbifold surface and we give a full classification of the formal generators which arise in this way. This shows in particular that the partially wrapped Fukaya category of an orbifold surface can always be described as the perfect derived category of a graded associative algebra. We conjecture that associative algebras obtained from dissections of orbifold surfaces form a new class of associative algebras closed under derived equivalence.

math.SG↗

arXiv2022

A$_\infty$ deformations of extended Khovanov arc algebras and Stroppel's conjecture

Extended Khovanov arc algebras $\mathrm{K}_m^n$ are graded associative algebras which naturally appear in a variety of contexts, from knot and link homology, low-dimensional topology and topological quantum field theory to representation theory and symplectic geometry. C. Stroppel conjectured in her ICM 2010 address that the bigraded Hochschild cohomology groups of $\mathrm{K}_m^n$ vanish in a certain range, implying that the algebras $\mathrm K_m^n$ admit no nontrivial A$_\infty$ deformations, in particular that the algebras are intrinsically formal. Whereas Stroppel's Conjecture is known to hold for the algebras $\mathrm K_m^1$ and $\mathrm K_1^n$ by work of Seidel and Thomas, we show that $\mathrm K_m^n$ does in fact admit nontrivial A$_\infty$ deformations with nonvanishing higher products for all $m, n \geq 2$. We describe both $\mathrm K_m^n$ and its Koszul dual concretely as path algebras of quivers with relations and give an explicit algebraic construction of A$_\infty$ deformations of $\mathrm K_m^n$ by using the correspondence between A$_\infty$ deformations of a Koszul algebra and filtered associative deformations of its Koszul dual. These deformations can also be viewed as A$_\infty$ deformations of Fukaya--Seidel categories associated to Hilbert schemes of surfaces based on recent work of Mak and Smith.

math.RT↗

arXiv2022

Strict quantization of polynomial Poisson structures

We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson structures of arbitrary degree. We give several examples of nonlinear Poisson structures and construct explicit formal star products whose deformation parameter can be evaluated to any real value of $\hbar$, giving strict quantizations on the space of analytic functions on $\mathbb R^d$ with infinite radius of convergence. We also address further questions such as continuity of the classical limit $\hbar \to 0$, compatibility with *-involutions, and the existence of positive linear functionals. The latter can be used to realize the strict quantizations as *-algebras of operators on a pre-Hilbert space which we demonstrate in a concrete example.

math.QA↗

arXiv2021

Towards a categorification of scattering amplitudes

Categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential is reported. Amplitudes for cubic theories are directly written down in terms of projectives of hearts of intermediate $t$-structures restricted to the cluster category of quiver representations, without recourse to geometry. It is shown that for theories with $ϕ^{m+2}$ potentials those corresponding to $m$-cluster categories are to be used. The case of generic polynomial potentials is treated and our results suggest the existence of a generalization of higher cluster categories which we call pseudo-periodic categories. An algorithm to obtain the projectives of hearts of intermediate $t$-structures for these types is presented.

hep-th↗

arXiv2021

Learning scattering amplitudes by heart

The canonical forms associated to scattering amplitudes of planar Feynman diagrams are interpreted in terms of masses of projectives, defined as the modulus of their central charges, in the hearts of certain $t$-structures of derived categories of quiver representations and, equivalently, in terms of cluster tilting objects of the corresponding cluster categories.

hep-th↗

arXiv2021

Deformations of categories of coherent sheaves via quivers with relations

We give an explicit combinatorial description of the deformation theory of the Abelian category of (quasi)coherent sheaves on any separated Noetherian scheme $X$ via the deformation theory of path algebras of quivers with relations, by using any affine open cover of $X$, or any tilting bundle on $X$, if available. We also give sufficient criteria for obtaining algebraizations of formal deformations, in which case the deformation parameters can be evaluated to a constant and the deformations can be compared to the original Abelian category on equal terms. We give concrete examples as well as applications to the study of noncommutative deformations of singularities.

