Invariants of stable quasimaps with fields

Abstract

For arbitrary smooth hypersurface X ⊂ P n we construct moduli of quasimaps with P fields. Apply Kiem-Li's cosection localization we obtain a virtual fundamental class. We show the class coincides, up to sign, with that of moduli of quasimaps to X. This generalizes Chang-Li's [CL] numerical identity to the cycle level, and from Gromov Witten invariants to quasimap invariants.

The logarithmic gauged linear sigma model

“…In this case, we have r = 1 and may chose P = P(E ∨ ⊗ L ω ⊕ O). Combining with the results in [15,39,19], and more generally in [24,44], we have Corollary 1.9 (Proposition 4.1). Notations as above, we have…”

“…As discovered in [15] and further developed in [39,19,24], GLSM can be viewed as a deep generalization of the hyper-plane property of Gromov-Witten (GW) theory for arbitrary genus. However, comparing to GW theory, a major difference as well as a main difficulty of GLSM is the appearance of an extra torus action on the target, called the R-charge, which makes the moduli stacks in consideration for defining the GLSM virtual cycles non-proper in general.…”

Splitting of the virtual class for genus one stable quasimaps

“…First we recall the moduli stack of stable quasimaps with fields introduced in [8]. To simplify the notation, we will focus on the genus one case.…”

“…The authors [8] have constructed a cosection of Ob Y/D1,1 by using the defining polynomial q(x) of Q. Namely a homomorphism (3.2) σ :…”

“…The proof of Theorem 1.2 in this paper uses the strategy which used in [2]. The invariants of stable quasimaps to Q can be realized as the invariants of stable quasimaps to P n with p-field developed in [8]. Since the structure of the moduli of stable quasimaps to P n with fields is much easier to analyze, it allows us to separate the "primary" and the "ghost" contributions algebraically.…”