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Journal Article

Higher Kac-Moody algebras and moduli spaces of G-bundles


Faonte,  Giovanni
Max Planck Institute for Mathematics, Max Planck Society;

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Faonte, G., Hennion, B., & Kapranov, M. (2019). Higher Kac-Moody algebras and moduli spaces of G-bundles. Advances in Mathematics, 346, 389-466. doi:10.1016/j.aim.2019.01.040.

Cite as: https://hdl.handle.net/21.11116/0000-0004-856A-F
We provide a generalization to the higher dimensional case of the construction of the current algebra g((z)), of its Kac-Moody extension and of the classical results relating them to the theory of G-bundles over a curve.
For a reductive algebraic group G with Lie algebra g, we define a dg-Lie algebra g_n of n-dimensional currents in g. We show that any symmetric G-invariant polynomial P on g of degree n+1 determines a central extension of
g_n by the base field k that we call higher Kac-Moody algebra g_{n,P} associated to P. Further, for a smooth, projective variety X of dimension n>1, we show that g_n acts infinitesimally on the derived moduli space RBun_G(X,x) of G-bundles over X trivialized at the formal neighborhood of a point x of X.
Finally, for a representation \phi: G-->GL_r, we construct an associated determinantal line bundle on RBun_G(X,x) and prove that the action of g_n extends to an action of g_{n,P_\phi} on such bundle for P_\phi the (n+1)-st Chern character of \phi.