User dan kneezel - MathOverflow

Hirzebruch's motivation of the Todd class

2011-04-03T19:26:16Z

In Prospects in Mathematics (AM-70), Hirzebruch gives a nice discussion of why the formal power series $f(x) = 1 + b_1 x + b_2 x^2 + \dots$ defining the Todd class must be what it is. In particular, the key relation $f(x)$ must satisfy is that

($\star$) the coefficient of $x^n$ in $(f(x))^{n+1}$ is 1 for all $n$.

As Hirzebruch observes, there is only one power series with constant term 1 satisfying that requirement, namely $$f(x) = \frac{x}{1-e^{-x}} = 1 + \frac{x}{2}+\sum_{k\geq 2}{B_{k}\frac{x^{k}}{k!}} = 1 + \frac{x}{2} + \frac{1}{6}\frac{x^2}{2} - \frac{1}{30}\frac{x^4}{24} + \dots,$$ where the $B_k$ are the Bernoulli numbers.

The only approach I see to reach this conclusion is:

Use ($\star$) to find the first several terms: $b_1 = 1/2, b_2 = 1/12, b_3 = 0, b_4 = -1/720$.
Notice that they look suspiciously like the coefficients in the exponential generating function for the Bernoulli numbers, so guess that $f(x) = \frac{x}{1-e^{-x}}$.
Do a residue calculation to check that this guess does satisfy ($\star$).

My question is whether anyone knows of a less guess-and-check way to deduce from ($\star$) that $f(x) = \frac{x}{1-e^{-x}}$.

Answer by Dan Kneezel for What makes the stable module category stable?

2011-08-26T02:07:56Z

The answer to the first question is yes, the two categories are nontrivially related: Both the traditional stable homotopy category and the stable module category are examples of stable homotopy categories in the sense of Hovey-Palmieri-Strickland, Axiomatic Stable Homotopy Theory. See the first few pages of Schwede's Stable model categories are categories of modules for a convenient overview. Of special interest are Definition 2.1.1 and the subsequent paragraph on page 107, and Example 2.4.(v) on page 111. You may also find it useful to consult Hovey's Model Categories, especially Chapters 2 and 7, as well as Chapter I.2 in Quillen's Homotopical Algebra.

As for the second question, observe the following extract from Example 2.4.(v):

Fortunately, the two different meanings of ‘stable’ fit together nicely; the stable module category is the homotopy category associated to an underlying stable model category structure [21, Section 2].

so it sounds like Schwede would agree with Mariano and Tom that, at least originally, the selection of the word 'stable' in stable module category likely had nothing to do with stable homotopy and all that.

Corrections are welcome.

Computing the q-series of the j-invariant

2011-07-31T03:19:12Z

It is a fundamental fact, often quoted these days in its connection with Monstrous Moonshine, that the q-expansion (i.e., the Laurent expansion in a neighborhood of $\tau = i\infty$) of the j-invariant is given by

$$j(\tau) = \frac{1}{q} + 744 + 196884q + 21493760q^2+\dots, \quad q = e^{2 \pi i \tau}.$$

I do not recall, however, ever seeing a modern treatment, nor even a hint, of how one might go about obtaining this expansion. Does anybody know a nice way to compute these coefficients? (I mean a way which does not invoke Moonshine, not that I'd expect that to make the computation more pleasant.) Is there a standard way to do it?

I did find one approach published by H.S. Zuckerman in the late 1930s*, which makes use of a "fifth order multiplicator equation" for $j(\tau)$ -- distilled from Fricke and Klein's Vorlesungen uber die Theorie der elliptischen Modulfunktionen -- and an identity of Ramanujan for the generating function of partition numbers of the form $p(25n + 24)$. Is this typical?

*Zuckerman, Herbert S., The computation of the smaller coefficients of $J(\tau)$. Bull. Amer. Math. Soc. 45, (1939). 917–919.}

What are D-branes, really?

