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  1. On one hand, I know that the NSR superstring is described by a map $\Phi: \Sigma \to X$, where $\Sigma$ is a supermanifold with local coordinates $(\sigma,\theta)=(\sigma^0,\sigma^1 | \bar{\theta},\theta)$, and where $X$ is the target space. Then the map $\Phi^\sharp: \mathscr{C}^\infty(X) \to \mathscr{C}^\infty(\Sigma)$ of function rings is of the form $$ \Phi(x^j)(\sigma,\theta) = \phi^j(\sigma) + \psi_-^j(\sigma)\bar{\theta} + \psi_+^j(\sigma)\theta + \frac{1}{2}F_-^j(\sigma)\bar{\theta}\theta + \frac{1}{2}F_+^j(\sigma)\theta\bar{\theta} $$ for some $\phi^j,\psi_\pm^j,F_\pm^j: |\Sigma| \to \mathbb{R}$.

  2. On the other hand, (e.g. drawing on (2.57) from this paper of Freed) I know that $\phi$ should be a map $\Sigma \to X$, and that $\psi_-$ and $\psi_+$ should be spinor fields with coefficients in $\phi^*\mathsf{T}X$, e.g. $\psi_+$ is a section of $(\Pi S_+) \otimes_{\mathbb{R}} \phi^*\mathsf{T}X \to \Sigma$ for some spinor bundle $S_+ \to \Sigma$.

My question is, assuming that I have fixed spinor bundles $S_-,S_+$ on $\Sigma$, how do I define the structure sheaf of $\Sigma$ so as to make (1) and (2) both valid? (For the moment, assume the $F_\pm$'s are zero.) (This is related to this question, but I'm mainly interested in the exact description of the structure sheaf...) Do I set $$ \mathscr{C}^\infty_\Sigma := \mathscr{C}_{|\Sigma|}^\infty \otimes \Pi\varGamma(S_-) \otimes \Pi\varGamma(S_+) $$ where $\varGamma(S_\pm)$ is the sheaf of sections of $S_\pm \to \Sigma$? I feel like this is incorrect, since each section of this structure sheaf will be just a component $\phi^j: \Sigma \to \mathbb{R}$ for the bosonic part, but an entire spinor $\psi_\pm: \Sigma \to S_\pm$ for the fermionic part...

A follow up question is, how do I modify $\mathscr{C}^\infty_\Sigma$ above to include the $F_\pm$'s?

Any help is really appreciated, thanks!

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    $\begingroup$ First, remember that the + and - indices are indicating that they are spinors in two dimensions. Then, the structure sheaf on a supermanifold has the exterior algebra of the odd parts which gets you the F (not sure why you have two Fs since the $\theta$s anticommute). $\endgroup$
    – Aaron Bergman
    Commented Sep 4, 2022 at 2:45
  • $\begingroup$ I see, I forgot that the entire exterior algebra was already part of the structure sheaf. But that being said, I'm still confused how I bring the spinor bundles on $X$ into the picture. Is there a way to define the structure sheaf for the worldsheet with a formula that includes the spinor bundle? Thanks! $\endgroup$
    – Alec
    Commented Sep 4, 2022 at 22:17
  • $\begingroup$ The spinor bundles are part of the definition of the supermanifold because it has to have a local action of the superpoincare group. The thetas are the coordinates on the odd spinor bundle. $\endgroup$
    – Aaron Bergman
    Commented Sep 5, 2022 at 1:16
  • $\begingroup$ Ah ok, I see. So the spinor bundles are essentially encoded by the odd part of the supermanifold's structure sheaf. Would you happen to know of a reference that talks about how to recover these bundles from the supermanifold structure or vice-versa? Thanks so much! $\endgroup$
    – Alec
    Commented Sep 6, 2022 at 22:53
  • $\begingroup$ Deligne and Freed might help. $\endgroup$
    – Aaron Bergman
    Commented Sep 7, 2022 at 1:32

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