Spherical Twists Research Articles

Weyl symmetry for curve counting invariants via spherical twists

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of A 1 A_1 -singularities, our results match the rationality of the associated Calabi–Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.

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

We study actions of spherical twists on 2-Calabi–Yau categories with a Bridgeland stability condition. In these categories, we describe how to reduce the phase spread of a spherical object using stable spherical twists. In 2-Calabi–Yau quiver categories, we describe how to construct all spherical stable objects by applying simple spherical twists to the simple objects. As an application of this idea, we prove that for 2-Calabi–Yau categories associated to ADE quivers, all spherical objects lie in the braid group orbit of a simple object. We also give a new proof of the fact that the space of Bridgeland stability conditions is connected for these categories.

On the Composition of Two Spherical Twists

Abstract Segal [17] proved that any autoequivalence of an enhanced triangulated category can be realised as a spherical twist. However, when exhibiting an autoequivalence as a spherical twist one has various choices for the source category of the spherical functor. We describe a construction that realises the composition of two spherical twists as the twist around a single spherical functor whose source category semi-orthogonally decomposes into the source categories for the spherical functors we started with. We give a description of the cotwist for this spherical functor and prove, in the special case when our starting twists are around spherical objects, that the cotwist is the Serre functor (up to a shift). We finish with an explicit treatment for the case of $\mathbb {P}$-objects.

Computation of categorical entropy via spherical functors

We study the relationship between the categorical entropy of the twist and cotwist functors along a spherical functor. In particular, we prove the categorical entropy of the twist functor coincides with that of the cotwist functor if the essential image of the right adjoint functor of the spherical functor contains a split-generator. We also see our results to generalize the computations of the categorical entropy of spherical twists and P $\mathbb {P}$ -twists by Ouchi and Fan. As an application, we apply our results to the Gromov–Yomdin-type conjecture by Kikuta–Takahashi.

Groups generated by two twists along spherical sequences

AbstractWe describe all groups that can be generated by two twists along spherical sequences in an enhanced triangulated category. It will be shown that with one exception such a group is isomorphic to an abelian group generated by not more than two elements, the free group on two generators or the braid group of one of the types $A_2$ , $B_2$ and $G_2$ factorised by a central subgroup. The last mentioned subgroup can be nontrivial only if some specific linear relation between length and sphericity holds. The mentioned exception can occur when one has two spherical sequences of length 3 and sphericity 2. In this case the group generated by the corresponding two spherical twists can be isomorphic to the nontrivial central extension of the symmetric group on three elements by the infinite cyclic group. Also we will apply this result to give a presentation of the derived Picard group of selfinjective algebras of the type $D_4$ with torsion 3 by generators and relations.

Shuffling functors and spherical twists on [formula omitted

For a semisimple complex Lie algebra g, the BGG category O is of particular interest in representation theory. It is known that Irving's shuffling functors Shw, indexed by elements w∈W of the Weyl group, induce an action of the braid group BW associated to W on the derived categories Db(Oλ) of blocks of O.We show that for maximal parabolic subalgebras p of sln corresponding to the parabolic subgroup Wp=Sn−1×S1 of Sn, the derived shuffling functors LShsi are instances of Seidel and Thomas' spherical twist functors. Namely, we show that certain parabolic indecomposable projectives Pp(w) are spherical objects, and the associated twist functors are naturally isomorphic to LShw[1] as auto-equivalences of Db(Op).We give an overview of the main properties of the BGG category O, the construction of shuffling and spherical twist functors, and give some examples how to determine images of both. To this end, we employ the equivalence of blocks of O and the module categories of certain path algebras.

Categorical polynomial entropy

For classical dynamical systems, the polynomial entropy serves as a refined invariant of the topological entropy. In the setting of categorical dynamical systems, that is, triangulated categories endowed with an endofunctor, we develop the theory of categorical polynomial entropy, refining the categorical entropy defined by Dimitrov, Haiden, Katzarkov, and Kontsevich. We justify this notion by showing that for an automorphism of a smooth projective variety, the categorical polynomial entropy of the pullback functor on the derived category coincides with the polynomial growth rate of the induced action on cohomology. We also establish in general a Yomdin-type lower bound for the categorical polynomial entropy of an endofunctor in terms of the induced endomorphism on the numerical Grothendieck group of the category. As examples, we compute the categorical polynomial entropy for some standard functors like shifts, Serre functors, tensoring line bundles, automorphisms, spherical twists, P-twists, and so on, illustrating clearly how categorical polynomial entropy refines the study of categorical entropy and enables us to study the phenomenon of categorical trichotomy. A parallel theory of polynomial mass growth rate is developed in the presence of Bridgeland stability conditions.

