The String Coffee Table

The setup.

The issue at hand is that of gauge theories involving abelian connection $p$-form fields for arbitrary $p$.

For $p=1$ this includes ordinary electromagnetism. More general examples from $p=0$ (remarkably) up to $p=5$ play an important role in various contexts.

In special cases, these connection $p$-form fields can be understood as connections on $(p-1)$-gerbes and can alternatively be described by Deligne cohomology or by Cheeger-Simons differential characters.

Without a connection on them, these gerbes are classified by $(p+1)$st integral cohomology. Adding the connection to them can be regarded as passing to a differential version of integral cohomology.

But in fact, there is a differential version [FMS I, p. 13] of every generalized cohomology theory ($\to$), not just for simplicial integral cohomology, but also for K-theory and elliptic cohomology, for instance.

A large chunk of examples for higher $p$-form gauge theories involves RR-forms, which correspond not to ordinary cohomology, but to K-theory. Hence we ultimately need the differential version of K-theory, too.

From this point of view, it are the Cheeger-Simons differential characters which are the more natural tool to study connections on higher gerbes.

A rather nice introduction to these differential characters is given in [FMS II, section 2].

Briefly, it works like this:

We want to describe an abelian $p$-connection $\hat A$, hence something that associates volume holonomies to $p$-cycles:

But we also want this to be smooth in a certain sense. Since curvatures of abelian $p$-connections are supposed to be globally defined $p+1$-forms, we simply demand that the homomorphism $\hat A$ does indeed come from a curvature $(p+1)$-form whenever it is evaluated on a boundary:

Together with the curvature $(p+1)$-form, one can extract a characteristic class in $H^{p+1}(X,\mathbb{Z})$ from the differential character $\hat A$. The image of real deRham cohomology of these two is required to coincide. This is the sense in which the differential character is a differential refinement of integral cohomology.

Freed-Moore-Seiberg mainly set up their formalism to make the point that

Fact. In the presence of a nontrivial torsion subgroup of $H^{d}(X)$ the Hilbert space of $p$-form connections does not admit a simultaneous grading by units of electric and magnetic flux.

There is much more in their papers, but here I’ll just briefly indicate how this result comes about.

The space of all connections.

We denote

$\bullet$ the space of all $U(1)$ $(n-1)$-bundles with connection on $M$, modulo gauge transformations

otherwise known as

$\bullet$ the space of abelian $(n-2)$-gerbes with connection on $M$, modulo gauge transformations

or

$\bullet$ the space of differential characters of degree $n-1$ of $M$

or equivalently as

$\bullet$ the $n$th Deligne cohomology of $M$

by

For generalized abelian gauge theories, this space is our configuration space. Clearly.

A more detailed understanding of this space is obtained by realizing that it sits inside two different exact sequences, namely

Here $A^p(X)$ denotes the space of $p$-forms on $X$, and $A^p_\mathbb{Z}(X)$ the subspace of forms with integral periods.

(A) The map

is that which assigns to each $(n-1)$-connection $\hat A$ its curvature $n$-form $F_{\hat A}$.

Hence the first exact sequence says that the space of connections with vanishing curvature can be identified with $H^{n-1}(X,\mathbb{R}/\mathbb{Z})$. More on that in the next subsection.

(B) The map

is that which assigns to each $(n-1)$-connection $\hat A$ the characteristic class $[\hat A]$ of the $(n-2)$-gerbe that the connection lives on.

The second exact sequence hence says that connections on trivial $(n-2)$-gerbes are precisely those given by globally defined $(n-1)$-forms (module gauge transformations).

The fact, mentioned above, that the image in deRham cohomology of curvature and characteristic class coincide can hence be expressed by the commutativity of the following diagram [Gom II, (4)]

A still better picture of the space of all $p$-form connections is obtained by noticing (e.g. [Gom II, lemma 3.3, p. 9]) that the second of the above two exact sequences is actually split, meaning that the space of all $(n-1)$-form connections can be realized as the direct product of the space of all globally defined $(n-1)$-form connections and the space of all characteristic classes of $(n-2)$-gerbes, up to isomorphism:

As FMS emphasize ([FMS II, p. 11]) this justifies the procdure familiar in physics, where one thinks about an arbitrary connection $\hat A$ as a topologically nontrivial part $\hat A_0$ plus a globally defined part $A_1$, as $\hat A = \hat A_0 + A_1$. (Here the hat $\hat \cdot$ always inicates that the letter in question denotes a connection which is not necessarily given by a globally defined form.)

