The String Coffee Table

We mainly want to understand and proof the following theorem:

Theorem: The partition function of a supersymmetric $C_n$-linear conformal field theory $E$, with $n$ even, is a weak integral modular form of weight $n/2$.

The plan of J. Ebert’s talk was as follows:

1) modular forms

2) recall elements of CFTs

3) the partition function on the Teichmüller space of tori, the Pfaffian line bundle

4) the semigroup of $E$

5) supersymmetry and the proof of the theorem

1) modular forms

Definition: A modular function of weight $k$ is a holomorphic function $f : \mathbb{H} \to \mathbb{C}$ on the upper half plane $\mathbb{H}$, such that for all $\left( \array{ a & b \\ c & d } \right) \in \mathrm{SL}(2,\mathbb{Z})$ we have

It follows that for every modular function $f$

which again implies that $f$ only depends on

and hence has a Laurent expansion (“$q$-expansion”)

Definition: A modular function $f$ is called a weak modular form if for all but a finite number of $n \lt 0$ we have that $a_n = 0$ in the above expansion.

Furthermore, $f$ is called a modular form if all $a_n$ for $n \lt 0$ vanish.

Finally, $f$ is called integral if all of the coefficients $a_n$ are integral.

Example: The crucial modular forms to know about are

i)

which has weight 12 and is known as the discriminant,

ii) the Eisenstein series

where $\sigma_r(k) = \sum_{d|k}d^r$. These have weight $4$ and $6$, respectively.

Obviously, modular forms can be added and multiplied and yield further modular forms. This gives us a (graded) ring of modular forms. It turns out to be equivalent to the ring of polynomials over three indeterminates given by the above three modular forms, $\mathbb{Z}[c_4,c_6,\Delta]$ , divided by the ideal generated by $(c_4^3 - c_6^2 - 1728\Delta)$. For the ring $MF_*$ of weak modular forms we have instead

2) review of CFT

We need some concepts from Stolz-Teichner’s functorial description of CFT. I will not give the detailed definition of the following items. The main point is that we want to consider a bicategory whose 2-morphisms are (super-)conformal spin cobordisms, decorated with certain generalized Fock spaces on which certain Clifford algebras act. The crucial part of this definition for our purposes ist that for closed surfaces this generalized “Fock space” is not a Fock space but just the top exterior power of the kernel of some Dirac operator on the surface - the Pfaffian line.

Slightly more precisely we have the following.

Denote by $\mathcal{SB}^2_n$ the category whose

- objects are 1-dimensiona spin manifolds $Y$

- morphisms are conformal spin bordisms from $Y_1$ to $Y_2$ labelled by a vector $\psi \in F_\mathrm{alg}(\Sigma)^{-1}$ from some sort of fermionic Fock space on which Clifford algebras cooked up from $\Sigma,Y_1$ and $Y_2$ are represented. (The reader should either look this up in Stolz/Teichner or trust that for a rough understanding of the main point to be discussed here the details can can be safely ignored.)

A CFT of degree $n$ (similar remarks apply to the precise meaning of this degree) is (defined to be) a continuous functor

compatible with certain extra structure floating around, such as monoidal structure and certain involutions.

As mentioned above, by construction we have for closed surfaces that

is the Pfafiian line of some canonical Dirac operator $D^+$ defined on $\Sigma$.

It follows that the CFT $E$ defines for any Riemannian spin surface $\Sigma$ an element

by

where we use

The map $Z_E$ is of course the partition function.

Before continuing, it proves convenient to pass from the language of spin structures and Dirac operators on conformal surfaces to the analogous concepts in complex analysis. Johannes Ebert noted the following small dictionary:

Consider in particular the torus. Here we have four spin structures and $K$ is the trivial holomorphic line bundle. Three of its square roots are nontrivial line bundles. For these we have $\mathrm{ker}(\bar \partial) = 0$.

The fourth spin structure, the periodic one, has the property that $\mathrm{dim}\mathrm{ker}(\bar\partial) = 1$, since this is just the trivial holomorphic line bundle itself.

The important point here is that on any torus there is a canonical spin structure with $\mathrm{dim}\mathrm{ker}(\bar \partial) = 1$.

3) the partition function on the Teichmüller space of tori, the Pfaffian line bundle

For every point in the upper half plane there is a complex torus

and every complex torus is of this form. Two such tori $\Sigma_\tau$, $\Sigma_{\tau'}$ are equivalent if and only if there is a $g \in \mathrm{SL}(2,\mathbb{Z})$ such that

In such a case the biholomorphic map relating both tori is given by

For each such torus $\Sigma$ we have its Pfaffian line $\mathrm{Pf}_{[\Sigma]} := \mathrm{ker}(\bar \partial_{\Sigma,\mathrm{per}\Sigma})^*$ discussed above. These fit together into a holomorphic line bundle

on the torus Teichmüller space $\Tau_1$.

The bundle $\mathrm{Pf}^{-2}$ is always trivial. Its fibers are $\mathrm{Pf}_{\Sigma_\tau}^{-2} = \mathrm{ker}\bar\partial_K$, which are nothing but the spaces of holomorphic 1-forms in one complex dimension. Hence $dz_{\tau}$ with $t_\tau$ the canonical coordinate on $\Sigma_\tau$ is a canonical basis for $\mathrm{Pf}_{\Sigma_\tau}^{-2}$. Hence an explicit trivialization is given by mapping

We have an action of $\mathrm{SL}(2,\mathbb{Z})$ not only on $\Tau_1$ but also on $\mathrm{Pf}^{-2}$ over it. In terms of coordinates used in the above trivialization the action on the fibers is

Therefore for any point $(\tau,x)$ in the Pfaffian bundle over Teichmüller space we have the $\mathrm{SL}(2,\mathbb{Z})$ action

The reason for going through all of this is the following crucial consequence:

An equivariant section of $\mathrm{Pf}^{\otimes 2k}$ has the transformation behaviour of a modular form of weight $k$.

