An Infinite Dimensional Symmetry Algebra in String Theory

TL;DR

The paper identifies a new infinite-dimensional symmetry algebra in string theory, called the weighted tensor algebra, which has well-defined commutators and expands understanding of gauge symmetries.

Contribution

It introduces the weighted tensor algebra, a novel infinite-dimensional symmetry algebra with non-singular commutators in string theory.

Findings

Defined the structure constants of the algebra

Established it as a subalgebra of gauge symmetry

Connected it to existing W-algebras but distinguished it

Abstract

Symmetry transformations of the space-time fields of string theory are generated by certain similarity transformations of the stress-tensor of the associated conformal field theories. This observation is complicated by the fact that, as we explain, many of the operators we habitually use in string theory (such as vertices and currents) have ill-defined commutators. However, we identify an infinite-dimensional subalgebra whose commutators are not singular, and explicitly calculate its structure constants. This constitutes a subalgebra of the gauge symmetry of string theory, although it may act on auxiliary as well as propagating fields. We term this object a {\it weighted tensor algebra}, and, while it appears to be a distant cousin of the $W$-algebras, it has not, to our knowledge, appeared in the literature before.