  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.
