$\mathrm{SL}(2,\text{Z})$ modular forms and Witten genus in odd dimensions

Abstract

Using $\mathrm{SL}(2,\text{Z})$ modular forms introduced in Liu (1996) and Chen–Han–Zhang (2011), we construct some modular forms over $\mathrm{SL}(2,\text{Z})$ and some modular forms over ${\Gamma }^0(2)$ and ${\Gamma }_0(2)$ in odd dimensions. In parallel, we obtain some new cancellation formulas for odd-dimensional spin manifolds and odd-dimensional ${\mathrm{spin}}^c$ manifolds, respectively. As corollaries, we get some divisibility results of index of the Toeplitz operators on spin manifolds and ${\mathrm{spin}}^c$ manifolds.