# Geometric Properties of Real Gromov-Witten Invariants

Penka Georgieva and Aleksey Zinger Partially supported by ERC Grant
ROGW-864919Partially supported by NSF grant DMS 2301493

###### Abstract

We collect geometric properties of the all-genus real Gromov-Witten theory and
provide updates on its development since its introduction in 2015. We bring
attention to a modification of the original construction of this theory which
is applicable in a broader setting, obtain properties of the orientations of
the moduli spaces of real maps under this modification which had previously
been obtained for the original construction, and compare such orientations
obtained via the two constructions when both constructions are applicable.’ In
particular, the orientation of the moduli space of real maps from a disjoint
union of domains may not be the product orientation of the moduli spaces of
real maps from its components, a possibility overlooked in our past work.

###### Contents

  1. 1 Introduction
  2. 2 Notation and terminology
  3. 3 Comparisons of orientations
    1. 3.1 Distinguished orientations
    2. 3.2 Induced orientations
    3. 3.3 Orientations of moduli spaces
    4. 3.4 Orientations of nodal strata
    5. 3.5 Spaces of rational maps
  4. 4 Updates on past work and corrections
    1. 4.1 Real-orientable symplectic manifolds, [19, Sections 1.1,2.1,2.2]
    2. 4.2 Associated relative spin structures, [18, Corollary 3.8]
    3. 4.3 Equivariant localization, [19, Sections 4,5]
    4. 4.4 Real GW vs. curve counts, [25]
    5. 4.5 Real GW-theory of local 3-folds, [10]
    6. 4.6 Wrapup of proof of (4.15)

##  1 Introduction

The foundations of complex Gromov-Witten (or GW-) theory, i.e. of counts of
$J$-holomorphic curves in symplectic manifolds, were established in the 1990s
and have been spectacularly applied in symplectic topology, algebraic
geometry, and string theory since then. On the other hand, the progress in
real GW-theory, i.e. of counts of $J$-holomorphic curves in symplectic
manifolds preserved by anti-symplectic involutions, has been much slower, as
its mathematical foundations in positive genera did not even exist until the
mid 2010s. Since the introduction of real positive-genera GW-theory in [17],
its has been extended to broader settings in [10] by modifying the approach of
[17] to orienting the moduli spaces of stable real maps. The present paper
obtains geometric properties of the resulting orientations analogous to the
geometric properties obtained in [18] for the orientations constructed in [17]
and compares the orientations obtained via the approaches of [17] and [10]
when both are applicable. We also update on developments in real GW-theory
since its introduction and correct some oversight in past work. We hope that
the present paper and its more algebraic companion [20] will facilitate
further development of real GW-theory.

A real manifold is a pair $(X,\phi)$ consisting of a (smooth) manifold $X$ and
an involution $\phi$ on $X$, i.e. $\phi$ is a diffeomorphism of $X$ such that
$\phi^{2}\\!=\\!\textnormal{id}_{X}$. In such a case, we denote by
$X^{\phi}\\!\subset\\!X$ the fixed locus of $\phi$. A real bundle pair
$(V,\varphi)$ over a real manifold $(X,\phi)$ consists of a complex vector
bundle $V\\!\longrightarrow\\!X$ and a conjugation $\varphi$ on $V$ lifting
$\phi$, i.e. $\varphi^{2}\\!=\\!\textnormal{id}_{V}$ and
$\varphi\\!:V_{x}\\!\longrightarrow\\!\overline{V}_{\phi(x)}$ is a
$\mathbb{C}$-linear isomorphism for every $x\\!\in\\!X$. In such a case,
$V^{\varphi}\\!\longrightarrow\\!X^{\phi}$ is a real vector bundle. For a real
bundle pair $(V,\varphi)$ over $(X,\phi)$, we denote by

$\Lambda_{\mathbb{C}}^{\textnormal{top}}(V,\varphi)=\big{(}\Lambda_{\mathbb{C}}^{\textnormal{top}}V,\Lambda_{\mathbb{C}}^{\textnormal{top}}\varphi\big{)}$

the top exterior power of $V$ over $\mathbb{C}$ with the induced conjugation.
Direct sums, duals, and tensor products over $\mathbb{C}$ of real bundle pairs
over $(X,\phi)$ are again real bundle pairs over $(X,\phi)$.

If $(X,\phi)$ is a real manifold and $L\\!\longrightarrow\\!X$ is a complex
vector bundle, the homomorphisms

$\displaystyle\phi_{L}^{\oplus}\\!:L\\!\oplus\\!\phi^{*}\overline{L}$
$\displaystyle\longrightarrow L\\!\oplus\\!\phi^{*}\overline{L},$
$\displaystyle\quad\phi_{L}^{\oplus}(v,w)$
$\displaystyle=(w,v),\qquad\hbox{and}$
$\displaystyle\phi_{L}^{\otimes}\\!:L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L}$
$\displaystyle\longrightarrow
L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L},$
$\displaystyle\quad\phi_{L}^{\otimes}(v\\!\otimes\\!w)$
$\displaystyle=w\\!\otimes\\!v,$

are conjugations covering $\phi$. The homomorphism

$\big{(}L\\!\oplus\\!\phi^{*}\overline{L}\big{)}^{\phi_{L}^{\oplus}}\longrightarrow
L|_{X^{\phi}},\quad(v,w)\longrightarrow v,$

of real vector bundles over $X^{\phi}$ is then an isomorphism. We will call
the orientation on the domain of this isomorphism induced from the complex
orientation of the target the projection orientation of
$(L\\!\oplus\\!\phi^{*}\overline{L})^{\phi_{L}^{\oplus}}$. If
$\textnormal{rk}_{\mathbb{C}}\\!L\\!=\\!1$, then

$\big{(}L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L},\phi_{L}^{\otimes}\big{)}\longrightarrow\Lambda_{\mathbb{C}}^{\textnormal{top}}\big{(}L\\!\oplus\\!\phi^{*}\overline{L},\phi_{L}^{\oplus}\big{)},\qquad
v\\!\otimes\\!w\longrightarrow-\mathfrak{i}(v,0)\\!\wedge\\!(0,w),$ (1.1)

is an isomorphism of real bundle pairs over $(X,\phi)$. The induced
isomorphism

$\begin{split}&\big{\\{}rv\\!\otimes_{\mathbb{C}}\\!v\\!\in\\!L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L}\\!:v\\!\in\\!L|_{X^{\phi}},\,r\\!\in\\!\mathbb{R}\big{\\}}\\!=\\!\big{(}L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L}\big{)}^{\phi_{L}^{\otimes}}\\\
&\hskip
144.54pt\longrightarrow\big{(}\Lambda_{\mathbb{C}}^{\textnormal{top}}\big{(}L\\!\oplus\\!\phi^{*}\overline{L}\big{)}\\!\big{)}^{\Lambda_{\mathbb{C}}^{\textnormal{top}}\phi_{L}^{\oplus}}\\!=\\!\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}\\!\big{(}L\\!\oplus\\!\phi^{*}\overline{L}\big{)}^{\phi_{L}^{\oplus}}\big{)}\end{split}$

of real line bundles over $X^{\phi}$ intertwines the canonical orientation on
the domain (given by the standard orientation of $\mathbb{R}$) and the
projection orientation on the target.

######  Definition 1.1.

Let $(X,\phi)$ be a real manifold. A real orientation
$(L,[\psi],\mathfrak{s})$ on a real bundle pair $(V,\varphi)$ over $(X,\phi)$
consists of

  1. (RO1)

a complex line bundle $L\\!\longrightarrow\\!X$ such that

$w_{2}(V^{\varphi})=w_{2}(L)|_{X^{\phi}}\qquad\hbox{and}\qquad\Lambda_{\mathbb{C}}^{\textnormal{top}}(V,\varphi)\approx\big{(}L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L},\phi_{L}^{\otimes}\big{)},$
(1.2)

  2. (RO2)

a homotopy class $[\psi]$ of isomorphisms of real bundle pairs in (1.2), and

  3. (RO3)

a spin structure $\mathfrak{s}$ on the real vector bundle
$V^{\varphi}\\!\oplus\\!L^{*}|_{X^{\phi}}$ over $X^{\phi}$ compatible with the
orientation induced by $[\psi]$.

By (RO2), a real orientation on $(V,\varphi)$ determines a homotopy class of
isomorphisms

$\Lambda_{\mathbb{R}}^{\textnormal{top}}V^{\varphi}=\big{(}\Lambda_{\mathbb{C}}^{\textnormal{top}}V\big{)}^{\Lambda_{\mathbb{C}}^{\textnormal{top}}\varphi}\approx\big{(}L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L}\big{)}^{\phi_{L}^{\otimes}}=\big{\\{}rv\\!\otimes_{\mathbb{C}}\\!v\\!\in\\!L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L}\\!:v\\!\in\\!L|_{X^{\phi}},\,r\\!\in\\!\mathbb{R}\big{\\}}$
(1.3)

and thus an orientation on $V^{\varphi}$. In particular, the real vector
bundle $V^{\varphi}\\!\oplus\\!L^{*}|_{X^{\phi}}$ over $X^{\phi}$ in (RO3) is
oriented. By the first condition in (1.2), it admits a spin structure. By [14,
Theorems 1.1,1.2], a real orientation $(L,[\psi],\mathfrak{s})$ on a rank $k$
real bundle pair $(V,\varphi)$ over a symmetric surface $(\Sigma,\sigma)$,
possibly nodal and disconnected, determines a homotopy class of isomorphisms

$\big{(}V\\!\oplus\\!(L^{*}\\!\oplus\\!\sigma^{*}\overline{L}^{*}),\varphi\\!\oplus\\!\sigma_{L^{*}}^{\oplus}\big{)}\approx\big{(}\Sigma\\!\times\\!\mathbb{C}^{k+2},\sigma\\!\times\\!\mathfrak{c}\big{)}$
(1.4)

of real bundle pairs over $(\Sigma,\sigma)$, where $\mathfrak{c}$ is the
standard conjugation on $\mathbb{C}^{k+2}$; see Section 2 for the terminology.
This homotopy class of isomorphisms is one of the key ingredients in the
construction of real GW-invariants in [17]; see also [15] and Section 3.3.

A real symplectic manifold is a triple $(X,\omega,\phi)$ consisting of a
symplectic manifold $(X,\omega)$ and an anti-symplectic involution $\phi$ on
$X$, i.e. $\phi^{*}\omega\\!=\\!-\omega$. The fixed locus $X^{\phi}$ of $\phi$
is then a Lagrangian submanifold of $(X,\omega)$ and

$TX^{\phi}=\big{\\{}\dot{x}\\!\in\\!TX|_{X^{\phi}}\\!:\textnormal{d}\phi(\dot{x})\\!=\\!\dot{x}\big{\\}}.$

A real orientation on a real symplectic manifold $(X,\omega,\phi)$ is a real
orientation on the real bundle pair $(TX,\textnormal{d}\phi)$ over $(X,\phi)$
in the sense of Definition 1.1. We call a real symplectic manifold admitting a
real orientation real-orientable. The anti-holomorphic involutions

$\displaystyle\tau_{n}\\!:\mathbb{P}^{n-1}$
$\displaystyle\longrightarrow\mathbb{P}^{n-1},$
$\displaystyle\tau_{n}\big{(}[[Z_{1},\ldots,Z_{n}]\big{)}$
$\displaystyle=\begin{cases}[\overline{Z}_{2},\overline{Z}_{1},\ldots,\overline{Z}_{n},\overline{Z}_{n-1}],&\hbox{if}~{}n\\!\in\\!2\mathbb{Z};\\\
[\overline{Z}_{2},\overline{Z}_{1},\ldots,\overline{Z}_{n-1},\overline{Z}_{n-2},\overline{Z}_{n}],&\hbox{if}~{}n\\!\not\in\\!2\mathbb{Z};\end{cases}$
$\displaystyle\eta_{2m}\\!:\mathbb{P}^{2m-1}$
$\displaystyle\longrightarrow\mathbb{P}^{2m-1},$
$\displaystyle\eta_{2m}\big{(}[Z_{1},\ldots,Z_{2m}]\big{)}$
$\displaystyle=\big{[}\overline{Z}_{2},-\overline{Z}_{1},\ldots,\overline{Z}_{2m},-\overline{Z}_{2m-1}\big{]},$

are anti-symplectic with respect to the Fubini-Study symplectic forms
$\omega_{n}$ and $\omega_{2m}$ on the complex projective spaces
$\mathbb{P}^{n-1}$ and $\mathbb{P}^{2m-1}$, respectively. The real symplectic
manifolds $(\mathbb{P}^{n-1},\omega_{n},\tau_{n})$ with
$n\\!\in\\!2\mathbb{Z}$ and $(\mathbb{P}^{2m-1},\omega_{2m},\eta_{2m})$ admit
real orientations $(L,[\psi],\mathfrak{s})$ with
$L\\!=\\!\mathcal{O}_{\mathbb{P}^{n-1}}(n/2)$ and
$L\\!=\\!\mathcal{O}_{\mathbb{P}^{2m-1}}(m)$, respectively. Many other
examples of compact real symplectic manifolds with real orientations are
described in [19, Section 1.1]; see also Section 4.1 in the present paper.
These include many projective complete intersections, such as the real quintic
threefolds, i.e. the smooth hypersurfaces in $\mathbb{P}^{4}$ cut out by
degree 5 homogeneous polynomials on $\mathbb{C}^{5}$ with real coefficients;
they play a prominent role in the interactions of symplectic topology with
string theory and algebraic geometry.

For a symplectic manifold $(X,\omega)$, we denote by $\mathcal{J}_{\omega}$
the contractible space of $\omega$-tamed almost complex structures on $X$ and
by $c_{1}(X,\omega)\\!\in\\!H^{2}(X;\mathbb{Z})$ the first Chern class of the
complex vector bundle $(TX,J)$ over $X$. For a real symplectic manifold
$(X,\omega,\phi)$, let

$\mathcal{J}_{\omega}^{\phi}=\big{\\{}J\\!\in\\!\mathcal{J}_{\omega}\\!:\phi^{*}J\\!=\\!-J\big{\\}}.$

For $g\\!\in\\!\mathbb{Z}$, $\ell\\!\in\\!\mathbb{Z}^{\geq 0}$,
$B\\!\in\\!H_{2}(X;\mathbb{Z})$, and $J\\!\in\\!\mathcal{J}_{\omega}^{\phi}$,
we denote by $\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$ the moduli
space of stable real $J$-holomorphic degree $B$ maps from closed, possibly
nodal and disconnected, symmetric Riemann surfaces of arithmetic genus $g$
with $\ell$ conjugate pairs of marked points.

