This project is based on recent developments at the interface between geometry and physics. There are a number of intricate conjectures about central objects in mathematics (“Hitchin moduli spaces†and “K3 spacesâ€) coming from theoretical physicists studying particular quantum systems at low energies. The PI uses new tools coming out of geometric analysis which are well-suited for verifying the extremely delicate conjectures about the geometry of the Hitchin moduli space and K3 spaces. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these physics conjectures. Because Hitchin moduli spaces occupy a distinguished position in geometry at the crossroads of a number of mathematical fields including algebraic geometry, differential geometry, representation theory, and geometric analysis, this work has cross-disciplinary impact.
This project is centered on the asymptotic geometry of Hitchin moduli spaces and families of K3 surfaces. A main goal is to verify the beautiful conjectural description of the hyperkaehler metric on the Hitchin moduli space by the physicists Gaiotto, Moore, and Neitzke. A second goal is to verify a similar conjectural description of hyperkaehler metrics on elliptically-fibered K3 surfaces by the physicists Kachru, Tripathy, and Zimet. The Hitchin moduli spaces and elliptically-fibered K3 surfaces are both algebraic completely integrable systems admitting a hyperkaehler metric. Both are fibered over a half-dimensional base and the generic fibers are abelian varieties. Some fibers are singular, and these conjectures are most interesting and most difficult near these singular fibers. Constructive analytic techniques and the methods of geometric microlocal analysis seem to be the most appropriate way to verify conjectures from physics because they are well-suited to analyzing the singular differential operators that naturally appear.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
