# Intrinsic mirror symmetry and Frobenius structure theorem via Gromov-Witten
theory of root stacks

Samuel Johnston  Samuel Johnston, Imperial College London, South Kensington
Campus, London SW7 2AZ, UK<EMAIL_ADDRESS>
###### Abstract.

Using recent results of Battistella, Nabijou, Ranganathan and the author, we
compare candidate mirror algebras associated with certain log Calabi-Yau pairs
constructed by Gross-Siebert using log Gromov-Witten theory and Tseng-You
using orbifold Gromov-Witten theory of root stacks. Although the structure
constants used to defined these mirror algebras do not typically agree, we
show that any given structure constant involved in the construction the
algebra of Gross and Siebert can be computed in terms of structure constants
of the algebra of Tseng and You after a sequence of log blowups. Using this
relation, we provide another proof of associativity of the log mirror algebra,
and a proof of the weak Frobenius Structure Theorem in full generality. Along
the way, we introduce a class of twisted punctured Gromov-Witten invariants of
generalized root stacks induced by log étale modifications, and use this to
study the behavior of log Gromov-Witten invariants under ramified base change.

###### 2010 Mathematics Subject Classification:

14J33,14N3F,14N10

##  1\. Introduction

The past few years have seen much progress in constructing mirrors to log
Calabi-Yau varieties directly using algebro-geometric enumerative geometry.
Work of Gross and Siebert in [GS19] has shown that a particular class of log
Gromov-Witten invariants satisfy relations reminicent of the classical WDVV
relations in the theory of stable maps, which in turn encode an algebra
associated with the log Calabi Yau pair ($X,D$). However, the arguments used
to prove the required relations in log Gromov-Witten theory are subtler than
one might naively expect, as the required relations do not follow from a
straightforward application of WDVV as in the setting of standard Gromov-
Witten theory. Seperately, Tseng and You in [TY20] defined a similar ring
using “large r” orbifold Gromov-Witten invariants. As the orbifold theory does
have access to most of the standard techniques from Gromov-Witten theory, the
proofs that their algebras satisfy the necessary relations is more
straightforward.

As both the logarithmic and the orbifold Gromov-Witten theories are versions
of Gromov-Witten theory relative to a simple normal crossings divisor, it is
natural to ask what the relationship is between the theories, and in our
specific context understand the relationship between the two mirror algebras
constructed using the different relative curve counting theories. Recent work
of Battistella, Nabijou and Ranganathan in [BNR22] and [BNR24] addresses the
first of these questions in genus $0$. Roughly speaking, they identify the
difference between the theories to be a result of the fact that log Gromov-
Witten theory is invariant under log étale modifications, by results in
[AW13], a relation which does not hold in the large $r$ orbifold theory.
Moreover, for any given log invariant of interest associated with a log smooth
target $X$, there exists a log étale modification
$\pi\colon\tilde{X}\rightarrow X$ such that the log invariant on $X$ is equal
to an orbifold invariant on $\tilde{X}$. Separately, work of the author in
[Joh22a] has extended the main result of [AW13] to the negative contact
setting, and explicitly described how the log Gromov-Witten theory of a target
changes under log étale modifications, both at the level of virtual cycles and
moduli stacks. This result in particular demonstrated a birational invariance
result for the intrinsic mirror algebra constructed in [GS19].

Using these results, we initiate a study into the second of the questions
above, in the process observing relations in log Gromov-Witten theory
analogous to standard relations in ordinary Gromov-Witten theory. More
precisely, after first providing simple examples demonstrating the structure
constants defining these algebras differ in general, the following theorem
makes clear how these classes of invariants are related nonetheless:

######  Theorem 1.1.

For $(X,D)$ a log Calabi-Yau variety in the sense of [GS19], with log and
orbifold mirror algebras $R^{log}_{(X,D)}$ and $R^{orb}_{(X,D)}$ respectively,
then any structure constants $N_{p,q,r}^{\textbf{A}}$ defining the product on
$R^{log}_{(X,D)}$ in the theta basis is equal to a sum of orbifold structure
constants on a log étale modification.

Using this fact, we give a new proof of the associativity of the mirror
algebra introduced by Gross and Siebert in [GS19], and a proof of the weak
Frobenius structure theorem in full generality i.e. Conjecture $9.2$ of
[GS19], which relates the natural trace form on $R_{(X,D)}$ with certain
decendent log Gromov-Witten invariants. We recall both theorems below:

######  Corollary 1.2.

Given a log Calabi-Yau pair $(X,D)$, then the structure constants
$N_{p,q,r}^{\textbf{A}}$ defined in [GS19] satisfy the following relation:

$\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}\text{, }s\in
B(\mathbb{Z})}N_{p_{1},s,r}^{\textbf{A}_{1}}N_{p_{2},p_{3},s}^{\textbf{A}_{2}}=\sum_{\textbf{A}=\textbf{A}_{1}+\textbf{A}_{2}\text{,
}s\in
B(\mathbb{Z})}N_{p_{1},p_{2},s}^{\textbf{A}_{1}}N_{p_{3},s,r}^{\textbf{A}_{2}}.$

Hence, the structure constants define an associative product on $R_{(X,D)}$.

######  Theorem 1.3 (Conjecture $9.2$ [GS19]).

Let $\beta$ be a type of tropical map with curve class A and $k+1$ marked
points $x_{out},x_{1},\ldots,x_{k}$ with contact order $0,p_{1},\ldots,p_{k}$
respectively associated with integral points of $B$. Define the invariant:

$N_{p_{1},\ldots,p_{k}}^{\textbf{A}}=\int_{[\mathscr{M}(X,\beta)]^{vir}}\psi_{x_{out}}^{k-2}ev_{x_{out}}^{*}(pt)$

Then the $\vartheta_{0}$ coefficient of
$\vartheta_{p_{1}}\cdots\vartheta_{p_{k}}$ is:

$\sum_{\textbf{A}}N_{p_{1},\ldots,p_{k}}^{\textbf{A}}z^{\textbf{A}}$

We emphasize that key to the proofs of the results above are exploiting the
distinct natures of the combinatorial data which govern the log and orbifold
Gromov-Witten theories. Invariants in log Gromov-Witten theory frequently
decompose into contributions coming from fixed tropical data, but due to
subtleties which include gluing of log maps, we typically cannot pullback
relations between invariants coming from moduli of stable maps. Orbifold
Gromov-Witten theory does possess relations derived from moduli of stable
curves, but typically lacks the full tropical refinement present in the
logarithmic theories. The relative disadvantages of either theory of
circumvented by exploiting the log/orbifold comparison. The full proof for the
results above must utilize the distinct advantages of each theory; it is not
simply a matter of translating relations from orbifold Gromov-Witten theory
into log Gromov-Witten theory.

To begin with, we will recall the relative Gromov-Witten invariants used to
define the orbifold and log mirror algebras, and highlight through example how
these classes of invariants typically do not coincide. We then introduce
twisted log Gromov-Witten theory for root stacks defined by certain log étale
modifications and generalize the main result of [Joh22a] which will be useful
in this setting. Following this, we proceed to a proof of Theorem 1.1, and
prove Corollary 1.2 assuming a semipositivity condition used by Tseng and You
in [TY20]. To remove this assumption, we provide an argument for the vanishing
of certain orbifold invariants based on the vanishing of certain log
invariants, modifying an argument which appeared in [GS19]. This vanishing
result in turn allows us to use the WDVV and TRR relations in the orbifold
theory to conclude the desired relations in the log theory.

