Ochanine's theorem

This is a record of my attempt to try and understand parts of the book Manifolds and Modular Forms. It follows on from this previous post.

Jacobi Forms and Elliptic Genera

Definition For $\tau \in \mathbb{H}$, let $\Psi(\tau,x)$ be an elliptic function in $x$ (i.e., meromorphic and doubly periodic with respect to $2\pi i (\mathbb{Z} \tau + \mathbb{Z}))$. Let $k \in \mathbb{Z}$, and let $\Gamma \subset \text{SL}_2(\mathbb{Z})$ be a congruence subgroup. Then we call $\Psi$ a \textit{Jacobi form} on $\Gamma$ of weight $k$ if
\[
    \Psi \left( \gamma(\tau), \frac{x}{c\tau + d} \right) (c\tau + d)^{-k} = \Psi(\tau, x),
\]
for $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma$.

Notice that the Weierstrass $\wp$-function,
\[
    \wp(\tau, x) = \frac{1}{x^2} + \sum_{\omega \in 2\pi i (\mathbb{Z} \tau + \mathbb{Z}), \, \omega \neq 0}
    \left( \frac{1}{(x - \omega)^2} - \frac{1}{\omega^2} \right),
\]
is a Jacobi form on $\Gamma = \text{SL}_2(\mathbb{Z})$ of weight 2.
Applying Cauchy's integral formula, one can show the following:

Theorem  Let \( n \in \mathbb{Z} \), \( \alpha, \beta \in \mathbb{R} \), and let \( g_n(\tau) \)
be the \( n \)-th coefficient in the Laurent expansion of \( \Psi(\tau, \cdot) \) at \( 2\pi i(\alpha \tau + \beta) \). Then we have
\[
g_n[\gamma]_{k+n} = g_n
\]
for all \( \gamma \in \Gamma \) with \( (\alpha, \beta) \gamma \equiv (\alpha, \beta) \mod \mathbb{Z}^2 \).

In particular, the coefficients are modular forms.

Definition For each $\tau \in \mathbb{H}$, take
\[
    f(x) = f(\tau, x) = (\wp(\tau, x) - e_1(\tau))^{-1/2}
\]
with the choice of normalization
\[
    (\wp(\tau, x) - e_1(\tau))^{1/2} = \frac{1}{x} + O(1).
\]
Then $f$ is an odd function, and
\[
    Q(x) = x f(x)
\]
defines a genus for a compact oriented differentiable manifold of dimension $4k$:
\[
    \varphi(X) = \left\{ \prod_{i=1}^{2k} \frac{x_i} {f(x_i)} \right\}[X],
\]
where the Pontryagin class $p(TX)$ is given by
\[
    p(TX) = (1 + x_1^2) \cdots (1 + x_{2k}^2).
\]
Notice that $\varphi(X) = \varphi(X)(\tau)$ can be written as a function of $\tau$.

Lemma For a compact oriented differentiable manifold $X$ of dimension $4k$, the elliptic genus $\varphi(X)(\tau)$ is a modular form of weight $2k$ on $\Gamma_0(2)$.
Proof Notice that the elliptic genus is defined by the even power series $x \varphi_1(\tau, x)$. We can see that this is even by the action of $-I$:
\[
    \varphi_1(\tau, -x) = -\varphi_1(\tau, x).
\]
Thus, we formally factorize $p(X) = p(TX)$ and write
\[
    \varphi(X)(\tau) = \left\{ \prod_{i=1}^{2k} x_i \varphi_1(\tau, x_i) \right\}_{4k} [X],
\]
where
\[
    \prod_{i=1}^{2k} (1 + x_i^2).
\]
We will quote below the fact that $x\varphi_1(\tau, x)$ is a Jacobi form of weight 0. For $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \Gamma_0(2)$, we compute:
\[
    \varphi(X) [\gamma]_{2k}(\tau) = \left\{ \prod_{i=1}^{2k} x_i \varphi_1(\gamma(\tau), x_i) \right\}_{4k} (c\tau + d)^{-2k} [X].
\]
Rewriting this as
\[
    \left\{ \prod_{i=1}^{2k} x_i \frac{1}{c\tau + d} \varphi_1\left(\gamma(\tau), \frac{x_i}{c\tau + d}\right) \right\}_{4k} [X],
\]
we obtain
\[
    \left\{ \prod_{i=1}^{2k} x_i \varphi_1(\tau, x_i) \right\}_{4k} [X] = \varphi(X)(\tau).
\]
 It is holomorphic on $\mathbb{H}$ and at $\infty$ since from the expansion of $\varphi_1(\tau, x_i)$, we see that $\varphi(X)(\tau)$ is a power series in $q$ converging for $|q| < 1$.

