Similar to this question, I'm trying to figure out why one would be interested in Lagrangian submanifolds. But from a more geometric point of view. My best find so far is Exercise 12.4 in McDuff, Salamon, which is to

... construct a symplectic embedding $$B^{2n+2}(r)\hookrightarrow \mathbb{T}(1)\times\mathbb{R}^{2n}$$ in the following way. Find a linear Lagrangian subspace $L$ of $\mathbb{R}^2\times \mathbb{R}^{2n}$ whose $\epsilon$-neighbourhood $L_\epsilon$ projections injectively into $\mathbb{R}^2/\mathbb{Z}^2\times \mathbb{R}^{2n}=\mathbb{T}^2(1)\times \mathbb{R}^{2n}$. Then consider a composite mapping $$B^{2n+2}(r)\hookrightarrow L_\epsilon\hookrightarrow \mathbb{T}^2(1)\times\mathbb{R}^{2n}.$$

A bit earlier they explained that one may embed $B^{2n}(1)\hookrightarrow B^n(\delta)\times \mathbb{R}^n$ by $(x,y)\mapsto (\delta x,\delta^{-1}y)$.

I suspect that the reason people care about Lagrangian submanifolds is that there are more situations where one may move a construction from one place to another by finding appropriate Lagrangian submanifolds.

So my question is: What are other geometric problems in which the existence of a certain Lagrangian manifold is used in a manner similar to the above? That is, when do you go hunting for Lagrangian submanifolds?

Edit: To clarify, I'm not that much interested in generalities, but in concrete geometrical problems where a Lagrangian does the trick.

Lagrangian submanifolds (not necessarily closed) always exist locally in $(M,\omega)$, as you can easily see from Darboux coordinate theorem, so its existence in general is quite trivial. The non-trivial part is the existence of Lagrangian submanifolds satisfying certain constraints, e.g. closed exact Lagrangian submanifolds (which means $\omega=d\theta$ and $\theta|_L$ is exact and special Lagrangian submanifolds in Calabi-Yau manifolds (which means the imaginary part of the holomorphic volume form $\mathrm{Im}\Omega|_L=0$).

People care about Lagrangian submanifolds because a lot of geometric information of the symplectic maifold $(M,\omega)$ can be extracted by studying the geometry of Lagrangian submanifolds.

At first glance symplectic geometry is very hard to study, because it is very flexible. However, if one also takes care of the almost complex structure $J$, then certain geometric invariant called (genus 0) Gromov-Witten invariant can be defined by (virtually) counting how many $J$-holomorphic $\mathbb{P}^1$s are there inside a fixed a homology class $\beta\in H_2(M,\mathbb{Z})$. This is obviously a very elegant invariant, because in the classical theory (like characteristic classes) we only care about homology classes, but now we are able to tell how many geometric objects a given homology class contains, at least in some cases. Such a consideration will give you a quantum cohomology ring $QH^\ast(X)$, which is the ordinary cohomology ring equipped with an additional product structure called quantum product. Such an algebraic structure is useful in solving classical problems in enumerative geometry (just as the enhancement from a Banach space to a Banach algebra is fundamental in the development of functional analysis).

If you take further consideration into the Lagrangian submanifolds of $M$, then more algebraic structures can be obtained. For example, if the Lagrangian submanifold $L\subset M$ is good enough (satisfying a bunch of technical assumptions), then we can equip the Floer cochain complex $CF^\ast(L,L)$ the structure of an $A_\infty$ algebra. It's a kind of non-associative algebra and you can simply think of it as a curved version of a differential graded algebra, then of course it's a very delicate algebraic structure and contains very rich geometric informations. When you pass from $CF^\ast(L,L)$ to its cohomology level, we obtain the Lagrangian Floer cohomology group $HF^\ast(L,L)$, and it's a differential graded algebra. So the process of taking cohomology can be realized algebraically as a linearization. I think this already provides an answer to your question: given a sufficiently good Lagrangian submanifold $L\subset M$, then you can attach to it many interesting algebraic structures.

These algebraic structures are of course very useful in solving classical geometric problems, e.g. Arnold conjecture (Fukaya, et al.), Lagrangian embedding (Seidel), nearby Lagrangian problem (Abouzaid, Fukaya-Seidel-Smith), and recently Joyce et al. used it to study special Lagrangian submanifolds in $\mathbb{C}^n$ and obtained deep uniqueness results.

  • This is a great answer to a lot of questions i had. Thanks for that! I doesn't provide other geometric problems where a Lagrangian is involved though. – Jan-David Salchow Sep 28 '14 at 3:46

One answer to your question "why one would be interested in Lagrangian submanifolds" is to quote Weinstein's Symplectic Creed (from a 1981 article), which says everything is Lagrangian --- i.e. (1) most constructions in symplectic geometry have an interpretation involving a Lagrangian, and (2) this is a profitable point of view. For instance, if $M$ is symplectic and $M//G$ is its quotient by the Hamiltonian action of $G$ with moment map $\mu$, there is a Lagrangian $\Lambda$ in $M^- \times M//G$ that remembers $\mu^{-1}\{0\}$ and the quotient map $\mu^{-1}\{0\} \to M//G$. This answer is explained in more detail by Stefan Waldmann in the question you link to.

