On Costello’s Construction of the Witten Genus: L∞ Spaces and Dg-manifolds

Definition 1.1. A curved L∞ algebra over A consists of a locally free finitely generated graded A]-module V, together with a cohomological degree 1 and square zero derivation d : Ŝym(V∨[−1])→ Ŝym(V∨[−1]) where V∨ is the A]-linear dual and the completed symmetric algebra is also over A]. There are two additional requirements on the derivation d: (1) The derivation d makes Ŝym(V∨[−1]) into a dga over the dga A; (2) Reduced modulo the nilpotent ideal I ⊂ A, the derivation d preserves the ideal in Ŝym(V∨[−1]) generated by V. Note that our dualizing convention is such that V∨[−1] = V[1]∨. We can decompose the derivation d into its constituent pieces dn : V∨[−1]→ Symn(V∨[−1]) and after dualizing and shifting we obtain maps ln : ΛnV[2− n]→ V. The maps {ln} satisfy higher Jacobi relations. In particular, if ln = 0 for all n 6= 2, then V is just a graded Lie algebra. Similarly, if ln = 0 for all n 6= 1, 2, then V is a differential graded Lie algebra. If ln = 0 for n 6= 1, 2, 3 then l3 is a contracting homotopy for the Jacobi relation, i.e. (−1)l2(l2(x, y), z) + (−1)l2(l2(z, x), y) + (−1)l2(l2(y, z), x) = (−1)(l1l3(x, y, z) + l3(l1x, y, z) + (−1)|x|l3(x, l1(y), z) + (−1)|x|+|y|l3(x, y, l1z)). If V is a L∞ algebra over A, then C∗(V) will denote the differential graded A-algebra Ŝym(V∨[−1]). Our convention will be that V is concentrated in non-negative degrees, so that C>0(V) is a (maximal) ideal of C∗(V).