2.1 J-holomorphic polygons

Let $D \subset \mathbf {C}$ be the closed unit disk in the complex plane. We denote by $D^{\circ }$ the open unit disk and $\partial D = D - D^{\circ }$ the boundary of D. Let $\mathbf {z} = (z_0,\ldots ,z_n)$ denote an ordered $(n+1)$ -tuple of points in $\partial D$ . We require that the points of $\mathbf {z}$ are pairwise distinct and that the counterclockwise arc subtending $z_{i-1}$ and $z_i$ (or $z_n$ and $z_0$ ) does not contain any other point of $\mathbf {z}$ .

Let $L_0,L_1,\ldots ,L_n$ be an $(n+1)$ -tuple of 1-dimensional submanifolds of T and let $(x_0,\ldots ,x_n)$ be a tuple of points in T with

$$ \begin{align*} x_0 \in L_0 \cap L_n, \qquad x_1 \in L_1 \cap L_0, \qquad \ldots,\qquad x_n \in L_n \cap L_{n-1}. \end{align*} $$

We write $\mathcal {W}(x_0,\ldots ,x_n)$ for the set of pairs $(\mathbf {z},u)$ where $u:D \to T$ is a continuous map, smooth away from $\mathbf {z}$ , that carries $z_i$ to $x_i$ and that maps the counterclockwise arc between $z_i$ and $z_{i+1}$ into $L_i$ . It carries a topology in which a sequence $(\mathbf {z}_i,u_i)$ converges $(\mathbf {z},u)$ if $\mathbf {z}_i \to \mathbf {z}$ in $(\partial D)^{\times (n+1)}$ and $u_i \to u$ in a suitable Sobolev space.

Fix an almost complex structure J on T. A polygon $(\mathbf {z},u) \in \mathcal {W}(x_0,\ldots ,x_n)$ is called J-holomorphic if the differential of u is $\mathbf {C}$ -linear on each tangent space of the interior $D^{\circ }$ . Write $\widetilde {\mathcal {M}}(x_0,\ldots ,x_n)\subset \mathcal {W}(x_0,\ldots ,x_n)$ for the subspace of J-holomorphic polygons. The group $\mathrm {PSL}_2(\mathbf {R})$ of biholomorphisms of $D^{\circ }$ acts on $\widetilde {\mathcal {M}}$ by reparametrisation and we denote the quotient by $\mathcal {M}(x_0,\ldots ,x_n)$ .

2.2 Transversely cut criteria

Each connected component of $\mathcal {M}(x_0,\ldots ,x_n)$ is labelled by a nonnegative integer called the analytic index of the component (or of any map in the component). In case the conditions that cut $\widetilde {\mathcal {M}}$ out of $\mathcal {W}$ are transverse in a sense that we will not review here, each component is a topological manifold and the analytic index coincides with the dimension of this component. A formula for this dimension is given below (2.3.1). When the ‘transversely cut’ condition is satisfied, we call the components of dimension 0 rigid polygons.

The ‘transversely cut’ condition is satisfied whenever the $(L_0,\ldots ,L_n)$ are in general position – that is, whenever the $L_i$ are pairwise transverse and $L_i \cap L_j \cap L_k$ is empty. On T or another surface, this triple intersection condition can be relaxed [Reference SeidelSe2, Lem. 13.2]; for instance, $\mathcal {M}$ is transversely cut in a neighbourhood of u as soon as u is not constant. Even some constant maps u are transversely cut; for instance, if $(L_0,L_1,L_2)$ are pairwise transverse, then at any triple intersection point $x \in L_0 \cap L_1 \cap L_2$ , the tangent lines $(T_x L_0,T_x L_1,T_x L_2)$ come in either clockwise or counterclockwise order. A constant map $D \to L_0 \cap L_1 \cap L_2$ contributes to $\mathcal {M}(x,x,x)$ if and only if they come in clockwise order:

An example of a nontransversely cut quadrilateral in T (necessarily constant) is analyzed in [Reference Lekili and PerutzLPe1, Thm. 8]. We will encounter some nontransversely cut triangles in Subsection 4.9.

2.3 Maslov index of an intersection point

Let $(L,L')$ be an ordered pair of connected 1-dimensional submanifolds of T and fix an orientation of both L and $L'$ . Suppose that L and $L'$ meet transversely at the point x; then we define $\mathrm {mas}(x) \in \mathbf {Z}/2$ by the rule indicated in the diagram:

If L and $L'$ are not homologous to zero, one may lift the Maslov index to a $\mathbf {Z}$ -valued invariant by equipping L and $L'$ (and T) with gradings; see [Reference Lekili and PerutzLPe2, §6]. Let us denote this $\mathbf {Z}$ -valued Maslov index by $\mathrm {mas}_{\mathbf {Z}}(x)$ . A formula for the dimension near $u \in \mathcal {M}(x_0,\ldots ,x_n)$ is

(2.3.1) $$ \begin{align} \mathrm{mas}_{\mathbf{Z}}(x_0) - \mathrm{mas}_{\mathbf{Z}}(x_1) - \cdots - \mathrm{mas}_{\mathbf{Z}}(x_n). \end{align} $$

2.4 Sign of a rigid polygon

By making some additional choices, one may attach a sign to each rigid polygon with boundary on $(L_0,\ldots ,L_n)$ – in other words, one may define a map

(2.4.1) $$ \begin{align} \mathcal{M}(x_0,\ldots,x_n) \to \{1,-1\}. \end{align} $$

We recall the recipe for (2.4.1) given in [Reference SeidelSe1, §7] – it depends on the choice of orientation for each $L_i$ and on the additional data of a basepoint $\star _i \in L_i$ in each $L_i$ . One requires that $\star _i \notin L_j$ for any $j \neq i$ . The point $\star _i$ endows $L_i$ with a nontrivial spin structure (also known as bounding or Neveu–Schwarz spin structure) which is trivialised away from $\star _i$ .

If $u\vert _{\partial D}:\partial D \to \cup _{i = 0}^n L_i$ preserves the counterclockwise orientation of D, the sign is $+1$ or $-1$ according to whether one encounters an even or odd number of stars going around $\partial D$ – that is, it is $(-1)^{\#u^{-1}\{\star _0,\ldots ,\star _n\}}$ . Changing the orientation of $L_0$ does not change this sign, changing the orientation of $L_n$ changes the signs by $(-1)^{\mathrm {mas}(x_0) + \mathrm {mas}(x_n)}$ and changing the orientation of any of the other $L_i$ changes the sign by $(-1)^{\mathrm {mas}(x_i)}$ .

For short, we will sometimes write

(2.4.2) $$ \begin{align} (-1)^{|x|}:=(-1)^{\mathrm{mas}(x)}. \end{align} $$

2.5 Floer cochain complexes

Let t be a formal variable and let $\mathbf {Z}[t^{\mathbf {R}_{\geq 0}}]$ denote the semigroup algebra of $\mathbf {R}_{\geq 0}$ ; that is, the group of finite $\mathbf {Z}$ -linear combinations of symbols of the form $t^a$ , where $a \in \mathbf {R}_{\geq 0}$ , with the multiplication $t^a t^b = t^{a+b}$ . Let $\Lambda $ be a $\mathbf {Z}[t^{\mathbf {R}_{\geq 0}}]$ -algebra. In a moment we will take $\Lambda $ to be the Novikov completion of $\mathbf {Z}[t^{\mathbf {R}_{\geq 0}}]$ , but the differential in $\mathrm {CF}(L,L')$ is given by a finite sum, so that in this section it might as well be $\mathbf {Z}[t^{\mathbf {R}_{\geq 0}}]$ itself.

When L and $L'$ intersect transversely, then we let

(2.5.1) $$ \begin{align} \mathrm{CF}(L,L') := \bigoplus_{x \in L \cap L'} \Lambda. \end{align} $$

The choice of orientation for L and $L'$ endows this group with a $\mathbf {Z}/2$ -grading,

$$ \begin{align*} \mathrm{CF} = \mathrm{CF}^0 \oplus \mathrm{CF}^1, \qquad \text{where } \mathrm{CF}^i(L,L') := \bigoplus_{x | \mathrm{mas}(x) = i} \Lambda. \end{align*} $$

Further equipping L and $L'$ with stars (Subsection 2.4) gives us the bigon differential

(2.5.2) $$ \begin{align} \mu_1:\mathrm{CF}^i(L,L') \to \mathrm{CF}^{i+1}(L,L'): x \mapsto \sum_{y \mid \mathrm{mas}(y) = i+1} y \left(\sum_{u \in \mathcal{M}(x,y)} \pm t^{\mathrm{area}(u)}\right) \end{align} $$

where the sign is given in Subsection 2.4 and $\mathrm {area}(u):=\int _D u^* (dx\, dy)$ . The inner sum is finite. A general argument using nonrigid bigons shows that $\mu _1 \mu _1 = 0$ – this is a case of the $A_{\infty }$ -relations (Subsection 2.11). In T, this can be proved more simply by lifting the grading from $\mathbf {Z}/2$ to $\mathbf {Z}$ : the $\mathbf {Z}$ -grading is always concentrated in only two degrees.

We have just described the ‘absolute’ Floer cochain complex. After fixing a point $D \in T$ , not on L or $L'$ , we also have a ‘relative to D’ complex in which $\Lambda $ in (2.5.1) can be shrunk to $\mathbf {Z}[t]$ (or another $\mathbf {Z}[t]$ -algebra) and the expression $t^{\mathrm {area}(u)}$ in is replaced by $t^{\#u^{-1}(D)}$ .

2.6 Example – special Lagrangians

We will call a circle $L \subset T$ a ‘special Lagrangian’ if it is the image under $\mathbf {R}^2 \to T$ of a straight line. If that straight line has the form $y+mx = b$ , then we will call m the slope of the special Lagrangian; otherwise, we say L has slope $\infty $ . Thus, the possible slopes are $m \in \mathbf {Q} \cup \{\infty \}$ . If L is special with finite slope, let us call the orientation under the parametrisation $x \mapsto (x,b-mx)$ the ‘default orientation’.

If $L \neq L'$ are two special Lagrangians, of finite slopes m and $m'$ , then they meet transversely in a set of cardinality $|nd' - n'd|$ , if $m = n/d$ and $m' = n'/d'$ . All of the intersection points are in a single Maslov degree; with the default orientations, these degrees are

$$ \begin{align*} \mathrm{CF}(L,L') = \begin{cases} \mathrm{CF}^0(L,L') & \text{if}\ m'> m \\ \mathrm{CF}^1(L,L') & \text{if}\ m' < m. \end{cases} \end{align*} $$

There are no bigons and the differential (2.5.2) is zero.

2.7 Polygon maps

Suppose that $(L_0,\ldots ,L_n)$ are in sufficiently general position that all of the $\mathcal {M}(x_0,\ldots ,x_n)$ are transversely cut. The $(n+1)$ -gon map is a multilinear map

(2.7.1) $$ \begin{align} \mu_n:\mathrm{CF}^{i_n}(L_{n-1},L_n) \times \cdots \times \mathrm{CF}^{i_1}(L_0,L_1) \to \mathrm{CF}^{i_1+\cdots+i_n + 2 - n}(L_0,L_n) \end{align} $$

which carries $(x_n,x_{n-1},\ldots ,x_1)$ to

(2.7.2) $$ \begin{align} \sum_{y} y \left(\sum_{u \in \mathcal{M}(y,x_1,x_2,\ldots,x_n)} \pm t^{\mathrm{area}(u)}\right). \end{align} $$

It is not defined until the $L_i$ are equipped with orientations and stars. Furthermore, the inner sum in (2.7.2) is usually infinite, so $\Lambda $ should carry a topology in which it converges. The standard choice for $\Lambda $ is one of $\Lambda ^0$ or $\Lambda ^0[t^{-1}]$ , where $\Lambda ^0$ is the Novikov ring

(2.7.3) $$ \begin{align} \Lambda^0 = \Lambda_{\mathbf{Z}}^0 =\left\{\sum_{i = 0}^{\infty} a_i t^{\lambda_i} \mid a_i \in \mathbf{Z}, \lambda_i \in \mathbf{R}_{\geq 0} \text{ and } \lim_{i \to \infty} \lambda_i = \infty\right\}. \end{align} $$

The fact that $\Lambda ^0$ can be taken to have $\mathbf {Z}$ -coefficients is a reflection of the fact that the moduli spaces $\mathcal {M}$ are not orbifolds – this holds for T and more generally for semipositive symplectic manifolds.

In the relative setting, as long as D does not lie on any $L_i$ , we replace $t^{\mathrm {area}(u)}$ with $t^{\#u^{-1}(D)}$ and (2.7.1) is multilinear over $\mathbf {Z}[\![t]\!]$ .

2.8 Example – some triangle maps

Suppose $L_0,\ldots ,L_k$ are special Lagrangians of slopes

(2.8.1) $$ \begin{align} m_0 < m_1 < \cdots < m_k < \infty. \end{align} $$

If $k \neq 2$ , then the set of rigid $(k+1)$ -gons with boundary on $L_0,\ldots ,L_k$ is empty and the maps

$$ \begin{align*} \mu_k:\mathrm{CF}(L_{k-1},L_k) \times \cdots \times \mathrm{CF}(L_0,L_1) \to \mathrm{CF}(L_0,L_k) \end{align*} $$

are zero – this is a consequence of Subsection 2.3.1. In other words, when (2.8.1) is satisfied, $\mu _2$ is the only interesting polygon map. In this section we explain how to compute $\mu _2$ in detail for Lagrangians of the form $L_{(m_0)}, L_{(m_1)},L_{(m_2)}$ (Subsection 1.3). These are the maps that can be packed into a product structure on $\bigoplus \mathrm {CF}(L_{(0)},L_{(m)})$ (Subsection 1.4), and our notation is adapted to describing this product structure.

One consequence of the vanishing of the $\mu _k$ for $k \neq 2$ is that this product is strictly associative (Subsection 2.11), and we can describe the product in terms of the theta functions. If (2.8.1) is not satisfied, then the maps $\mu _k$ do not all vanish for $k \geq 3$ – they can be written in terms of Hecke’s indefinite theta series [Reference PolishchukPoli].

