The n-Category Café

1) $n$-dimensional gauge theory: the $n$-particle on the $n$-group

One advantage of conceiving $n$-vector bundles with connection on base spaces $$X$$ as functors # $$\mathrm{tra} : P_n(X) \to T_n$$ to the structure $n$-groupoid $$T_n \,,$$ and of conceiving all spaces in terms of categories, is that it allows to regard gauge theory on $n$-dimensional spaces $X$ as the dynamics of $n$-particles of “shape” $$\mathrm{par} = P_n(X)$$ (a suitable $n$-category of $n$-paths in $X$) propagating on the target space $$\mathrm{tar} = Gr_n \,.$$

Notice the simple but important and potentially delusive fact that an $n$-bundle with connection here is regarded as a configuration of the $n$-particle in the structure $n$-group (or rather in a suitable transport groupoid).

This is quite familiar from Chern-Simons theory (“the membrane propagating on $B G$”) and its toy version, Dijkgraaf-Witten theory (those puzzled by this might find alleviation by looking at the review in section 4.2 of Bruce Bartlett’s thesis) as well as maybe the 2-dimensional version of these (the “string on $B G$”).

2) Gauge Theory with Generalized Connections

One of the most fundamental gauge theories, both theoretically as well as for practical purposes, is Yang-Mills theory. This is a theory whose “fields” are smooth principal $G$-bundles with connection on Riemannian base spaces, for $G$ some Lie group.

Another example is gravity. Since the Riemannian metric appearing in General Relativity may be encoded in a connection on the tangent bundle, it is possible to also conceive gravity as a theory whose fields are smooth principal $\mathrm{SO}(n)$-bundles with connection.

The observation of

Abhay Ashtekar
New Variables for Classical and Quantum Gravity
Phys. Rev. Lett. 57, 2244 - 2247 (1986)

that encoding the metric in terms of its Levi-Civita connection seems to simplify the notoriously elusive quantization of gravity, gave rise to an entire field: “Loop quantum gravity”.

For much more on this see John’s TWF 18 and many other TWFs.

Curiously, the gauge-theoretic reformulation of gravity did not lead to a convergence with the familiar quantization methods usually used in the highly-developed field of quantum Yang-Mills theory.

While the latter is usually handled in terms of the smooth 1-forms encoding the connection on a principal bundle, researchers in the loop quantum gravity program chose to focus on the equivalent description of bundles with connection in terms of smooth parallel transport functors.

Locally, these are functors $$\mathrm{tra} : P_1(X) \to \Sigma(G)$$ which send paths to elements of the group $G$.

For some reasons not really known to me (I am hoping that John will help me out here as he reads this!), there was apparently early on agreement reached in the LQG community that it would be useful to drop the assumption of smoothness on these functors, and pass to a larger realm of “generalized connections”.

(My impression is that the motivation behind this is a) the fact that it makes the space of all connections more tractable and b) that this was regarded as harmonizing well with the expectations of what quantum gravity should be like. But I do not really know.)

As Christian Fleischhack briefly reviews on p. 12 , these generalized connections are functors $$\mathrm{tra} : P_1(X) \to \Sigma(G)$$ obtained as inductive limits of continuous maps from the collection of edges of graphs embedded in $X$ to $G$, under the obvious inclusion relation of graphs. This induces a topology on the space of (field) configurations $$\mathrm{conf} = [P_1(X),G]$$ induced from that on $G$.

(I am not, at the moment, sure what that generalization really amounts to, with respect to the situations the theory based on it applies to. Another generalization of smooth transport functors to a merely continuous setup appears in the theory of directed homotopy. It would be useful to understand if and how this is related to the generalized connections considered here.)

As usual in quantum theory, a measure on this configuration space has to be chosen by hand (which is a little sad – but we are working on it I, II).

The choice made here is called the Ashtekar-Lewandowski measure, $$\mu_0 \,.$$ It is specified by the property that its push-forward along any generalized connection yields the standard Haar-measure on the group $G$.

Finally, this allows to define the space of states of the quantum theory to be that of square integrable functions $$L^2(\mathrm{conf},\mu_0)$$ on configuration space $\mathrm{conf}$.

Notice that this corresponds to saying that our $n$-particle propagating on $\Sigma(G)$ couples to the trivial line bundle on $\Sigma(G)$.

3) The Weyl Algebra on the Configuration Space of Generalized Connections

The aim of math-ph/0407006 is to construct and study the Weyl algebra “of abservables” in the present setup. Technical details are abundant. I will try to extract the main idea and point out the connection to the discussion in QFT of Charged n-Particle: Algebra of Observables and Isham on Arrow Fields.

The main point is the identification of morphisms in configuration space.

The morphisms in the functor category $$\mathrm{conf} \subset [P_1(X),\Sigma(G)]$$ are natural transformations of parallel transport functors, hence gauge transformation of connections.