###  1.1. Future Work

Along with constructing a deformation of the Stanley-Reisner ring of
$\Sigma(X)$ from large $r$ orbifold invariants of a log Calabi-Yau pair
$(X,D)$, Tseng and You also define a quantum product on relative quantum
cohomology for arbitrary simple normal crossings pairs $(X,D)$. This is a
graded algebra structure on the cohomology of the inertia stack for
arbitrarily large $r$. In the log Calabi-Yau setting, this product rule
restricts to define a product on the degree $0$ part of the ring. The
relations in this paper suggest an analogous construction in log Gromov-Witten
theory, and a more thorough treatment of the state space should enable this to
be accomplished.

Additionally, You in [You22] has previously studied in various settings how
the relative orbifold quantum cohomology ring changes under birational
modifications. Motivated by the crepant transformation conjecture, he predicts
that while individual invariants might not satisfy any reasonable
interpretation of birationally invariance, their generating series might be
better behaved, with their difference controlled by analytic continuation in
Novikov parameters. A more thorough understanding of the behavior of orbifold
invariants under strata blowups, when combined with the results in this paper,
should yield additional tools and techniques for computing and studying the
intrinsic mirror construction.

The relations in this paper should also hold for more general log Calabi-Yau
DM stacks. The main missing ingredient needed to carrying this out presently
is the development of a well-behaved theory of twisted log maps to log DM
stacks. Once this is accomplished though, the arguments in this paper should
generalize to the DM stack setting, and in particular produce mirror algebras
for these targets. A generalized mirror construction will likely be useful in
studying mirrors to skew-symmetrizable cluster varieties in the same vein as
[AB23], as well as mirror LG models to smooth Fano orbifolds. We will return
to these questions in future work.

##  2\. Acknowledgements

I would like to thank Robert Crumplin, Mark Gross, Ajith Kumaran, Navid
Nabijou, Dhruv Ranganathan and Fenglong You for many useful discussions
related to this work. In particular, I would like to thank Mark Gross for
reviewing an earlier draft of this document, and Navid Nabijou and Dhruv
Ranganathan for keeping me updated on their progress with their joint work
with Luca Battistella. This work was done with the financial support of the
ERC grant “Mirror Symmetry in Algebraic Geometry” as well as the Heilbronn
Institute for Mathematical Research and Imperial College London.

##  3\. Tropical and Logarithmic preliminaries

We suppose throughout that we are working over an algebraically closed field
$\mathds{k}$ with characteristic $0$. We begin by briefly reviewing the
necessary tropical and logarithmic geometry we require, as well as review the
main results of [BNR22] and [BNR24]. Given an algebraic stack with log
structure $Y$, we denote the underlying stack over schemes by $\underline{Y}$
throughout.

###  3.1. Tropicalization and Artin Fans

Consider a pair $(X,D)$ with $X$ a smooth projective variety and $D$ a simple
normal crossings divisor with $D=D_{1}+\ldots+D_{k}$ a decomposition of the
divisor $D$ into irreducible components. Each $D_{i}$ is determined by a
section $s_{i}$ of a line bundle $L_{i}\in Pic(X)$ up to scaling by
$\mathds{k}^{*}$. By definition, this induces a morphism
$(L_{i},s_{i}):X\rightarrow[\mathbb{A}^{1}/\mathbb{G}_{m}]$, and taking the
product of these morphisms induces a morphism
$X\rightarrow[\mathbb{A}^{k}/\mathbb{G}_{m}^{k}]$. We let $\mathcal{X}$ be the
minimal open substack of $[\mathbb{A}^{k}/\mathbb{G}_{m}^{k}]$ through which
the above morphism factors. Each of the stacks above admit natural divisorial
log structures and the log algebraic stack $\mathcal{X}$ is called the Artin
fan for the pair $(X,D)$.

The use of Artin Fans in logarithmic geometry allows us to translate
constructions and calculations in a more combinatorial/tropical setting into
the geometric setting. In this paper, we consider the tropical setting to be
the category of rational polyhedral cone complexes, or RPCC. Its objects
consist of rational polyhedral cone complexes, which are given by collections
of integral convex cones glued along face inclusions. For a cone $\sigma$, we
denote its integral points by $\sigma_{\mathbb{N}}$. For a cone complex
$\Sigma$, we denote by $\Sigma(\mathbb{Z})$ the set of morphisms
$\mathbb{Z}\rightarrow\sigma_{\mathbb{N}}^{gp}$ for some $\sigma\in\Sigma$. By
working in the more general context of cone stacks, defined in [CCUW20]
Definition $2.7$, a useful result of [CCUW20] is the following equivalence of
categories:

######  Lemma 3.1.

There is an equivalence of $2$-categories between the categories of Artin fans
and cone stacks. On a cone $\sigma\in Ob(\textbf{RPCC})$, the equivalence is
given by sending $\sigma$ to $\mathcal{A}_{\sigma}$.

Given a cone complex $\Sigma$, we will refer to its corresponding Artin fan as
$\mathcal{A}_{\Sigma}$. We will also occasionally refer to idealized Artin
cones in this paper, denoted by $\mathcal{A}_{\sigma,K}$ for
$K\subset\sigma^{\vee}_{\mathbb{N}}$ an ideal. This is an idealized log stack
whose underlying stack is given by:

$\mathcal{A}_{\sigma,K}=[(Spec\mathds{k}[\sigma^{\vee}_{\mathbb{N}}]/K)/Spec\mathds{k}[\sigma_{\mathbb{N}}^{gp}]],$

and which carry canonical log and idealized log structures.

When $(X,D)$ is a simple normal crossings pair, forgetting the morphism
$X\rightarrow\mathcal{X}$ and remembering only the target is the same
information as the dual cone complex of $(X,D)$, which we denote by
$\Sigma(X)$. The forgetful morphism
$\mathcal{X}\rightarrow[\mathbb{A}^{1}/\mathbb{G}_{m}]$ remembering only a
single divisor $D_{i}$ corresponds under the correspondence to a morphism of
rational polyhedral cone complexes $\Sigma(X)\rightarrow\mathbb{R}_{\geq 0}$
i.e. a PL function on the complex $\Sigma(X)$. We call the function induced by
a divisor $D$ by $D^{*}$. If we restrict to a cone $\sigma\in\Sigma(X)$, this
morphism induces a group homomorphism on integral tangent vectors at any point
$p\in int(\sigma)$ contained in the interior of $\sigma$, which we may
identify with a group homomorphism
$\sigma_{\mathbb{N}}^{gp}\rightarrow\mathbb{Z}$.

One important input coming from this correspondence above is the relation
between subdivisions and refinements of the Artin fan and log étale
modifications of $X$. Given a cone $\sigma$ and a conic subdivision
$\tilde{\sigma}\rightarrow\sigma$, we have a corresponding morphism of Artin
fans $\mathcal{A}_{\tilde{\sigma}}\rightarrow\mathcal{A}_{\sigma}$. By [AW13]
Corollary $2.6.7$, this morphism is log étale and proper. Moreover, if we have
a strict smooth map $X\rightarrow\mathcal{A}_{\sigma}$, we may pullback this
map to produce a proper log étale and birational map $\tilde{X}\rightarrow X$.
Alternatively, if we consider a morphism of cones
$\sigma^{\prime}\rightarrow\sigma$ induced by a refinement of lattices, we may
pullback as before to produce a log étale modification. In the setting of a
pair $(X,D)$ above, these produce orbifold enhancements of the target which we
call generalized root stack constructions along the boundary divisor $D$. More
generally, pulling back subdivisions and refinements of cone complex
$\mathcal{X}$ induce log étale modifications of $X$. By the birational
invariance of punctured log invariance, we are free to choose a convenient
birational model of $X$ of the form above to study relations in the log
Gromov-Witten invariants of $(X,D)$ of interest.

