The n-Category Café

In its simplest incarnation (which shall be all I’m going to consider this moment, not to get distracted by other technicalities), what is called BF-theory is the study of a family of functionals $L_{\mathrm{BF}}$ on the space

$$\Omega^1(X,g) \times \Omega^2(X,g)$$

of pairs $(A,B)$ consisting of a Lie-algebra valued 1-form $A$ and a Lie algebra valued 2-form $B$ equipped with a bilinear invariant form $$\langle \cdot, \cdot \rangle : g \otimes g \to \mathbb{R}$$ on a 4-dimensional manifold $X$,

given by

$$L : (A,B) \mapsto \int_X \left( \alpha \langle F_A \wedge F_A\rangle + \beta \langle F_A \wedge B\rangle + \gamma \beta \langle B \wedge B\rangle \right) \,,$$

where $\alpha, \beta, \gamma \in \mathbb{R}$ are three real parameters.

Very often people assume $\alpha = 0$ and a little less often they assume also $\gamma = 0$. That’s also what the authors of the above paper do. But for instance John Baez in his BF-Theory as a topological QFT keeps general $\gamma$ and makes remarks concerning the $\alpha$ term.

More recently, the $\gamma$-term has caught more attention as people found that incorporating it there are ways to extract the Einstein-Hilbert action functional from a BF-theoretic functional. See for instance the discussion by Jacques Distler, BF, and the references given there.

As far as I am aware (but I might be wrong about this, corrections are welcome), Florian Girelli and Hendryk Pfeiffer had been the first (not that there are many others now…) to search for a way to interpret the BF functional in a more unified way as a functional on a single connection with values in a Lie 2-algebra, back in

Florian Girelli, Hendryk Pfeiffer
Higher gauge theory – differential versus integral formulation
hep-th/0309173.

They regard the special class of strict Lie 2-algebras that come from infinitesimal crossed modules of the form $$h \stackrel{t}{\to} g$$

of the special form where $g$ is an ordinary Lie algebra and $h$ is the abelian Lie algebra based on the same vector space that underlies $g$, with the Lie algebra homomorphism $t : h \to g$ being the trivial one that sends everything to 0.

It is clear that (when our bilinear form is non-degenerate) the critical points of $$L : (A,B) \mapsto \int_X \langle F_A, B\rangle$$ are precisely those pairs for which $A$ is flat, $F_A = 0$ and $B$ is flat in the sense that $d_A B = 0$.

This means that to each critical solution one can associate a certain constrained flat parallel transport 2-functor.

In their latest paper they pick up this idea and propose a generalization of the functional such that it knows about all transport 2-functors with values in a strict 2-group.

I plan to give a more detailed description of their article here in a short while (hopefully). But right now I would like to close this entry here with an observation and a question:

In Lie $\infty$-algebra connections we develop a general concept of invariant polynomials on general Lie $\infty$-algebras, together with the related concept of characteristic forms of Lie $\infty$-algebra valued connections with respect to these invariant polynomials.

As discussed in section 2.6.1 every invariant polynomial on the ordinary Lie algebra $g$ gives rise to an invariant polynomial on the (arbitrary) strict Lie 2-algebra $(t : h \to g)$ (no restriction on $h$ and $t$ here). We find that, when starting with any degree 2 invariant polynomial $\langle \cdot, \cdot\rangle$ on $g$, the integral over the corresponding characteristic form is precisely the BF-theory functional, for

$$\alpha = \gamma$$ and $$\beta = 2\alpha$$

and generalized to arbitrary Lie 2-algebras, in that $B$ appears everywhere under the image of $t$:

$$L : (A,B) \mapsto \int_X \left( \alpha \langle F_A \wedge F_A\rangle + \beta \langle F_A \wedge t(B)\rangle + \gamma \beta \langle t(B) \wedge t(B)\rangle \right) \,.$$

The critical points of this functional are precisely all transport 2-functors, namely all $(h \stackrel{t}{\to} g)$-valued connections whose 2-form curvature vanishes (which is the necessary and sufficient condition for them to integrate up to strict transport 2-functors) and whose 3-form curvature is unconstrained (apart from the constraints imposed by the fact that it is the 3-form curvature of a $(h \stackrel{t}{\to}g )$-valued connection).

I have to run now, leaving this entry somewhate less detailed than I intended it to. But until I manage to return to it and add more details, maybe you enjoy thinking about the obvious question:

what is all this possibly telling us about the true nature of BF-theory?

Update. Here is an addendum, written after I found more time to sneak away from holidy family issues ;-)

From BF to BFCG

A quick look at

F. Girelli, H. Pfeiffer, E. M. Popsecu
Topological Higher Gauge Theory - from BF to BFCG theory
arXiv:0708.3051

There is a well known expression due to Yetter, which

- counts the number of flat discrete 2-connections with values in a strict finite 2-group one can put of a 3- dimensional manifold for a chosen trianulation

- and weights this by a factor that depends on the chosen triangulation

- such that the result is a topological invariant of the manifold.

$$Z = (triangulation weight) \times (number of flat discrete 2-connections)$$

Apparently the analogous expression for 4-dimensional manifolds had been considered before, but not previously shown to yield an invariant.

Girelli, Pfeiffer and Popescu do two things:

1) they remark that the invariance in 4-dimensions is a straightforward corollary of existing results

and

2) discuss a functional (which they call the functional of “BFCG theory”) on tuples of Lie algebra valued differential forms, with the property that a path integral over it has a chance of reproducing the discrete expression for $Z$.

No attempt is made at defining the path integral beyond the usual plausibility considerations, and in particular the origin of the triangulation weight from the path integral is not addressed, but the authors do present a whole list of other functionals studied in the literature which are obtained as special cases of the one they find.

Questions:

1) While not relevant for the main point they make, the general setup makes one wonder about the following:

if ordinary BF theory alone is already governed by a Lie 2-algebra, shouldn’t the “BFCG” Lagrangian in fact be related to a functional on a Lie $n$-algebra valued connection, for $n \gt 2$?

2) The discrete expression for $Z$ looks suspiciously like the result of a direct generalization of the Fukuma-Hosono-Kawai model with the algebra being a group algebra, generalized from groups to 2-groups. The FHK state sum is, in direct analogy, the sum over all ordinary flat discrete 1-connections one can put on a certain triangulation, times precisely the kind of triangulation weight one sees here. Is Z therefore possibly best understood in terms of something like 2-group 2-algebras?

Both these questionshave little impact on the concrete results of the article, which can be understood without any notion of “higher gauge theory”. But from the point of view of higher gauge theory itself, they seem to be urgent.

I know that John Baez has thought about question 2 here, maybe he can point me to some notes.