The String Coffee Table

As I mentioned last time, the claim is that quantizing an abelian chiral $p$-form connection field (meaning one with Hodge self-dual curvature) on a $2p$-dimensionsal space of the form $M = X\times \mathbb{R}$ leads to the phase space

of $p-1$-form connections on $X$ modulo gauge tranformations (aka differential characters, aka Deligne cohomology, aka connections on $p-2$-gerbes), whose quantization amounts to passing from the group $\hat H^p$ to a Heisenberg group central extension controlled by the (“cup product”) pairing

which on topologically trivial conections is the Chern-Simons term

In fact, one important point made in [FMS II] (p. 33) is that in order to get “level 1” central extensions this way, one needs to correct this pairing slightly.

In order to see this, first notice that the cup product pairing is graded commutative. For $\hat A_1 \in \hat H^{p_1}$ and $\hat A_2 \in \hat H^{p_2}$ we have

which can immediately be checked for topologically trivial connections.

Next, one needs some facts about Heisenberg groups [FMS II, section 3.1] .

Heisenberg groups.

Heisenberg groups are maximally noncommutative central extensions $\tilde G$ of abelian groups $G$ by $U(1)$

in the following sense.

All these central extensions have elements that are pairs $(g,a) \in G\times A$, with the product essentially that of $G\times A$, but twisted by a function

as

Of course for this to be associative $c$ has to satisfy the cocycle equation

and $c$ may differ by a coboundary

without changing the central extension.

There is a more invariant object than $c$ which controls these extensions, namely the quotient

which appears as the group commutator of $\tilde G$:

This $s$ is in particular invariant under transformations of $c$ by group coboundaries. Furthermore, it has the following three properties. It is

1) alternating: $s(g,g) = 1$

2) bimultiplicative: $s(g_1 g_2, g_3 g_4) = s(g_1,g_3) s(g_1,g_3)s(g_2,g_3) s(g_2,g_3)$.

As a consequence of these two properties, it is also skew: $s(g_2,g_1) = s(g_1,g_2)^{-1}$.

The point of all this is that there is a theorem [FMS II, p. 23] whose proof is given in [FMS I, prop A.1, p. 22], that central extensions by $U(1)$ of topological abelian groups are, up to noncanonical isomorphism, bijectively specified by such continuous maps $s : G\times G \to U(1)$ which are alternating and bimultiplicative.

Hence it’s the bihomomorphisms $s$ which allow us to conveniently describe central extensions.

The standard example to keep in mind is that of textbook quantum mechanics, where we have some vector space $V$ thought of as an addtitive group of translations, as well as its dual $V^*$, thought of as the group of translations in momentum space, and the central extension of $V \otimes V^*$ is given by the canonical symplectic form $\Omega$ as the cocycle $(q,p) \mapsto e^{2pi\Omega(q,p)}$.

This gives rise to a Heisenberg group, and of course this is the example that gave the name to Heisenberg groups. In general, a Heisenberg group is a $U(1)$-central extension of an abelian group, for which the above map $s : G\times G \to U(1)$ is nondegenerate, in the obvious sense.

Coming back to the cases we are interested in, one is tempted to define a central extension of the group of connections $\hat H^p$ by taking $s$ to be the cup product pairing.

However, this is not alternating! While the cup product pairing is indeed skew in $4k + 2$ dimensions, this only implies that

which, in the presence of torsion subgroups, does not suffice to make $\hat A \star \hat A$ vanish, but only says that $\hat A \star \hat A$ is two torsion. In fact, Gomi has shown in

Kiyonori Gomi
Differential characters and the Steenrod squares
math.AT/0411043

that

where $\nu_{p-1}$ is some characteristic class known as the Wu class of $X$, and $[\hat A]$ is the characteristic class of $\hat A$.

There are two ways out of this. One is the one followed by Gomi, the other is the one followed by FMS.

1) Instead of letting the cup product define the map $s$, we let it define directly the group cocylce of the central extension [Gom, Lemma 3.2].

Due to the graded commutativity of the cup-product, one finds from this [Gom, Lemma 3.4]

where (as hopefully before) $p$ is the degree of the curvature form of $\hat A_1$ and $\hat A_2$, with $X$ being $(2p-1)$-dimensional.

