Tuesday, October 11, 2011

Starting my PhD - Noncommutative Geometry and Black Holes

I've had a great summer holiday. I've done a few programming projects. I've restarted my position with Universal Pictures International. It's time to start writing about my mathematics consistently. My PhD supervisor has given me a small project based on this paper: he has this noncommutative geometric model of a black hole, and I'm suppose to see what happens if I re-do each step over a finite field. It's been two weeks and, as usual, I've made very little progress on it. I don't actually understand what's going on. So let's spend a moment or two trying to clear the fog and see what a ``noncommutative geometric model'' means.

1. Noncommutative Geometry Re-visited

Regular readers (ha! as if there are any...) will know that my previous experience with NCG was with the Bost-Connes system, a topic I hope to take up again soon. My supervisor's work; however, is considerably different. We keep the general philosophy of starting from an algebra and trying to construct a geometry from it, but instead of working over Operator Algebras we're sticking to less-analytic algebraic structures and imposing additional objects on them that are meant to emmulate geometric notions. Let's go over this in detail and discuss some of these objects - keeping in mind that I don't know any geometry, canot provide any motivation for these concepts, and generally don't have a clue what I'm doing.

1.1. Differential Forms

Most readers will know that one can impose a Differentiable Structure on a topological manifold and use it to start doing some geometry. We'll dispose of the topological manifold and replace it with an algebra $ {A}$ and impose on it the notion of a 1-form. NCG 1-forms live in ``differential calculus`` $ {\Omega_1}$, which we define to be a bimodule (id est, a module on both sides) over $ {A}$. The ``differentiable structure'' comes in the form of a linear map $ {d: A \rightarrow \Omega_1}$ that obeys the product rule:

$ \displaystyle d(ab) = d(a) \cdot b + a \cdot d(b) \; \forall a, b \in A$

Additionally, we require that $ { (a, b) \mapsto a \cdot d(b) }$ spans the bimodule $ {\Omega_1}$.
Some readers may wonder what sort of algebras $ {A}$ can have a differential calculus. Actually, each algebra $ {A}$ necessarily has one, but we won't discuss that in this post.

1.2. Vector Bundles & Connections

My knowledge of modern geometry ends at this point and I am solely trusting the NCG literature. Initially, when seeing these terms, I think of the long definitions needed in Differential Geometry and start chasing down the terms in various textbooks. We don't need that here, and can describe these objects with simple algebraic ideas. That said, the next tool in our discussion is the NCG notion of a Vector Bundle. The definition here is a chain of algebraic structures: a NCG vector bundle is a finitely generated projective module over $ {A}$. A projective module is the image of a free module under a projection/idempotent. A free module is a module with basis vectors. E.g., $ {A^n}$ is a free module and if $ {E \in M_n(A)}$ is idempotent, then $ {E A^n}$ is a vector bundle.
Now if $ {\mathcal{E}}$ is an NCG vector bundle, and $ {\Omega_1}$ a differential calculus, both over $ {A}$, we can also define the NCG notion of a connection: A connection is a linear map

$ \displaystyle \nabla : \mathcal{E} \rightarrow \Omega_1 \otimes \mathcal{E}$

with the following condition:

$ \displaystyle \nabla (as) = d(a) \otimes s + a \nabla(s) \; \forall a\in A, \; s \in \mathcal{E}$

Next time we'll (Lord willing) discuss metrics, curvature, and how they all fit together to get a model for a black hole.

2. Sources

  1. LTCC Lecture notes in NCG by S Majid
  2. Section 4 of Almost commutative Riemannian geometry: wave operators
  3. Section 2 of Noncommutative Riemannian geometry on graphs

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