Geometric Aspects of Field Theories and Lattice Models

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These research projects form part of an ongoing vigorous interaction between geometry and theoretical physics. The engagement of mathematics with other sciences, and of science with mathematics, is a continual source of fruitful ideas with long-term beneficial consequences for society. We can measure these consequences by looking backwards: our current technology and economy rely on the foundation of basic research from past decades and centuries. The mixing of different disciplines is mutually beneficial. As a mathematician the PI has a particular goal to bring structures and intuitions from physics into mathematics. The research has many facets designed to do just that. Specifically, we will work on foundational issues in geometric formulations of quantum field theory. The projects are a mix of specific problems and general structural investigations. The techniques are mathematical, often with inspiration from physics. This work has ramifications for pure geometry as well as applications to questions in physics, such as classification of phases of matter. This award supports graduate students working with the PI participate in some of these projects, and they also carry out their own separate projects within this broad field.We work within the Axiom System for field theories initiated by Segal and Atiyah in the 1980s. These geometric axioms have been refined and extended in many directions since, and we seek to continue this process. For example, the usual axioms evaluate a theory on a single manifold or bordism, whereas many computations involve evaluation on a family of manifolds, the axiomatics of which we will investigate further. In some ways a field theory is akin to a representation of a Lie group. In Lie theory unitary structures play a prominent role, and so too does unitarity in ordinary quantum field theory. While the geometric axioms, especially for topological field theories, include locality in a strong form, there is no corresponding extended notion of unitarity. This is an area we will investigate further. Other aspects of general theory to pursue relate to non-topological invertible field theories and theories which are topological modulo invertible theories. This project also has several lines of inquiry related to specific theories. For example, we have a plan to construct three-dimensional topological Chern-Simons theory as a fully extended theory. We also aim to investigate dynamics in a geometric incarnation of the two-dimensional Ising model. Finally, we aim to construct lattice models which correspond to invertible field theories, part of a larger effort to develop a geometric theory of discrete models.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.