A theory of generalized Donaldson–Thomas invariants

By D. Joyce, Yinan Song. 2008

436 citations recorded by the source corpus.

This book studies generalized Donaldson-Thomas invariants $\bar{DT}{}^\alpha(\tau)$. They are rational numbers which 'count' both $\tau$-stable and $\tau$-semistable coherent sheaves with Chern character $\alpha$ on $X$; strictly $\tau$-semistable sheaves must be counted with complicated rational weights. The $\bar{DT}{}^\alpha(\tau)$ are defined for all classes $\alpha$, and are equal to $DT^\alpha(\tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wall-crossing formula under change of stability condition $\tau$. To prove all this, the authors study the local structure of the moduli stack $\mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $\mathfrak M$ may be written locally as $\mathrm{Crit}(f)$ for $f:U\to{\mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_\mathfrak M$. They compute the invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $\mathrm{mod}$-$\mathbb{C}Q\backslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/232be2a208cccb7fb09bc13dfd5e5fa18bc11032

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Citations in the corpus

Curves on K3 surfaces and modular forms (2010, methodology): “the proof of Theorem 33 is the expression of the moduli space P → Blocally as Crit(φ) for a C-invariant holomorphic function φ. There are two approaches to the question: via the methods of Joyce-Song [16] or via the announced work of BehrendGetzler. Since we do not need Theorem 33, we omit the discu”
Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds (2015, methodology): “algebraic and symplectic Gromov–Witten invariants, and Donaldson–Thomas invariants [26, 33]).”
Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants (2010, background): “cuss in the paper. Finally, we remark that it would be highly desirable to relate the classical limits of our motivic Donaldson-Thomas invariants with the invariants introduced by Joyce and Song (see [29]). More precisely, we expect that the 5 category of coherent sheaves used in the loc. cit. carri”
Quiver algebras as Fukaya categories (2013, methodology): “4 of [5] implies that the Donaldson-Thomas invariants of Yφ, defined with respect to the stability condition associated to φ, count such special Lagrangian submanifolds, the existence and numerics of which are therefore governed by the Joyce-Song and Kontsevich-Soibelman wall-crossing formulae [22, ”
Scattering diagrams, Hall algebras and stability conditions (2016, methodology): “For the most general and abstract form of our result this is defined using the motivic Hall algebra [2, 10, 12, 16] of the category of representations of (Q, I).”
A Darboux theorem for derived schemes with shifted symplectic structure (2013, neutral): “Related results are important in Donaldson–Thomas theory [18–20].”
Wall crossing from Boltzmann black hole halos (2010, methodology): “In their work on Donaldson-Thomas invariants for coherent sheaves on a Calabi-Yau threefold X [14] (which presumably count D6-D4-D2-D0 bound states in type IIA string theory compactified on X ), Joyce and Song give a fully explicit expression for the rational DT invariants Ω̄− on one side of the wal”
Symmetries and stabilization for sheaves of vanishing cycles (2012, background): “In a paper written independently of our programme [9, 11, 23], Kiem and Li [31] have recently proved an analogue of Corollary 6.11 by complex analytic methods, beginning from Joyce and Song’s result [24, Th. 5.4], proved using gauge theory, thatMαst(τ) is locally isomorphic to Crit(f) as a complex a”
Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, methodology): “weighted) Euler characteristic in deﬁning the invariants In,β, Pn,β. As discussed in the arXiv version of [37, Theorem 8.11], the formulas (28), (29) can be proved by combining the work of Joyce-Song [19] and Behrend-Getzler’s announced result [6]. The latter result is the 22 YUKINOBUTODA derived ca”
Curve counting theories via stable objects Ⅱ : DT/ncDT flop formula (2009, background): “ Main result Let f: X→Y be a ﬂopping contraction from a smooth projective Calabi-Yau 3-fold X to a singular 3-fold Y, and φ: X+ 99K Xits ﬂop as in the diagram (2). In order to apply Joyce-Song’s work [12] to our situation, we assume the following conjecture. Conjecture 1.2. Let Mbe the moduli stack ”