Quantum $\operatorname{SL}(2)$ and logarithmic vertex operator algebras at $(p,1)$-central charge

Terry Gannon; Cris Negron

doi:10.4171/jems/1489

JournalsjemsVol. 28, No. 2pp. 777–842

Quantum $SL (2)$ and logarithmic vertex operator algebras at $(p, 1)$ -central charge

Terry Gannon
University of Alberta, Edmonton, Canada
Cris Negron
University of Southern California, Los Angeles, USA
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Abstract

We provide a ribbon tensor equivalence between the representation category of small quantum $SL (2)$ , at parameter $q = e^{πi / p}$ , and the representation category of the triplet vertex operator algebra at integral parameter $p > 1$ . We provide similar quantum group equivalences for representation categories associated to the Virasoro and singlet vertex operator algebras at central charge $c = 1 - 6 (p - 1)^{2} / p$ . These results resolve a number of fundamental conjectures coming from studies of logarithmic CFTs in type $A_{1}$ .

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Terry Gannon, Cris Negron, Quantum $SL (2)$ and logarithmic vertex operator algebras at $(p, 1)$ -central charge. J. Eur. Math. Soc. 28 (2026), no. 2, pp. 777–842

DOI 10.4171/JEMS/1489