2010-10-22T23:32:02Z

In the past couple years, I've read many words pertaining to "D-branes" without feeling I have fully comprehended them. In broad terms, I think I get what they're about: They're supposed to serve as habitats for the ends of open strings and can be conceived of as submanifolds (of the target manifold in a sigma model), possibly augmented with a vector bundle, or a sheaf of [somethings], or maybe some other kind of label/data. (Corrections welcome.)

In the hopes of tightening my grasp on the concept, here are some of the questions that have been nagging me during my reading.

What specifically is the definition of a D-brane, say in the context of a topological field theory? (Or what are the most promising provisional definitions?) What references are most accessible to a mathematical audience?
What picture should I have in my head when an author talks about "the moduli space of D-branes"?
What is the idea behind the "dynamics of D-branes" that researchers sometimes talk about? (Perhaps when I understand better how to think about these gadgets, it will be easier to conceive of how they should change over time.)
What goes into verifying (or at least asserting/conjecturing) that the elements of twisted K-theory classify "D-brane charges"?

(Question reposted from here.)

Comment by Dan Kneezel

2012-09-28T02:30:19Z

jmac, I think you're getting dinged mainly because it is considered Bad Form to ask questions in the Answers or Comments fields. In general, if a thread inspires you to ask a new question, you should click "Ask Question" (grey button, top right of page). Before you do so, though, you should make sure to explain the context of your question and demonstrate you've already made some effort to resolve it on your own. See here for tips on how to formulate a good MO question. mathoverflow.net/howtoask My 2 cents: I think your question is reasonable, I had wondered about it myself.

Comment by Dan Kneezel

2011-10-24T05:05:02Z

For further discussion of this perspective, see en.wikipedia.org/wiki/…

Comment by Dan Kneezel

2011-09-14T21:08:11Z

Is anyone else having trouble getting audio from the videos? I'm pretty sure that the audio worked just fine for me back in '07 when I originally watched these, but now I don't hear anything. I tried streaming the videos in Chrome, in Firefox, downloading and opening with the standalone QuickTime application, as well as opening with VLC. The video shows up just fine but no audio from any of them. Running Windows 7.

Comment by Dan Kneezel

2011-08-01T19:57:27Z

@W. Zudilin: Thanks!

Comment by Dan Kneezel

2011-08-01T19:34:16Z

Interesting! Thanks!

Comment by Dan Kneezel

2011-08-01T19:16:17Z

@N. Elkies: Thank you for your detailed comments. They have been quite enlightening!

Comment by Dan Kneezel

2011-07-31T07:52:39Z

@N. Elkies: Oh! The multiplicator in the Zuckerman reference is a relation between $h_5$ and $j$. In particular, $$j(\tau)=h_5+6\cdot 5^3+\frac{63\cdot 5^5}{h_5}+\frac{52\cdot 5^8}{h_5^2}+\frac{63\cdot 5^{10}}{h_5^3}+\frac{63\cdot 5^{13}}{h_5^4}+\frac{5^{15}}{h_5^5}.$$

Comment by Dan Kneezel

2011-07-31T04:14:22Z

Can this approach be extended to Thompson series $T_g(q)$ away from $g = id$?

Comment by Dan Kneezel

2011-07-31T04:05:23Z

Nice. Thank you!

Comment by Dan Kneezel

2011-04-03T21:05:42Z

@Eric Interesting. Thanks for the link.

Comment by Dan Kneezel

2011-04-03T20:45:58Z

That is very nice indeed. Thanks!

Comment by Dan Kneezel

2010-10-28T22:09:51Z

@Greg, @Urs - Thank you both for your explanations. I was really quite divided over which one to accept as an answer -- if I could pick two, I would -- but I decided to go with Urs' for including the observation that in 2d RCFTs, D-branes are Frobenius algebra modules.

Comment by Dan Kneezel

2010-10-23T02:55:37Z

@José, @Laie - Thank you both for your reference suggestions. @José - Thanks also for your comments. Refinements to my question are under construction.