The String Coffee Table

There is a math way and a physics way to understand the motivation for the question addressed in this paper.

Consider first the more mathematical point of view.

Given the group $G=U(1)$, we may consider its loop group, i.e. the group of maps

There is a canonical group cocycle on this loop group

Of course such a cocycle exist for every simple, simply connected compact Lie group, but for our present purposes we restrict attention to the case $G=U(1)$.

Gomi notices (somewhat paraphased by myself here) the following:

$•$ Being a group valued function, we may regard, trivially, every element $f:S^1\to G$ as a connection on a (-1)-gerbe on $S^1$ (a 0-bundle).

$•$ Accordingly, we may regard $\mathrm{df}$ as the corresponding curvature 1-form of that (-1)-gerbe.

$•$ Hence the integrand of the above cocycle looks somewhat like a Chern-Simons term for a (-1)-gerbe.

$•$ Representations of the centrally extended loop group play a major role for conformal 2-dimensional quantum field theory.

That’s the initial observation, which serves as a motivation to regard this situation as a special case of the following much more general concept:

$•$ Given any $n$-gerbe with connection, we can represent it equivalently by

- a cocycle in Deligne hypercohomology ($\to$)

- a Cheeger-Simons differential character

Either way, there is a product operation on these elements, called the cup product, which is essentially a globally well defined version of forming a Chern-Simons term.

So if we have two abelian $(n-1)$-gerbes with connection on a $(2n+1)$-dimensional manifold, represented by Deligne cochains ${\omega }_1=(f_{i_0,\cdots ,i_n},\cdots ,A_i)$ and $\omega \prime =(f{\prime }_{i_1,\cdots ,i_{n+1}},\cdots ,A{\prime }_i)$, with local connection $n$-forms $A_i,A{\prime }_i\in \Omega (U_i)$, we can form the cup product $2n$-gerbe represented by

You can find the detailed formula on p. 6 of Gomi’s paper, or alternatively in Brylinski’s book Loop spaces, characteristic classes and geometric quantization.

For the present purpose, the important point to notice is that the cup product gerbe has a connection form which is locally precisely the Chern-Simons form (of $A$ if $A\prime =A$) obtained from the local connection forms of the two original gerbes, and that one can extend this Chern-Simons cup product consistently globallly by appropriate pairings on the other entries of the Deligne cochains.

The analogous cup product of course exists also for Cheeger-Simons differential characters.

Good, so given a $(n-1)$-gerbe with connection on a $(2n+1)$-dimensional space $X$, we can always pass to the corresponding Chern-Simons $(2n-1)$-gerbe, consistently.

For $X=S^1$ the circle and hence $n=0$, we get a Chern-Simons 1-gerbe from a $(-1)$-gerbe. Above we noticed that its $(2n+1=1)$-holonomy around the circle is nothing but a cocycle on the group of all $(-1)$-gerbes on the circle, hence a cocycle on the loop group.

$•$ The starting point for Gomi’s considerations is the observation (his Lemma 3.2 on p. 10) that this generalizes. More precisely, that in the above situation the holonomy of the Chern-Simons $2n$-gerbe obtained from a $(n-1)$-gerbe is always a cocycle on “the group of these $(n-1)$-gerbes”, where this group is taken to be the corresponding Deligne cohomology group.

$•$ As the next step, Gomi proposes (bottom of p. 3) to study the representation theory of these centrally extended gerbe groups, remarking that this would be a generalization of the representations of the centrally extended loop group, which is known to be so tremendously important for 2-dimensional quantum field theory, in particular for the WZW model.

In fact, there is more physical motivation. Gomi mentions relations of Chern-Simons functionals to the theory of chiral $p$-forms. But there is still more.

All this is related to a complex of ideas which relates self-dual (“chiral”) $p$-form quantum fields in $2p$-dimensions to Chern-Simons action functionals on $2p\pm 1$ dimensions.

From a physical point of view, aspects of this are for instance in particular discussed in

Edward Witten
AdS/CFT correspondence and topological field theory
hep-th/9812012

and

Erik Verlinde
Global aspects of electric-magnetic duality
hep-th/9506011.

I am very grateful to Jens Fjelstad for immediately recalling these two papers.

In the first paper, Witten discusses in detail aspects of the Chern-Simons term that appears for type II string theory compactified on $X\times S^5$ in the presence of

units of 5-form flux. The action functional reads

where $B_{\text{RR}}$ is the RR-2-form and $B_{\text{NS}}$ the Kalb-Ramond 2-form.

