Spectral flow calculations for reducible solutions to the massive Vafa-Witten equations

Abstract

The Vafa-Witten equations (with or without a mass term) constitute a non-linear, first order system of differential equations on a given oriented, compact, Riemannian 4-manifold. Because these are the variational equations of a functional, the linearized equations at any given solution can be used to define an elliptic, first order, self-adjoint differential operator. The purpose of this article is to give bounds (upper and lower) for the spectral flow between respective versions of this operator that are defined by the elements in diverging sequences of reducible solutions. (The spectral flow is formally the difference between the respective Morse indices of the solutions when they are viewed as critical points of the functional.) In some cases, the absolute value of the spectral flow is bounded along the sequence, whereas in others it diverges. This is a curious state of affairs. In any event, the analysis introduces localization and excision techniques to calculate spectral flow which may be of independent interest.