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K-State Today

November 19, 2013

Soibelman discusses moduli spaces of Higgs bundles at the mathematics colloquium lecture this afternoon

Submitted by Reta McDermott

Yan Soibelman, professor of mathematics, will present "Moduli Spaces of Higgs Bundles in Mathematics and Physics" as part of the mathematics department colloquium series at 2:30 p.m. today in 122 Cardwell Hall.

In 1987, Nigel Hitchin in his study of self-dual Yang-Mills equations on Riemann surfaces introduced the notion of Higgs bundle. The latter is a pair consisting of a smooth vector bundle with connection --gauge field -- and a matrix-valued differential (1,0)-form on the Riemann surface, satisfying some compatibility conditions. He named the (1,0)-form "Higgs field," motivated by an analogy with the "Higgs field" in physics, which is the scalar particle predicted by the physics standard model.

From the point of view of complex analysis, or algebraic geometry, Higgs bundle on a Riemann surface is the same as a pair consisting of a holomorphic vector bundle on the corresponding complex curve and a matrix-valued holomorphic differential form.

Higgs bundles depend on parameters. In other words, they have "moduli." Moduli spaces of Higgs bundles play a very important role in modern mathematics and mathematical physics. It is partly explained by the fact that they are total spaces of complex integrable systems, or "Hitchin integrable systems." 

Hitchin integrable systems are the main subject of study in Geometric Langlands program. They form an important class of examples for Homological Mirror Symmetry program. They appear as moduli spaces of vacua of certain supersymmetric gauge theories. They appear in the recent proof of Kac conjecture in representation theory of quivers, among others.  

In his purely mathematical talk, Soibelman will explain the role of Hitchin integrable systems in a very recent development in mathematics and mathematical physics, which is the theory of wall-crossing formulas and Donaldson-Thomas invariants developed jointly with Maxim Kontsevich.

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