Of course this expression, as written, is ill defined, since both 2-forms in general only exist locally. The true meaning of this integral is as the 5-volume holonomy of a Chern-Simons 5-gerbe obtained from two 1-gerbes.

Witten mentions, in footnote 2 on p. 3 of the above paper, that the right way to interpret the above is hence as the holonomy integral of the cup product of two Cheeger-Simons differential characters, which is equivalent to the fomulation in terms of cup products of Deligne cocycles discussed above.

In fact this must still be a slight lie, since $B_{\text{RR}}$ is not, as far as I am aware, the 2-form connection of a gerbe, in contrast to $B_{\text{NS}}$, which is. I guess we have to/want to assume $B_{\text{RR}}$ here to be globally defined.

Witten discusses in great detail the quantization of these 2-form fields, which does involve a Heisenberg group central extension of the additive group of 2-forms ($\simeq$ 1-gerbes) (equation 3.6). While not unrelated, this particular central extension is however not quite the one that Gomi is considering.

In order to make the connection, we should look at the relation between the Chern-Simons kind of central extension appearing in the canonical loop group cocycle, and similar central extensions as they appear in $p$-form quantum field theory - where $p=0$ in this case.

For the simplest toy example, that of 2-dimensional conformal field theory, this has, in a manner which is very suitable for the formalism of the discussion used here, nicely been spelled out for instance on pp. 5 of

Robert C. Helling, Giuseppe Policastro
String quantization
hep-th/0409182.

We have a self-dual “1-form field-strength” $(\partial X)\mathrm{dz}$ in 2 dimensions, the “left moving worldsheet $u(1)$-current” and when we consider the exponentiated operators

for any suitably nice

they live in the Heisenberg group with group commutator

where

is the Chern-Simons-like cocycle (at least after you adjust all the signs and prefactors correctly, which I won’t do here).

As far as I understand, the program outlined by Gomis would serve the purpose of better understanding situations similar to this one, but for CFTs in dimension 4, 6, etc.

In Gomi’s discussion, the intimate relationship between chiral $p$-forms in $2p+2$ dimensions and Chern-Simons theories of $p$-forms in $2p+1$ dimensions is reflected in the generalization of the Segal-Witten reciprocity law from $p=0$ to arbitrary $p$ (pp. 15 in his paper).

So let $W$ be a manifold of even (real) dimension

with boundary $\partial W$ of dimension ${\dim }_{ℝ}\partial W=4k+1.$

Consider local $2k$-forms $B^{2k}$ on $W$, being the local connection forms of $(2k-1)$-gerbes on $W$. Call the complexified (Deligne cohomology) group of these $(2k-1)$-gerbes $G(W{)}_{ℂ}$.

We can restrict a gerbe on $W$ to the boundary. This induces a restriction map

Now, given the previous discussion, we can similarly consider the group $G(\partial W)$ of $(2k-1)$-gerbes on the boundary $\partial W$, and in particular the central extension $\tilde{G}(\partial W)$ of that group by that Chern-Simons-like cocycle. Moreover, we can pull back that centrally extended group of gerbes on the boundary to the group of gerbes on the bulk to obtain

and thus a central extension also of the group $G(W)$ of gerbes living over the even-dimensional bulk

where I am suppressing the subscript ${\cdot }_{ℂ}$ everywhere.

Gomi shows that the cocycle $S_{\text{bulk}}$ for the bulk extension $r^{*}\tilde{G}(\partial W)$ is - just as you would expect from the physics - given no longer by a Chern-Simons-like term, but by something of the form of the action functional for a self-dual form field

where $B$ and $B\prime$ denote local connection $2k$-forms, and $H_B$, $H_{B\prime }$ the corresponding $(2k+1)$-form curvatures.

Let there be a Riemann metric on $W$ and denote by $G^{+}(W)$ the subgroup of $(2k-1)$-gerbes whose field strength are self-dual, i.e. such that

Gomi then proceeds to show that, restricted to the space of self-dual $2k$-forms in the bulk, the central extension splits, in the sense that we can find an injective group homomorphism

thus finding a subgroup where the central extension vanishes:

for $B,B\prime$ self-dual.

I expect that this has some deep interpretation in terms of conformal field theories involving self-dual $p$-forms. But I do not know which one.