Abstract
It is believed that any classical gauge symmetry gives rise to an L∞ algebra. Based on the recently realized relation between classical \( \mathcal{W} \) algebras and L∞ algebras, we analyze how this generalizes to the quantum case. Guided by the existence of quantum W algebras, we provide a physically well motivated definition of quantum L∞ algebras describing the consistency of global symmetries in quantum field theories. In this case we are restricted to only two non-trivial graded vector spaces X 0 and X −1 containing the symmetry variations and the symmetry generators. This quantum L∞ algebra structure is explicitly exemplified for the quantum \( {\mathcal{W}}_3 \) algebra. The natural quantum product between fields is the normal ordered one so that, due to contractions between quantum fields, the higher L∞ relations receive off-diagonal quantum corrections. Curiously, these are not present in the loop L∞ algebra of closed string field theory.
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ArXiv ePrint: 1706.09034
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Blumenhagen, R., Fuchs, M. & Traube, M. On the structure of quantum L∞ algebras. J. High Energ. Phys. 2017, 163 (2017). https://doi.org/10.1007/JHEP10(2017)163
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DOI: https://doi.org/10.1007/JHEP10(2017)163