The String Coffee Table

Fix once and for all

$\bullet$ an integer $k \in \mathbb{N}$

$\bullet$ a $(4k+1)$-dimensional manifold $X$ .

$\bullet$ some Riemannian metric $g$ on $X$.

Denote

$\bullet$ by $\hat H^n(X)$ the group of gauge equivalence classes of $(n-1)$-form connections, here to be thought of as realized as the $n$th Deligne hypercohomology of $X$

$\bullet$ by

the group of $2k$-form connections on $X$.

The strategy.

The strategy is to decompose the space $\hat H^n$ of all connections into a product of subgroups, according to the general analysis reviewed in the section “The space of all connections” in I. As described there, one finds

The last factor contains the global gauge sectors and the first two factors encode the connections given by globally defined $2k$-forms, the first factor containing the flat ones (compare the section “The space of flat connections” in I).

For the purpose of finding representations, Gomi goes one step further and also decomposes $H^{2k+1}(X,\mathbb{Z})$ into a free part and a pure torsion part.

This is possible, but not canonical. Hence Gomi starts his construction by making a choice

choice number 1) a decomposition

of the integral cohomology group into a free and a torsion part.

With this choice performed, $\mathbf{G}(X)$ decomposes (p. 17) into four groups as

The strategy is to represent this piecewise.

For that Gomi makes

choice number 2: some homomorphism

representing the group of harmonic $2k$-forms up to integral harmonic $2k$-forms

and

choice number 3) a finite dimensional projective unitary representation of the torsion subgroup

coming from the restriction of the Chern-Simons cocycle on torsion connections

Here

$\bullet$ $T^{2k+1} = T^{2k+1}(X)$ is the torsion subgroup (p. 12) of the integral cohomology group $H^{2k+1}(X,\mathbb{Z})$.

$\bullet$ $F^{2k+1}$ is a group isomorphic to $H^{2k+1}/T^{2k+1}$, whose rank

is the $(2k+1)$st Betti number of $X$ (def 4.4, p. 13).

$\bullet$ $\mathbb{H}^{2k}(X)$ is the group of harmonic $2k$-forms on $X$ (prop. 3.1, p. 8) with respect to the chosen metric $g$

$\bullet$ $\mathbb{H}^{2k}(X)_\mathbb{Z}$ is accordingly the group of harmonic $2k$-forms on $X$ with integral periods

$\bullet$ $A^{2k+1}(X)$ is the group of globally defined $(2k+1)$-forms on $X$

$\bullet$ $\mathbf{d}^*(A^{2k+1}(X))$ is accordingly the image under $\mathbf{d}^* \propto * \mathbf{d} *$ of all $(2k+1)$-forms.

The strategy is then to

1) construct the Heisenberg extension first only on $\mathbf{d}^*(A^{2k+1})$,

2) then tensor the result with the 1-dimensional rep of $\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}$ using $\lambda$,

3) then obtain from this an induced representation of $\mathbf{G}_\omega(X)$, which is the group of connections whose characteristic class is in the free part $F^{2k+1}$ of $H^{2k+1}(X,\mathbb{Z})$

4) finally tensor this result with the finite dimensional rep of the extension of $T^{2k+1}$.

This strategy is built on the known example $k=0$.

Example. Let $k=0$ and $X = S^1$, the circle. An abelian $k$-form connection on $X$ is now simply a circle valued function on the circle - in the context of $p$-form gauge theory to be thought of as the worldsheet boson of a string compactified on a circle.

Accordingly, the group of all these 0-form connections is simply the loop group of $U(1)$

This decomposes as above, with

$\bullet$

the space of constant 0-forms on the circle, i.e. maps that send the entire string to a single point

$\bullet$

the space of maps that are total derivatives and hence have no winding around the circle

$\bullet$

the space of linear maps with winding

$\bullet$

Applied to this motivating example, Gomi’s more general construction is supposed to reproduce the well-known positive energy reps of $\hat LU(1)/\mathbb{Z}_2$.

