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)\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 $\mathrm{Vect}$.

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 $A$, 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)$\text{2dRCFT} \stackrel{extract Frobenius algebra}{\to} (A \in \mathrm{Rep}(V)) \stackrel{construct 2dRCFT}{\to} \text{2dRCFT}$

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

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

Posted at January 3, 2007 6:55 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1098

Read the post FFRS on Uniqueness of CFT: Morphisms into Transport Functors
Weblog: The n-Category Café
Excerpt: On morphisms into functors and states and sections in the FFRS description of conformal field theory.
Tracked: January 3, 2007 9:51 PM
Read the post FFRS on Uniqueness of CFT: Sewing as Natural Transformation
Weblog: The n-Category Café
Excerpt: On FFRS's formulation of sewing constraints in terms of natural transformations.
Tracked: January 4, 2007 9:15 PM
Read the post The Globular Extended QFT of the Charged n-Particle: Definition
Weblog: The n-Category Café
Excerpt: Turning a classical parallel transport functor on target space into a quantum propagation functor on parameter space.
Tracked: January 24, 2007 8:12 PM
Read the post CFT in Oberwolfach
Weblog: The n-Category Café
Excerpt: An Oberwolfach meeting on conformal field theory.
Tracked: January 29, 2007 6:32 PM
Read the post QFT of Charged n-particle: Chan-Paton Bundles
Weblog: The n-Category Café
Excerpt: Chan-Paton bundles from the pull-push quantization of the open 2-particle.
Tracked: February 7, 2007 8:24 PM
Read the post Amplimorphisms and Quantum Symmetry, II
Weblog: The n-Category Café
Excerpt: A remark on the appearance of iterated module n-categories in quantum field theory.
Tracked: February 25, 2007 4:45 PM
Read the post Extended QFT and Cohomology II: Sections, States, Twists and Holography
Weblog: The n-Category Café
Excerpt: How transformations of extended d-dimensional quantum field theories are related to (d-1)-dimensional quantum field theories. How this is known either as twisting or as, in fact, holography.
Tracked: June 11, 2007 1:10 PM
Weblog: The n-Category Café
Excerpt: The last session of Recent Developments in QFT in Leipzig was general discussion, which happened to be quite interesting for various reasons....
Tracked: July 22, 2007 7:33 PM