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.




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(*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 $str\_M(q^{L\_0-c/24})\not =0$?
> 
> 
> 



Character of parity-twisted supersymmetric VOA module -- question inspired by the Stolz-Teichner program
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](http://arxiv.org/abs/hep-th/9406190)*.


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):


1. $V\_L^\theta$ - the fixed point algebra, with character $\frac{J(\tau)+\Delta(\tau)/\Delta(2\tau)}{2} + 12 = q^{-1} + 98580q + 10745856q^2 + \cdots$
2. $V\_L^{\theta = -1}$ - the $-1$ eigenspace, with character $\frac{J(\tau)-\Delta(\tau)/\Delta(2\tau)}{2} + 12 = 24 + 98304q + 10747904 q^2 + \cdots$
3. $V\_L(\theta)^\theta$ - part of the monster vertex algebra, with character $\frac{J(\tau)-\Delta(\tau)/\Delta(2\tau)}{2} - 12 = 98304q + 10747904 q^2 + \cdots$
4. $V\_L(\theta)^{\theta = -1}$ - 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.


