2005/08/18 by Aaron D. Lauda, Lauda, Aaron D. · 1 citation
Mathematics · Physics and Astronomy · #18D05 #57M99 #57R56 #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.GT #math.QA #msc:18D05 #msc:57M99 #msc:57R56
paper · pdf · doi:10.48550/arxiv.math/0508349
66 pages, diagrams in xypic and pstricks
arxiv created 2005/08/18 · openalex publication_date 2005/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the Moore-Segal axioms for an open-closed topological field theory, we consider planar open string topological field theories. We rigorously define a category 2Thick whose objects and morphisms can be thought of as open strings and diffeomorphism classes of planar open string worldsheets. Just as the category of 2-dimensional cobordisms can be described as the free symmetric monoidal category on a commutative Frobenius algebra, 2Thick is shown to be the free monoidal category on a noncommutative Frobenius algebra, hence justifying this choice of data in the Moore-Segal axioms. Our formalism is inherently categorical allowing us to generalize this result. As a stepping stone towards topological membrane theory we define a 2-category of open strings, planar open string worldsheets, and isotopy classes of 3-dimensional membranes defined by diffeomorphisms of the open string worldsheets. This 2-category is shown to be the free (weak monoidal) 2-category on a `categorified Frobenius algebra', meaning that categorified Frobenius algebras determine invariants of these 3-dimensional membranes.