Obstruction theory and the level $n$ elliptic genus

TL;DR

This paper proves that certain elliptic genera can be uniquely lifted to highly structured orientations in complex cobordism, using obstruction theory and properties of Landweber exact ring spectra.

Contribution

It establishes a new obstruction-theoretic approach to lifting elliptic genera to $ ext{E}_$-orientations, simplifying proofs of their existence.

Findings

Unique lift of level n elliptic genus to $ ext{E}_$-orientation $ ext{MU} o ext{tmf}_1(n)$ for all $n  2.

Obstruction theory applies to height $ 2$ Landweber exact $ ext{E}_$-rings with even-degree homotopy.

Simplified proof of the lift of elliptic genera to $ ext{tmf}_1(n)$.

Abstract

Given a height $\leq 2$ Landweber exact $\mathbb{E}_\infty$-ring $E$ whose homotopy is concentrated in even degrees, we show that any complex orientation of $E$ which satisfies the Ando criterion admits a unique lift to an $\mathbb{E}_\infty$-complex orientation $\mathrm{MU} \to E$. As a consequence, we give a short proof that the level $n$ elliptic genus lifts uniquely to an $\mathbb{E}_\infty$-complex orientation $\mathrm{MU} \to \mathrm{tmf}_1 (n)$ for all $n \geq 2$.