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Joyce-Song wall crossing as an asymptotic expansion. (English) Zbl 1300.14057

In a general form one can view the Donaldson-Thomas invariants \(DT(\alpha, \sigma)\) as virtual counts of Calabi-Yau threefold categories with fixed class \(\alpha\), and Bridgeland stability condition \(\sigma\). The Joyce-Song formula gives the jump in \(DT(\alpha, \sigma)\) when \(\sigma\) crosses certain real codimension \(1\) subvarieties in the space of stability conditions. It is structured like an asymptotic expansion with sums over trees, but contains no coupling constant, which would discount contributions of trees on many vertices.
The author conjectures that the formula can be obtained from an actual asymptotic expansion of exponential holomorphic Darboux coordinates on a moduli space \(\mathcal{M}\) of singular Higgs bundles around the so-called semiflat coordinates. This expansion appears in the work of Gaiotto, Moore and Neitzke with terms indexed by labeled trees, and the coupling constant is the volume of Hitchin fibers of \(\mathcal{M}\) over the affine space of meromorphic quadratic differentials. The Donaldson-Thomas invariants arise in this context from the physically defined BPS spectrum \(\Omega(\gamma,u)\), where \(\gamma\) is a homology class and \(u\) parametrizes the base. The expansion is only valid away from the so-called wall of marginal stability, and the author conjectures that taking that into account would yield the Joyce-Song formula. This is verified for a class of \(SU(2)\) Seiberg-Witten gauge theories with \(0\leq N_f\leq3\), where \(\Omega(\gamma,u)\) can be defined rigorously in terms of semistable representations of a quiver related to \(\mathcal{M}\).
In addition, Gaiotto, Moore and Neitzke argued that the Kontsevich-Soibelman wall-crossing formula follows from a continuity condition for the Darboux coordinates, when they are described as a solution to an infinite-dimensional Riemann-Hilbert problem. Under the author’s conjecture this continuity condition written in terms of the asymptotic expansion gives exactly the Joyce-Song formula.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)

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