###  3.2. Tropical maps and punctured log maps

Given a log Calabi-Yau pair $(X,D)$, the enumerative geometry used to
construct its mirror algebra is encoded in the theory of punctured log maps,
constructed by Abramovich, Chen, Gross and Siebert in [ACGS20], which we
briefly recall below, omiting details which will not be important for us in
this context:

######  Definition 3.2.

Let $Y=(\underline{Y},\mathcal{M}_{Y})$ be an fs log stack, together with a
decomposition
$\mathcal{M}_{Y}=\mathcal{M}\oplus_{\mathcal{O}_{Y}^{\times}}\mathcal{P}$. A
puncturing of $Y$ along $\mathcal{P}$ is a fine subsheaf of monoids
$\mathcal{M}_{Y^{\circ}}\subset\mathcal{M}\oplus_{\mathcal{O}_{Y}^{\times}}\mathcal{P}^{gp}$,
containing $\mathcal{M}_{Y}$, together with a structure map
$\alpha_{Y^{\circ}}:\mathcal{M}_{Y^{\circ}}\rightarrow\mathcal{O}_{Y}$,
satisfying:

  1. (1)

The inclusion $p^{\flat}:\mathcal{M}_{Y}\rightarrow\mathcal{M}_{Y^{\circ}}$ is
a morphism of log structures on $\underline{Y}$.

  2. (2)

For any geometric point $x\in Y$, let
$s_{x}\in\mathcal{M}_{Y^{\circ},x}\setminus\mathcal{M}_{Y,x}$. Representing
$s_{x}=(m_{x},p_{x})\in\mathcal{M}_{x}\oplus_{\mathcal{O}^{\times}_{Y_{x}}}\mathcal{P}^{gp}_{x}$,
we have $\alpha_{Y^{\circ}}(s_{x})=\alpha_{\mathcal{M}}(m_{x})=0$ in
$\mathcal{O}_{Y,x}$.

A punctured log curve is a log curve $C\rightarrow S$, together with a
puncturing along $\mathcal{P}$ induced by marked sections of $C\rightarrow S$.
A stable punctured log map $C^{\circ}\rightarrow X$ is a logarithmic morphism
from a prestable punctured log curve to $X$ such that the underlying morphism
of schemes is a stable map.

Similar to the theory of ordinary log stable maps, every punctured log map has
a “combinatorial shadow” induced by tropicalization. More precisely, to any
punctured log stable map $f:C\rightarrow X$ over a logarithmic point $S$ and
letting $\sigma=Hom_{mon}(\overline{\mathcal{M}}_{S},\mathbb{R}_{\geq 0})$,
the log structure encodes a metrization of the dual graph of $C$ by a length
function $l:E(G)\rightarrow\overline{\mathcal{M}}_{S}$, and a PL function
$\Sigma(f)_{p}:G\rightarrow\Sigma(X)$ for every point $p\in\sigma$. We note
such data gives the following discrete invariants:

  1. (1)

A function $\boldsymbol{\sigma}(-):E(G)\cup L(G)\cup V(G)\rightarrow\Sigma(X)$
assigning to a feature of $G$ the smallest cone of $\Sigma(X)$ which contains
it’s image under $\Sigma(f)$.

  2. (2)

For every edge or leg $e\in E(G)\cup L(G)$ an integral tangent vector
$u_{e}\in\boldsymbol{\sigma}(e)^{gp}_{\mathbb{N}}$.

We may also keep track of the genus distribution on the dual graph by a
function $g:V(G)\rightarrow\mathbb{N}$. In this paper, we will always be
working with genus $0$ domains, so this additional data will be forgotten from
now on. The data $(G,l,\boldsymbol{\sigma},u)$ defines the tropical type of
the log stable map $f$. A choice of the tuple of data above is called
realizable if there is a PL map from meterized dual graph $G$ to $\Sigma(X)$
of the given tropical type. In such a situation, there exists a cone $\tau$
and family of family of tropical maps $(f,\Gamma)$ defined over $\tau$ which
are the moduli spaces of tropical curves of type $\tau$ and the universal
family lying over it respectively. The primary utility that punctured log maps
provide tropically is that the legs are possibly bounded.

Applying functorial tropicalization to a punctured log map
$C^{\circ}\rightarrow X$ produces a map of cone complexes
$\Sigma(C)\rightarrow\Sigma(X)$, which is a punctured tropical map defined as
follows:

######  Definition 3.3.

Given a family of ordinary tropical curves
$\pi_{(G,l)}:\Gamma_{\sigma}\rightarrow\sigma$, a puncturing
$\Gamma_{\sigma}^{\circ}\rightarrow\sigma$ of $\Gamma_{\sigma}$ is an
inclusion of a sub cone complex
$\Gamma_{\sigma}^{\circ}\rightarrow\Gamma_{\sigma}$ over $\sigma$ which is an
isomorphism away from cones of $\Gamma_{\sigma}$ associated with legs of $G$.
Moreover, for cones $\sigma_{l}$ associated with punctured legs of $G$,
$\sigma_{l}$ must intersect the interior of the target cone, and there exists
a section $\sigma\rightarrow\sigma_{l}$ which when composed with
$\sigma_{l}\rightarrow\Gamma_{\sigma}$ is a section associated with the unique
vertex of $G$ contained in the leg $l$.

We bundle the data of the contact orders associated to legs $u_{l}$ as well as
a degree $\textbf{A}\in H_{2}(X)$ into a pair $\beta=((u_{l}),\textbf{A})$
which we call a type of punctured log map. The main result of [ACGS20] is a
construction of a finite type DM stack $\mathscr{M}(X,\beta)$ with log
structure parameterizing punctured log stable maps to $X$ of type $\beta$.
There is also an algebraic stack $\mathfrak{M}(\mathcal{X},\beta)$ with log
structure parameterizing prestable punctured log maps of type $\beta$ to the
Artin fan $\mathcal{X}$, and composition induces a map
$\mathscr{M}(X,\beta)\rightarrow\mathfrak{M}(\mathcal{X},\beta)$. This
morphism carries a perfect obstruction theory, which facilitates virtual
pullback of classes on $\mathfrak{M}(\mathcal{X},\beta)$ in the formalism of
[Man08].

It will be useful for various arguments in this paper to enrich the target of
morphism carrying the perfect obstruction theory. Specifically, given a
punctured log curve $C\rightarrow\mathcal{X}$, recall that a marked point $p$
of a puncture with contact order $u_{p}$ is forced to map to a stratum
$\mathcal{D}_{p}$ of the log stack $\mathcal{X}$. Thus, on the level of
underlying stacks, we have a morphism
$ev_{p}:\underline{\mathfrak{M}(\mathcal{X},\beta)}\rightarrow\underline{\mathcal{D}_{p}}$.
Moreover, the stratum of the Artin fan of $X$ specifies a stratum $D_{p}$ of
$X$, and the strict morphism $X\rightarrow\mathcal{X}$ restricts to a morphism
$D_{p}\rightarrow\mathcal{D}_{p}$. Hence, for a collection of marked points
$p_{1},\ldots p_{k}$, of the type $\beta$, we define:

$\mathfrak{M}^{ev}(\mathcal{X},\beta):=\mathfrak{M}^{ev(p_{1},\ldots,p_{k})}(\mathcal{X},\beta)=\mathfrak{M}(\mathcal{X},\beta)\times_{\prod{i=1}^{k}\underline{\mathcal{D}_{p_{i}}}}\prod_{i=1}^{k}\underline{D_{p_{i}}}$

The morphism $\mathscr{M}(X,\beta)\rightarrow\mathfrak{M}(\mathcal{X},\beta)$
certainly factors through the projection
$\mathfrak{M}^{ev}(\mathcal{X},\beta)$, and the resulting morphism
$\mathscr{M}(X,\beta)\rightarrow\mathfrak{M}^{ev}(\mathcal{X},\beta)$ also
carries a perfect obstruction theory.

