Let $G$ be a simply connected, semi-simple affine algebraic group and $C$ be a smooth projective curve with $g \geq 3$. Let $\mathcal{M}_G$ be a moduli stack of principal G-bundles on the curve $C$ and let $M_G$ be a moduli space of semi-simple principal G-bundles.

Let $V$ be a representation of the group $G$, $\mathcal{U}$ be a universal $G$-bundle on $\mathcal{M}_G \times C$, and let $\mathcal{U}_V := \mathcal{U}\times_G V$. Consider derived pushforward $R^* (\pi_1)_* \mathcal{U}_V$ then we have a bounded complex on $\mathcal{M}_G$. When we take a determinant functor, we obtain a line bundle $\mathcal{L}_V = Det(R^* (\pi_1)_* \mathcal{U}_V)$ on $\mathcal{M}_G$. Fiberwisely, over a principal $G$-bundle $E$, $\mathcal{L}_V|_{[E]} = \wedge^{top} H^0(C, E_V)^{\vee} \otimes \wedge^{top} H^1(C,E_V)$ where $E_V := E\otimes_G V$.

On the other hand, it is well-known that there is a line bundle $\Theta_V(C)$ on $M_G$ so-called Theta bundle, fiberwisely $\Theta_V(C)|_{[E]} = \wedge^{top} H^0(C, E_V)^{\vee} \otimes \wedge^{top} H^1(C,E_V)$. So fiberwisely, $\mathcal{L}_V$ and $\Theta_V(C)$ are the same(of course they lies on the other space).

Does $\Theta_V(C)$ is a descent of $\mathcal{L}_V$?

  • $\begingroup$ I think an answer of this question is well-explained in (3.7) of S. Kumar, Infinite Grassmannians and moduli spaces of G-bundles(New directions). $\endgroup$ – keaton Aug 12 at 9:33

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