Lattice Topological Field Theory on Non-Orientable Surfaces

Abstract

The lattice definition of the two-dimensional topological quantum field theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that there is a one-to-one correspondence between real associative $\ast$-algebras and the topological state sum invariants defined on such surfaces. The partition and $n$-point functions on all two-dimensional surfaces (connected sums of the Klein bottle or projective plane and $g$-tori) are defined and computed for arbitrary $\ast$-algebras in general, and for the the group ring $A=\mathbb{R}[G]$ of discrete groups $G$, in particular.

Corrected Latex file, 39 pages, 28 figures available upon request

References in corpus (2)

• Lattice Topological Field Theory in Two Dimensions
• Topological Models on the Lattice and a Remark on String Theory Cloning

Cited by in corpus (10)

• TFT construction of RCFT correlators I: Partition functions
• Symmetries and Strings of Adjoint QCD${}_2$
• TFT construction of RCFT correlators II: Unoriented world sheets
• Matrix product states and equivariant topological field theories for bosonic symmetry-protected topological phases in (1+1) dimensions
• Twenty-five years of two-dimensional rational conformal field theory
• Dijkgraaf-Witten invariants of surfaces and projective representations of groups
• Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf-Witten theory
• Non-local Order Parameters and Quantum Entanglement for Fermionic Topological Field Theories
• Dilaton shifts, probability measures, and decomposition
• Mednykh's Formula via Lattice Topological Quantum Field Theories