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

TL;DR

This paper explores the structure of quantum L$_ abla$ algebras, extending classical gauge symmetry concepts to the quantum realm, exemplified through quantum ${ m W}_3$ algebra, and discusses quantum corrections in higher relations.

Contribution

It introduces a physically motivated definition of quantum L$_ abla$ algebras and analyzes their structure, especially in relation to quantum ${ m W}$ algebras and string field theory.

Findings

Quantum L$_ abla$ algebras describe symmetry consistency in quantum field theories.

Higher L$_ abla$ relations receive off-diagonal quantum corrections due to normal ordering.

Quantum corrections differ from those in loop L$_ abla$ algebras of string theory.

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.