# Theta bundles on moduli space of principal G-bundles

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 $$\mathcal{L}_V$$?

• 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). – keaton Aug 12 at 9:33