UAlberta Math and Statistics Colloquium: Steve Rayan

The moduli space of Higgs bundles on a complex algebraic curve — an object that originates from Yang-Mills theory in theoretical physics, but which now lies at the interface of algebraic geometry and geometric representation theory — is perhaps best understood structurally through the spectral correspondence, which abelianizes each Higgs bundle at the cost introducing an additional curve typically of higher genus. This curve is called the spectral curve of the Higgs bundle as it encodes the spectrum of the Higgs field. This abelianization leads to the now famous Hitchin fibration. At the same time, algebraic curves of arbitrary genus and moduli spaces associated with them have recently become key objects in a different part of physics, namely condensed matter physics, due to the advent of 2-dimensional hyperbolic quantum matter, whose electronic band theory was developed by J. Maciejko and myself and which has been further explored both theoretically and experimentally by various groups internationally, including that of I. Boettcher. These two pictures overlap when the position curve of the band theory coincides with the spectral curve of a Higgs bundle, as elicited in my joint work with E. Kienzle. This talk will attempt to introduce the two pictures (without any assumptions of knowledge in either algebraic geometry or theoretical physics) and I will point out a few intriguing things that happen in the overlap. One of these observations is an expansion of the Hitchin fibration that anticipates a higher spectral correspondence that I have been exploring in joint work with K. Banerjee. There will be lots of visual illustrations to support the talk.