nLab Cardy condition




The Cardy condition is part of the sewing constraint consistency condition on data that gives an open-closed 2d CFT or 2d TQFT.

It is the algebraic reflection of the fact that the following two cobordisms are equivalent (as topological as well as as conformal cobordisms):

  1. an open string coming in, closing to a closed string and then opening up again to an open string (the “zip-unzip cobordism”);

  2. an open string coming in, splitting into two open strings, these crossing each other (with the endpoints of one of them at the same time making a full rotation), then both merging again to one open string.

(e.g. Lauda-Pfeiffer 05 (3.44), Kong 06, figure 3)

cardy condition screen captured from Lauda  Pfeiffer 2006

For instance in the classification of open-closed 2d TQFT with coefficients in Vect via Frobenius algebras, this means that for OO the Frobenius algebra of open string states and for CC the commutative Frobenius algebra of closed string states, then the canonical linear function

OCO O \longrightarrow C \longrightarrow O

is equal to the canonical map

OΔOOτOOμO, O \stackrel{\Delta}{\longrightarrow} O \otimes O \stackrel{\tau}{\longrightarrow} O \otimes O \stackrel{\mu}{\longrightarrow} O \,,

where μ\mu is the coproduct, μ\mu the product and τ\tau the braiding (e.g. Lauda-Pfeiffer 05 (2.14)).


Discussion in 2d TQFT includes

Discussion in 2d CFT includes

Last revised on August 5, 2023 at 21:23:58. See the history of this page for a list of all contributions to it.