For conventions used in this note, see my other blog here.

In lecture 2 we talked about classic symmetry and their re-interpretation in the language of differential (exterior) form. We have made the connection between the so-called symmetry defect operator (SDO) and a charged, point (0-dimensional) operator. In this note we try to generalized this concept to charged operators defined on manifolds of dimension more than zero, such as a line or a surface, etc.

Let’s start with the ordinary symmetry once again on a D-dimensional manifold. We first upgraded the global symmetry parametrized by ϵ to a local symmetry parametrized by ϵ(x), and write the action variation as

(1)δS=dDxJμμϵ(x)

under eiQΣ the charged operators with non-trivial linking Link(Σ,x) transform as

(2)ϕ(x)ϵ(x)Δϕ(x).

In our convention, Δ stands for a small but finite change while ϵ(x) stands for an infinitesimal parameter. Eq. (1) can be regarded as the definition of the Noether current Jμ, which tells us how much the action changed under the transformation in question. But, being a symmetry of the system, of course the action must remains unchanged under the transformation, thus we have the Noether current

(3)μJμ=0dJ=0.

In terms of differential forms, function ϵ(x) is a 0-form and constant function ϵ is a closed form. The variation of action reads

(4)δS=M(D)(J)dϵ,

where J is a (D1)-form, dϵ is a 1-form (since ϵ is a zero form), hence their wedge product is a D-form, something can be integrated over D-dimensional manifold M (whose boundary is Σ).


The advantage of Eq. (2) is that it can be generalized to higher forms. Assume now the symmetry is parametrized by a 1-form ξ=ξμdxμ. Then dξ is a 2-form, as a result J is a (D2)-form thus J is a 2-form. The conservation law becomes

(5)dJ=0μJμν=0.

Since (D2)-form can be integrated over a (D2) manifold, we can define the charge operator as

(6)Q(ΣD2):=ΣJ.