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


The stack $\mathrm{Higgs}\_{\omega}$ 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 $\phi \in \Gamma(C \times S, \operatorname{ad}(E) \otimes\_C D)$. Here, how, given this torsor $E$ on $C \times S$, does $\operatorname{ad}(E)$ refer 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 $T^\* \mathrm{Bun}\_G$ can be identified with $\mathrm{Higgs}\_{\omega}$?



Identifying $T^* \mathrm{Bun}_G$ with Higgs bundles
The tangent complex to $\operatorname{Bun}\_G(C)$ can be identified with $T\_{\operatorname{Bun}\_G(C)}=\mathbf{R}\pi\_\*{\operatorname{ad} P[1]}$, where $\pi:\operatorname{Bun}\_G(C)\times C\rightarrow \operatorname{Bun}\_G(C)$ is the natural projection and $P$ is the universal bundle.


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


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


