(Pseudo-)Synthetic BRST quantisation of the bosonic string and the higher quantum origin of dualities

Abstract

In this article I am arguing in favour of the hypothesis that the origin of gauge and string dualities in general can be found in a higher-categorical interpretation of basic quantum mechanics. It is interesting to observe that the Galilei group has a non-trivial cohomology, while the Lorentz/Poincare group has trivial cohomology. When we constructed quantum mechanics, we noticed the non-trivial cohomology structure of the Galilei group and hence, we required for a proper quantisation procedure that would be compatible with the symmetry group of our theory, to go to a central extension of the Galilei group universal covering by co-cycle. This would be the Bargmann group. However, Nature didn't choose this path. Instead in nature, the Galilei group is not realised, while the Lorentz group is. The fact that the Galilei group has topological obstructions leads to a central charge, the mass, and a superselection rule, required to implement the Galilei symmetry, that forbids transitions between states of different mass. The topological structure of the Lorentz group however lacks such an obstruction, and hence allows for transitions between states of different mass. The connectivity structure of the Lorentz group as opposed to that of the Galilei group can be interpreted in the sense of an ER=EPR duality for the topological space associated to group cohomology. In string theory we started with the Witt algebra, and due to similar quantisation issues, we employed the central extension by co-cycle to obtain the Virasoro algebra. This is a unique extension for orientation preserving diffeomorphisms on a circle, but there is no reason to believe that, at the high energy domain in physics where this would apply, we do not have a totally different structure altogether and the degrees of freedom present there would require something vastly more general and global.