######  Theorem 1.2.

Suppose $(X,\omega,\phi)$ is a real-orientable $2n$-manifold with
$n\\!\not\in\\!2\mathbb{Z}$ and $J\\!\in\\!\mathcal{J}_{\omega}^{\phi}$.

  1. (1)

A real orientation $(L,[\psi],\mathfrak{s})$ on $(X,\omega,\phi)$ induces
orientations on the moduli spaces
$\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$ for all
$g,\ell\\!\in\\!\mathbb{Z}^{\geq 0}$ and $B\\!\in\\!H_{2}(X;\mathbb{Z})$.

  2. (2)

The induced orientations satisfy the properties stated in Propositions
3.12-3.16 and 3.18.

Under the assumptions of Theorem 1.2, the (virtual) dimensions of all moduli
spaces $\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$,

$\dim^{\textnormal{vir}}\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)\equiv(1\\!-\\!g)(n\\!-\\!3)\\!+\\!2\ell\\!+\\!\big{\langle}c_{1}(X,\omega),B\big{\rangle},$
(1.5)

are even. In particular, the product orientation on the Cartesian product of
two such spaces is independent of their ordering.

The notion of real orientation provided by [17, Definition 5.1] and used in
[17] to orient moduli spaces of stable real maps replaces
$(L\\!\otimes_{\mathbb{C}}\\!\phi^{*}\overline{L},\phi_{L}^{\otimes})$ and
$(L^{*}\\!\oplus\\!\phi^{*}\overline{L}^{*},\phi_{L^{*}}^{\oplus})$ in (1.2)
and (1.4) by $(L,\widetilde{\phi})^{\otimes 2}$ and
$2(L,\widetilde{\phi})^{*}$, respectively, for a real line bundle pair
$(L,\widetilde{\phi})$ over $(X,\phi)$. The real vector bundle
$(2L)^{2\widetilde{\phi}}\\!=\\!2L^{\widetilde{\phi}}$ over $X^{\phi}$ is then
canonically oriented by taking the same orientation on both copies of
$L^{\widetilde{\phi}}$. The homomorphism

$\Phi_{L}\\!:2(L,\widetilde{\phi})\longrightarrow\big{(}L\\!\oplus\\!\phi^{*}\overline{L},\phi_{L}^{\oplus}\big{)},\quad\Phi_{L}(v,w)=\big{(}v\\!+\\!\mathfrak{i}w,\widetilde{\phi}(v\\!-\\!\mathfrak{i}w)\\!\big{)},$
(1.6)

is an isomorphism of real bundle pairs over $(X,\phi)$. The restriction

$\Phi_{L}\\!:(2L)^{2\widetilde{\phi}}\longrightarrow\big{(}L\\!\oplus\\!\overline{L}\big{)}^{\phi_{L}^{\oplus}}$
(1.7)

intertwines the canonical orientation on its domain and the projection
orientation on its target. Definition 1.1, which is a slightly reworded
version of the notion of twisted (real) orientation data introduced in [10,
Definition A.1], thus weakens (broadens) the notion of real orientation of
[17, Definition 5.1]. In particular,
$(\mathbb{P}^{2m-1},\omega_{2m},\eta_{2m})$ with $m\\!\not\in\\!2\mathbb{Z}$
admits a real orientation in the sense of Definition 1.1 in the present paper,
but not in the sense of [17, Definition 5.1]. If
$(X,\phi)\\!=\\!(\Sigma,\sigma)$ is a symmetric surface, the compositions of
the isomorphisms $\textnormal{id}\\!\oplus\\!\Phi_{L^{*}}$ and (1.4) yields a
homotopy class of isomorphisms

$\big{(}V\\!\oplus\\!2L^{*},\varphi\\!\oplus\\!2\widetilde{\sigma}^{*}\big{)}\approx\big{(}\Sigma\\!\times\\!\mathbb{C}^{k+2},\sigma\\!\times\\!\mathfrak{c}\big{)}$
(1.8)

of real bundle pairs over $(\Sigma,\sigma)$.

As noted in [10], the construction of orientations on the moduli spaces of
real maps in [17] goes through almost verbatim with the weaker notion of real
orientation of Definition 1.1; see also Section 3.3. However, the resulting
orientations of the moduli spaces are generally different when both
constructions apply; the orientations on
$\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$ arising from the two
constructions in such cases are compared via Lemma 3.1 with

$\textnormal{rk}_{\mathbb{C}}L\\!=1\qquad\hbox{and}\qquad\deg
L=-\big{\langle}c_{1}(X,\omega),B\big{\rangle}/2.$

Propositions 3.12, 3.13, and 3.18 compare the orientations of various moduli
spaces under the orientating procedure described in Sections 3.3. Propositions
3.14 and 3.16 do the same for the codimension 2 strata of maps with a
conjugate pair of nodes in the domain and the codimension 1 strata of maps
with an isolated real node, respectively.

Propositions 3.13, 3.14, and 3.18 are analogues of Theorems 1.4, 1.2, and 1.5,
respectively, in [18], which apply to the orientating procedure used in [17]
(the statements of these theorems are also incorporated into the statements of
Propositions 3.13, 3.14, and 3.18). On the other hand, Proposition 3.12
concerns the behavior of the orientations of the moduli spaces of real maps
under the disjoint union of the domains of the maps. Contrary to the
expectation, the orientation at the disjoint union of two maps may not be the
product of the orientations at the two maps. This possibility was overlooked
in our past work, resulting in oversights in some proofs, but fortunately
these oversights do not impact any material statements; see Section 4.
Proposition 3.16 likewise has no analogue in our past work; it had not been
needed for the previously completed work in real positive-genus GW-theory, but
is used in [20].

Similarly to the GW-invariants of a compact symplectic manifold $(X,\omega)$,
the real GW-invariants of a compact real symplectic manifold $(X,\omega,\phi)$
with a real orientation $(L,[\psi],\mathfrak{s})$ are rational numbers, but an
invertible transformation determined by the topology of $(X,\phi)$ is expected
to convert these numbers into integers. If the (real) dimension of $X$ is 6,
such a transformation is expected to be linear. This is confirmed in [25] for
“Fano” classes in a compact real symplectic sixfold with a real orientation,
adapting the result of [28, Theorem 1.5] from the complex GW-theory to the
real GW-theory; see also Section 4.4. The structure of such a transformation
for the complex GW-invariants of a compact Calabi-Yau symplectic sixfold is
predicted in [21] based on string theory considerations, mathematically
motivated in [3, 4], and confirmed in [22]. The analogue of this
transformation for the real GW-invariants of a compact Calabi-Yau symplectic
sixfold with an anti-symplectic involution is predicted in [2, 27] based on
string theory considerations and mathematically motivated in [10]; its
mathematical confirmation is the subject of [12]. Such a transformation for
the complex genus 1 GW-invariants of a compact Calabi-Yau symplectic tenfold
is mathematically motivated in [26], but its adaptation to the real setting is
yet to be obtained. Other important geometric problems in real GW-theory
include the mirror symmetry predictions of [27]. More algebraic directions for
future research are indicated in [20].

We gather the most frequently used notation and terminology in Section 2. In
Section 3.1, we describe the behavior of natural orientations on the
determinants of real Cauchy-Riemann operators on real bundle pairs of the form
$(L\\!\oplus\\!\sigma^{*}\overline{L},\sigma_{L}^{\oplus})$ over symmetric
Riemann surfaces $(\Sigma,\sigma)$ under various operations. We use this to
study the behavior of the orientations on the determinants of real Cauchy-
Riemann operators induced by real orientations in Section 3.2 and the
orientations on the moduli space of real maps in Sections 3.3 and 3.4,
establishing Propositions 3.12-3.16 in the process. In Section 3.5, we
specialize to genus 0 and establish Proposition 3.18. In Section 4, we extend
other results from our past work to the broader setting of Definition 1.1 and
correct some oversights.

##  2 Notation and terminology

For $\ell\\!\in\\!\mathbb{Z}^{\geq 0}$, let
$[\ell]\\!=\\!\\{1,2,\ldots,\ell\\}$. We define the arithmetic genus $g$ of a
closed, possibly nodal and disconnected, Riemann surface
$(\Sigma,\mathfrak{j})$ by

$g\\!-\\!1=\sum_{i=1}^{m}(g_{i}\\!-\\!1)$

if $g_{1},\ldots,g_{m}$ are the arithmetic genera of the topological
components of $\Sigma$. A symmetric Riemann surface
$(\Sigma,\sigma,\mathfrak{j})$ is a closed, possibly nodal and disconnected,
Riemann surface $(\Sigma,\mathfrak{j})$ with an anti-holomorphic involution
$\sigma$. For example, there are two topological types of such involutions on
$\mathbb{P}^{1}$:

$\tau,\eta\\!:\mathbb{P}^{1}\longrightarrow\mathbb{P}^{1},\qquad\tau(z)=\frac{1}{\bar{z}},\quad\eta(z)=-\frac{1}{\bar{z}}\,.$
(2.1)

If $\phi$ is an involution on another space $X$, a map
$u\\!:\\!\Sigma\\!\longrightarrow\\!X$ is $(\phi,\sigma)$-real (or
$\phi$-real, or just real) if $u\\!\circ\\!\sigma\\!=\\!\phi\\!\circ\\!u$. For
$g\\!\in\\!\mathbb{Z}$ and $\ell\\!\in\\!\mathbb{Z}^{\geq 0}$, we denote by
$\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet}$ the Deligne-Mumford
moduli space of stable closed, possibly nodal and disconnected, symmetric
Riemann surfaces $(\Sigma,\sigma,\mathfrak{j})$ of arithmetic genus $g$ with
$\ell$ conjugate pairs $(z_{i}^{+},z_{i}^{-})$ of marked points. This space is
a smooth compact orbifold of dimension $3(g\\!-\\!1)\\!+\\!2\ell$; it is empty
if $g\\!+\\!\ell\\!\leq\\!1$. Let
$\mathbb{R}\overline{\mathcal{M}}_{g,\ell}\\!\subset\\!\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet}$
be the topological component parametrizing connected marked curves. If in
addition $(X,\omega,\phi)$, $J$, and $B$ are as in Theorem 1.2, we denote by

$\overline{\mathfrak{M}}_{g,\ell}^{\phi}(B;J)\subset\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$

the subspace of maps from connected domains.

Let $(\Sigma,\mathfrak{j})$ be a closed, but possibly nodal and disconnected,
Riemann surface. For $k\\!\in\\!\mathbb{Z}^{\geq 0}$, we denote by
$k\bar{\partial}_{\Sigma}$ the standard CR-operator on the complex vector
bundle $\Sigma\\!\times\\!\mathbb{C}^{k}$ over $\Sigma$. If $V$ is any complex
vector bundle over $\Sigma$ and

$D_{V}\\!:\Gamma(\Sigma;V)\longrightarrow\Gamma_{\mathfrak{j}}^{0,1}(\Sigma;V)\\!\equiv\\!\Gamma\big{(}\Sigma;(T^{*}\Sigma,\mathfrak{j})^{0,1}\\!\otimes_{\mathbb{C}}\\!V\big{)}$

is a generalized Cauchy-Riemann (or CR-) operator on $V$ as in [28, Section
2.2], the determinant line

$\det D_{V}\equiv\Lambda_{\mathbb{R}}^{\textnormal{top}}(\ker
D_{V})\otimes\big{(}\Lambda^{\textnormal{top}}_{\mathbb{R}}(\text{cok}\,D_{V})\big{)}^{*}$

of $D_{V}$ has a canonical complex orientation; see [24, Appendix A.2] and
Proposition 5.9 in the 5th arXiv version of [29]. The space of generalized CR-
operators on a complex vector bundle $V$ is affine. By [29], the determinant
lines of these operators form a line bundle over this space; its topology
depends on the coherent system of determinant line bundles used.

If $(\Sigma,\mathfrak{j})$ is a disjoint union of closed Riemann surfaces
$(\Sigma_{1},\mathfrak{j}_{1})$ and $(\Sigma_{2},\mathfrak{j}_{2})$ and
$V_{i}\\!=\\!V|_{\Sigma_{i}}$ for $i\\!=\\!1,2$, then
$D_{V}\\!=\\!D_{V_{1}}\\!\oplus\\!D_{V_{2}}$ for some generalized CR-operators
$D_{V_{1}}$ and $D_{V_{2}}$ on $V_{1}$ and $V_{2}$, respectively. By the
Direct Sum property in Section 2.2 of the 5th arXiv version of [29], this
decomposition induces an isomorphism

$\det D_{V}\approx\big{(}\\!\det D_{V_{1}}\big{)}\\!\otimes\\!\big{(}\\!\det
D_{V_{2}}\big{)}$ (2.2)

between the corresponding determinant lines which depends continuously on
$D_{V_{1}}$ and $D_{V_{2}}$. If instead $V\\!=\\!V_{1}\\!\oplus\\!V_{2}$ for
some complex vector bundles $V_{1},V_{2}$ over $\Sigma$ and
$D_{V}\\!=\\!D_{V_{1}}\\!\oplus\\!D_{V_{2}}$ for some generalized CR-operators
$D_{V_{1}}$ and $D_{V_{2}}$ on $V_{1}$ and $V_{2}$, respectively, we similarly
obtain an isomorphism (2.2) which depends continuously on $D_{V_{1}}$ and
$D_{V_{2}}$. However, the isomorphism in (2.2) in either case depends on the
coherent system of determinant line bundles used, unless $D_{V_{1}}$ and
$D_{V_{2}}$ are either ($\mathbb{C}$-linear) CR-operators or surjective. By
the continuous dependence, there is a canonical homotopy class of isomorphisms
(2.2) in both cases even if $D_{V}\\!\neq\\!D_{V_{1}}\\!\oplus\\!D_{V_{2}}$.
Since the isomorphism (2.2) respects the complex orientations if
$D_{V}\\!=\\!D_{V_{1}}\\!\oplus\\!D_{V_{2}}$ and $D_{V_{1}},D_{V_{2}}$ are CR-
operators, this is also the case for all generalized CR-operators
$D_{V},D_{V_{1}},D_{V_{2}}$ on $V,V_{1},V_{2}$, respectively.

