Let $E_4(q)=1+ 240q+ 2160q^2+ 6720q^3+\ldots $ be the Eisenstein series of weight 4, also known as the theta-series of the $E_8$-lattice.

I'm looking for a $\mathbb N$-graded vector space $V$ of graded dimension $(1, 240, 2160, 6720,\ldots )$.
Moreover, I would like $V$ to be a module for the Virasoro algebra of central charge $c=4$.

I think that this is indeed possible:
Take $1$ times the Verma module of character $1+q+2q^2+3q^3+5q^4+7q^5+\ldots$
plus $239$ times the Verma module of character $q(1+q+2q^2+3q^3+5q^4+\ldots)$
plus $1919$ times the Verma module of character $q^2(1+q+2q^2+3q^3+\ldots)$, etc.
But that's not what I'm looking for.

I'm wondering whether there exists a natural construction that produces the above Virasoro module. Does anybody know of such a construction?

Variant: In the above, I said that I wanted central charge $c=4$ and minimal energy $h=0$.
But I'm flexible: if you know a natural construction of a $Vir_c$-module (pick you $c$) whose character is $q^hE_4(q)$, I'll be happy to hear about it.


What I was asking for cannot be realized... for a simple and stupid reason:

The coefficients of the Verma module of the Virasoro algebra are given by the partition function $p(n)=\{1,1,2,3,5,7,11,...\}$. Now, the growth of the partition function is given by $$p(n) \sim \frac {1} {4n\sqrt{3}} e^{\pi \sqrt {\frac{2n}{3}}} \mbox { as } n\rightarrow \infty.$$ (see e.g. wikipedia) and is much faster than the growth of the coefficients of the theta function of the $E_8$-lattice (which is polynomial).

So my prescription "take one times the Verma module [...] plus 239 times the Verma module [...] plus 1919 times ..." will run into negative numbers at some point.

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Would you be happy with a supercharacter $str \ q^{L(0)-c/24}$ as opposed to the usual graded dimension ${\rm tr} \ q^{L(0)-c/24}$? Also, from the $q$-exapansion of $E_4$, it seems to me that the correct central charge should be zero, not $4$.

You could take $V(E_8)$, the $E_8$ lattice vertex algebra of central charge $c=8$, and tensor it with the $\mathbb{Z}_2$-graded symplectic fermion vertex superalgebra $\mathcal{SF}^8$ of central charge $c'=-8$. The tensor product $V(E_8) \otimes \mathcal{SF}^8$ is $\mathbb{Z}_2$-graded by the fermionic part.

Since ${\rm tr}({\mathcal{SF}^8}) q^{L(0)-c'/24}=q^{1/3} \prod_{i=1}^\infty (1+q^i)^8$, after taking the supertrace of the tensor product, you will clear the denominator in $\frac{\theta_{E_8}}{\eta^8}$, and end up with the supertrace equal $\theta_{E_8}$. Actually you can think of it as a virtual character (the Virasoro generator is even).

This construction is perhaps silly, but it has some motivation. In rational CFT, all characters are of weight zero. In rational SCFT there are also supercharacters, again of weight zero. Symplectic fermions $\mathcal{SF}^{2d}$, $d \geq 1$ are not rational, but "quasi-rational" or sometimes called "logarithmic", and their supercharacters exhibit "higher weight" phenomena.

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