There are at least two ways to construct affine Kac-Moody algebras, namely by their Cartan algebra giving a presentation by generators and relations, and as (an extension of) a central extension of the loop algebra, i.e. the Lie algebra \(\mathfrak{g}((t))=\mathfrak{g}\otimes_kk((t))\) for a finite dimensional Lie algebra \(\mathfrak{g}\) of a semisimple algebraic group \(G\) over a field of characteristic zero \(k\). This central extension is given by the Kac-Moody cocycle \[ c(x\otimes f,y\otimes g)=\kappa(x,y)\mathrm{Res}_0(f\partial g), \] where \(\kappa\) is the Killing form of \(\mathfrak{g}\). This theory is well-established and documented (see [V. G. Kac, Infinite dimensional Lie algebras. 3rd ed. Cambridge etc.: Cambridge University Press (1990; Zbl 0716.17022)] and [R. V. Moody and A. Pianzola, Lie algebras with triangular decompositions. New York, NY: John Wiley & Sons (1995; Zbl 0874.17026)]).
In the present article, the authors generalize this theory to higher dimensions, i.e. instead of functions on the circle (corresponding to \(k((t))\)), they want to consider functions on higher dimensional manifolds/varieties. The main new feature of their approach is the systematic use of derived algebraic geometry. The functions on \(D^{\circ}_n:=\widehat{\mathbb A}\setminus \{0\}\), i.e. a formal neighborhood of zero in the affine space \({\mathbb A}^n\), have the cohomology \[ H^p(D^{\circ}_n,{\mathcal O})=\begin{cases} k[[t_1,\ldots,t_n]] & \mathrm{if } p=0 \\ (t_1\ldots t_n)^{-1}k[t_1^{-1},\ldots,t_n^{-1}] & \text{if } p=n-1 \\ 0 & \text{otherwise} \end{cases} . \] This shows that in cohomology, one obtains (in degree \(n-1\)) the missing polar part. The idea from the derived point of view is to replace \(k((t))\) by the complex \(R\Gamma(D^{\circ}_n,{\mathcal O})\). The loop algebra \(\mathfrak{g}((t))\) becomes then \(\mathfrak{g}^\bullet_n:=R\Gamma(D^{\circ}_n,\mathfrak{g}\otimes{\mathcal O})\), the \(n\)th derived current algebra. The Killing form will be replaced by an invariant polynomial \(P\in S^{n+1}(\mathfrak{g}^*)^{\mathfrak g}\). The first main theorem reads then.
Theorem 1. The functional \(\gamma_P:(\mathfrak{g}^\bullet_n)^{\otimes(n+1)}\to k\) given by \[ \gamma_P(x_0\otimes f_0,\ldots,x_n\otimes f_n)=P(x_0,\ldots,x_n)\mathrm{Res}(f_0 df_1\wedge\ldots\wedge df_n) \] is of total degree \(2\) and a cocycle defining a class in \({\mathbb H}_{\mathrm{Lie}}^2(\mathfrak{g}^\bullet_n)\). This class is non-zero for \(\mathfrak{g}\) simple.
Here \({\mathbb H}_{\mathrm{Lie}}^2(\mathfrak{g}^\bullet_n)\) is the total cohomology of the differential graded Lie algebra \(\mathfrak{g}^\bullet_n\), and \(\mathrm{Res}\) is a certain residue normalized on the Bochner-Martinelli form \(\Omega(z,z^*)\).
In the classical setting, it is well-known that the central extension of the loop algebra plays a role in the construction of the determinant line bundle on the moduli space of \(G\)-bundles on curves, as principal \(G\)-bundles on a smooth projective curve \(X\) may be glued from bundles on a formal disc and bundles on the complement by using a gluing function on the boundary of the disc with values in \(G\) (Beauville-Lazlo construction), see for example [C. Sorger, ICTP Lect. Notes 1, 1–57 (2000; Zbl 0989.14009)].
In the present article, the authors generalize the Beauville-Lazlo construction to the derived higher dimensional setting. Their second main theorem (Theorem 5.3.8 and Theorem 5.5.9) is the construction of an action of the differential graded Lie algebra \(\mathfrak{g}^\bullet_n\) on the derived moduli stack of \(G\)-bundles by changes of the local trivialization, the construction of a determinant line bundle on the moduli stack and an action of the extended dg Lie algebra on the determinant line bundle in parallel to the classical theory.