Let $(\Sigma,\sigma,\mathfrak{j})$ be a symmetric Riemann surface. For
$k\\!\in\\!\mathbb{Z}^{\geq 0}$, we denote by
$k\bar{\partial}_{(\Sigma,\sigma)}$ the standard CR-operator on the real
bundle pair
$(\Sigma\\!\times\\!\mathbb{C}^{k},\sigma\\!\times\\!\mathfrak{c})$; this is
the restriction of $k\bar{\partial}_{\Sigma}$ to the subspace of
$\sigma$-invariant sections with the range restricted to the subspace of
$\sigma$-invariant $(0,1)$-forms. If $(V,\varphi)$ is a real bundle pair over
a symmetric Riemann surface $(\Sigma,\sigma)$ and

$\begin{split}D_{(V,\varphi)}\\!:\Gamma(\Sigma;V)^{\varphi}\equiv&\big{\\{}\xi\\!\in\\!\Gamma(\Sigma;V)\\!:\,\xi\\!\circ\\!\sigma\\!=\\!\varphi\\!\circ\\!\xi\big{\\}}\\\
&\qquad\longrightarrow\Gamma_{\mathfrak{j}}^{0,1}(\Sigma;V)^{\varphi}\equiv\big{\\{}\zeta\\!\in\\!\Gamma_{\mathfrak{j}}^{0,1}(\Sigma;V)\\!:\,\zeta\\!\circ\\!\textnormal{d}\sigma=\varphi\\!\circ\\!\zeta\big{\\}}\end{split}$

is a real CR-operator on $(V,\varphi)$ as in [18, Section 2.2], let

$\det D_{(V,\varphi)}\equiv\Lambda_{\mathbb{R}}^{\textnormal{top}}(\ker
D_{(V,\varphi)})\otimes\big{(}\Lambda^{\textnormal{top}}_{\mathbb{R}}(\text{cok}\,D_{(V,\varphi)})\big{)}^{*}$

be the determinant line of $D_{(V,\varphi)}$. The space of real CR-operators
on a real bundle pair $(V,\varphi)$ is affine. By [29], the determinant lines
of these operators form a line bundle over this space. Thus, an orientation on
the determinant line of one real CR-operator on $(V,\varphi)$ determines an
orientation of the determinant lines for all other such operators.

If $(\Sigma,\sigma,\mathfrak{j})$ is a disjoint union of symmetric Riemann
surfaces $(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$ and
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$ and
$(V_{i},\varphi_{i})\\!=\\!(V,\varphi)|_{\Sigma_{i}}$ for $i\\!=\\!1,2$, then

$D_{(V,\varphi)}=D_{(V_{1},\varphi_{1})}\\!\oplus\\!D_{(V_{2},\varphi_{2})}$
(2.3)

for some real CR-operators $D_{(V_{1},\varphi_{1})}$ and
$D_{(V_{2},\varphi_{2})}$ on $(V_{1},\varphi_{1})$ and $(V_{2},\varphi_{2})$,
respectively. By the Direct Sum property for determinant line bundles, this
decomposition induces an isomorphism

$\det D_{(V,\varphi)}\approx\big{(}\\!\det
D_{(V_{1},\varphi_{1})}\big{)}\\!\otimes\\!\big{(}\\!\det
D_{(V_{2},\varphi_{2})}\big{)}$ (2.4)

between the corresponding determinant lines which depends continuously on
$D_{(V_{1},\varphi_{1})}$ and $D_{(V_{2},\varphi_{2})}$. If instead

$(V,\varphi)=(V_{1},\varphi_{1})\\!\oplus\\!(V_{2},\varphi_{2})$ (2.5)

for some real bundle pairs $(V_{1},\varphi_{1}),(V_{2},\varphi_{2})$ over
$(\Sigma,\sigma)$ and (2.3) holds for some real CR-operators
$D_{(V_{1},\varphi_{1})}$ and $D_{(V_{2},\varphi_{2})}$ on
$(V_{1},\varphi_{1})$ and $(V_{2},\varphi_{2})$, respectively, we similarly
obtain an isomorphism (2.4) which depends continuously on
$D_{(V_{1},\varphi_{1})}$ and $D_{(V_{2},\varphi_{2})}$. There is again a
canonical homotopy class of isomorphisms (2.4) in both cases even if (2.3)
does not hold.

##  3 Comparisons of orientations

###  3.1 Distinguished orientations

Suppose $(\Sigma,\sigma,\mathfrak{j})$ is a symmetric Riemann surface of
arithmetic genus $g$ and $L$ is a complex vector bundle over $\Sigma$. If
$D_{L}$ is a generalized CR-operator on $L$, then

$\displaystyle
D_{\sigma^{*}\overline{L}}\\!:\Gamma(\Sigma;\sigma^{*}\overline{L})\longrightarrow\Gamma_{\mathfrak{j}}^{0,1}\big{(}\Sigma;\sigma^{*}\overline{L}\big{)},\quad
D_{\sigma^{*}\overline{L}}(\xi)=D\xi\\!\circ\\!\textnormal{d}\sigma,\quad\hbox{and}$
(3.1) $\displaystyle
D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}\\!\equiv\\!D_{L}\\!\oplus\\!D_{\sigma^{*}\overline{L}}\\!:\Gamma\big{(}\Sigma;L\\!\oplus\\!\sigma^{*}\overline{L}\big{)}^{\sigma_{L}^{\oplus}}\longrightarrow\Gamma_{\mathfrak{j}}^{0,1}\big{(}\Sigma;L\\!\oplus\\!\sigma^{*}\overline{L}\big{)}^{\sigma_{L}^{\oplus}}$
(3.2)

are a generalized CR-operator on $\sigma^{*}\overline{L}$ and a real CR-
operator on $(L\\!\oplus\\!\sigma^{*}\overline{L},\sigma_{L}^{\oplus})$,
respectively. The homomorphisms

$\Gamma(\Sigma;L)\longrightarrow\Gamma(\Sigma;\sigma^{*}\overline{L}),~{}~{}\xi\longrightarrow\xi\\!\circ\\!\sigma,\quad\Gamma_{\mathfrak{j}}^{0,1}\big{(}\Sigma;L)\longrightarrow\Gamma_{\mathfrak{j}}^{0,1}(\Sigma;\sigma^{*}\overline{L}),~{}~{}\zeta\longrightarrow\zeta\\!\circ\\!\textnormal{d}\sigma,$

are isomorphisms and determine an isomorphism

$\det D_{L}\approx\det D_{\sigma^{*}\overline{L}}$ (3.3)

which depends continuously on $D_{L}$. By the continuous dependence, there is
a canonical homotopy class of isomorphisms (3.3) even if $D_{L}$ and
$D_{\sigma^{*}\overline{L}}$ are not related by (3.1). By the stabilization to
the surjective case as in the proof of Lemma 3.1 below, this homotopy class
respects the complex orientations of the two determinant lines if and only if

$(1\\!-\\!g)\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg L\in 2\mathbb{Z}\,,$
(3.4)

as is immediate if $D_{L}$ is a ($\mathbb{C}$-linear) CR-operator; the
expression on the left-hand side above is the complex index of a
($\mathbb{C}$-linear) CR-operator on $L$ or $\sigma^{*}\overline{L}$.

Furthermore, the projections

$\displaystyle\Gamma\big{(}\Sigma;L\\!\oplus\\!\sigma^{*}\overline{L}\big{)}^{\sigma_{L}^{\oplus}}$
$\displaystyle\longrightarrow\Gamma(\Sigma;L),$
$\displaystyle\quad(\xi,\xi^{\prime})$ $\displaystyle\longrightarrow\xi,$
(3.5)
$\displaystyle\Gamma_{\mathfrak{j}}^{0,1}\big{(}\Sigma;L\\!\oplus\\!\sigma^{*}\overline{L}\big{)}^{\sigma_{L}^{\oplus}}$
$\displaystyle\longrightarrow\Gamma_{\mathfrak{j}}^{0,1}(\Sigma;L),$
$\displaystyle\quad(\zeta,\zeta^{\prime})$
$\displaystyle\longrightarrow\zeta,$

are isomorphisms and determine an isomorphism

$\det D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}\approx\det
D_{L}$ (3.6)

which depends continuously on $D_{L}$. There is a canonical homotopy class of
isomorphisms (3.6) even if $D_{L}$ and
$D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$ are not related by
(3.2). Via this homotopy class, the complex orientation on $\det D_{L}$
induces an orientation on $\det
D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$, which we call the
projection orientation of the latter; it varies continuously with
$D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$.

For any real bundle pair $(L,\widetilde{\sigma})$ over
$(\Sigma,\sigma,\mathfrak{j})$, (2.4) implies that the real line

$\det D_{2(L,\widetilde{\sigma})}\approx\big{(}\\!\det
D_{(L,\widetilde{\sigma})}\big{)}\\!\otimes\\!\big{(}\\!\det
D_{(L,\widetilde{\sigma})}\big{)}$

has a canonical orientation; it is obtained by taking the same orientation on
each of the two factors on the right-hand side above.

######  Lemma 3.1.

Let $(L,\widetilde{\sigma})$ be a real bundle pair over a symmetric Riemann
surface $(\Sigma,\sigma,\mathfrak{j})$ of arithmetic genus $g$. The homotopy
class of isomorphisms

$\det D_{2(L,\widetilde{\sigma})}\approx\det
D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$

induced by the isomorphism $\Phi_{L}$ of real bundle pairs as in (1.6)
respects the canonical orientation on the left-hand side and projection
orientation on the right-hand side if and only if

$\frac{\big{(}\\!(1\\!-\\!g)\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg
L\big{)}\\!\big{(}\\!(1\\!-\\!g)\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg
L\\!-\\!1\big{)}}{2}\in 2\mathbb{Z}.$

###### Proof.

By the continuity of the canonical and projection orientations, it is
sufficient to establish the claim for any pair of real CR-operators on
$2(L,\widetilde{\sigma})$ and
$(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})$. We assume that
$D_{2(L,\widetilde{\sigma})}$ is two copies of the restriction
$D_{(L,\widetilde{\sigma})}$ of a CR-operator $D_{L}$ on $L$ to the subspaces
of the domain and target invariant under the conjugation induced by
$\widetilde{\sigma}$ and
$D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$ is given by (3.2).

Suppose first that the CR-operator $D_{L}$ on $L$ is surjective and thus

$k\equiv\dim\ker D_{(L,\widetilde{\sigma})}=\dim_{\mathbb{C}}\ker
D_{L}=\textnormal{ind}_{\mathbb{C}}D_{L}=(1\\!-\\!g)\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg
L\,.$

If $\xi_{1},\ldots,\xi_{k}$ is a basis for $\ker D_{(L,\widetilde{\sigma})}$,
then

$(\xi_{1},0),\ldots,(\xi_{k},0),(0,\xi_{1}),\ldots,(0,\xi_{k})\quad\hbox{and}\quad\Phi_{L}(\xi_{1},0),\Phi_{L}(0,\xi_{1}),\ldots,\Phi_{L}(\xi_{k},0),\Phi_{L}(0,\xi_{k})$

are a basis for $\ker D_{2(L,\widetilde{\sigma})}$ determining the canonical
orientation and a basis for $\ker
D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$ determining the
projection orientation. This establishes the claim if $D_{L}$ is surjective.

If $(L,\widetilde{\sigma})$ is the direct sum of real bundle pairs
$(L_{1},\widetilde{\sigma}_{1})$ and $(L_{2},\widetilde{\sigma}_{2})$ over
$(\Sigma,\sigma)$ and $D_{L}\\!=\\!D_{L_{1}}\\!\oplus\\!D_{L_{2}}$, the
isomorphisms $\Phi_{L},\Phi_{L_{1}},\Phi_{L_{2}}$ as in (1.6) induce an
isomorphism

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(3.7)

of exact triples of real CR-operators. Since direct sums of complex vector
bundles commute with the projections as in (3.5) and the corresponding
isomorphisms as in (2.2) respect the complex orientations, the isomorphism

$\det
D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}\approx\big{(}\\!\det
D_{(L_{1}\oplus\sigma^{*}\overline{L_{1}},\sigma_{L_{1}}^{\oplus})}\big{)}\\!\otimes\\!\big{(}\\!\det
D_{(L_{2}\oplus\sigma^{*}\overline{L_{2}},\sigma_{L_{2}}^{\oplus})}\big{)}$
(3.8)

induced by the bottom row in (3.7) respects the projection orientations. By
the Exact Squares property of determinant line bundles as in Section 2.2 in
the 5th arXiv version of [29], the isomorphism

$\det D_{2(L,\widetilde{\sigma})}\approx\big{(}\\!\det
D_{2(L_{1},\widetilde{\sigma}_{1})}\big{)}\\!\otimes\\!\big{(}\\!\det
D_{2(L_{2},\widetilde{\sigma}_{2})}\big{)}$ (3.9)

induced by the top row in (3.7) respects the canonical orientations if and
only if

$\big{(}\textnormal{ind}\,D_{(L_{1},\widetilde{\sigma}_{1})}\big{)}\big{(}\textnormal{ind}\,D_{(L_{2},\widetilde{\sigma}_{2})}\big{)}\in
2\mathbb{Z}.$

Since
$\textnormal{ind}\,D_{(L,\widetilde{\sigma})}\\!=\\!\textnormal{ind}\,D_{(L_{1},\widetilde{\sigma}_{1})}\\!+\\!\textnormal{ind}\,D_{(L_{2},\widetilde{\sigma}_{2})}$,
the claim holds for $(L_{1},\widetilde{\sigma}_{1})$ if it holds for
$(L,\widetilde{\sigma})$ and $(L_{2},\widetilde{\sigma}_{2})$.

