In this paper we define and examine the truncated Witten genus. It is defined as the equivariant index of the Dirac operator on the manifold Map(C_p,M) with its natural C_p-action. Here, Map(C_p,M) is the space of maps from the cyclic group of order p into a closed, connected, spin manifold, M. By applying the Atiyah-Singer index theorem we give a topological formula for the truncated Witten genus which is related to the formula for the Witten genus by truncation of the infinite products. We also show that the equivariant index of the Dirac operator on the projective space ℙMap(C_p,ℂⁿ⁺¹) is closely related to the truncated Witten genus of ℂℙⁿ. The spaces ℙMap(C_p,ℂⁿ⁺¹) define a filtration of the space ℙMap(S¹,ℂⁿ⁺¹) which has been used to study equivariant objects on the smooth loop space of ℂℙⁿ.

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• Published version in Mathematische Zeitschrift vol 249(3), 2005
• MathSciNet review MR2121741