A Portal Special Presentation- Geometric Unity: A First Look: Difference between revisions

Now, this is a tremendous amount of freedom that we've just gained. Normally, we keep losing freedom, but this is the first time we actually begin to see that we have a lot of freedom, and we're going to actually retain some of this freedom to the end of the talk. But the idea being that I can now start to define operators which correspond to the "Ship in the Bottle" problem. I can take field content \(\epsilon\) and \(\pi\), where these are elements of the inhomogeneous gauge group. In other words, \(\epsilon\) is a gauge transformation, and \(\pi\) is a gauge potential.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ (\epsilon, \pi) \in \mathcal{G} $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon \eta = [\text{Ad}(\epsilon^{-1}, \Phi), \eta] $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon \Omega^i(ad) \rightarrow \Omega^{d-3+i}(ad) $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ T_{\epsilon, \pi} = \Pi - h^{-1}d_{A_0}h $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon F_\pi +[\bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon T_\omega, T_{\epsilon, \pi}] + *T_{\epsilon, \pi} = 0 $$</div>

I'm going to do that again on the other side. There are going to be plus and minus signs, but it's a magic bracket that knows whether or not it should be a plus sign or a minus sign and I apologize for that, but I'm not able to keep that straight. And then there's going to be one extra term, where all these \(T\)s have the \(\epsilon\) and \(\pi\)s.

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\epsilon d_A \zeta + [\bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\epsilon\zeta, T] + [T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\epsilon\zeta] + *\zeta = F_A \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu + [[T, \nu], T] + [T, [T, \nu]] + [[T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu], T] + *\nu $$</div>

<div style="text-align: center; margin-left: auto; margin-right: auto;">$$ *(d_A^* \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em^* + * + ...)(\bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\epsilon F_{A_\pi} + ...) $$</div>