The Multiplicative Hitchin System in Supersymmetric Gauge Theory

Multiplicative Higgs bundles are an analogue of ordinary Higgs 
bundles where the Higgs field takes values in a Lie group instead of its 
Lie algebra.  In this talk I’ll discuss two contexts where 
multiplicative Higgs bundles appear in supersymmetric gauge theory.  
I’ll explain how hyperkähler structures on these moduli spaces arise 
physically and mathematically and relate to the theory of Poisson Lie 
groups, and finally I’ll introduce a speculative multiplicative analogue 
of the geometric Langlands conjecture.  This is based on joint work in 
progress with Vasily Pestun.