For a pseudoholomorphic polygon $u:\Sigma\to X$ with inputs at $p_{1},\ldots p_{k}$ and an output at $q$, arranged in clockwise order, the index is $\deg q-\sum_{1}^{k}\deg p_{k}$, where the degree convention is (63). The index amounts to a Maslov number computation, and an alternative topological description is as follows: take a section $s$ of the complex line bundle $\Lambda^{n}TM\to\Sigma$, which restricts on $\partial\Sigma$ to a section of the real line bundle $\Lambda^{n}TL$. (When several Lagrangians are involved, it is understood that $TL$ refers to the appropriate Lagrangian on the portion of $\partial\Sigma$.) We assume $s$ has isolated zeros up to the boundary and the corner (aka. strip like ends). We may also regard $s$ as a function $\Omega(s)$ on the polygon, by contraction with $\Omega$. Then

Here the order of zeros is computed from winding numbers, and for general sections $s$ may take positive and negative values. At a corner where $\partial\Sigma$ passes from $L_{+}$ to $L_{-}$ in the clockwise direction, we can put the tangent spaces $TL_{\pm}\subset TX$ into the standard form respecting the complex structure

so if $\Omega(s)\sim z^{\alpha}$ in the complex coordinate of the upper half plane model, the excess vanishing order at the corner is $\frac{1}{\pi}(\alpha-\sum_{1}^{n}\phi_{i})$. Formula (23) is equivalent to the standard index formula by a topological version of the Cauchy residue formula.

We first consider the holomorphic strip case with input $p$ and output $q$, and the integrability of the complex structure will be important. The first order deformations of the holomorphic strips $\Sigma$ are given by holomorphic sections of $TX|_{\Sigma}$, which takes boundary value in $TL$ over $\partial\Sigma$, and decays at the corners.

We now move on to holomorphic polygons with $k+1$ corner points for $k\geq 2$. The extended linearized Cauchy-Riemann equation (cf. [69, Chapter 9]) involves a vector field $v\in C^{\infty}(\Sigma,u^{*}TX)$ decaying at the ends, and $\rho\in\Omega^{0,1}(\Sigma,T\Sigma)$ representing a tangent vector of the Stasheff associahedron (i.e. the deformation of Riemann surface structure on the domain $\Sigma$), satisfying

where $J_{X}$ is the complex structure on $X$. Here $\rho$ can be taken to be compactly supported, so $\bar{\partial}v=0$ near the corners. It immediately follows that

We impose that the input at one of the $k$ corners maps to an intersection point in $CF^{0}(L,L^{\prime})$, and the other inputs map to degree one self intersections of $L$ or $L^{\prime}$. The output maps to $q\in CF^{*}(L,L^{\prime})$.

We now discuss solutions to linearized Cauchy-Riemann equations in weighted Sobolev spaces $W^{1,2;\mu}$ (cf. [70, section 2]). These spaces agree with their unweighted counterparts along the strip like input ends, but at the strip like output end $s\gg 0$, a vector field $v\in W^{l,2;\mu}$ means that $\exp(-\mu s)v$ lies in $W^{l,2}$. Generally we choose $\mu$ to avoid a discrete set of indicial values. The main point of these weighted Sobolev spaces is that they allow for holomorphic vector fields with prescribed exponential growth along the output end, which is conceptually similar to allowing for meromorphic functions in Riemann surface theory. If we think of the strip like end $q$ as the infinity (resp. the origin) in the upper half plane model of $\Sigma$, then the natural coodinate is $z=e^{\pi(s+it)}$ (resp. $z=e^{-\pi(s+it)}$), and the exponential growth $o(e^{\mu s})$ becomes $o(|z|^{\mu/\pi})$ (resp. $o(|z|^{-\mu/\pi})$.

For larger $\mu$ more vector fields are included in the Sobolev space, and the index increases by one each time $\mu$ crosses an indicial value (counted with multiplicity). In our problem, the indicial values are

where $\phi_{1},\ldots\phi_{n}$ are the characterizing angles at the Lagrangian intersection point $q$ at the output end. Then the index for the linearized Cauchy-Riemann operator $W^{1,2;\mu}\to L^{2,\mu}$ is

where $k$ is the number of input ends. In particular, for holomorphic strips with $\deg p=\deg q$ (resp. $\deg q-\deg p=1$), then the index for $\mu=\pi$ is equal to $n$ (resp. $n+1$). In contrast, the ordinary index (for the $\mu=0$ case) is zero, and the moduli space obtained by taking $\mathbb{R}$-quotient has virtual dimension $-1$ (resp. zero). There are in fact sufficient conditions to rule out the negative dimension moduli spaces, and constrain the zero dimensional moduli spaces:

A very analogous statement holds in the polygon case, and is left to the reader:

A similar statement applies to teardrop curves:

The above lemma describes the optimal case for teardrop curves. Such $\deg q=2$ teardrop curves arise in isolated zero dimensional moduli spaces after taking the $Aut(D^{2},q)$ quotient, and the counting contribution to $m_{0}$ are $\pm 1$ depending on the spin structure and the orientation issues.

