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

## TL;DR

This paper explores algebraic topology aspects of String^c structures, extending Witten genera to these structures and providing vanishing results, thereby deepening understanding of their mathematical properties.

## Contribution

It introduces extended Witten genera for String^c structures of various levels and analyzes their algebraic topology, offering new insights and vanishing theorems.

## Key findings

- Extended Witten genera for String^c structures of various levels.
- Vanishing results for these generalized Witten genera.
- Analysis from Whitehead tower and loop group perspectives.

## Abstract

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