Since $\Gamma_0(2) = \Gamma_1(2)$ has another cusp at $0$, we consider:
\[
    \varphi(X) \left[\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \right]_{2k} (\tau) =
    \left\{ \prod_{i=1}^{2k} x_i \frac{1}{\tau} \varphi_1 \left( -\frac{1}{\tau}, \frac{x_i}{\tau} \right) \right\}_{4k} [X].
\]
This simplifies to
\[
    \left\{ \prod_{i=1}^{2k} x_i \varphi_2(\tau, x_i) \right\}_{4k} [X],
\]
which is the elliptic genus defined by $x \varphi_2(\tau, x)$. Thus, it has a power series expansion in $q^{1/2}$. We conclude that $\varphi(X)(\tau)$ is a modular form of weight $2k$ on $\Gamma_0(2)$. $\blacksquare$

It is a recurring theme that given the extra datum of a vector bundle one can 'twist' various invariants by the vector bundle to take into account this structure. Here is one incarnation of this:

Definition    Let \( X \) be a differentiable manifold of dimension \( 2k \), and let \( W \) be a complex vector bundle over \( X \).  
Then the twisted \(\hat{A}\)-genus \( \hat{A}(X, W) \) is defined as:

\[
\hat{A}(X, W) = \left( \prod_{i=1}^{k} \frac{x_i /2}{\sinh(x_i /2)} \cdot \text{ch}(W) \right)[X],
\]



where the characteristic classes are given by:
\[
c(T_{\mathbb{C}} X) = (1 + x_1) \cdots (1 + x_k)(1 - x_1) \cdots (1 - x_k),
\]
\[
p(TX) = (1 + x_1^2) \cdots (1 + x_k^2),
\]
and \( \text{ch} \) is the Chern character.
An important theorem due to Atiyah and Hirzebruch is

Theorem Let \( W \) be the complex extension of a real vector bundle over a compact, oriented, differentiable spin manifold \( X \) with
\[
\dim X \equiv 4 \pmod{8}.
\]
Then
\[
\hat{A}(X, W) \in 2\mathbb{Z}.
\]


One can also run the same construction with Chern classes on stably almost complex manifolds, and use $N$-division points rather than just 2-division points like we did above, to get \emph{complex elliptic genera of level $N$}. These map out of the complex cobordism ring rather than the real one. A similar proof shows that these complex elliptic genera are modular forms for the $n^{\text{th}}$ congruence subgroup.
In general, for a fixed $N$ division point $\alpha$, we (after a suitable normalisation) get a power series $f$ such that $f^N$ is an elliptic function for the lattice $L$ with divisor $(h)=N \cdot (0)-N \cdot(\alpha)$. The power series can be explicitly determined, and when one does so (I think this is where Atiyah-Singer is used) the $q-$expansion of the elliptic genus $\tilde{\varphi}$ is a power series in $q^{\frac{1}{N}}$, so to get a modular form on $\Gamma_1(N)$ we replace $q$ with $q^N$ to get $\tilde{\varphi}$, which one can show has integral $q-$expansion provided the manifold $X$ that we plug in has $c_1(X) \equiv 0 \mod{N}$.

