The n-Category Café

This was a little bit of a déjà-vu to me, since it addressed questions in 2-dimensional conformal field theory, a simplified version of which originally, back then, made me run into higher gauge theory while looking into strings, then contact John Baez, and – the rest is history – mine at least.

The question is simple: given a 2-dimensional quantum field theory coming from a $\sigma$-model which describes the propagation of string-shaped objects in some space, one may wonder if there are any quantum observables of this theory which come, classically, from the holonomy of some connection around the string.

The way I had originally encountered this question was in terms of Pohlmeyer invariants: Klaus Pohlmeyer was interested in understanding if the quantum observable algebra of the (bosonic) string (propagating on a Minkowski target space) could be understood entirely in terms of holonomies of constant $\mathrm{gl}(n)$-connections on target space around the string, for arbitrary $n$.

In fact, in his work the question was motivated by an attempt to apply the method of Lax pairs to the bosnic string, which has the feature of an infinite-dimensional integrable system.

While, classically, these holonomy-observables for the string are easily written down, using the standard formulas for path-ordered exponentials, one immediately runs into the problem of how to regularize these observables (i.e. how to normal order them) such as to yield admissible quantum observables.

Pohlmeyer, his collaborators and students had followed a certain iterative approach to solve this problem. The current endpoint of the developments in these directions, as far as I am aware, is

C. Meusburger, K.-H. Rehren
Algebraic quantization of the closed bosonic string math-ph/0202041 ,

which achieves a partial understanding of a possible quantization of the algebra of Pohlmeyer invariants.

For some reason, back then, I had gotten interested in this and was wondering why people didn’t simply re-express the Pohlmeyer invariants in terms of the well-known DDF invariants.

After I didn’t seem to get a satisfactory answer to this question, I made this observation a research article

U. S.
DDF and Pohlmeyer invariants of (super)string hep-th/0403260 .

But, alas, it turned out that this insight had in fact been published twenty years before, but apparently forgotten, as described here.

In any case, I kept thinking about these string observables. In most parts of the 2d CFT community these entities were considered a lot, but to a rather different end than in Pohlmeyer’s original program: their quantum version played the role of generators of boundary states. These are, heuristically, to be thought of as states which describe condensates of closed strings which form a D-brane, i.e. a boundary condition for open strings.

The particular boundary operators that the Pohlmeyer observables corresponded to, coming from the Wilson line observable of a connection on the target space, describe D-branes with Chan-Paton bundles that carry a connection

U.S.
Super-Pohlmeyer invariants and boundary states for non-abelian gauge fields
hep-th/0408161

Strikingly, the structures appearing when dealing with these super Pohlmeyer observables have a close resemblance to structures which one encounters in the theory 2-connections. Back then I tried to understand this using a certain formalism for Deformations of 2d SCFTs, since that seemd to admit a very natural way to include the data provided by a 2-connection with values in a 2-group.

This is how I originally entered the world of categorified gauge theory. With hindsight, the time at this point wasn’t really ripe to seriously try to address boundary states for D-branes using this relation. And in fact I have so far not come back to this issue, being busy thinking about various other things.

But today I was starkly reminded of these times, when Konnier talked about of what is, essentially, the problem of quantizing the Pohlmeyer invariants, not for the simple case of a string on Minkowski space, but for the Wess-Zumino-Witten model, describing a string propagating on a compact group manifold.

This investigation had been initiated in

Constantin Bachas, Matthias Gaberdiel
Loop Operators and the Kondo Problem
hep-th/0411067

These authors very much emphasize that inserting the Wilson loop operator (the “Pohlmeyer observable”) is an instance of inserting a defect line on the worldsheet.

Alekseev and Monnier succeed in finding a quantum regularization of the Wilson line operator for the string propagating on a group, and they extract a bunch of interesting results from that.