Let $\Sigma$ be a holomorphic polygon with $k\geq 0$ input ends $p_{i}$, and one output end at $q$. The case $k=0$ corresponds to teardrops, and $k=1$ corresponds to strips. We consider the ordinary linearized Cauchy-Riemann operator in weighted Sobolev spaces $W^{1,2;\mu}$, to classify the structure of the first order deformation theory. As usual, the complex structure is integrable. Since $\bar{\partial}$ is elliptic, its cokernel in $L^{2}$ is finite dimensional, represented by holomorphic 1-forms on $\Sigma$, which must have finite order of vanishing at $q$. For large enough $\mu$, the dual space $L^{2;-\mu}$ for $L^{2;\mu}$ imposes an exponential decay condition $O(e^{-\mu s})$ at $q$, so the cokernel evantually vanishes for $\mu\gg 1$. Then the kernel dimension in $W^{1,2;\mu}$ is equal to the index, computed by (25). For convenience, we use $\mu\in\pi\mathbb{N}$, which avoids the indicial values. Then

It is convenient to view the domain $\Sigma$ of the holomorphic polygon as the upper half plane with coordinate $z$, with corners $p_{i}$ on the real line and $q$ at infinity.

Let $p\in\partial\Sigma\simeq\partial\mathbb{H}$, and let $K$ be the maximal number depending on $p$, such that there exist $\mathbb{R}$-linearly independent $v_{1},\ldots v_{K}\in\ker\bar{\partial}$, satisfying

• •

  In case $p$ is not a corner point, then $v_{1}(p),\ldots,v_{K}(p)$ are $\mathbb{R}$-linearly independent vectors,

• •

  In case $p=p_{i}$ is a corner point, then the nonzero elements in the $\mathbb{R}$-span of $v_{1},\ldots v_{K}$ are vector fields vanishing minimally at $p$.

We can also apply a Möbius transform to make the output end $q$ lie at the origin. The growth condition translates to $o(|z|^{-\mu/\pi})$ at zero. Let $K$ be maximal, such that there are $\mathbb{R}$-linearly independent vector fields $z^{-\mu/\pi}v_{1},\ldots,z^{-\mu/\pi}v_{K}\in\ker\bar{\partial}\subset W^{1,2;\mu}$, and any nonzero element in the $\mathbb{R}$-span of $v_{1},\ldots v_{K}$ vanishes mimimally at $q=0$. As a caveat, this does not assume $v_{1},\ldots v_{K}$ satisfy the growth constraints at infinity to lie in $\ker\bar{\partial}\subset W^{1,2;\mu}$. Minor adaptions give

We view the boundary $\partial\Sigma\simeq\mathbb{P}^{1}(\mathbb{R})$. By the above lemmas, there is a real algebraic vector bundle $\mathcal{E}$ of rank $K$ over $\mathbb{P}^{1}(\mathbb{R})$ such that $v_{1},\ldots v_{K}$ provide the basis of local sections. By Grothendieck’s classification of vector bundles,

Since the rank $K$ is nondecreasing in $\mu$, it evantually stabilizes for $\mu\gg 0$. Since around any given point, the same choice of $v_{1},\ldots v_{K}$ is valid for all large $\mu$, the algebraic vector bundle $\mathcal{E}$ is independent of $\mu\gg 0$. The elements of $\ker\bar{\partial}\subset W^{1,2;\mu}$ can be interpreted as global sections of $\mathcal{E}\otimes\mathcal{O}(\frac{\mu}{\pi})$. Thus for all large $\mu$,

Contrasting with the index formula (26),

The structure of $\mathcal{E}\simeq\oplus_{1}^{K}\mathcal{O}(n_{i})$ provides meromorphic sections $v_{1},\ldots v_{n}$ which are a basis of local sections on $\mathbb{R}\subset\mathbb{P}^{1}(\mathbb{R})$, and have excess vanishing orders $n_{1},\ldots n_{n}$ at $q$. Consider the function $\Omega(v_{1},\ldots v_{n})$. By construction, it has no boundary zero, and its excess corner vanishing order is $\sum_{1}^{n}n_{i}$. Comparing with the index formula (23),

Since all interior vanishing orders are nonnegative by holomorphicity,

The significance is that the algebraic vector bundle structure on $\mathcal{E}\to\mathbb{P}^{1}(\mathbb{R})$ now extends over the entire $\Sigma$. The $v_{1},\ldots v_{n}$ now provide the basis of local sections for the vector bundle $u^{*}TX|_{\Sigma}$. One upshot is that an algebraic structure arises on $u^{*}TX\to\Sigma$ from solving the Cauchy-Riemann equation with Lagrangian boundary:

In contrast, the Lagrangians are only assumed to be smooth, not necessarily real analytic.

