# Torsor descriptions of $Bun_G$

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 $$SL_n$$?)

• 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? – Will Sawin Feb 6 at 21:17
• 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. – Macu Feb 7 at 0:21
• @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. – Will Sawin Feb 7 at 1:43
• OK, so the situation of $SL_n$ is more of an exception than a general pattern? – Macu Feb 7 at 1:48
• 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. – Roman Fedorov Feb 8 at 2:58