Witten Genus and String Complete Intersections

Let M be a 4k-dimensional closed oriented smooth manifold. Let E be a complex vector bundle over M . For any complex number t , set 3t(E)= C | M + t E + t232(E)+ · · · , St(E)= C | M + t E + t2S2(E)+ · · · , where for any integer j ≥ 1, 3 j (E) is the j-th exterior power of E and S j (E) is the j-th symmetric power of E ; see [Atiyah 1967]. Set Ẽ = E −Crk(E). Let q = e iτ with τ ∈ H, the upper half plane. Witten [1988] defined 2q(E)= ⊗