If $m_1 < m_2$ , there are $m_2 - m_1$ intersection points between $L_{(m_1)}$ and $L_{(m_2)}$ , each contributing a basis element to $\mathrm {CF}(L_{(m_1)},L_{(m_2)})$ . Let us index those intersection points in the following way:

(2.8.2) $$ \begin{align} \tau^{m_1}(x_{m_2 - m_1,\kappa}) := (\kappa,-m_2\kappa) \end{align} $$

where $\kappa \in \{0,\frac {1}{m_2 - m_1},\ldots , \frac {m_2 - m_1 - 1}{m_2 - m_1}\}$ and $\tau $ denotes the Dehn twist map

$$ \begin{align*} \tau:(x,y) \mapsto (x,y-x) \end{align*} $$

Fix an irrational number $\varepsilon $ and equip each $L_{(m)}$ with a star §2.4

(2.8.3) $$ \begin{align} \star_{(m)} = \star_{(m),\varepsilon} := (\varepsilon,-m\varepsilon). \end{align} $$

The Dehn twist carries $L_{(m)}$ to $L_{(m+1)}$ and preserves the stars. As $\varepsilon $ is irrational, the stars avoid the intersection points $L_{(m_1)} \cap L_{(m_2)}$ . We will compute the triangle maps

$$ \begin{align*} \mathrm{CF}(L_{(m_1)},L_{(m_1+m_2)}) \times \mathrm{CF}(L_{(0)},L_{((m_1))}) \to \mathrm{CF}(L_{(0)},L_{(m_1+m_2)}) \end{align*} $$

by computing $\mu _2(\tau ^{m_1} x_{m_2,\kappa _2}, x_{m_1,\kappa _1})$ . For example,

(2.8.4) $$ \begin{align} \mu_2(\tau x_{1,0},x_{1,0}) = \begin{cases} x_{2,0}\Big(\sum_{i \in \mathbf{Z}} t^{i^2}\Big) + x_{2,\frac{1}{2}}\Big(\sum_{i \in \mathbf{Z}} t^{(i+\frac{1}{2})^2}\Big) & \text{(absolute setting)} \\ x_{2,0}\Big(\sum_{i \in \mathbf{Z}} t^{i^2}\Big) + x_{2,\frac{1}{2}}\Big(\sum_{i \in \mathbf{Z}} t^{i(i+1)}\Big) & \text{(relative setting)}. \end{cases} \end{align} $$

Theorem. Let

(2.8.5) $$ \begin{align} E(\kappa_1,\kappa_2) = E_{m_1,m_2}(\kappa_1,\kappa_2) := \frac{m_1 \kappa_1 + m_2 \kappa_2}{m_1 + m_2} \end{align} $$

(which carries $\frac {1}{m_1} \mathbf {Z} \times \frac {1}{m_2} \mathbf {Z}$ into $\frac {1}{m_1+m_2} \mathbf {Z}$ ). Then in the absolute setting, $\mu _2(\tau ^{m_1} x_{m_2,\kappa _2}, x_{m_1,\kappa _1})$ is given by

(2.8.6) $$ \begin{align} \mu_2(\tau^{m_1} x_{m_2,\kappa_2}, x_{m_1,\kappa_1}) = \sum_{\ell \in \mathbf{Z}} x_{m_1 + m_2,E(\kappa_1,\kappa_2 + \ell)} t^{(\ell+\kappa_2 - \kappa_1)^2/(2(\frac{1}{m_1} + \frac{1}{m_2}))}. \end{align} $$

Let the functions $\phi (s)$ and $\lambda (u,v) = \lambda _{m_1,m_2}(u,v)$ be given by (cf. [Reference Lekili and PerutzLPe2, p. 83])

(2.8.7) $$ \begin{align} \phi(s):= \lfloor s \rfloor s - \frac{1}{2} \lfloor s \rfloor \lfloor s+1 \rfloor; \qquad \lambda(u,v) := m_1 \phi(u) + m_2 \phi(v) - (m_1 + m_2) \phi(E(u,v)). \end{align} $$

Then in the relative setting, $\mu _2(\tau ^{m_1} x_{m_2,\kappa _2}, x_{m_1,\kappa _1})$ is given by

(2.8.8) $$ \begin{align} \mu_2(\tau^{m_1} x_{m_2,.\kappa_2},x_{m_1,\kappa_1}) = \sum_{\ell \in \mathbf{Z}} x_{m_1 + m_2,E(\kappa_1,\kappa_2 + \ell)} t^{\lambda(\kappa_1,\kappa_2+\ell)}, \end{align} $$

where we understand $x_{m_1+m_2,E(\kappa _1,\kappa _2+\ell )} := x_{m_1 + m_2,\kappa }$ if $E(\kappa _1,\kappa _2+\ell ) = \kappa $ modulo $1$ .

Proof. The index $\ell $ in either sum (2.8.6), (2.8.8) determines a triangle, two of whose vertices are at $x_{m_1,\kappa _1}$ and $\tau ^{m_1} x_{m_2,\kappa _2}$ . In the universal cover, the coordinates of all three vertices are

$$ \begin{align*} (\kappa_1,0), \qquad (E(\kappa_1,\kappa_2+\ell),0), \qquad (\kappa_2+\ell,-m_1(\kappa_2 + \ell - \kappa_1)) \end{align*} $$

as in the diagram

On any of these triangles there are an even number of stars (2.8.3): for each $i \in \mathbf {Z}$ with $\kappa _1 < \varepsilon + i < \kappa _2 + \ell $ there are exactly two stars whose x-coordinate (in the universal cover) is $\varepsilon +i$ , one along $L_{(m_1)}$ and the other along either $L_{(0)}$ or $L_{(m_1+m_2)}$ . Thus, every summand in the triangle has sign $+1$ .

The exponent of t in (2.8.6) is the area of the $\ell $ th triangle; that is,

$$ \begin{align*} \frac{1}{2} m_1(\kappa_2+\ell-\kappa_1) (E(\kappa_1, \kappa_2+\ell) - \kappa_1) =(\ell+\kappa_2-\kappa_1)^2 / (2(\frac{1}{m_1}+\frac{1}{m_2})). \end{align*} $$

The more complicated exponent of t in (2.8.8) is the lattice area (Subsection 1.6) of the same triangle, which coincides with the cardinality of $u^{-1}(D)$ when D is in the first quadrant very close to $(0,0)$ – see [Reference Lekili and PerutzLPe2].

2.9 Theta functions

Let $\theta _{m,k}$ , $\theta ^{\mathrm {abs}}_{m,k}$ be as in (1.8.1):

$$ \begin{align*} \theta_{m,k} := \sum_{i = -\infty}^{\infty} t^{m \frac{i (i-1)}{2}+ k i}\, z^{mi + k}, \qquad \theta^{\mathrm{abs}}_{m,k} := \sum_{i = -\infty}^{\infty} t^{(mi+k)^2/(2m)} z^{mi+k}. \end{align*} $$

The Jacobi theta function is $\theta _{1,0}$ and the others are obtained by a simple change of variables:

$$ \begin{align*} \theta_{m,k}(t,z) = z^k \theta_{1,0}(t^m,t^k z^m), \qquad \theta_{m,k}(t,t^{\frac{1}{2}} z) = t^{{k(m-k)}/(2m)}\theta_{m,k}^{\mathrm{abs}}(t,z). \end{align*} $$

Although these series are doubly infinite in z, when formally expanding the product of $\theta _{m,k}$ and $\theta _{m',k'}$ , only finitely many terms contribute to the coefficient of any monomial $z^e t^f$ – the same goes for $\theta ^{\mathrm {abs}}_{m,k}$ and $\theta ^{\mathrm {abs}}_{m',k'}$ . This is a consequence of the convexity of the functions $i \mapsto m \binom {i}{2}+ki$ and $i \mapsto (mi+k)^2/(2m)$ in the exponent of t. That the resulting series $\theta _{m,k} \cdot \theta _{m',k'}$ or $\theta ^{\mathrm {abs}}_{m,k} \cdot \theta ^{\mathrm {abs}}_{m',k'}$ can be written as a linear combination of $\theta _{m+m',0},\cdots ,\theta _{m+m',m+m'-1}$ is a standard but nontrivial fact about the theta functions. The formulas for these coefficients are the same as (2.8.8) and (2.8.6): putting $\kappa _1 = k_1/m_1$ and $\kappa _2 = k_2/m_2$ ,

$$ \begin{align*} \theta_{m_2,k_2} \cdot \theta_{m_1,k_1} = \sum_{\ell \in \mathbf{Z}} \theta_{m_1 + m_2,E(\kappa_1,\kappa_2 + \ell)} t^{\lambda(\kappa_1,\kappa_2 + \ell)} \end{align*} $$

and

$$ \begin{align*} \theta^{\mathrm{abs}}_{m_2,k_2} \cdot \theta^{\mathrm{abs}}_{m_1,k_1} = \sum_{\ell \in \mathbf{Z}} \theta^{\mathrm{abs}}_{m_1 + m_2,E(\kappa_1,\kappa_2 + \ell)} t^{(\ell+\kappa_2 - \kappa_1)^2/(2(\frac{1}{m_1} + \frac{1}{m_2}))}. \end{align*} $$

In other words, the map $x_{m,k} \mapsto \theta _{m,k}$ or $\theta ^{\mathrm {abs}}_{m,k}$ is a ring homomorphism. One may verify this directly (and we will do so in the next section when we turn on an F-field), but it is natural to ask what is the Floer-theoretic origin of these series. Each summand of (1.8.1) is indexed by a right triangle, with one vertex at $x_{m,k/m}$ and sides along $L_{(0)},L_{(m)},L_{(\infty )}$ . The exponent of t carries the area (or lattice area) of this right triangle and the exponent of z carries the number of times the vertical edge of the triangle wraps around $L_{(\infty )}$ – this z can be interpreted as the monodromy of a rank 1 local system (Subsection 3.3). For instance, the right triangles contributing to $\theta _{2,1}$ have the form (for i positive)

2.10 The punctured torus, the large complex structure limit

Part of the motivation for relative Floer theory is to make sense of the specialisation $t = 0$ in the polygon sums. Setting $t = 0$ has the effect of discarding the polygons that contribute a positive power of t, which in the relative case is the same as discarding the polygons that touch D. One gets the same effect by doing Floer theory for Lagrangians in the punctured torus $T - D$ . When $T-D$ is equipped with the right symplectic structure, one with infinite area in a neighbourhood of D, this is called the ‘large volume limit’ of Floer theory.

We can treat the Lagrangians $L_{(m)}$ as boundary conditions for triangles $u:\Delta ^2 \to T - D$ and sum over them to obtain a map $\mu _2$ as before, this time defined on the free $\mathbf {Z}$ -modules spanned by $L_{(m_1)} \cap L_{(m_2)}$ . For instance,

$$ \begin{align*} \mu_2(\tau x_{1,0},x_{1,0}) = x_{2,0} + x_{2,1/2} \cdot 2, \end{align*} $$

where $x_{2,0}$ comes from a constant map and the two copies of $x_{2,1/2}$ come from the two shaded triangles in the figure

We also record

$$ \begin{align*} \mu_2(\tau^2 x_{1,0},x_{2,0}) & = x_{3,0} + x_{3,1/3} + x_{3,2/3} \\ \mu_2(\tau^2 x_{1,0},x_{2,1/2}) & = x_{3,1/3} + x_{3,2/3} \\ \mu_2(\tau^3 x_{3,1/3},x_{3,2/3}) & = x_{6,3/6} \\ \end{align*} $$

and

$$ \begin{align*} \mu_2(\tau^4 x_{2,1/2}, \mu_2(\tau^2 x_{2,1/2},x_{2,1/2})) = x_{6,3/6}. \end{align*} $$

If one sets $z = x_{1,0}$ , $x = x_{2,1/2}$ , $y = x_{3,2/3}$ , these equation imply in particular that

$$ \begin{align*} y^2 z + x^3 = xyz. \end{align*} $$

This is the equation for a nodal plane cubic curve – the ‘large complex structure limit’ that matches the large volume limit under mirror symmetry.

2.11 $A_{\infty }$ -relations

The moduli spaces $\mathcal {M}$ that parametrise nonrigid polygons are not usually compact. For example, suppose $L_0,L_1,L_2,L_3$ are as in the diagram

Denote the red-black, black-purple and purple-blue intersection points by f, g and h respectively and the blue-red intersection point by $hgf$ . Then $\mathcal {M}(hgf,f,g,h)$ includes the following one-parameter family of quadrilaterals:

(2.11.1)

They all have the same image closure but along the boundary may back-track along either the blue or the red line. Near the $\ast $ s, the map u is biholomorphic to the map from the upper half-plane to $\mathbf {C}$ that sends z to $z^2$ . At the extreme parameters, where the $\ast $ reaches all the way to the black or purple line, there is no such J-holomorphic quadrilateral – so $\mathcal {M}$ is not compact – but each of those extremes is occupied by a pair of J-holomorphic triangles.

Since the quadrilaterals are not rigid, they do not contribute to $\mu _3(h,g,f)$ . But the triangles at the extremes are rigid; at one end they contribute to $\mu _2(g,f)$ and $\mu _2(h,gf)$ and at the other end to $\mu _2(h,g)$ and $\mu _2(hg,f)$ . The interpolating family of quadrilaterals exhibits a relation between them.