So we need to understand the four steps of the above strategy

Step 1) Projective irrrep of $\mathbf{d}^*(A^{2k+1})$

On the globally defined $2k$-forms in $\mathbf{d}^*A^{2k+1}$, the Chern-Simons-like cocycle that controls the entire business is simply the integral

There is a general theorem (from Pressley/Segal, prop. 5, p. 18 in Gomi’s paper) that

$\bullet$ given a group cocycle on a vector space $V$, regarded as an abelian group

$\bullet$ and given a continuous complex structure $J : V \to V$ on $V$ which is compatible with the cocycle and turns it into a positive definite form

$\bullet$ then there is a continuous unitary rep

of the central extension $\tilde V$

$\bullet$ such that the center acts by scalar multiplication

$\bullet$ which is an irrep if V is seperable and complete with respect to $S_X(J\cdot,\cdot)$.

One of the more technical problems that Gomi indicates how to solve (prop. 3.1, p. 8) is the

construction of such a complex structure on the space of $2k$-forms.

The idea is to start with the Hodge inner product

and find a complex structure $J$ such that we get the Chern-Simons cocycle term from this, i.e. such that

Clearly, this would require setting $J$ equal to $* \mathbf{d} = -\mathbf{d}^* *$. However, this operator is not a complex structure, since it does not square to $-1$.

In order for this guy to square to $-1$ we need to divide it (using functional calculus) by its norm. So we set

This now is a complex structure, but no longer relates the inner product with the cocycle. In order to remedy this we need to modify the inner product, too.

As Gomi describes (p. 9) the right solution is to take the inner product $(\cdot,\cdot)_V$ induced from the modified norm

where $\{\psi_i\}$ is an orthonormal system of eigenvectors of $\Delta = \mathbf{d}\mathbf{d}^* + \mathbf{d}^* \mathbf{d}$ with eigenvalues $\{\ell_i\}$, of which $b = b_{2k}(X)$ vanish.

It is at this point that use is made of the fact that we are currently restricting attention to the space $\mathbf{d}^* A^{2k+1}(X)$. For, on this space the operator $*\mathbf{d}$ squares to minus the Laplace operator

Therefore, if we restrict the eigenbasis $\{\psi_i\}$ to an eigenbasis $\{\phi_j\}$ only of $\mathbf{d}^* A^{2k+1}(X)$, it becomes a simultaneous eigenbasis of $\Delta$ and $*\mathbf{d}$, where $*\mathbf{d}$ has eigenvalues $\{\sqrt{|\ell_i|}\}$.

The norm $\Vert\cdot\Vert_V$ is built in such a way that $J = \frac{*\mathbf{d}}{|*\mathbf{d}|}$ is an isometry on $\mathbf{d}A^{2k+1}$ and such that it relates the inner product with the cocycle

as desired. Hence if we take

to be the completion of $\mathbf{d}^*A^{2k+1}$ with respect to this norm, Pressly and Segal guarantee us a representation

of the centrally extended group $\tilde V$. Here the representation space $H$ is (prop. 5.1, p. 18, due to Pressley-Segal) essentially the symmetric algebra of the $(+i)$-eigenspace of $J$ (the “positive energy eigenspace”).

Step 2) tensor the result with the 1-dimensional rep $\lambda$ of $\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}$

Simply set (proof of lemma 5.3, p. 19)

Then use the fact that the cocycle $S_X$ vanishes on $\mathbb{H}^{2k}$ to deduce that this is still irreducible if $\rho$ is.

Step 3) form the unitary irrep induced from the subgroup rep

We obtained above a projective irrep for

This is a group of globally defined $2k$-form connections. Hence the corresponding characteristic classes all vanish. In particular, they are not torsion. So this is a subgroup of

the froup of $2k$-form connections whose characteristic class is in the free part $F^{2k+1} \subset H^{2k+1}$ (with respect to the choice of isomorphism $\omega$).

