Some new categorical invariants

TL;DR

This paper introduces new categorical invariants, explores stability conditions on derived categories of Kronecker quivers, and proposes a framework for non-commutative curve counting with connections to number theory and geometry.

Contribution

It defines a new notion of a norm on triangulated categories, studies stability spaces of Kronecker quivers, and develops a non-commutative curve-counting theory with potential links to classical mathematics.

Findings

Explicit description of stability space for $K(l)$ quivers.

Introduction of a norm on categories with phase gaps.

Proposal of a topology on categories and non-commutative curve counting.

Abstract

We introduce several notions and give examples. We prove that ${\rm Stab}(D^b(K(l)))\cong {\mathbb C}\times \mathcal H$ for $l\geq 3$, where $K(l)$ is $l$-Kronecker quiver. This is an example of SOD, where ${\rm Stab}( \langle \mathcal T_1,\mathcal T_2\rangle )\not \cong{\rm Stab}(\mathcal T_1)\times {\rm Stab}(\mathcal T_2)$. This example suggest a new notion of a norm, strictly increasing on $\{D^b(K(l))\}_{l\geq 2}$. To a triangulated category $\mathcal T$ which has property of a phase gap we attach a non-negative number $\Vert \mathcal T \Vert_{\varepsilon}$. Natural assumptions on a SOD imply $ \Vert \langle \mathcal T_1,\mathcal T_2\rangle \Vert_{\varepsilon}\geq {\rm max}\{ \Vert \mathcal T_1 \Vert_{\varepsilon}, \Vert\mathcal T_2 \Vert_{\varepsilon}\}$. Using this we define a topology on the set of equivalence classes of triangulated categories with a phase gap, where the set of discrete derived categories is a discrete subset and the rationality of a smooth surface $S$ ensures that $[D^b(point)] \in {\rm Cl}([D^b(S)])$. Viewing $D^b(K(l))$ as a non-commutative curve, we observe that it is reasonable to count non-commutative curves in any category in a small neighborhood of $D^b(K(l))$. Examples show that this idea (non-commutative curve-counting) opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi-Yau curve-counting, where the entities we count are a Calabi-Yau modification of $D^b(K(l))$. Finally we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm playing a role similar to the classical notion of degree of an extension in Galois theory.