math.AG↗

arXiv2021

Quantizations of local surfaces and rebel instantons

We construct explicit deformation quantizations of the noncompact complex surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons. We then show that the quantum instanton moduli space can be viewed as the étale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.

math.AG↗

arXiv2020

Deformation-obstruction theory for diagrams of algebras and applications to geometry

Let $X$ be a smooth complex algebraic variety and let $\operatorname{Coh} (X)$ denote its Abelian category of coherent sheaves. By the work of W. Lowen and M. Van den Bergh, it is known that the deformation theory of $\operatorname{Coh} (X)$ as an Abelian category can be seen to be controlled by the Gerstenhaber-Schack complex associated to the restriction of the structure sheaf $\mathcal O_X \vert_{\mathfrak U}$ to a cover of affine open sets. We construct an explicit $L_\infty$ algebra structure on the Gerstenhaber-Schack complex controlling the higher deformation theory of $\mathcal O_X \vert_{\mathfrak U}$ in case $X$ can be covered by two acyclic open sets, giving an explicit deformation-obstruction calculus for such deformations. Deformations of complex structures and deformation quantizations of $X$ are recovered as degenerate cases, as is shown by means of concrete examples.

math.QA↗

arXiv2020

Deformations of path algebras of quivers with relations

Let $A = \Bbbk Q / I$ be the path algebra of any finite quiver $Q$ modulo any two-sided ideal $I$ of relations and let $R$ be any reduction system satisfying the diamond condition for $I$. We introduce an intrinsic notion of deformation of reduction systems and show that there is an equivalence of deformation problems between deformations of the associative algebra $A$ and deformations of the reduction system $R$, the latter being controlled by a natural, explicit L$_\infty$ algebra. It follows in particular that any formal deformation of the associative multiplication on $A$ can, up to gauge equivalence, be given by a combinatorially defined star product, and the approach via reduction systems can be used to give a concrete and complete description of the deformation theory of $A$. For the polynomial algebra in a finite number of variables, this combinatorial star product can be described via bidifferential operators associated to graphs, which we compare to the graphs appearing in Kontsevich's universal quantization formula. Using the notion of admissible orders on the set of paths of the quiver $Q$, we give criteria for the existence of algebraizations of formal deformations, which we also interpret geometrically via algebraic varieties of reduction systems. In this context the Maurer-Cartan equation of the L$_\infty$ algebra can be viewed as a generalization of the Braverman-Gaitsgory criterion for Poincar\'e-Birkhoff-Witt deformations of Koszul algebras.

math.QA↗

arXiv2018

Classical deformations of noncompact surfaces and their moduli of instantons

We describe semiuniversal deformation spaces for the noncompact surfaces $Z_k := \operatorname{Tot} (\mathcal O_{\mathbb P^1} (-k))$ and prove that any nontrivial deformation $Z_k (τ)$ of $Z_k$ is affine. It is known that the moduli spaces of instantons of charge $j$ on $Z_k$ are quasi-projective varieties of dimension $2j-k-2$. In contrast, our results imply that the moduli spaces of instantons on any nontrivial deformation $Z_k (τ)$ are empty.

math.AG↗

arXiv2018

A Lie theoretical construction of a Landau-Ginzburg model without projective mirrors

We describe the Fukaya-Seidel category of a Landau-Ginzburg model LG(2) for the semisimple adjoint orbit of sl(2,C). We prove that this category is equivalent to a full triangulated subcategory of the category of coherent sheaves on the second Hirzebruch surface. We show that no projective variety can be mirror to LG(2), and that this remains so after compactification.

math.SG↗

arXiv2013

Deformations of the discrete Heisenberg group

We study deformations of the discrete Heisenberg group acting properly discontinuously on the Heisenberg group from the left and right and obtain a complete description of the deformation space.

math.GR↗

arXiv2013

Isomorphisms of Moduli Spaces

We give infinitely many new isomorphisms between moduli spaces of bundles on local surfaces and on local Calabi--Yau threefolds.

math.AG↗