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

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''[https://youtu.be/Z7rd04KzLcg?t=5741 01:35:41]''<br>
''[https://youtu.be/Z7rd04KzLcg?t=5741 01:35:41]''<br>
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.
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 \(\varepsilon\) and \(\pi\), where these are elements of the inhomogeneous gauge group. In other words, \(\varepsilon\) 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;">$$ (\varepsilon, \pi) \in \mathcal{G} $$</div>




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<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;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\varepsilon \eta = [\text{Ad}(\varepsilon^{-1}, \Phi), \eta] $$</div>




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<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;">$$ \bigcirc\kern-0.940em\circ\kern-0.58em·\text{}\kern0.3em_\varepsilon \Omega^i(ad) \rightarrow \Omega^{d-3+i}(ad) $$</div>




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<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;">$$ T_{\varepsilon, \pi} = \Pi - h^{-1}d_{A_0}h $$</div>




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




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''[https://youtu.be/Z7rd04KzLcg?t=6677 01:51:17]''<br>
''[https://youtu.be/Z7rd04KzLcg?t=6677 01:51:17]''<br>
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.
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 \(\varepsilon\) and \(\pi\)s.


''[https://youtu.be/Z7rd04KzLcg?t=6710 01:51:50]''<br>
''[https://youtu.be/Z7rd04KzLcg?t=6710 01:51:50]''<br>
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<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;">$$ \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\varepsilon d_A \zeta + [\bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\varepsilon\zeta, T] + [T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em_\varepsilon\zeta] + *\zeta = F_A \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu + [[T, \nu], T] + [T, [T, \nu]] + [[T, \bigcirc\kern-0.96em\circ\kern-0.60em·\text{}\kern0.3em \nu], T] + *\nu $$</div>




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