## 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
math.CT/0602079

and

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
pdf

I. Runkel, J. Fjelstad, J. Fuchs, Ch. Schweigert
Topological and conformal field theory as Frobenius algebras
math.CT/0512076.

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

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
hep-th/0204148

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators II: Unoriented world sheets
hep-th/0306164

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators III: Simple currents
hep-th/0403157

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators IV: Structure constants and correlation functions
hep-th/0412290

J. Fuchs, I. Runkel, Ch. Schweigert
TFT construction of RCFT correlators V: Proof of modular invariance and factorisation
hep-th/0503194

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 ${}_{C}\mathrm{Mod}$ 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 ${ℂ}^{n}$.

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 ${ℤ}_{2}$. 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

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/747

Read the post (R)CFT on more general 2-Categories
Weblog: The String Coffee Table
Excerpt: RCFTs can apparently be defined as 2-functors on 2-paths in Riemann surfaces. Can this be generalized to other domain 2-categories?
Tracked: April 18, 2006 11:42 AM
Read the post A Note on RCFT and Quiver Reps
Weblog: The String Coffee Table
Excerpt: A note on how the structure of RCFT is encoded in something very similar to a quiver representation.
Tracked: April 18, 2006 6:06 PM
Weblog: The String Coffee Table
Excerpt: Porter and Turaev on formal HQFT.
Tracked: May 4, 2006 9:13 PM
Read the post Eigenbranes and CatLinAlg
Weblog: The String Coffee Table
Excerpt: The concept of eigenbranes, which appears in the context of geometric Langlands, and the categorified linear algebra of FRS.
Tracked: May 7, 2006 4:27 PM
Read the post The FRS Theorem on RCFT
Weblog: The String Coffee Table
Excerpt: Statement of the FRS theorem in rational conformal field theory.
Tracked: May 19, 2006 2:33 PM
Read the post Searching for 2-Spectral Theory
Weblog: The String Coffee Table
Excerpt: Does a categorified spectral theorem connect duality defects in CFT with diagonizable Hecke-like 2-operators?
Tracked: May 29, 2006 2:57 PM
Read the post Bulk Fields and induced Bimodules
Weblog: The n-Category Café
Excerpt: Bulk field insertions in 2D CFT in terms of 2-transport: endomorphisms of 2-monoids.
Tracked: September 27, 2006 5:29 PM
Read the post Puzzle Pieces Falling Into Place
Weblog: The n-Category Café
Excerpt: On the 3-group which should be underlying Chern-Simons theory.
Tracked: September 28, 2006 7:18 PM
Read the post A 3-Category of twisted Bimodules
Weblog: The n-Category Café
Excerpt: A 3-category of twisted bimodules.
Tracked: November 3, 2006 2:21 PM
Read the post Postdoctoral Position at the interface of Algebra, Conformal Field Theory and String Theory
Weblog: The n-Category Café
Excerpt: Postdoc position in algebra, CFT and strings.
Tracked: December 5, 2006 3:02 PM
Read the post FFRS on Uniqueness of Conformal Field Theory
Weblog: The n-Category Café
Excerpt: A strengthening of the FFRS theorem on 2-dimensional rational conformal field theory.
Tracked: January 3, 2007 7:43 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 4:43 PM
Read the post Some Notes on Local QFT
Weblog: The n-Category Café
Excerpt: Some aspects of the AQFT description of 2d CFT.
Tracked: April 1, 2007 5:35 AM
Read the post QFT of Charged n-Particle: Towards 2-Functorial CFT
Weblog: The n-Category Café
Excerpt: Towards a 2-functorial description of 2-dimensional conformal field theory. A project description.
Tracked: August 3, 2007 10:51 PM
Read the post Sigma-Models and Nonabelian Differential Cohomology
Weblog: The n-Category Café
Excerpt: Notes on nonabelian differential cohomology and its application to classical and qantum parallel transport.
Tracked: April 11, 2008 8:52 AM

Post a New Comment