Stability conditions on triangulated categories
Motivated by the results obtained by M. R. Douglas for Dirichlet branes in string theory, in the present paper the author introduces a general notion of stability on triangulated categories: If \(\mathcal{D}\) is a triangulated category, a stability condition on \(\mathcal{D}\) is a pair \((Z,\mathcal{P})\), where \(Z:K(\mathcal{D})\to \mathbb{C}\) is a group homomorphism, and \(\mathcal{P}=\{\mathcal{P}(\varphi)\mid \varphi\in\mathbb{R}\}\) is a set of full additive subcategories of \(\mathcal{D}\) which satisfy some natural conditions (e.g., it is proved in Lemma 5.2 that all subcategories \(\mathcal{P}(\varphi)\) are abelian). In the hypothesis that \(\mathcal{D}\) is essentially small, the class of all stability conditions on \(\mathcal{D}\) is a set. In Section 6 the author constructs a natural topology (which is induced by a natural metric, as it is proved in Section 8) on the set \(\text{Stab}(\mathcal{D})\) of locally-finite stability conditions on \(\mathcal{D}\) (this kind of stability conditions are defined in Definition 5.7). The main result of the paper is Theorem 1.2. This theorem states that if \(\mathcal{D}\) is a locally small triangulated category, for each connected component \(\Sigma\subset \text{Stab}(\mathcal{D})\) there are linear spaces \(V(\Sigma)\subset \text{Hom}_{\mathbb{Z}}(K(\mathcal{D}),\mathbb{C})\), with a well defined linear topology, and a local homeomorphism \(\mathcal{Z}:\Sigma\to V(\Sigma)\) which maps a stability condition \((Z,\mathcal{P})\) into \(Z\). This theorem is proved in Proposition 6.3 and Section 7 (Theorem 7.1).
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