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We prove several vanishing theorems for a class of generalized elliptic genera on foliated manifolds, by using classical equivariant index theory. The main techniques are the use of the Jacobi theta-functions and the construction of a new class of elliptic operators associated to foliations.

Kawakubo and Uchida showed that, if a closed oriented 4k-dimensional manifold M admits a semi-free circle action such that the dimension of the fixed point set is less than 2k, then the signature of M vanishes. In this note, by using G-signature theorem and the rigidity of the signature operator, we generalize this result to more general circle actions. Combining the same idea with the remarkable WittenTaubesBott rigidity theorem, we explore more vanishing results on spin manifolds admitting such circle actions. Our results are closely related to some earlier results of ConnerFloyd, LandweberStong and HirzebruchSlodowy.

In this paper, we generalize the anomaly cancellation formulas given by Alvarez-Gaum and Witten (1983), Liu (1995) and Han and Zhang (2004)[1],[2],[7] to the cases where an auxiliary bundle W and a complex line bundle are involved with no conditions on the first Pontryagin forms being assumed.

In this paper, we give a simple proof of Mirzakhanis recursion formula of Weil-Petersson volumes of moduli spaces of curves using the Witten-Kontsevich theorem. We also briefly describe a very general recursive phenomenon in the intersection theory of moduli spaces of curves. In particular, we present several new recursion formulas for higher degree classes.