Categorical vs Topological Entropy of Autoequivalences of Surfaces

In this paper, we give an example of an autoequivalence with positive categorical entropy (in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich) for any surface containing a (-2)-curve. Then we show that this equivalence gives another counter-example to a conjecture proposed by Kikuta and Takahashi. In a second part, we study the action on cohomology induced by spherical twists composed with standard autoequivalences on a surface S and show that their spectral radii correspond to the topological entropy of the corresponding automorphisms of S.

Exceptional sequences and spherical modules for the Auslander algebra of TEXTBACKSLASHmathboldTEXTBACKSLASHk[x]∕(xt)

We classify spherical modules and full exceptional sequences of modules over the Auslander algebra of $k[x]/(x^t)$. We categorify the left and right symmetric group actions on these exceptional sequences to two braid group actions: of spherical twists along simple modules, and of right mutations. In particular, every such exceptional sequence is obtained by spherical twists from a standard sequence, and likewise for right mutations.

On entropy of spherical twists

In this paper, we compute categorical entropy of spherical twists. In particular, we prove that the Gromov–Yomdin-type conjecture holds for spherical twists. Moreover, we construct counterexamples of Gromov–Yomdin type conjecture for K3 surfaces modifying Fan’s construction for even higher-dimensional Calabi–Yau manifolds. The appendix, by Arend Bayer, shows the nonemptiness of complements of a number of spherical objects in the derived categories of K3 surfaces.

Perverse sheaves of categories and some applications

We study perverse sheaves of categories their connections to classical algebraic geometry. We show how perverse sheaves of categories encode naturally derived categories of coherent sheaves on P1 bundles, semiorthogonal decompositions, and relate them to a recent proof of Segal that all autoequivalences of triangulated categories are spherical twists. Furthermore, we show that perverse sheaves of categories can be used to represent certain degenerate Calabi–Yau varieties.

Stability and local minimality of spherical harmonic twists u={mathbf {Q}}(|x|) x|x|^{-1}, positivity of second variations and conjugate points on mathbf{SO}(n)

In this paper we discuss the stability and local minimising properties of spherical twists that arise as solutions to the harmonic map equation HME[u;Xn,Sn-1]:=Δu+|∇u|2u=0inXn,|u|=1inXn,u=φon∂Xn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathbf{HME}[u;\\, \\mathbb {X}^n, \\mathbb {S}^{n-1}] :=\\left\\{ \\begin{array}{ll} \\Delta u + |\\nabla u|^2 u =0 &{} \\qquad \\text { in } \\mathbb {X}^n, \\\\ |u|=1 &{}\\qquad \\text { in } \\mathbb {X}^n , \\\\ u = \\varphi &{}\\qquad \\text { on } \\partial \\mathbb {X}^n, \\end{array}\\right. \\end{aligned}$$\\end{document}by way of examining the positivityof the second variation of the associated Dirichlet energy. Here, following [31], by a spherical twist we mean a map u in mathscr {W}^{1,2}(mathbb {X}^n, mathbb {S}^{n-1}) of the form x mapsto {mathbf {Q}}(|x|)x|x|^{-1} where {mathbf {Q}}={mathbf {Q}}(r) lies in mathscr {C}([a, b], mathbf{SO}(n)) and {mathbb {X}}^n={x in mathbb {R}^n : a<|x|<b} (nge 2). It is shown that subject to a structural condition on the twist path the energy at the associated spherical twist solution to the system has a positive definite second variation and subsequently proven to furnish a strong local energy minimiser.A detailed study of Jacobi fields and conjugate points along the twist path {mathbf {Q}}(r)=mathrm{exp}(mathscr {G}(r) {mathbf {H}}) and geodesics on mathbf{SO}(n) is undertaken and its remarkable implication and interplay on the minimalityof spherical harmonic twists exploited.

