Coherent orientations in symplectic field theory revisited

Given a decorated Riemann surface $(\Sigma,j,\boldsymbol{p},\boldsymbol{r})$, we regard its marked points as punctures and find holomorphic cylindrical coordinates around them. Let $\phi_{i}^{\pm}:D\subset\mathbb{C}\to\mathcal{U}_{p_{i}^{\pm}}\subset\Sigma$ be a biholomorphic map from the unit disc $D$ to a neighborhood $\mathcal{U}_{p_{i}^{\pm}}$ of $p^{\pm}_{i}$ such that $\phi_{i}^{\pm}(o)=p_{i}^{\pm}$ and $d\phi_{i}^{\pm}(\frac{\partial}{\partial x})=r^{\pm}_{i},$ where $o\in D$ is the origin and $\frac{\partial}{\partial x}\in T_{o}D.$ Let $\dot{\mathcal{U}}_{p_{i}^{\pm}}={\mathcal{U}}_{p_{i}^{\pm}}-\{p_{i}^{\pm}\},$ and $h_{i}^{\pm}:\mathbb{R}^{\geq 0(\leq 0)}\times S^{1}\to\dot{\mathcal{U}}_{p_{i}^{\pm}}$ be the biholomorphic map defined by

Given an admissible loop $S$, we obtain a path of symplectic matrices $B(t)\in\operatorname{Symp}(2n-2,\mathbb{R})$ that solves $\dot{B}(t)=J_{0}S(t)B(t)$ and $B(0)=\operatorname{Id}.$ We define the Conley-Zenhder index of $S$ by $\mu_{\operatorname{CZ}}(S)=\mu(\{B(t)\}_{t})$, where $\mu$ is the Maslov index, and we grade $S$ over $\mathbb{Z}_{2}$ by $|S|=\mu_{\operatorname{CZ}}(S)+(n-1)\mod 2.$

For an admissible loop $S$, we define $\lambda_{S}=\min\{-\lambda_{-1},\lambda_{1}\}>0$, where $\lambda_{1}$ is the smallest positive eigenvalue of $J_{0}\frac{\partial}{\partial t}+S$, and $\lambda_{-1}$ is the largest negative eigenvalue of $J_{0}\frac{\partial}{\partial t}+S$.

Fix $k\in\mathbb{Z}^{\geq 0}$, and $p>1$ such that $kp>2.$

Note that the third term $\widetilde{S}_{i}^{\pm}(s,t)\eta\otimes(ds-\sqrt{-1}dt)$ on the right hand side of Formula (2.1.4) is the $0$-th order operator as in (3).

We define the $\operatorname{mod}2$ indices

and

It is convenient to state the following lemma:

Given a continuous family of Fredholm operators $\{\phi_{\tau}\}_{\tau\in[0,1]}$, we can find a finite-dimensional space $F\subset W$ and a subspace $U\subset W$ such that $W=F\oplus U$ and $W=\operatorname{im}(\phi_{\tau})+F$. Let $\pi_{U}:W\to U$ be the projection map. Then the map $\pi_{U}\circ\phi_{\tau}:V\to U$ is surjective, and $\ker(\pi_{U}\circ\phi_{\tau})=\phi_{\tau}^{-1}(F).$ By Lemma 2.11, we have $\det\phi_{\tau}\simeq\wedge^{\operatorname{top}}(\phi_{\tau}^{-1}(F))\otimes\wedge^{\operatorname{top}}F^{*}\simeq\wedge^{\operatorname{top}}\ker(\pi_{U}\circ\phi_{\tau})\otimes\wedge^{\operatorname{top}}F^{*}$. Since both $\ker(\pi_{U}\circ\phi_{\tau})$ and $F^{*}$ form vector bundles over $[0,1]$, $\det\phi_{\tau}$ forms a line bundle over $[0,1]$. More generally, a homotopy of CR tuples $\{L_{\tau}\}_{0\leq\tau\leq 1}$ induces an isomorphism $\det L_{0}\simeq\det L_{1}$.

