Coulomb branch localization, quasimaps, and surface counting in Calabi--Yau fourfolds

We present a string theoretic approach to surface counting in local Calabi--Yau fourfolds via supersymmetric localization in topologically twisted four-dimensional gauge theories. This approach is based on a spectral correspondence between PT1-stable pairs on local fourfolds and twisted quasimaps with fixed two-dimensional domain associated to the ADHM quiver, or, equivalently, ADHM sheaves. For local toric fourfolds, we derive a conjectural residue formula for the K-theoretic quasimap partition function via Coulomb branch localization. As a result, in this case, we obtain a conjectural prescription fixing all usual sign ambiguities in the equivariant computation of such invariants. We present some explicit computations for local P2, extending the results available in the literature, and describe the formalism in general. This is the first instance of Coulomb branch localization for a quasimap theory in the context of four-dimensional gauge theories.