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conformal field theory

Recall that a TQFT is an FQFT defined on the $(\mathrm{\infty },n)$-category of cobordisms whose morphisms are plain cobordisms and diffeomorphisms between these.

In a conformal quantum field theory the cobordisms are equipped with a conformal structure (a Riemannian structure modulo a pointwise rescaling).

A conformal field theory (CFT) is accordingly a functor on such a richer category of conformal cobordisms.

See the discussion at FQFT for more details.

The low dimensional case of $2$-dimensional CFT is of course the best understood. It is in fact so predominant that in much of the literature CFT is understood to be $2$-dimensional.

$2$-dimensional conformal field theories have two major applications:

• they describe critical phenomena on surfaces in condensed matter physics;

• they are the building blocks used in string theory

In the former application it is mostly the local behaviour of the CFT that is relevant. This is encoded in vertex operator algebras.

In the string theoretic applications the extension of the local theory to a full representation of the 2d conformal cobordism category is crucial. This extension is called solveing the sewing constraints .

Remarks

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• The target space structure corresponding to defect line?s in 2d CFT are bi-branes.

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References

Useful references are at vertex operator algebra.

The special case of rational conformal field theory has been essentially entirely formalized and classified. The classification result for full rational 2d CFT was given by Fjelstad–Fuchs–Runkel–Schweigert

• FRS reviews

• The FRS theorem of RCFT