Quasimap wall-crossing for GIT quotients

Abstract

In this paper, we prove a wall-crossing formula for ǫ-stable quasimaps to GIT quotients conjectured by Ciocan-Fontanine and Kim, for all targets in all genera, including the orbifold case. We prove that adjacent chambers give equivalent invariants, provided that both chambers are stable. In the case of genus-zero quasimaps with one marked point, we compute the invariants in the left-most stable chamber in terms of the small I-function. Using this we prove that the quasimap J-functions are on the Lagrangian cone of the Gromov-Witten theory. The proof is based on virtual localization on a master space, obtained via some universal construction on the moduli of weighted curves. The fixed-point loci are in one-to-one correspondence with the terms in the wall-crossing formula. Contents 1. Introduction 1 2. Curves with entangled tails and calibrated tails 10 3. Quasimap invariants with entangled tails 22 4. The master space and its virtual cycle 25 5. The properness of the master space 28 6. Localization on the master space 37 7. The wall-crossing formula 47 Appendices 52 Appendix A. Intersection theory on inflated projective bundles 52 References 55

Quasimaps to moduli spaces of sheaves on a K3 surface

“…Sketch of the proof. As in the case of [Nes21, Theorem 6.4] we have to refer mostly to [Zho,Section 6]. The difference with is that we use reduced classes now.…”

Gromov-Witten Theory of $A_n$ type quiver varieties and Seiberg Duality

“…By wall-crossing in [CFK14] [Zho19], we only have to identify the quasimap I-functions of pairs of geometries before and after mutation in order to prove that the genus zero GW invariants are equal.…”

Towards a mirror theorem for GLSMs

“…A more modern formulation of the mirror theorem uses the language of Givental's symplectic formalism and applies to more general targets [5,18] and to big I-functions [11,7,49,51]. In this context, a mirror theorem for Y amounts to finding an explicit function I Y (known as an I-function) which lies on the overruled Lagrangian cone L Y .…”

“…A fundamental result from the theory of quasimaps is that I-functions for Y may be defined directly in terms of 0 + -stable quasimap invariants [12,11,51]. More precisely, I-functions for Y often take the form (1.3.1)…”