###  3.3. Construction of virtual classes on $\mathscr{M}(X,\beta)$

An initially troubling feature of theory of punctured log maps is the fact
that if we fix only a type $\beta$, unlike in the theory of ordinary log maps,
we do not have a canonical virtual fundamental class on
$\mathscr{M}(X,\beta)$. Indeed, the moduli stack
$\mathfrak{M}(\mathcal{X},\beta)$ might not be equidimensional. There are at
least two paths for dealing with this issue. One is to note that by
Proposition $3.28$ of [ACGS20], if we fix additional tropical data in the form
of a realizable tropical type $\tau$, then the moduli stack
$\mathfrak{M}(\mathcal{X},\tau)$ of punctured log maps of type $\tau$ is
equidimensional. Moreover, the moduli stack $\mathfrak{M}(\mathcal{X},\beta)$
is stratified by realizable tropical types of type $\beta$, so we produce
classes on $\mathscr{M}(X,\beta)$ associated with every such tropical type.
Note that the dimension of these classes will differ, and can be computed
using the dimension of the associated tropical moduli problem

An alternative approach to equipping this moduli stack with an equidimensional
class is suggested by Battistella, Nabijou and Ranganathan in [BNR24]. This
assumes the target is _slope sensitive_ , a property originally introduced in
[BNR22] Section $4.1$ and $4.2$. Critically, for any discrete data $\beta$,
this property can be assured for a corresponding set of discrete data
$\beta^{\prime}$ on a log étale modification $\tilde{X}\rightarrow X$.
Assuming this condition, we may also equip $\mathfrak{M}(\mathcal{X},\beta)$
with a virtual fundamental class. By taking the virtual pullback of this
class, we produce a class on $\mathscr{M}(X,\beta)$. Letting $D_{i}$ be an
irreducible component of $D$, and $e_{i}=|\\{p\text{ marked
point}|D_{i}^{*}(u_{p}))<0\\}|$, the dimension of this class is given by:

(3.1) $vdim\text{ }\mathscr{M}(X,\beta)=deg\text{
}c_{1}(T_{X}^{log})+(dim\text{ }X-3)(1-g)+n-\sum_{i=1}^{k}e_{i}.$

##  4\. Intrinsic Mirror constructions

Associated to the pair $(X,D)$ is the rational polyhedral cone complex
$\Sigma(X)$. When $X$ is smooth and $D$ is simple normal crossings, this is
the dual cone complex of the divisor $D$. We recall from [GS19] two possible
restrictions on the geometry of pair $(X,D)$ that are sufficient to construct
mirror algebras:

######  Assumption 4.1.

An admissible pair is a log smooth pair $(X,D)$ which satisfies one of two of
the following conditions:

  1. (1)

$c_{1}(T_{X}^{log})$ is nef or antinef.

  2. (2)

$K_{X}+D=_{\mathbb{Q}}\sum_{i}a_{i}D_{i}$, with $a_{i}\geq 0$.

If $(X,D)$ satisfies the second of the above conditions, we say $(X,D)$ is log
Calabi-Yau.

If the pair $(X,D)$ is log Calabi-Yau in the second sense, we call a divisor
$D_{i}$ _good_ if $a_{i}=0$. We adopt the terminology that divisors which are
not good are called _bad_. We define $(B,\mathscr{P})$ to be the subcomplex of
$\Sigma(X)$ consisting of the cones of $\Sigma(X)$ associated to strata which
are contained only in good divisors. If $(X,D)$ satisfies condition $1$ above,
then we define $(B,\mathscr{P})$ to be the cone complex $\Sigma(X)$.

To define the logarithmic mirror algebra, we first pick a finitely generated
monoid $Q\subset H_{2}(X)$ containing all effective curve classes, and
$I\subset\mathds{k}[Q]$ a monomial ideal such that $\sqrt{I}$ is the maximal
monomial ideal of $\mathds{k}[Q]$. We write $\mathds{k}[Q]/I=S_{X,I}$. The
mirror algebra $R^{log}_{(X,D)}$ first will be a free $S_{X,I}$ module:

$R^{log}_{(X,D)}=\oplus_{p\in B(\mathbb{Z})}\vartheta_{p}\cdot S_{X,I}.$

The generators of the free $S_{X,I}$ module above are called theta functions.
We equip $R^{log}_{(X,D)}$ with an $S_{X,I}$ algebra structure via structure
constants for the product of theta functions. For $p,q,s\in B(\mathbb{Z})$,
the structure constant
$N_{p,q,s}:=\sum_{\textbf{A}}N_{p,q,s}^{\textbf{A}}z^{\textbf{A}}\in S_{X,I}$
given by the coefficient appearing in front of $\vartheta_{s}$ in the product
$\vartheta_{p}\vartheta_{q}$ will be a certain series of $3$-pointed punctured
Gromov-Witten invariants indexed by curve classes, with two positive contact
orders $p$ and $q$, and one negative contact order $-s$.

More precisely, we first consider the evaluation space for log points with
contact order $-s$. This is a logarithmic stack, whose underlying stack is
given by $X_{s}\times B\mathbb{G}_{m}$, with $X_{s}$ the closure of the
locally closed stratum $X_{s}^{\circ}\subset X$ associated with the smallest
cone $\sigma\in\Sigma(X)$ containing $s$, and whose log structure is a sub log
structure of the pullback of the product log structure
$X\times[\mathbb{A}^{1}/\mathbb{G}_{m}]$, whose ghost sheaf is given by:

(4.1) $\Gamma(U,\overline{\mathcal{M}}_{\mathscr{P}(X,s)})=\\{(m,s(m))\text{
}|\text{
}m\in\Gamma(U,\overline{\mathcal{M}}_{Z})\\}\subset\Gamma(U,\overline{\mathcal{M}}_{Z})\oplus\mathbb{N}=\Gamma(U,\overline{\mathcal{M}}_{\widetilde{\mathscr{P}(X,s)}})$

In the terminology of [ACGS20], for every curve class A associated with a non-
zero element of $S_{X,I}$, there is a non-realizable decorated tropical type
$\beta$ with a single vertex decorated by A, and $3$ legs with contact orders
$p_{1}$, $p_{2}$, and $-s$. There is a corresponding moduli space of punctured
log maps $\mathscr{M}(X,\beta)$, which by [GS19] Proposition $3.3$ admits an
evaluation map $\mathscr{M}(X,\beta)\rightarrow\mathscr{P}(X,s)$. We let
$W=B\mathbb{G}_{m}^{\dagger}$ be the log stack with underlying stack
$B\mathbb{G}_{m}$ and log structure induced by pullback along the closed
immersion $B\mathbb{G}_{m}\rightarrow[\mathbb{A}^{1}/\mathbb{G}_{m}]$. Letting
$z\in X_{s}$, then by Proposition $3.8$ of [GS19], there is a map
$W\rightarrow\mathscr{P}(X,s)$ which maps $W$ to $z\times
W\subset\mathscr{P}(X,s)$. We then consider the fine and saturated fiber
product
$\mathscr{M}(X,\beta,z):=\mathscr{M}(X,\beta)\times_{\mathscr{P}(X,s)}^{fs}W$.
We may similarly construct
$\mathfrak{M}^{ev}(\mathcal{X},\beta,z):=\mathfrak{M}^{ev}(\mathcal{X},\beta)\times_{\mathscr{P}(X,s)}^{fs}W$,
and the morphism
$\mathscr{M}(X,\beta,z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X},\beta,z)$
pulled back from
$\mathscr{M}(X,\beta)\rightarrow\mathfrak{M}^{ev}(\mathcal{X},\beta)$ carries
the pulled back obstruction theory. In particular, when $(X,D)$ is log Calabi-
Yau, $\mathscr{M}(X,\beta,z)$ comes with a virtual fundamental class of
virtual dimension $0$. The structure constant $N_{p,q,r}^{\textbf{A}}$ are
then defined as:

$N_{p_{1},p_{2},s}^{\textbf{A}}=deg[\mathscr{M}(X,\beta,z)]^{vir}.$

If $(X,D)$ satisfies condition $1$ of Assumption 4.1, then the structure
constant is as above if $vdim\text{ }\mathscr{M}(X,\beta,z)=0$, and is defined
to be zero otherwise. The main result of [GS19] is that these structure
constants define an associative product on $R^{log}_{(X,D)}$.

