Decoding the Graph-Wall-Tome Connection

Questions by Eric Weinstein

What is $$F_A$$ geometrically?

$$F_A$$ is the curvature tensor associated to the connection or vector potential $$A$$.

What are $$R_{\mu v}$$ and $$R$$ geometrically?

Explanation of $$R$$

$$R$$ is a scalar value, describing the "curvature of the spacetime manifold" at each point along the manifold. It's based on a concept of 'parallel transport', where you move a vector around some path on the manifold.

$$R$$ can be computed at each point on the manifold, and describes the difference in the vector's angle after following an infinitesimally small path around the neighborhood of that point, vs. what it was originally. The video does a great job of visualizing when and why that vector angle change would happen, with flat vs. curved manifolds.

In the video, they focus first on the curvature of space. Hopefully they incorporate back in curvature in time, because that's less obvious.

The same video then proceeds to explain $$R_{\mu v}$$. It progresses through some concepts.

How do they relate?

Cohomology

What does this have to do with Penrose Stairs?

• Penrose stairs
• Spinor

We’ve heard Eric talk about Penrose stairs and spinors - essentially phenomena where you cannot return to the original state through a 360 degree rotation, but require a 720 degree rotation.

What are “Horizontal Subspaces” and what do they have to do with Vector Potentials or Gauge fields?

• Vertical and horizontal bundles
• Introduction to gauge theory
• Symmetry

From theplebistocrat:

Generally, we're wanting to understand how fermions arise from - or are embedded within / upon - topological "spaces" that have distinct rules which govern operations within those topological spaces, and then how those rules produce higher dimensional operations in corresponding spaces.

Just intuitively, and geometrically speaking, the image that I'm getting when describing all of this and trying to hold it in my head is the image of a sort of Penrose Tower of Babel, where the fundamental underlying structures reach upwards (but also downwards and inwards?) before reaching a critical rotation that corresponds to a collapse of structure into a higher dimensional fiber bundle.

But doesn't this require the symmetry break? How is left and right rotation in a subspace transformed into verticality? This is a crazy rabbit hole, friends. Keep your chins up. Let me know if this was helpful or leading astray.