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

### APA

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

### MLA

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

### BibTex

@misc{ 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}} }

Mykola Dedushenko Stony Brook University

## 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.