# Open-closed TQFT The axiomatic definition of an open-closed TQFT was first given by [[2000#Moore (Talk)|Moore]] and [[1999#Segal (Lectures)|Segal]]. The open-closed 2d TQFT axioms are equivalent to the BCFT sewing conditions of Lewellen. (See [Moore's slide](https://www.on.kitp.ucsb.edu/online/mp01/moore1/oh/128.html) for this claim.) However, the Lewellen sewing conditions work more generally than 2d TQFT: they work for 2d CFT (TQFT is only a special CFT). ## Refs - 3d TQFT and 2d BCFT - original - [[1999#Felder, Frohlich, Fuchs, Schweigert (Sep)]] - [[1999#Felder, Frohlich, Fuchs, Schweigert (Dec)]] - important series of 5 papers - [[2002#Fuchs, Runkel, Schweigert]] - [[2003#Fuchs, Runkel, Schweigert]] - [[2004#Fuchs, Runkel, Schweigert (Mar)]] - [[2004#Fuchs, Runkel, Schweigert (Dec)]] - [[2005#Fjelstad, Fuchs, Runkel, Schweigert]] - a uniqueness result (oriented, same boundary condition) - [[2006#Fjelstad, Fuchs, Runkel, Schweigert]] - [[2013#Kong, Li, Runkel]]: relation to Lewellen: relation (a) there amounts to R16–19, (b) to R32, (c) to R6–9, (d) to R28, (e) to R29, (f) to R31. - [[2020#Fuchs, Schweigert]] - alternative formulation - [[2020#Traube]]: alternative formulation of Kong-Li-Runkel using string net - Virasoro TQFT - [[2025#Jafferis, Rozenberg, Wang]]: open-closed extension of Virasoro TQFT - [[2026#Jafferis, Wang]]: the purely open sector and relation to the closed sector - [[2025#Liu, Ming, Sun, Wu, Yang (Aug)]]: convergence of the purely open sector (with totally geodesic end-of-the-world branes) - 2d open/closed TQFT - [[2000#Lazaroiu]] - [[2005#Lauda, Pfeiffer]] - [[2006#Moore, Segal]]: use Morse theory to show 2d open-closed TQFT axioms are equivalent to sewing theorems ## The extended double In [[2013#Kong, Li, Runkel]], they construct the TQFT diagrams using the doubling trick. The open-string world sheet is first doubled to a closed worldsheet by glueing two copies. Each boundary operator becomes a bulk operator, and each bulk operator becomes two.