Tilting and refined Donaldson-Thomas invariants (Q405842)

Let \(Q\) be a quiver without loops or \(2\)-cycles and let \(W\) be a non degenerate potential. The refined Donaldson-Thomas invariant associated to the Ginzburg algebra of \((Q,W)\) is defined in terms of stable representations of \(Q\), hence in principle it depends on the stability condition, i.e., on the central charge in terms of Bridgeland stability.NEWLINENEWLINEGiven that the category of representations of \(Q\) is the heart of the standard \(t\)-structure on its derived category we can apply Bridgeland theory to study tilting for this heart, performing a mutation method coming from physics. Then, given a heart \(\mathcal{A}\) with finitely many simple objects and a discrete central charge, Proposition 4.1 shows that we can construct a sequence of simple tilts from \(\mathcal{A}\) to \(\mathcal{A}[-1]\). This argument is applied for the quiver with potential \((Q,W)\) in Section \(5\), showing in addition that its Jacobi algebra is finite-dimensional.NEWLINENEWLINEFinally it is proven that refined DT invariants, conjectured to be independent of the chosen central charge by \textit{M. Kontsevich} and \textit{Y. Soibelman} [``Stability structures, motivic Donaldson--Thomas invariants and cluster transformations'', preprint, \url{arXiv:0811.2435}].NEWLINENEWLINEIt is expected that this approach can be generalized to a wider class of 3-dimensional Calabi-Yau categories considered in [\url{arXiv:0811.2435}], are indeed independent for discrete central charges with finitely many stable objects.