Since the space $\mathfrak{D}(\mathcal{T})$ is contractible, $\det L$ and $\det L^{\prime}$ are canonically isomorphic for any $L,L^{\prime}\in\mathfrak{D}(\mathcal{T})$. For this reason, we also write $\det\mathcal{T}$, $o(\mathcal{T})$, and $o_{\operatorname{can}}(\mathcal{T})$.

The disjoint union $\mathcal{T}\coprod\mathcal{T}^{\prime}$ of two CR tuples $\mathcal{T}$ and $\mathcal{T}^{\prime}$ is defined in the obvious way. The punctures of $\dot{\Sigma}\coprod\dot{\Sigma}^{\prime}$ are ordered so that the punctures of $\dot{\Sigma}$ come before those of $\dot{\Sigma}^{\prime}$, and the relative orders of the punctures of $\dot{\Sigma}$ and $\dot{\Sigma}^{\prime}$ are preserved, respectively. For any $L\in\mathfrak{D}(\mathcal{T})$ and $L^{\prime}\in\mathfrak{D}(\mathcal{T}^{\prime})$, the disjoint union map induces an isomorphism

where $L\coprod L^{\prime}\in\mathfrak{D}(\mathcal{T}\coprod\mathcal{T}^{\prime})$ is defined as follows: Let $\{e_{1},\dots,e_{n}\}$ be a basis of $\ker L$, $\{f_{1},\dots,f_{m}\}$ be a basis of $\operatorname{coker}L$, $\{e^{\prime}_{1},\dots,e^{\prime}_{n^{\prime}}\}$ be a basis of $\ker L^{\prime}$, $\{f^{\prime}_{1},\dots,f^{\prime}_{m^{\prime}}\}$ be a basis of $\operatorname{coker}L^{\prime}$. Then the isomorphism is given by:

In the case when both $L$ and $L^{\prime}$ are surjective, $L\coprod L^{\prime}$ is also surjective, and the isomorphism simplifies to

The isomorphism (2.2.1) is continuous with respect to the homotopy of CR operators $L$ and $L^{\prime}$, hence induces a map on CR tuples $\mathcal{T}$ and $\mathcal{T}^{\prime}$:

and we denote the image of $v\otimes v^{\prime}$ by $v\coprod v^{\prime}$.

Note that $\det(\mathcal{T}\coprod\mathcal{T}^{\prime})$ can be canonically identified with $\det(\mathcal{T}^{\prime}\coprod\mathcal{T})$ by identifying the disconnected Riemann surface $\Sigma\coprod\Sigma^{\prime}$ with $\Sigma^{\prime}\coprod\Sigma$, and identifying the bundles $E\coprod E^{\prime}$ with $E^{\prime}\coprod E$ in the obvious way. Under such identification, for any $v\in\det\mathcal{T}$ and $v^{\prime}\in\det\mathcal{T}^{\prime}$, $v\coprod v^{\prime}$ and $v^{\prime}\coprod v$ lie in the same vector space. The following lemma is clear from the above definition.

Given two CR tuples $\mathcal{T}$ and $\mathcal{T}^{\prime}$, and $L\in\mathfrak{D}(\mathcal{T})$ and $L^{\prime}\in\mathfrak{D}(\mathcal{T}^{\prime})$, a positive integer $\tau\leq\operatorname{min}(k_{-},k_{+}^{\prime})$, and a sufficiently large gluing parameter $R>0$, if the first $\tau$ negative ends of $\dot{\Sigma}$ match the last $\tau$ positive ends of $\dot{\Sigma}^{\prime}$, i.e., for $1\leq i\leq\tau$,

