Fukaya categories 1: What is Floer theory?

[2019-10-11 Fri]

If you have never come across the amazing ideas of Andreas Floer, the definition of a Fukaya category will seem almost completely unmotivated. In the words of the famous joke, "If I were going there, I wouldn't start from here." So for those of you who haven't seen Floer theory before, I'm going to start by reviewing the basics. To avoid getting hung up on technicalities, the review will be at a cartoon level of rigour; I will become more rigorous later when I specialise to the setting I want to use (exact Lagrangians in cotangent bundles).

Hamiltonian systems

Suppose we have a particle of mass \(m\) moving in a 1-dimensional space. Let's write \(q(t)\) for its position at time \(t\) (so velocity is \(\dot{q}(t)\)) and \(p(t)=m\dot{q}(t)\) for its momentum. Suppose the particle is acted on by a conservative force \(F=-d\phi/dq\) (for some potential function \(\phi\), which might be time-dependent); for example, if the particle is on a spring with stiffness \(k\) then \(\phi(q)=\frac{1}{2}kq^2\) (giving Hooke's law \(F=-kq\)). Newton's second law tells us that \(\dot{p}(t)=F(q(t))=-d\phi/dq\). The equations of motion are therefore \[\dot{p}=-d\phi/dq,\qquad\dot{q}=p/m.\] Hamilton observed that this can be recast in terms of the total energy (Hamiltonian) \[H=\frac{1}{2}m\dot{q}^2+\phi(q)=\frac{p^2}{2m}+\phi(q)\] of the system, considered as a function of \(p\) and \(q\): \[\dot{p}=-\frac{\partial H}{\partial q},\qquad\dot{q}=\frac{\partial H}{\partial p}.\] Given an initial condition \((p(0),q(0))\), you can solve these equations for some time (it's just a system of ODEs). Rather than specifying the initial condition, we're going to impose boundary conditions and look for solutions. For example, one simple choice is to ask for \((p(t),q(t))\) to be periodic in \(t\) (with some fixed period, say \(1\)). So we can ask: given a Hamiltonian \(H\), how many 1-periodic orbits are there? Historically, a lot of work in dynamical systems has focused on existence of periodic or almost periodic orbits (e.g. the Poincaré-Birkhoff theorem, Poincaré recurrence, KAM theory) and it's one of the simplest things you can ask about a dynamical system (not necessarily simple to answer, though).

Hamilton observed that his equations had a variational formulation. In other words, let \(\Omega\) be the set of all 1-periodic loops \(\gamma(t)=(p(t),q(t))\) in \(\mathbf{R}^2\) and define the action functional \(A\colon\Omega\to\mathbf{R}\) by \[A(\gamma)=\oint(p(t)\dot{q}(t)-H(p(t),q(t)))dt.\] The integrand here is the quantity usually called the Lagrangian of the system.

The path \(\gamma\) is a critical point of \(A\) if and only if \(\gamma\) solves Hamilton's equations.

A tangent vector to \(\Omega\) at \(\gamma\) is a vector field \((v(t),w(t))\) defined along \(\gamma\) (so \((v(t+1),w(t+1))=(v(t),w(t))\)). Consider the action of the deformed loop \(\gamma_\epsilon(t)=(p(t)+\epsilon v(t),q(t)+\epsilon w(t))\): \[A(\gamma_\epsilon)=A(\gamma)+\epsilon\oint\left(v\dot{q}+p\dot{w}-\frac{\partial H}{\partial p}v-\frac{\partial H}{\partial q}\right)dt+O(\epsilon^2)\] The order \(\epsilon\) part of this expression is called the Gateaux derivative \(d_\gamma A(v,w)\). Because of the periodic boundary conditions, we can integrate by parts and we get \[d_\gamma A(v,w)=\oint\left(v\left(\dot{q}-\frac{\partial H}{\partial p}\right)-w\left(\dot{p}+\frac{\partial H}{\partial q}\right)\right)dt.\] The loop \(\gamma\) is a critical point of \(A\) if and only if the Gateaux derivative vanishes for all \(v,w\), which then implies Hamilton's equations.

More generally...

