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In this paper, we study some algebraic topology aspects of String$^c$structures, more precisely, from the perspective of Whitehead tower and theperspective of the loop group of $Spin^c(n)$. We also extend the generalizedWitten genera constructed for the first time in \cite{CHZ11} to correspond toString$^c$ structures of various levels and give vanishing results for them.

String$\mathbf{^c}$ Structures and Modular Invariants