The n-Category Café

Please provide links to the sections so we don’t have to download the whole thing.

[John Baez: Done. You still need to download the whole thing, but you’ll go straight to the section you want.]

John wrote:

Tim van Beek wanted a heads-up when it was done…

Thanks! This way I don’t have to perform busy waiting.

…so he could make another round of corrections.

Actually I read it because I find it interesting, and because I’m allowed to ask questions about it here, that one or two of those may help you to improve the paper is a tradeoff.

(Please cut me some slack if it takes me a couple of days to report back, time management is a challenge right now).

My duty to mathematical physics thus fulfilled…

Don’t know if it goes without saying: I hope you are going to visit math phys often for the fun of it…

On p. 31 you have $G = d B$. The notation most people will be familiar with for the field strength of the Kalb-Ramond field and will recognize in other articles is $H = d B$. Wouldn’t that be better?

The curvature of a Cartan connection $A$ is

$$\mathrm{d} A + \frac{1}{2}[A, A]$$

with

$$[A, B] (X, Y) = [A (X), B(Y)] - [A (Y), B(X)]$$

with the commutator evaluated in a Lie algebra.

For example, if a group is,

$$(a, b)(c, d) = (a + b c, b d)$$

its Lie algebra is

$$[(\epsilon, 1), (0, 1 + \epsilon)] = (\epsilon, 1 + \epsilon) - (\epsilon + \epsilon^2, 1 + \epsilon) = (-\epsilon^2, 0)$$

$$[X, Y] = -X$$

The Maurer–Cartan form is $$A^0 = \frac{1}{y} \mathrm{d}x$$

$$A^1 = \frac{1}{y} \mathrm{d}y$$

$$\mathrm{d} A^0 = \frac{1}{y^2} \mathrm{d}x \wedge \mathrm{d}y$$

$$\mathrm{d} A^1 = 0$$

The horizontal component of the curvature is,

$$\mathrm{d}A^0 (X, Y) + [A (X) , A (Y)] = \frac{1}{y^2} + [\frac{1}{y} X, \frac{1}{y} Y]$$

$$= \frac{1}{y^2} + \frac{1}{y^2} [X, Y] =\frac{1}{y^2} - \frac{1}{y^2} = 0$$

The vertical component of the curvature is

$$\mathrm{d}A^1(X, Y) + [A (X) , A (Y)] = 0 + \frac{1}{y^2} [X, Y] = 0$$

So the curvature is

$$\mathrm{d} A + \frac{1}{2}[A, A] = 0$$

Finally made my way through the version I got yesterday afternoon (CET), and it looks pretty much completed.

Repeating myself: It is a joy to read! Here are some aspects of what I like most:

• simple yet helpful pictures,

• motivational handwaving, clearly pointed out as such, plus the information if it can be made precise plus references,

• simple analogies that pick up the reader where she/he stands, to new concepts that are stated in a way where the analogy is even mirrored in the very phrasing…(list could go on).

In fact, I have only two suggestions for additional references: If you like to, you could mention the Aharonov–Bohm effect on page 28 or 34. Likewise, you could mention the term “anyon” on page 34 where you mention particles with “exotic statistics” (once people know the keyword, they can look it up in wikipedia, which already has a nice article with references).

I’m completly unperturbed by typos as long as I still get the meaning (you may have noticed that I cannot write one single post without one), but some readers will infer from the existence of typos that the authors are generally sloppy - I mean, more serious readers than those who managed to raise item 20 on the crackpot index (“if he can’t spell Einstien, what are the chances he understands relativity?”).

So here we go:

• Abstract: “which has topological topological gravity”

• p3: “You should think of of objects”

• p6 bottom: “We define [the] path groupoid”

• p12: the horizontally composition picture: right side should go from x to z, but goes to y instead.

• p13: interchange law, equation, right side: both indexes are 1 in the first bracket, the first one should be 2.

• p13: “the path groupoid of a spacetime manifold describes all the possible motions of a point particle”: Maybe “phase space” would be better than “spacetime”, because you did not exclude spacelike paths :-)

• p14: “The key idea [is] to define a concept of ‘thin homotopy’…”

• p30: “This also [has] a higher version, which dates back to 1935…”

• p38: “…the previous section is close[ly] related…”

• p38: “…conditions of Theorem [broken link]”

• p.39: “it suffices suffices”, and alpha is missing some tex tags twice

• p40: “…a vector space equipped with [a] nongenenerate…”

• p43: “Then [a] gauge transformation…”

• p46: “…the holonomy over a surface will not changee if [we] apply a smooth homotopy…”

• p47: “…if the the transition functions satisfy a…” and “readibility” instead of “readability”

• p52: “Conversely, any closed, integral 2-form F on M [is] the curvature of come…”

P.S.: If you feel the need to balance your karma, may I ask you to proofread some pages on the nLab about axiomatic and constructive QFT in a few months? (Even superheroes need some distraction from time to time, from the consuming task of saving the planet .)

Our paper has finally been published, but what makes this worth mentioning is that it’s available for free, open-access:

• John C. Baez and John Huerta, An invitation to higher gauge theory, General Relativity and Gravitation 43 (2011), 2335-2392.