More generally, there is a compactification (the Deligne–Mumford–Stasheff compactification) of $\mathcal {M}$ . When everything is transversely cut (Subsection 2.2), the compactification is a topological manifold-with-corners whose corners are indexed by tuples of rigid polygons. Equipping the $L_i$ with orientations and stars induces an orientation on $\mathcal {M}$ and its compactification – (2.4.1) is a special case of this orientation. The oriented compactification of $\mathcal {M}$ is used in the proof (it essentially is the proof) of the following equations among the polygon maps $\mu _n$ :

(2.11.2) $$ \begin{align} \sum_{i+j = n+1} \sum_{\ell < i} (-1)^{|f_1|+\cdots+|f_\ell|-\ell} \mu_i(f_n,,\ldots,f_{\ell + j + 1},\mu_j(f_{\ell+j},\ldots,f_{\ell+1}),f_{\ell},\ldots,f_1) = 0. \end{align} $$

The algebraic structure formed by the $\mathrm {CF}(L,L')$ and the maps $\mu _n$ , subject to the relations (2.11.2), is called an ‘ $A_{\infty }$ -precategory’ in [Reference Kontsevich and SoibelmanKS, §4.3]. Each L is like an object and each $f \in \mathrm {CF}(L,L')$ is like a morphism between objects. (2.11.2) expresses the fact that these morphism spaces are cochain complexes and that the composition law is associative up to chain homotopy in a strong sense. It falls short of being an $A_{\infty }$ -category; for instance, because $\mathrm {CF}(L,L')$ is defined only when L and $L'$ meet transversely, so $\mathrm {CF}(L,L)$ is undefined and there is no ‘morphism’ that could play the role of the identity map. A standard way to address this problem is by analysing the sense in which $\mathrm {CF}(L,L')$ are invariant under Hamiltonian isotopies – Floer’s theory of continuation.

2.12 Example – identity maps

There is a Floer cochain complex $\mathrm {CF}(L,L')$ , well-defined up to quasi-isomorphism, even if L and $L'$ do not intersect transversely. In case $L'$ meets both $L_0$ and $L_1$ transversely, in the absolute setting (respectively relative setting) any Hamiltonian isotopy (respectively any Hamiltonian isotopy supported on the complement of D) induces a quasi-isomorphism between $\mathrm {CF}(L_0,L')$ and $\mathrm {CF}(L_1,L')$ (Subsection 2.13). One accesses $\mathrm {CF}(L,L')$ by perturbing L.

We illustrate this in an example that fills in the zeroth graded piece of (1.4.1). Let $\phi ^s$ denote the flow of $H(x,y) = \sin (2 \pi x)$ ; that is,

(2.12.1) $$ \begin{align} \phi^s(x,y) = (x,y - s \cos(2\pi x)). \end{align} $$

Then for $0 < s < 1$ , $\phi ^s L_{(0)}$ meets $L_{(0)}$ transversely at the points $x = (.25,0)$ and $y = (.75,0)$ , so that $\mathrm {CF}(\phi ^s L_{(0)},L_{(0)}) = \Lambda x \oplus \Lambda y$ . In a suitable fundamental domain, the picture is this:

Equipping $L_{(0)}$ with its default orientation and $\phi ^s L_{(0)}$ with the orientation induced by $\phi ^s:L_{(0)} \cong \phi ^s L_{(0)}$ , the Maslov degrees are

(2.12.2) $$ \begin{align} \mathrm{CF}^0(\phi^s L_{(0)},L_{(0)}) = x \Lambda \qquad \mathrm{CF}^1(\phi^s L_{(0)},L_{(0)}) = y\Lambda. \end{align} $$

There are two bigons contributing to the differential $\mu _1$ , of equal area A. Both bigons have input x and output y and their signs (Subsection 2.4) are opposite to each other (no matter where the stars are placed), so that the differential is

$$ \begin{align*} \mu_1(x) = y(t^A - t^A) = 0. \end{align*} $$

Some variations of this computation are made in Subsection 2.14 and Subsection 3.5.

2.13 Continuation

Let $X_H$ denote the Hamiltonian vector field of a function $H:[0,1] \times T \to \mathbf {R}$ and write $\phi ^s$ for its time s flow, $\phi ^s:T \to T$ . Let us review how the quasi-isomorphism

(2.13.1) $$ \begin{align} \mathrm{CF}(L,L') \to \mathrm{CF}(\phi^s L, L') \end{align} $$

works in the absolute setting. The map (2.13.1) goes back to [Reference FloerFl, Thm. 4]; our notation is closer to the appendix of [Reference AurouxAur1]. It is defined in terms of a set $\mathcal {M}(x,y,\phi ,\beta )$ of maps

$$ \begin{align*} u:[-\infty,\infty] \times [0,1] \to T \end{align*} $$

that obey the boundary conditions

$$ \begin{align*} \begin{array}{rcl} u([-\infty,\infty] \times \{0\}) & \subset & L \\ \quad u([-\infty,\infty] \times \{1\}) & \subset & L' \end{array} \qquad u(\{\infty\} \times [0,1]) = \{x\} \qquad u(-\infty,\tau) = \phi^{s \cdot \tau}(y) \end{align*} $$

and (with analytic index zero; Subsection 2.2) Floer’s $X_H$ -perturbed J-holomorphic curve equation:

(2.13.2) $$ \begin{align} \partial u/\partial \sigma + J(\partial u/\partial \tau - \beta(\sigma) s X_H) = 0. \end{align} $$

Here $\beta $ (the ‘profile function’) is a monotone decreasing $\mathbf {R}$ -valued function on $[-\infty ,\infty ]$ with $\beta (\sigma ) = 1$ for $\sigma \ll 0$ and $\beta (\sigma ) = 0$ for $\sigma \gg 0$ . The formula for (2.13.1) is

(2.13.3) $$ \begin{align} x \mapsto \sum_{y \in \phi^s L \cap L'} y \sum_{u \in \mathcal{M}(x,y,\phi)} \pm t^{\text{topological energy of}\ u} \end{align} $$

where the ‘topological energy’ is (see Lemma 14.4.5 [Reference OhOh])

$$ \begin{align*} \int u^* \omega + \int_{0}^{1} H(\tau, u(\infty, \tau)) d\tau - \int_{-\infty}^{\infty} \beta'(\sigma) s \int_0^1 (H_\tau \circ u) d\tau d\sigma. \end{align*} $$

This is sometimes a negative quantity, so we must allow $t^{-1} \in \Lambda $ .

The homotopy inverse to (2.13.1) is just the continuation map for the reversed flow $\phi ^{-s}$ . To describe the chain homotopy between the composite

(2.13.4) $$ \begin{align} \mathrm{CF}(\phi^s L,L') \to \mathrm{CF}(\phi^{-s} \phi^s L,L') = \mathrm{CF}(L,L') \to \mathrm{CF}(\phi^s L,L') \end{align} $$

and the identity map, let $B_r(\sigma )$ (for each $r> 0$ ) be a function that vanishes on an interval of length r and that agrees up to translation of $\sigma $ with $\beta (\sigma )$ when $\sigma $ is to the left of that interval and with $\beta (-\sigma )$ when $\sigma $ is to the right of that interval. Then

(2.13.5) $$ \begin{align} \Delta(y) = \sum_x x \sum_u \pm t^{\text{topological energy of}\ u}, \end{align} $$

where the inner sum is over strips $u:[-\infty ,\infty ] \times [0,1] \to T$ that solve (for some r, with index $-1$ )

$$ \begin{align*} \partial u/\partial \sigma + J(\partial u/\partial \tau - B_r(\sigma) s X_H) = 0 \qquad r \in \mathbf{R}_{\geq 0} \end{align*} $$

and that have $u(-,0) \subset L$ , $u(-,1) \subset L'$ and $u(-\infty ,\tau ) = \phi ^{s\cdot \tau }(x)$ and $u(\infty ,\tau ) = \phi ^{s\cdot \tau }(y)$ .

2.14 Example

Suppose that L and $L'$ are a pair of parallel, horizontal circles, at distance c apart. With $\phi ^s$ as in (2.12.1), $\phi ^s(L) \cap L'$ is empty unless $|s|> c$ . If $|s|$ only slightly exceeds c, then $\phi ^s(L) \cap L'$ has two intersection points, say, x and y, as in the diagram:

Thus, $\mathrm {CF}(L,L') = 0$ , while (noting that there are two bigons in the right picture, a small one of area A and a large one of area $A+c$ ) and $\mathrm {CF}(\phi ^s L, L') = x \Lambda \oplus y\Lambda $ , with differential

$$ \begin{align*} \mu_1(x) = y (t^{A} - t^{A+c}) \end{align*} $$

(see Subsection 2.4 for the signs).

As $\mathrm {CF}(L,L') = 0$ , the composite (2.13.4) is zero. By Floer’s theorem, the identity map on $\mathrm {CF}(\phi ^s L,L')$ is chain homotopic to zero, with (2.13.5) supplying the contracting homotopy. Indeed, the contracting homotopy is the geometric series

$$ \begin{align*} \Delta(y) = x \cdot (t^{-A} + t^{-A + c} + \cdots + t^{-A + nc} + \cdots). \end{align*} $$

The strip $u_n$ contributing the nth term in the series (of topological energy $nc - A$ ) stretches horizontally across $n+2$ fundamental domains, crossing the boundary of the fundamental domain exactly $n+1$ times. Here is a picture of $u_0\vert _{\partial ([-\infty ,\infty ] \times [0,1])}$ and $u_1\vert _{\partial ([-\infty ,\infty ] \times [0,1])}$ :

3.1 Cochain complex

If L and $L'$ are two embedded circles in T, meeting transversely, let us write (just as in (1.11.1))

(3.1.1) $$ \begin{align} \mathrm{CF}(L,L';\underline{\Lambda}) := \bigoplus_{x \in L \cap L'} \underline{\Lambda}_x. \end{align} $$

If $a \in \underline {\Lambda }_x$ and we wish to regard it as an element of (3.1.1), we will write it as $x \cdot a$ . We equip (3.1.1) with a $\mathbf {Z}/2$ -grading by equipping L and $L'$ with orientations, just as in Subsection 2.5. After further equipping L and $L'$ with stars, we define a differential

$$ \begin{align*} \mu_1:\mathrm{CF}^i(L,L';\underline{\Lambda}) \to \mathrm{CF}^{i+1}(L,L';\underline{\Lambda}) \end{align*} $$

by the following analog of (2.5.2):

(3.1.2) $$ \begin{align} x\cdot a \mapsto \sum_{y | \mathrm{mas}(y) = i+1} y \left(\sum_{u \in \mathcal{M}(y,x)} \pm t^{\mathrm{area}(u)} \nabla \gamma'\left(a \nabla \gamma(1_{\underline{\Lambda}_y})\right)\right), \end{align} $$

where

• • $\gamma :I \to L$ is the path along the L-side of the bigon u starting at y and ending at x,

• • $\gamma ':I \to L'$ is the path along the $L'$ -side of u starting at x and ending at y.

This differential does not obey anything like $\mu _1(xa ) = \mu _1(x)a$ – in fact, $ \mu _1(x)a$ is typically undefined and, in general, $\mu _1(x a)$ and $\mu _1(x)$ do not have any useful relationship with each other.

We obtain a $\underline {\Lambda }$ -version of the continuation map (2.13.1)

(3.1.3) $$ \begin{align} \mathrm{CF}(L,L';\underline{\Lambda}) \to \mathrm{CF}(\phi^s L,L';\underline{\Lambda}) \end{align} $$

by multiplying each summand of (2.13.3) by $\nabla \gamma '(a \nabla \gamma (\nabla (\phi ^{-\tau s}(y))(1_{\underline {\Lambda }_y})))$ , where $\gamma = u(\tau ,0)$ and $\gamma ' = u(-\tau ,1)$ are the paths along L and $L'$ respectively and $\phi ^{-\tau s}(y)$ is the reverse of the trajectory from y to $\phi ^s y$ ; that is,

(3.1.4) $$ \begin{align} \sum_{y \in \phi^s L \cap L'} y \sum_{u \in \mathcal{M}(x,y,\phi)} \pm t^{\text{topological energy of}\ u} \nabla \gamma'(a \nabla \gamma(\nabla (\phi^{-\tau s}(y))(1_{\underline{\Lambda}_y}))). \end{align} $$

The same recipe as Subsection 2.13 gives the homotopy inverse to (3.1.3), only replacing (2.13.5) by

(3.1.5) $$ \begin{align} a \cdot y \mapsto \sum_x x \sum_u \pm t^{\text{topological energy of}\ u} \nabla \gamma' \left(a \cdot \nabla (\phi^{\tau s} (y)) \nabla \gamma \nabla(\phi^{-\tau s}(x)) (1_{\underline{\Lambda}_x})\right). \end{align} $$

3.2 Polygon maps

Let $L_0,L_1,\ldots ,L_n$ be oriented, starred submanifolds of T as in Subsection 2.7. We define a variant of (2.7.1)

(3.2.1) $$ \begin{align} \mu_n:\mathrm{CF}^{i_n}(L_{n-1},L_n;\underline{\Lambda}) \times \cdots \times \mathrm{CF}^{i_1}(L_0,L_1;\underline{\Lambda}) \to \mathrm{CF}^{i_1+\cdots+i_n + 2 - n}(L_0,L_n;\underline{\Lambda}) \end{align} $$

carrying $(x_n \cdot a_n,x_{n-1} \cdot a_{n-1},\ldots , x_1 \cdot a_1)$ (with each $a_i \in \underline {\Lambda }_{x_i}$ ) to

(3.2.2) $$ \begin{align} \sum_y y \left(\sum_u \pm t^{\mathrm{area}(u)} \nabla \gamma_n(a_n \nabla \gamma_{n-1} (a_{n-1} \cdots \nabla \gamma_2(a_2 \nabla \gamma_1(a_1\nabla(\gamma_0(1))\cdots))))\right), \end{align} $$

where u runs over the same set of rigid polygons as (2.7.1), the signs are just the same and $\gamma _i$ is the path along the boundary of u going from $x_i$ to $x_{i+1}$ or from $x_{n-1}$ to $x_0$ .

