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  • Writer's pictureAlex Cassem

Relationship Between Group Theory & Riemannian Geometry

Recently, I have been looking more and more into how you can show/develop explicitly a field theory that started form a group construction into a geometric theory based on Riemannian geometry (or perhaps something with more complex, in the mathematical sense, structure).

I found the following resource/reference, Cartan Geometry, to be extremely insightful on where to start. The link displays a great table showing how the underlying Riemann structure relates to not just the gauge group, but local geometry, global geometry, connection coefficients, and even the differential cohomology. The example I shall take from the table, but also add a few of my own details, is of Einstein Gravity.

As many of us have learned in first learning GR, we construct the geometry from Lorentz invariant quantities (vectors, tensors...), however, the fundamental group underpinning the action of Einstein gravity is actually the Poincare gauge group, Iso(d − 1, 1). Locally, at each point in this group gives another sub group, specifically a stabilizer group, which is given by what leaves each point invariant in the action. for Iso(d − 1, 1), this is the Orthogonal group, O(d − 1, 1).

The actual space that can be constructed from this/these groups will locally model Minkowski spacetime R^{d-1,1} (think about what spacetime I just stated...). This gives a local geometry that appears Lorentzian, and globally will be pseudo-Riemannian. Now, there is one more level of geometry needed to use Einstein gravity. I will be describing it in terms of physics and not so much mathematical language (since I find it easier that way). To relate different quantities or fields (couple them) to gravity, we need a new way to connect them. Think about how the derivative is the relationship between the tangent space along a curve living in cartesian space. Well, the tangent space is now called a tangent bundle, and the derivative in our analogy is called the spin connection. This is from a field called differential cohomology (and that is the extent to what I know specifically about that field of mathematics, although I am in the process of learning it in-depth).

Finally, the summation of all these components and constitutes are what make up a first-order formulation of gravity, being Einstein Gravity.

Now, as some historians of physics will mention, well, wasn't this one of our first attempts to quantize gravity? Well, yes, and it was semi-successful as you could guess if you noticed one caveat I mentioned above. This formulation is specifically for Minkowski space (flat)! You can quantize this formulation, but get very limited results and predictions (such as when we expect QM effects from gravity), and compute some low-order correlation functions (Lorentzian Wormholes, Matt Visser, 1996).

So, I guess the real question is whether or not out of a gauge group formulation, we can generalize the local space and thus the global geometry.... Or what gauge group is required and has that much structure (if there even is one)?

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