PIRSA:20020019 Scientific Series Quantum Fields and Strings

DOI10.48660/20020019

Series: Quantum Fields and Strings

Dedushenko, M. (2020). Noncommutative Zhu algebra and quantum field theory in four and three dimensions. Perimeter Institute. https://pirsa.org/20020019

Dedushenko, Mykola. Noncommutative Zhu algebra and quantum field theory in four and three dimensions. Perimeter Institute, Feb. 25, 2020, https://pirsa.org/20020019 

@misc{ pirsa_PIRSA:20020019,
  doi = {10.48660/20020019},
  url = {https://pirsa.org/20020019},
  author = {Dedushenko, Mykola},
  keywords = {Quantum Fields and Strings},
  language = {en},
  title = {Noncommutative Zhu algebra and quantum field theory in four and three dimensions},
  publisher = {Perimeter Institute},
  year = {2020},
  month = {feb},
  note = {PIRSA:20020019 see, \url{https://pirsa.org}}
}

Abstract

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.

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