We now induce on $\mathbf{G}_\omega(X)$ a projective irrep from the projective irrep of the subgroup $\mathbf{G}^0(X)$.

This works by

$\bullet$ forming the bundle $\tilde \mathbf{G}_\omega(X) \to \tilde\mathbf{G}_\omega(X)/\tilde\mathbf{G}_0(X)$

$\bullet$ using the subgroup rep $\rho_\lambda : \tilde \mathbf{G}_0(X) \to \mathrm{Aut}(H)$ to associate a vector bundle $\tilde \mathbf{G}_\omega(X) \times_{\rho_\lambda} H \to \tilde\mathbf{G}_\omega(X)/\tilde\mathbf{G}_0(X)$

$\bullet$ noticing that (because all groups are abelian) this still has a left $\tilde \mathbf{G}_\omega(X)$-action

$\bullet$ hence finding a rep of $\tilde \mathbf{G}_\omega(X)$ on the space

of sections of this vector bundle.

Gomi gives a more explicit description of this induced rep for the present case (below equation (13), p. 20), which plays the crucial role for the classification of irreps later. I’ll discuss this below.

4) finally add the rep of the torsion part

The above establishes a unitary projective rep of all but the torsion part $T^{2k+1}$ of $\mathbf{G}(X)$. It turns out one can construct lifts $\sigma$ of every torsion class to a representing connection. (I’ll discuss these $\sigma$ below). Using this we can pull back the Chern-Simons cocycle on connections to $T^{2k+1}$ (lemma 4.1, p. 12) and get the finite dimensional irrep

which we already assumed to be given above.

We combine this with the rep described above to a projective rep of all of $\mathbf{G}(X)$ by tensoring the two representation spaces and sending every connection $\hat A$ to the rep of its torsion part times the rep of $\hat A$ minus the lift $\sigma(t(\hat A))$ of the torsion part $t(\hat A)$ of its characteristic class:

That’s, in outline, the construction of the projective unitary rep of all of $\mathbf{G}(X)$.

It is claimed to be independent of the choice of lift $\sigma$ (remark 3, p. 21) and that different choices of $\omega$ lead to unitarily equivalent reps (prop. 6.2, p. 21).

It is also claimed that the rep obtained this way, when restricted to connections on trivial gerbes, decomposes nicely into a direct sum of reps $H_\lambda$ of these. This property is called admissability (def. 1.2, p. 3). This is supposed to make contact with Freed-Hopkins-Telemann.

The main result (theorem 1.3, p. 3, proven in section 6) obtained from this construction is a classification of the reps constructed above, in particular a characterization of their irreps (hence, physically, of the superselection sectors of the theory).

The claim is that

$\bullet$ all these reps decompose into irreps;

$\bullet$ and there are (up to equivalence)

different irreps, where $b = b_{2k+1}(X)$ is the $(2k+1)$th Betti number of $X$ and $r$ is the number of elements in $H^{2k+1}$ which are their own inverse.

As Gomi notes (p. 4) this coincides (as a special case) in particular with the result of a quantization analysis of 2-form gauge theory obtained by Henningson (see end of II).

While in view of the claims and results of Freed-Moore-Segal there seems to be room for discussion as to what the correct quantization procedure for chiral $p$-form field theories is (I talked about that in II), this does not affect the mathematical result presented. The precise relation to the quantization performed by Henningson would however seem like an interesting question.

Anyway, in order to understand the factor $2^b$ in the above number of irreps, one needs the explit form of the induced irrep $\rho_\lambda^\omega$ of $\mathbf{G}_\omega(X)$ on the space of sections of that vector bundle, which I mentioned above.

The appearance of the Betti numbers.

First of all, it is shown that this vector bundle is in fact trivial and equivalent to $F^{2k+1}\times H_\lambda$. Hence sections are just functions

Gomi finds (p. 20), for $\hat A \in \mathbf{G}_\omega(X)$ the action of $\hat A$ on such a $\Phi$ is given by

Here, as before, $\sigma$ is a lift that associates to characteristic classes a connection on a gerbe with that class. So in particular $\hat A - \sigma([\hat A])$ is a connection on a trivial gerbe.