Each connected component of the Deligne–Mumford–Stasheff compactification of the space of nonrigid polygons has the same vertices $x_0,\ldots ,x_n$ – that is, every u in that component has those same vertices. Moreover, the path from $x_i$ to $x_{i+1}$ or from $x_n$ to $x_0$ along u belongs to the same homotopy class, so that $\nabla (\gamma _i):\Lambda _{x_i} \to \Lambda _{x_{i+1}}$ is locally constant in u. Thus, the $A_{\infty }$ -relations among the (3.2.1) hold for the usual reasons:

(3.2.3) $$ \begin{align} \sum_{i+j = n+1} \sum_{\ell < i} (-1)^{|x_1| + \cdots+ |x_{\ell}| - \ell} \mu_i(x_n a_n,\ldots, x_{\ell+j+1}a_{\ell+j+1},\mu_j(\cdots), x_\ell a_\ell,\ldots x_1 a_1) = 0. \end{align} $$

3.3 Local systems

We can put the formulas in Subsection 3.2 in context by considering local systems on the $L_i$ . For each i, let $\mathcal {E}_i$ be a local system of $\underline {\Lambda }\vert _{L_i}$ -modules on $L_i$ . Then we define

(3.3.1) $$ \begin{align} \mathrm{CF}((L_i,\mathcal{E}_i),(L_{i+1},\mathcal{E}_{i+1});\underline{\Lambda}) = \bigoplus_{x \in L_i \cap L_{i+1}} \operatorname{Hom}(\mathcal{E}_{i,x},\mathcal{E}_{i+1,x}). \end{align} $$

The differential is modified by

(3.3.2) $$ \begin{align} \mu_1(xf:\mathcal{E}_{i,x} \to \mathcal{E}_{i+1,x}) = \sum y \sum_u \pm t^{\mathrm{area}(u)} \nabla \gamma' \circ f \circ \nabla \gamma \end{align} $$

(where $xf$ denotes f placed in the xth summand of (3.3.1)). In case $\mathcal {E}_i = \underline {\Lambda }\vert _{L_i}$ is the trivial sheaf of modules, then $\operatorname {Hom}(\mathcal {E}_{i,x},\mathcal {E}_{i+1,x}) = \operatorname {Hom}(\underline {\Lambda }_x,\underline {\Lambda }_x)$ is naturally identified with $\underline {\Lambda }_x$ and (3.3.2) coincides with (3.1.2).

The polygon maps

(3.3.3) $$ \begin{align} &\mu_n:\mathrm{CF}((L_{n-1},\mathcal{E}_{n-1}),(L_n,\mathcal{E}_n);\underline{\Lambda}) \times \cdots \times \mathrm{CF}((L_0,\mathcal{E}_0),(L_1,\mathcal{E}_1);\underline{\Lambda}) \nonumber\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\to \mathrm{CF}((L_0,\mathcal{E}_0),(L_n,\mathcal{E}_n);\underline{\Lambda}) \end{align} $$

are defined by sending $(f_n,f_{n-1},\ldots ,f_1)$ to the formal expression

(3.3.4) $$ \begin{align} \sum_{y} y \left(\sum_{u \in \mathcal{M}(y,x_1,x_2,\ldots,x_n)} \pm t^{\mathrm{area}(u)}\nabla \gamma_n\circ f_n\circ \nabla\gamma_{n-1} \circ f_{n-1} \circ \cdots \circ \nabla \gamma_1 \circ f_1 \circ \nabla \gamma_0 \right). \end{align} $$

We have left the degrees in (3.3.3) off for typesetting reasons; they are the same as in (2.7.1). The formula (3.3.4) specialises to (3.2.2) in case $\mathcal {E}_i = \underline {\Lambda }\vert _{L_i}$ . In general, (3.3.4) can fail to converge, unless the following ‘unitarity’ condition is imposed on the $\mathcal {E}_i$ :

3.4 F-field

We would like to package the $\underline {\Lambda }$ -coupled triangle products among the $L_{(m)}$ into a graded ring, as in Subsection 1.4. The necessary natural isomorphism between $\mathrm {CF}(L_{(m)},L_{(n)};\underline {\Lambda })$ and $\mathrm {CF}(L_{(0)},L_{(n-m)};\underline {\Lambda })$ exists only when $\underline {\Lambda }$ is pulled back along the projection map

(3.4.1) $$ \begin{align} \mathfrak{f}:T \to S^1 :(x,y) + \mathbf{Z}^2 \mapsto x + \mathbf{Z}. \end{align} $$

To make this explicit, let $\sigma :C \to C$ be a ring automorphism, where C is commutative. We also let $\sigma $ denote induced automorphism of $C[\![t]\!]$ or of $\Lambda _C$ , with $\sigma (t^a) = t^a$ . We are mainly interested in the case that

(3.4.2) $$ \begin{align} C\ \text{perfect of characteristic}\ p, \quad \sigma(c) = c^{1/p}, \end{align} $$

in which case, if $f(t) = \sum c_a t^a$ belongs to $C[\![t]\!]$ or to $\Lambda _C$ , then we can write $\sigma (f)(t) = f(t^p)^{1/p}$ . The quotient

(3.4.3) $$ \begin{align} (\mathbf{R} \times C)/\sim \end{align} $$

of $\mathbf {R} \times C$ by the equivalence relation $(x,c) \sim (x+1,\sigma (c))$ is the étalé space of a locally constant sheaf of rings on $S^1$ – as are $(\mathbf {R} \times C[\![t]\!])/\sim $ and $(\mathbf {R} \times \Lambda _C)/\sim $ . We denote the pullback-to-T of these sheaves of rings by $\underline {C}$ , $\underline {C}[\![t]\!]$ and $\Lambda _{\underline {C}}$ .

We will call (3.4.1) an ‘F-field’ on T. In diagrams, we keep track of it with a red line – the inverse image of a point close to the right edge of the fundamental domain $[0,1)$ of $S^1$ , as in the figure on the left below. On the right we have drawn, in a different scale, the preimage of the red line in part of the universal cover of T, along with a triangle that contributes to $\mu _2(x_2 \cdot b,x_1 \cdot a)$ . One understands that $\sigma $ or $\sigma ^{-1}$ is to be applied every time one crosses this ‘danger line’ – $\sigma $ if one crosses it from right to left and $\sigma ^{-1}$ if one crosses it from left to right.

3.5 Example

Let $\underline {\Lambda } = \Lambda _{\underline {C}}$ be as in Subsection 3.4 and let $L_{(0)}$ and $\phi ^s L_{(0)}$ be as in Subsection 2.12.

We will compute the differential on $\mathrm {CF}(\phi ^s L_{(0)},L_{(0)};\underline {\Lambda })$ – this specialises to the example of Subsection 2.12 in case $\sigma $ is trivial. As in that example, we still have $\mathrm {CF}^0(\phi ^s L_{(0)},L_{(0)};\underline {\Lambda }) = x \Lambda $ and $\mathrm {CF}^1(\phi ^s L_{(0)},L_{(0)};\underline {\Lambda }) = y\Lambda $ , but the map $\mu _1$ is not $\Lambda $ -linear, so we must compute not just $\mu _1(x)$ but $\mu _1(x a)$ for all $a \in \Lambda $ . The same two bigons in Subsection 2.12, of area A, contribute to $\mu _1(xa)$ , but only of them crosses the ‘danger line’:

The left bigon contributes $y \cdot (-t^A a)$ and the right bigon contributes $y \cdot (t^A \sigma (a))$ , so that the differential is given by

$$ \begin{align*} \mu_1(x a) = y t^A(\sigma(a) - a). \end{align*} $$

If we make the change of basis $(x,y) \to (x,yt^A)$ , then

$$ \begin{align*} \mathrm{HF}^0 \cong \operatorname{ker}(a \mapsto \sigma(a) - a) \qquad \mathrm{HF}^1 \cong \operatorname{coker}(a \mapsto \sigma(a) -a). \end{align*} $$

More suggestively, $\mathrm {HF}^i(\phi ^s L_{(0)},L_{(0)};\underline {\Lambda })$ is isomorphic to $H^i(L_{(0)};\underline {\Lambda }\vert _{L_{(0)}})$ , and the cohomology of the circle $L_{(0)}$ with coefficients in $\underline {\Lambda }$ .

3.6 Example

With $\underline {\Lambda }$ and $\phi $ as in the previous example, let us compute $\mathrm {CF}(\phi ^s L,L';\underline {\Lambda })$ when L and $L'$ are two special Lagrangians that are parallel to $L_{(0)}$ and to each other but that do not intersect. Let c be the distance between (and therefore also the area between) L and $L'$ . Suppose that $|s|$ slightly exceed c, so that $\phi ^s(L) \cap L'$ has two intersection points that we again denote by x and y.

Then the differential on $\mathrm {CF}(\phi ^s L,L';\underline {\Lambda })$ is $\mu _1(x a) = y(t^a \sigma (a) - t^{A+c} a)$ . If c is not zero, the complex is acyclic, but it is interesting to note that the formal series

(3.6.1) $$ \begin{align} x \cdot \sum_{n \in \mathbf{Z}} \sigma^{-n}(a) t^{nc} \qquad a \in \underline{C}_x \end{align} $$

is killed by $\mu _1$ . Since (3.6.1) has an infinite ‘tail’, it does not lie in $\underline {\Lambda }_x$ and does not contribute to $\mathrm {CF}(\phi ^s L,L';\underline {\Lambda })$ .

Since $L \cap L'$ is empty, the continuation maps associated to $\phi ^s$ and its reverse $\phi ^{-s}$

$$ \begin{align*} \mathrm{CF}(L,L';\underline{\Lambda}) \to \mathrm{CF}(\phi^s L,L';\underline{\Lambda})\quad \text{and} \quad \mathrm{CF}(\phi^s L,L';\underline{\Lambda}) \to \mathrm{CF}(L,L';\underline{\Lambda}) \end{align*} $$

both vanish. But the explicit contracting homotopy on $\mathrm {CF}^1(\phi ^s L,L';\underline {\Lambda }) \to \mathrm {CF}^0(\phi ^s L,L';\underline {\Lambda })$ is interesting. It is given by the series

(3.6.2) $$ \begin{align} \Delta(a \cdot y) = x (t^{-A} \sigma^{-1}(a) + t^{-A+c} \sigma^{-2}(a) + \cdots + t^{-A+nc} \sigma^{-n-1}(a) + \cdots). \end{align} $$

The strip $u_n$ contributing the $t^{-A+nc}$ term in this series is the same as in Subsection 2.14, but that contribution is now multiplied by $\nabla \gamma ' \left (a \cdot \nabla (\phi ^{\tau s} (y)) \nabla \gamma \nabla (\phi ^{-\tau s}(x)) (1_{\underline {\Lambda }_x})\right )$ , which simplifies to $\sigma ^{-n-1}(a)$ .

3.7 Computing the triangle maps

Let $L_{(m)}$ (equipped with the same orientations and stars), $x_{m,\kappa }$ and $\tau $ be as in Subsection 2.8. Let $\underline {\Lambda }$ and $\underline {C}$ be pulled back along $\mathfrak {f}$ , with $\sigma $ denoting the nontrivial monodromy, as in Subsection 3.4. We will compute

$$ \begin{align*} \mathrm{CF}(L_{(m_1)},L_{(m_1+m_2)};\underline{\Lambda}) \times \mathrm{CF}(L_{(0)},L_{(m_1)};\underline{\Lambda}) \to \mathrm{CF}(L_{(0)},L_{(m_1+m_2)};\underline{\Lambda}) \end{align*} $$

by computing $\mu _2(\tau ^{m_1}x_{m_2,\kappa _2} \cdot b, x_{m_1,\kappa _1} \cdot a)$ .

Theorem. Let E be as in (2.8.5). Then in the absolute case, $\mu _2(\tau ^{m_1} x_{m_2,\kappa _2},x_{m_1,\kappa _1})$ is given by

(3.7.1) $$ \begin{align} \sum_{\ell \in \mathbf{Z}} x_{m_1+m_2,E(\kappa_1,\kappa_2+\ell)} t^{(\ell+\kappa_2 - \kappa_1)^2/(2(\frac{1}{m_1} + \frac{1}{m_2}))} \sigma^{\ell-\lfloor E(\kappa_1,\kappa_2+\ell)\rfloor}(b) \sigma^{-\lfloor E(\kappa_1,\kappa_2+\ell)\rfloor}(a), \end{align} $$

where we understand $x_{m_1+m_2,E(\kappa _1,\kappa _2+\ell )} := x_{m_1+m_2,\kappa }$ if $E(\kappa _1,\kappa _2+\ell ) = \kappa $ modulo $1$ . In the relative case, $\mu _2(\tau ^{m_1} x_{m_2,\kappa _2},x_{m_1,\kappa _1})$ is given by

(3.7.2) $$ \begin{align} \sum_{\ell \in \mathbf{Z}} x_{m_1+m_2,E(\kappa_1,\kappa_2+\ell)} t^{\lambda(\kappa_1,\kappa_2+\ell)} \sigma^{\ell-\lfloor E(\kappa_1,\kappa_2+\ell)\rfloor}(b) \sigma^{-\lfloor E(\kappa_1,\kappa_2+\ell)\rfloor}(a). \end{align} $$

Proof. The triangles that contribute are exactly as in the proof in Subsection 2.8; we will index them again by $\ell \in \mathbf {Z}$ . The $\pm $ sign and the exponent of t in (1.11.4) are the same as in Subsection 2.8, but it remains to compute $\nabla \gamma _2(b \nabla \gamma _1(a \nabla \gamma _0(1)))$ . If I is an interval and $\gamma :I \to T$ is a path in T, write $\mathfrak {f}(\gamma )$ for the number of times $\gamma $ crosses the ‘danger line’ (Subsection 3.4), counted with sign. Then

(3.7.3) $$ \begin{align} \hskip-104pt\nabla \gamma_2(b \nabla \gamma_1(a \nabla \gamma_0(1))) = \sigma^{\mathfrak{f}(\gamma_2)}(b \sigma^{\mathfrak{f}(\gamma_1)}(a)) \end{align} $$

(3.7.4) $$ \begin{align} & \hskip7pt= \sigma^{\mathfrak{f}(\gamma_2)}(b) \sigma^{\mathfrak{f}(\gamma_1)+\mathfrak{f}(\gamma_2)}(a) \end{align} $$

(3.7.5) $$ \begin{align} & \hskip-7pt= \sigma^{\mathfrak{f}(\gamma_2)}(b) \sigma^{-\mathfrak{f}(\gamma_0)}(a). \end{align} $$

The $\ell $ th triangle (pictured below) has $\mathfrak {f}(\gamma _2) = \lfloor E(\kappa _1,\kappa _2+\ell )\rfloor $ and $\mathfrak {f}(\gamma _0) = \ell - \lfloor E(\kappa _1,\kappa _2+\ell ) \rfloor $ .

