Higher Spin Curves and Cohomological Field Theories

Brigham Young University, Provo UT

Investigators

Abstract

The Investigator studies intersection theory on and the cohomological field theory arising from the moduli of algebraic spin curves and stable spin maps. Especially important in this study is the exploration of the plethora of similarities between Gromov-Witten theory and cohomological field arising from higher spin curves and spin maps. Also important in this research is the study of the many relations between these cohomological field theories and integrable hierarchies, including significant recent progress toward proving some remarkable conjectures like the W-algebra conjecture and Generalized Witten Conjecture. The research in this project is concerned with connections between algebraic geometry and physics, and it has implications for both subjects. Algebraic geometry, which is the study of solutions to polynomial equations, and especially the sub-discipline of algebraic curves, which is one of the main foci of this research, have many applications. The most notable of these applications are in secure electronic communications (cryptography and error correcting codes). Recent research has shown that algebraic geometry also has significant ties to high-energy physics and plays an important role in helping us understand the fundamental nature of the universe. Conversely, physics has helped us better understand some important aspects of algebraic geometry and its applications. This research is focused on studying and further developing some of these important links.