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I looked closer and figured out the typo. (the 9 in the idea section was a ] )
Urs,
there’s a point I’m missing in #102: you write
This is an oo-groupoid over $U(1)$, so we are entitled to the oo-groupoid cardinality of it
but $U(1)$ is not a $\mathbb{Q}$-module, so I’m note sure about taking this oo-groupoid cardiality. Should we choose a lift from $U(1)$ to $\mathbb{R}$?
Yes, what I mean is regarding $U(1)$ as a subset of $\mathbb{C}$.
It think a better way to say this is, again, that we pass to the canonical representation of $U(1)$ and use this to turn an $\infty$-groupoid over $U(1)$ into a vector in that representation space (if the $\infty$-groupoid cardinality converges, at least).
Yes, makes sense: the “one categorical degree below” version of the canonical morphism $\mathbf{B} U(1)\to Vect_{\mathbb{C}}$ should be the inclusion $U(1)\hookrightarrow \mathbb{C}$. now I should try to understand in which sense taking sections is the same thing as computing an integral with the oo-groupoid measure…
on second thought, Urs prescription “regarding $U(1)$ as a subset of $\mathbb{C}$ was the obvious one: when one writes down a dear old ill defined path-integral,one writes an expression like
$\int e^{iS(x)} dx$so it is clearly integrating the $U(1)$-valued function $e^{iS}$ as a $\mathbb{C}$-valued function. I got confused in looking too much at $U(1)$ as an abelian group :-)
Yes, exactly.
In that notorious entry exercise in groupoidification - the path integral it is crucially oo-groupoids over the associated vector bundle, associated to the background field, that are being pull-pushed. This way in each fiber we have an oo-groupoid over the representing vector space, which under groupoid cardinality we may identify with a genuine vector. The $\exp(i S)$-term that appears towards the end in the action funcitonal is accordingly the image under the representation of the $U(1)$-valued action functional.
…
I was thinking back to this period in Bakalov-Kirillov’s book introduction: “under some (not too restrictive) assumptions, the notions of modular tensor category, 3D TQFT and 2D modular functor (topological and complex-analytic) are essentially equivalent”
As pointed out by Bruce about one month ago (sorry, Bruce, if I took so much time to really understand what you were saying..) in modern language we would say this as follows: “the notions of 3-2-1 TQFT, 2-1 modular functor and modular tensor category are equivalent. the obvious maps establishing the equivalences are given by forgetting the data on top-dimensional manifolds”
but this is a “down to level 1” version of Hopkins-Lurie classification of extended TQFTs! and naturally leads to the following version of the cobordism hypothesis: “the datum of an $n$-extended $d$-dimensional TQFT (i.e., a $d$-to-$d-n$ TQFT) is equivalent to the datum the theory associates with connected $(d-n)$-dimensional manifolds”. a statement like this can be found in extended topological quantum field theory, but since I’m quite sure I wrote that, this does not supports what I’m saying too much.. :-) on the other hand, at cobordism hypothesis only the fully extended case seems to be considered, i.e., $d$-to-$0$ TQFTs, and that is Hopkins-Lurie.
so I was wondering whether the $d$-to-$d-n$ case is already there in Lurie’s notes or one should work it out (at first sight the cobordism category $Bord_n$ considered by Lurie begins with $0$-dimensional manifolds, so it it fully extended, but I have not gone into the details of the proofs to see if this hypothesis is actually used).
But this can’t be entirely true: in the case of $(d,d-1)$ TFT it would mean that an ordinary functor on a category of cobordisms is fixed by the vector spaces that it assigns to boundaries. But that’s not quite right.
There is something in “modular functor” that provides more information, I guess.