In other words, a morphism between functors $$\mathrm{tra} \stackrel{g}{\to} \mathrm{tra}'$$ is a $G$-valued functions $g : X \to G$ such that $$\array{ \bullet &\stackrel{g(x)}{\to}& \bullet \\ \mathrm{tra}(\gamma)\downarrow\;\;\; && \;\;\;\downarrow \mathrm{tra}'(\gamma) \\ \bullet &\stackrel{g(y)}{\to}& \bullet }$$ for all paths $x \stackrel{\gamma}{\to} y$

For comparison, in the smooth setup, an object in $\mathrm{conf} \subset [P_1(X),\Sigma(G)]$ would be a smooth 1-form $A$ on $X$ with valued in the Lie algebra of $G$, and a morphism $$A \stackrel{g}{\to} A'$$ would be a smooth $G$-valued function $g$ in $X$ such that $$A' = g A g^{-1} + g d g^{-1} \,.$$

Every “arrow field” on configuration space, i.e. every suitable assignment of a gauge transformation at each point in configuration space, induces a corresponding translation operator on $L^2(\mathrm{conf},\mu_0)$.

But, as familiar from the Weyl algebra of the ordinary point particle, among all those translation operators we want to single out a special subgroup.

The subgroup that Christian Fleischhack chooses is that of those gauge transformations $$w^S_g$$ (compare prop 3.19 and def. 3.21 on pp. 22,23) which, on every point of configuration space (i.e. on every parallel transport functor $\mathrm{tra} : P_1(X) \to \Sigma(G)$) is given by one and the same function $g$, which is required to have support on a codimension-one surface $S \subset X$.

This isn’t exactly the way Christian Fleischhack puts it, but up to plenty of technical details that I won’t bother with here, it is true that he is considering translations on configuration space that are given by gauge transformations supported on codimension 1 surfaces.

(The deeper reason for exactly this choice, which, from the point of view of general gauge theory, comes out of thin air, is really to be found in the choice of dynamics that Christian Fleischhack secretly has in mind: namely the Einstein-Hilbert action regarded as an action functional on $G = \mathrm{SU}(2)$-connections. It is this implicit choice of dynamics which governs the design choice in the Weyl algebra in this paper – and justifies its title.)

In the language of flows on categories, we hence find, for each hypersurface $S$ in $X$ and each 1-parameter familiy $t \mapsto g(-,t)$ of functions $g(t) : S \to G$ a flow $$w^S_g : \Sigma(\mathbb{R}) \ni (\bullet \stackrel{t}{\to} \bullet) \mapsto \array{ & \nearrow \searrow^{\mathrm{Id}\;\;} \\ \mathrm{conf} & \;\,\Downarrow^{g(t)} & \mathrm{conf} \\ & \searrow \nearrow_{\mathrm{Ad}_{g(t)}} } \in \mathrm{Flow}(\mathrm{conf}) \,.$$

Translation of functions in $L^2(\mathrm{conf},\mu_0)$ along these flows is the analog of the action of the translational Weyl operators $$\exp(i \lambda \, p)$$ for the 1-particle on the real line $\mathrm{tar} = P_1(\mathbb{R})$.

Christian Fleischhack shows that the Weyl algebra on the configuration space of generalized connections found this way has, with respect to the chosen Ashtekar-Lewandowski measure $\mu_0$ pretty much the same nice properties as the ordinary Weyl algebra for the particle on the line.

In particular, he shows that $H = L^2(\mathrm{conf},\mu_0)$ is, up to isomorphism, the unique irreducible representation of this algebra.

As discussed in great detail in Christian Fleischhack’s paper, that this is supposed to be equivalent to, or at least to play an equivalent role as, the so-called “LOST theorem”, due to

Lewandowski, Okolow, Sahlmann, Thiemann,
Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
gr-qc/0504147.

This is regarded to be central, as it uniquely fixes the space of states of the gauge theory one wishes to quantize.

The crucial assumptions going into this is, as far as I understand at least,

a) that we pass from a space of connections to a space of generalized conections

b) that the translation group in the Weyl algebra is taken to come from gauge transformations supported on hypersurfaces.

4) comparison with other work

A while ago Freed, Moore and Segal had a series of two papers on Hilbert spaces of $L^2$-functions on configuration spaces of connections. So something very closely related to the issue in loop quantum gravity.

They consider different flavors of connections, namely just abelian connections, but also “higher” versions of them: we would say that they consider smooth $n$-transport with values in the $n$-groupoid $$\Sigma^n(U(1)) \,.$$

Of course Freed-Moore-Segal don’t use these words, they are instead thinking of these functors, equivalently, as Cheeger-Simons differential characters (whose definition comes, in fact, pretty close to that of a smooth $n$-functor on closed $n$-paths).

Apart from these technical issues, the problem they study is very much the same as in the kinematical part of lqg: they investigate the $L^2$-Hilbert spaces of functions on the space of all these connections.

Their main point is a pretty interesting, because rather subtle, cohomological effect one sees here. And of course, instead of “cohomological effect”, we might say “quantum effect”!

This concerns a certain uncertainty relation, if you wish, concerning the “electric” and “magnetic” parts of our connections.

I once went through the trouble of summarizing their work:

Freed, Moore, Segal on p-Form Gauge Theory, I

Freed, Moore, Segal on p-Form Gauge Theory, II

This is very closely related to some work by Gomi, which appeared around the same time:

Gomi on Chern-Simons terms and central extensions of gerbe gauge groups

Gomi on Reps of p-Form Connection Quantum Algebras

It is remarkable

a) how very closely related this is to the starting point of lqg

b) how little interaction there has, apparently, been.

But it is all part of one idea. One could give a discussion about how Freed-Moore-Segal’s description of $p$-form connections is an example for a “quantum theory of a charged $n$-particle”, too.