The same thing works for a particle in \(\mathbf{R}^n\): we have momenta \(p_1,\ldots,p_n\) and coordinates \(q_1,\ldots,q_n\) and the action is \(\oint(\sum_i p_i\dot{q}_i-H)dt\). More generally, if our particle lives on a manifold \(Q\) with local coordinates \(q_i\), we can work in the cotangent bundle \(T^*Q\) with local coordinates \(p_i,q_i\). The cotangent bundle has a "canonical 1-form" \(\lambda=\sum p_idq_i\), and the action of a loop in the cotangent bundle is really \(\oint(\gamma^*\lambda-H(\gamma(t))dt)\).

Even more generally, we can work on an exact symplectic manifold with symplectic form \(\omega=d\lambda\) for some 1-form and use the same formula. Even more generally, we can work on any symplectic manifold, but the action functional is no longer well-defined there (instead, the 1-form \(dA(v)=\oint(\omega(\dot{\gamma}(t),\xi(t))-dH(\xi(t)))dt\) is well-defined, where \(\xi\) is a vector field along \(\gamma\); this is enough for talking about critical points of \(A\)).

Morse theory

Can we use this variational formulation to get lower bounds on the number of periodic orbits? In variational calculus/Morse theory, there's a general idea that if you have a function \(F\colon M\to\mathbf{R}\) then the critical points of \(F\) are related to the topology of \(M\). Is there something about the topology of \(\Omega\) that will let us deduce the existence of critical points of the symplectic action functional?

For example, if \(M\) is a finite-dimensional compact manifold and all the critical points of \(F\) are nondegenerate, then you get a cell decomposition of \(M\): each critical point \(p\) of index \(i\) defines an \(i\)-dimensional downward manifold (consisting of points which flow up to \(p\) under the gradient flow of \(F\)). The cellular chain complex computes the ordinary cohomology of \(M\), so there have to be at least \(b_i(M)\) critical points of index \(i\), where \(b_i(M)\) is the \(i\)th Betti number of \(M\).

Now, \(\Omega\) is not finite-dimensional, but sometimes Morse theory works for infinite-dimensional spaces too. In Milnor's book on the subject, for example, he uses Morse theory of the length functional on loop space (whose critical points are geodesics) to compute the topology of the loop spaces of various things (Lie groups and homogeneous spaces).

Consider the space \(\Omega_{x,y}S^n\) of paths in \(S^n\) from \(x\) to \(y\) (two generic points). The critical points of the length functional on \(\Omega_{x,y}S^n\) are the geodesics from \(x\) to \(y\). There is a unique shortest geodesic (index zero), a unique next-shortest geodesic (index \(n-1\)), a unique next-next-shortest (index \(2(n-1)\)) etc. We deduce that \(H^*(\Omega_{x,y}S^n;\mathbf{Z})=\mathbf{Z}[t]\) where \(|t|=n-1\). Note that this is the same as the topology of the based loop space.

One important point here is that every critical point has finite index (recall that the index of a critical point \(p\) is the dimension of the space of downward directions, i.e. the dimension of the negative eigenspace of \(Hess(F)\) at \(p\), i.e. the number of ways you can make your geodesic shorter).

Another important point, if you try to actually fill in the details, is that the length functional satisfies the Palais-Smale condition (a sequence \(x_n\) of points with \(F(x_n)\) bounded and \(dF(x_n)\to 0\) converges to a critical point).

Sadly, neither of these facts is true for the symplectic action functional \(A\). Indeed, it's not even clear that the gradient flow of the symplectic action functional is well-defined.

Gradient flow

To define the gradient flow of \(A\) on \(\Omega\), we need to pick a metric \(G\) on \(\Omega\). Recall that \(T_\gamma\Omega\) consists of vector fields \(\eta\) along \(\gamma\). This can be achieved by picking a metric on \(\mathbf{R}^2\) (say the standard Euclidean metric) and defining \[G(\eta_1,\eta_2)=\oint \eta_1(t)\cdot\eta_2(t)dt.\] The Gateaux derivative \(d_\gamma A\) lives in the dual space to \(T_\gamma\Omega\), and our metric gives us an injection \[\flat\colon T_\gamma\Omega\to T^*_\gamma\Omega,\qquad\eta\mapsto G(\eta,\cdot).\] The gradient \(Grad(A)\), if it exists, is then the unique vector field on \(\Omega\) satisfying \(\flat Grad(A)=d_\gamma A\). We have \[d_\gamma A(v,w)=\oint\left(v\left(\dot{q}-\frac{\partial H}{\partial p}\right)-w\left(\dot{p}+\frac{\partial H}{\partial q}\right)\right)dt\] so we can take \[Grad(A)=\left(\dot{q}-\frac{\partial H}{\partial p},-\dot{p}-\frac{\partial H}{\partial q}\right).\] If we let \(J(a,b)=(b,-a)\) then we get \[(\nabla A)(t)=J\dot{\gamma}+(\nabla H)(t)\]