But I haven’t thought about this hard enough.
well, I guess one has to say what is the “datum” the theory associates with connected $(d-n)$-dimensional manifolds. think to a (2,1) TFT. if I say that this theory is completely determined by giving the vector space $Z(S^1)$, then I’m clearly saying something totally false. but if I say that the theory is completely determined by giving the Frobenius algebra $Z(S^1)$ then I’m saying something true.
the same happens for (3,2,1) TQFTs: one has to specify the tensor category structure on $Z(S^1)$ in order to specify the theory, and not only the abelian category structure.
so I think some sort of suitably formalized statement “lowest dimensional data fix it all” could be done. but I have yet no clear idea of what to ask these data to be.. :-)
I see what you mean now.
I was also thinking that a modular tensor category is a categorized version of a Frobenius algebra, and that if one looks at the trace of a Frobenius algebra $tr: A\to k$ as to a morphism $A\to 0Vect_k$, then this pattern is found in modular tensor categories in the form $\mathcal{C}\to 1Vect_k$, as $X\mapsto Hom(\mathbf{1},X)$, where $\mathbf{1}$ is the unit object. so a suitable notion of $k$-Frobenius algebra could be the kind of answer I was looking for: something like “an $n$-to$1$ TQFT is equivalent to an $(n-1)$-Frobenius algebra”.
thinking more carefully to what I wrote, I’m still convinced that for any dimension $k$ there should be a suitable notion of “$k$-dimensional datum” such that any $n$-to-$k$ TQFT is completely determined by the $k$-dimensional datum. however, this abstract nonsense is probably doomed to have no concrete description for all $k\geq 3$, since a classification of compact oriented $k$-manifolds is non available (or not even possible) for such $k$’s. and even for $k=2$, where one has an infinite number of genera, I’m not confident a clear description of the 2-dimensional datum can be given. so we are left with the $k=0$ case, where we have Hopkins-Lurie answer: the 0-dimensional datum is a fully dualizable object, and the $k=1$ case, where I suspect the answer is a suitable higher version of Frobenius algebras.
a tentative formalization could be the following: let $Vect_2$ denote the 2-category of $Vect$-enriched additive categories, and $(\infty,1)Bord_2$ the $\infty$-category of 2-1 cobordism; what is a representation $(\infty,1)Bord_2 \to Vect_2$? the tentative answer is, of course: a modular tensor category.
In case some of you are interested, Turaev’s book on Homotopy Quantum Field Theory has now been published. It looks very nice. He tackles various 3D analogues of stuff I mentioned some time ago.
Thanks, Tim, for the reference!
concerning #214 above, the more I think to it the more I’m convinced that representations $(\infty,1)Bord_2\to Vect_2$ are the same thing as modular tensor categories. this seems to be a classical statement which is well known at least in folklore. can anyone address me to a reference? (neither bakalov and kirillov’s book nor funar-gelca’s arXiv:math/9907022 seem to say this in a fully explicit form). or should I work out all the details by miself and post them here? :)
or should I work out all the details by miself and post them here? :)
Is that a rhetorical question? :)
no, really! :)
the fact is I am sure what I’m sayng should be well known, and at the same time I’m unaware of a reference :(
in any case I will write something about this here and in the lab..
ok, I got it. it’s just the well known fact that the structure of modular tensor category can be expressed in terms of Moore-Seiberg data. However I feel the language of $(\infty,1)$-two-dimensional cobordism makes everything cleaner here, so I will write a survey note on this (anyone interested in collaborating is invited :-) ). in particular, Moore-Seiberg data are seen to be just structure constants for a modular tensor category seen as a Frobenius algebra in $Vect_k$-enriched additive categories.
by the way, I’ve added a Moore-Seiberg data stub to the Lab.
it seems that notes of this kind already exist.. they are Segal’s Lecture Notes for the July-August 1999 ITP Workshop on Geometry and Physics. In particular what I was aiming at is the subject of section 3.2 in Lecture 3.
Domenico,
thanks. Somebody should contact Bruce Bartlett about this. He tells me that his recent result with Jamie Vicary fixes a mistake/gap in the literature on the statement of getting MTCs from TFTs. He says it was unclear until recently how the condition arises that the cateory assigned to the circle has all duals for objects. Or something like that, I might be misremembering the details.