Fredholm Property of Operators from 2D String Field Theory

In a recent study of Landau-Ginzburg model of string field theory by Gaiotto, Moore and Witten, there appears a type of perturbed Cauchy-Riemann equation, i.e. the ζ -instanton equation. Solutions of ζ -instanton equation have degenerate asymptotics. This degeneracy is a severe restriction for obtaining the Fredholm property and constructing relevant homology theory. In this article, we study the Fredholm property of a sort of differential operators with degenerate asymptotics. As an application, we verify certain Fredholm property of the linearized operator of ζ -instanton equations.