For any real bundle pair $(L_{1},\widetilde{\sigma}_{1})$ on
$(\Sigma,\sigma)$, we can find a real bundle pair
$(L_{2},\widetilde{\sigma}_{2})$ on $(\Sigma,\sigma)$ so that the complex
vector bundles $L_{2}$ and $L\\!\equiv\\!L_{1}\\!\oplus\\!L_{2}$ admit
surjective CR-operators. The claim of this lemma then holds for the real
bundle pairs $(L_{2},\widetilde{\sigma}_{2})$ and
$(L,\widetilde{\sigma}_{1}\\!\oplus\\!\widetilde{\sigma}_{2})$ and thus for
$(L_{1},\widetilde{\sigma}_{1})$. ∎

######  Lemma 3.2.

Suppose $(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$ and
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$ are symmetric Riemann surfaces of
arithmetic genera $g_{1}$ and $g_{2}$, respectively,
$(\Sigma,\sigma,\mathfrak{j})$ is the disjoint union of
$(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$ and
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$, and $L$ is a complex vector bundle
over $\Sigma$. If $L_{i}\\!=\\!L|_{\Sigma_{i}}$ for $i\\!=\\!1,2$, then the
associated homotopy class of isomorphisms (3.8) respects the projection
orientations on the three determinant lines. If in addition
$\widetilde{\sigma}$ is a conjugation on $L$ and
$\widetilde{\sigma}_{i}\\!=\\!\widetilde{\sigma}|_{L_{i}}$ for $i\\!=\\!1,2$,
then the associated homotopy class of isomorphisms (3.9) respects the
canonical orientations on the three determinant lines if and only if

$\big{(}\\!(1\\!-\\!g_{1})\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg
L_{1}\big{)}\\!\big{(}\\!(1\\!-\\!g_{2})\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg
L_{2}\big{)}\in 2\mathbb{Z}.$

###### Proof.

The first claim follows from the homotopy class of isomorphisms (2.2) with
$V\\!=\\!L$ respecting the complex orientation. The second claim follows from
the Exact Squares property of determinant line bundles as in Section 2.2 in
the 5th arXiv version of [29], since

$\textnormal{ind}\,D_{(V_{i},\widetilde{\sigma}_{i})}=(1\\!-\\!g_{i})\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg
L_{i}$

for $i\\!=\\!1,2$. This claim also follows from the first claim and Lemma 3.1.
∎

Suppose $\sigma$ is an anti-holomorphic diffeomorphism between closed Riemann
surfaces $(\Sigma_{1},\mathfrak{j}_{1})$ and $(\Sigma_{2},\mathfrak{j}_{2})$
of arithmetic genus $g$ each, so that

$(\Sigma,\sigma,\mathfrak{j})\equiv\big{(}\Sigma_{1}\\!\sqcup\\!\Sigma_{2},\sigma,\mathfrak{j}_{1}\\!\sqcup\\!\mathfrak{j}_{2}\big{)}$
(3.10)

is a symmetric Riemann surface of arithmetic genus $2g\\!-\\!1$, and
$(V,\varphi)$ is a real bundle pair over $(\Sigma,\sigma)$. Let
$V_{i}\\!=\\!V|_{\Sigma_{i}}$ for $i\\!=\\!1,2$. The restrictions

$\Gamma(\Sigma;V\big{)}^{\varphi}\longrightarrow\Gamma(\Sigma_{1};V_{1}),~{}~{}\xi\longrightarrow\xi|_{\Sigma_{1}},\quad\Gamma_{\mathfrak{j}}^{0,1}(\Sigma;V)^{\varphi}\longrightarrow\Gamma_{\mathfrak{j}}^{0,1}(\Sigma_{1};V_{1}),~{}~{}\zeta\longrightarrow\zeta|_{\Sigma_{1}},$
(3.11)

then induce isomorphisms between real CR-operators on $(V,\varphi)$ and
generalized CR-operators on $V_{1}$ and thus an isomorphism

$\det
D_{(V,\varphi)}\approx\det\\!\big{(}D_{(V,\varphi)}|_{\Sigma_{1}}\big{)}$
(3.12)

which depends continuously on $D_{(V,\varphi)}$. The
$\textnormal{rk}_{\mathbb{C}}L\\!=\\!1$ case of the first claim below is [10,
Lemma 3.2].

######  Lemma 3.3.

Suppose $(\Sigma_{1},\mathfrak{j}_{1})$, $(\Sigma_{2},\mathfrak{j}_{2})$, $g$,
and $(\Sigma,\sigma,\mathfrak{j})$ are as above and $L$ is a complex line
bundle over $\Sigma$. Let $\sigma_{i}\\!=\\!\sigma|_{\Sigma_{i}}$ and
$L_{i}\\!=\\!L|_{\Sigma_{i}}$ for $i\\!=\\!1,2$. The homotopy class of
isomorphisms

$\det D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}\approx\det
D_{L_{1}\oplus\sigma_{1}^{*}\overline{L_{2}}}$ (3.13)

as in (3.12) respects the projection orientation on the left-hand side and the
complex orientation on the right-hand side if and only if

$(1\\!-\\!g)\textnormal{rk}_{\mathbb{C}}L\\!+\\!\deg L_{2}\in 2\mathbb{Z}\,.$

If in addition $\widetilde{\sigma}$ is a conjugation on $L$ lifting $\sigma$,
then the homotopy class of isomorphisms

$\det D_{2(L,\widetilde{\sigma})}\approx\det D_{2L_{1}}$ (3.14)

as in (3.12) respects the canonical orientation on the left-hand side and the
complex orientation on the right-hand side.

###### Proof.

Since the restrictions to $\Sigma_{1}$ as in (3.11) commute with direct sums,
the second claim follows immediately from the definition of the canonical
orientation on the left-hand side of (3.14). The first claim follows from the
sentence containing (3.4) with $L$ replaced by $L_{2}$, since
$D_{L}\\!=\\!D_{L_{1}}\\!\oplus\\!D_{L_{2}}$ for some generalized CR-operators
$D_{L_{1}},D_{L_{2}}$ on $L_{1},L_{2}$, respectively. If $L$ admits a
conjugation $\widetilde{\sigma}$ lifting $\sigma$, then this claim also
follows from the second claim and Lemma 3.1. ∎

Suppose $(\Sigma,\sigma,\mathfrak{j})$ is a symmetric Riemann surface with a
conjugate pair of nodes $\\{x^{+},x^{-}\\}$. Let
$(\widehat{\Sigma},\widehat{\sigma},\widehat{\mathfrak{j}})$ be the symmetric
Riemann surface obtained from $(\Sigma,\sigma,\mathfrak{j})$ by replacing
$x^{+}$ and $x^{-}$ by 2 smooth points each. We denote by
$q\\!:\widehat{\Sigma}\\!\longrightarrow\\!\Sigma$ the holomorphic projection
and by $\widehat{x}_{1}^{\pm},\widehat{x}_{2}^{\pm}\\!\in\\!\widehat{\Sigma}$
the preimages of $x^{\pm}$ so that
$\widehat{\sigma}(\widehat{x}_{i}^{\pm})\\!=\\!\widehat{x}_{i}^{\mp}$ for
$i\\!=\\!1,2$. We will then call the tuple

$\big{(}q\\!:\widehat{\Sigma}\\!\longrightarrow\\!\Sigma,\widehat{\sigma},\widehat{\mathfrak{j}},(\widehat{x}_{1}^{+},\widehat{x}_{1}^{-}),(\widehat{x}_{2}^{+},\widehat{x}_{2}^{-})\\!\big{)}$
(3.15)

the normalization of $(\Sigma,\sigma,\mathfrak{j})$ at $\\{x^{+},x^{-}\\}$.

Let $(V,\varphi)$ be a real bundle pair over $(\Sigma,\sigma)$. Define

$\displaystyle\textnormal{ev}_{x}^{+}\\!:\Gamma(\widehat{\Sigma},q^{*}V)^{q^{*}\varphi}\longrightarrow
V_{x^{+}},~{}~{}\textnormal{ev}_{x}^{+}(\xi)=\xi(\widehat{x}_{1}^{+})\\!-\\!\xi(\widehat{x}_{2}^{+}),$
$\displaystyle\textnormal{ev}_{x}^{+}\\!:\Gamma^{0,1}_{\widehat{\mathfrak{j}}}(\widehat{\Sigma},q^{*}V)^{q^{*}\varphi}\longrightarrow\\{0\\},\qquad\mathbf{0}_{V_{x^{+}}}\\!:V_{x^{+}}\longrightarrow\\{0\\}\,.$

A real CR-operator $D_{(V,\varphi)}$ on $(V,\varphi)$ lifts to a real CR-
operator $q^{*}D_{(V,\varphi)}$ on $q^{*}(V,\varphi)$ so that the triple

$0\longrightarrow D_{(V,\varphi)}\longrightarrow
q^{*}D_{(V,\varphi)}\stackrel{{\scriptstyle\textnormal{ev}_{x}^{+}}}{{\longrightarrow}}\mathbf{0}_{V_{x^{+}}}\longrightarrow
0$

of operators is exact and thus induces an isomorphism

$\det q^{*}D_{(V,\varphi)}\approx\big{(}\\!\det
D_{(V,\varphi)}\big{)}\\!\otimes\\!\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}V_{x^{+}}\big{)}$
(3.16)

which depends continuously on $D_{(V,\varphi)}$; see the Exact Triples
property in Section 2.2 in the 5th arXiv version of [29]. The
$\textnormal{rk}_{\mathbb{C}}L\\!=\\!1$ case of the first claim below is [11,
Lemma 7.5].

######  Lemma 3.4.

Suppose $L$ is a complex vector bundle over a symmetric Riemann surface
$(\Sigma,\sigma,\mathfrak{j})$ with a conjugate pair of nodes
$\\{x^{+},x^{-}\\}$ and (3.15) is the normalization of
$(\Sigma,\sigma,\mathfrak{j})$ at $\\{x^{+},x^{-}\\}$. If
$D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$ is a real CR-
operator on the real bundle pair
$(L\\!\oplus\\!\sigma^{*}\overline{L},\sigma_{L}^{\oplus})$ over
$(\Sigma,\sigma)$, then the associated isomorphism (3.16) respects the
projection orientations on the two determinant lines and the complex
orientation on the last factor if and only if
$\textnormal{rk}_{\mathbb{C}}\\!L\\!\in\\!2\mathbb{Z}$. If in addition
$\widetilde{\sigma}$ is a conjugation on $L$ lifting $\sigma$ and
$D_{2(L,\widetilde{\sigma})}$ is a real CR-operator on the real bundle pair
$2(L,\widetilde{\sigma})$ over $(\Sigma,\sigma)$, then the associated
isomorphism (3.16) respects the canonical orientations on the two determinant
lines and the complex orientation on the last factor.

###### Proof.

By the continuous dependence of (3.16) on $D_{(V,\varphi)}$, it is sufficient
to establish each claim for a single real CR-operator on the relevant real
bundle pair. Since the real dimension of $L_{x^{+}}$ is even, the second claim
for
$D_{2(L,\widetilde{\sigma})}\\!=\\!D_{(L,\widetilde{\sigma})}\\!\oplus\\!D_{(L,\widetilde{\sigma})}$
follows from the Exact Squares property applied to the commutative square of
operators in Figure 1.

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{(L,\widetilde{\sigma})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{q^{*}D_{(L,\widetilde{\sigma})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{0}_{L_{x^{+}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{2(L,\widetilde{\sigma})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{q^{*}D_{(2L,\widetilde{\sigma})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{0}_{2L_{x^{+}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{(L,\widetilde{\sigma})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{q^{*}D_{(L,\widetilde{\sigma})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{0}_{L_{x^{+}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$
Figure 1: Exact square of operators for the proof of Lemma 3.4

Let $D_{L}$ be a CR-operator on the complex vector bundle $L$ and
$D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$ be as in (3.2).
Define

$\displaystyle\textnormal{ev}_{x}\\!:\Gamma(\widehat{\Sigma},q^{*}L)\longrightarrow
L_{x^{+}}\\!\oplus\\!L_{x^{-}},~{}~{}\textnormal{ev}_{x}(\xi)=\big{(}\xi(\widehat{x}_{1}^{+})\\!-\\!\xi(\widehat{x}_{2}^{+}),\xi(\widehat{x}_{1}^{-})\\!-\\!\xi(\widehat{x}_{2}^{-})\\!\big{)},$
$\displaystyle\textnormal{ev}_{x}\\!:\Gamma^{0,1}_{\widehat{\mathfrak{j}}}(\widehat{\Sigma},q^{*}L)\longrightarrow\\{0\\}.$

The CR-operator $D_{L}$ lifts to a CR-operator $q^{*}D_{L}$ on $q^{*}L$ so
that the diagram

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(3.17)

is an isomorphism of exact triples of Fredholm operators. The bottom row
respects the complex orientations of the three determinant lines, while the
left and middle vertical isomorphisms intertwine the projection orientations
on the domains with the complex orientations on the targets. The right
vertical isomorphism is orientation-preserving with respect to the complex
orientations on its domain and target if and only if the complex dimension of
$L_{x^{-}}$ is even. This establishes the first claim. ∎

Suppose now that $(\Sigma,\sigma,\mathfrak{j})$ is a symmetric Riemann surface
with an isolated real node $x$, called $E$-node in [17] and elsewhere. Let
$(\widehat{\Sigma},\widehat{\sigma},\widehat{\mathfrak{j}})$ be the symmetric
Riemann surface obtained from $(\Sigma,\sigma,\mathfrak{j})$ by replacing $x$
by 2 smooth points. We denote by
$q\\!:\widehat{\Sigma}\\!\longrightarrow\\!\Sigma$ the holomorphic projection
and by $\widehat{x}^{+},\widehat{x}^{-}\\!\in\\!\widehat{\Sigma}$ the
preimages of $x$; they are interchanged by the involution $\widehat{\sigma}$.
We will then call the tuple

$\big{(}q\\!:\widehat{\Sigma}\\!\longrightarrow\\!\Sigma,\widehat{\sigma},\widehat{\mathfrak{j}},(\widehat{x}^{+},\widehat{x}^{-})\big{)}$
(3.18)

the normalization of $(\Sigma,\sigma,\mathfrak{j})$ at $x$.

Let $(V,\varphi)$ be a real bundle pair over $(\Sigma,\sigma)$. Since
$x\\!\in\\!\Sigma^{\sigma}$, $V_{x}^{\varphi}\\!\subset\\!V_{x}$ is a half-
dimensional totally real subspace. An orientation on $V_{x}^{\varphi}$ and the
complex orientation on $V_{x}$ induce an orientation on
$V_{x}/V_{x}^{\varphi}$ via the exact sequence

$0\longrightarrow V_{x}^{\varphi}\longrightarrow V_{x}\longrightarrow
V_{x}/V_{x}^{\varphi}\longrightarrow 0.$

If
$(V,\varphi)\\!=\\!(L\\!\oplus\\!\sigma^{*}\overline{L},\sigma_{L}^{\oplus})$
for a complex vector bundle $L\\!\longrightarrow\\!X$, we call the orientation
on $V_{x}/V_{x}^{\varphi}$ induced by the associated projection orientation of
$V_{x}^{\varphi}$ the projection orientation. If
$(V,\varphi)\\!=\\!2(L,\widetilde{\sigma})$ for some real bundle pair
$(L,\widetilde{\sigma})$ over $(\Sigma,\sigma)$, then
$V_{x}^{\varphi}\\!=\\!2L_{x}^{\widetilde{\sigma}}$. We call the orientation
on $V_{x}/V_{x}^{\varphi}$ induced by the associated canonical orientation of
$2L_{x}^{\widetilde{\sigma}}$ the canonical orientation. The isomorphism (1.7)
with $\phi\\!=\\!\sigma$ intertwines the canonical orientation on its domain
and the projection orientation on its target if and only if
$\binom{\textnormal{rk}_{\mathbb{C}}\\!L-1}{2}$ is even. The same applies to
the isomorphism induced by (1.6) on the corresponding quotients
$V_{x}/V_{x}^{\varphi}$. Thus, Lemma 3.5 below is consistent with Lemma 3.1.