To analyze obstructions, Serre duality motivates us to consider the dualized cokernel to the ordinary (unweighted, unextended) linearized Cauchy-Riemann operator. A dualized cokernel element $\eta$ is represented by a holomorphic 1-forms in $\Omega^{1,0}(\Sigma,u^{*}T^{*}X)$ with $L^{2}$ integrablity, and its $T^{*}X$ factor lies in the annilator of the $TL$ boundary condition. Equivalently, for all test vector fields $v\in W^{1,2}(\Sigma,TX,TL)$,

where $\langle,\rangle$ is the pairing of $TX$ with $T^{*}X$, and the wedge takes care of the forms on $\Sigma$. In the canonical form (28), this dualized cokernel is isomorphic to $\Gamma(\mathbb{RP}^{1},\oplus_{1}^{n}\mathcal{O}(-2-n_{i})).$ In particular,

For $k\geq 3$ the deformation of the holomorphic polygons is governed instead by the extended Cauchy-Riemann equation, since the punctured Riemann surface structure on $\Sigma$ is allowed to vary. The dualized cokernel of the extended Cauchy-Riemann operator, is the subspace of the dualized cokernel of the ordinary Cauchy-Riemann operator, which pairs trivially with $J_{X}\circ du\circ\rho$ for any $\rho$ representing some tangent vector of the Stasheff associahedron.

We now consider the parametrized moduli space of holomorphic curves over the infinite dimensional space of Hamiltonian deformations for the Lagrangian $L$. Infinitesimally around a holomorphic curve $\Sigma$, we have a Hamiltonian vector field $X_{H}$ defined by $\omega(X_{H},\cdot)=dH$, viewed as a $TX$-valued vector field over $\Sigma$. We are interested in whether the Hamiltonian deformation kills the cokernel of the ordinary Cauchy-Riemann operator. This question was first addressed by Oh [64]. The following account follows a similar strategy but differs in details.

Recall the ordinary Cauchy-Riemann operator maps $W^{1,2}(\Sigma,u^{*}TX,u^{*}TL)$ to $L^{2}(\Sigma,u^{*}TX\otimes T^{*(1,0)}\Sigma)$. The effect of Hamiltonian deformation is to enlarge the domain of the $\bar{\partial}$ operator, by including the vector fields $X_{H}$ for all the allowed Hamiltonians $H$. The question is to analyze the pairing of $\bar{\partial}X_{H}$ with the dualized cokernel elements.

The upshot is that by the Sard-Smale theorem, provided we can always ensure ‘somewhere boundary injectivity’ for any holomorphic disc in a given moduli space, then generic Hamiltonian perturbation would be able to achieve regularity for the moduli space.

We now comment on the gap between what we proved and the (weak version of) automatic transversality that we will later assume.

1. 1.

   Prop. 3.8, Lemma 3.5 and Cor. 3.6 establish the dichotomy for holomorphic discs $u:\Sigma\to X$ arising in virtual dimension $n-1$ moduli spaces, that either $u$ is automatically transverse, or $\Omega(\cdot,v_{1},\ldots v_{n-1})$ vanishes for any $(n-1)$ first order deformation vectors. This argument does not establish unperturbed regularity for the lower dimensional moduli spaces, so it is not completely clear if complex structure perturbations can be removed in the arguments for $\partial\mathcal{C}=L-L^{\prime}$ in section 2.9.

2. 2.

   For the bad curves, $\Omega(\cdot,v_{1},\ldots v_{n-1})$ vanishes identically as a 1-form on $\Sigma$, so at any point on the boundary, $v_{1},\ldots v_{n-1}$ and the tangent vector to $\partial\Sigma$ are $\mathbb{R}$-linearly dependent. Suppose for the moment that the moduli spaces are regular, then the boundary evaluation to $L\cup L^{\prime}$ for the bad curves arise in Hausdorff dimension at most $n-1$. Morever, since the Solomon functional is defined through $\int_{\mathcal{C}}\lambda\wedge\Omega$, and $\Omega$ vanishes around the bad curves, smoothness assumptions imply that the bad curves cannot contribute.

   When regularity assumptions are dropped, one needs to appeal to virtual techniques, so these conclusions require further justification. One problem is that the standard virtual perturbation techniques based on Kuranishi structures do not necessarily produce virtual cycles inside the original moduli spaces, but only inside their small neighbourhoods. This perturbation step destroys the identical vanishing of $\Omega$, by a small amount corresponding to the size of the perturbation. As one shrinks the size of the perturbations, one needs uniform mass bound on the virtual chains to justify that the integral contribution to $\int_{\mathcal{C}}\lambda\wedge\Omega$ from the bad curves actually converges to zero.

3. 3.

   Alternatively, one can hope to replace Lagrangians by arbitrarily small Hamiltonian perturbations to achieve transversality. This is mostly adequate for our purpose, except that one needs to justify the ‘somewhere boundary injectivity’ property (cf. Remark 3.8).