Abstract
Multiplicative Hitchin systems are analogues of Hitchin's integrable system
based on moduli spaces of G-Higgs bundles on a curve C where the Higgs field is
group-valued, rather than Lie algebra valued. We discuss the relationship
between several occurences of these moduli spaces in geometry and
supersymmetric gauge theory, with a particular focus on the case where C = CP1
with a fixed framing at infinity. In this case we prove that the identification
between multiplicative Higgs bundles and periodic monopoles proved by
Charbonneau and Hurtubise can be promoted to an equivalence of hyperkähler
spaces, and analyze the twistor rotation for the multiplicative Hitchin system.
We also discuss quantization of these moduli spaces, yielding the modules for
the Yangian Y(g) discovered by Gerasimov, Kharchev, Lebedev and Oblezin.
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