Define

$\displaystyle\textnormal{ev}_{x}\\!:\Gamma(\widehat{\Sigma},q^{*}V)^{q^{*}\varphi}\longrightarrow
V_{x}/V_{x}^{\varphi},~{}~{}\textnormal{ev}_{x}(\xi)=\xi(\widehat{x}^{+})\\!+\\!V_{x}^{\varphi},$
$\displaystyle\textnormal{ev}_{x}\\!:\Gamma^{0,1}_{\widehat{\mathfrak{j}}}(\widehat{\Sigma},q^{*}V)^{q^{*}\varphi}\longrightarrow\\{0\\},\qquad\mathbf{0}_{V_{x}/V_{x}^{\varphi}}\\!:V_{x}/V_{x}^{\varphi}\longrightarrow\\{0\\}\,.$

A real CR-operator $D_{(V,\varphi)}$ on $(V,\varphi)$ lifts to a real CR-
operator $q^{*}D_{(V,\varphi)}$ on $q^{*}(V,\varphi)$ so that the triple

$0\longrightarrow D_{(V,\varphi)}\longrightarrow
q^{*}D_{(V,\varphi)}\stackrel{{\scriptstyle\textnormal{ev}_{x}}}{{\longrightarrow}}\mathbf{0}_{V_{x}/V_{x}^{\varphi}}\longrightarrow
0$

of operators is exact and thus induces an isomorphism

$\det q^{*}D_{(V,\varphi)}\approx\big{(}\\!\det
D_{(V,\varphi)}\big{)}\\!\otimes\\!\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}V_{x}/V_{x}^{\varphi}\big{)}$
(3.19)

which depends continuously on $D_{(V,\varphi)}$.

######  Lemma 3.5.

Suppose $L$ is a complex vector bundle over a symmetric Riemann surface
$(\Sigma,\sigma,\mathfrak{j})$ with an $E$-node $x$ and arithmetic genus $g$.
Let (3.18) be the normalization of $(\Sigma,\sigma,\mathfrak{j})$ at $x$. If
$D_{(L\oplus\sigma^{*}\overline{L},\sigma_{L}^{\oplus})}$ is a real CR-
operator on the real bundle pair
$(L\\!\oplus\\!\sigma^{*}\overline{L},\sigma_{L}^{\oplus})$ over
$(\Sigma,\sigma)$, then the associated isomorphism (3.19) respects the
projection orientations on the three lines if and only if
$\textnormal{rk}_{\mathbb{C}}\\!L\\!\in\\!2\mathbb{Z}$. If in addition
$\widetilde{\sigma}$ is a conjugation on $L$ lifting $\sigma$ and
$D_{2(L,\widetilde{\sigma})}$ is a real CR-operator on the real bundle pair
$2(L,\widetilde{\sigma})$ over $(\Sigma,\sigma)$, then the associated
isomorphism (3.19) respects the canonical orientations on the three lines if
and only if $\textnormal{rk}_{\mathbb{C}}\\!L(g\\!+\\!\deg L)$ is even.

###### Proof.

For the purposes of the second claim, we replace $L_{x^{+}}$ by
$L_{x}/L_{x}^{\widetilde{\sigma}}$ in the exact square of Fredholm operators
in Figure 1. The canonical orientation on $(2L)/(2L)^{2\widetilde{\sigma}}$ as
defined above is the same as the canonical orientation with respect to the
isomorphism
$(2L)/(2L)^{2\widetilde{\sigma}}\\!\approx\\!2(L/L)^{\widetilde{\sigma}}$ if
and only if $\textnormal{rk}_{\mathbb{C}}\\!L$ is even. By the Exact Squares
property, the middle row in the diagram in Figure 1 respects the canonical
orientations on the two determinant lines and the latter orientation on
$(2L)/(2L)^{2\widetilde{\sigma}}$ if and only if

$\big{(}\textnormal{ind}\,\mathbf{0}_{L_{x}/L_{x}^{\varphi}}\big{)}\big{(}\textnormal{ind},D_{(L,\widetilde{\sigma})}\big{)}=\textnormal{rk}_{\mathbb{C}}\\!L\big{(}(1\\!-\\!g)\textnormal{rk}_{\mathbb{C}}\\!L\\!+\\!\deg
L\big{)}$

is even. Combining the two statements, we obtain the second claim.

For the purposes of the first claim, we define

$\textnormal{ev}_{x}\\!:\Gamma(\widehat{\Sigma},q^{*}L)\longrightarrow
L_{x},~{}~{}\textnormal{ev}_{x}(\xi)=\xi(\widehat{x}^{+})\\!-\\!\xi(\widehat{x}^{-})\quad\textnormal{ev}_{x}\\!:\Gamma^{0,1}_{\widehat{\mathfrak{j}}}(\widehat{\Sigma},q^{*}L)\longrightarrow\\{0\\}.$

With the assumptions as in (3.17), the diagram

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0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$

is an isomorphism of exact triples of Fredholm operators. The bottom row
respects the complex orientations of the three determinant lines, as before,
while the left and middle vertical isomorphisms intertwine the projection
orientations on the domains with the complex orientations on the targets. The
right vertical isomorphism, which is given by taking the difference of the two
components of $L_{x}\\!\oplus\\!L_{x}$, is orientation-preserving with respect
to the projection orientation on its domain and the complex orientation on its
target if and only if the complex dimension of $L_{x}$ is even. This
establishes the first claim. ∎

###  3.2 Induced orientations

Let $(\Sigma,\sigma)$ be a symmetric Riemann surface and $D_{(V,\varphi)}$ be
a real CR-operator on a rank $k$ real bundle pair $(V,\varphi)$ over
$(\Sigma,\sigma)$. A real orientation $(L,[\psi],\mathfrak{s})$ for
$(V,\varphi)$ determines a homotopy class of isomorphisms

$\begin{split}\big{(}\\!\det D_{(V,\varphi)}\big{)}\\!\otimes\\!\big{(}\\!\det
D_{(L^{*}\oplus\sigma^{*}\overline{L}^{*},\sigma_{L^{*}}^{\oplus})}\big{)}&\approx\det
D_{(V\oplus
L^{*}\oplus\sigma^{*}\overline{L}^{*},\varphi\oplus\sigma_{L^{*}}^{\oplus})}\\\
&\approx\det\\!\big{(}\\!(k\\!+\\!2)\bar{\partial}_{(\Sigma,\sigma)}\big{)}\approx\det\\!\big{(}k\bar{\partial}_{(\Sigma,\sigma)}\big{)}\\!\otimes\\!\det\\!\big{(}2\bar{\partial}_{(\Sigma,\sigma)}\big{)}\end{split}$
(3.20)

via isomorphisms for direct sums of real bundle pairs as in (2.4) and the
homotopy class of isomorphisms (1.4). Combined with the projection orientation
on the second determinant line above and the canonical orientation on the last
determinant line, (3.20) determines a homotopy class of isomorphisms

$\det
D_{(V,\varphi)}\approx\det\\!\big{(}k\bar{\partial}_{(\Sigma,\sigma)}\big{)}\approx\big{(}\\!\det\bar{\partial}_{(\Sigma,\sigma)}\big{)}^{\otimes
k}$ (3.21)

of real lines. If in addition $\widetilde{\sigma}$ is a conjugation on $L$
lifting $\sigma$, isomorphisms as in (2.4) and the homotopy class of
isomorphisms (1.8) determine a homotopy class of isomorphisms

$\begin{split}\big{(}\\!\det
D_{(V,\varphi)}\big{)}\\!\otimes\\!\big{(}\\!\det\,D_{2(L^{*},\widetilde{\sigma}^{*})}\big{)}&\approx\det
D_{(V\oplus 2L^{*},\varphi\oplus 2\widetilde{\sigma}^{*})}\\\
&\approx\det\\!\big{(}\\!(k\\!+\\!2)\bar{\partial}_{(\Sigma,\sigma)}\big{)}\approx\det\\!\big{(}k\bar{\partial}_{(\Sigma,\sigma)}\big{)}\\!\otimes\\!\det\\!\big{(}2\bar{\partial}_{(\Sigma,\sigma)}\big{)}\,.\end{split}$
(3.22)

Along with the canonical orientations on the second and last determinant lines
above, (3.22) determines another homotopy class of isomorphisms (3.21).

The next four corollaries follow readily from the corresponding lemmas. The
$\textnormal{rk}_{\mathbb{C}}V\\!=\\!1$ case of the first claim of Corollary
3.8 is [11, Lemma 7.5]. The second claims of Corollaries 3.7 and 3.8 are Lemma
3.1 and Corollary 4.5, respectively, in [18].

######  Corollary 3.6 (of Lemma 3.2).

Suppose $(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$,
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$, $(\Sigma,\sigma,\mathfrak{j})$,
and $g_{1},g_{2}$ are as in Lemma 3.2, $(V,\varphi)$ is a rank $k$ real bundle
pair over $(\Sigma,\sigma)$, and $(L,[\psi],\mathfrak{s})$ is a real
orientation for $(V,\varphi)$. If

$(V_{i},\varphi_{i})=(V,\varphi)\big{|}_{\Sigma_{i}}\quad\hbox{and}\quad\big{(}L_{i},[\psi_{i}],\mathfrak{s}_{i}\big{)}=\big{(}L,[\psi],\mathfrak{s}\big{)}\\!\big{|}_{\Sigma_{i}}$

for $i\\!=\\!1,2$, then the diagram

$\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
82.17583pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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112.17583pt\raise-42.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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(3.23)

with the homotopy classes of the horizontal isomorphisms determined by the
real orientations $(L,[\psi],\mathfrak{s})$ and
$(L_{i},[\psi_{i}],\mathfrak{s}_{i})$, commutes if and only if
$(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)\\!\in\\!2\mathbb{Z}$. If in addition
$\widetilde{\sigma}$ is a conjugation on $L$ lifting $\sigma$ and
$\widetilde{\sigma}_{i}\\!=\\!\widetilde{\sigma}|_{L_{i}}$ for $i\\!=\\!1,2$,
then the diagram (3.23) with (3.20) replaced by (3.22) commutes if and only if

$(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)\\!+\\!\big{(}g_{1}\\!-\\!1\\!+\\!\deg
L_{1}\big{)}\\!\big{(}g_{2}\\!-\\!1\\!+\\!\deg L_{2}\big{)}\in 2\mathbb{Z}.$

###### Proof.

Since the indices of
$D_{(L_{i}^{*}\oplus\sigma_{i}^{*}\overline{L}_{i}^{*},\sigma_{L_{i}^{*}}^{\oplus})}$,
$D_{2(L_{i}^{*},\widetilde{\sigma}_{i}^{*})}$, and
$2\bar{\partial}_{(\Sigma_{i},\sigma_{i})}$ are even, the outer isomorphisms
in (3.20) and (3.22) commute with the isomorphisms (2.4) associated with the
disjoint union of $(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$ and
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$. Since the restrictions of the
homotopy classes of isomorphisms (1.4) and (1.8) determined by
$(L,[\psi],\mathfrak{s})$ to $\Sigma_{i}$ are determined by
$(L_{i},[\psi_{i}],\mathfrak{s}_{i})$, the homotopy classes of the middle
isomorphisms in (3.20) and (3.22) also commute with the isomorphisms (2.4)
associated with the disjoint union of
$(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$ and
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$. The first claim of the corollary
thus follows from the first statement of Lemma 3.2 with $L$ replaced by
$L^{*}$ and the second statement of this lemma with
$L\\!=\\!\Sigma\\!\times\\!\mathbb{C}$. The second claim of the corollary
similarly follows from the second statement of Lemma 3.2 with $L$ replaced by
$L^{*}$ and with $L\\!=\\!\Sigma\\!\times\\!\mathbb{C}$. ∎

######  Corollary 3.7 (of Lemma 3.3).

Suppose $(\Sigma_{1},\mathfrak{j}_{1})$, $(\Sigma_{2},\mathfrak{j}_{2})$, $g$,
$(\Sigma,\sigma,\mathfrak{j})$, and $\sigma_{1}$ are as in Lemma 3.3. Let
$(V,\varphi)$ and $(L,[\psi],\mathfrak{s})$ be as in Corollary 3.6. If
$L_{i}\\!=\\!L|_{\Sigma_{i}}$ for $i\\!=\\!1,2$, then the composition

$\det
D_{(V,\varphi)}|_{\Sigma_{1}}\underset{\approx}{\xleftarrow{(\ref{Dblorient_e})}}\det
D_{(V,\varphi)}\underset{\approx}{\xrightarrow{(\ref{ROrientCompl_e2})}}\det\big{(}k\bar{\partial}_{(\Sigma,\sigma)}\big{)}\underset{\approx}{\xrightarrow{(\ref{Dblorient_e})}}\det\big{(}k\bar{\partial}_{\Sigma_{1}}\big{)}\,,$
(3.24)

with the middle isomorphism determined by $(L,[\psi],\mathfrak{s})$,
intertwines the complex orientations on the two ends if and only if
$g\\!-\\!1\\!+\\!\deg L_{2}\\!\in\\!2\mathbb{Z}$. If in addition
$\widetilde{\sigma}$ is a conjugation on $L$ lifting $\sigma$, then the
composition (3.24) with (3.20) replaced by (3.22) intertwines the complex
orientations on the two ends.

