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, we generalise a result of Macr\`{i}--Mehrotra--Stellari by establishing a homeomorphism 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.Comment: 30 pages. Comments more than welcomed

A Thurston compactification of the space of stability conditions

We propose a compactification of the moduli space of Bridgeland stability conditions of a triangulated category. Under mild conditions on the triangulated category, we conjecture that this compactification is a real manifold with boundary, on which the action of the auto-equivalence group extends continuously. The key ingredient in the compactification is an embedding of the stability space into an infinite-dimensional projective space. We study this embedding in detail in the case of 2-Calabi--Yau categories associated to quivers, and prove our conjectures in the $A_2$ and $\widehat{A_1}$ cases. Central to our analysis is a detailed understanding of Harder--Narasimhan multiplicities and how they transform under auto-equivalences. We achieve this by introducing a structure called a Harder--Narasimhan automaton and constructing examples for $A_2$ and $\widehat{A_1}$.Comment: 40 pages, 11 figures. Comments welcome

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).Comment: 42 pages, eps figure

The Elliptic Hall algebra and the deformed Khovanov Heisenberg category

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann, specialized at $\sigma = \bar\sigma^{-1} = q$. A key step in the proof may be of independent interest: we show that the sum (over $n$) of the Hochschild homologies of the positive affine Hecke algebras $\mathrm{AH}_n^+$ is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the $q$-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.Comment: 49 pages, numerous figure