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. If times allow, I will discuss the relation

between the Kac-Moody extension and the determinantal line

bundle on RBun_G(X,x).

### Deformation Theory Seminar

Wednesday, June 21, 2017 - 2:00pm

#### Giovanni Faonte

Max Planck Inst - Bonn and Yale