###### Proof.

Isomorphisms for direct sums of complex vector bundles as in (2.2) and the
restrictions of the isomorphisms (1.4) to the bundles over $\Sigma_{1}$
determine a homotopy class of isomorphisms

$\begin{split}\big{(}\\!\det
D_{(V,\varphi)}|_{\Sigma_{1}}\big{)}\\!\otimes\\!\big{(}\\!\det
D_{(L^{*}\oplus\sigma^{*}\overline{L}^{*},\sigma_{L^{*}}^{\oplus})}|_{\Sigma_{1}}\big{)}&\approx\det
D_{(V\oplus
L^{*}\oplus\sigma^{*}\overline{L}^{*},\varphi\oplus\sigma_{L^{*}}^{\oplus})}|_{\Sigma_{1}}\\\
&\approx\det\\!\big{(}\\!(k\\!+\\!2)\bar{\partial}_{\Sigma}\big{)}\approx\det\\!\big{(}k\bar{\partial}_{\Sigma}\big{)}\\!\otimes\\!\det\\!\big{(}2\bar{\partial}_{\Sigma}\big{)}.\end{split}$
(3.25)

Since the restrictions of isomorphisms (1.4) are complex trivializations
$V\\!\oplus\\!L^{*}\\!\oplus\\!\sigma^{*}\overline{L}^{*}|_{\Sigma_{1}}$,
(3.25) is orientation-preserving with respect to the complex orientations on
all determinant lines above. The isomorphisms (3.12) intertwine the
isomorphisms in (3.20) and (3.25). The first claim of the corollary thus
follows from the first statement of Lemma 3.3 with $L$ replaced by $L^{*}$ and
the second statement of this lemma with
$L\\!=\\!\Sigma\\!\times\\!\mathbb{C}$. The second claim of the corollary
similarly follows from the second statement of Lemma 3.3 with $L$ replaced by
$L^{*}$ and with $L\\!=\\!\Sigma\\!\times\\!\mathbb{C}$. ∎

######  Corollary 3.8 (of Lemma 3.4).

Suppose $(\Sigma,\sigma,\mathfrak{j})$, $\\{x^{+},x^{-}\\}$,
$(\widehat{\Sigma},\widehat{\sigma})$, and $q$ are as in Lemma 3.4. If
$(V,\varphi)$ and $(L,[\psi],\mathfrak{s})$ are as in Corollary 3.6, the
diagram

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(3.26)

with the homotopy classes of the horizontal isomorphisms determined by the
real orientations $q^{*}(L,[\psi],\mathfrak{s})$ and $(L,[\psi],\mathfrak{s})$
and by the complex orientations of $V_{x^{+}}$ and $\mathbb{C}^{k}$, does not
commute up to homotopy. If in addition $\widetilde{\sigma}$ is a conjugation
on $L$ lifting $\sigma$, then the diagram (3.26) with (3.20) replaced by
(3.22) commutes up to homotopy.

###### Proof.

Since the indices of
$D_{(L^{*}\oplus\sigma^{*}\overline{L}^{*},\sigma_{L^{*}}^{\oplus})}$,
$D_{2(L^{*},\widetilde{\sigma}^{*})}$, and $2\bar{\partial}_{(\Sigma,\sigma)}$
are even, the outer isomorphisms in (3.20) and (3.22) commute with the
corresponding isomorphisms (3.16). Since the pullbacks of the homotopy classes
of isomorphisms (1.4) and (1.8) determined by $(L,[\psi],\mathfrak{s})$ to
$\widehat{\Sigma}$ are determined by $q^{*}(L,[\psi],\mathfrak{s})$, the
homotopy classes of the middle isomorphisms in (3.20) and (3.22) also commute
with the corresponding isomorphisms (3.16). The first claim of the corollary
thus follows from the first statement of Lemma 3.4 with $L$ replaced by
$L^{*}$ and the second statement of this lemma with
$L\\!=\\!\Sigma\\!\times\\!\mathbb{C}$. The second claim of the corollary
similarly follows from the second statement of Lemma 3.4 with $L$ replaced by
$L^{*}$ and with $L\\!=\\!\Sigma\\!\times\\!\mathbb{C}$. ∎

######  Corollary 3.9 (of Lemma 3.5).

Suppose $(\Sigma,\sigma,\mathfrak{j})$, $x$, $g$,
$(\widehat{\Sigma},\widehat{\sigma})$, and $q$ are as in Lemma 3.5. If
$(V,\varphi)$ and $(L,[\psi],\mathfrak{s})$ are as in Corollary 3.6, the
diagram

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105.144pt\raise-42.89444pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\det\\!\big{(}k\bar{\partial}_{(\Sigma,\sigma)}\big{)}\\!\otimes\\!\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}\mathbb{C}^{k}/\mathbb{R}^{k}\big{)}\,,}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(3.27)

with the homotopy classes of the horizontal isomorphisms determined by the
real orientations $q^{*}(L,[\psi],\mathfrak{s})$ and
$(L,[\psi],\mathfrak{s})$, the induced orientation of $V_{x}^{\varphi}$, and
the standard orientation of $\mathbb{C}^{k}/\mathbb{R}^{k}$, commutes up to
homotopy if and only if $g\\!-\\!1\\!\in\\!2\mathbb{Z}$. If in addition
$\widetilde{\sigma}$ is a conjugation on $L$ lifting $\sigma$, then the
diagram (3.27) with (3.20) replaced by (3.22) commutes up to homotopy if and
only if $\deg L\\!\in\\!2\mathbb{Z}$.

###### Proof.

The proof is identical to that of Corollary 3.8, with Lemma 3.4 replaced by
3.5. ∎

By Definition 1.1(RO3), a real orientation $(L,[\psi],\mathfrak{s})$ on
$(V,\varphi)$ determines a relative spin structure on the real bundle
$V^{\varphi}$ over $X^{\phi}\\!\subset\\!X$ in the spirit of [8, Definition
8.1.2], which we call the the associated relative spin structure on
$V^{\varphi}$. If in addition $\widetilde{\phi}$ is a conjugation on $L$
lifting $\phi$ and $L^{\widetilde{\phi}}\\!\longrightarrow\\!X^{\phi}$ is
orientable, then $2(L^{*})^{\widetilde{\phi}^{*}}$ has a canonical homotopy
class of trivializations as in the proof of [17, Corollary 5.6]. Such a real
orientation on $(V,\varphi)$ thus determines a spin structure on
$V^{\varphi}$, which we call the the associated spin structure on
$V^{\varphi}$.

Let $D_{(V,\varphi)}$ be a real CR-operator on a real bundle pair
$(V,\varphi)$ over $(\mathbb{P}^{1},\tau)$. By [23, Lemma 6.37], a (relative)
spin structure on the real vector bundle $V^{\varphi}$ over the $\tau$-fixed
locus $S^{1}\\!\subset\\!\mathbb{P}^{1}$ and a choice of a disk
$\mathbb{D}\\!\subset\\!\mathbb{P}^{1}$ determine an orientation on $\det
D_{(V,\varphi)}$. If $(V,\varphi)$ admits a real orientation, then $\deg
V\\!\in\\!2\mathbb{Z}$ and this orientation does not depend on the choice of
$\mathbb{D}$. Since the kernel of the surjective real CR-operator
$\bar{\partial}_{(\mathbb{P}^{1},\tau)}$ consists of the constant
$\mathbb{R}$-valued functions on $\mathbb{P}^{1}$,
$\det\bar{\partial}_{(\mathbb{P}^{1},\tau)}$ is canonically oriented. A
homotopy class of isomorphisms as in (3.21) thus also determines an
orientation on $\det D_{(V,\varphi)}$. Any complex vector bundle $L$ over
$(\mathbb{P}^{1},\tau)$ admits a conjugation $\widetilde{\tau}$ lifting
$\tau$. The real vector bundle $L^{\widetilde{\tau}}$ is orientable if and
only if $\deg L\\!\in\\!2\mathbb{Z}$.

######  Corollary 3.10.

Suppose $(V,\varphi)$ is a real bundle pair over $(\mathbb{P}^{1},\tau)$,
$(L,[\psi],\mathfrak{s})$ is a real orientation for $(V,\varphi)$, and
$D_{(V,\varphi)}$ is a real CR-operator. The orientations on $\det
D_{(V,\varphi)}$ induced by the associated relative spin structure on
$V^{\varphi}$ and by $(L,[\psi],\mathfrak{s})$ via (3.20) are the same if and
only if $\deg V\\!\in\\!4\mathbb{Z}$. If in addition $\widetilde{\tau}$ is a
conjugation on $L$ lifting $\tau$, the former orientation and the orientation
induced by $(L,[\psi],\mathfrak{s})$ via (3.22) with
$\widetilde{\sigma}\\!=\\!\widetilde{\tau}$ are the same if and only if $\deg
V\\!\cong\\!0,-2$ $\mod 8$. If $\deg V\\!\in\\!4\mathbb{Z}$, the
orientation on $\det D_{(V,\varphi)}$ induced by the associated spin structure
and the latter orientation are the same.

###### Proof.

The third claim follows from [18, Lemma 3.3] with replaced by $L^{*}$. The
$\deg V\\!\in\\!4\mathbb{Z}$ and $\deg V\\!-\\!2\\!\in\\!4\mathbb{Z}$ cases of
the second claim follow from [18, Lemma 3.4] and [18, Corollary 3.6],
respectively, with $L$ replaced by $L^{*}$ and thus with $\deg L$ replaced by
$-(\deg V)/2$; they are established via technical computations. The first
claim follows from the second and Lemma 3.1 with $L$ replaced by $L^{*}$. ∎

###  3.3 Orientations of moduli spaces

Let $g,\ell\\!\in\\!\mathbb{Z}$ with $g\\!+\\!\ell\\!\geq\\!2$. Denote by
$\mathbb{R}\mathcal{M}_{g,\ell}^{\bullet}\\!\subset\\!\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet}$
the subspace of smooth curves. The determinants of the operators
$\bar{\partial}_{(\Sigma,\sigma)}$ on symmetric Riemann surfaces form a real
line orbi-bundle $\det\bar{\partial}_{\mathbb{C}}$ over
$\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet}$. By [17, Propositions
5.9,6.1], the real line orbi-bundle

$\Lambda^{\textnormal{top}}_{\mathbb{R}}\big{(}T(\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet})\\!\big{)}\otimes\big{(}\\!\det\bar{\partial}_{\mathbb{C}}\big{)}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet}$
(3.28)

has a natural orientation. The construction of this orientation over the
equivalence class
$[\mathcal{C}]\\!\in\\!\mathbb{R}\mathcal{M}_{g,\ell}^{\bullet}$ of a smooth
marked curve $\mathcal{C}$ with the underlying symmetric Riemann surface
$(\Sigma,\sigma,\mathfrak{j})$ involves:

  1. ($\mathcal{M}1$)

the Kodaira-Spencer and Dolbeault isomorphisms that identify
$T_{[\mathcal{C}]}(\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet})$ with
the cokernel $H^{1}(\Sigma;T\mathcal{C})^{\varphi}$ of an injective real CR-
operator on a real line bundle pair $(T\mathcal{C},\varphi)$ over
$(\Sigma,\sigma)$;

  2. ($\mathcal{M}2$)

the Serre Duality that identifies $H^{1}(\Sigma;T\mathcal{C})^{\varphi}$ with
the dual
$(H^{0}(\Sigma;T\mathcal{C}^{*}\\!\otimes\\!T^{*}\Sigma)^{\varphi*\otimes\textnormal{d}\sigma^{*}})^{*}$
of the kernel of a surjective real CR-operator
$\bar{\partial}_{(T\mathcal{C}^{*}\otimes
T^{*}\Sigma,\varphi*\otimes\textnormal{d}\sigma^{*})}$ on the real bundle pair
$(T\mathcal{C}^{*}\\!\otimes\\!T^{*}\Sigma,\varphi^{*}\\!\otimes\\!\textnormal{d}\sigma^{*})$
over $(\Sigma,\sigma)$;

  3. ($\mathcal{M}3$)

an exact triple of Fredholm operators that identifies
$\det\bar{\partial}_{(T\mathcal{C}^{*}\otimes
T^{*}\Sigma,\varphi*\otimes\textnormal{d}\sigma^{*})}$ with the tensor product
of $\det\bar{\partial}_{(T^{*}\Sigma^{\otimes
2},\textnormal{d}\sigma^{*\otimes 2})}$ and the top exterior power of a
complex vector space and thus determines an orientation on the tensor product
of the two determinant lines;

  4. ($\mathcal{M}4$)

a homotopy class of isomorphisms $\det\bar{\partial}_{(T^{*}\Sigma^{\otimes
2},\textnormal{d}\sigma^{*\otimes
2})}\\!\approx\\!\det\bar{\partial}_{(\Sigma,\sigma)}$ obtained via (3.22)
from a canonical real orientation on $(T^{*}\Sigma^{\otimes
2},\textnormal{d}\sigma^{*\otimes 2})$ with $L\\!=\\!T^{*}\Sigma$ and
involution $\textnormal{d}\sigma^{*}$;

  5. ($\mathcal{M}5$)

twisting the resulting homotopy class of isomorphisms
$\Lambda^{\textnormal{top}}_{\mathbb{R}}(T_{[\mathcal{C}]}(\mathbb{R}\overline{\mathcal{M}}_{g,\ell}^{\bullet})\\!)\\!\approx\\!\det\bar{\partial}_{(\Sigma,\sigma)}$
by $(-1)^{(g-1)+|\sigma|}$, where $|\sigma|$ is the number of connected
components of $\Sigma^{\sigma}$.

The parity of each of the two summands in the twisting exponent in
($\mathcal{M}5$) is invariant under disjoint unions of symmetric surfaces and
under the first node-identifying immersion in (3.36) below. This exponent is
even if the domain is a doublet as in (3.10) or $(\mathbb{P}^{1},\tau)$. Thus,
the twisting in ($\mathcal{M}5$) can be ignored for the purposes of Lemma 3.11
and Propositions 3.12-3.14 and 3.18 below.

Let $(X,\omega,\phi)$ be a real symplectic manifold of real dimension $2n$ and
$J\\!\in\\!\mathcal{J}_{\omega}^{\phi}$. We denote by

$\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)\subset\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$

the subspace of maps from smooth stable domains and by

$\mathfrak{f}\\!:\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)\longrightarrow\mathbb{R}\mathcal{M}_{g,\ell}^{\bullet}$

the forgetful morphism dropping the map component. The determinants of the
linearizations of the $\bar{\partial}_{J}$-operators on the space of real maps
from symmetric surfaces to $(X,\phi)$ form a real line orbi-bundle $\det
D_{(TX,\textnormal{d}\phi)}$ over $\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)$;
see [17, Section 4.3]. The forgetful morphism $\mathfrak{f}$ induces an
isomorphism

$\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}T\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)\\!\big{)}\approx\big{(}\\!\det
D_{(TX,\textnormal{d}\phi)}\big{)}\\!\otimes\\!\mathfrak{f}^{*}\\!\big{(}\Lambda^{\textnormal{top}}_{\mathbb{R}}(T\mathbb{R}\mathcal{M}_{g,\ell}^{\bullet})\\!\big{)}$
(3.29)

of real line orbi-bundles over $\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)$; the
line orbi-bundle on the left-hand side is the top exterior power of the
virtual tangent bundle for the moduli space
$\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)$. A real orientation
$(L,[\psi],\mathfrak{s})$ on $(X,\omega,\phi)$ determines a homotopy class of
isomorphisms

$\det
D_{(TX,\textnormal{d}\phi)}\approx\mathfrak{f}^{*}\\!\big{(}\\!(\det\bar{\partial}_{\mathbb{C}})^{\otimes
n}\big{)}$ (3.30)

of real line orbi-bundles over $\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)$ via
(3.20), with $(L,[\psi],\mathfrak{s})$ replaced by
$u^{*}(L,[\psi],\mathfrak{s})$ for each stable map $u$ representing an element
of $\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)$. If in addition
$\widetilde{\phi}$ is a conjugation on $L$ lifting $\phi$, a homotopy class of
isomorphisms (3.30) is similarly determined via (3.22).

