Mutations of noncommutative crepant resolutions in geometric invariant theory

Abstract

Let $X$ be a generic quasi-symmetric representation of a connected reductive group $G$. The GIT quotient stack $\mathfrak{X}=[X^{\rm ss}(\ell)/G]$ with respect to a generic $\ell$ is a (stacky) crepant resolution of the affine quotient $X/G$, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of $X/G$. Halpern-Leistner and Sam showed that the derived category $\mathrm{D}^b(\mathrm{coh}~\mathfrak{X})$ is equivalent to certain subcategories of $\mathrm{D}^b(\mathrm{coh}~[X/G])$, which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When $G$ is a torus, it turns out that the exchanges are nothing but iterated Iyama--Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama--Wemyss mutations induce a group action of the fundamental group $\pi_1(\mathbb{P}^1 \backslash\{0,1,\infty\})$ on the derived category of a Calabi-Yau complete intersection in a weighted projective space.