There are a good number of quantum topology papers in which a TQFT-like set-up is constructed as a functor to the category of vector spaces from some category of cobordisms which satisfy some "Lagrangian" condition. For example, Cheptea-Habiro-Massuyeau consider a category whose morphisms are cobordisms $M$ between closed oriented surfaces $F_+$ and $F_-$, where we choose Lagrangian subgroups $A_{\pm}$ of $H_1(F_\pm)$ correspondingly, and where we require that $H_1(M)=m_-(A_-)+m_+H_1(F_+)$ and that $m_+(A_+)\subseteq m_-(A_-)$ in $H_1(M)$ (the $m_\pm$ are inclusion maps). Similar conditions are imposed in many other papers.
I have never understood why such conditions are imposed. One half-thought I have is that it is related to Wall's result that the kernel of the inclusion of $H_1(\partial M)$ in $H_1(M)$ is a Lagrangian subgroup of $H_1(\partial M)$. Another half-idea is that it might be some weak 2-framing condition for the cobordism or something.

What is the conceptual explanation for this Lagrangian condition and its variants? Does it have anything to do with framing? (or orientation?) Why do "symplectic" and "Lagrangian" have anything to do with TQFT? (is it all just Wall's result in some guise?)

Every time I see a Lagrangian condition I feel very stupid for not knowing what it's doing there.

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    $\begingroup$ To me this looks very much like some version of weak 2-framings, as you put it. There seem to be two problems in 3D TQFT: (1) you usually don't get a well defined oriented theory. Instead you get a theory that is defined for some other kind of manifold. By making certain auxiliary choices you can get a well defined TQFTs, and (2) everybody has their own method of making these choices. Another common way to deal with these choices is to choose (cob. classes of) bounding 4-manifolds for your 3-manifold and bounding 3-manifolds for you surfaces. (cont) $\endgroup$ – Chris Schommer-Pries Nov 9 '10 at 9:37
  • $\begingroup$ (cont) Many of these approaches are (nearly) equivalent, but I don't know a good reference which compares them and gives details of the comparison. $\endgroup$ – Chris Schommer-Pries Nov 9 '10 at 9:38
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    $\begingroup$ Without having read the paper you reference, my guess is that this is equivalent to the gauge fixing condition for path-integral BV quantization. The ref is here dx.doi.org/10.1016/0920-5632(90)90647-D. Basically, one has a space of fields that is a symplectic manifold. A gauge choice corresponds to a Lagrangian submanifold. Physically equivalent gauge choices correspond to cobordant Lagrangian submanifolds. $\endgroup$ – Kelly Davis Apr 5 '11 at 6:35
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    $\begingroup$ This question turns out in fact to be a special case of this question: mathoverflow.net/questions/389/… $\endgroup$ – Daniel Moskovich Oct 23 '14 at 1:09

A historical reason for the importance of things Lagrangian in a TQFTy setting comes from Floer Homology and a conjecture of Atiyah that ``the two'' Floer theories are the same for 3-manifolds. Floer defined one homology (monopole Floer homology) based on the Chern Simons action functional, with the generators of its chain complex (the analogues of critical points in Morse Theory) being flat connections. He built another homology theory (the more standard one, today) to deal with the Arnol'd conjecture in Hamiltonian mechanics. If you take a Heegegard splitting of the 3-manifold, then, associated to the dividing surface you have the moduli space of flat connections, which is a symplectic manifold. You can then start to set up a Hamiltian Floer theory on this moduli space. Those flat connections over the surface which extend into the three-manifold on one side of the surface define a Lagrangian submanifold of the moduli space. Those flat connections which extend to the other side define another Lagrangian submanifold. The symplectic Floer business on the moduli space, set up properly, gives you a count the intersections of these two Lagrangian submanifolds, and hence of the set flat connections which extend to the whole 3-manifold. Atiyah conjectured these two Floer theories are the same. I suggest as an entry the Math Reviews article, See MR1283871, by Donaldson on the proof of a special case of this conjecture by Deitmar Salamon and Stamatis Dostoglou.

A philosophical reason is Weinstein's "symplectic creed": everything is Lagrangian. The classical version of a quantum state is a Lagrangian submanifold. So if one's TQFT yields a "semi-classical" Hilbert space, say a symplectic vector space, then the states (wavefunctions) of the "Hilbert space" be Lagrangian subspaces

  • $\begingroup$ I adjusted your quotes a bit; I hope you don't mind. The parsing of the "grave" accent is a bit wonky, since it's Markdown's verbatim marker. $\endgroup$ – Ben Webster Nov 9 '10 at 7:22
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    $\begingroup$ @Richard,change "monopole" to "instanton." To connect more to Daniel's question, the observation he attributes to Wall is the reason the flat connections on the surface which extend to the 3-manifold are Lagrangian: the differential of the restriction map on flat connections is the cohomology map $H^1(X)\to H^1(\partial X)$ (suitably twisted). Incidentally, a useful but not as well-known fact is that the CS function lifts this restriction to a Legendrian submanifold of a S^1 bundle over the flat connections on a surface. $\endgroup$ – Paul Nov 9 '10 at 14:16
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    $\begingroup$ (cont.) There are other possible answers to the question; Daniel should look at Kevin Walker's old and new notes on his web page. $\endgroup$ – Paul Nov 9 '10 at 14:17

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