MTd2 said:
Sorry... But I don't get where you are trying to go.
I have this idea cooking in my head, so I am bouncing it off here. It concerns a general approach to framing fields. This is just the preliminary parts here, and I am slamming out some of the notation --- I hope I get the indices etc more or less right.
A differential form and its dual vector \vec v, \underline f is seen in the product
<br />
\vec{v} \underline{f}~=~v^i\vec{\partial_i}\underline{dx}^j f_j~=~v^if_i<br />
What I did was to assume that the differential form had the form {\underline f}~=~e^{V}{\underline dx}. The contraction is then
<br />
v^i\vec{\partial_i}\underline{dx}^je^{V_j}~=~v^ie^{2V_j}{\partial_i}\underline{dx}^j~+~v_i(\partial_iV_j){\underline dx}^j.<br />
Then consider a transformation e^{V}~\rightarrow~e^{-i\chi^\dagger}e^{V}e^{i\chi} which gives a variation in V as
<br />
\delta V~=~i(\chi~-~\chi^\dagger)~-~{i\over 4}[(\chi~+~\chi^\dagger),~V]<br />
and the deviation in the contraction is
<br />
\langle v,~F\rangle~\rightarrow~v_ie^{V_i}~+~e^{-i\chi^\dagger}\big(\partial_iV~+~i\partial_i(\chi~-~\chi^\dagger)\big)e^{i\chi}<br />
We might now want a form of this contraction which is gauge covariant. So to do this we back track and consider y^i~=~x^i~+~\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}\lambda^{\dot\beta}. We then have that
<br />
dy^i~=~dx^i~+~ d\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}\lambda^{\dot\beta}~+~\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}d\lambda^{\dot\beta}<br />
The differential operators dual to this system are
<br />
\partial_i,~D_\alpha~=~\partial_\alpha~+~i\sigma^i_{\alpha{\dot\beta}}{\bar\lambda}^{\dot\beta}\partial_i<br />
Then for the vector {\underline f}~=~e^{V}{\underline dy} there exists a differential form contraction will result in
<br />
\omega_A~=~\sigma^i_{\alpha{\dot\beta}}\partial_i V d\lambda^{\dot\beta}~+~d\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}\lambda^{\dot\beta}~+~\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}d\lambda^{\dot\beta},<br />
where the left index A runs over i and \alpha. This is analogous to the differential forms \omega^0~=~\gamma(dx^0~-~u^0 dx) and \omega^i~=~\gamma(dx^i~-~u^idt) corresponding to special relativity. This is a Finsler bundle, and from this a generalized lifting condition (an Euler-Lagrange equation constraint) will be derived for the framing of fields.
An invariant vector for the contraction v^\alpha (D_\alpha\Phi) {\underline d\lambda}) for \Phi~=~(1/4) {\bar D}{\bar D}V will define under the contraction
<br />
\Phi_\alpha~=~D_\alpha\Phi~=~\frac{1}{4}{\bar D}{\bar D}D_\alpha V<br />
which if we impose the holomorphic condition D_\alpha\chi^\dagger~=~0 then this is gauge invariant for V~\rightarrow~V~+~\chi~+~\chi^\dagger. In general
<br />
\Phi_\alpha~=~\frac{1}{4}{\bar D}{\bar D}e^{-V}D_\alpha e^V<br />
which is also gauge covariant as \Phi_\alpha~\rightarrow~e^{-i\chi}\Phi_\alpha e^{i\chi}
