Equivariant cohomology and current algebras

Alekseev, Anton; Ševera, Pavol

Confluentes Mathematici, Tome 4 (2012) no. 2.

This paper touches upon two big themes, equivariant cohomology and current algebras. Our first main result is as follows: we define a pair of current algebra functor which assigns Lie algebras (current algebras) $\mathcal{C}\mathcal{A}(M,A)$ and $\mathcal{S}\mathcal{A}(M,A)$ to a manifold M and a differential graded Lie algebra (DGLA) A. The functors  $\mathcal{C}\mathcal{A}$ and $\mathcal{S}\mathcal{A}$ are contravariant with respect to M and covariant with respect to A. If A = C𝔤, the cone of a Lie algebra 𝔤 spanned by Lie derivatives L(x) and contractions I(x)(x ∈ 𝔤) and satisfying the Cartan's magic formula [d, I(x)] = L(x), the corresponding current algebras coincide, and they are equal to $\mathcal{C}\mathcal{A}(M,C\mathfrak{g})=\mathcal{S}\mathcal{A}(M,C\mathfrak{g})\cong C^{\mathrm{\infty }}(M,\mathfrak{g})$, the space of smooth 𝔤-valued functions on M with the pointwise Lie bracket. Other examples include affine Lie algebras on the circle and Faddeev–Mickelsson–Shatashvili (FMS) extensions of higher-dimensional current algebras. The second set of results is related to the construction of a new DGLA D𝔤 assigned to a Lie algebra 𝔤. It is generated by L(x) and I(x) (similar to C𝔤) and by higher contractions I(x²), I(x³) etc. Similar to C𝔤, D𝔤 can be used in building differential models of equivariant cohomology. In particular, twisted equivariant cohomology (including twists by 3-cocycles and higher odd cocycles) finds a natural place in this new framework. The DGLA D𝔤 admits a family of central extensions D_p𝔤 parametrized by homogeneous invariant polynomials $p\in {\left(S{\mathfrak{g}}^{\ast }\right)}^{\mathfrak{g}}$. There is a Lie homomorphism from $\mathcal{C}\mathcal{A}(M,D_p\mathfrak{g})$ to the FMS current algebra defined by p. Let G be a Lie group integrating the Lie algebra 𝔤. The current algebras $\mathcal{S}\mathcal{A}(M,D\mathfrak{g})$ and $\mathcal{S}\mathcal{A}(M,D_p\mathfrak{g})$ integrate to groups DG(M) and D_pG(M). As a topological application, we consider principal G-bundles, and for every homogeneous polynomial $p\in {\left(S{\mathfrak{g}}^{\ast }\right)}^{\mathfrak{g}}$ we pose a lifting problem (defined in terms of DG(M) and D_pG(M)) with the only obstruction the Chern–Weil class cw(p). When M is a sphere, we study integration of the current algebra $\mathcal{C}\mathcal{A}(M,D_p\mathfrak{g})$. It turns out that the corresponding group is a central extension of DG(M). Under certain conditions on the polynomial p, this is a central extension by a circle.

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