Super Teichmüller Theory and the Super Conformal Action Functional
Final Report Abstract
In this project, I have used the component field formalism for maps from super Riemann surfaces to Riemannian manifolds to study critical points of the superconformal action functional and associated supergeometric moduli spaces. In collaboration with Shing-Tung Yau and Artan Sheshmani, I have identified super J-holomorphic curves as an interesting class of critical points of the superconformal action. A super J-holomorphic curve is a map from a super Riemann surface to an almost Kähler manifold satisfying an odd first order differential equation of Cauchy–Riemann type. When expressed in component fields this supergeometric differential equation yields a system of partial differential equations coupling classical equations of J-holomorphic curves with a Dirac equation for a twisted spinor. We have constructed the moduli space of super J-holomorphic curves under certain conditions on the target manifold using an implicit function theorem and compactified this moduli space in the case of genus zero via super stable maps. We expect that interesting super Gromov–Witten invariants can be constructed from the compactified moduli space. Together with Jürgen Jost, Ruijun Wu and Miaomiao Zhu I have investigated the Yang–Mills–Dirac–Higgs model, a variant of the superconformal action where the domain has only commuting variables and a gauge symmetry. We have shown that the weak critical points are smooth up to a gauge transformation and obtained a bubbling result.
Publications
- (2019). Geometry and analysis of the Yang-Mills-Higgs-Dirac model
Jost, Jürgen, Enno Keßler, Ruijun Wu, and Miaomiao Zhu
(See online at https://doi.org/10.48550/arXiv.1908.00430) - (2020). Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures
Keßler, Enno, Artan Sheshmani, and Shing-Tung Yau
(See online at https://doi.org/10.48550/arXiv.2010.15634) - (2021). “Super J-holomorphic Curves: Construction of the Moduli Space.” In: Mathematische Annalen
Keßler, Enno, Artan Sheshmani, and Shing-Tung Yau
(See online at https://doi.org/10.1007/s00208-021-02260-0)
