Identifying $T^* Bun_G$ with Higgs bundles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:07:46Z http://mathoverflow.net/feeds/question/116709 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116709/identifying-t-bun-g-with-higgs-bundles Identifying $T^* Bun_G$ with Higgs bundles Vinoth 2012-12-18T14:48:10Z 2012-12-18T19:53:37Z <p>Let $G$ be a semisimple algebraic group, $C$ be a smooth projective curve, and $\omega$ be canonical line bundle.</p> <p>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 $E \times_G \mathfrak{g}$?</p> <p><strong>Main question:</strong> 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 $Higgs_{\omega}$?</p> http://mathoverflow.net/questions/116709/identifying-t-bun-g-with-higgs-bundles/116733#116733 Answer by Pavel Safronov for Identifying $T^* Bun_G$ with Higgs bundles Pavel Safronov 2012-12-18T19:53:37Z 2012-12-18T19:53:37Z <p>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.</p> <p>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.</p> <p>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 $\phi\in H^0(U\times C, \mathrm{ad} P\otimes \omega_C)$.</p>