In the orbifold setting, assuming in addition that the boundary divisor $D$
has simple normal crossings and is an admissible pair in the sense of
Definition 4.1(1), Tseng and You in [TY20] define analogous invariants giving
these structure constants. In loc cit., the authors observe that the contact
orders $p,q,-s$ and curve class A may also be used to construct a sequence of
moduli stack $\mathscr{M}^{orb}(X_{r},\beta)$, with $r=(r_{i})$ a vector of
pairwise relatively prime positive integers indexed by the connected
components of the boundary and $X_{r}$ the corresponding stack theoretic root
of the boundary divisor associated with $r$. In the orbifold setting, the
negative contact orders are interpreted as a high age orbifold point. The
moduli space $\mathscr{M}^{orb}(X_{r},\beta)$ is a Deligne-Mumford stack with
a perfect obstruction theory with $vdim\text{
}\mathscr{M}^{orb}(X_{r},\beta)=dim\text{ }D_{s}$.

To fix notation which will be used throughout, we let $H^{*}(-)$ be the
singular cohomology functor, and pick a basis $\\{T_{i,s}\\}\in H^{*}(D_{s})$
for the singular cohomology of the stratum associated with
$s\in\mathbb{Z}^{k}$ a contact order with $D=\sum_{i=1}^{k}D_{i}$. Since we
are assuming $X$ is simple normal crossings, there is a Poincaré dual basis
$T_{s}^{i}$. Moreover, given a collection of contact orders
$p_{i}\in\mathbb{Z}^{k}$, $\alpha_{i}\in H^{*}(D_{p_{i}})$, $\textbf{A}\in
NE(X)$, and setting $m_{i,-}$ be the number of negative entries appearing in
the contact order $m_{i}$, we define:

(4.2)
$\langle\prod_{i}\psi^{a_{i}}[\alpha_{i}]_{p_{i}}\rangle_{\textbf{A}}:=\prod_{i}r^{m_{i,-}}\int_{[\mathscr{M}^{orb}(X_{r},\textbf{A},\textbf{p})]^{vir}}\prod_{i}\psi_{x_{i}}^{a_{i}}\cup
ev_{x_{i}}^{*}([\alpha_{i}]).$

By Corollary $18$ of [TY20], this invariant stabilizes for $r_{i}>>0$. In
light of this stabilization, these invariants are usefully thought of as a
Gromov-Witten invariant of the infinite root stack associated with the pair
$(X,D)$. The authors of loc. cit. define the orbifold structure constants by:

$N_{p_{1},p_{2},s}^{\textbf{A},orb}=\langle[1]_{p_{1}},[1]_{p_{2}},[pt]_{-s}\rangle_{\textbf{A}}.$

In Theorem $37$ of [TY20], the authors show that these also define an
associative product rule on $R^{log}_{(X,D)}$, using arguments involving the
WDVV relations in orbifold Gromov-Witten theory.

As logarithmic and orbifold Gromov-Witten theory are distinct methods at
probing the enumerative geometry of the pair $(X,D)$, it is natural to ask
about the relationship between these two enumerative theories. In genus $0$,
the relationship was described in [BNR22] in the setting of non-negative
contact orders, and [BNR24] in the setting of general contact orders. We
record their main result below:

######  Theorem 4.2.

Let $\beta$ be a tropical type which fixes a curve class $\textbf{A}\in NE(X)$
and tangency orders $p_{1},\ldots,p_{l}$. Then assuming $\beta$ is slope
sensitive and letting $[\mathscr{M}(X,\beta)]^{vir}$ be the refined virtual
class on the punctured log moduli space, we have:

$\int_{[\mathscr{M}(X,\beta)]^{vir}}\prod_{i}\psi_{x_{i}}^{a_{i}}ev_{x_{i}}^{*}([\alpha_{i}])=\langle\prod_{i}\psi_{x_{i}}^{a_{i}}[\alpha_{i}]\rangle.$

In the context of this paper, we will be interested in the following two
corollaries:

######  Corollary 4.3.

Given a type $\beta$ of log stable map to $X$ which is slope sensitive with
only one negative contact order, then $\mathscr{M}(X,\beta)$ is virtually
equidimensional. Moreover, the log Gromov-Witten and large $r$ orbifold
Gromov-Witten invariants with respect to $\beta$ coincide.

######  Corollary 4.4.

Given a type $\beta$ of log stable map to $X$ which is slope sensitive and for
all realizable tropical types $\tau$ marked by $\beta$ we have
$[\mathscr{M}(X,\tau)]^{vir}=0$, then $[\mathscr{M}(X,\beta)]^{vir}=0$.

Before proceeding, we record the following relation between curve classes and
contact orders, which holds in both the settings of log Gromov-Witten theory
and orbifold Gromov-Witten theory, which follows from the weak balancing
conditions in both settings as well as Remark $2.11(2)$ of [GS19]:

######  Proposition 4.5.

Suppose we have a type $\beta=(\textbf{A},(s_{i}))$, hence $\textbf{A}\in
NE(X)$ and $s_{i}\in\sigma_{i,\mathbb{N}}^{gp}$ for cones
$\sigma_{i}\in\Sigma(X)$. Letting $\mathscr{M}(X,\beta)$ refer to either the
corresponding space of log or orbifold stable maps, then
$\mathscr{M}(X,\beta)\not=\emptyset$ only if
$D_{j}\cdot\textbf{A}=\sum_{i}D_{j}^{*}(s_{i})$ for all irreducible components
$D_{j}\subset D$. In particular, if all but one contact order is fixed except
for a final contact order $s_{l}$, then there are only finitely many possible
choices for a contact order $s_{l}$.

##  5\. Counterexample to equality of theta basis

In order to highlight how these two constructions differ, we consider the
following simple example:

Let $(X_{t},D_{t})$ the the log Calabi-Yau pair of $\mathbb{P}^{2}$ and the
toric boundary, $D_{t}=D_{t,1}+D_{t,2}+D_{t,3}$ and $X=Bl_{x}\mathbb{P}^{2}$,
with $x\in D_{t,1}$ a point which is not a zero strata of $(X_{t},D_{t})$.
Finally, let $D$ be the strict transform of $D_{t}$. Then $(X,D)$ is a log
Calabi-Yau pair satisfying $K_{X}+D=0$ and tropicalization $\Sigma(X)$ which
can be identified with $\Sigma(X_{t})$. Hence, both $R_{(X,D)}$ and
$R_{(X,D)}^{orb}$ exist, and are algebras over $S_{X}=\mathds{k}[NE(X)]$ which
are free as $S_{X}$ modules with basis integral points of $\Sigma(X)$.
Moreover, $D$ is an ample divisor, hence $X\setminus D$ is affine. We present
two phenomenon which obstruct the equality of structure constants without
passing to appropriate blowups:

Case 1: Non-equality at the level of moduli stacks

Let $L,E\in NE(X)$ be the class of a generic line and the exceptional curve
class respectively, $\textbf{A}=3L-2E$, $p_{1}=D_{3}+D_{1}$ and
$p_{2}=3D_{2}+2D_{3}$. We wish to compute the $\vartheta_{0}$ term of
$\vartheta_{p_{1}}\vartheta_{p_{2}}$ of class $\textbf{A}\in NE(X)$, i.e.
$N_{p_{1},p_{2},0}^{\textbf{A}}$ and $N_{p_{1},p_{2},0}^{\textbf{A},orb}$.

