The Langlands Program for 3-Manifolds

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The Langlands program is a vast network of ideas, theorems, and conjectures in number theory, algebra, geometry, and physics. At the heart of the program is a duality principle (a form of electric-magnetic duality in physics), which predicts surprising connections between a priori unrelated mathematical objects. Understanding the nature of this duality is a fundamental problem in mathematics and physics. Rather than study these phenomena in the traditional contexts of number fields or algebraic curves, the PI sets the Langlands program in the realm of 3-dimensional topology - a rich and active area of research in its own right. In this setting, these deep ideas become much more accessible than was previously possible. For example, one component of the project is about observing Langlands phenomena by comparing different ways to count knots in a given 3-dimensional space. Such methods open up the Langlands program to a wider mathematical world and provide pathways to involve a new generation of students in this fundamental research area. This project provides research and training opportunities for students. In more detail, the PI will define (mathematically) and study the space of states associated to a 3-manifold in the family of geometric Langlands topological quantum field theories of Kapustin and Witten. The proposed research is broadly in the realm of geometric representation theory, intersecting with concepts and tools from gauge theory, quantum topology (skein theory), and derived algebraic geometry. Some particular goals of the research include: formulating a duality conjecture for the dimensions of skein modules, as well as tools to explicitly verify in a number of important cases; a proof of the Betti and de Rham geometric Langlands conjecture for elliptic curves at generic level by developing an elliptic/quantum variant of generalized Springer theory; a refined construction of the state space of the B-model using tools from derived algebraic geometry; and a computation of the homology of gauge groups of 3-manifolds in terms of the Langlands dual spectral Whittaker theory. This project is jointly funded by the Algebra and Number Theory program in the Division of Mathematical Sciences, the Established Program to Stimulate Competitive Research (EPSCoR), and the Topology program in the Division of Mathematical Sciences.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.