then we can glue $\mathcal{T}$ and $\mathcal{T}^{\prime}$ to obtain a new CR tuple $\mathcal{T}^{\prime\prime},$ and glue $L$ and $L^{\prime}$ to obtain $L^{\prime\prime}\in\mathfrak{D}(\mathcal{T}^{\prime\prime})$. The gluing construction is straightforward: for each $1\leq i\leq\tau$ we “chop off” the end $(-\infty,-2R]\times S^{1}$ from $\dot{\Sigma}$ around $p_{i}^{-}$, and the end $[2R,\infty)\times S^{1}$ from $\dot{\Sigma}^{\prime}$ around $p^{+}_{k^{\prime}_{+}-\tau+i}$ , identify the regions $[-2R,-R]\times S^{1}\subset\dot{\Sigma}$ and $[R,2R]\times S^{1}\subset\dot{\Sigma}^{\prime}$, and the bundles $E$ and $E^{\prime}$ above these regions, and interpolate $L$ and $L^{\prime}$ over the identified regions. We now explain how $\boldsymbol{p}^{\prime\prime}$ is ordered. The positive (negative) marked points of $\boldsymbol{p}^{\prime}$ are ordered before the positive (negative) marked points of $\boldsymbol{p}$, more precisely, $\boldsymbol{p}^{\prime\prime}_{+}=({p^{\prime}}_{+}^{1},\dots,{p^{\prime}}_{+}^{k^{\prime}_{+}-\tau},{p}_{+}^{1},\dots,{p}_{+}^{k_{+}})$ and $\boldsymbol{p}_{-}^{\prime\prime}=({p^{\prime}}_{-}^{1},\dots,{p^{\prime}}_{-}^{k^{\prime}_{-}},p_{-}^{\tau+1},\dots,p_{-}^{k_{-}})$. We denote $\mathcal{T}^{\prime\prime}=\mathcal{T}\sharp_{\tau,R}\mathcal{T}^{\prime}$ and $L^{\prime\prime}=L\sharp_{\tau,R}L^{\prime}$.

We follow the complete gluing convention in [HT09] by restricting the gluing to the case when $k_{-}=k_{+}^{\prime}=\tau$, i.e., all the negative punctures of $\dot{\Sigma}$ match with all the positive punctures of $\dot{\Sigma}^{\prime}$ and we glue along all of them. For complete gluing we write $\mathcal{T}^{\prime\prime}=\mathcal{T}\sharp_{R}\mathcal{T}^{\prime}$ and $L^{\prime\prime}=L\sharp_{R}L^{\prime}$. To get a non-complete gluing from complete gluing, one can add a few trivial CR tuples before gluing in the obvious way. See Figure (1).

From now on, all gluings are assumed to be complete unless otherwise specified.

Because of Lemma 2.14, we write $\sharp$ instead of $\sharp_{R}$, meaning gluing with some unspecified gluing parameter. The gluing map induces an isomorphism

and we denote the image of $v\otimes v^{\prime}$ under the isomorphism as $v\sharp v^{\prime}$.

To construct a coherent orientation system for all CR tuples $\mathcal{T}$ with loops of admissible symmetric matrices from $\{S_{1},S_{2},\dots\}$, we follow the method outlined in [BM04]. This involves the following steps:

1. (1)

   For each loop of admissible symmetric matrices $S$, we perform the following:

   1. (a)

      Choose a CR tuple $\mathcal{T}^{+}_{S}$ with one positive puncture, zero negative punctures, and the associated admissible loop $S$.

   2. (b)

      Select an orientation $o(\mathcal{T}^{+}_{S})$.

   3. (c)

      Pick a CR tuple $\mathcal{T}^{-}_{S}$ with zero positive puncture and one negative puncture and the associated admissible loop being $S$.

   4. (d)

      Determine the orientation $o(\mathcal{T}^{-}_{S})$ by gluing $\mathcal{T}^{+}_{S}$ and $\mathcal{T}^{-}_{S}$ together. Specifically, the equation $o(\mathcal{T}^{+}_{S})\sharp o(\mathcal{T}^{-}_{S})=o_{\operatorname{can}}(\mathcal{T}^{+}_{S}\sharp\mathcal{T}^{-}_{S})$ uniquely determines $o(\mathcal{T}^{-}_{S})$.

