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January 3, 2007

FFRS on Uniqueness of Conformal Field Theory

Posted by Urs Schreiber

J. Fjelstad, J. Fuchs, I. Runkel, Ch. Schweigert
Uniqueness of open/closed rational CFT with given algebra of open states
hep-th/0612306

As I mention from time to time, J. Fjelstad, J. Fuchs, I. Runkel & C. Schweigert (“FFRS”) are developing an algebraic framework, deeply rooted in category-theoretic notions, supposed to “solve” 2-dimensional conformal field theory - or at least the special case of these theories that are known as “rational”.

Solving 2-dimensional conformal field theory” means:

classifying representations of the category of 2-dimensional conformal cobordisms

(at least up to some technical subtleties concerning the precise definition of this cobordism category).

While the representations of topological 2-dimensional cobordisms are rather tractable, for conformal cobordisms the situation is much more interesting – hence also much more involved.

The powerful insight on which the FFRS approach is based is that the problem of understanding representations of conformal 2d cobordisms may be split into a complex analytic part and a topological part.

(1)(R)CFT =complex analytic+topological =chiral data+sewing constraints =vertex operator algebraV+Frobenius algebra internal toC=Rep(V) \begin{aligned} \text{(R)CFT} &= \text{complex analytic} + \text{topological} \\ &= \text{chiral data} + \text{sewing constraints} \\ &= \text{vertex operator algebra}\;V + \text{Frobenius algebra internal to}\; C =\mathrm{Rep}(V) \end{aligned}

Roughly, one could say that this splitting allows to regard 2d conformal field theory as 2d topological field theory, but internalized in a modular tensor category other than Vect\mathrm{Vect}.

However, that’s an oversimplification. 3-dimensional topological field theory plays an important role, too, for instance.

In fact, the entire theorem, and the formalism underlying it, is quite voluminous and already fills a couple of pages. (A self-contained introduction that I can highly recommend is math.CT/0512076. For a 2-page summary see maybe section 2.2 of this.)

Accordingly, construction on this edifice is an ongoing project and we see parts of the formulation being refined, and more partial results added to the total picture.

The latest paper wants to close the following gap:

It was proven so far that any rational 2D CFT (representation of the category of 2d conformal cobordisms) gives rise to a Frobenius algebra AA, with certain properties, internal to some modular tensor category – and that from any such internal algebra a 2-dimensional rational CFT can be reconstructed.

What was not known was to which degree this converse step was the inverse of the former, i.e. how much the composition

(2)2dRCFTextractFrobeniusalgebra(ARep(V))construct2dRCFT2dRCFT \text{2dRCFT} \stackrel{extract Frobenius algebra}{\to} (A \in \mathrm{Rep}(V)) \stackrel{construct 2dRCFT}{\to} \text{2dRCFT}

differs from the identity.

The new paper now presents mild conditions under which this composition is in fact the identity, up to isomorphism. Therefore, under these conditions, we have that

the Frobenius algebra ARep(V)A \in \mathrm{Rep}(V) specifies uniquely (up to isomorphism) a 2d rational conformal field theory with chiral data encoded by the vertex operator algebra VV.

I might comment on some of the details involved in following entries.

Posted at January 3, 2007 6:55 PM UTC

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