TY - BOOK

T1 - Quantum Group Structures in Chern-Simons Theory

AU - Aamand, Nanna Havn

PY - 2023

Y1 - 2023

N2 - In this thesis I demonstrate how certain quantum group structures can be realised from Wilson line operators in perturbative Chern-Simons theory. The relevant theory for this purpose is a so called split Chern-Simons theory on a manifold with boundaries, for which the associated Lie algebra g admits a decomposition into the direct sum of a dual pair of maximal isotropic subalgebras (a Manin triple). The thesis consists of an introduction and two papers.The first paper is joint work with Dani Kaufman. We study the limit when two parallel Wilson lines carrying different representations of g come together. By explicitly computing the corresponding Feynman integrals we show, up to leading order in perturbation theory, that this operation produces a single Wilson line associated to the tensor product representation in the quantized universal enveloping algebra Uℏ(g). In combination with a known result of recovering the classical r-matrix from crossing Wilson lines in the same theory, this suggests that the category of Wilson lines is equivalent to the category of representations of Uℏ(g) as a braided tensor category. We point out a connection of this theory with the Fock-Goncharov moduli spaces of local systems.In the second paper, I study Chern-Simons theory for a Lie algebra that decomposes into the direct sum of a pair of dual subalgebras of which one of them is abelian. The resulting theory can be identified with a topologically twisted 3d N = 4 gauge theory. For this Lie algebra Feynman diagrams become particularly simple and I show, at all orders in perturbation theory, that the expectation value of a pair of crossing Wilson line operators is a solution to the quantum YangBaxter equation. My proof is based on a known technique for constructing knot invariants from Wilson loops in Chern-Simons perturbation theory, using the Axelrod-Singer compactification of the configuration space of Feynman diagram vertices.

AB - In this thesis I demonstrate how certain quantum group structures can be realised from Wilson line operators in perturbative Chern-Simons theory. The relevant theory for this purpose is a so called split Chern-Simons theory on a manifold with boundaries, for which the associated Lie algebra g admits a decomposition into the direct sum of a dual pair of maximal isotropic subalgebras (a Manin triple). The thesis consists of an introduction and two papers.The first paper is joint work with Dani Kaufman. We study the limit when two parallel Wilson lines carrying different representations of g come together. By explicitly computing the corresponding Feynman integrals we show, up to leading order in perturbation theory, that this operation produces a single Wilson line associated to the tensor product representation in the quantized universal enveloping algebra Uℏ(g). In combination with a known result of recovering the classical r-matrix from crossing Wilson lines in the same theory, this suggests that the category of Wilson lines is equivalent to the category of representations of Uℏ(g) as a braided tensor category. We point out a connection of this theory with the Fock-Goncharov moduli spaces of local systems.In the second paper, I study Chern-Simons theory for a Lie algebra that decomposes into the direct sum of a pair of dual subalgebras of which one of them is abelian. The resulting theory can be identified with a topologically twisted 3d N = 4 gauge theory. For this Lie algebra Feynman diagrams become particularly simple and I show, at all orders in perturbation theory, that the expectation value of a pair of crossing Wilson line operators is a solution to the quantum YangBaxter equation. My proof is based on a known technique for constructing knot invariants from Wilson loops in Chern-Simons perturbation theory, using the Axelrod-Singer compactification of the configuration space of Feynman diagram vertices.

M3 - Ph.D. thesis

BT - Quantum Group Structures in Chern-Simons Theory

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -