On the Structure of Quantum L$_\infty$ algebras

Ralph Blumenhagen; Michael Fuchs; Matthias Traube

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arXiv · 1706.09034

On the Structure of Quantum L$_\infty$ algebras

Ralph Blumenhagen·Michael Fuchs·Matthias Traube

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Abstract

It is believed that any classical gauge symmetry gives rise to an L$_\infty$ algebra. Based on the recently realized relation between classical ${\cal W}$ algebras and L$_\infty$ algebras, we analyze how this generalizes to the quantum case. Guided by the existence of quantum ${\cal W}$ algebras, we provide a physically well motivated definition of quantum L$_\infty$ 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$_\infty$ algebra structure is explicitly exemplified for the quantum ${\cal W}_3$ algebra. The natural quantum product between fields is the normal ordered one so that, due to contractions between quantum fields, the higher L$_\infty$ relations receive off-diagonal quantum corrections. Curiously, these are not present in the loop L$_\infty$ algebra of closed string field theory.

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Ralph Blumenhagen, Michael Fuchs, Matthias Traube. 2017-06-27. On the Structure of Quantum L$_\infty$ algebras. https://arxiv.org/abs/1706.09034

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