One should notice that extended CFT hasn't been fully formalized yet. ("TCFT" is only superficially conformal, in fact it axiomatizes topological strings. Also the Moore-Segal discussion is about 2d TFT, as far as I am aware.)

While not fully formalized yet, there are a few results that already come very close. For *rational* 2d CFT the FRS formalims and its related constructions probably goes the furthest: it gives constructions of open-closed rational CFTs in terms of "state sum constructions" that are entirely analogous to those for 2d extended TFTs, notably entirely analogous to the Fukuma-Hosono-Kawai construction. In both cases there is an algebra of open string states assigned to the interval, and from it the rest of the structure is induced. The crucial difference is that for TFT this algebra is an ordinary algebra (albeit $A_\infty$), while for CFT in the FRS formulation it is an algebra object *internal* to the modular tensor category of representations of the vertex operator algebra, that describes the CFT to be described locally.

From this one reads off the following use of extended CFT for string theory:

the refinement of the VOA to a "full CFT" in the sense of a full representation of the conformal cobordism 1-category is precisely a solution to the *sewing constraints*. This involves modular invariance, but also all its higher genus analogs. This is already a step not always correctly done in the physics literature. In the FRS articles you find examples of "modular invariant CFTs" to which corresponds one, none or several full CFTs.

the further refinement to a extended conformal cobordisms representation (to the extent that it has been formalized) takes care of taking also all possble boundary conditions hence all possible D-brane configurations into account.

A modern (re)view of the state of the art of such "extended" 2d CFT is in

Liang Kong, *Conformal field theory and a new geometry*.

Related aspects are in

Stephan Stolz, Peter Teichner, *Supersymmetric field theories and generalized cohomology*

There the non-topological QFT is regarded explicitly as a cobordism representation, albeit not yet fully extended.

Structures that are expected to eventually serve as ingredients for fully extended CFT have been discussed in

Chris Douglas, André Henriques, *Topological modular forms and conformal nets* .

They discuss a 3-category of conformal data such that the 3d extended TQFT induced by fully dualizabe objects in there is holographic dual to the given 2d CFT.

André has recently been giving talks on more of the story of how to more explicitly obtain the extnded CFT from this data. See the video of his talk at the recent *Mathematical foundations of quantum field theory* workshop.

This post has been migrated from (A51.SE)