Character of parity-twisted supersymmetric VOA module -- question inspired by the Stolz-Teichner program I'll begin with some background that is unnecessary for the actual question, but that might be interesting to the reader:
Topological modular forms ($TMF$) is a generalized cohomology theory whose coefficient ring $TMF^*(pt)$ is closely related to the ring $MF_*:=\mathbb Z[c_4,c_6,\Delta]/c_4^3-c_6^2-1728\Delta$ of classical modular forms.
The main goal of the Stolz-Teichner program is to construct $TMF$ via methods of functorial quantum field theory.
More precisely, Stolz-Teichner want to realize cocycles for $TMF^*(X)$ as extended supersymmetric field theories over $X$, that is, functors from the 2-category of 0-, 1-, and 2-dimensional supermanifolds over $X$ (i.e., equipped with a map to $X$) to some algebraic target 2-category (e.g. algebras and bimodules). 
In particular, an element in $TMF^*(pt)$ should be represented by a functor $Z$ from the 2-category of 0-, 1-, and 2-dimensional supermanifolds (no map to $X$) to the 2-category of algebras.
There is a natural map from $TMF^*(pt)$ to $MF_*$ (a $\mathbb Q$-isomorphism),
and, correspondingly, there is a natural way of extracting a modular form from a supersymmetric field theory $Z$. The construction goes roughly as follows.
Given a supersymmetric field theory, consider the value $V:=Z(S^1_{per})$ of $Z$ on the manifold $S^1$, equipped with the periodic supermanifold structure (Ramond sector). The vector space $V$ comes equipped with an action of the semigroup of annuli, which, at the infinitesimal level, means that there are two operators $L_0$ and $\bar L_0$ acting on $V$. Moreover,
the theory being supersymmetric, there is also an odd square root of $\bar L_0$, called $\bar G_0$ (the theories considered here are only half-supersymmetric: no square root of $L_0$).
The modular form associated to $Z$ is given by evaluating the field theory on elliptic curves (with their Ramond-Ramond supermanifold structure). Letting $E_q:= \mathbb C^\times/\mathbb Z^q$, the value of the modular form at the point $q$ is the supertrace of the operator $q^{L_0}+\bar q^{\bar L_0}$ on $V$. A priori, this doesn't look holomorphic... this is where supersymmetry comes to help: the existence of an odd square root of $\bar L_0$ implies that the coefficient of $\bar q^n$ is zero whenever $n\not =0$, and so $Z(E_q)=str(q^{L_0}+\bar q^{\bar L_0})$ is indeed holomorphic as a function of $q$.
My question is about the existence of situations (ignoring $q$) where $str(\bar q^{\bar L_0})$ is a non-zero constant.

(Notational warning: $q$, $L_0$, $M$ below correspond to $\bar q$, $\bar L_0$, and $V$ above)

The question:

Let $V$ be an $N=1$ super vertex algebra that is holomorphic, in the sense that $V$ has a unique irreducible module (namely $V$ itself). Let $M$ be its unique irreducible parity-twisted module (the Ramond sector of $V$). The Ramond algebra (spanned by $L_n$ and $G_n$, $n\in\mathbb Z$) acts on $M$. In particular, we get an even operator $L_0$ and an odd operator $G_0$ acting on $M$, subject to the relation $G_0^2 = L_0 - c/24$. 
That relation implies that the supertrace $str_M(q^{L_0-c/24})$ is a constant (as opposed to a power series in $q$).

Is there an example of a vertex algebra $V$ as above such that $str_M(q^{L_0-c/24})\not =0$?

 A: Such an object is described in Dixon, Ginsparg, Harvey, Beauty and the Beast: superconformal symmetry in a monster module Comm. Math. Phys. Volume 119, Number 2 (1988), 221-241.  A reasonably explicit construction is given in Huang's paper A nonmeromorphic extension of the moonshine module vertex operator algebra.
In short: the Leech lattice vertex algebra $V_L$ has a canonical involution $\theta$, and the $\theta$-fixed point subalgebra has a semisimple category of modules with 4 simple objects (I think this is a result of Dong):


*

*$V_L^\theta$ - the fixed point algebra, with character $\frac{J(\tau)+\Delta(\tau)/\Delta(2\tau)}{2} + 12 = q^{-1} + 98580q + 10745856q^2 + \cdots$

*$V_L^{\theta = -1}$ - the $-1$ eigenspace, with character $\frac{J(\tau)-\Delta(\tau)/\Delta(2\tau)}{2} + 12 = 24 + 98304q + 10747904 q^2 + \cdots$

*$V_L(\theta)^\theta$ - part of the monster vertex algebra, with character $\frac{J(\tau)-\Delta(\tau)/\Delta(2\tau)}{2} - 12 = 98304q + 10747904 q^2 + \cdots$

*$V_L(\theta)^{\theta = -1}$ - the odd bit, with character $\frac{\Delta(2\tau)/\Delta (\tau) - \Delta(2\tau+1)/\Delta(\tau+1/2)}{2} = 4096q^{1/2} + 1228800q^{3/2} + 74244096 q^{5/2} + \cdots$


The sum of the first and fourth modules has a canonical product structure that gives you the holomorphic vertex superalgebra you want.  Its parity-twisted module is the sum of the second and third modules, whose characters differ by the nonzero constant 24.