The partition function $Z_E$ mentioned above obviously defines a section of this bundle:

This section is equivariant by construction, since we defined it from a functor $E$ that is by definition insensitive to the transformation $(\Sigma,\psi) \mapsto (g\Sigma,g_*\psi)$ induced by an element $g\in\mathrm{SL}(2,\mathbb{Z})$.

In order to show that the partition function $Z_E$ has a chance of being a modular form we need to show that it is a holomorphic function of $\tau$. In order to get there we first need to re-express the partition function in terms of the transport over the annulus. This is the content of the next section.

4)

We can build the partition function on the torus from a supertrace over the propagator of the annulus. More precisely, let

be the annulus defined by $\tau \in \mathbb{H}$. $A_\tau$ is a spin bordism from the circle $S^\mathrm{per}$ with periodic spin structure to itself. We can evaluate our CFT functor $E$ on this morphism to obtain

where $\Omega$ is a certain canonical vacuum state in the above mentioned (but not explained) algebraic Fock spaces with which our morphisms are supposed to be labelled.

Lemma: The partition function evaluated on the torus is the supertrace of the transport over the annulus

This can be proven by cutting the torus in two equal halfs, each an annulus andf then using various properties satisfied by $E$.

The map

defined by propagation over the annulus, has, by the properties of our CFT functor $E$, the following properties:

$\bullet$ $\rho$ is a semigroup homomorphism

$\bullet$ $\rho(\tau)$ is a $C_n$-linear even operator (but I haven’t explained this yet)

$\bullet$ $\rho(\tau)$ is a Hilbert-Schmidt operator,

$\bullet$ $\rho$ is weakly differentiable, meaning that the function $f_v : \tau \mapsto \left\lbrace \array{ v & \mathrm{if} \tau \in \mathbb{R} \\ \rho(\tau)\cdot v & \mathrm{otherwise} } \right.$ is differentiable for all $v$.

Using this, we can prove the following

Lemma: Any semigroup homomorphisms $\tau$ from the upper half plane to self-adjoint $C_n$-linear Hilbert-Schmidt operators on some Hilbert space $H$ is of the form

where $L$ and $\bar L$ are commuting, even $C_n$-linear unbounded operators on $H$.

Proof: Define

and

(Notice that the notation $\rho(\tau)$ is not supposed to imply that $\rho$ is holomorphic, which it is not. You might prefer to write $\rho(\tau,\bar\tau)$.)

The fact that $\mathbb{H}$ is a commutative semigroup and that $\rho$ is a semigroup homomorphism implies that

Some further important properties of $\rho$ are

i) $\rho(\tau+1) = \rho(\tau)$ (because $A_{\tau+1}$ and $A_\tau$ define the same annulus)

ii) $\rho(-\bar \tau) = \rho(\tau)^*$ (\because $\bar A_\tau = A_{-\bar \tau}$), this implies that both $L$ and $\bar L$ are self-adjoint

It also follows that

and hence that the eigenvalues of $L-\bar L$ are all integral.

5) supersymmetry and the proof of the theorem

With all this information available, we can now quite easily show that the partition function is a modular form – if we introduce supersymmetry.

In stolz/Teichner’s work there is a little gap left which is filled by a hypothesis. We really want to define a susy CFT functor to be one which is defined not on the conformal spin bordisms mentioned above, but on superconformal bordisms. While it seems to be obvious how to do this, people are cautious about this point because the rigorous fortmulation has possibly not be written down yet. Anyway, we expect that when we introduce (1,0) superconformal symmetry (called $1/2$ supersymmetry by Stolz/Teichner and called the heterotic case by physicists) our CFT functor will be defined by $L$ and $\bar L$ such that there is an odd operator $\bar G$ with

The “hypothesis” is that this is the case. So instead of honestly studying CFT functor $E$ on superconformal worldsheets, we apply the above analysis and then restrict attention to the case where the above condition holds.

Now, by the content of section 4) we have that the partition function is given by

Since $\bar L$ has, by assumption of supersymmetry, a symmetric spectrum, it follows by the usual argument that only the kernel of $\bar D$ can contribute nontrivially to the above graded trace

Let $H_k$ be the $k$th Eigenspace of $L$. Then this can be rewritten as

That the sum here is only over integers follows from the “level mathcing condition”, derived above, which says that $\mathrm{spec}(L-\bar L) \subset \mathbb{Z}$.

Finally, note that there cannot be an infinite number of negative $k$ contributing to this sum, since if there were the propagator were not Hilbert-Schmidt, contrary to our definition of the CFT functor. Hence

$Z_E$ is a weak integral modular form.

Note that for each fixed $k$, we can interpret the above sum expression for the partition function as the partition function of a SQM system, whose Hamiltonian is given by the restriction of $\bar L$ to $H_k$.

Hence we should really think of the CFT partition function as a power series with coefficients in the K-theory of a point!

Judgig from the situation sketched in the previous entryI bet there is a neat way to express this in a way that manifestly realizes the CFT partition function as the dimension of a virtual 2-vector space which is the 2-kernel minus the 2-cokernel of some 2-operator. But that’s speculation.