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Link: Help me! I’m trying to understand Bakalov and Kirillov
If I may borrow a parlance from mathoverflow: +1 for the descriptive title, Bruce!
This is an example that I think would make more sense on this forum than the nLab. It is more of a discussion. Once you understand Bakalov and Kirillov, then we can transfer that newly found knowledge to the nLab.
We just need to figure out a way to attract eyes over here instead.
From the nLab:
How does the “s” map work?
Bruce Bartlett: I don’t understand the $s$ map in Example 5.1.11 in the online version of the book. It’s supposed to be the automorphism
$s : \Gamma_{1,1} \rightarrow \Gamma_{1,1}$of the torus with one boundary circle which is the analogue of a map which I do understand, the automorphism
$s_t : \Gamma_{1,0} \rightarrow \Gamma_{1,0}$of the ordinary torus (no boundary circle) to itself, which sends
$(\theta, \phi) \mapsto (\phi, -\theta).$But the map $s_t$ doesn’t seem to work if there is a boundary circle involved. Because an automorphism of $\Gamma_{1,1}$ is supposed to be the identity on the boundary circle, but the map $s_t$ isn’t: it rotates the boundary circle a quarter rotation (right?). The reason I say so is the following picture: we think of the torus as $\mathbb{R}^2 / \mathbb{Z}^2$, as Bakalov and Kirillov encourage us to do. Then the presence of the boundary circle can be thought of as having a little tangent vector pointing to the right at all the lattice positions (this uses their alternative Definition 5.1.10 the extended surface category). The map $(x,y) \mapsto (y, -x)$ rotates the plane by a quarter rotation and takes the lattice to itself, hence it descends to the quotient. But this map rotates the tangent vector by a quarter rotation, and we’re supposed to fix the tangent vectors!
So how does the map $s$ work?
Eric,
the map $s$ should be precisely as you describe it, but you keep fixes the $S^1$. see it this way: realize your torus with boundary circle as a rubber square with identifications on the opposite sides, with a circular hole in the middle. then turn the square of a 1/4 rotation, while keeping the boundary of the hole pointwise fixed (the homotopy class of this map is well defined).
This is an example that I think would make more sense on this forum than the nLab. It is more of a discussion. Once you understand Bakalov and Kirillov, then we can transfer that newly found knowledge to the nLab.
I second that.
At least I suggest the following: if you do want such discussion to be had on an nLab page, create a page with a title that will be the correct title after the entry has stabilized. Here for instance it might be Bakalov-Kirillov TFT or something like that. Then create in that entry a section “Discussion”.
Because I think we should have in mind the usability of the nLab in, say, five yearss from when a given entry is created. Probably an entry titled “Help me understand xyz” is completely obsolete then, because somebody will have helped, somebody will have understood and some stable material will have been written up elsewhere.
at the level of the plane, you can think it as a two-steps process: first, you turn all $S^1$’s of a 1/4 clockwise (this is not an automorphism of your structure, since it moves a basepoint in the $S^1$); then turn all the plane of a 1/4 rotation anticlockwise around the origin. the composition of these two is the transformation you’re interested in.
(no need to apologize :-) )
I somewhat disagree with 6. I mean, my past experience is that it is pretty hard to use mixed pages with blocks of normal content with large pages-long sections of discussion. Discussion is useful only in certain mode and having lots of it in regular page is kind of distracting from seeing the main part. I mean, the pages with more work character like informal write ups of seminars, which appeared before in nlab should stay I think in such a format, and concept pages should be created as more clean pages separately. Thus I would not like the learning discussion being a part of future page on some approach to modular functors and CFT but rather a separate accompanying page; thus a variant of the present page title like "Journal club on Bakalov-Kirillov book" is good enough for me; while both the general reference/book entry with toc as well as entries for concepts/topics partly covered in that book like modular functor, Kirillov-Bakalov RCFT etc. should also be separated. Thus I advocate not only making different pages according to different topics as Urs does but also with respect to different purposes.
Sure, whatever works. Whatever we do, let’s make sure that pages become such that they can actually be used as decent sources of information after the dust has settled and the activity has gone elsewhere.
Right, long term purpose is important. If it were just for daily usage, nlab would not be anything special...
don’t know.. in the published version it is correct (and a sequence of moves showing the equivalence of RHS and LHS is showed)
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