Non-semisimple quantum invariants of three and four manifolds

Purdue University, West Lafayette IN

Investigators

Abstract

Quantum topology emerged from the interaction of classical topology and quantum physics. As a foundation in modern physics, quantum field theory remains the best experimentally-verified model of the real world. A topological quantum field theory (TQFT) is a quantum field theory whose correlation functions do not depend on the geometries of the spacetime. Hence it computes quantities which are not sensitive to the shape of the spacetime; this is essentially what topology is about. The quantities computed from a TQFT are called quantum invariants. They provide powerful tools to understand curved spacetimes -- manifolds. Mathematically, TQFTs bridge several subjects including tensor categories, low dimensional topology, and quantum computing. This project aims to study quantum invariants of three- and four-dimensional manifolds. The PI will use innovative methods to construct interesting quantum invariants. The broader impacts of the project contain mentoring and outreach. The PI will continue mentoring students and postdocs to work on topics related to the project. The PI will also organize conferences as a platform to foster collaboration and broaden participation of research. The objective of the project is to investigate non-semisimple quantum invariants of three and four manifolds. Previously, the PI initiated the program of constructing Kuperberg-type invariants of 4-manifolds utilizing trisection diagrams and Hopf triplets. The Hopf triplets, serving as the algebraic input to the construction, were initially assumed to be semisimple and this was recently generalized to the non-semisimple case, but specific examples of such triplets have not been found. This project will continue the development of the program. Firstly, the PI plans to systematically search for non-semisimple Hopf triplets that produce invariants of 4-manifolds. Besides theoretical studies of its properties, extensive computations will be conducted on exotic manifolds to assess the strength of the invariant. Secondly, the PI will refine the procedure to derive an invariant of complex spin structures of 4-manifolds by using Hopf triplets consisting of Hopf superalgebras. Thirdly, the construction will be extended to its full generality by introducing the notion of trialgebras. In dimension three, the project aims to extend the Kuperberg invariant on the basis of weak Hopf algebras, and thus provides a unification of various 3-dimensional quantum invariants. Additionally, the PI will mathematically explore a novel program of constructing modular tensor categories from 3-manifolds using techniques from geometric topology. Specifically, the PI will develop methods to produce the F- and R-matrices of the category which are currently not achievable. 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.