Homological Quantum Field Theory

Abstract

We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal $d$-algebra. We construct a structure of transversal 1-category on the space of chains of maps from a suspension space $S(Y)$, with certain boundary restrictions, into a fixed compact oriented manifold. We define homological quantum field theories \HL and construct several examples of such structures. Our definition is based on the notions of string topology of Chas and Sullivan, and homotopy quantum field theories of Turaev.

18 figures

References in corpus (18)

• Introduction to string field theory
• Closed String Field Theory: Quantum Action and the BV Master Equation
• Operads and Motives in Deformation Quantization
• String Topology
• Homotopy field theory in dimension 3 and crossed group-categories
• Two-dimensional topological gravity and equivariant cohomology
• Homotopy field theory in dimension 2 and group-algebras
• Geometric construction of modular functors from Conformal Field Theory
• Notes on string topology
• Topological Field Theory Interpretation of String Topology
• Examples of categorification
• Rational Combinatorics
• TQFT's and gerbes
• On the combinatorics of hypergeometric functions
• Formal Homotopy Quantum Field Theories, I: Formal Maps and Crossed C-algebras
• Symmetric Boolean Algebras
• Homology and manifolds with corners
• Graphical introduction to classical Lie algebras

Cited by in corpus (2)

• Super, quantum and non-commutative species
• Homology and manifolds with corners