A gradient flowline of \(A\) is therefore a path of loops \(\gamma_s(t)\) (tracing out a cylinder \(u(s,t)=\gamma_s(t)\)) satisfying \[\frac{\partial u}{\partial s}=J\frac{\partial u}{\partial t}+\nabla H.\] This is called Floer's equation. Note that if \(H\equiv 0\) then it reduces to the condition that the cylinder is \(J\)-holomorphic (i.e. its tangent spaces are preserved by \(J\)).

Note that if \(\gamma_0\) is a random loop, there's no reason to expect that you can solve Floer's equation with \(\gamma_0\) as an initial condition (in particular, I think \(\gamma_0\) has to be an analytic loop when \(H\equiv 0\)?).

Floer's insight

In Morse theory, you can use the cell structure described earlier to write down the cellular chain complex of the ambient manifold. The chain groups in degree \(i\) are the free abelian groups on the critical points of index \(i\); the cellular differential is the map \(\partial\colon C_i\to C_{i-1}\) defined by \[\partial x=\sum_{y\in crit_{i-1}} n(x,y)y\] where \(n(x,y)\) is the number of gradient flowlines which asymptote to \(x\) upwards and \(y\) downwards.

This definition depends only on the set of flowlines between two critical points of index difference 1 (rather than knowing the full gradient flow everywhere).

You can mimic this for the symplectic action functional: rather than trying to do gradient flow on the whole of \(\Omega\), we can fix two periodic orbits \(\gamma_\pm\) and look at the set of Floer cylinders \(u(s,t)\) such that \(\lim_{s\to\pm\infty} u(s,t)=\gamma_{\pm}(t)\). Let's fix a homotopy class \(\beta\) of such cylinders and write \(\mathcal{M}(\gamma_+,\gamma_-;\beta)\) for the set of Floer cylinders with these asymptotes and this homotopy class. Note that I don't care about the parametrisation of the cylinder in this definition (I'm identifying cylinders with the same image). The space \(\mathcal{M}(\gamma_+,\gamma_-;\beta)\) is called a moduli space of Floer cylinders. We now appeal to a theorem of the following form (as stated, this is not a theorem because it doesn't say what the assumptions are):

(TRANSVERSALITY) Under suitable assumptions, for generic choice of \(J\), \(\mathcal{M}(\gamma_+,\gamma_-;\beta)\) is a finite-dimensional manifold (its dimension is given by an index formula where the terms depend only on \(\gamma_\pm\) and the homotopy class \(\beta\)). Moreover, the set of homotopy classes for which this space is nonempty is finite.

To define the analogue of \(n(x,y)\), we need to be able to count the points in \(\mathcal{M}(\gamma_+,\gamma_-;\beta)\), so we only look at those \(\gamma_\pm,\beta\) for which the moduli space has dimension zero (this is the analogue in Morse theory of only looking at critical points with index difference 1). I haven't told you how the grading/index works in Floer theory, and I'm not going to, so let's elide that point for now.

Using this, we can define a graded \(\mathbf{Z}/2\)-module \(CF^*(H,J)\) (freely generated by 1-periodic orbits of \(H\)) and a map \(\partial\colon CF^i(H,J)\to CF^{i+1}(H,J)\), given by \[\partial \gamma_-=\sum_{\gamma_+}n(\gamma_+,\gamma_-)\gamma_+,\] where \(n(\gamma_+,\gamma_-)\) is the count mod 2 of elements in \(\mathcal{M}(\gamma_+,\gamma_-;\beta)\) summed over those \(\beta\) which give a zero-dimensional moduli space. Of course you can do homology, but I prefer cohomology.

