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Noncommutative Zhu algebra and quantum field theory in four and three dimensions

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For any vertex operator algebra V, Y. Zhu constructed an associative algebra Zhu(V) that captures its representation theory (more generally, given a finite order automorphism g of V, there exists an algebra Zhu_g(V) that captures g-twisted representation theory of V). 

To a 4d N=2 superconformal theory T, one assigns a vertex algebra V[T] by the construction of Beem et al. We explain one role of Zhu algebra in this context. Namely, we show that a certain quotient of the Zhu algebra describes what happens to the Schur sector of the theory T under the dimensional reduction on S^1. This connects the VOA construction in 4d N=2 SCFT to the topological quantum mechanics construction in 3d N=4 SCFT, with the latter being given by the aforementioned quotient of the Zhu algebra. In the process, we will discuss how to reformulate the VOA construction on an S^3 x S^1 geometry.