What is the significance of the tautological line bundle in twistor theory?

[chrispb]

Hi all,

I'm a physicist attempting to learn twistor theory. I'm confused by this notion of a tautological line bundle. So far, the most accessible source has been http://en.wikipedia.org/wiki/Tautological_line_bundle" . In the definition, they say v in x. Do they mean a point in RP^n again? If so, why didn't they write RP^n x RP^n in the first place?

So, what I'm getting out of the second paragraph is RP^n is the base space, and we're adding a line acting as a fiber at every point in it. More specifically, we're adding the line that passes through the point. If so, why do they say RP^n x R^(n+1) instead of RP^n x R^n in the definition? Regardless, it seems that after doing this, my total space will just be R^(n+1). This statement, however, disagrees with their claim at the bottom that I'll recover a Mobius strip for n=1.

In twistor theory, I'm primarily concerned with "O(1), the dual of the tautological line bundle O(-1) over CP^1". Where does this O(-1) notation come from? Is it part of some more exciting area of math?

Thanks in advance! Chris

 

[lavinia]

Hi all,
I'm a physicist attempting to learn twistor theory. I'm confused by this notion of a tautological line bundle.

Euclidean space can be decomposed into straight lines through the origin. Removing the origin- a point that they all share - makes it possible to create a quotient space by identifying each line to a single point. This is the base space of the tautological lin bundle. As a topological space it is homeomorphic to project space - real or complex, depending on whether the lines are real or complex.

Above each point is the line that it came from. The collection of all of these lines is the total space of the tautological line bundle.

This total space is not the same as euclidean space minus the origin because the origins of each of these lines are distinct.

 

[chrispb]

Ah, I see. Thank you for the clarification. One quick question; are we associating the entire line minus the origin (as in, both rays) with a single point in the base space? In other words, both the positive and negative sides of the x-axis (in R^2), say? In other words, I could take the unit circle and identify antipodal points and get back the base space?

 

[tmccullough]

I think you would be better served by getting a handle on projective space before worrying about what a bundle on projective space looks like. Projective space isn't terribly confusing, but it's worth taking a bit of time to wrap your mind around it...

To answer your question - yes, projective n-space is the n-sphere with antipodal points identified. Note that a circle with antipodal points identified is still a circle, but this is a bad example. For $$n>1$$, it is its own distinct thing. It's worth emphasizing that $$\mathbb{RP}^n$$ is the topological space which parameterizes lines in $$\mathbb{R}^{n+1}.$$ It may be handy to consider the generalization for this idea is called a Grassmanian, this may come up in your readings - it's just handy anyways.

From the definition of bundle it's probably not terribly clear why it's important. In general, bundles are fancy things - they arise as fundamental objects for algebraic topology (where bundles have a fundamental property) and algebraic geometry (the O(1) is a reference to the "twisting sheaf" - a fundamental algebraic geometry idea for projective varieties).

Learning much about either of these may interest you, but will definitely take you FAR afield from your stated intentions. The crossroads of physics and math is frustrating. In retrospect, this was a much more verbose response than I had planned - sorry.

 

[chrispb]

I appreciate the reply; it was very helpful. It seems I should just check out a few books on algebraic geometry. Hopefully my limited algebra and differential geometry background will be sufficient :)

A lot of physicists in their papers throw fancy math lingo around, but don't care that physicists (at least, grad students) tend not to be familiar with the intricacies of algebraic geometry. Oh well. Thanks again!

 

[lavinia]

Ah, I see. Thank you for the clarification. One quick question; are we associating the entire line minus the origin (as in, both rays) with a single point in the base space? In other words, both the positive and negative sides of the x-axis (in R^2), say? In other words, I could take the unit circle and identify antipodal points and get back the base space?

yes.

 

[tmccullough]

I definitely don't want to discourage you, but be aware that algebraic geometry and algebraic topology both require significant background before they make much sense. This really will take you far afield.

If you really intend to do theoretical physics for your life, then it might be worth it. It's a significant time investment, though.

 

[lavinia]

BTW: There is a wonderful description in Milnor's Characteristic Classes of the tautological line bundle and its relatives - the bundles of K-planes in euclidean n-space over the Grassmann manifold of k-planes in n-space. These manifolds are key in mathematics and physics and can be understood geometrically without algebraic topology or algebraic geometry.

 

[chrispb]

BTW: There is a wonderful description in Milnor's Characteristic Classes of the tautological line bundle and its relatives - the bundles of K-planes in euclidean n-space over the Grassmann manifold of k-planes in n-space. These manifolds are key in mathematics and physics and can be understood geometrically without algebraic topology or algebraic geometry.

I'll take a look at it; thanks!

I definitely don't want to discourage you, but be aware that algebraic geometry and algebraic topology both require significant background before they make much sense. This really will take you far afield.

If you really intend to do theoretical physics for your life, then it might be worth it. It's a significant time investment, though.

I appreciate the heads up. It really comes down to knowing enough to do the physics and understand the literature (namely, Witten). Unfortunately, the particular applications I'm interested in are new enough that no one quite knows what's really the heart of it and what's just fluff. I'm hoping to decide for myself, but for the time being, most of this isn't key to understanding the physics.