Combining (3.28)-(3.30), we obtain a homotopy class of isomorphisms

$\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}T\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)\\!\big{)}\approx\mathfrak{f}^{*}\\!\big{(}\\!(\det\bar{\partial}_{\mathbb{C}})^{\otimes(n+1)}\big{)}$
(3.31)

of real line orbi-bundles over $\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)$. If
$n\\!\not\in\\!2\mathbb{Z}$, we can break the right-hand side into
$(n\\!+\\!1)/2$ products of
$\mathfrak{f}^{*}(\\!(\det\bar{\partial}_{\mathbb{C}})^{\otimes 2})$, each of
which is canonically oriented, and thus obtain an orientation on
$\mathfrak{M}_{g,\ell}^{\phi;\star}(B;J)$. Via the stabilization of the domain
as in the proof of [17, Corollary 5.10], this orientation extends over the
subspace

$\mathfrak{M}_{g,\ell}^{\phi;\bullet}(B;J)\subset\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$

of all maps from smooth domains. By the proof of [17, Theorem 1.3], the
extended orientation further extends over the rest of
$\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$.

For $g_{1},g_{2}\\!\in\\!\mathbb{Z}$ and
$\ell_{1},\ell_{2}\\!\in\\!\mathbb{Z}^{\geq 0}$, let

$\iota_{\sqcup}\\!:\mathbb{R}\overline{\mathcal{M}}_{g_{1},\ell_{1}}^{\bullet}\\!\times\\!\mathbb{R}\overline{\mathcal{M}}_{g_{2},\ell_{2}}^{\bullet}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{g_{1}+g_{2}-1,\ell_{1}+\ell_{2}}^{\bullet}$

be the open embedding obtained by taking the disjoint union of two marked
curves and re-ordering their conjugate pairs of marked points in some fixed
way. If
$[\mathbf{u}_{i}]\\!\in\\!\overline{\mathfrak{M}}_{g_{i},\ell_{i}}^{\phi;\bullet}(B_{i};J)$
for $i\\!=\\!1,2$, we similarly define

$\big{[}\mathbf{u}_{1}\\!\sqcup\\!\mathbf{u}_{2}\big{]}\in\overline{\mathfrak{M}}_{g_{1}+g_{2}-1,\ell_{1}+\ell_{2}}^{\phi;\bullet}(B_{1}\\!+\\!B_{2};J)$

to be the equivalence class of the stable map obtained by taking the disjoint
union of the two maps and ordering the conjugate pairs of marked points of the
two maps in some fixed way.

######  Lemma 3.11.

Suppose $g_{1},g_{2}\\!\in\\!\mathbb{Z}$ and
$\ell_{1},\ell_{2}\\!\in\\!\mathbb{Z}^{\geq 0}$. Let

$\pi_{1},\pi_{2}\\!:\mathbb{R}\overline{\mathcal{M}}_{g_{1},\ell_{1}}^{\bullet}\\!\times\\!\mathbb{R}\overline{\mathcal{M}}_{g_{2},\ell_{2}}^{\bullet}\longrightarrow\mathbb{R}\overline{\mathcal{M}}_{g_{1},\ell_{1}}^{\bullet},\mathbb{R}\overline{\mathcal{M}}_{g_{2},\ell_{2}}^{\bullet}$

be the component projections. The diagram

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(3.32)

with the homotopy classes of the horizontal isomorphisms determined by the
orientations of the line orbi-bundles (3.28), commutes.

###### Proof.

Let
$([\mathcal{C}_{1}],[\mathcal{C}_{2}])\\!\in\\!\mathbb{R}\overline{\mathcal{M}}_{g_{1},\ell_{1}}^{\bullet}\\!\times\\!\mathbb{R}\overline{\mathcal{M}}_{g_{2},\ell_{2}}^{\bullet}$
and $[\mathcal{C}]\\!=\\!\iota_{\sqcup}([\mathcal{C}_{1}],[\mathcal{C}_{2}])$.
It can be assumed that the unmarked domains
$(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$ and
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$ of $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$, respectively, are smooth and thus so is their disjoint
union $(\Sigma,\sigma,\mathfrak{j})$. We denote the top exterior powers of the
relevant tangent spaces of $[\mathcal{C}_{i}]$ and $[\mathcal{C}]$ by
$\mathcal{M}_{i}$ and $\mathcal{M}$, respectively, and the left vertical
isomorphism in (3.32) by $\Lambda_{\mathbb{R}}^{\textnormal{top}}$. The
horizontal isomorphisms in (3.32) factor through the horizontal isomorphisms
in the diagram

$\textstyle{\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\approx}$$\scriptstyle{\Lambda_{\mathbb{R}}^{\textnormal{top}}}$($\mathcal{M}1$)-($\mathcal{M}3$)$\scriptstyle{\approx}$$\textstyle{\det\bar{\partial}_{2(T\Sigma,\textnormal{d}\sigma)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{Runionorient_e0})}$$\scriptstyle{\approx}$$\scriptstyle{(\ref{reldetdfn_e0})}$$\scriptstyle{\approx}$$\textstyle{\big{(}\\!\det\bar{\partial}_{(\Sigma,\sigma)}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{Runionorient_e0})}$$\scriptstyle{\approx}$$\textstyle{\mathcal{M}_{1}\\!\otimes\\!\mathcal{M}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$($\mathcal{M}1$)-($\mathcal{M}3$)$\scriptstyle{\approx}$$\textstyle{\big{(}\\!\det\bar{\partial}_{2(T\Sigma_{1},\textnormal{d}\sigma_{1})}\big{)}\\!\otimes\\!\big{(}\\!\det\bar{\partial}_{2(T\Sigma_{2},\textnormal{d}\sigma_{2})}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{reldetdfn_e0})}$$\scriptstyle{\approx}$$\textstyle{\big{(}\\!\det\bar{\partial}_{(\Sigma_{1},\sigma_{1})}\big{)}\\!\otimes\\!\big{(}\\!\det\bar{\partial}_{(\Sigma_{2},\sigma_{2})}\big{)}\,.}$

Since each of Steps ($\mathcal{M}1$)-($\mathcal{M}3$) commutes with disjoint
unions, the left rectangle in this diagram commutes. By the second claim of
Corollary 3.6, the right rectangle in this diagram commutes up to homotopy
because

$(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)\\!+\\!\big{(}g_{1}\\!-\\!1\\!+\\!\deg
T\Sigma_{1}\big{)}\\!\big{(}g_{2}\\!-\\!1\\!+\\!\deg T\Sigma_{2}\big{)}\in
2\mathbb{Z}.$

Combining these two statements, we obtain the claim. ∎

######  Proposition 3.12.

Suppose $(X,\omega,\phi)$, $n$, $J$, and $(L,[\psi],\mathfrak{s})$ are as in
Theorem 1.2 and $g_{i},\ell_{i}\\!\in\\!\mathbb{Z}$,
$B_{i}\\!\in\\!H_{2}(X;\mathbb{Z})$, and
$[\mathbf{u}_{i}]\\!\in\\!\overline{\mathfrak{M}}_{g_{i},\ell_{i}}^{\phi;\bullet}(B_{i};J)$
for $i\\!=\\!1,2$. The natural isomorphism

$T_{[\mathbf{u}_{1}\sqcup\mathbf{u}_{2}]}\big{(}\overline{\mathfrak{M}}_{g_{1}+g_{2}-1,\ell_{1}+\ell_{2}}^{\phi;\bullet}(B_{1}\\!+\\!B_{2};J)\\!\big{)}\approx
T_{[\mathbf{u}_{1}]}\big{(}\overline{\mathfrak{M}}_{g_{1},\ell_{1}}^{\phi;\bullet}(B_{1};J)\\!\big{)}\\!\oplus\\!T_{[\mathbf{u}_{2}]}\big{(}\overline{\mathfrak{M}}_{g_{2},\ell_{2}}^{\phi;\bullet}(B_{2};J)\\!\big{)}$
(3.33)

is orientation-preserving with respect to the orientations induced by
$(L,[\psi],\mathfrak{s})$ if and only if

$(n\\!-\\!1)(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)/2\in 2\mathbb{Z}.$

If in addition $\widetilde{\phi}$ is a conjugation on $L$ lifting $\phi$ and
the homotopy class of isomorphisms (3.30) is obtained via (3.22), then this
isomorphism is orientation-preserving if and only if

$(n\\!-\\!1)(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)/2\\!+\\!\big{(}g_{1}\\!-\\!1\\!+\\!\langle
c_{1}(X,\omega),B_{1}\rangle/2\big{)}\big{(}g_{2}\\!-\\!1\\!+\\!\langle
c_{1}(X,\omega),B_{2}\rangle/2\big{)}\in 2\mathbb{Z}\,.$

###### Proof.

The second claim follows from the first and Lemma 3.1 applied three times. It
can be assumed that the domains $(\Sigma_{1},\sigma_{1},\mathfrak{j}_{1})$ and
$(\Sigma_{2},\sigma_{2},\mathfrak{j}_{2})$ of $[\mathbf{u}_{1}]$ and
$[\mathbf{u}_{2}]$, respectively, are smooth and stable with their markings
and thus so is their disjoint union $(\Sigma,\sigma,\mathfrak{j})$. Let
$\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}$ be the marked domains of
$\mathbf{u}_{1},\mathbf{u}_{2},\mathbf{u}_{1}\\!\sqcup\\!\mathbf{u}_{2}$,
respectively, and $u_{1},u_{2},u$ their map components. We denote the top
exterior powers of the relevant tangent spaces of
$[\mathcal{C}_{i}],[\mathcal{C}]$ and
$[\mathbf{u}_{i}],[\mathbf{u}_{1}\\!\sqcup\\!\mathbf{u}_{2}]$ by
$\mathcal{M}_{i},\mathcal{M}$ and $\mathfrak{M}_{i},\mathfrak{M}$,
respectively, and the determinants of the real CR-operators
$D_{u_{i}^{*}(TX,\textnormal{d}\phi)},D_{u^{*}(TX,\textnormal{d}\phi)}$ and
$\bar{\partial}_{(\Sigma_{i},\sigma_{i})},\bar{\partial}_{(\Sigma,\sigma)}$ by
$D_{i},D$ and $\bar{\partial}_{i},\bar{\partial}$, respectively.

$\textstyle{\mathfrak{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{RfMisom_e})}$$\textstyle{D\\!\otimes\\!\mathcal{M}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{Runionorient_e0})}$$\scriptstyle{(\ref{reldetdfn_e0})}$$\scriptstyle{(\ref{RcMisom_e})}$$\textstyle{\bar{\partial}^{\otimes
n}\\!\otimes\\!\bar{\partial}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{Runionorient_e0})}$$\textstyle{(D_{1}\\!\otimes\\!D_{2})\\!\otimes\\!(\mathcal{M}_{1}\\!\otimes\\!\mathcal{M}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{reldetdfn_e0})}$$\scriptstyle{(\ref{RcMisom_e})}$$\textstyle{\big{(}\bar{\partial}_{1}^{\otimes
n}\\!\otimes\\!\bar{\partial}_{2}^{\otimes
n}\big{)}\\!\otimes\\!\big{(}\bar{\partial}_{1}\\!\otimes\\!\bar{\partial}_{2}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{M}_{1}\\!\otimes\\!\mathfrak{M}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{RfMisom_e})}$$\textstyle{(D_{1}\\!\otimes\\!\mathcal{M}_{1})\\!\otimes\\!(D_{2}\\!\otimes\\!\mathcal{M}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{reldetdfn_e0})}$$\scriptstyle{(\ref{RcMisom_e})}$$\textstyle{\big{(}\bar{\partial}_{1}^{\otimes
n}\\!\otimes\\!\bar{\partial}_{1}\big{)}\\!\otimes\\!\big{(}\bar{\partial}_{2}^{\otimes
n}\\!\otimes\\!\bar{\partial}_{2}\big{)}}$

$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{Runionorient_e0})}$$\scriptstyle{(\ref{reldetdfn_e0})}$$\textstyle{\bar{\partial}^{\otimes
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{Runionorient_e0})}$$\textstyle{D_{1}\\!\otimes\\!D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\ref{reldetdfn_e0})}$$\textstyle{\bar{\partial}_{1}^{\otimes
n}\\!\otimes\\!\bar{\partial}_{2}^{\otimes n}}$

Figure 2: Isomorphisms inducing the orientations of Proposition 3.12

The tangent spaces in the statement of the proposition are oriented by the top
and bottom rows of the top diagram in Figure 2 from

  1. ($\mathfrak{o}1$)

the canonical orientations on the $(n\\!+\\!1)/2$ factors of
$\bar{\partial}^{\otimes 2},\bar{\partial}_{i}^{\otimes 2}$,

  2. ($\mathfrak{o}2$)

the projection (resp. canonical) orientations on the determinants of real CR-
operators on the real bundle pairs
$u^{*}(L^{*}\\!\oplus\\!\phi^{*}\overline{L}^{*},\phi_{L^{*}}^{\oplus}),u_{i}^{*}(L^{*}\\!\oplus\\!\phi^{*}\overline{L}^{*},\phi_{L^{*}}^{\oplus})$
in the first (resp.
$2u^{*}(L^{*},\widetilde{\phi}^{*}),2u_{i}^{*}(L^{*},\widetilde{\phi}^{*})$ in
the second) claim of the proposition.

The vertical isomorphisms at the bottom of this diagram interchange the middle
factors in their domains and multiply the result by $(-1)$ to the power of the
index of $D_{2}$ times the dimension of $\mathcal{M}_{1}$. Since
$n\\!\not\in\\!2\mathbb{Z}$ and $\langle
c_{1}(X,\omega),B_{2}\rangle\\!\in\\!2\mathbb{Z}$,

$\begin{split}\big{(}\textnormal{ind}\,D_{2}\big{)}\big{(}\\!\dim\mathcal{M}_{1}\big{)}&\cong(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)\\\
&\cong\big{(}\textnormal{ind}\,n\bar{\partial}_{(\Sigma_{2},\sigma_{2})}\big{)}\\!\big{(}\textnormal{ind}\,\bar{\partial}_{1}\big{)}\mod
2\end{split}$

The two rectangles with solid arrows in this diagram commute.