To compute this invariant in the log setting, we recall from [GHS22] Theorem
$3.24$ that $\vartheta_{p_{1}}\vartheta_{p_{2}}[\vartheta_{0}]$ can be
computed by summing over all pairs of broken lines in the canonical wall
structure associated with $(X,D)$ with initial slopes given by $p_{1}$ and
$p_{2}$ which meet at a fixed generic point of $\Sigma(X)$ with final slopes
which sum to zero. In this setting, the number of broken lines with fixed
initial slope which end at a fixed generic point is finite. In particular, if
we pick our generic point in the cone of $\Sigma(X)$ spanned by $D_{1}$ and
$D_{3}$, then there is a unique broken line associated with $p_{1}$, and two
broken lines associated with $p_{2}$, $\beta_{1}$ and $\beta_{2}$, see the
diagram below for an illustration of these broken lines, together with
labeling by the slope of their final linear segments. However, neither of the
two final slopes of $\beta_{i}$ is $-p_{1}$. Since $p_{1}$ is the slope of the
unique linear segment associated with the unique broken line with initial
slope $p_{1}$, it follows that $N_{p_{1},p_{2},0}^{\textbf{A}}=0$.

$D_{3}$$D_{2}$$D_{1}$$(-2,-1)$$(-3,-2)$$(-1,0)$$(-2,-1)$ Figure 5.1. The
broken lines contributing to $\vartheta_{p_{2}}$ and $\vartheta_{p}$, which
all have a unique bending vertex, colored violet if it contributes to
$\vartheta_{p_{2}}$ and colored blue if it contributes to $\vartheta_{p}$. We
color the unique ray in the scattering diagram red, and final rays of the
broken lines by blue, violet, yellow and green in order to differentiate.

Before moving on to computing the corresponding orbifold invariant, we also
note the following broken line expansion for $\vartheta_{p}$, with
$p=2D_{2}+D_{3}$.

$\vartheta_{p}=x^{-D_{1}-D_{3}}z^{2L-E}+x^{-2D_{1}-D_{3}}z^{2L}$

In particular, $\vartheta_{p_{1}}\vartheta_{p}[\vartheta_{0}]=z^{2L-E}$. By
the main theorem of [Joh22b], this implies there is a unique log curve with
$3$ marked points of contact orders $p_{1},p$ and $0$ of curve class $2L-E$
which maps the contact order $0$ point to a general point of $X\setminus D$.

To investigate the corresponding structure constant of $R_{(X,D)}^{orb}$, we
note that by Corollary $2.10$ of [BNR22], it is equivalent to consider an
analogous integral on the corresponding moduli space of naive log maps
$\mathscr{M}^{naive}(X,\beta)$. By the main theorem of [Joh22b], the
contribution to the invariant of interest coming from the locus of curves with
log enhancements vanishes. Hence, any non-zero contribution must come from the
locus of curves which do not admit log enhancements. To produce one such
curve, we take the unique curve $C_{1}$ which contributes to
$N_{p_{1},p}^{2L-E}$ above, glue a chain of rational curves $C_{0}\cup C_{2}$
to the point of $C_{1}$ associated with the contact order $p$ point, and equip
$C_{0}$ with an extra marked point. Calling the resulting curve $C$, we extend
$f|_{C_{1}}:C_{1}\rightarrow X$ to a map $f:C\rightarrow X$ by contracting
$C_{0}$, and letting $C_{2}$ map isomorphically onto the strict transform of
the line between $x$ and the zero stratum $D_{2}\cap D_{3}$.

There is a unique way of extending this map to naive log map with contact
orders $p_{1}$ and $p_{2}$ and $0$ at marked points, but no logarithmic
enhancement exists. Indeed, the obstruction to the log enhancement can be seen
tropically; the tropical curve would have three vertices, two of which map to
the zero cone and are connected to the third vertex $v_{0}$ lying in the cone
spanned by $D_{2}$ and $D_{3}$ by a distinct edge. The slopes of these edges
are given by $2D_{2}+D_{3}$ and $D_{2}+D_{3}$, hence cannot intersect in the
interior of $\boldsymbol{\sigma}(v_{0})$, which obstructs the required
tropical curve from existing. On the other hand, using the fact that both the
linear functions $D_{3}^{*}$ and $D_{2}^{*}$ do not restrict to zero on each
edge, it is straightforward to show that a naive log enhancement exists, see
[BNR22] Example $3.10$ for a similar example, with Example $3.13$ of loc. cit.
explaining the obstruction to a log enhacement.

By recalling no irreducible curve satisfies the required constraints, combined
with considering the restriction on combinatorial types imposed by the curve
class and contact orders of marked points, one can show that all curves
contributing to the naive invariant have combinatorial type as above. Using
the fact that the conic is isolated among solutions to the point constraint in
its relevant moduli space, it is straightforward to show that $C$ is the
unique curve potentially contributing to the invariant.

To prove that $C$ contributes to the orbifold trace, we need to show the curve
is in addition unobstructed. By the description of the obstruction theory for
naive log maps described in Section $2.2$ of [BNR22], it suffices to show that
$\mathfrak{M}^{naive}(\mathcal{X},\beta)$ is locally smooth and dimension $0$
around the image of
$[C]\in\mathfrak{M}^{naive}(\mathcal{X},\beta)(\mathds{k})$ and
$H^{1}(C,f^{*}T_{X}^{log})=0$. If these conditions hold, then locally around
$[C]$, $\mathscr{M}^{naive}(X,\beta)$ is smooth and $2$ dimensional by
Riemann-Roch. Since $2$ is the expected dimension of the moduli space, the map
$C\rightarrow X$ would be unobstructed.

For the first property, let $\beta_{i}$ be the tropical type of log stable map
to $(X,D_{i})$ induced by the naive tropical type $\beta$. By the
classification of strata of $\mathfrak{M}^{naive}(\mathcal{X},\beta)$ as fiber
products of locally closed strata of maps to
$\mathfrak{M}(\mathcal{A}_{\mathbb{N}},\beta_{i})$ over the moduli stack of
prestable curves, we derive the stratum of interest which contains $[C]$ is
smooth and dimension $0$. Moreover, since the nodes of $\beta_{i}$ must smooth
simultaneously for $i=2,3$, the only possible distinct stratum whose closure
contains the stratum of interest is the non-degenerate stratum . However,
since no curve in the stratum of interest has a log enhancement, we conclude
that the stratum is open. Since the virtual dimension of
$\mathfrak{M}^{naive}(\mathcal{X},\beta)$ is $0$, we conclude that
$\mathfrak{M}^{naive}(\mathcal{X},\beta)$ is smooth and of dimension $0$
locally around the image of $[C]$.

