My research lies at the interface of theoretical physics and pure mathematics, inspired by the dual questions: What algebraic and geometric structures govern the behavior of elementary quantum particles and fields? Conversely, how can physics intuition be used to derive new predictions and results in algebra, geometry, and other branches of mathematics?
Correspondingly, my methods tend to be interdisciplinary, combining traditional physics methodology with modern ideas from algebra, geometry, and topology.
Some specific problems that interest me include:
1. How higher algebraic structures arising from the mathematical formalism of TQFT show up in supersymmetric quantum field theory — especially in supersymmetric gauge theory.
- For an introduction, see my StringMath 2017 talk (Hamburg) on Koszul Duality Patterns in Physics; lectures at ICTS Bangalore (July 2018) on En algebras in SUSY field theory, G actions in quantum mechanics, and the Fukaya category of point/G; and a series of lectures at QFT for Mathematicians (June 2019) on boundary conditions.
2. The interplay of extended operators (lines, surfaces, boundaries, interfaces) with dualities in supersymmetric gauge theories.
- Recently I’ve been particularly interested in line operators and boundary conditions in 3d N=4 theories, which are closely related to the notion of Symplectic Duality in mathematics, and more generally to a vast set of ideas in geometric representation theory. My students/collaborators and I constructed categories of line operators in topological twists of 3d N=4 theories, which we are now applying to constructions of HOMFLY-PT link homology (cf. talks at KITP December 2018, and Perimeter October 2019).
- I have also been studying boundary conditions in 3d N=2 theories, which support some exotic sorts of chiral algebras, and behave in interesting ways under particle-vortex dualities and level-rank dualities. My collaborator Natalie Paquette recently spoke about them at Strings 2018 in Okinawa. I recently began to systematically construct boundary chiral algebras in gauge theories.
3. Using QFT to produce new topological invariants (and organize old invariants) of 3-manifolds and 4-manifolds. My older work on this concerns triangulations of 3-manifolds, SL(N,C) Chern-Simons theory, and the 3d-3d correspondence. Recently, I have been looking at 2d QFTs related to triangulations of 4-manifolds.
4. I am also involved in a broader effort to develop a practical, versatile dictionary between modern concepts in mathematics and quantum field theory, allowing any researcher to fluently translate powerful constructions and intuition back and forth.
- To further these goals, some colleagues and I are organizing a series of workshops on QFT for Mathematicians, at UC Davis (Chiral Algebras for the 21st Century) and the Perimeter Institute (QFT for Mathematicians 2019, 2020).