Under suitable assumptions, \(\partial\circ\partial = 0\) so that \((CF^*(H,J),\partial)\) is a chain complex, and the Floer cohomology group \(HF^*(H,J)\) is actually independent of \(J\) and \(H\) (just as cellular homology is independent of the choice of cell structure).

Independence of \(H\) is a bit subtle if we're working on a noncompact manifold like \(\mathbf{R}^2\) (we have to specify the asymptotic behaviour of \(H\) near infinity, e.g. it should be asymptotically linear/quadratic/cubic/etc in the radius, or something like that). Quadratic at infinity is nice for examples because the usual kinetic energy is quadratic in \(p\) and the simple harmonic oscillator potential is quadratic in \(q\). Later, when we talk about wrapped Fukaya categories, we will basically be working with Hamiltonians that are quadratic at infinity.

All I will do here is to sketch the proof that \(\partial^2=0\). Note that \[\partial^2\gamma_-=\sum_{\gamma_+}\sum_{\delta}n(\gamma_+,\delta)n(\delta,\gamma_-)\gamma_+.\] In other words, \(\partial^2\gamma_-\) counts "broken cylinders": pairs of cylinders \(u_1,u_2\) such that the negative asymptote of \(u_1\) is \(\gamma_-\), the positive asymptote of \(u_2\) is \(\gamma_+\), and the other ends of the cylinders are both asymptotic to some intermediate orbit \(\delta\). If \(ind(\gamma)\) denotes the index of \(\gamma\) then we are interested in the case \(ind(\gamma_+)=ind(\gamma_-)+2\).

If you look at the moduli space of orbits connecting \(\gamma_+\) to \(\gamma_-\) with index difference 2 then it turns out to be a 1-dimensional space. That is, Floer cylinders connecting \(\gamma_\pm\) come in 1-parameter families. These families need not be compact: a cylinder from \(\gamma_+\) to \(\gamma_-\) could "break" into two cylinders at some intermediate orbit \(\delta\), yielding the kinds of broken cylinders that are counted by \(\partial^2\gamma_-\). In fact:

in good situations, this is the only way in which compactness of these 1-parameter families can fail (COMPACTNESS), and
any broken cylinder does occur as the "endpoint" of a 1-parameter family (GLUING).

This implies that broken cylinders occur in pairs (for each broken cylinder, glue to get a 1-parameter family of Floer cylinders; at the other end of the 1-parameter family there's another broken cylinder). This implies that the coefficient of \(\gamma_+\) in \(\partial^2\gamma_-\) is even, so if you work over \(\mathbf{Z}/2\), you get \(\partial^2=0\).

To get the same result over \(\mathbf{Z}\), you have to think about how the counts \(n(\gamma_+,\gamma_-)\) are to be made into signed counts (by orienting the moduli spaces in a coherent way). This is a bit of a faff (and is not always possible).

Now you get lower bounds on the number of 1-periodic orbits for an arbitrary \(H\) by computing \(HF(H,J)\) for your favourite choice of \(H,J\) (where you know the periodic orbits and the Floer cylinders) because the rank of \(HF(H,J)\) is a lower bound on the number of 1-periodic orbits for any \(H\).

As I mentioned before, this is something of a fantasy situation: I've been talking about noncompact manifolds like \(\mathbf{R}^2\) the whole time for simplicity, but then we need conditions on \(H\) near infinity to ensure invariance. To ground ourselves, I will state an actual theorem to illustrate what we've been talking about.

(Floer) Let \(H_t\colon X\to\mathbf{R}\) be a time-dependent Hamiltonian on a symplectic manifold \((X,\omega)\) where \(\int_C\omega=0\) for any class \(C\in\pi_2(X)\) (for example, \(X=T^{2n}\)). The number of 1-periodic orbits is at least \(\sum b_i(X)\).

The condition on \(\pi_2(X)\) rules out problems that can occur with the COMPACTNESS theorem that I stated if you're on a non-exact manifold (a strip can "bubble off a holomorphic sphere"). This result was conjectured (for arbitrary \(X\)) by Arnold. There are many proofs of the completely general Arnold conjecture in the literature, which have varying levels of acceptance by the community. I suspect that nowadays, for each symplectic geometer who cares about the conjecture in full generality, there is a proof (either in the literature or in preparation) which satisfies that individual. The difficulty of the general conjecture is mostly to do with the fact that, in the general compactness theorem, you have to allow bubbling as well as cylinder-breaking, and that to handle this, you need slightly different (much harder) transversality and gluing theorems.

