Talk:Connection (algebraic framework)

Question

Question about the definition for non-commutative connection: Where is the ring A defined for the bimodule definition, and how does it relate to the rings R,S? More specifically, what does the derivation u in D(A) do the the elements r,s in the rings R,S respectively? — Preceding unsigned comment added by 128.104.191.52 (talk) 22:14, 17 May 2011

This basic idea is a development of an earlier observation, if we consider the commutative ring of continuous functions $C(X)$ on a topological space $X$...then we can detect all the topological information by analysing the algebraic structure of the ring. Of course, this is only topological information. The natural question: what about geometric data?

Well, we need more than just our ring $C(X)$! We need to consider additional data. To consider vector fields, we need to work with the projective module $M$ over $C(X)$. Why?

Lets consider for simplicity $\mathbb {R} ^{3}$ from vector calculus. Recall that a vector field is just a triple of smooth functions $(f(x,y,z),g(x,y,z),h(x,y,z))$ and to each point we assign a vector. How? Well, the components are obtained by evaluating each function $f,g,h$ at the given point. So if we consider the free module $C^{\infty }(\mathbb {R} )\oplus C^{\infty }(\mathbb {R} )\oplus C^{\infty }(\mathbb {R} )$, then we are really considering all vector fields on $\mathbb {R} ^{3}$.

Of course, we are a little bit sloppy, because this idea generalizes the notion of a Fiber bundle and instead of vector fields we have Sections over the fibration.

The idea the article was really getting at requires us to consider some further development to work with Fermions. This requires developing the algebraic framework thus explained to handle Spinors and Spin bundles (intuitively, a Principal-spin Bundle). What to do?

Well, the idea is articulated best in an article on the nLab. I will quote it:

A spectral triple is algebraic data that mimics the geometric data provided by a smooth Riemannian manifold $X$ with spin structure and generalizes it to noncommutative geometry. It consists of

1. An associative algebra A, to be thought of as the algebra of smooth functions on X;

2. a $\mathbb {Z} _{2}$-graded Hilbert space ${\mathcal {H}}$, to be thought of as the space of (square integrable) sections of the spinor bundle of $X$;

These two items encode the topology and smooth structure.

3. A Fredholm operator ${\mathcal {D}}$ acting on ${\mathcal {H}}$, to be thought of as the Dirac operator acing on the spinors.

This item encodes the Riemannian metric and possibly a connection.

Now in direct response to "What's going on in the section on noncommutative connections", well it basically considers a mapping from a derivation $u\in \mathrm {Der} (A)$ of the noncommutative associative algebra $A$, then takes it to a differential operator $\nabla _{u}\in \mathrm {Diff} (P,P)$ where $P$ is the projective module which is the algebraic counter-part to the space of vector fields.

The confusion lies in using a $R-S$ bimodule, and the author of that section did not specify the generalization clearly. I have found Landi's eprint, section 7.5, pp 122 et seq, very useful on the matter...

Pqnelson (talk) 20:23, 2 February 2012 (UTC)