| dc.description.abstract | Quantum field theories (QFTs) are geometric and analytic in nature. With enough symmetry, some QFTs may admit partial or fully algebraic descriptions. Topological and conformal field theories are prime examples of such QFTs. In this thesis, the algebraic structure of 2+1D Topological Quantum Field Theories (TQFTs) and associated Conformal Field Theories (CFTs) is studied. The line operators of 2 + 1D TQFTs and their correlation functions are captured by an algebraic structure called a Modular Tensor Category (MTC). A basic property of line operators is their operator product expansion. This is captured by the fusion rules of the MTC. We study the existence and consequences of special fusion rules where two line operators fuse to give a unique outcome. There is a natural action of a Galois group on MTCs which allows us to jump between points in the space of TQFTs. We study how the physical properties of a TQFT like its symmetries and gapped boundaries transform under Galois action. We also study how Galois action interacts with other algebraic operations on the space of TQFTs like gauging and anyon condensation. Moreover, we show that TQFTs which are invariant under Galois action are very special. Such Galois invariant TQFTs can be constructed from gauging symmetries of certain simple abelian TQFTs. TQFTs also admit gapless boundaries. In particular, 1+1D Rational CFTs (RCFTs) and 2+1D TQFTs are closely related. Given a chiral algebra, the consistent partition functions of an RCFT are classified by surface operators in the bulk 2 + 1D TQFT. On the other hand, Narain RCFTs can be constructed from quantum error-correcting codes (QECCs). We give a general map from Narain RCFTs to QECCs. We explore the role of topological line operators of the RCFT in this construction and use this map to give a quantum code theoretic interpretation of orbifolding. | en_US |
