A de Rham model for complex analytic equivariant elliptic cohomology (Q2227276)

Let \(G\) be a compact Lie group, not necessarily connected. The goal of the paper is to construct a version of equivariant elliptic cohomology using differential forms. Definition resembles the Cartan model for equivariant cohomology. The output is a sheaf of commutative differential graded algebras over certain stack Bun\(_G(\mathcal E)\) classifying \(G\)-bundles over the universal elliptic curve. If \(G=T\) is a torus of rank \(r\) then Bun\(_T(\mathcal E)\) is the fiber product of \(r\) copies of the dual elliptic universal curves over the moduli space of elliptic curves. For a smooth manifold \(M\) the elliptic cohomology is the sheaf \(\widehat{\mathrm{Ell}}^\bullet_G(M)\) glued from local data containing the following information: for each pair of commuting elements of \(h_1,h_2\in G\) one considers the fixed point set \(M^{\langle h_1,h_2\rangle}\) and the differential forms with values in function \(U\subset\mathrm{Bun}_T(\mathcal E)\) satisfying certain coherence constrains. For a representation \(G\to\mathrm{Spin}(2n)\) the Euler class in this model is a section (a Mathai-Quillen type cocycle) of the sheaf \(\widehat{\mathrm{Ell}}^\bullet_{\mathrm{Spin}(2n)}(pt)\) twisted by the Looijenga line bundle \(\mathcal L_1\). It is essentially given by the Weierstrass sigma function. (Similarly for a representation in \(U(n)\).) It is shown that these classes give a complex analytic equivariant refinement of the MString-orientation in elliptic cohomology. Although the torsion in the de Rham model is lost, the current definition unifies the construction of \textit{I. Grojnowski} [Lond. Math. Soc. Lect. Note Ser. 342, 114--121 (2007; Zbl 1236.55008)] for connected groups and the approach of \textit{J. A. Devoto} [Mich. Math. J. 43, No. 1, 3--32 (1996; Zbl 0871.55004)] for finite groups.