One can still say more by giving a more detailed picture of the space of all $(n-1)$-forms modulo closed forms. It turns out ([FMS II, p. 12]) that this space is a torus bundle

over the vector space of $(n-1)$-forms modulo closed forms. The fiber of this bundle looks like

the space of closed forms modulo integral forms. This space if the space of topologically trivial and flat connections (which plays a special role below). We can think of this space as a higher-dimensional torus, so that the above is indeed a torus bundle.

That the base of this bundle is indeed a vector space can be seen by choosing any Riemannian metric on $X$, and noticing that then then Hodge decomposition theorem says that the space of forms modulo closed forms can be identified with the image $\mathbf{d}^* (A^{n})$ of

In summary, we find that the space $\hat H^n(X)$ of $(n-1)$-form connections can be imagined as consisting of one connected component for each class in $H^n(X,\mathbb{Z})$, each of which looks like a torus bundle over a vector space.

In [Gom II, cor. 3.2] this vector bundle is identified, after a choice of metric, with the trivial bundle

The space of flat connections.

The space of flat connections is

namely the space of group homomorphisms from $(n-1)$-chains to $\mathbb{R}/\mathbb{Z}$ which vanish on cycles - up to gauge transformations.

Inside this space we have the flat connections which are in addition topologically trivial, meaning that they can be given by a globally defined connection $(n-1)$-form on $M$. These are given by

As usual, we can identify this with the space $\mathbb{H}^{n-1}(M)$ of harmonic forms, if we take care to divide out by gauge equivalent connections:

The nontrivial part is to characterize those flat connections, which are not topologically trivial. Clearly, these must have characteristic classes with vanishing image in deRham cohomology, hence they must be torsion classes. In fact, the space of nontrivial flat $n$-form connections can be identified with the torsion subgroup $\mathrm{Tors}(H^n(M)) \subset H^n(M,\mathbb{Z})$.

More precisely, $H^{n-1}(M,\mathbb{R}/\mathbb{Z})$ sits in this exact sequence

where $\beta$ is the Bockstein map ($\to$).

The Hilbert space of connections…

is hence naturally taken to be the space of square integrable functions on configuration space, hence of the space of connections modulo gauge transformations,

… its grading by units of magnetic flux…

One of the subtleties of the entire business here is that we are doing quantum mechanics on a disconnected configuration space.

By the above, each connection in $\hat H^n(X)$ comes with a curvature $n$-form $F$ and a characteristic class $c$.

The characteristic classes clearly label the connected components of configuration space. Accordingly, we have a grading of our Hilbert space by these classes:

Notice that these characteristic classes correspond to the curvature of our connection, which measures the magnetic flux. Hence this is a grading by magnetic flux.

…, its grading in units of electric flux…

In the full quantum theory, however, due to Dirac charge quantization both magnetic and electric flux are quantized. The characteristic class for the electric charge lives in $H^{d-n}(X,\mathbb{Z})$.

By a direct generalization of the way momentum eigenstates are characterized in elementary quantum mechanics, we can characterize a state $\psi \in \mathcal{H}$ as a state of definite electric flux $\hat E \in \hat H^{d-n}$ by the property that under translation in the space of connections it transforms by a phase

If we want to identify states that are supported on a given characteristic class of electric flux, it turns out ([FMS II], p.20) that we have to demand the above for all flat connections $\hat \phi$.

Hence we could contemplate grading our Hilbert space also by units of electric flux

… and the incompatibility between both.

But we cannot in general use both gradings at the same time. The reason is that, as mentioned above, the space of flat connections contains topologically nontrivial connections, namely those in the image of the Bockstein homomorphism $\beta$. Hence translation by these does not respect the classes of magnetic flux. As noted above, these nontrivial flat connections are given by the torsion subgroup of $H^n(X,\mathbb{Z})$.

So we can grade the Hilbert space of connections by electric and magnetic flux only up to torsion.

A more precise statement.

The failure of the joint diagonalizability of electric and magnetic flux can be expressed more quantitatively.

For definiteness, restrict attention for the moment to standard electromagnetism, hence to a $(n = 1+1)$-form connection on a (3+1)-dimensional space of the form $M = Y \times \mathbb{R}$.

