**Title:** The Rigidity Theorems of Witten via Equivariant Elliptic Cohomology

**Speaker:** Professor Ioanid Rosu

**Speaker Info:** MIT

**Brief Description:**

**Special Note**:

**Abstract:**

The rigidity of the elliptic genus for spin manifolds with $S^1$ action was conjectured by Witten and proved by Taubes, with later improvements by Bott \& Taubes. We give a new proof of the rigidity theorem by using $S^1$ equivariant elliptic cohomology. This is defined as a coherent sheaf over an elliptic curve, as suggested by I. Grojnowski. The elliptic genus, which a priori is a germ of a holomorphic function at the identity of the elliptic curve, will be shown to give a global holomorphic function on the elliptic curve. This means precisely that the elliptic genus is rigid. This rigidity can be then interpreted as enabling us to construct Gysin maps in equivariant elliptic cohomology.Also, a construction similar to Grojnowski's, but performed over C^*, is shown to give the usual $S^1$ equivariant K-theory. Various generalizations are also discussed.

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