Hence we find two things:

i) Using the above theorem on how $s$ classifies central extensions, we see immediately that for $p$ even the central extension is trivial [Gom, Theorem 3.5].

ii) Moreover, for $p$ odd the central extension is “level 2” in the sense that we have an extra factor of 2 in the exponent. As we noticed above, this factor ensures that the form $s$ is indeed alternating.

2) In [FMS II, p. 35] it is argued that for physical applications we do need Heisenberg groups with “level one” central extensions. Hence FMS remove the above factor of 2 and instead add another term to the exponent, such that the form $s$ becomes alternating.

The failure of the original guess for $s$ to be alternating is measured by the Wu class $\nu_{p-1}$ of $X$, and the claim is that switching the sign of $s$ by

where

makes $s$ a nondegenerate alternating bihomomorphism $s : \hat H^p \times \hat H^p \to U(1)$, and hence gives rise to a Heisenberg group extension of the phase space $\hat H^p$ of self-dual connections.

The FMS claim is that this is the group of observables for chiral $(p-1)$-form gauge theory in $2p$ dimensions, where $p$ is odd.

As a consistency check, one finds that for $p=1$ the above prescription does in fact reproduce the worldsheet quantization of the bosonic string compactified on a circle [FMS II, p. 34], namely here the Heisenberg group is precisely the level-1 Kac-Moody central extension $\hat LU(1)$ of the loop group of $U(1)$.

Accordingly, using the “level 2”-quantization Gomi instead arrives at $\hat LU(1)/\mathbb{Z}_2$ [Gom, prop. 3.6].

In fact, in remark 5 on p.12 of [Gom] the author mentions the problem of finding a central extension for arbitrary $p$ that would be to the level-2 central extenion of $\hat H^p$ like $\hat LU(1)$ is to $\hat LU(1)/\mathbb{Z}_2$.

As far as I understand, this problem is solved by FMS as described above.

If the above claim is right, that the Heisenberg central extension of $\hat H^p$ is the algebra of observables for chiral $(p-1)$-form gauge theory in $2p$-dimensions, then we need to find representations of the Heisenberg group extension of $\hat H^p$, because these will be the Hilbert spaces of states of the theory.

Representations of Heisenberg groups.

One central question we may want to ask is this:

Given the quantization of flux observables as above, what are the superselection sectors ($\to$) of the quantum theory?

In other words

What are the irreducible representations of the Heisenberg central extension of $\hat H^p$?

The general answer to that is given in prop. A.5 (p. 26) in [FMS I].

Roughly, the answer is that unitary irreps of the central extension $\tilde G$ are in bijection with irreps of the center of $\tilde G$.

Applied to finite-dimensional Heisenberg groups, this is essentially the Stone-von Neumann theorem ($\to$).

For infinite-dimensional groups like the extensions of $\hat H^p$ we need to have a polarization on $\tilde G$ and demand the irrep to be of “positive energy”.

Now, for a Heisenberg group we demanded the bihomomorphism defining the central extension $s : G\times G \to \mathbb{R}/\mathbb{Z}$ to be non-degenerate. This nondegeneracy implies [FMS I, p. 22] that the center of $\tilde G$ is precisely $Z = U(1)$. But this means that there is only one group homomorphism $Z \to U(1)$, hence only one (positive energy) irrep (up to a choice of polarization on $G$) of $\tilde G$.

The situation discussed by Gomi is different. As discussed above, Gomi uses a bihomomorphism $s$ different from that proposed by FMS. As a result, he finds not one irrep (up to equivalence), but

of them [Gom II, theorem 1.3, p. 3]. Here

is the $(p-1)$st Betti number of $X$ and

is the number of elements $t \in H^p(X,\mathbb{Z})$ with $2t = 0$.

In fact, for the case $p=3$ and assuming that the second and third integral cohomology of $X$ are torsion free, the same result had been obtained, by rather different looking means, in

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

By what I said, it seems that this is in contradiction to what FMS are claiming (not on the math side, but regarding which cocycles to use in applications to physics). But maybe I am missing something.

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