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 -dimensional manifold. We first upgraded the global symmetry parametrized by to a local symmetry parametrized by , and write the action variation as
under the charged operators with non-trivial linking transform as
In our convention, stands for a small but finite change while stands for an infinitesimal parameter. Eq. (1) can be regarded as the definition of the Noether current , 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
In terms of differential forms, function is a -form and constant function is a closed form. The variation of action reads
where is a -form, is a -form (since is a zero form), hence their wedge product is a -form, something can be integrated over -dimensional manifold (whose boundary is ).
The advantage of Eq. (2) is that it can be generalized to higher forms. Assume now the symmetry is parametrized by a -form . Then is a 2-form, as a result is a -form thus is a -form. The conservation law becomes
Since -form can be integrated over a manifold, we can define the charge operator as