From invariant hyperplane arrangements to Coulomb branches

by Roger Bielawski

Thursday 27 Jun 2024, 16:15 → 17:15 Europe/Zurich

Sitzungszimmer (MI)

Hyperkaehler manifolds are a special type of geometry based on quaternions. Many examples of such manifolds arose first in physics. In particular, supersymmetric gauge theories predict existence of a class of such (generally singular) manifolds, called Coulomb branches. Despite years of effort by many mathematicians, and some well-known hyperkaehler manifolds having been identified as Coulomb branches, a general mathematical definition is still lacking. In my talk, I shall describe how to construct Coulomb branches from certain W-invariant arrangements of hyperplanes in a Cartan algebra of a compact Lie group (W denotes the Weyl group). The construction associates to such an arrangement first a hyperkaehler analogue X of a toric variety. This X is also acted upon by W, and the Coulomb branch parametrises certain (but not all) W-orbits on X. The talk is based on joint work with Lorenzo Foscolo.