Another answer, which is maybe more relevant given that you write about moving a construction from one place to another, is that Lagrangian correspondences from $M_0$ to $M_1$, i.e. $L_{01} \subset M_0^- \times M_1 = (M_0\times M_1,(-\omega_M) \oplus \omega_N)$ Lagrangian, is the right notion of "a map from $M_0$ to $M_1$". A more obvious notion of map would be smooth maps $\varphi: M_0 \to M_1$ with $\varphi^*\omega_1 = \omega_0$, but all such $\varphi$ are embeddings of symplectic submanifolds, so there really aren't enough for this to be a good notion of morphism.

Alan Weinstein kicked off this point of view (I think the first reference is the same article from above?). He called it the "symplectic `category'". The reason "category" is in quotes is that there's a notion of composing Lagrangian correspondences, i.e. $$L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{M_1} L_{12}),$$ but the resulting thing may not be a(n immersed) Lagrangian if the intersection defining the fiber product is not transverse.

This point of view fits well with the Fukaya category, as Katrin Wehrheim and Chris Woodward (and Sikimeti Ma'u on one of them) showed in a series of five papers, one of them being this paper.

So, my answer to your question "when do you go hunting for Lagrangian submanifolds?" is: whenever you care about the morphisms between two symplectic manifolds.

EDIT: Since you want a concrete situation, here is an example of a result whose statement doesn't involve Lagrangians but whose proof involves using a Lagrangian correspondence to move from one place to another.

In this amazing paper of Ivan Smith's, Smith proves that the natural representation $$\tilde{\Gamma}_g^{\text{hyp}} \to \pi_0\text{Symp}(Q_0 \cap Q_1)$$ is faithful. Here $Q_0 := Z(\sum \lambda_jz_j^2)$ and $Q_1 := Z(\sum \lambda_jz_j^2)$ are quadric hypersurfaces in $\mathbb{P}^{2g+1}$, and varying the $\lambda_j$'s in $\mathbb{C}$ and parallel-transporting induces an action on $Q_0 \cap Q_1$ of the hyperelliptic mapping class group $\Gamma^{\text{hyp}}_{g,1}$ of once-pointed genus-$g$ curves. $\tilde{\Gamma}_g^{\text{hyp}}$ is a certain non-split extension of $(\mathbb{Z}/2\mathbb{Z})^{2g}$ and $\Gamma_g^{\text{hyp}}$.

This is a corollary of the main theorem, which is an identification of Fukaya categories $$D^\pi\mathcal{F}(\Sigma_g) \simeq D^\pi\mathcal{F}(Q_0^{2g} \cap Q_1^{2g}; 0),$$ where the RHS side is the full subcategory of $L$'s on which quantum multiplication by $c_1(TM)$ acts by zero. The way Smith gets this identification is by embedding both categories into the Fukaya category of the total space $Z := \text{Bl}_{Q_0\cap Q_1}(\mathbb{P}^{2g+1})$ of the pencil generated by $Q_0$ and $Q_1$: $$D^\pi\mathcal{F}(\Sigma_g) \hookrightarrow D^\pi\mathcal{F}(Z) \hookleftarrow D^\pi\mathcal{F}(Q_0^{2g} \cap Q_1^{2g}; 0).$$ Then he identifies the images.

The way he gets the second embedding is by considering a (formal summand of a) Lagrangian $\Lambda$ in $(Q_0\cap Q_1)^- \times Z$, then associating to it an $A_\infty$ functor between the Fukaya categories via Mau--Wehrheim--Woodward's machinery (see YHBKJ's comment on this question). It's easy to describe $\Lambda$: it's the composition of a correspondence $\Lambda_1 \subset (Q_0\cap Q_1)^- \times E$ with $\Lambda_2 \subset E^- \times Z$, where $E$ is the exceptional fiber in $Z$. $E$ is isomorphic to $(Q_0 \cap Q_1) \times \mathbb{P}^1$, so define $\Lambda_1 := \Delta_{Q_0 \cap Q_1} \times S^1_{\text{eq}}$ (so the functor associated to $\Lambda_1$ just multiplies a Lagrangian in $Q_0 \cap Q_1$ by the equator). To get $\Lambda_2$, note that $E$ is the symplectic quotient of $Z$ by $U(1)$ (as long as we're careful about symplectic forms); set $\Lambda_2$ to be the correspondence I mentioned at the very top of my answer.