Lagrangian boundary conditions

In all of this, we settled on some boundary conditions (1-periodicity) to get a finite number of solutions to Hamilton's equations. What other kind of boundary conditions could we try to impose? One of my favourites is "start at \(q(0)=q_0\) and finish at \(q(1)=q_1\)", leaving \(p(0)\) and \(p(1)\) undetermined). In terms of the space \(\mathbf{R}^2\), this means we are looking for time-1 trajectories which connect the line \(q=q_0\) to the line \(q=q_1\). These lines are the cotangent fibres at \(q_0\) and \(q_1\) (thinking of \(\mathbf{R}^2\) as the cotangent bundle of \(\mathbf{R}\)). These boundary conditions still let us do the crucial integration by parts in the symplectic action functional: we allow variations \((v(t),w(t))\) of \((p(t),q(t))\) with \(w(0)=w(1)=0\), so \[\int p\dot{w}dt+\int\dot{p}wdt=\int\frac{d}{dt}(pw)dt=\left[pw\right]_0^1=0.\]

Another way to think about this is the following. Take the line \(L_0:=\{q=q_0\}\) and flow it along the flow of the Hamiltonian vector field \((-\partial H/\partial q,\partial H/\partial p)\) to get a new (wiggly) line \(L'_0\). The Hamiltonian trajectories starting on \(L_0\) and ending on \(L_1\) correspond 1-to-1 with the intersection points \(L'_0\cap L_1\). Two generic 1-dimensional submanifolds intersect at a discrete set of points, so we expect to get a discrete set of Hamiltonian trajectories this way.

If we work in higher dimensions (with coordinates \((p_1,\ldots,p_n,q_1,\ldots,q_n)\)) then the same trick works, using boundary conditions fixing \(\mathbf{q}(0)\) and \(\mathbf{q}(1)\) (i.e. looking for Hamiltonian trajectories that connect these two cotangent fibres). More generally, we could try to impose an \(n\)-dimensional boundary condition \(L_0\) at time \(0\) and an \(n\)-dimensional boundary condition \(L_1\) at time \(1\) and we'd expect to get a discrete set of Hamiltonian trajectories (because two generic \(n\)-dimensional submanifolds in \(\mathbf{R}^{2n}\) intersect at a discrete set of points).

To make integration by parts work, we need \(\sum_ip_idq_i\) to vanish along \(L_0\) and along \(L_1\). This works when:

\(p\equiv 0\) (i.e. \(L\) is the zero-section),
\(dq\equiv 0\) (i.e. \(L\) is a cotangent fibre),
some of the \(p_i\)s vanish and some of the \(q_i\)s are fixed (i.e. \(L\) is the conormal bundle of a subspace in \(\mathbf{R}^n\).

These three possibilities are the "canonical examples" of Lagrangian submanifolds: \(n\)-dimensional submanifolds on which the 2-form \(\omega=\sum dp_i\wedge dq_i\) vanishes. Indeed, \(\lambda:=\sum p_i dq_i\) is a primitive for \(\omega\), so if \(\lambda\) vanishes on \(L\) then \(\omega\) vanishes on \(L\).

For completely general Lagrangian boundary conditions, integration by parts won't work the way we did it, but you can tweak the symplectic action by a total differential to make it work. In other words, \(\lambda\) is not the only primitive for \(\omega\): \(\lambda+df\) is also a primitive, for any function \(f(p,q)\).

So instead of looking for 1-periodic orbits, we can look for time-1 Hamiltonian trajectories connecting pairs of Lagrangian submanifolds. We can try and set up a Floer cohomology theory, which counts Hamiltonian trajectories from \(L_0\) to \(L_1\) (or intersection points \(L'_0\cap L_1\) for some Hamiltonian pushoff \(L'_0\) of \(L_0\)) and whose differential involves counts of Floer strips with boundary on \(L_0,L_1\) (or holomorphic strips if we look at the intersection picture).

Now we have arrived at the starting point for talking about Fukaya categories. In the next talk, I will be more precise about exactly what assumptions we want to impose.