3.8 Theta functions with F-field coupling

Keeping the notation of the previous section, where $\underline {\Lambda }$ and $\underline {C}$ are pulled back along $\mathfrak {f}$ , we may give

(3.8.1) $$ \begin{align} \bigoplus_{m = 1}^{\infty} \mathrm{CF}(L_{(0)},L_{(m)};\underline{\Lambda}) \end{align} $$

the structure of a graded ring without unit. One may equip it with a unit by taking the degree 0 piece to be $\Lambda _{C^{\sigma }} = \mathrm {HF}^0(L_{(0)},L_{(0)};\underline {\Lambda })$ (Subsection 3.5). Here is a description of (3.8.1) analogous to that of Subsection 2.9.

Theorem §1.12. For each $a \in C$ and each pair of integers $m,k$ with $m> k \geq 0$ , let $\theta _{m,k}[a]$ denote the formal series

(3.8.2) $$ \begin{align} \theta_{m,k}[a] := \sum_{i = -\infty}^{\infty} t^{m\frac{i(i-1)}{2} + ki} z^{mi + k} \sigma^i(a). \end{align} $$

Let $\theta ^{\mathrm {abs}}_{m,k}[a]$ denote the formal series

(3.8.3) $$ \begin{align} \theta^{\mathrm{abs}}_{m,k}[a] := \sum_{i= -\infty}^{\infty} t^{\frac{1}{2m}(mi + k)^2} z^{mi+ k} \sigma^i(a). \end{align} $$

Then the relative (respectively absolute) version of (3.8.1) is isomorphic (as a graded ring-without-unit) to the $\mathbf {Z}[\![t]\!]$ -linear span of the $\theta _{m,k}[a]$ (respectively $\Lambda _{\mathbf {Z}}$ -linear span of the $\theta ^{\mathrm {abs}}_{m,k}[a]$ ) via the map

(3.8.4) $$ \begin{align} x_{m,k/m} \cdot a \mapsto \theta_{m,k} [a]. \end{align} $$

As in Subsection 2.9, the summands of the $\theta [a]$ are indexed by right triangles. The factor of $\sigma ^i(a)$ plays the same role as the $\nabla \gamma _2(b \nabla \gamma _1(a \nabla (\gamma _0(1)))$ factor in (1.11.4).

This triangle contributes $t^9 z^5 \sigma ^5(a)$ to the relative version of $\theta _{2,1}[a]$ (3.8.2) and $t^{6.25} z^5 \sigma ^5(a)$ to the absolute version (3.8.3). Presumably, a ‘family Floer’ argument along these lines would prove the theorem, but we will give a proof in terms of the explicit formulas.

Proof. Let us give the proof first in the relative case. Fix $m_1,m_2 \in \mathbf {Z}_{\geq 0}$ , $k_1 \in \{0,\ldots ,m_1 - 1\}$ , $k_2 \in \{0,\ldots ,m_2 - 1\}$ and $a,b \in C[\![t]\!]$ . The product of $\theta _{m_2,k_2}[b]$ and $\theta _{m_1,k_1}[a]$ is, by definition,

(3.8.5) $$ \begin{align} \sum_{i_1,i_2 \in \mathbf{Z} \times \mathbf{Z}} \sigma^{i_2}(b)\sigma^{i_1}(a) t^{m_2\binom{i_2}{2} + m_1 \binom{i_1}{2} + k_2i_2 + k_1i_1} z^{m_2 i_2 + m_1 i_1 + k_2 + k_1}. \end{align} $$

We may also index the sum by triples $(r,c,d)$ , where $(c,d) \in \mathbf {Z} \times \mathbf {Z}$ and $r \in \left \{0,1,\ldots , \frac {m_1 + m_2}{\mathrm {gcd}(m_1,m_2)}-1\right \}$ . First, for $\ell \in \mathbf {Z}$ , we define d and r via

$$ \begin{align*} \ell = \frac{m_1+m_2}{\mathrm{gcd}(m_1,m_2)}d + r. \end{align*} $$

It then follows that if we set

$$ \begin{align*} e(r) = e_{m_1,m_2,k_1,k_2}(r) := \left\lfloor \frac{m_2 r + k_1 + k_2}{m_1 + m_2} \right\rfloor \in \left\{0,1,\ldots, \frac{m_2}{\mathrm{gcd}(m_1,m_2)}-1\right\} \end{align*} $$

and $\kappa _1 = k_1/m_1, \kappa _2 = k_2/m_2$ , we have

(3.8.6) $$ \begin{align} \lfloor E(\kappa_1,\kappa_2+\ell) \rfloor = \frac{m_2 d}{\mathrm{gcd}(m_1,m_2)} + e(r). \end{align} $$

The triple $(r,c,d)$ (and the integer $\ell $ ) is determined as the unique solution to

(3.8.7) $$ \begin{align} \left(\begin{array}{r} i_1 \\ i_2 \end{array} \right) = \left(\begin{array}{rr} 1 & -m_2/\mathrm{gcd}(m_1,m_2) \\ 1 & m_1/\mathrm{gcd}(m_1,m_2) \end{array} \right) \left(\begin{array}{c} c -e(r) \\ d \end{array} \right)+ \left(\begin{array}{r} 0 \\ r \end{array} \right). \end{align} $$

To save space in the exponents, let us write $g := \gcd (m_1,m_2)$ . After reindexing the sum (3.8.5) is

(3.8.8) $$ \begin{align} \sum_r \sum_c \sum_d \sigma^{c-e(r) + m_1d/g + r}(b) \sigma^{c - e(r) - dm_2/g}(a) t^{\square_1}z^{\square_2}, \end{align} $$

where

(3.8.9) $$ \begin{align} \begin{array}{rcc} \square_1 & = & m_1\binom{c-e(r) - dm_2/g}{2}+ k_1(c-e(r) - dm_2/g) \\ & & + m_2\binom{c-e(r) + dm_1/g + r}{2} + k_2(c-e(r) + dm_1/g + r) \end{array} \end{align} $$

and

(3.8.10) $$ \begin{align} \square_2 = m_1(c-e(r) - dm_2/g) + m_2(c-e(r) + dm_1/g + r) + k_1 + k_2. \end{align} $$

We note that $\square _2 = (m_1 + m_2)c + (m_2 r + k_1 + k_2) - (m_1+m_2)e(r)$ does not depend on d and, furthermore, that

$$ \begin{align*} k(r) = k_{m_1,m_2,k_1,k_2}(r) := m_2 r + k_1 + k_2 - (m_1+m_2)e(r) \end{align*} $$

belongs to $\{0,\ldots ,m_1+m_2-1\}$ . (Note that $k(r)/(m_1+m_2)$ is the fractional part of $E(\kappa _1,\kappa _2+\ell )$ .) Thus, the sum (3.8.8) is the same as

$$ \begin{align*} \sum_r \sum_c \left(\sum_d \sigma^{c -e(r)+ dm_1/g + r}(b) \sigma^{c -e(r) - dm_2/g}(a) t^{\square_1} \right)z^{(m_1 + m_2)c + k(r)}. \end{align*} $$

Since $\sigma $ acts trivially on t, this is the same as

$$ \begin{align*} \sum_r \sum_c \sigma^c\left(\sum_d \sigma^{-e(r)+dm_1/g + r}(b) \sigma^{-e(r) - dm_2/g}(a) t^{\square_1} \right)z^{(m_1 + m_2)c + k(r)}. \end{align*} $$

Now we claim

(3.8.11) $$ \begin{align} \square_1 = \lambda\left(\kappa_1,\kappa_2+\ell \right) + (m_1 + m_2) \binom{c}{2} + k(r) c, \end{align} $$

where $\square _1$ is as in (3.8.9) and $\lambda $ is as in (2.8.7) and $\ell = \frac {m_1+m_2}{g}d + r$ .

Taking (3.8.11) for granted, we obtain that $\theta _{m_2,k_2}[b]\cdot \theta _{m_1,k_1}[a]$ is equal to

$$ \begin{align*} \sum_{r = 0}^{\frac{m_1+m_2}{g}-1} \sum_c \sigma^c\left(\sum_d \sigma^{-e(r) - dm_2/g}(a) \sigma^{-e(r)+dm_1/g + r}(b) t^{\lambda(\kappa_1, \kappa_2+\ell)} \right)t^{(m_1 + m_2)\binom{c}{2} + k(r) c}z^{(m_1 + m_2)c + k(r)}, \end{align*} $$

which is equal to

(3.8.12) $$ \begin{align} \sum_r \theta_{m_1+m_2,k(r)}\left[\sum_d \sigma^{-e(r)+dm_1/g + r}(b) \sigma^{-e(r) - dm_2/g}(a) t^{\lambda(\kappa_1, \kappa_2+\ell)}\right]. \end{align} $$

We now compare (3.8.12) to $(b \cdot x_{m_2,k_2/m_2}) (a \cdot x_{m_1,k_1/m_1})$ , which is given by (3.7.2)

$$ \begin{align*} \sum_{\ell \in \mathbf{Z}} \left(\sigma^{\ell - \lfloor E(\kappa_1,\kappa_2 + \ell) \rfloor}\!(b) \, \sigma^{-\lfloor E(\kappa_1,\kappa_2 + \ell)\rfloor}\!(a) \, t^{\lambda(\kappa_1,\kappa_2 + \ell)}\right) x_{m_1+ m_2,E(\kappa_1,\kappa_2 + \ell)}. \end{align*} $$

Writing $\ell = \frac {m_1+m_2}{\mathrm {gcd}(m_1,m_2)} d + r$ and using the formula (3.8.6), we can rewrite this as

(3.8.13) $$ \begin{align} \sum_r \left(\sum_d \sigma^{-e(r)+dm_1/g + r}(b) \sigma^{-e(r) - dm_2/g}(a) t^{\lambda(\kappa_1, \kappa_2+ \ell)} \right) x_{m_1+ m_2,k(r)}. \end{align} $$

It is now evident that under our map (3.8.4), the two expressions (3.8.12) and (3.8.13) agree.

It remains to verify the claim in (3.8.11). This will be a direct computation. We have

$$ \begin{align*} &\lambda(\kappa_1,\kappa_2+ \ell) = m_1 \phi(\kappa_1) + m_2 \phi(\kappa_2 + \ell) - (m_1+m_2) \phi(\frac{m_2d}{g}+e(r) + \frac{k(r)}{m_1+m_2}) \\ &= m_2 (\ell(\ell+\kappa_2) - \frac{\ell(\ell+1)}{2} ) \\ & - (m_1+m_2) \left((\frac{dm_2}{g}+e(r))(\frac{dm_2}{g}+e(r) + \frac{k(r)}{m_1+m_2}) - \frac{(\frac{dm_2}{g}+e(r))(\frac{dm_2 }{g}+e(r)+1)}{2}) \right) \\ &= m_2 (\ell \kappa_2 + \frac{\ell(\ell-1)}{2}) - (\frac{dm_2 }{g}+e(r)) k(r) - (m_1+m_2) \frac{(\frac{dm_2 }{g}+e(r)) (\frac{dm_2 }{g}+e(r)-1)}{2} \\ &= m_2 \kappa_2 (\frac{d(m_1+m_2)}{g} +r) + \frac{m_2 (\frac{d(m_1+m_2)}{g}+r) (\frac{d(m_1+m_2)}{g}+r-1)}{2} \\ &- (\frac{dm_2 }{g}+e(r)) k(r) - (m_1+m_2) \frac{(\frac{d m_2 }{g}+e(r)) (\frac{d m_2 }{g}+e(r)-1)}{2}. \end{align*} $$

Substituting in $k(r) = m_2 r + k_1 + k_2 - (m_1+m_2)e(r)$ , we get

$$ \begin{align*} \lambda(\kappa_1,\kappa_2+\ell) &= m_2 \kappa_2 (\frac{d(m_1+m_2)}{g} +r) + \frac{m_2 (\frac{d(m_1+m_2)}{g}+r) (\frac{d(m_1+m_2)}{g}+r-1)}{2} \\ &- (\frac{dm_2}{g}+e(r)) (m_1\kappa_1 + m_2 \kappa_2 + m_2r - e(r)(m_1+m_2)) \\ &- (m_1+m_2) \frac{ (\frac{dm_2 }{g}+e(r)) (\frac{dm_2}{g}+e(r)-1)}{2} \\ &= \frac{(m_1+m_2) m_1 m_2}{2 g^2} d^2 + \frac{m_1 m_2}{g} (r+ \kappa_2 -\kappa_1) d \\ &+ m_2 \kappa_2 r + \frac{m_2 r (r-1)}{2} + \frac{(m_1+m_2) e(r)(e(r)+1)}{2} - (m_1\kappa_1 +m_2 \kappa_2 + m_2 r)e(r). \end{align*} $$

We can rewrite this in a symmetric form as follows:

$$ \begin{align*} \lambda(\kappa_1,\kappa_2+\ell) &= \frac{(m_1+m_2) m_1 m_2}{2 g^2} d^2 + \frac{m_1 m_2}{g} (r+ \kappa_2 -\kappa_1) d \\ &\quad+ \frac{m_2(e(r)-r)(e(r)-r+1)}{2} + \frac{m_1e(r)(e(r)+1)}{2} - k_2(e(r)-r) - k_1e(r), \end{align*} $$

and this in turn can be seen to be equal to

$$ \begin{align*} \lambda(\kappa_1,\kappa_2+\ell) &= m_1\binom{-e(r) - m_2d/g}{2}+ k_1(-e(r) - m_2d/g) \\ &\quad+ m_2\binom{-e(r) + m_1d/g + r}{2} + k_2(-e(r) + m_1d/g + r). \end{align*} $$

This completes the proof of the claim (3.8.11) and hence the proof of the theorem in the relative case.