Turning now to the computation of $H^{1}(C_{i},f^{*}|_{C_{i}}T_{X}^{log})$,
since $C_{0}$ is contracted, clearly
$f|_{C_{0}}^{*}T_{X}^{log}=\mathcal{O}_{C_{0}}^{2}$. Moreover, since $C_{1}$
contributes to the naive curve count $N_{p_{1},p}^{2L-E}=\eta(p_{1},p,2L-E)$
from [KY23] Definition $1.1$, by loc. cit. Proposition $3.13$, we have
$f^{*}|{C_{1}}T_{X}^{log}=\mathcal{O}_{C_{1}}^{2}$. For $C_{2}$, note that
since $c_{1}(T_{X}^{log})\cdot(L-E)=0$, Riemann-Roch gives

$dim_{\mathds{k}}\text{
}H^{0}(f_{C_{2}}^{*}T_{X}^{log})-dim_{\mathds{k}}\text{
}H^{1}(f_{C_{2}}^{*}T_{X}^{log})=2.$

Since the only deformations of the log map come from infinitesimal
automorphisms of the domain curve which fix the unique marked point, we have
$H^{0}(f_{C_{2}}^{*}T_{X}^{log})=\mathds{k}^{2}$, implying
$H^{1}(f_{C_{2}}^{*}T_{X}^{log})=0$

With these facts in mind, we now consider the following normalization
sequence:

$0\rightarrow
f^{*}(T_{X}^{log})\rightarrow\oplus_{i}f|_{C_{i}}^{*}(T_{X}^{log})\rightarrow\mathds{k}_{e_{1}}^{2}\oplus\mathds{k}_{e_{2}}^{2}\rightarrow
0$

As usual, the middle term above is considered a sheaf on $C$ via pushforward,
and $e_{i}$ refer to the nodes of $C$. Taking the long exact sequence in
cohomology gives the sequence:

$H^{0}(\oplus_{i}f|_{C_{i}}^{*}(T_{X}^{log}))\rightarrow\mathds{k}_{e_{1}}^{2}\oplus\mathds{k}_{e_{2}}^{2}\rightarrow
H^{1}(f^{*}(T_{X}^{log}))\rightarrow
H^{1}(\oplus_{i}f|_{C_{i}}^{*}(T_{X}^{log}))$

By our previous computation of $f|_{C_{i}}^{*}(T_{X}^{log})$ for each
irreducible component $C_{i}$, the rightmost term vanishes. Hence, to show
$H^{1}(f^{*}(T_{X}^{log}))=0$, it suffices to show that the left most arrow in
the sequence above is surjective.

To do this, we consider the image of each component of
$\oplus_{i}f|_{C_{i}}^{*}(T_{X}^{log})$ in
$\mathds{k}_{e_{1}}^{2}\oplus\mathds{k}_{e_{2}}^{2}$ for $i=0,1$. First, since
$f|_{C_{0}}^{*}T_{X}^{log}=\mathcal{O}^{2}_{C_{0}}$, we have the restriction
map is given by the diagonal
$\Delta:\mathds{k}^{2}\rightarrow\mathds{k}_{e_{1}}^{2}\oplus\mathds{k}_{e_{2}}^{2}$.
For $C_{1}$, since $f^{*}|_{C_{1}}T_{X}^{log}$ is the trivial bundle, it is in
particular globally generated. Hence, the restriction map to
$\mathds{k}_{e_{1}}^{2}$ is surjective. Since the intersection of the images
of these linear maps has trivial intersection, it follows that the sum of the
two maps is surjective, and hence the leftmost morphism in the exact sequence
above is surjective as required.

Case $2$: Non-equality due to refined virtual class:

In this second example, we consider $p=D_{2}+D_{3}$, and we are interested in
the structure constant
$(\vartheta_{p}^{orb})^{2}[\vartheta_{p}^{orb}z^{L-E}]$. The intrinsic mirror
structure constant is easily computed to vanish, once again using a broken
line argument.

To compute the orbifold structure constant, we again use a naive log/orbifold
correspondence, this time coming from Theorem $5.1$ of [BNR24] to account for
the high age/negative contact orders which must be present. In the naive log
setting, letting $\beta$ be the combinatorial type of twisted map contributing
to the invariant of interest, it is straightforward to see that the domain of
such maps consists of two components $C_{1}\cup C_{2}$ joined along a node $q$
with $C_{2}$ containing three special points, with the map mapping $C_{1}$
isomorphically onto the strict transform of the line considered in case $1$
and $C_{2}$ contracted. Each such stable map admits a naive log enhacement of
the correct type. In fact, unlike in case $1$, each admits a logarithmic
enhacement. A map of this type is thus determined by a choice of cross-ratio
of the $4$ special points on $C_{2}$, hence as stacks, we have
$\mathscr{M}^{naive}(X,\beta)\cong\overline{\mathscr{M}}_{0,4}\cong\mathbb{P}^{1}$.
However, by equation 3.1, we have $vdim\text{
}\mathscr{M}^{naive}(X,\beta)=0$. By the naive log/orbifold correspondence,
the structure constant in question can be computed as the degree of this
virtual cycle.

To compute the virtual cycle, observe that the map
$\mathscr{M}^{naive}(X,\beta)\rightarrow\mathfrak{M}^{naive}(\mathcal{X},\beta)$
has image entirely contained in a stratum associated with the realizable
tropical type $\tau$ described above. By essentially the same calculation
which appears in example $1.12$ of [BNR24], we have

$[\mathscr{M}^{naive}(X,\beta)]^{ref}=[\mathscr{M}^{naive}(X,\beta)]\cap(-\psi_{q\in
C_{1}}-\psi_{q\in C_{2}})$

where the cotangent lines above are associated with the two flags of
$G_{\tau}$. The component of the universal family over
$\mathscr{M}^{naive}(X,\beta)$ associated with $C_{1}$ is a trivial family of
once-marked genus $0$ curves, hence
$[\mathscr{M}^{naive}(X,\beta)]\cap(-\psi_{q\in C_{1}})=0$. On the other hand,
under the isomorphism
$\mathscr{M}^{naive}(X,\beta)\cong\overline{\mathscr{M}}_{0,4}$, $\psi_{q\in
C_{2}}$ is the pullback of the associated cotangent line on
$\overline{\mathscr{M}}_{0,4}$. Hence,
$[\mathscr{M}^{naive}(X,\beta)]\cap(-\psi_{q\in C_{2}})=-1$. In summary,
$(\vartheta_{p}^{orb})^{2}[\vartheta_{p}^{orb}z^{L-E}]=-1$. We contrast this
calculation with the fact that all logarithmic structure constants in this
setting are non-negative.

We note however that we nonetheless have a $\mathds{k}[NE(X)]$ algebra
isomorphism $f\colon R_{(X,D)}^{orb}\rightarrow R_{(X,D)}$, with
$f(\vartheta_{D_{i}}^{orb})=\vartheta_{D_{i}}$ for all $i$. Indeed, by [GS19]
Example $1.21$, we have the presentation:

$R_{(X,D)}\cong\mathds{k}[NE(X)][\vartheta_{D_{1}},\vartheta_{D_{2}},\vartheta_{D_{3}}]/(\vartheta_{D_{1}}\vartheta_{D_{2}}\vartheta_{D_{3}}-z^{L}-\vartheta_{D_{1}}z^{L-E})$

In order to show that the assignment
$f(\vartheta_{D_{i}}^{orb})=\vartheta_{D_{i}}$ extends to an isomorphism, just
as in the calculation of $R_{(X,D)}$ from loc. cit. it suffices to compute
$\vartheta^{orb}_{D_{1}}\vartheta^{orb}_{D_{2}}\vartheta^{orb}_{D_{3}}$. The
problems highlighted above in the relevant log and orbifold invariants do not
occur for calculating the relevant naive/orbifold invariants, and we derive
the analogous expression for the triple product. Hence $R_{(X,D)}^{orb}$ has
exactly the same presentation as $R_{(X,D)}$. In particular, in this example,
the log and orbifold bases can be expressed as $\mathds{k}[NE(X)]$ linear
combinations of each other.