We refer to $\mathcal{T}_{S}^{\pm}$ as the positive (negative) capping CR tuple and $o(\mathcal{T}^{\pm}_{S})$ as the positive (negative) capping orientation.

1. (2)

   For an arbitrary CR tuple $\mathcal{T}$, we define $\mathcal{T}^{\pm}_{\boldsymbol{S}^{\pm}}$ as the CR tuple obtained by taking the iterated disjoint union of $\mathcal{T}^{\pm}_{S^{\pm}_{1}}\coprod\dots\coprod\mathcal{T}^{\pm}_{S^{\pm}_{k^{\pm}}}$, where $S^{\pm}_{i}$ is the admissible loop of symmetric matrices at the $i$-th positive (negative) puncture. Then, the orientation $o(\mathcal{T}^{\pm}_{\boldsymbol{S}^{\pm}})$ is determined by the equation:

   (2.4.1)

   $$o(\mathcal{T}^{\pm}_{\boldsymbol{S}^{\pm}})=\epsilon^{\coprod}\cdot o(\mathcal{T}^{\pm}_{S_{1}^{\pm}})\coprod\dots\coprod o(\mathcal{T}^{\pm}_{S_{k^{\pm}}^{\pm}}),$$

   where $\epsilon^{\coprod}=\epsilon^{\coprod}(\mathcal{T}^{\pm}_{S_{1}^{\pm}},\dots,\mathcal{T}^{\pm}_{S_{k^{\pm}}^{\pm}})\in\{\pm 1\}$ is to be chosen.

2. (3)

   Finally, we glue $\mathcal{T}^{+}_{\boldsymbol{S}^{+}}$, $\mathcal{T}$, and $\mathcal{T}^{-}_{\boldsymbol{S}^{-}}$ together to obtain a CR tuple with no punctures. The orientation $o(\mathcal{T})$ is determined by the equation:

   (2.4.2)

   $$o(\mathcal{T}_{\boldsymbol{S}^{+}}^{+})\sharp o(\mathcal{T})\sharp o(\mathcal{T}^{-}_{\boldsymbol{S}^{-}})=\epsilon^{\sharp}\cdot o_{\operatorname{can}}(\mathcal{T}_{\boldsymbol{S}^{+}}^{+}\sharp\mathcal{T}\sharp\mathcal{T}^{-}_{\boldsymbol{S}^{-}}),$$

   where $\epsilon^{\sharp}=\epsilon^{\sharp}(\mathcal{T}^{+}_{\boldsymbol{S}^{+}},\mathcal{T},\mathcal{T}^{-}_{\boldsymbol{S}^{-}})\in\{\pm 1\}$ is to be chosen.

The choices of $\epsilon^{\coprod}$ and $\epsilon^{\sharp}$ in [BM04] and in [HT09] are different, resulting in different properties of the coherent orientations. We list the properties and the choices of $\epsilon^{\coprod}$ and $\epsilon^{\sharp}$ of coherent orientations $o_{bm}$ in [BM04] and $o_{ht}$ in [HT09] below. The signs $\epsilon^{\coprod}$ and $\epsilon^{\sharp}$ in Formulas (2.4.1) and (2.4.2) come from disjoint union and gluing of multiple (possibly greater than two) CR tuples, respectively. They are determined inductively by the signs that come from disjoint union and gluing two CR tuples.

The gluing convention in this paper is different from that of [BM04]²²2The matching ordering between the negative punctures of $\dot{\Sigma}$ and the positive punctures of $\dot{\Sigma}^{\prime}$ is reversed in [BM04]., which is source of the extra sign $\epsilon^{\sharp}_{bm}$.

Let $\mathcal{T}^{\prime}$ be the same CR tuple as $\mathcal{T}$ except that the $i$-th and the $(i+1)$-th positive (or negative) punctures are swapped. Note that $\det\mathcal{T}$ does not depend on the ordering of the punctures of $\mathcal{T}$, so $\det\mathcal{T}=\det\mathcal{T}^{\prime}$. On the other hand, the coherent orientation depends on the ordering of the punctures. Indeed,