Elliptic genus and string cobordism at dimension $24$.

It is known that spin cobordism can be determined by Stiefel-Whitney numbers and index theory invariants, namely $KO$-theoretic Pontryagin numbers. In this paper, we show that string cobordism at dimension 24 can be determined by elliptic genus, a higher index theory invariant. We also compute the image of 24 dimensional string cobordism under elliptic genus. Using our results, we show that under certain curvature conditions, a compact 24 dimensional string manifold must bound a string manifold.