The moduli stack $Bun_{SL_n}X$ of $SL_n$-principal bundles over a projective curve $X$ is a $\mathbb{G}_m$-torsor over the sublocus of $Bun_{GL_n}X$ corresponding to vector bundles with zero determinant.

Is there a similar description for $Bun_{Sp_{2n}}X$? Edit: by this I mean a description as a torsor over something (e.g. a substack of $Bun_{GL_n}X$ for $SL_n$, which is not the same as the descriptions of its objects as vector spaces + something)

Also, by a theorem of Grothendieck the map of stacks $Bun_{O_n}X\to Bun_{GL_n}X$ induces an injection at the level of moduli spaces, and I think this essentially means that $Bun_{O_n}$ is a relative gerbe over its image (is this correct?). Is it possible to describe $Bun_{O_n}$ as a torsor over something (maybe a substack of $Bun_{GL_n}X$, as for $SL_n$?)

  • $\begingroup$ Aren't these the stacks of vector bundles together with a chosen map $V \otimes V \to \mathcal O_X$ that is symplectic, resp., orthogonal? $\endgroup$ – Will Sawin Feb 6 at 21:17
  • $\begingroup$ Yes, so that gives you a map $Bun_G\to Bun_{GL_n}$, but I don't think it is a $\mathbb{G}_m$-torsor. On the other hand I am interested in a description of $Bun_G$ as a torsor over something. $\endgroup$ – Macu Feb 7 at 0:21
  • $\begingroup$ @Macu The only group I know that acts on $\operatorname{Bun}_G$ is $\operatorname{Bun}_{Z(G)}$, which is $\operatorname{Bun}_{\mathbb Z/2}$ in both cases. So this represents it as a $\mathbb Z/2$-gerbe on a $(\mathbb Z/2)^{2g}$-torsor over an open and closed component of $\operatorname{Bun}_{G/Z(G)}$, or something like that. $\endgroup$ – Will Sawin Feb 7 at 1:43
  • $\begingroup$ OK, so the situation of $SL_n$ is more of an exception than a general pattern? $\endgroup$ – Macu Feb 7 at 1:48
  • $\begingroup$ It is a general pattern: you can relate $Bun_G$ with $Bun_H$ if $G/Z(G)$ is isomorphic to $H/Z(H)$ ($Z$ stands for the center). But probably it does not help that much. $\endgroup$ – Roman Fedorov Feb 8 at 2:58

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.