The String Coffee Table

One way to define a topological field theory is to specify a certain functor from cobordisms to vector spaces. The content of the previous paper by Lauda and Pfeiffer was the precise classification of the functors that correspond to open/closed topological 2D field theories.

It was proven in particular, for the first time, that the well-known description of the corresponding cobordism category in terms of generators and relations does indeed work. Then it was shown that the functors to vector spaces on these cobordisms are specified by certain pairs consisting of a symmetric and a commutative Frobenius algebra. This slightly generalizes old “folk knowledge” about TFTs which assumed that the commutative part has to be the center of the symmetric part.

Defining a TFT in terms of certain functors is very slick (Lauda&Pfeiffer call this the “global approach”), but one would like to have explicit realizations of such functors in terms of “local physics”, too.

A well known prescription for such a local description is the “state sum model” by Fukuma-Hosono-Kawai. Here edges of a dual triangulation of the worldsheet are labelled by a semisimple algebra and vertices are labelled by product and coproduct operations in this algebra. The “state sum” is obtained by “contracting all products”.

In the existing literature this model has been discussed for closed oriented and closed unoriented strings, and for ordinary algebras living in $\mathbf{Vect}$.

Meanwhile, the basic idea of Fukuma-Hosono-Kawai has been adapted and vastly generalized to a detailed description of conformal field theories within what I currently call the FRS formalism.

In that context, it has been found that, more generally, there is nothing in this construction which is special to $\mathbf{Vect}$ and that more generally one can work with algebra objects in tensor categories more general than $\mathbf{Vect}$. (For CFTs one has to.) A detailed dictionary between algebraic phenomena in these categories and the corresponding physics has been established in this context.

In particular, it was found that the algebra object itself describes the space of open string states and that the product in the algebra realizes the OPE of boundary field insertions.

Lauda&Pfeiffer’s new paper can maybe be described as a new look at the old Fukuma-Hosono-Kawai construction, now using the more powerful category-theoretic perspective promoted by FRS.

As with the previous paper, this one is in large parts concerned with carefully making “folk theorems” or at least “folk knowledge” about 2D topological field theory precise.

Maybe the main result of the paper is the proof that the obvious way to define the Fukuma-Hosono-Kawai state sum for open/closed worldsheets does indeed work, in that it encodes a suitable functor from cobordisms to vector spaces, independent of the choice of triangulation.

While the general idea is rather straightforward, some details require special attention. Lauda&Pfeiffer provide a setup in which this construction works for every algebra object $A$ in a symmetric monoidal category $C$ which is what they call strongly seperable.

For $C = \mathbf{Vect}$ strong seperability of algebra objects reduces to the ordinary concept. The main point is that strong seperability is weaker than the property called speciality, which is what is used in FRS formalism and what was implicitly used by Fukuma-Hosono-Kawai. (For the case of matrix algebras this is illustrated nicely in example 4.8 of Lauda&Pfeiffer’s paper).

As a consequence, the bubble move, which is one of to moves to go from one dual triangulation to another, now has a slightly more sophisticated TFT image than originally. The corresponding identity in $C$ is generalized to Lauda&Pfeiffers equation (2.48).

Essentially what this means is that the speciality condition, which says that a “vacuum bubble” is proportional to the identity is generalized to replacing proportionality by proportionality up to multiplication of a central element of $A$, the so-called window element (2.42).

Given these generalizations to common “folk knowledge”, it is interesting to note that not all of them survive the state sum construction.

In example 2.19 Lauda&Pfeiffer emphasize that there are 2D open/closed TFT functors (and hence such TFTs) whose commutative closed sector Frobenius algebra is not the center of the symmetric open sector algebra. However, in the central theorem 4.7 it then turns out that none of these are obtained from state sum models.

For the state sum construction presented, the cylinder always acts as a projection onto the center of the Frobenius algebra. (See for instance slide 3 of this poster for the basic mechanism behind this phenomenon).

It thus appears that one of the key points of the previous paper seems to be curiously irrelevant for the state sum construction. One might want to better understand this. For instance, it might suggest that it could be worthwhile to take a close look at the equivalence relations between different TFTs. It might be that state sum models do not see the entire space of TFTs. But couldn’t it also be that every 2D open/closed TFT is nontrivially ismorphic to one coming from a state sum model? Equivalently, that every knowledgeable Frobenius algebra is isomorphic to one whose commutative part is the center of the symmetric part?

The authors emphasize also that, to them, an important result is that the state sum model really leads to a TFT functor which associates the algebra object $A$ to the open interval. This confirms the statement known from FRS (there in the CFT context and obtained from comparison with more pedestrian descriptions of CFTs) that the algebra object $A$ indeed does describe the space of open string states.

These indications that the formalisms do match might be taken as motivation to realize further phenomena known from FRS in the topological context. For instance one could have a closer look at boundary conditions in the TFT context in terms of internal $A$ modules, a major theme in the FRS approach.

Related to this is the phenomenon of “defect lines”. These are like double-sided boundary conditions where two different “phases” of the conformal field theory touch. Algebraically these very naturally arise in the form of internal bimodules. It is also known that for instance the Ising model can realize such defect lines. An interpretation of defect lines in string theory however is still missing. Maybe it would be easier to identitfy defect lines in topological string theory. This might motivate to investigate these structures along the lines of Lauda&Pfeiffer’s work.

(I should say that I find it natural to speculate that defect lines are related to worldsheet fields which are not globally defined on the world sheet, for instance because they have branch cuts. From the 2-transport perspective defect lines play a role analogous to that of transition functions in a gerbe. I hence like to hope that they might actually be related to the gluing of locally defined CFTs, such as are currently discussed in the context of the chiral deRham complex and the pure spinor string.)

Another important lesson that might be learned from FRS is how to transport the TFT state sum formalism from oriented to unoriented worldsheets. For the description of closed topological strings in terms of matrix algebras this has already been done (by Karimipour and Mostafazadeh). In the more general situations that Lauda&Pfeiffer are looking at the FRS formalism suggests that one has to use Jandl structures on the algebra objects involved.