$S^1$ actions

Manifolds which admit an $S^1$ action have additional structure, and we often want some way of seeing this, as we did with twisting vector bundles. Here is what we will do.
\begin{definition}
    The freee loop space $\mathfrak{L}X$ of $X$ is the infinite dimensional manifold $\{g: S^1 \to X \text{ differentiable} \}$
\end{definition}
This admits an $S^1$ action by translating along the loops, and the fixed point set is the set of constant loops, i.e. points of $X$. The tangent space at a loop $g$ is the space of sections $\Gamma(g^*TX)$, which implies that for a constant loop at a point $p \in X$ $T_p(\mathfrak{L}X)\cong \Gamma(S^1 \times T_pX) =\mathfrak{L}T_pX$, so this decomposes into eigenspaces for the $S^1$-action.
We define the signature function by:
\[
\text{sign}(q, \mathfrak{L}X) =
\prod_{i=1}^{2k} x_i \frac{(1 +e^{-x_i})}{(1 - e^{-x_i})}
\prod_{n=1}^{\infty}
\left( \frac{(1 + q^n e^{-x_i})(1 + q^n e^{x_i})}{(1 - q^n e^{-x_i})(1 - q^n e^{x_i})} \right) [X].
\]
although the only motivation we can give is that analogous formulae work in other settings that have been omitted for reasons of time and space. One can compute directly that the constant term of the power series is the signature of $X$.

Ochanine's theorem

Define the \( \tilde{\cdot} \) operator for a function \( f(\tau, x) \) by
\[
\tilde{f}(\tau, x) = f\left(-\frac{1}{2\tau}, \frac{x}{2\tau}\right)(2\tau)^{-k}
\]
for an appropriate weight \( k \) for \( f \). If \( f(\tau) \in M_k(\Gamma_0(2)) \), a function only in \( \tau \), we give it weight \( k \) (the weight as a modular form), then
\[
\tilde{f}(\tau) = f\left(-\frac{1}{2\tau}\right)(2\tau)^{-k} = f\left[\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]_k (2\tau).
\]


Recall that \( \Gamma_0(2) \) has two cusps, \( \infty \) and \( 0 \). Thus, the operator on \( f \) is simply computing the expansion at \( 0 \),
and composing with \( \tau \to 2\tau \) since \( 0 \) has width 2. One can easily verify that
\[
f\left[\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]_k (\tau) \in M_k(\Gamma_0(2))
\]
and
\[
\tilde{f}(\tau) = f\left[\begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right]_k (2\tau) \in M_k(\Gamma_0(2)),
\]
and in fact, \( \tilde{\cdot} \) is a graded algebra automorphism of \( M^*(\Gamma_0(2)) \).

We can define two modular forms \( \delta \in M_2(\Gamma_0(2)) \) and \( \epsilon \in M_4(\Gamma_0(2)) \) and compute their Fourier coefficients:
\[
\delta = -\frac{3}{2} e_1 = \frac{1}{4} + 6 \sum_{n=1}^{\infty} \sigma_{\text{odd}, 1}(n) q^n = \frac{1}{4} + 6q + \dots
\]
\[
\epsilon = (e_1 - e_2)(e_1 - e_3)
\]
\[
= \left(-\frac{1}{4} - 4 \sum_{n=1}^{\infty} \sigma_{\text{odd}, 1}(n) q^n - 2 \sum_{n=1}^{\infty} \sigma_{\text{odd}, 1}(n) q^{n/2} \right)
\]
\[
\cdot \left(-\frac{1}{4} - 4 \sum_{n=1}^{\infty} \sigma_{\text{odd}, 1}(n) q^n - 2 \sum_{n=1}^{\infty} (-1)^n \sigma_{\text{odd}, 1}(n) q^{n/2} \right)
\]
\[
= \left(-\frac{1}{4} - 2q^{1/2} - 6q + \dots \right) \left(-\frac{1}{4} + 2q^{1/2} - 6q + \dots \right)
\]
\[
= \frac{1}{16} - q + \dots
\]
Notice that for \( \gamma \in \Gamma_0(2) \), \( [\gamma]_2 \) fixes \( e_1 \) and either fixes \( e_2, e_3 \) or interchanges \( e_2 \) with \( e_3 \) (by equation (1.3)), thus indeed
\[
\epsilon \in M_4(\Gamma_0(2)).
\]

Next, applying the \( \tilde{\cdot} \) operator, we get \( \tilde{\delta} \in M_2(\Gamma_0(2)) \) and \( \tilde{\epsilon} \in M_4(\Gamma_0(2)) \):
\[
\tilde{\delta}(\tau) = -\frac{3}{2} e_2(2\tau) = \frac{3}{4} e_1(\tau) = -\frac{1}{8} - 3 \sum_{n=1}^{\infty} \sigma_{\text{odd}, 1}(n) q.
\]