In the absolute case, the product of $\theta ^{\mathrm {abs}}_{m_2,k_2}[b]$ and $\theta ^{\mathrm {abs}}_{m_1,k_1}[a]$ is given by

(3.8.14) $$ \begin{align} \sum_{i_1,i_2 \in \mathbf{Z} \times \mathbf{Z}} \sigma^{i_2}(b)\sigma^{i_1}(a) t^{\frac{(m_2 i_2 + k_2)^2}{2m_2} + \frac{(m_1i_1 +k_1)^2}{2m_1}} z^{m_2 i_2 + m_1 i_1 + k_2 + k_1}. \end{align} $$

Performing the same re-indexing using (3.8.7), we arrive at

(3.8.15) $$ \begin{align} \sum_r \sum_c \sum_d \sigma^{c-e(r) + m_1d/g + r}(b) \sigma^{c - e(r) - dm_2/g}(a) t^{\square^{abs}_1}z^{\square^{abs}_2}, \end{align} $$

where

(3.8.16) $$ \begin{align} \square^{abs}_1 = \frac{(m_1(c-e(r) - dm_2/g) + k_1)^2}{2m_1} + \frac{(m_2(c-e(r) + dm_1/g + r) + k_2)^2}{2m_2} \end{align} $$

and $\square _2^{abs}=\square _2$ given as before by (3.8.10). Following the same steps, the only difference in the calculation is the verification of the analogue of equation (3.8.11), which now takes the form:

(3.8.17) $$ \begin{align} \square^{abs}_1 = \frac{(\ell+\kappa_2-\kappa_1)^2 m_1 m_2}{2(m_1+m_2)} + \frac{((m_1 + m_2)c + k(r))^2}{2(m_1+m_2)}. \end{align} $$

Recalling that $\ell = (m_1+m_2)d/g + r$ , $k(r) = m_2 r + k_1 + k_2 - (m_1+m_2)e(r)$ and $\kappa _i = k_i/m_i$ for $i=1,2$ , we can compare the equations (3.8.16) and (3.8.17) directly to verify the claim. This completes the proof in the absolute case.

4.1 At $\boldsymbol{t = 0}$

The specialisation $t = 0$ renders uninteresting the absolute version of the maps $\mu _n$ , at least if we also set $t^a = 0$ for every $a>0$ . But it is a standard part of relative Floer theory. In fact, it is part of the motivation for relative Floer theory – in any sum over triangles (say), the contribution from triangles which are not disjoint from D vanishes, so that working with $t = 0$ is closely related to replacing the closed symplectic manifold T with the open $T -D$ . See [Reference Lekili and PerutzLPe2, §6.1] for some more context.

For short, let us write $S_n$ for the nth graded piece of (3.8.1). Let us also put $S_0 := C^{\sigma }[\![t]\!]$ – here $C^{\sigma }$ denotes the $\sigma $ -fixed subring of C. Then $S_{\bullet }$ is a graded $C^{\sigma }[\![t]\!]$ -algebra – it is associative and commutative by Theorem 1.12, Subsection 3.8. If C is a perfect field and $\sigma $ is the pth root map, then $C^{\sigma } = \mathbf {F}_p$ . In any case, there is an isomorphism in the category of $C^{\sigma }$ -schemes

where $i_0$ and $i_{\infty }$ are the inclusions of C-schemes $\operatorname {Spec}(C) \to \mathbf {P}^1_{/C}$ with coordinates $0$ and $\infty $ , respectively.

If C is a field, then $\operatorname {Proj}(S \times _{S_0} C^{\sigma })$ is a 1-dimensional scheme, which can be covered by two affine charts. It fails to be regular at a unique point and the complement of this point is isomorphic to $\operatorname {Spec}(C[x,x^{-1}])$ . For the other chart, take the complement of any other point – one obtains an affine Zariski neighbourhood of the nonregular point that is isomorphic to the spectrum of a subring of $C[y]$ , namely,

$$ \begin{align*} \{f \in C[y] : \sigma(f(0)) = f(1)\}. \end{align*} $$

This ring is in some sense an order in a Dedekind domain, but if C has infinite degree over $C^{\sigma }$ , then it is not of finite type.

4.2 Floer cochains at $\boldsymbol{t = 1}$

Let C and $\sigma $ be as in Subsection 3.4, with $\underline {C}$ pulled back along $\mathfrak {f}$ from the sheaf whose étalé space is (3.4.3). If L and $L'$ are 1-dimensional submanifolds that intersect transversely, we will write (similar to (2.5.1))

(4.2.1) $$ \begin{align} \mathrm{CF}(L,L';\underline{C}) = \bigoplus_{x \in L \cap L'} \underline{C}_x. \end{align} $$

This supports a $\mathbf {Z}/2$ -grading and a bigon differential

(4.2.2) $$ \begin{align} \mu_1(x \cdot a) = \sum_{y | \mathrm{mas}(y) = \mathrm{mas}(x)+1} y \left(\sum_{u \in \mathcal{M}(y,x)} \pm \nabla \gamma'\left(a \nabla \gamma(1_{\underline{\Lambda_y}})\right) \right) \end{align} $$

with $\gamma $ and $\gamma '$ as in (3.1.2). (4.2.2) is a finite sum.

If we further endow C with a topology, for which $\sigma $ is continuous, we can investigate the algebraic structures on (4.2.1) induced by (3.2.1). That is, we study the sums

(4.2.3) $$ \begin{align} \sum_y y \left(\sum_{u} \pm \nabla \gamma_n(a_n \nabla \gamma_{n-1}(a_{n-1} \cdots \nabla \gamma_2(a_2 \nabla(\gamma_1(a_1 \nabla \gamma_0(1))\cdots)))) \right). \end{align} $$

(4.2.2) and (4.2.3) are simply the specialisations one obtains by setting t and every power $t^a$ to $1$ in the formulas from Section 3. The sums $\sum _u$ in (4.2.3) might diverge or converge in the topological ring C, so that at best the map

(4.2.4) $$ \begin{align} \mathrm{CF}(L_{n-1},L_n;\underline{C}) \times \cdots \times \mathrm{CF}(L_0,L_1;\underline{C}) \dashrightarrow \mathrm{CF}(L_0,L_n;\underline{C}) \end{align} $$

is only partially defined. In many cases, the domain of convergence is reduced to a point, but we will see that the triangle maps are not trivial.

4.3 The triangle products at $\boldsymbol{t = 1}$

Suppose that C is complete with respect to a non-Archimedean norm $|\cdot |$ and that

$$ \begin{align*} \sigma(c) = |c|^{1/p} \end{align*} $$

for some $p> 1$ . With p prime and $C,\sigma $ as in (3.4.2), the pair $(C,|\cdot |)$ is a perfectoid field of characteristic p [Reference ScholzeSc, §3]. The maps $\nabla \gamma $ for the sheaf of rings $\underline {C}$ are continuous, but (crucially) they d not preserve the norms.

Write

$$ \begin{align*} \mathcal{O}_C := \{c \in C : |c|\leq 1\} \qquad \mathfrak{m}_C:=\{c \in C:|c|<1\}; \end{align*} $$

then $\mathcal {O}_C$ is the ring of integers in C and $\mathfrak {m}$ is the unique maximal ideal of $\mathcal {O}_C$ . They are both stable by the $\sigma $ -action so that they determine locally constant subsheaves of $\underline {C}$ that we denote by $\underline {\mathcal {O}}_C$ and $\underline {\mathfrak {m}}_C$ . The fibre of $\underline {\mathfrak {m}}$ at x is the set of topologically nilpotent elements in $\underline {C}_x$ .

Following the notation of (4.2.1), set

(4.3.1) $$ \begin{align} \mathrm{CF}(L,L';\underline{\mathfrak{m}}):=\bigoplus_{x \in L \cap L'} \underline{\mathfrak{m}}_x. \end{align} $$

It is an open subgroup of $\mathrm {CF}(L,L';\underline {C})$ .

Suppose $L_0$ , $L_1$ and $L_2$ are special of finite slopes $m_0$ , $m_1$ and $m_2$ , in the sense of Subsection 2.6. The contribution of a triangle with vertices

$$ \begin{align*} x_1 \in L_0 \cap L_1 \qquad x_2 \in L_1 \cap L_2 \qquad y \in L_2 \cap L_0 \end{align*} $$

to $\mu _2(x_2 \cdot b, x_1 \cdot a)$ has the form (3.7.5) $\sigma ^{\mathfrak {f}(\gamma _2)}(b) \sigma ^{-\mathfrak {f}(\gamma _0)}(a)$ , where $\gamma _0$ and $\gamma _2$ are the two edges of u incident with the output vertex y. If (and only if) $m_0 < m_1 < m_2$ , then $\mathfrak {f}(\gamma _2)$ and $\mathfrak {f}(\gamma _0)$ all have the same sign – with perhaps finitely many exceptions where one of $\mathfrak {f}(\gamma _0)$ and $\mathfrak {f}(\gamma _2)$ is zero – so that when $|a|< 1$ and $|b|<1$ ,

$$ \begin{align*} |\sigma^{\mathfrak{f}(\gamma_2)}(b) \sigma^{-\mathfrak{f}(\gamma_0)}(a)| = |b|^{p^{-\mathfrak{f}(\gamma_2)}}|a|^{p^{\mathfrak{f}(\gamma_0)}} \end{align*} $$

is very rapidly decreasing as the side lengths of the triangles go to infinity. The triangle product

$$ \begin{align*} \mu_2:\mathrm{CF}(L_1,L_2;\underline{\mathfrak{m}}) \times \mathrm{CF}(L_0,L_1;\underline{\mathfrak{m}}) \to \mathrm{CF}(L_0,L_2;\underline{\mathfrak{m}}) \end{align*} $$

is therefore convergent when $m_0 < m_1 < m_2$ . In particular, we have a graded ring (for now, without unit)

(4.3.2) $$ \begin{align} \bigoplus_{m = 1}^{\infty} \mathrm{CF}(L_{(0)},L_{(m)};\underline{\mathfrak{m}}). \end{align} $$

4.4 The irrelevant ideal in the Fargues–Fontaine graded ring

Let C be an algebraically closed field of characteristic p that is complete with respect to a norm $|\cdot |$ . Let B and $\varphi $ be as in Subsection 1.14; that is,

(4.4.1) $$ \begin{align} B = \left\{\sum_{i \in \mathbf{Z}} b_i z^i \mid \forall r \in (0,1), |b_i|r^i \to 0\text{ as }|i| \to \infty\right\}. \end{align} $$

This appears in [Reference Kontsevich and SoibelmanKS, Def. 21] and in [Reference Fargues and FontaineFF, Ex. 1.6.5]. Fargues and Fontaine defined a version of B for every local field E; (4.4.1) is the case when $E = \mathbf {F}_p(\!(z)\!)$ . Below, we are taking advantage of the fact that when E has equal characteristic, each element of B has a unique series expansion, something that is not clear when E has mixed characteristic [Reference Fargues and FontaineFF, Rem. 1.6.7].

Let $\varphi $ be as in (1.14.2); that is, the automorphism of B given by $\varphi (\sum c_i z^i) = \sum c_i^p z^i$ . The homogeneous coordinate ring of ${\mathit {FF}}_E(C)$ is

(4.4.2) $$ \begin{align} \mathbf{F}_p(\!(z)\!) \oplus B^{\varphi = z} \oplus B^{\varphi = z^2} \oplus \cdots \end{align} $$

We will prove the theorem of Subsection 1.14; that is, that (4.3.2) is isomorphic to the irrelevant ideal of this ring.

Proof of (1.14.4)

Suppose that $a \in C$ has $|a| < 1$ . Then the sequence $|\sigma ^i(a)| = |a|^{p^{-i}}$ of real numbers is bounded as $i \to \infty $ and very rapidly decreasing as $i \to -\infty $ and $r^{mi+k} |a|^{p^{-i}} \to 0$ as $|i| \to \infty $ for any $r \in (0,1)$ . Therefore, for any m and k the expression

(4.4.3) $$ \begin{align} \sum_{i \in \mathbf{Z}} z^{mi+k} \sigma^i(a) \end{align} $$

belongs to B (1.14.1). Applying $\varphi $ to (4.4.3) gives $\sum _{i \in \mathbf {Z}} z^{mi+k} \sigma ^{i-1}(a)$ – re-indexing this series gives

$$ \begin{align*} \sum_{i \in \mathbf{Z}} z^{m(i+1) + k} \sigma^i(a) = z^m \sum_{i \in \mathbf{Z}} z^{mi + k} \sigma^i(a), \end{align*} $$

so that (4.4.3) belongs to $B^{\varphi = z^m}$ . But (4.4.3) is $\theta _{m,k}[a]\vert _{t = 1}$ , so that by Subsection 3.8 the map

(4.4.4) $$ \begin{align} x_{m,k/m} \cdot a \mapsto \theta_{m,k}[a]\vert_{t = 1} \end{align} $$

intertwines $\mu _2$ with the ring structure on B.

If $f = \sum b_i z^i$ belongs to $B^{\varphi = z^m}$ , then $b_{mi+k} = \sigma ^m(b_k)$ , so f is determined by $b_0,\ldots ,b_{m-1}$ . To obey (1.14.1), the elements $b_0,\ldots ,b_{m-1}$ must all belong to $\mathfrak {m}$ . The map

$$ \begin{align*} f \mapsto \sum_{k = 0}^{m-1} x_{m,k/m} \cdot b_k \end{align*} $$

gives the inverse isomorphism to $\mathrm {CF}(L_{(0)},L_{(m)};\underline {\mathfrak {m}}) \cong B^{\varphi = z^m}$ .

