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Cheung, Elliot

University of British Columbia

This dissertation demonstrates the utility of homotopical ideas arising from derived geometry and homotopical Lie algebras (L-infinity algebras), by describing how one can apply finite dimensional Batalin-Vilkovisky formalism to the study of Chern-Simons theory. In 1989, Witten famously demonstrated that one can use quantum Chern-Simons theory to produce topological invariants. Unfortunately, many of the ideas behind quantum gauge theory remain enigmatic to this day. In particular, the mathematically notorious path integral has yet to be fully understood. BV formalism provides a framework for understanding perturbative path integrals in a mathematically rigorous manner. However, with BV formalism, one often has to assume that the gauge theory contexts are finite dimensional. It is by now a matter of conventional wisdom, that deformation theoretic problems in characteristic zero are "governed" by L-infinity algebras. The conventional L-infinity governing the deformation theory of flat connections on a 3-manifold, which are the objects of study in Chern-Simons theory, is infinite dimensional on the chain level but has finite dimensional cohomology. By triangulating a 3-manifold, one can produce a finite-dimensional homotopically equivalent L-infinity algebra (the Whitney L-infinity algebra) that also describes the moduli of flat connections. In this thesis, we study geometric L-infinity algebras that are homotopic to ones present in conventional BV formalism. We refer to these as BV L-infinity algebras. A BV L-infinity algebra describes a certain analytic space X that can locally be embedded in some ambient smooth space as a critical locus. Although the embeddings may not assemble in a way that describes X as a global critical locus, the different embeddings can be seen being "homotopically coherent" with each other. We show that the ideas from BV formalism can be extended this homotopical setting. When equipped with an "orientation", one can produce perturbative path integral invariants in this homotopical setting.

Text

2025-05-28

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/91221

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/