A student asked a question in the string theory class today to which my answer was a suggestion of how to think about the issue raised in order to go about answering the question themselves. There’d be a few minutes of diagram drawing, and all would (hopefully) be clear. I thought that might not be an unreasonable thing to ask of a student, particularly in a graduate class, where they are ultimately trying to develop skills to do research. Well, it all went a bit pear-shaped as the student seemed to get quite strongly annoyed by this suggestion. I’ve still no idea why.
Anyway, on the bus home I thought I’d do some idle doodling, and ended up doing the exercise I’d suggested… and sure enough what I suggested doing works nicely and does (I think) make it clear.
I’m sure it was all a misunderstanding… Probably my fault.
Over late night dinner just now, since I can’t put down this lovely brush pen I’ve been drawing with recently, I scribbled the figures out from the notebook for you to look over my shoulder, as it were, and see what we’re up to in the class. Don’t worry so much about what it all means. It is sometimes nice to just look at the shapes*. (Actually, one of the students brought his mother to visit the class today. She sat through the whole hour and fifty minutes of the lecture. That was nice. I hope she enjoyed it all!)
I find these diagrams and the computations they represent rather pretty.
Wednesday will be the big climactic lecture of a sequence I’ve been leading them through showing how to actually compute the sum over random surfaces of arbitrary genus required by string theory (see here)…and what the results mean. The scribblings in the picture are parts of the precision tools being used for this, and we’ve been meeting nifty things like Wigner’s semicircle law and the Dyson Gas…
These are a few of my favourite things…”
Yeah. I had some wine with dinner. Can you tell?
-cvj
*For another look over the shoulder at fun diagrams, see e.g. here.
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See the sidebar for how to do latex. It is fully functional here.
Thanks for writing a bit out about the definitions. So indeed, we have different terms, and so the meaning is modified.
Sounds like a nice book, in any case. I hope to take a look at it some time.
Best,
-cvj
Hmm… Latex ain’t the coin of the realm here. Anyway, you see what I mean. 🙂
I hope Latex works here…
This is, more-or-less, Definition 4.1.1 from the aforementioned book.
For finite integers $k$ and $n$ such that $0 \le k \le n$, an “$n$-dimensional $k$-handle” $h$ is a copy of $D^k \times D^{n – k}$ (Here $D^r$ is a standard $r$-dimensional disk.), attached to the boundary of a smooth $n$-manifold $X$ along $\partial D^k \times D^{n – k}$ by an embedding $\phi : \partial D^k \times D^{n – k} \rightarrow \partial X$.
So, formally, your ribbons are 2-dimensional 1-handles and they are being attached to the boundary of $D^2$. As you say, these ribbons, 2-dimensional 1-handles, are exactly the things that “change the topology of a closed Riemann surface”.
A handle slide Definition 4.2.10 which I’m to lazy to type up 🙂 will as I said, define an equivalence of your top left and top right figures.
Well, i do not have the text in front of me, and i don’t doubt that there are diagrams that can be drawn that end up looking the same, and I am very happy that you find that it looks familiar, and maybe once the terms you referred to are defined and the context explained it’ll all be the same as you say, but…
There are no handles in those figures you referred to. They are ribbons.
They look like handles, but they are not. I’ve slid no handles, I’ve just shown tht those diagrams can be drawn on a sphere in the first case, and in the second case, on a torus. I have slid no handles, but maybe the issue is that I do not know what handle-sliding is, and indeed you hae not told me what you are defining a handle to be. Maybe your handles are ribbons? My handles are the things thar change the topology of a closed Riemann surface, and with that definition nothing I have done changes them.
No tori were harmed in the making of this post. 🙂
Best,
-cvj
Figure 4.8 in Gompf and Stipsicz is basically your first figure. Figure 4.5 is your second figure. A handle slide, Definition 4.2.10, defines the equivalence of your top left top right figures, see Figure 4.8.
I realize you are doing this for string world-sheets, but such handle slides have been formalized in arbitrary finite dimension.
Thanks…!
-cvj
As someone with rather poor math skills, those kind of diagrams are my favorite of your drawings. They have the arcane fascination of Chinese characters or cave paintings: I know they have meaning, but I’m entirely without the tools to decode it. It seems to represent sartorial details of the Universe.
(Although, I should say that this is not really what that is, as far as I know. But good to have the reference.)
Best,
-cvj
Great! Thanks.
-cvj
I guess you know this is all formalized in arbitrary dimension.
It’s the handle slides used in Smale’s proof of the h-cobordisim theorem. There’s a whole book on doing this stuff in 4 dimensions, “4-Manifolds and Kirby Calculus” by Gompf and Stipsicz.