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Stability conditions and Stokes factors. (English) Zbl 1239.14008

Let \(\mathcal A\) be an abelian category of finite-dimensional modules over a finite-dimensional associative \(\mathbb{C}\)-algebra and \(\mathrm{Stab}({\mathcal A})\) the manifold of stability conditions on it. D. Joyce [“Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds”, Geom. Topol. 11, 667–725 (2007; Zbl 1141.14023)] constructed holomorphic generating functions for \(\mathrm{Stab}({\mathcal A})\) combining invariants for stability conditions.
In this paper, the authors explore this construction further, by showing that a stability condition \(Z\) on \(\mathcal A\) can be naturally interpreted as defining Stokes data for an irregular connection on \({\mathbb P}^1\) with values in the Ringel–Hall Lie algebra of \(\mathcal A\). The wall and chamber structure on the space of stability conditions \(\mathrm{Stab}({\mathcal A})\) corresponds to the discontinuous behaviour, with respect to Stokes rays, of the Stokes factors of an isomonodromic family of irregular connections.
The authors prove indeed that, for an element \(Z\) in \(\mathcal A\), there exists a connection \(\nabla_{Z}\) whose Stokes factor corresponding to a Stokes ray \(l = {\mathbb R}_{>0}e^{i \pi \phi}\) is the characteristic function of \(Z\)-semistable modules of phase \(\phi\). Equivalently, the Stokes multipliers \(S_+\) and \(S_-\) relative to the ray \(r = {\mathbb R}_{>0}\) are the function \(1_{\mathcal A}\) (taking the value 1 on any module) and the identity respectively. In particular, under this perspective, the holomorphic functions defined by Joyce arise as defining the residues of the family of connections.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)

Citations:

Zbl 1141.14023
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References:

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