December 29, 2007

BF-Theory as a Higher Gauge Theory

Posted by Urs Schreiber

I’ll quickly say a couple of words on the occasion of

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

about the interpretation of the class of topological field theories known as BF-Theory, motivated also by the results Jim, Hisham and myself are talking about in Lie $\infty$-Connections and their Application to String- and Chern-Simons $n$-Transport.

It’s about if and how exactly to interpret the BF-theory Lagrangian as a functional on Lie $n$-algebra valued forms.

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.

Posted at December 29, 2007 7:15 PM UTC

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Re: BF-Theory as a Higher Gauge Theory

Dear Urs,

you wrote:

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

Tim Porter and I wrote an article where we prove, among other things, that Yetter’s state sum yields and invariant of n-dimensional manifolds for any positive integer n. In fact what we prove is that it yields a homotopy invariant of simplicial complexes, and that it can be extended to handle the more general case of crossed complexes as well.

See “On Yetter’s invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups”, in TAC.

As far as I can remember Tim’s article “TQFT’s from homotopy $n$-types” is also valid for manifolds of any dimension.

Happy new year!!

Joao

Posted by: Joao Martins on December 30, 2007 12:14 PM | Permalink | Reply to this

Re: BF-Theory as a Higher Gauge Theory

I wrote:

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

João Martins kindly replied

Tim Porter and I wrote an article where we prove, among other things, that Yetter’s state sum yields and invariant of $n$-dimensional manifolds for any positive integer $n$. In fact what we prove is that it yields a homotopy invariant of simplicial complexes, and that it can be extended to handle the more general case of crossed complexes as well.

Thanks for letting me know. So what is the status of the paragraph on the top of p. 6 of Girelli-Pfeiffer-Popesu, where it says

It seems to have gone unnoticed that the same result [independence of the choice of triangulation] for $d = 4$ is in fact already implied by [11] in combination with [22], again by using Alexander moves:

Theorem III.2. Let $\Lambda$ be a closed and oriented combinatorial $d$-manifold, $d \in \{3,4\}$, and $(G,H,\lt,t)$ be a finite crossed module. The partition function [$Z$] is invariant under Pachner moves and therefore well defined on equivalence classes of combinatorial manifolds.

Since the original references may not be very accessible to readers interested in higher gauge theory, we sketch in Appendix A how one can obtain a self contained proof of triangulation independence using Pachner moves [23].

Are they just giving a new (“self contained”) proof of a known fact, or do they indeed generalize known results?

Posted by: Urs Schreiber on December 30, 2007 1:33 PM | Permalink | Reply to this
Read the post Dijkgraaf-Witten and its Categorification by Martins and Porter
Weblog: The n-Category Café
Excerpt: On Dijkgraaf-Witten theory as a sigma mode, and its categorification by Martns and porter.
Tracked: January 5, 2008 3:23 AM

Re: BF-Theory as a Higher Gauge Theory

In some thread, which I cannot find right now, I had a discussion with John Baez about how 4-dimensional BF theory is to 3-dimensional Chern-Simons theory roughly like Chern-Simons is to 2-dimensional CFT.

I was reminded of that further today while thinking more about the interpretation of the BF-theory action functional as the integral over an invariant polynomial on a strict Lie 2-algebra, mentioned above.

The curious thing about invariant polynomials $P$ on higher Lie algebras is, as we discuss in our article that their image under a Lie infty-algebra valued connection $A$ with curvature $F_A$ $P(F_A) \in \Omega^\bullet(Y)$ is not necessarily a closed form. Unlike the situation for ordinary algebras, where it always is.

In fact, for the invariant polynomial $\langle \cdot, \cdot \rangle$ on a strict Lie 2-algebra $t : h \to g$ coming from an invariant bilinear symmtric form on $g$, we have, unless I am dreaming $d \langle \beta \wedge \beta \rangle = 2 \langle \beta \wedge t(H)\rangle \,,$ where $F_A = (\beta, H)$ i.e. $\beta$ is the 2-form curvature (“fake” curvature) and $H$ the 3-form curvature of a $(h \to g)$-valued connection on $Y$.

In particular this says that the 5-form $\langle \beta \wedge t(H)\rangle$ is closed. It hence plays a role rather similar (but somewhat different) to the 4-form $\langle F_A \wedge F_A\rangle$ of an ordinary $g$-connection $A$, which is the exterior derivative of the corresponding Chern-Simons 3-form $d CS(A) = \langle F_A \wedge F_A\rangle \,.$

Certainly this is supposed to tell us something. And I am trying to find out what.

Potentially there is a BF $b^4 u(1)$-4–bundle arising as an obstruction to lifting something to something, somehow involving Chern-Simons 3-bundles (which themselves obstruct lifts of $g$-bundles to $string(g)$-2-bundles).

Hm…

Posted by: Urs Schreiber on January 8, 2008 6:45 PM | Permalink | Reply to this
Read the post Smooth 2-Functors and Differential Forms
Weblog: The n-Category Café
Excerpt: An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
Tracked: February 6, 2008 11:02 AM

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