4.5 SYZ duality

The degree 1 part of (1.14.4) is an isomorphism

(4.5.1) $$ \begin{align} \mathrm{CF}(L_{(0)},L_{(1)};\underline{\mathfrak{m}}) \cong \operatorname{Hom}_{{\mathit{FF}}}(\mathcal{O},\mathcal{O}(1)) \end{align} $$

where $\mathcal {O}(1)$ is the Serre line bundle on (1.14.3). In general, it seems that $\mathrm {CF}(L,L';\underline {\mathfrak {m}})$ captures the set of homomorphisms between two vector bundles on ${\mathit {FF}}$ whenever L and $L'$ are (or are just isotopic to, if we replace $\mathrm {CF}$ by $\mathrm {HF}$ ) special Lagrangians (Subsection 2.6) of finite slopes m and $m'$ with m strictly less than $m'$ . But for other kinds of homomorphisms or Ext groups in $\operatorname {Coh}({\mathit {FF}})$ , another construction must be necessary – one that we only partially understand. In the next two subsections (Subsection 4.6 and Subsection 4.7), we illustrate this in terms of skyscraper sheaves on ${\mathit {FF}}$ .

The closed points of ${\mathit {FF}}_E(C)$ are naturally parametrised by the $\mathbf {Z}$ -orbits of E-untilts of the perfectoid field C. When $E = \mathbf {F}_p(\!(z)\!)$ , an ‘E-untilt’ is just a continuous homomorphism $i:E \to C$ – such a homomorphism must carry z to a nonzero element of $\mathfrak {m}$ and, conversely, every nonzero element of $\mathfrak {m}$ extends to a map from $\mathbf {F}_p(\!(z)\!)$ , and the $\mathbf {Z}$ -action is generated by $i \mapsto \sigma \circ i$ . There is a map

(4.5.2) $$ \begin{align} \left(\text{closed points of }{\mathit{FF}}(E,C)\right) \to \mathbf{R}/\mathbf{Z}. \end{align} $$

It is defined for any E. When $E = \mathbf {F}_p(\!(z)\!)$ , it carries the $\mathbf {Z}$ -orbit of $\iota :\mathbf {F}_p(\!(z)\!) \to C$ to the $\mathbf {Z}$ -coset of $\log _p(\log (|i(z)|^{-1}))$ . We expect that (4.5.2) is the SYZ dual to (3.4.1) and that the skyscraper sheaves have something to do with fibres of (3.4.1).

4.6 Skyscraper sheaves and $L_{(\infty )}$

If $\zeta \in C$ is invertible, let us denote by $L_{(\infty )}^{\zeta }$ the special Lagrangian $L_{(\infty )}$ equipped with the rank 1 local system of $\underline {C}\vert _{L_{(\infty )}}$ -modules (i.e., a local system of C-modules) whose fibre at $(0,0)$ is $\underline {C}_{(0,0)} = C$ and whose monodromy (in the direction of the default orientation, top to bottom) is multiplication by $\zeta $ . Let $e_m$ denote $(0,0)$ regarded as the unique intersection point of $L_{(m)}$ and $L_{(\infty )}$ , so that

(4.6.1) $$ \begin{align} \mathrm{CF}(L_{(m)},L_{(\infty)}^{\zeta};\underline{C}) = \mathrm{CF}^0(L_{(m)},L_{(\infty)}^{\zeta};\underline{C}) = e_m \cdot C. \end{align} $$

If C is algebraically closed, then one also has $\operatorname {Hom}_{{\mathit {FF}}}(\mathcal {O}(m),\delta ) \cong C$ for any skyscraper sheaf $\delta $ . If $|\zeta | < 1$ and $\delta $ is the skyscraper sheaf supported at the $\mathbf {Z}$ -orbit of the untilt $\mathbf {F}_p(\!(z)\!) \to C$ , then we expect that for any surjection $q:\mathcal {O}(1) \to \delta $ , there is an isomorphism making the diagram

(4.6.2)

commute; instead of constructing this isomorphism here, let us verify that the two rows of (4.6.2) have the same kernel. We may find $i:\mathcal {O} \to \mathcal {O}(1)$ such that

$$ \begin{align*} 0 \to \mathcal{O} \xrightarrow{i} \mathcal{O}(1) \xrightarrow{q} \delta \to 0 \end{align*} $$

is exact, so that the kernel of the bottom row in (4.6.2) is isomorphic to $\operatorname {Hom}(\mathcal {O},\mathcal {O}) = \mathbf {F}_p(\!(z)\!)$ , the ground field of (1.14.3). We will show that the kernel of $\mu _2(e_1 \cdot 1,-)$ has the structure of a 1-dimensional $\mathbf {F}_p(\!(z)\!)$ -module.

In general, the triangle map $\mu _2(e_1 \cdot b, x_{1,0} \cdot a)$ (4.2.4) is given by

(4.6.3)

with the figure at the right illustrating the triangle that contributes the $i = 5$ term (for the sign, see Subsection 2.4). This is just $b \cdot \theta _{1,0}[a]$ at $t = 1$ and $z = -\zeta $ , and it converges whenever $|\zeta |$ and $|a|$ are both less than one.

Thus, the top row of (4.6.2) is isomorphic to the map $\mathfrak {m} \to C$ sending a to $\vartheta (a) := \sum _{n \in \mathbf {Z}} (-\zeta )^n a^{p^{-n}}$ . This map obeys

$$ \begin{align*} \vartheta(a^p) = (-\zeta) \vartheta(a); \end{align*} $$

that is, it intertwines the $\mathbf {F}_p(\!(z)\!)$ -module structure on C given by the homomorphism $z \mapsto -\zeta $ with the $\mathbf {F}_p(\!(z)\!)$ -module structure on $\mathfrak {m}$ given by $(z,a) \mapsto a^p$ . The kernel is therefore an $\mathbf {F}_p(\!(z)\!)$ -module. The image of this kernel under the isomorphism $\mathfrak {m} \cong B^{\varphi = z}$ (given by $a \mapsto \sum a^{p^i} z^{-i}$ (1.14.4)) is the set of $b \in B^{\varphi = z}$ whose set of zeroes is exactly $\{(-\zeta )^{p^n}\}_{n \in \mathbf {Z}}$ . The function

$$ \begin{align*} h(z) = \left(\sum_{i \in \mathbf{Z}} a^{p^i} z^{-i}\right)\left(\prod_{n = 0}^{\infty} (1 + \zeta^{p^n}/z)\right)^{-1} \end{align*} $$

(the meromorphic part of the Weierstrass factorisation [Reference Fargues and FontaineFF, Ch. 2]) belongs to $C(\!(z)\!)$ and obeys the functional equation $h(z^{1/p})^p = (\zeta + z) h(z)$ ; that is, its coefficients obey the recursion

(4.6.4) $$ \begin{align} h_n^p - \zeta h_n = h_{n-1}; \end{align} $$

for each $h_{n-1}$ there are exactly p solutions in $h_n$ to (4.6.4), so the set of such $h(z)$ is a 1-dimensional $\mathbf {F}_p(\!(z)\!)$ -submodule of $C(\!(z)\!)$ .

4.7 Ore adjoint

Let $L_{(\infty )}^{\zeta }$ be as in Subsection 4.6. If we swap the order of $L_{(\infty )}$ and $L_{(m)}$ in (4.6.1), the Maslov index of the intersection point is $1$ , so that

$$ \begin{align*} \mathrm{CF}(L_{(\infty)}^{\zeta},L_{(m)};\underline{C}) = \mathrm{CF}^1(L_{(\infty)}^{\zeta},L_{(m)};\underline{C}) = e_m \cdot C. \end{align*} $$

The triangle sum

$$ \begin{align*} \mathrm{CF}^1(L_{(\infty)}^{\zeta},L_{(0)};\underline{C}) \times \mathrm{CF}^0(L_{(1)},L_{(\infty)}^{\zeta};\underline{C}) \dashrightarrow \mathrm{CF}^1(L_{(1)},L_{(0)};\underline{C}) \end{align*} $$

is formally given by

(4.7.1) $$ \begin{align} \sum_{n \in \mathbf{Z}} (-1)^{3n} \sigma^{-n}(b \zeta^n a) = \sum_{n \in \mathbf{Z}} (-\zeta)^{n p^n}{(ab)^{p^n}}. \end{align} $$

It is the same triangles as (4.6.3) that contribute to (4.7.1), but they are decorated differently. For instance, the triangle contributing the $n = 5$ summand is

Even if $|\zeta | < 1$ , the $n \to -\infty $ tail of (4.7.1) does not converge unless $ab = 0$ . Even so, it is interesting in a formal way. In [Reference PoonenPoon], Poonen following [Reference OreOre] attached to each series of the form $f(a) = \sum u_n a^{p^n}$ an ‘adjoint’ series $f^{\dagger }(a) := \sum u_{-n}^{p^n} a^{p^n}$ – let us call it the Ore adjoint. Evidently, (4.7.1) is exactly $\vartheta ^\dagger [ba]$ .

Under some hypotheses on f, Poonen showed that the kernels of f and $f^\dagger $ are Pontrjagin dual to each other in a canonical fashion. These hypotheses are not satisfied by $\vartheta (a)$ , but as $\operatorname {ker}(\vartheta )$ (being the additive group of a local field) is Pontrjagin self-dual and as $\vartheta ^\dagger $ does not converge in any case, we are perhaps free to speculate that ‘ $\operatorname {ker}(\vartheta ^\dagger )$ ’ is somehow morally isomorphic to $\mathbf {F}_p(\!(z)\!)$ . This speculation is consistent with mirror symmetry: on the Fargues–Fontaine curve there indeed is a short exact sequence

(4.7.2) $$ \begin{align} 0 \to \operatorname{Hom}(\mathcal{O}(1),\mathcal{O}(1)) \to \operatorname{Hom}(\mathcal{O}(1),\delta) \to \operatorname{Ext}^1(\mathcal{O}(1),\mathcal{O}) \to 0 \end{align} $$

coming from the resolution $\mathcal {O} \to \mathcal {O}(1)$ of the skyscraper sheaf $\delta $ and the vanishing of $\operatorname {Ext}^1(\mathcal {O}(1),\mathcal {O}(1)) = H^1({\mathit {FF}};\mathcal {O})$ . The kernel of (4.7.2) is naturally isomorphic to $H^0({\mathit {FF}};\mathcal {O})$ ; that is, to $\mathbf {F}_p(\!(z)\!)$ . The middle group is isomorphic to C and to $\mathrm {HF}^0(L_{(1)},L_{(\infty )}^\zeta ;\underline {C})$ . But we emphasise that $\operatorname {Ext}^1(\mathcal {O}(1),\mathcal {O})$ is not isomorphic to $\mathrm {HF}(L_{(1)},L_{(0)};\underline {C})$ , nor to any open subgroup of it.

4.8 Loud Floer cochains on $L_{(0)}$

Let $\{\phi ^s\}_{s \in \mathbf {R}}$ be as in (2.12.1):

(4.8.1) $$ \begin{align} \phi^s(x,y) = (x,y - s \cos(2\pi x)). \end{align} $$

We can try to compare $\mathrm {CF}(L,L';\underline {C})$ and $\mathrm {CF}(\phi ^s L,L';\underline {C})$ by specialising to $t = 1$ in (3.1.4). In some cases – for instance, if L and $L'$ are parallel to $L_{(0)}$ as in Subsection 3.6 – the summation (3.1.4) is finite and defines a map

$$ \begin{align*} \mathrm{CF}(L,L';\underline{C}) \to \mathrm{CF}(\phi^s L,L';\underline{C}) \end{align*} $$

without any problems. But this map is not always a quasi-isomorphism. The series defining the homotopy (3.1.5) may not converge at $t = 1$ – (3.6.2) is a vivid example of this.

We will analyse the continuation maps between the groups $\mathrm {CF}(\phi ^s L_{(0)},L_{(0)};\underline {C})$ . If n is an integer and $n < s < n+1$ , then $L_{(0)}$ meets $\phi ^s L_{(0)}$ in $2n+2$ points. In the fundamental domain $[0,1] \times [0,1]$ , half of them have x-coordinate $<0.5$ and half of them have x-coordinate $>0.5$ . They are linearly ordered by the x-coordinate, and after listing them in that order we will name them

$$ \begin{align*} z_{n}^{(s)},\ldots,z_{-n}^{(s)},\xi_{-n}^{(s)},\ldots,\xi_{n}^{(s)}. \end{align*} $$

More explicitly, $z_i^{(s)}$ and $\xi _i^{(s)}$ are the two solutions to $ i - s \cos (2 \pi x) = 0 $ ; that is, for a suitable branch of the inverse cosine function,

$$ \begin{align*} z_i^{(s)} = \frac{1}{2\pi} \arccos(i/s) \qquad \xi_i^{(s)} = 1- \frac{1}{2\pi} \arccos(i/s). \end{align*} $$

The case $1 < s < 2$ is shown in the diagram, along with orientations and stars:

The rules of Subsection 2.3 give each point $z_i^{(s)}$ the Maslov index $0$ and each $\xi _i^{(s)}$ the Maslov index $1$ , so that

$$ \begin{align*} \mathrm{CF}^0(\phi^s L_{(0)},L_{(0)};\underline{C}) = \bigoplus_{i = -n}^n z_i^{(s)}\cdot C \qquad \mathrm{CF}^1(\phi^s L_{(0)},L_{(0)};\underline{C}) = \bigoplus_{i = -n}^n \xi_i^{(s)} \cdot C. \end{align*} $$

Each $\xi _i^{(s)}$ is the output vertex of exactly two bigons and the other vertex of both bigons in $z_i^{(s)}$ . There is an ‘upward’ bigon whose boundary passes through $(0.5,s)+\mathbf {Z}^2$ and a ‘downward’ bigon whose boundary passes through $(0,-s)+\mathbf {Z}^2$ . If one places a star at (or close to, as in the figure above) $(0.5,s)+\mathbf {Z}^2$ and $(0,-s)+\mathbf {Z}^2$ , then the sign of every downward bigon is $1$ and the sign of every upward bigon is $-1$ . The downward bigons cross the danger line exactly once and the upward bigons exactly never, so that

$$ \begin{align*} \mu_1(z_i^{(s)} \cdot a) = \xi_i^{(s)} \cdot (-a + \sigma(a) ), \end{align*} $$

with the upward bigon contributing $-a$ and the downward bigon contributing $\sigma (a)$ .