The crucial part is the exponential term, which involves the Chern-Simons cocycle $S_X$ evaluated on various objects:

Related to the discussion reviewed in II, it is the factor of 2 in the last term which gives rise to certain degeneracies.

In order to make this more explicit, it is noted that if we choose the lift $\sigma$ to take values in harmonic forms, then the last term is precisely a 1-dimensional rep of $\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}$ that enters the definition of $\rho_\lambda$. Hence we can absorb this term by replacing in the above formula

where

sends a harmonic form to the Chern-Simons pairing with that form (def. 4.11, p. 16).

Again, the crucial point is the factor of 2.

Namely (Lemma 4.12, p. 16), we can identify the space of all homomorphisms $\mathrm{Hom}(\mathbb{H}^{2k}/\mathbb{H}^{2k}_\mathbb{Z}, \mathbb{R}/\mathbb{Z} )$ with the free part $H^{2k+1}/T^{2k+1}$ of the integral cohomology, which looks like $\mathbb{Z}^b$. This is where the Betti numbers come in.

Now, as we vary $\xi$ over $H^{2k+1}/T^{2k+1}$ we hit with $\lambda + 2s(\xi)$ (for fixed $\lambda$) only every second homomorphism, due to the factor of 2. This means that there are $2^b$ choices for $\lambda$ which lead to different reps $\rho_{\lambda + 2s(\xi)}$, as $\xi$ varies. This is the key phenomenon that govers the classifications of irreps.

The number of irreps for topologically trivial connections.

Let’s first again restrict attention to the globally defined connections $[\hat A] = [A] = 0$ in $\mathbf{G}_0(X)$. Precisely for these, the funny shift in the last term in the above equation for $\rho_\lambda^\omega$ vanishes. It follows that $\rho_\lambda^\omega(A)$ acts within each fiber of our bundle seperately as (proof of lemma 5.5, p. 20)

In other words, when restricted to connections with trivial classes in $\mathbf{G}_0(X)$, $\rho_\lambda^\omega$ sees the direct sum of all fibers of our bundle, and acts on the fiber over the class $\xi$ as the rep $\rho_{\lambda+2s(\xi)}$. Hence (prop. 5.7, p.21)

If we also take the rep $V$ of the torsion part into account this reads

It is in expressions like these that the remarks on Betti numbers in the previous section apply to: Clearly, there are $2^b$ choices for $\lambda$ such that the representation spaces $\mathcal{H}_{\lambda,V}^\omega |_{\mathbf{G}_0(X)}$ given above differ. In other words, $\lambda$ and $\lambda'$ lead to the same representation space iff they differ by $2s(\xi)$, for some $\xi$.

The number of irreps for the full group.

Gomi now claims, that this situation essentially carries over to the full group of connections.

More precisely, he claims

$\bullet$ (theorem 6.11, p. 24) that the rep $\rho_{\lambda,V}^\omega$ is irreducible precisely if the finite dimensional rep on $V$ of the torsion part is;

$\bullet$ and (theorem 6.10, p. 24) that two such irreps $\rho_{\lambda,V}^\omega$, $\rho_{\lambda',V'}^{\omega'}$ are equivalent precisely if the reps of the torsion parts are and if there is $\xi \in F^{2k+1}$ such that $\lambda' = \lambda + 2s(\xi)$.

As discussed above, this means that for fixed $V$ there are $2^b$ inequivalent irreps. It hence remains to count the equivalence classes of irreps for the group of torsion connections.

Gomi (pp. 24-25) cites a standard fact about projective reps of finite groups, which says that their number is the same of the number of regular elements of the cocycle. In the present case an element is regular with respect to the Chern-Simons cocycle if an only if it is its own inverse. The number of these elements we called $r$.

Hence, in total, there are

equivalence classes of irreps $\rho_{\lambda,V}^\omega$ of $\tilde \mathbf{G}(X)$.