= Canonical supercurrent bracket of free RNS matter
{title2=$\theta_{++}=(\partial_+X)^2+\frac i2\psi_1\cdot\partial_+\psi_1$}
For <RNS string> matter normalized by $[\dot X^\mu(\sigma),X^\nu(\sigma')]=-i\pi\eta^{\mu\nu}\delta(\sigma-\sigma')$ and $\{\psi_1^\mu(\sigma),\psi_1^\nu(\sigma')\}=\pi\eta^{\mu\nu}\delta(\sigma-\sigma')$, put $A=\partial_+X$ and $J=\psi_1\cdot A$. Then $[A^\mu(\sigma),A^\nu(\sigma')]=(i\pi/2)\eta^{\mu\nu}\partial_\sigma\delta(\sigma-\sigma')$. The noncentral part of $\{J(\sigma),J(\sigma')\}$ is $\pi\delta A^2+(i\pi/2)\partial_\sigma\delta\,\psi_1(\sigma)\cdot\psi_1(\sigma')$. The <distributional derivative> identity $f(\sigma,\sigma')\delta'=f(\sigma,\sigma)\delta'+(\partial_{\sigma'}f)(\sigma,\sigma)\delta$, with a vanishing coincident Grassmann bilinear, converts this to $\pi\delta\theta_{++}$. Use the chiral <equation of motion> to identify $\partial_\sigma\psi_1=\partial_+\psi_1$. Quantum <normal ordering> adds a central contact term proportional to $\delta''$, the local form of the <N=1 super-Virasoro algebra> central extension.
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