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W-algebras as coset vertex algebras

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Abstract

We prove the long-standing conjecture on the coset construction of the minimal series principal W-algebras of ADE types in full generality. We do this by first establishing Feigin’s conjecture on the coset realization of the universal principal W-algebras, which are not necessarily simple. As consequences, the unitarity of the “discrete series” of principal W-algebras is established, a second coset realization of rational and unitary W-algebras of type A and D are given and the rationality of Kazama–Suzuki coset vertex superalgebras is derived.

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Notes

  1. We have changed slightly the notation for the parametrization of simple \(\mathscr {W}^k(\mathfrak {g})\)-modules from [7]. Namely we have \(\chi _{\lambda }=\gamma _{\lambda -(k+h^{\vee })\rho ^{\vee }}\), where \(\gamma _{\lambda }\) is the central character used in [7].

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Acknowledgements

This work started when we visited Perimeter Institute for Theoretical Physics, Canada, for the conference “Exact operator algebras in superconformal field theories” in December 2016. We thank the organizers of the conference and the institute. The first named author would like to thank MIT for its hospitality during his visit from February 2016 to January 2018.

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Correspondence to Thomas Creutzig.

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This work was partially supported by JSPS KAKENHI Grants (#17H01086 and #17K18724 to T. Arakawa), an NSERC Discovery Grant (#RES0019997 to T. Creutzig), and a grant from the Simons Foundation (#318755 to A. Linshaw).

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Arakawa, T., Creutzig, T. & Linshaw, A.R. W-algebras as coset vertex algebras. Invent. math. 218, 145–195 (2019). https://doi.org/10.1007/s00222-019-00884-3

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