A genus-one FJRW invariant via two methods

Abstract

We calculate a genus-one FJRW invariant of an LG pair $(W_3=x_1^3+x_2^3+x_3^3, \mu_3)$ via two different methods. In the first method, we apply the cosection localization technique to get a genus-one three-spin virtual class explicitly and then calculate the target FJRW invariant via self-intersections of the three-spin virtual class. In the second method, we apply the Mixed Spin P-fields method for the pair and calculate the invariant using the localization formula. This invariant is the building block in establishing the all-genera LG/CY correspondence and its determination enables one to compute the all-genera FJRW invariants for the LG pair.