It is a fact that the ring of modular forms on $\Gamma_0(2)$ is a (graded) polynomial algebras on generators $\delta, \epsilon$, and since \( \tilde{\cdot} \) defines an automorphism we can write $\tilde{\varphi}(X)$ as a polynomial in $8\tilde{\delta}$ and $\tilde{\epsilon}$

\[\tilde{\varphi}(X) = \sum_{2a+4b=2k} c_{a,b} \cdot (8\tilde{\delta})^a \cdot \tilde{\epsilon}^b = P(8\tilde{\delta}, \tilde{\epsilon}),\]
where a priori \( c_{a,b} \in \mathbb{C} \) .

With all the machinery developed, we can now sketch a proof of the following theorem, important to topologists:

Theorem Let \( X \) be a compact, oriented, differentiable spin manifold with
\[
\dim X \equiv 4 \pmod{8}.
\]
Then
\[
\text{sign}(X) \equiv 0 \pmod{16}.
\]

This is a generalisation of a result of Rokhlin, who obtained the result for $4-$manifolds using the third stable homotopy group of spheres.

Proof The hypothesis of being spin implies vanishing of the second Stiefel-Whitney class, which is equivalent to $c_1 \equiv 0 \mod{2}$ since the Chern classes reduce mod 2 to the Stiefel-Whitney classes of appropriate dimension. By the previous discussion, this means that $\varphi(X)$ has an integral $q$-expansion.
By staring at the $q-$ expansions of $8\tilde{\delta}, \tilde{\epsilon}$ one can show that the coefficients \( c_{a,b} \) must also be integral by induction.  
Hence, \( \tilde{\varphi} \in \mathbb{Z}[8\tilde{\delta}, \tilde{\epsilon}] \).  
Applying the inverse of \( \tilde{\cdot} \), we get \( \varphi \in \mathbb{Z}[8\delta, \epsilon] \).
By direct computation one can show that

\[
\text{sign}(q, \mathfrak{L}X) = 2^{2k} \varphi(X) \cdot \prod_{n=1}^{\infty} \frac{ (1+q^n)^{2k}} {(1 - q^n)^{-2k}}.
\]

The product factor again has an integral \( q \)-expansion, and the other factor satisfies
\[
2^{2k} \varphi(X) = 2^{2k} P(8\delta, \epsilon) = P(32\delta, 16\epsilon),
\]
where we have computed the Fourier coefficients:
\[
32\delta = 8(1 + 24q + \dots), \quad 16\epsilon = 1 + 16q + \dots.
\]

If our manifold has dimension \( 4k \) with \( k \) odd, then the polynomial \( P(32\delta, 16\epsilon) \) is divisible by \( 32\delta \),  
otherwise the weight could not be \( 2k \) (note that \( \epsilon \) has weight 4). Hence, all \( q \)-coefficients of  
\( \text{sign}(q, \mathfrak{L}X) \) are divisible by 8:
\[
\text{sign}(q, \mathfrak{L}X) \equiv 0 \pmod{8}.
\]

Moreover, the genera \( \hat{A}(X, W) \in 2\mathbb{Z} \) since each \( W \) comes from real operations of  
\( T_C = T_X \otimes_{\mathbb{R}} \mathbb{C} \), thus forming a complex extension of a real vector bundle.  
We conclude that the coefficients \( c_{a,b} \) of \( P \) must be even, and that:
\[
\text{sign}(q, \mathfrak{L}X) \equiv 0 \pmod{16}.
\]

In particular, this holds for the constant term: the signature of \( X \). $\blacksquare$


A K3 surface is compact, 4-dimensional, and satisfies \( w_2(M) = 0 \). The signature of a K3 surface is \( -16 \), so 16 is the best possible number in Ochanine's theorem.

    
The $E_8$ manifold is the unique compact, simply connected, topological 4-manifold with intersection form given by the \( E_8 \) lattice. Ochanine's or Rokhlin's theorem shows that it has no smooth structure.

For spin $3-$manifolds, this leads to a new invariant which is known as the Rokhlin invariant and is an active area of research in low-dimensional topology.