The isomorphisms in the upper right rectangle in the top diagram are the
tensor products of the isomorphisms in the bottom diagram and in (3.32). For
the purposes of (2.4), $\bar{\partial}^{\otimes n}$ is viewed as the
determinant of the CR-operator $n\bar{\partial}_{(\Sigma,\sigma)}$ on
$(\Sigma\\!\times\\!\mathbb{C}^{n},\sigma\\!\times\\!\mathfrak{c})$ in these
diagrams. By Corollary 3.6, the bottom diagram would commute up to homotopy if
the dashed arrow were multiplied by $(-1)$ to the power of

$(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)\quad\Big{(}\hbox{resp.~{}}(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)\\!+\\!\big{(}g_{1}\\!-\\!1\\!+\\!\langle
c_{1}(L),B_{1}\rangle\\!\big{)}\\!\big{(}g_{2}\\!-\\!1\\!+\\!\langle
c_{1}(L),B_{2}\rangle\\!\big{)}\\!\Big{)}$ (3.34)

in the first (resp. second) claim of the proposition. Along with Lemma 3.11,
this implies that the upper right rectangle in the top diagram would commute
up to homotopy if the dashed arrow were multiplied by $(-1)$ to the power of
(3.34).

The pullback of the canonical orientation on
$\bar{\partial}_{1}^{\otimes(n+1)}\\!\otimes\\!\bar{\partial}_{2}^{\otimes(n+1)}$
by the bottom right vertical arrow in the top diagram (with the specified
sign) and the top right vertical arrow (without any sign) differs from the
canonical orientation on $\bar{\partial}^{\otimes(n+1)}$ by $(-1)$ to the
power of

$\binom{n\\!+\\!1}{2}\big{(}\textnormal{ind}\,\bar{\partial}_{1}\big{)}\big{(}\textnormal{ind}\,\bar{\partial}_{2}\big{)}\cong(n\\!+\\!1)(g_{1}\\!-\\!1)(g_{2}\\!-\\!1)/2\mod
2.$

Combining this with the correcting sign (3.34) for the top right vertical
arrow, we obtain the claim. ∎

For $g\\!\in\\!\mathbb{Z}$, $\ell\\!\in\\!\mathbb{Z}^{\geq 0}$,
$B\\!\in\\!H_{2}(X;\mathbb{Z})$, and $J\\!\in\\!\mathcal{J}_{\omega}$, we
denote by $\overline{\mathfrak{M}}_{g,\ell}^{\bullet}(B;J)$ the moduli space
of stable $J$-holomorphic degree $B$ maps from closed, possibly nodal and
disconnected, Riemann surfaces of arithmetic genus $g$ with $\ell$ marked
points. If in addition $[\ell]\\!=\\!S^{+}\\!\sqcup\\!S^{-}$, let

$\iota_{g;S^{+},S^{-}}\\!:\overline{\mathfrak{M}}_{g,\ell}^{\bullet}(B;J)\longrightarrow\overline{\mathfrak{M}}_{2g-1,\ell}^{\phi;\bullet}(B\\!-\\!\phi_{*}(B);J\big{)}$

be the open immersion sending the equivalence class of a stable map
$\mathbf{u}$ with the domain $(\Sigma,\mathfrak{j})$ and map component $u$ to
the equivalence class of the map

$u\\!\sqcup\\!\\{\phi\\!\circ\\!u\\}\\!:\big{(}\Sigma\\!\sqcup\\!\Sigma,\mathfrak{j}\\!\sqcup\\!(-\mathfrak{j})\\!\big{)}\longrightarrow(X,J)$

with the involution $\sigma$ interchanging the two copies of $\Sigma$ and the
first copy of $\Sigma$ carrying the marked points $z_{i}^{+}$ with
$i\\!\in\\!S^{+}$ and $z_{i}^{-}$ with $i\\!\in\\!S^{-}$ at the positions of
the corresponding marked points $z_{i}$ of $\mathbf{u}$.

######  Proposition 3.13.

Suppose $(X,\omega,\phi)$, $J$, $(L,[\psi],\mathfrak{s})$, $g$, $\ell$, and
$B$ are as in Theorem 1.2, $[\ell]\\!=\\!S^{+}\\!\sqcup\\!S^{-}$, and
$[\mathbf{u}]\\!\in\\!\overline{\mathfrak{M}}_{g,\ell}^{\bullet}(B;J)$. The
isomorphism

$\textnormal{d}_{[\mathbf{u}]}\iota_{g;S^{+},S^{-}}\\!:T_{[\mathbf{u}]}\big{(}\overline{\mathfrak{M}}_{g,\ell}^{\bullet}(B;J)\\!\big{)}\longrightarrow
T_{\iota_{g;S^{+},S^{-}}([\mathbf{u}])}\overline{\mathfrak{M}}_{2g-1,\ell}^{\phi;\bullet}\big{(}B\\!-\\!\phi_{*}(B);J\big{)}$

is orientation-preserving with respect to the orientation on the target
induced by $(L,[\psi],\mathfrak{s})$ via (3.20) and the complex orientation on
the domain if and only if $\langle
c_{1}(L),\phi_{*}(B)\rangle\\!+\\!|S^{-}|\\!\in\\!2\mathbb{Z}$. If in addition
$\widetilde{\phi}$ is a conjugation on $L$ lifting $\phi$ and the homotopy
class of isomorphisms (3.30) is obtained via (3.22), then this isomorphism is
orientation-preserving if and only if
$(g\\!-\\!1)\\!+\\!|S^{-}|\\!\in\\!2\mathbb{Z}$.

###### Proof.

The second claim is [18, Theorem 1.4]; it also follows from the first claim
and Lemma 3.1 with $g$, $\textnormal{rk}_{\mathbb{C}}\\!L$, and $\deg L$
replaced by $2g\\!-\\!1$, 1, and $-\langle c_{1}(X,\omega),B\rangle$,
respectively. The first claim stated for certain one-dimensional targets
$(X,\omega,\phi)$ is [10, Lemma 3.1], but its proof is a slight modification
of the proof of [18, Theorem 1.4] summarized below and applies in general. For
$g\\!\in\\!\mathbb{Z}$ and $\ell\\!\in\\!\mathbb{Z}^{\geq 0}$, we denote by
$\overline{\mathcal{M}}_{g,\ell}^{\bullet}$ the Deligne-Mumford moduli space
of stable closed, possibly nodal and disconnected, Riemann surfaces
$(\Sigma,\mathfrak{j})$ of arithmetic genus $g$ with $\ell$ marked points and
by

$\det\bar{\partial}_{\mathbb{C}}\longrightarrow\overline{\mathcal{M}}_{g,\ell}^{\bullet}$

the real line bundle formed by the determinants of the standard
$\bar{\partial}$-operators on Riemann surfaces.

As with Proposition 3.12, it is sufficient to assume that the domain $\Sigma$
of $[\mathbf{u}]$ is smooth and is stable with its markings. In both cases,
the proof then starts with the isomorphism (3.29). By [18, Lemma 3.2], the
isomorphism

$\Lambda^{\textnormal{top}}_{\mathbb{R}}\big{(}T(\mathbb{R}\mathcal{M}_{2g-1,\ell}^{\bullet})\\!\big{)}\\!\otimes\\!\big{(}\\!\det\bar{\partial}_{\mathbb{C}}\big{)}\\!\Big{|}_{\iota_{g;S^{+},S^{-}}([\mathbf{u}])}\longrightarrow\Lambda^{\textnormal{top}}_{\mathbb{R}}\big{(}T\mathcal{M}_{g,\ell}^{\bullet}\big{)}\\!\otimes\\!\big{(}\\!\det\bar{\partial}_{\mathbb{C}}\big{)}\\!\Big{|}_{\mathbf{u}}$

obtained by restricting to the first copy of $\Sigma$ similarly to (3.12)
respects the natural orientation of the domain provided by [17, Proposition
5.9] and the complex orientation on the target if and only if
$(g\\!-\\!1)\\!+\\!|S^{-}|$ is even. The summand $(g\\!-\\!1)$ arises from
Step ($\mathcal{M}2$) because the two natural complex orientations of the real
dual of a finite-dimensional complex vector space are the same if and only if
its complex dimension is even; see the beginning of [18, Section 3.1]. The
summand $|S^{-}|$ arises from Steps ($\mathcal{M}3$) and ($\mathcal{M}2$)
because the complex vector space in ($\mathcal{M}3$) is the direct sum of the
cotangent spaces at the marked points $z_{i}^{+}$ (and it is dualized in Step
($\mathcal{M}2$)). The analogous comparison of the orientation on the first
factor on the right-hand side of (3.29) induced by $(L,[\psi],\mathfrak{s})$
and the orientation obtained by restricting to the first copy of $\Sigma$ is
provided by Corollary 3.7. ∎

###  3.4 Orientations of nodal strata

Let $(X,\omega,\phi)$, $J$, $g,\ell$, and $B$ be as in Theorem 1.2. We denote
by $\mathcal{N}_{X}X^{\phi}$ the normal bundle of $X^{\phi}$ in $X$. The
symplectic orientation on $X$ and an orientation on $X^{\phi}$, such as one
induced by a real orientation $(L,[\psi],\mathfrak{s})$ on $(X,\omega,\phi)$,
determine an orientation on $\mathcal{N}_{X}X^{\phi}$ via the short exact
sequence

$0\longrightarrow TX^{\phi}\longrightarrow
TX|_{X^{\phi}}\longrightarrow\mathcal{N}_{X}X^{\phi}\longrightarrow 0$

of real vector bundles over $X^{\phi}$. For each $i\\!\in\\![\ell]$, let

$\textnormal{ev}_{i}\\!:\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)\longrightarrow
X\qquad\hbox{and}\qquad\mathcal{L}_{i},\mathcal{L}_{i}^{-}\longrightarrow\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$

be the natural evaluation map at the marked point $z_{i}^{+}$ and the
universal tangent line orbi-bundles at the marked points $z_{i}^{+}$ and
$z_{i}^{-}$, respectively. The involutions $\sigma$ on the domains of the
elements of $\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$ induce an
$\mathbb{C}$-antilinear isomorphism

$\Phi_{i}\\!:\mathcal{L}_{i}\longrightarrow\mathcal{L}_{i}^{-}$

covering the identity of
$\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$. Let

$\big{(}\mathcal{L}_{i}\\!\otimes\\!\mathcal{L}_{i}^{-}\big{)}^{\mathbb{R}}=\big{\\{}rv\\!\otimes_{\mathbb{C}}\\!\Phi_{i}(v)\\!:v\\!\in\\!\mathcal{L}_{i},\,r\\!\in\\!\mathbb{R}\big{\\}}.$

This real line bundle over
$\overline{\mathfrak{M}}_{g,\ell}^{\phi;\bullet}(B;J)$ is canonically oriented
by the standard orientation of $\mathbb{R}$.

Define

$\begin{split}\overline{\mathfrak{M}}_{g-2,\ell+2}^{\prime\phi;\bullet}(B;J)&=\big{\\{}[\mathbf{u}]\\!\in\\!\overline{\mathfrak{M}}_{g-2,\ell+2}^{\phi;\bullet}(B;J)\\!:\,\textnormal{ev}_{\ell+1}([\mathbf{u}])\\!=\\!\textnormal{ev}_{\ell+2}([\mathbf{u}])\\!\big{\\}},\\\
\overline{\mathfrak{M}}_{g-2,\ell+2}^{\prime\prime\phi;\bullet}(B;J)&=\big{\\{}[\mathbf{u}]\\!\in\\!\overline{\mathfrak{M}}_{g-1,\ell+1}^{\phi;\bullet}(B;J)\\!:\,\textnormal{ev}_{\ell+1}([\mathbf{u}])\\!\in\\!X^{\phi}\\!\big{\\}}.\end{split}$

The short exact sequences

$\begin{split}0&\longrightarrow
T\overline{\mathfrak{M}}_{g-2,\ell+2}^{\prime\phi;\bullet}(B;J)\longrightarrow
T\overline{\mathfrak{M}}_{g-2,\ell+2}^{\phi;\bullet}(B;J)\\!\big{|}_{\overline{\mathfrak{M}}_{g-2,\ell+2}^{\prime\phi;\bullet}(B;J)}\longrightarrow\textnormal{ev}_{\ell+1}^{\,*}TX\longrightarrow
0,\\\ 0&\longrightarrow
T\overline{\mathfrak{M}}_{g-1,\ell+1}^{\prime\prime\phi;\bullet}(B;J)\longrightarrow
T\overline{\mathfrak{M}}_{g-1,\ell+1}^{\phi;\bullet}(B;J)\\!\big{|}_{\overline{\mathfrak{M}}_{g-1,\ell+1}^{\prime\prime\phi;\bullet}(B;J)}\longrightarrow\textnormal{ev}_{\ell+1}^{\,*}\mathcal{N}_{X}X^{\phi}\longrightarrow
0\end{split}$

induce isomorphisms

$\begin{split}\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}T\big{(}\overline{\mathfrak{M}}_{g-2,\ell+2}^{\phi;\bullet}(B;J)\\!\big{)}\\!\big{)}\\!\big{|}_{\overline{\mathfrak{M}}_{g-2,\ell+2}^{\prime\phi;\bullet}(B;J)}&\approx\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}T\big{(}\overline{\mathfrak{M}}_{g-2,\ell+2}^{\prime\phi;\bullet}(B;J)\\!\big{)}\\!\big{)}\\!\otimes\\!\textnormal{ev}_{\ell+1}^{*}\big{(}\Lambda_{\mathbb{R}}^{\textnormal{top}}(TX)\\!\big{)},\\\
\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}T\big{(}\overline{\mathfrak{M}}_{g-1,\ell+1}^{\phi;\bullet}(B;J)\\!\big{)}\\!\big{)}\\!\big{|}_{\overline{\mathfrak{M}}_{g-1,\ell+1}^{\prime\prime\phi;\bullet}(B;J)}&\approx\Lambda_{\mathbb{R}}^{\textnormal{top}}\big{(}T\big{(}\overline{\mathfrak{M}}_{g-1,\ell+1}^{\prime\prime\phi;\bullet}(B;J)\\!\big{)}\\!\big{)}\\!\otimes\\!\textnormal{ev}_{\ell+1}^{*}\big{(}\Lambda_{\mathbb{R}}^{\textnormal{top}}(\mathcal{N}_{X}X^{\phi})\\!\big{)}\end{split}$
(3.35)

of real line bundles over
$\overline{\mathfrak{M}}_{g-2,\ell+2}^{\prime\phi;\bullet}(B;J)$ and
$\overline{\mathfrak{M}}_{g-1,\ell+1}^{\prime\prime\phi;\bullet}(B;J)$,
respectively.

We denote by