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In arXiv:1207.3497 - 4D partition function on $S^1 \times S^3$ and 2D Yang-Mills with nonzero area, Yuji Tachikawa explains the partition function for an 4d $\mathcal{N}=2$ sigma model on $S^3 \times S^1$ with target $T^*G_{\mathbb{C}}=G \times \mathfrak{g}\times \mathfrak{g}\times \mathfrak{g}$ is that of a q-deformed 2D Yang-Mills theory.

What do the modes of this σ-model look like? How does the geometry/topology of $S^3 \times S^1$ play a role here?

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