nLab
QFT with defects

Idea

Recall that an $n$-dimensional FQFT is a symmetric monoidal $(\mathrm{\infty },n)$-functor from the (infinity,n)-category of cobordisms.

$$Z:{\mathrm{Bord}}_n\to C.$$Z : Bord_n \to C \,.

If one replaces plain cobordisms here with cobordisms that are colored in that they are equipped with certain partitions labeled in a certain index set $\mathrm{Def}$, one calls a morphism

$$Z:{\mathrm{Bord}}_n^{\mathrm{Def}}\to C$$Z : Bord_n^{Def} \to C

a TQFT with defects.

The reason is that such a morphism will behave like encoding

• for each label in $\mathrm{Def}$ of codimension 0 regions one ordinary TQFT;

• for each label in $\mathrm{Def}$ of codimension 1 data on how to “connect” the two TQFTs on both sides

• etc.

So one may think of the codimension $k$ colors as defects where the TQFT that one is looking at changes its nature.

In particular, when the QFT on one side of the defect is trivial, then the defect behaves like a boundary condition for the remaining QFT. Since at least for $n=2$ QFT such bounary conditions are also called branes, defects are also called bi-branes.

Examples

• An old example is the calss of Turaev-Reshetikhin TQFT?, which is a functor on 3-dimensional cobordisms with codimension 3 and 2 defects. This is supposed to be the would-be result of Chern-Simons theory, where the defect lines are the original Wilson line?s in this context.

• Defects in 2-dimensional conformal field theory have a long history in real-world application, for instance

  • Kramers-Wannier duality from conformal defects

  • Frölich, Fuchs, Runkel, Schweigert, Duality and defects in rational conformal field theory (arXiv)

• Defects in 2-dimension TFT have been studied a lot in the context of genus-0 TFT, where they are described using the language of planar algebra?s.

  • See the discussion at Planar Algebras, TFTs with Defects for a start.

Formulation in terms of transformations of TQFTs

Based on techniques used in his work on the 2-category of 2-dimensional cobordisms, Chris Schommer-Pries shows that a TQFT with given collection of defects is the same as collection of natural transformations between ordinary functorial QFTs (functors on an (infinity,n)-category of cobordisms):

• Chris Schommer-Pries, Tpological Defects and Classification of Local TQFTs in Low Dimension, Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology (pdf)

This in particular connects to the attempt at

• Towards 2-functorial CFT

to encode the decoration prescription in the FFRS description of conformal field theory in terms of the components of transformations of the Reshetikhin-Turaev 3d TFT functor.