# Fukaya categories 1: What is Floer theory?

## 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.

## 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).

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):

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.

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).

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.

## 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 \(\mathbf{R}^n\).

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.