The basic observables are the electric and magnetic fluxes through arbitrary 2-cycles. By passing to dual currents, we may represent these cycles by closed 1-forms $\eta$ and define the observables

One finds for these the classical Poisson brackets

which vanish if the $\eta$s are indeed closed.

However, a globally defined closed 1-form $\eta$ is just the topologically trivial case of a flat connection. As discussed above, a major role in the quantum theory is played by those flat connections in $H^1(X,\mathbb{R}/\mathbb{Z})$ which are not topologically trivial.

As discussed last time ($\to$) in the context of a preprint by Gomi hep-th/0504075, there is a natural generalization of the pairing

to arbitrary (flat or non-flat) connections $\hat A$, deriving from the cup product on Deligne cohomology/differential characters.

whenever $\mathrm{dim} X = 2n-1$ and $\hat A_i \in \hat H^{n}$, and, moreover, this pairing serves as a cocycle on the group of connections.

In particular, when restricted to the space of flat connections $\omega_1,\omega_2 \in H^1(X,\mathbb{R}/\mathbb{Z})$ one finds (see [FMS I, p. 11] and [Gom II, Lemma 4.1 - 4.3]) that

(where $\beta$ is, as above, the Bockstein homomorphism) which denotes what is called the link pairing or torsion pairing

in cohomology.

This nontrivial pairing, then, leads to a nontrivial commutator, in the quantum theory, of the fluxes

which prevents the simulaneous diagonalization of electric and magnetic flux number in the presence of torsion classes.

I have (in as far as I didn’t gloss over lots of details and introduced plenty of imprecisions) followed FMS I in concentrating on the example of ordinary (3+1)-dimensional electromagnetism here. All these considerations generalize [FMS II, p. 25-26].

Applications.

Using this powerful formalism, one recovers various results obtained by more or less pedestrian means before.

An analysis “by hand” of the analogous quantization of 2-form electromagnetism on a (5+1)-dimensional spacetime has in particular been spelled out in

M. Henningson
The quantum Hilbert space of a chiral two-form in d = 5 + 1 dimensions
hep-th/0111150,

assuming, however, the absence of torsion subgroups.

[…]

The most interesting application is possibly that of

Quantization of self-dual (“chiral”) fields.

As I mentioned in the previous entry on Gomi’s paper ($\to$) a fascinating aspect of higher $p$-form gauge theories is that some of them, namely the chiral or self-dual ones, generalize several interesting aspects of the theory of conformal 0-forms in 2-dimensions, known as chiral bosons on the string worldsheet.

These theories play an important role in several places. In particular on the M5-brane there is a gauge theory of a self-dual 2-form, which is, when compactified on a torus, the geometrical mechanism behind the S-duality of 4-dimensional 1-form gauge theory.

This is in principle an old topic, with one canonical reference (purely on the general chiral aspect) probably being

X. Bekaert, M. Henneaux
Comments on Chiral p-Forms
hep-th/9806062

Freed-Moore-Segal, however, make a clean sweep and redo the theory of chiral $p$-forms in their framework, announcing future publication of discussion that this new way is indeed the right way [FMS II, p. 31].

Briefly, they present heuristic arguments which assert that the quantization of a connection $p$-form with Hodge self-dual $(p+1)$-form field strength in $2p + 2$ dimensions involves the central extension of a single copy of the space of connections $\hat H^{p+1}$, instead of two of them (namely “magnetic” and “electric”) as we had in the Maxwell-example above.

This is in fact more than plausible if compared to the situation of $p=0$, which is the conformal theory of a $U(1)$-valued boson living in 2-dimensions, otherwise known as the string compactified on a circle.

Here, our connection 0-form goes by the name $X$, and its “electric” field strength $\mathbf{d}X$ and “magnetic” field strength $*\mathbf{d}*X$ combine to the self-dual field strength $\partial X$, which contains all the relevant information. See the example on p. 34 of [FMS II] for more details.

Crucial properties of this standard example of chiral conformal field theory generalize to higher dimensions surprisingly well.

Assume again a gauge theory of $2k$-form connections on a manifold $M = X \times \mathbb{R}$ of dimension $4k+2$, with the standard action of the form

One finds that one can decompose tangent vectors $\delta \hat A$ to the solutions of the equations of motions of this action into “left moving” and “right moving” parts

[…]