## 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 $str_M(q^{L_0-c/24})\not =0$?

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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):

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.

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 Wonderful! I wonder if the non-zero constant that one gets by such a construction always has to be divisible by 24... – André Henriques Jul 5 at 7:20 That is probably true with central charge 24 - you may be able to extract a conjectural answer from Schellekens's table of 71 CFTs with some computation. I don't know enough to say much in the case of other central charges (e.g., 576). – S. Carnahan♦ Jul 5 at 8:50 Has this todo with the twisted orbifold construction of the moonshine from the Leech lattice VOA (e.g. arXiv:q-alg/9707008) I guess the sum of two of you modules gives the moonshine? – Marcel Bischoff Jul 12 at 17:24 @Marcel: Yes. Frenkel, Lepowsky, and Meurman constructed the twisted module for the explicit purpose of eliminating the 24 in the constant term of the character of the Leech lattice VOA. Incidentally, the paper you cite has nothing to do with the orbifold construction of the moonshine module. Instead, it describes an alternative construction using codes, in sufficiently general terms that you can use their methods to build other VOAs. – S. Carnahan♦ Jul 13 at 2:57