Quantum singularity theory via cosection localization

Abstract

We generalize the cosection localized Gysin map to intersection homology and Borel–Moore homology, which provides us with a purely topological construction of the Fan–Jarvis–Ruan–Witten invariants and some GLSM invariants.

Localizing virtual cycles for Donaldson-Thomas invariants of Calabi-Yau 4-folds

“…In this section, we construct a localized square root Euler class √ e(F, s) ∈ A n X (Y ) for an SO(2n)-bundle F and an isotropic section s ∈ H 0 (F ) with X = s −1 (0) by using a method in [16,17,18,19]. We will prove in §5 that the square root Euler class √ e(F, s) defined in this section coincides with the Oh-Thomas class constructed in [29].…”

“…Cosection localized Gysin map. A similar blowup construction was used in [17,18,19] for cosection localized Gysin maps, which we recall now. Let V be a vector bundle of rank r over Y and σ : V → O Y be a cosection.…”

“…So our first task in this paper is to provide an alternative construction of √ e(F, s) which allows a further localization by an extra section in a straightforward manner. This is achieved by a construction used in [16,17,18,19]. Letting ρ : Y → Y be the blowup along the zero locus X of s, we can decompose any ξ ∈ A * (Y ) into…”

Algebraic virtual cycles for quantum singularity theories

“…[12]). For the general case including broad sectors, the second and third named authors in [13] generalized the cosection localization of [12] to intersection homology and provided a direct construction of the cohomological field theories for both broad and narrow sectors. As the construction in [13] does not involve virtual cycles, one may wonder whether it is possible to construct the cohomological field theories by a Fourier-Mukai type integral transformation whose kernel is an algebraic virtual cycle.…”

“…The goal of this paper is to construct algebraic virtual cycles that give us the cohomological field theories of [13] by integral transformations.…”