Sorry, I'm sure there are much less complicated answers to your question. But this is the first example that came to mind of the "symplectic category" approach answering a question that's not phrased in terms of Lagrangians.

One concrete example alluded to in the above answer by YHBKJ is the theorem of M. Abouzaid in http://annals.math.princeton.edu/articles/4046: for some exotic spheres $\Sigma^{4k+1}$, the cotangent bundle is not symplectomorphic to $T^*S^{4k+1}$ although it is diffeomorphic. The reason is that $T^*\Sigma$ contains no Lagrangian sphere.

In algebraic geometry one can ask which projective varieties can occur as hyperplane sections. More precisely, given a pair of smooth varieties $X$ and $\Sigma$, does there exist a projective embedding $X \subset \mathbb{CP}^N$ and a hyperplane $H$ in $\mathbb{CP}^N$ transverse to $X$ such that $\Sigma=X \cap H$? Paul Biran approached this question from the point of view of symplectic topology and proved, among other results, the following theorem: Let $X \subset \mathbb{CP}^N$ be a projectively embedded smooth variety and let $\Sigma = X \pitchfork H$ be a hyperplane section. Then at least one of the following holds: (1) $X$ has a small dual; (2) $\Sigma$ has a Lagrangian sphere when considered as a symplectic manifold with the symplectic structure induced from $\mathbb{CP}^N$. The proof and a survey of related results can be found here: MR2185784 (2006g:53137) Biran, Paul: Symplectic topology and algebraic families. (English summary) European Congress of Mathematics, 827–836, Eur. Math. Soc., Zürich, 2005

Briefly: want a hyperplane section? Go hunt for a Lagrangian sphere first.

One goes hunting for Lagrangian submanifolds if one is interested in defining gauge theoretic 3-manifold invariants by cutting along a Heegaard surface.

Lagrangian submanifolds arise in gauge theory in the following important way (related to Nate's example from Ivan Smith's paper). Consider a 3-manifold $M$ with an $SU(2)$-bundle on it and the moduli space of flat connections on it; this has expected dimension zero so one should be able to count flat connections. One can do so in a Floer theoretic way and the resulting invariant of 3-manifolds is called instanton Floer homology.

Now we try to compute the instanton Floer homology. Pick a Heegaard splitting of $M$ into two handlebodies $H_1$ and $H_2$ and let $S$ be the Heegaard surface (common boundary of the handlebodies). Flat connections on $S$ form a symplectic manifold (well, really it's quite singular) and the subset of those which extend over $H_1$ (respectively $H_2$) is a Lagrangian submanifold of this moduli space. A flat connection on $M$ extends over both handlebodies, so corresponds to an intersection point of these two Lagrangians. The number of flat connections on $M$ (really the rank of the instanton Floer homology) should then be intersection (Floer homology) of these Lagrangians. This conjecture is the Atiyah-Floer conjecture; singularities of the moduli space of flat connections mean that it's not even well-formulated, though there is a (well-defined, proved) version for 3-manifolds fibring over the circle (with a certain nontriviality condition on the bundle) due to Dostoglou and Salamon.

Mimicking the Atiyah-Floer conjecture in a Seiberg-Witten setting, Ozsvath and Szabo were led to the definition of Heegaard-Floer homology (one of the dominant themes in current low-dimensional topology) - this is a 3-manifold invariant defined as the Lagrangian intersection Floer cohomology of a pair of Lagrangian tori in a symmetric product of a Heegaard surface.

So gauge theory is a really important motivation for studying Lagrangian submanifolds (and their intersection properties). In particular, Fukaya's introduction of an $A_{\infty}$-category of Lagrangians was motivated by these considerations (and an earlier, homological, version due to Donaldson) and the quest to put all these theories (Donaldson theory/instanton Floer) into an extended topological quantum field theory framework (I believe).

Another place where Lagrangian submanifolds arise naturally is in cotangent bundles. If $M\subset N$ is a submanifold then you can define the conormal bundle of $M$ to be the set of covectors which annihilate $TM$. This is Lagrangian (in fact an exact Lagrangian). You also get a Legendrian submanifold when you intersect this with the unit cosphere bundle (Legendrians are the contact-geometric cousins of Lagrangian submanifolds). Legendrians and Lagrangians are very intimately related and you might as well go hunting for both.

Now contact geometric invariants of this Legendrian (or symplectic invariants of the Lagrangian) conormal bundle are smooth invariants of the embedded submanifold $M$. In particular, Ng's knot contact homology (when $M\subset N$ is a knot in a 3-manifold) gives extremely interesting knot invariants related (see recent work of Aganagic-Ekholm-Ng-Vafa based on ideas of Witten) to Chern-Simons theory.

Just as in Jean-Claude Sikorav's answer (though in a completely orthogonal direction) this indicates that (exact) Lagrangian submanifolds of cotangent bundles remember much about smooth topology in the base.

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