Question

Let $G$ be a semisimple algebraic group, $C$ be a smooth projective curve, and $\omega$ be canonical line bundle.

The stack $Higgs_{\omega}$ is defined as the stack associating to each $S$ the groupoid consisting of $(E, \phi)$, where $E$ is a $G$-torsor over $X \times S$ and $\phi \in \Gamma(C \times S, ad(E) \otimes_C D)$. Here, how given this torsor $E$ on $C \times S$, does $ad(E)$ refers to the associated bundle $E \times_G \mathfrak{g}$?

Main question: Given a stack $X$, one can abstractly define its co-tangent stack. How does one show that the abstract definition of $T^* Bun_G$ can be identified with $Higgs_{\omega}$?

Answer 1

The tangent complex to $Bun_G(C)$ can be identified with $T_{Bun_G(C)}=\mathbf{R}\pi_*\mathrm{ad}\ P[1]$, where $\pi:Bun_G(C)\times C\rightarrow Bun_G(C)$ is the natural projection and $P$ is the universal bundle.

Then the cotangent stack is $T^* Bun_G(C) = Spec Sym (T_{Bun_G(C)})$. Maps from $U$ into the total space of the bundle $T^* Bun_G(C)\rightarrow Bun_G(C)$ are the same as maps $U\rightarrow Bun_G(C)$ together with a section of the dual sheaf of $T_{Bun_G(C)}$. Relative Serre duality identifies $\mathcal{Hom}(T_{Bun_G(C)}, \mathcal{O})$ with $\mathbf{R}\pi_*\mathcal{Hom}(\mathrm{ad}\ P, \omega_C)\cong\mathbf{R}\pi_*(\mathrm{ad}\ P\otimes\omega_C)$ using the Killing form.

So, maps $U\rightarrow H^0(T^* Bun_G(C))$ to the underlying ordinary stack are identified with $G$-bundles over $U\times C$ and a section $\phi\in H^0(U\times C, \mathrm{ad} P\otimes \omega_C)$.