= Jacobi abstruse identity
{c}
{title2=$\vartheta_3^4-\vartheta_4^4-\vartheta_2^4=0$}
The <Jacobi abstruse identity> is an identity of <theta constants>, with spatial argument zero:
$$
\vartheta_3(0,\tau)^4-\vartheta_4(0,\tau)^4-\vartheta_2(0,\tau)^4=0.
$$
It does not assert the same relation at a general nonzero spatial argument. With $q=e^{2\pi i\tau}$, the <Jacobi triple product> converts it into the oscillator-product identity
$$
\prod_{r\in\mathbb N-1/2}(1+q^r)^8-\prod_{r\in\mathbb N-1/2}(1-q^r)^8
=16q^{1/2}\prod_{n\ge1}(1+q^n)^8,
$$
where $\mathbb N=\{1,2,\ldots\}$ and $|q|<1$. In a critical <RNS string>, dividing by the eight-boson oscillator product proves equality of <GSO projection>-retained <Neveu–Schwarz sector> and <Ramond sector> state multiplicities at every mass level.
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