Elliptic Genera And Theta Functions Identities

In this paper, we study the elliptic genera of homogeneous manifolds and toric va-rieties. We use the Witten rigidity theorem and the Atiyah-Bott Lefschetz fixed pointformula to derive several classes of combinatorial identities of theta functions.As an analogue of the signature operator on finite dimensional manifolds, Wittenconsidered the signature operator on the loop space LM of a compact smooth manifoldM. By formally using the fixed point formula on LM, Witten introduced an operator(d|^)_s on M, whose index coincides with the Landeweber-Stong elliptic genus of M up to aconstant. By Witten rigidity theorem, (d|^)_s is a rigid operator when M is spin.Suppose M admits a smooth S^1-action. By the Atiyah-Bott Lefschetz fixed pointformula, the equivariant index of an elliptic operator on M can be expressed as a sumover the fixed point sets of the S^1-action. In particular, the equivariant index of (d|^)_s can beexpressed in terms of classical Jacobi theta functions. By Witten rigidity theorem, we canobtain combinatorial identities of theta functions.In this paper, we are mainly concerned with the elliptic genera of homogeneousmanifolds. Suppose G is a compact Lie group, H a closed subgroup of G with rank G =rank H. Let T be a common maximal torus of G and H, then M = G/H admits anatural T-action, with the fixed points in one-one correspondence with the left cosets inWG/WH. Here WG and WH are the Weyl groups of G and H respectively. Therefore,the fixed point formula on G/H has a nice combinatorial form by using the root systemsand Weyl groups of G and H. Meanwhile, a striking result of Hirzebruch and Slodowy isthat indexg (d|^)_s equals to the signature of G/H when G/H is spin. Thus we obtain severalclasses of combinatorial identities of theta functions. We also consider the possibility toderive theta functions identities from the combinatorial properties of toric varieties.