If $s'> s$ , then for a suitable choice of profile function $\beta $ (one that is very close to the ‘linear cascades’ limit considered in [Reference AurouxAur1]), the continuation map

(4.8.2) $$ \begin{align} \mathrm{CF}(\phi^s L_{(0)},L_{(0)};\underline{C}) \to \mathrm{CF}(\phi^{s'} L_{(0)},L_{(0)};\underline{C}) \end{align} $$

simply sends $z_i^{(s)} \cdot a$ to $z_i^{(s')} \cdot a$ and $\xi _i^{(s)} \cdot a$ to $\xi _i^{(s')} \cdot a$ . In particular, it defines a filtered diagram of cochain complexes (indexed by $s> 0$ , $s \notin \mathbf {Z}$ , with respect to the usual ordering of real numbers s). Let $\mathrm {CF}_{\mathrm {loud}}(L_{(0)},L_{(0)})$ denote the direct limit of this diagram

$$ \begin{align*} \mathrm{CF}_{\mathrm{loud}}(L_{(0)},L_{(0)};\underline{C}) := \varinjlim_{s> 0| s \notin \mathbf{Z}} \mathrm{CF}(\phi^s L_{(0)},L_{(0)};\underline{C}). \end{align*} $$

Since each map (4.8.2) is the inclusion of a direct summand of cochain complexes, $\mathrm {CF}_{\mathrm {loud}}$ is a model for the homotopy colimit of cochain complexes as well. Explicitly,

(4.8.3) $$ \begin{align} \mathrm{CF}_{\mathrm{loud}}^0 = \bigoplus_{i \in \mathbf{Z}} z_i \cdot C \qquad \mathrm{CF}_{\mathrm{loud}}^1 = \bigoplus_{i \in \mathbf{Z}} \xi_{i} \cdot C \qquad \mu_1(z_i \cdot a) = \xi_i \cdot (-a+\sigma(a)). \end{align} $$

4.9 Triangles between the $\phi ^s L_{(0)}$

Continuing with the notation of Subsection 4.8, let us suppose that none of s, $s'$ and $s+s'$ are in $\mathbf {Z}$ and describe the triangles between $L_{(0)},\phi ^{s} L_{(0)}$ and $\phi ^{s+s'}L_{(0)}$ . The ‘output’ corners of these triangles are

$$ \begin{align*}z_i^{(s+s')} \text{ and }\xi_i^{(s+s')}\text{ on }L_{(0)} \cap \phi^{s+s'} L_{(0)},\end{align*} $$

and the other two corners in counterclockwise order are

$$ \begin{align*} \phi^{s} z_i^{(s')}, \phi^{s} \xi_i^{(s')} \in \phi^{s} L_{(0)} \cap \phi^{s+s'} L_{(0)}, \qquad z_i^{(s)},\xi_i^{(s)} \in L_{(0)} \cap \phi^s L_{(0)}. \end{align*} $$

For each such triangle there is a unique pair of integers i and j so that the triangle lifts to $\mathbf {R}^2$ with boundary on the x-axis, the graph of $y = i - s \cos (2 \pi x)$ and the graph of $y = i+j - (s+s') \cos (2\pi x)$ . The only nonempty moduli spaces of triangles are

(4.9.1) $$ \begin{align} \mathcal{M}\left(z_{i+j}^{(s+s')},\phi^{s} z_j^{(s')},z_i^{(s)}\right) \quad \mathcal{M}\left(\xi_{i+j}^{(s+s')},\phi^{s} z_j^{(s')},\xi_i^{(s)}\right) \quad \mathcal{M}\left(\xi_{i+j}^{(s+s')},\phi^{s} \xi_j^{(s')},z_i^{(s)}\right) \end{align} $$

with $-s < i < s$ and $-s' < j < s'$ . When $i/s = j/s'$ , there is something tricky about the latter two moduli spaces (they are not transversely cut; Subsection 2.2), so let us for a moment assume that $i/s \neq j/s'$ . Then each space (4.9.1) contains exactly one triangle. The nature of this triangle depends on which of $i/s$ or $j/s'$ is larger.

The triangle of $\mathcal {M}\left (z_{i+j}^{(s+s')},\phi ^{s} z_j^{(s')},z_i^{(s)}\right )$ is the one bounded by

(4.9.2) $$ \begin{align} \max(0,(i+j) - (s+s')\cos(2\pi x)) \leq y \leq i - s \cos(2 \pi x) &\qquad \text{if}\ j/s' < i/s \end{align} $$

(4.9.3) $$ \begin{align} \min(0,(i+j) - (s+s') \cos(2\pi x)) \geq y \geq i - s\cos(2 \pi x) &\qquad \text{if}\ j/s'> i/s. \end{align} $$

For legibility, in the following illustration of these triangles the curves $\phi ^s L_{(0)}$ and $\phi ^{s+s'} L_{(0)}$ are drawn in a different aspect ratio than in the diagram of Subsection 4.8, vertically compressed. The figure is drawn in a union of $\sim s+s'$ fundamental domains, stacked on top of each other.

In this and the following diagrams, $\phi ^{s+s'}L_{(0)}$ is purple, $\phi ^s L_{(0)}$ is blue and $L_{(0)}$ is black. The left side shows the typical case when $i/s> j/s'$ and the right side shows the typical case when $i/s < j/s'$ .

In $\mathcal {M}\left (\xi _{i+j}^{(s+s')},\phi ^{s} z_j^{(s')},\xi _i^{(s)}\right )$ , we have the triangle

(4.9.4) $$ \begin{align} \begin{array}{cl} (4.9.2) \cup \{ \min(0,i - s\cos(2 \pi x)) \geq y \geq (i+j) - (s+s') \cos(2 \pi x) \} & \text{if } \frac{j}{s'}<\frac{i}{s} \\ (4.9.3) \cup \{ \max(0,i - s \cos(2 \pi x)) \leq y \leq (i+j) - (s+s') \cos(2 \pi x) \} & \text{if }\frac{j}{s'}>\frac{i}{s} \end{array} \end{align} $$

In $\mathcal {M}\left (\xi _{i+j}^{(s+s')},\phi ^{s} \xi _j^{(s')},z_i^{(s)}\right )$ , we have the triangle

(4.9.5) $$ \begin{align} \begin{array}{cl} (4.9.2) \cup \{ \min(i - s \cos(2 \pi x), i+j- (s+s') \cos(2\pi x))) \geq y \geq 0 \} & \text{if }\frac{j}{s'} <\frac{i}{s} \\ (4.9.3) \cup \{ \max(i - s\cos(2 \pi x), i+j - (s+s') \cos(2\pi x)) \leq y \leq 0 \} & \text{if } \frac{j}{s'}>\frac{i}{s} \end{array} \end{align} $$

Now we discuss the triangles with $i/s = j/s'$ . For generic s and $s'$ , it is only possible that $i/s = j/s'$ when $i = j = 0$ . In that case, $\mathcal {M}\left (z_0^{(s+s')},\phi ^s z_0^{(s')}, z_0^{(s)}\right )$ again contains a single point (the constant map with value $z_0 = (.25,0)$ ) and is again transversely cut, but these two assertions are not true for $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} z_0^{(s')},\xi _0^{(s)}\right )$ or for $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} \xi _0^{(s')},z_0^{(s)}\right )$ . These spaces each contain two points, which are degenerate triangles (they are at the boundary of the Deligne–Mumford–Stasheff compactification) which are not maps out of a triangle but out of a wedge sum of a triangle and a bigon:

The degenerate maps in $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} z_0^{(s')},\xi _0^{(s)}\right )$ and $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} \xi _0^{(s')},z_0^{(s)}\right )$ collapse the triangle part to a point (to $\xi _0 = (.75,0)$ ) but are nontrivial along the bigon:

(4.9.6)

The top two figures indicate the two points of $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} z_0^{(s')},\xi _0^{(s)}\right )$ and the bottom two are the two points of $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} \xi _0^{(s')},z_0^{(s)}\right )$ . Though they are not transversely cut, they have analytic index 0 – more precisely, they have index $+1$ along the constant triangle and index $-1$ along the bigon.

4.10 Triangle products on $\mathrm {CF}_{\mathrm {loud}}$

For short, let us put $A^{(s)} := \mathrm {CF}(\phi ^s L_{(0)},L_{(0)};\underline {C})$ . The triangles in the previous section, together with the identification $\mathrm {CF}(\phi ^{s+s'},\phi ^{s} L_{(0)};\underline {C})$ of and $\mathrm {CF}(\phi ^{s'} L_{(0)},L_{(0)};\underline {C})$ , give a multiplication $A^{(s)} \times A^{(s')} \to A^{(s+s')}$ . Specifically,

• • (Coming from (4.9.2) and (4.9.3))

  $$ \begin{align*} (z_i^{(s)} \cdot a, z_j^{(s')} \cdot b) \mapsto z_{i+j}^{(s+s')} \cdot ab\end{align*} $$

• • (Coming from (4.9.4))

  (4.10.1) $$ \begin{align} (\xi_i^{(s)} \cdot a,z_j^{(s')} \cdot b) \mapsto \xi_{i+j}^{(s+s')}\cdot \begin{cases} a\sigma(b) & \text{if}\ j/s' < i/s \\ ab & \text{if}\ j/s'> i/s \end{cases} \end{align} $$

• • (Coming from (4.9.5))

  (4.10.2) $$ \begin{align} (z_i^{(s)} \cdot a, \xi_j^{(s')} \cdot b) \mapsto \xi_{i+j}^{(s+s')} \cdot \begin{cases} ab & \text{if}\ j/s' < i/s \\ \sigma(a)b & \text{if}\ j/s'> i/s \end{cases} \end{align} $$

Since $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} z_0^{(s')},\xi _0^{(s)}\right )$ and $\mathcal {M}\left (\xi _{0}^{(s+s')},\phi ^{s} \xi _0^{(s')},z_0^{(s)}\right )$ are not transversely cut, they cannot be oriented in the usual way (Subsection 2.4). Instead, each space carries a virtual fundamental class which (depending on how it is constructed) may attach any integer to each of the four points of $\mathcal {M}$ . We will not develop the virtual fundamental class here but (to some extent following [Reference SeidelSe1, Section 7] addressing a similar issue) simply define

(4.10.3) $$ \begin{align} (\xi_0^{(s)} \cdot a, z_0^{(s')} \cdot b) \mapsto \xi_0^{(s+s')} \cdot ab \qquad (z_0^{(s)} a, \xi_0^{(s')} b) \mapsto \xi_0^{(s+s')} \cdot \sigma(a)b \end{align} $$

as though the left two degenerate triangles displayed in (4.9.6) contributed nothing. The definition (4.10.3) can be justified informally by introducing a Hamiltonian perturbation $\psi $ of $L_{(0)}$ (but not $\phi ^s L_{(0)}$ or $\phi ^{s+s'} L_{(0)})$ supported in a very small neighbourhood of $z_0$ and $\xi _0$ . In that case, all moduli spaces are transversely cut and the triangle products $\mu _2(\psi z_0^{(s)} \cdot a, \phi ^s \xi _0^{(s')} \cdot b)$ and $\mu _2(\psi \xi _0^{(s)} \cdot a,\phi ^s z_0^{(s')} \cdot b)$ are well-defined, though the specific formula will depend on $\psi $ – (4.10.3) is consistent with some of these $\psi $ .

Now we use the products $A^{(s)} \times A^{(s')} \to A^{(s+s')}$ to define a multiplication on $\lim _s A^{(s)}$ ; that is, on $ \mathrm {CF}_{\mathrm {loud}}(L_{(0)},L_{(0)};\underline {C})$ . It is not quite straightforward, because the products (4.10.1) and (4.10.2) are not eventually constant as s and $s'$ grow – it depends on which of $i/s$ and $j/s'$ is larger. The square

(4.10.4)

does not commute for all $s,s',S,S'$ with $s < S$ and $s' < S'$ . We address this in the following crude way: we choose an irrational number $e>0$ and note that since $i/s - j/(es)$ has constant sign for $s> 0$ , (4.10.4) does commute when $s' = es$ and $S' = eS$ . The induced multiplication on the colimit is explicitly $(z_i \cdot a, z_j \cdot b) \mapsto z_{i+j} \cdot ab$ and

$$ \begin{align*} (\xi_i \cdot a,z_j \cdot b) \mapsto \xi_{i+j} \cdot \begin{cases} a \sigma(b) & \text{if}\ j/e < i \\ ab & \text{if}\ j/e \geq i \\ \end{cases} \qquad (z_i \cdot a, \xi_j \cdot b) \mapsto \xi_{i+j} \cdot \begin{cases} ab & \text{if}\ j/e < i \\ \sigma(a)b & \text{if}\ j/e \geq i. \end{cases} \end{align*} $$

Thus, we get one binary operation on $\mathrm {CF}_{\mathrm {loud}}(L_{(0)},L_{(0)};\underline {C})$ for every irrational $e> 0$ . These multiplications are genuinely different for different e. Moreover, they are not associative; they do, however, obey the Leibniz rule $\mu _1(w w') = \mu _1(w) w' + w \mu _1(w')$ , with $\mu _1$ as in (4.8.3). It is likely that they can be extended to an $A_{\infty }$ -structure on $\mathrm {CF}_{\mathrm {loud}}(L_{(0)},L_{(0)};\underline {C})$ (and even more likely that there is such an $A_{\infty }$ -structure on a complex quasi-isomorphic to it, defined along the lines of [Reference Abouzaid and SeidelAS]), but we will not construct it. Instead, we simply note that the induced multiplication on $\mathrm {HF}^0_{\mathrm {loud}} := \operatorname {ker}(\mu _1)$ (and even on $\mathrm {HF}^0_{\mathrm {loud}} \oplus \mathrm {HF}^1_{\mathrm {loud}}$ , though the degree $1$ part vanishes if C is algebraically closed and $\sigma $ is the pth root map) is associative and independent of e. Indeed, it is simply the Laurent polynomial ring $C^{\sigma }[z^{\pm 1}]$ under the assignment $\sum c_i z^i \mapsto \sum z_i \cdot c_i$ .