Higher Zigzag Algebras

Given a Koszul algebra of finite global dimension we define its higher zigzag algebra as a twisted trivial extension of the Koszul dual. If our original algebra is the path algebra of a tree-type quiver, this construction recovers the zigzag algebras of Huerfano-Khovanov. We study examples of higher zigzag algebras coming from Iyama's type A higher representation finite algebras, give their presentations by quivers and relations, and describe relations between spherical twists acting on their derived categories. We connect this to the McKay correspondence in higher dimensions: if G is a finite abelian subgroup of SL_{d+1} then these relations occur between spherical twists for G -equivariant sheaves on affine (d+1) -space.

Contractible stability spaces and faithful braid group actions

We prove that any `finite-type' component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi--Yau-$N$ category $\mathcal{D}(\Gamma_N Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $\operatorname{Br}(Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $\operatorname{Br}(\Gamma_N Q)$ is isomorphic to $\operatorname{Br}(Q)$. This generalises Brav-Thomas' result for the $N=2$ case. Other classes of triangulated categories with finite-type components in their stability spaces include locally-finite triangulated categories with finite rank Grothendieck group and discrete derived categories of finite global dimension.

Geodesics on ${\bf SO}(n)$ and a class of spherically symmetric maps as solutions to a nonlinear generalised harmonic map problem

We address questions on existence, multiplicity as well as qualitative features including rotational symmetry for certain classes of geometrically motivated maps serving as solutions to the nonlinear system $$ \begin{cases} -\text{\rm div}[ F'(|x|,|\nabla u|^2) \nabla u] = F'(|x|,|\nabla u|^2) |\nabla u|^2 u &amp;\text{in } \mathbb{X}^n,\\ |u| = 1 &amp;\text{in } \mathbb{X}^n ,\\ u = \varphi &amp;\text{on } \partial \mathbb{X}^n. \end{cases} $$% Here $\varphi \in \mathscr{C}^\infty(\partial {\mathbb{X}}^n, \mathbb S}^{n-1})$ is a suitable boundary map, $F'$ is the derivative of $F$ with respect to the second argument, $u \in W^{1,p}(\mathbb{X}^n, \mathbb S}^{n-1})$ for a fixed $1&lt; p&lt; \infty$ and ${\mathbb{X}}^n=\{x \in \mathbb R^n : a&lt; |x|&lt; b\}$ is a generalised annulus. Of particular interest are spherical twists and whirls, where following \cite{Taheri2012}, a spherical twist refers to a rotationally symmetric map of the form $u\colon x \mapsto \rom{Q}(|x|)x|x|^{-1}$ with $\rom{Q}$ some suitable path in $\mathscr{C}([a, b], {\rm SO}(n))$ and a whirl has a similar but more complex structure with only $2$-plane symmetries. We establish the existence of an infinite family of such solutions and illustrate an interesting discrepancy between odd and even dimensions.

Spherical Twists as the σ2-Harmonic Maps from n-Dimensional Annuli into S n − 1 $\mathbb {S}^{n-1}$

Spherical Twists as the σ2-Harmonic Maps from n-Dimensional Annuli into S n − 1 $\mathbb {S}^{n-1}$

Torsion exceptional sheaves on weak del Pezzo surfaces of Type A

We investigate torsion exceptional sheaves on a weak del Pezzo surface of degree greater than two whose anticanonical model has at most An-singularities. We show that every torsion exceptional sheaf can be obtained from a line bundle on a (−1)-curve by spherical twists.

Braid groups and quiver mutation

We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A and D these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.

All Autoequivalences are Spherical Twists

In this short note we observe that, for purely formal reasons, any autoequivalence can be constructed as a twist around a spherical functor. As an example, we show how the P-twists constructed by Huybrechts and Thomas can be formulated as spherical twists.

Autoequivalences of derived categories via geometric invariant theory

We study autoequivalences of the derived category of coherent sheaves of a variety arising from a variation of GIT quotient. We show that these autoequivalences are spherical twists, and describe how they result from mutations of semiorthogonal decompositions. Beyond the GIT setting, we show that all spherical twist autoequivalences of a dg-category can be obtained from mutation in this manner.Motivated by a prediction from mirror symmetry, we refine the recent notion of “grade restriction rules” in equivariant derived categories. We produce additional derived autoequivalences of a GIT quotient and propose an interpretation in terms of monodromy of the quantum connection. We generalize this observation by proving a criterion under which a spherical twist autoequivalence factors into a composition of other spherical twists.