For the purposes of this question, a rank-$r$ (integral) *lattice* is a full-rank discrete subgroup $L \subset \mathbb R^r$ such that $\langle \ell, \ell' \rangle \in \mathbb Z$ for all $\ell \in L$. It is *even* if $\ell^2 = \langle \ell,\ell\rangle \in 2\mathbb Z$ for all $\ell \in L$ and *odd* if there is some $\ell$ such that $\ell^2$ is odd. Define $L^\* = \{ v\in \mathbb R^n \text{ s.t. } \langle v,\ell \rangle \in \mathbb Z \, \forall \ell \in L\}$. $L$ is *unimodular* if $L = L^\*$. 


Let $L$ be an odd unimodular lattice, and define $L\_{ev} = \{\ell \in L \text{ s.t. } \ell^2 \in 2\mathbb Z\}$. Then $L\_{ev}$ has index $2$ in $L$, and so index $4$ in $L\_{ev}^\*$. The cosets of $L\_{ev}$ in $L\_{ev}^\*$ are $L\_{ev}$, $L\_{odd} := L \smallsetminus L\_{ev}$, and two more that I will arbitrarily name $L\_+$ and $L\_-$. Vectors in $L\_\pm$ might have fractional length. (The structure of the group $L\_{ev}^\*/L\_{ev}$ and the lengths of vectors in $L\_\pm$ depend on the rank $r$ mod $8$.)


Given a discrete subset $S \subset \mathbb R^r$, the *Theta series* of $S$ is the $q$-series
$ \Theta\_S(q) = \sum\_{s\in S} q^{s^2/2}. $
Suppose that $L$ is odd and unimodular. I find myself interested in the following function:
$$ Z\_{RR}(L) := \Theta\_{L\_+}(q) - \Theta\_{L\_-}(q)$$
The letters "$RR$" stand for "Ramond-Ramond", because this this shows up as an RR-sector partition function of a spin conformal field theory built from $L$ (times a power of $\eta$). I'm pretty sure that $Z\_{RR}(L)$ is automatically a (level-1) modular form of weight $r$. [Edit: $r/2$.]


Let $\sqrt3\mathbb Z \subset \mathbb R$ denote the (non-unimodular) odd lattice of vectors $\ell$ such that $\ell^2 \in 3\mathbb Z$.
A unimodular lattice $L$ of rank $r$ is *3-framed* if it contains a copy of $(\sqrt3\mathbb Z)^r$. 3-framed lattices are in natural bijection with self-dual ternary codes; [Harada and Munemasa](https://arxiv.org/abs/0804.0637) used this bijection, together with Borcherds' classification of unimodular lattices of rank $\leq 25$, to classify self-dual ternary codes of rank $24$.


Recall the *discriminant* $\Delta(q) = q \prod\_{n=1}^\infty (1-q^n)^{24}$. It is a modular form of weight $12$. I'm pretty sure that


**Proposition:** If $L$ is 3-framed and unimodular, then $Z\_{RR}(L) \in \mathbb Z[\Delta]$. [Edit: $\mathbb Z[\Delta^{1/2}]$.]


I.e. I am claiming that $Z\_{RR}(L)$ vanishes unless $r = 12k$, in which case $\Delta^{-k}Z\_{RR}(L) \in \mathbb Z$.


By simply checking against Borcherds' classification, I know that the following conjecture is true in ranks $r = 12$ and $24$:


**Conjecture:** Suppose $L$ is a 3-framed unimodular lattice of rank $r = 12k$. Then the integer $\Delta^{-k}Z\_{RR}(L)$ is divisible by $24$. [Edit: $\Delta^{-k/2}$.]


Is the conjecture true in general? Is it known?


[Edit: And please continue to point out errors in the comments — I'm sure there are more.]



Is the "Ramond sector" invariant of a 3-framed lattice always divisible by 24?
The discussion in the comments establishes the conjecture when $r$ is divisible by $24$. When $r$ is merely divisible by $12$, the comments establish that $\Delta^{-r/24} Z\_{RR}$ is divisible by $12$. (The latter case follows from the former because $Z\_{RR}$ is easily seen to be multiplicative.)


Specifically, when $r$ is divisible by $24$, $Z\_{RR}$ is a difference of $\Theta$ series of two even unimodular lattices of rank $r$. Theorem 12.1 of <https://math.berkeley.edu/~reb/papers/on2/on2.pdf> establishes that every $\Theta$ series of an even unimodular lattice is a polynomial of $\Delta$ and $c\_4$, and the coefficient of $\Delta^{r/24}$ is divisible by $24$.


This answer is community wiki in case someone wants to expand it.


