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February 8, 2006

FRS Reviews

Posted by Urs Schreiber

Two new reviewes of (aspects of) the Fuchs-Runkel-Schweigert (“FRS”) approach to conformal field theory have recently appeared:

C. Schweigert, J. Fuchs & I. Runkel
Categorification and correlation functions in conformal field theory


C. Schweigert, J. Fuchs & I. Runkel
Twining characters and Picard groups in rational conformal field theory
math.QA/0602077 .

The first of these was, incidentally, also the basis for the colloquium talk preceeding the one on 2-vector bundles and elliptic cohomology which I mentioned recently. I conjecture that there is more to this conjunction of talks than meets the eye.

As introductions to the motivation and logic of FRS, I can stronly recommend the following texts

I. Runkel
Algebra in Braided Tensor Categories and Conformal Field Theory

I. Runkel, J. Fjelstad, J. Fuchs, Ch. Schweigert
Topological and conformal field theory as Frobenius algebras

J. Fuchs, I. Runkel & C. Schweigert
Open Strings and 3D Topological Field Theory

The full details can be found in this series of papers

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators I: Partition functions

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators II: Unoriented world sheets

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators III: Simple currents

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators IV: Structure constants and correlation functions

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators V: Proof of modular invariance and factorisation

These reviewes emphasize key notions of this approach, reflecting some recent refinements in the way one thinks about these issues. I list some of these from my own ideosyncratic point of view.

Think of a string as a categorified point as indicated here. This means that you first of all have to choose an abelian monoidal category C playing the role of the complex numbers. Accordings to FRS, if your string is to be described by (rational) conformal field theory then C has to be a modular tensor category, for instance the category of representations of a chiral vertex operator algebra.

From C we get a 2-category of categorified state spaces over C, i.e. a 2-category CMod of left C-modules (module categories over C).

The objects in that category are left C-modules. These are assigned to points of the worldsheet. According to FRS these modules enode the open string states at these points. If they lie on the boundary of the worldsheet these appear as boundary conditions.

At least locally, this can be made more concrete. Under suitable circumstances the 2-category of C modules is equivalent to that of algebra bimodules internal to C. Hence, locally, we can identify each left C-module with a category of right A-modules, where A is a Frobenius algebra object internal to C. Think of this as identifying the fiber of a vector bundle with a copy of n.

According to FRS, A is the “space” of open string states. The algebra structure on A is the OPE algebra of boundary fields.

The analogy with the vector bundle suggests that we cannot in general globally identify all the left C-modules assigned to our worldsheet with a given category of right A-modules. But assume we can do it on patches of the worldsheet. On double intersections of these patches two different identification with right A-module categories will be related by a weakly invertible A-A-bimodule - an element of the Picard group of A-bimodules. These hence act like transition functions.

According to FRS, the Picard group of A-bimodules can be identified with the symmetry group of the CFT describing our string. For instance, for the Ising model this group is isomorphic to 2. For the 3-states Potts model it is the nonabelian symmetric group S 3.

The lines where two different identifications of C-modules with categories of A-modules touch are called defect lines. For the Ising model these describe for instance boundaries between phases with ferromagnetic and those with antiferromagnetic coupling. (I mentioned defect lines recently in the context of topological strings.)

There might be triple intersections of patches where three of our local identifications with categories of A-modules touch. Hence there will be two defect lines that merge into a third. According to FRS, the merging of these will be described by a bimodule homomorphism, a 2-morphism in the category of bimodules internal to C.

Apart from those transformation which result from the way we have locally identified C-modules with categories of A-modules, there may be intrinsic assignments of C-module morphisms to pieces of string. In the vector bundle analogy these would correspond to a nontrivial connection on the bundle.

According to FRS, such morphisms encode field insertions on the string.

One can go through 1001 examples which illustrate this. See the very useful summary of all the main papers in section 6 of the first of the above two papers.

Posted at February 8, 2006 7:14 PM UTC

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