Z_2 harmonic spinors in gauge theory

Gauge-theoretic moduli spaces are often noncompact, and various techniques have been introduced to study their asymptotic features. Seminal work by Taubes shows that in many situations where the failure of compactness for sequences of solutions is due to the noncompactness of the gauge group, diverging sequences of solutions lead to what he called Z_2 harmonic spinors. These are multivalued solutions of a twisted Dirac equation which are branched along a codimension two subset. This leads to a number of new problems related to these Z_2 harmonic spinors as interesting geometric objects in their own right. I will survey this subject and talk about some recent work in progress with Haydys and Takahashi to compute the index of the associated deformation problem.

Speaker Biography

Rafe Mazzeo is an expert in PDEs and Microlocal analysis. He did his PhD at MIT, and was then appointed as Szegő Assistant Professor at Stanford University, where he is now Professor and Chair of the Department of Mathematics. He has served the mathematical community in many important ways, including as Director of the Park City Mathematics Institute.