Operads and moduli spaces

Abstract

This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of 'quantum invariants' of manifolds inspired by ideas originating from physics. We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We generalise open topological conformal field theories to open Klein topological conformal field theories and consider various related moduli spaces, in particular deducing a Mobius graph decomposition of the moduli spaces of Klein surfaces, analogous to the ribbon graph decomposition of the moduli spaces of Riemann surfaces. We also begin a study, in generality, of quantum homotopy algebras, which arise as 'higher genus' versions of classical homotopy algebras. In particular we study the problem of quantum lifting. We consider applications to understanding invariants of manifolds arising in the quantisation of Chern-Simons field theory.

155 pages, PhD thesis (University of Leicester)

References in corpus (17)

• Deformation quantization of Poisson manifolds, I
• On the structure of open-closed topological field theory in two dimensions
• On a theorem of Kontsevich
• Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
• Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair
• Deformation theory and rational homotopy type
• Some Comments on Branes, G-flux, and K-theory
• Remarks on Chern-Simons invariants
• Modular operads and Batalin-Vilkovisky geometry
• Renormalisation and the Batalin-Vilkovisky formalism
• Homotopy algebras and noncommutative geometry
• Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves
• The A-infinity operad and the moduli space of curves
• A dual point of view on the ribbon graph decomposition of moduli space
• Classes on compactifications of the moduli space of curves through solutions to the quantum master equation
• Unimodular L-infinity algebras
• Noncommutative geometry and compactifications of the moduli space of curves

Cited by in corpus (4)

• Unimodular homotopy algebras and Chern-Simons theory
• Koszul duality and homotopy theory of curved Lie algebras
• Minimal models of quantum homotopy Lie algebras via the BV-formalism
• Unbased rational homotopy theory: a Lie algebra approach