We prove wall-crossing formulas for the motivic invariants of the moduli spaces of framed objects in the ind-constructible abelian categories. Developed techniques are applied in the case of the motivic Donaldson-Thomas invariants of quivers with potentials. Another application is a new proof of the formula for the motivic invariants of smooth models of quiver moduli spaces.
We study wall-crossing phenomena in the McKay correspondence. Craw-Ishii show that every projective crepant resolution of a Gorenstein abelian quotient singularity arises as a moduli space of $θ$-stable representations of the McKay quiver. The stability condition $θ$ moves in a vector space with a chamber decomposition in which (some) wall-crossings capture flops between different crepant resolutions. We investigate where chambers for certain resolutions with Hilbert scheme-like moduli interpretations - iterated Hilbert schemes, or 'Hilb of Hilb' - sit relative to the principal chamber definin…
For an arbitrary integer $r\geq 1$, we compute $r$-framed motivic PT and DT invariants of small crepant resolutions of toric Calabi-Yau $3$-folds, establishing a "higher rank" version of the motivic DT/PT wall-crossing formula. This generalises the work of Morrison and Nagao. Our formulae, in particular their relationship with the $r=1$ theory, fit nicely in the current development of higher rank refined DT invariants.
We compare the $K$-theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure-Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the $K$-theory stable bases coincide with the monodromy matrices of the quantum cohomology o…
Wall-crossing phenomena are ubiquitous in many problems of algebraic geometry and theoretical physics. Various ways to encode the relevant information and the need to track the changes under the variation of parameters lead to rather complicated transformation rules and non-trivial combinatorial problems. In this paper we propose a framework, reminiscent of collections and plethysms in the theory of operads, that conceptualizes those transformation rules. As an application we obtain new streamlined proofs of some existing wall-crossing formulas as well as some new formulas related to attractor…
We prove, under suitable conditions, that there exist wall-crossing and reduction morphisms for moduli spaces of stable log pairs in all dimensions as one varies the coefficients of the divisor.
Families of Bridgeland stability conditions induce families of stability data, wall-crossing structures and scattering diagrams on the motivic Hall algebra. These structures can be transferred to the quantum torus if the stability conditions of the family have global dimension at most 2. We show that geometric stability conditions on surfaces with nef anticanonical bundle have global dimension 2 and we study the resulting family of stability data. We formulate a conjecture relating this family to the family of stability data associated with a quiver with potential and we verify this conjecture…
We give a new proof of the following theorem: moduli spaces of stable complexes on a complex projective K3 surface, with primitive Mukai vector and with respect to a generic Bridgeland stability condition, are hyperkähler varieties of $\mathrm{K3}^{[n]}$-type of expected dimension. We use derived equivalences, deformations and wall-crossing for Bridgeland stability to reduce to the case of the Hilbert scheme of points.
We compute I-functions and central charges for abelian GLSMs using virtual matrix factorizations of Favero and Kim. In the Calabi-Yau case we provide analytic continuation for the central charges by explicit integral formulas. The integrals in question are called hemisphere partition functions and we call the integral representation Higgs-Coulomb correspondence. We then use it to prove GIT stability wall-crossing for central charges.
Let $C$ be a curve on a K3 surface $X$ with Picard group $\mathbb{Z}.[C]$. Mukai's program seeks to recover $X$ from $C$ by exhibiting it as a Fourier-Mukai partner to a Brill-Noether locus of vector bundles on $C$. We use wall-crossing in the space of Bridgeland stability conditions to prove this for genus $\ge14$. This paper deals with the case $g-1$ prime left over from Paper I.