##  6\. Log Gromov-Witten theory of root stacks

At various stages of the arguments in this paper, we will use a generalization
of the main result from [Joh22a] to the setting of log étale modifications
given by root stacks. A technical issue occurs when considering this situation
as the theory of log stable maps to a log DM stack has not yet been fully
developed. However, when the orbifold structure arises via a log étale
modfication of a log smooth scheme, it is relatively straightforward to
construct the relevant virtually smooth moduli stacks which satisfy the
required properties. This has been constructed by Battistella, Naibjou and
Ranganathan in [BNR24] assuming only specified global contact orders. In this
section, we will provide the construction when we assume the tropical types
are realizable, from which the necessary generalization of the main result of
[Joh22a] will follow in a straightforward manner. For future purposes, we will
work in a slightly more general context than is strictly necessary for the
purpose of this paper.

Suppose we have a log smooth morphism $X\rightarrow B$ with $B$ log smooth,
and a morphism $\mathcal{A}_{X}\rightarrow\mathcal{A}_{B}$ making the obvious
square commute. We define the relative Artin fan
$\mathcal{X}:=B\times_{\mathcal{A}_{B}}\mathcal{A}_{X}$. Suppose in addition
we have a root stack $\tilde{X}\rightarrow X$ realized as a log étale
modification. To construct a moduli space of twisted log stable maps to
$\tilde{X}$ over $B$, we observe that we can construct the root stack
$\tilde{X}$ as a fiber product:

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{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
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{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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The bottom commuting square is the cartesian square defining $\mathcal{X}$ and
the top right vertical morphism is a log étale modification induced by a
lattice refinement. More precisely, the morphism
$\mathcal{A}_{\tilde{X}}\rightarrow\mathcal{A}_{X}$ is étale locally on
$\mathcal{X}$ induced by a lattice refinement of Artin cones
$\mathcal{A}_{\tilde{\sigma}}\rightarrow\mathcal{A}_{\sigma}$. Letting
$\Gamma_{\tau}\rightarrow\Sigma(X)$ be a family of punctured tropical maps, we
form the fiber product:

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{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.9143pt}{-24.59166pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
The induced morphism $\tilde{\Gamma}_{\tau}\rightarrow\Gamma_{\tau}$ is a
lattice refinement of cone complexes. By pushing forward the refinement to
$\tau$, we produce a cone $\gamma$. Moreover since the all the vertical maps
on cones are induced by lattice refinements, we have
$\gamma\otimes\mathbb{Q}\cong\tau\otimes\mathbb{Q}$. After pulling back
$\tilde{\Gamma}_{\tau}\rightarrow\tau$ along $\gamma\rightarrow\tau$, by the
same proof of Proposition $4.1$ of [Joh22a], we produce a family of punctured
tropical maps $\tilde{\Gamma}_{\gamma}\rightarrow\Sigma(\tilde{X})$. Moreover,
the previous family of maps is universal in the sense that any family of maps
$\Gamma^{\prime}_{\omega}\rightarrow\Sigma(X)$ induced by pulling back
$\Gamma_{\tau}$ along a map $\omega\rightarrow\tau$ which factors through the
refinement $\Sigma(\tilde{X})\rightarrow\Sigma(X)$ is induced by a unique
morphism $\omega\rightarrow\gamma$ which factors $\omega\rightarrow\tau$.
Using this fact, it is straightforward that refinements
$\gamma\rightarrow\tau$ glue to a determine a log étale modification of
$\mathcal{A}_{\mathfrak{M}(\mathcal{X})}$. We pullback this modification along
the strict morphism
$\mathfrak{M}(\mathcal{X},\tau)\rightarrow\mathcal{A}_{\mathfrak{M}(\mathcal{X})}$
to produce a log stack $\mathfrak{M}_{\tilde{\mathcal{X}},\tau^{\prime}}$.
Pulling
$\mathfrak{M}_{\tilde{\mathcal{X}},\tilde{\tau}}\rightarrow\mathfrak{M}(\mathcal{X},\tau)$
along the strict map
$\mathscr{M}(X,\tau)\rightarrow\mathfrak{M}(\mathcal{X},\tau)$ produces a log
DM stack $\mathscr{M}_{\tilde{X},\tilde{\tau}}$, which we record in the fiber
diagram below:

(6.1)
${\mathscr{M}_{\tilde{X},\tilde{\tau}}}$${\mathfrak{M}_{\tilde{\mathcal{X}},\tilde{\tau}}}$${\mathscr{M}(X,\tau)}$${\mathfrak{M}(\mathcal{X},\tau).}$

We claim that the stack $\mathscr{M}_{\tilde{X},\tau^{\prime}}$ represents a
moduli stack of twisted punctured log maps to $\tilde{X}$, a statement which
we will clarify shortly. Before this however, we recall a collection of
definitions and results from [Ols07].

######  Definition 6.1.

  1. (1)

A morphism $\mathcal{M}\rightarrow\mathcal{N}$ of locally free log structures
on a DM stack $X$ is called _simple_ if for every geometric point
$p\rightarrow X$, the monoids $\overline{\mathcal{M}}_{p}$ and
$\mathcal{\mathcal{N}}_{p}$ have the same rank, the morphism
$\phi:\overline{\mathcal{M}}_{p}\rightarrow\overline{\mathcal{N}}_{p}$ is
injective, and for every irreducible element $f\in\overline{\mathcal{N}}_{p}$
there exists an irreducible element $g\in\overline{\mathcal{M}}_{p}$ and a
positive integer $n$ such that $\phi(g)=nf$.

  2. (2)

A log twisted curve is the data
$(C/S,\sigma_{i},a_{i},\mathcal{M}_{S}^{min}\rightarrow\mathcal{M}_{S})$ with
$C/S$ a stable curve, $\sigma_{i}:S\rightarrow C$ sections of the curve,
$a_{i}$ positive integers indexed by the sections, and
$\mathcal{M}_{S}^{min}\rightarrow\mathcal{M}_{S}$ a simple morphism of fine
and saturated log structures on $S$ with $\mathcal{M}^{min}_{S}$ the minimal
log structure induced by the classifying morphism
$S\rightarrow\overline{\mathscr{M}}_{g,n}$.

  3. (3)

A twisted log curve is a prestable twisted curve $\mathcal{C}/S$ with log
structures on $\mathcal{C}$ and $S$ such that $\mathcal{C}\rightarrow S$ is
log smooth.

Note that the tropicalization of a simple log morphism
$(S,\mathcal{N}_{S})\rightarrow(S,\mathcal{M}_{S})$ is a morphism of cone
complexes which upon restriction to cones of the domain are given by a lattice
refinements. We call the lattice refinements arising in this way simple
lattice refinements. Both twisted curves and the enriched curves defined above
over some base scheme $S$ are objects in naturally defined groupoids. Theorem
$1.9$ of [Ols07] states the following:

######  Proposition 6.2.

For any scheme $S$, there is a natural equivalence of groupoids between the
groupoid of $n$-pointed twisted curves over $S$ and the groupoid of log
twisted $n$-pointed curves over $S$. Moreover, this equivalence is compatible
with base change $S^{\prime}\rightarrow S$.

In order to frame the definition of a twisted log curve in a way which will be
useful for us, we briefly recall the mechanics of the above equivalence. One
first shows that every family of twisted curves $\mathcal{C}\rightarrow S$
admits a canonical log enhancement, producing a twisted log curve over a log
scheme $(S,\mathcal{M}_{S}^{\prime})$, and the log structure on $\mathcal{C}$
pushes forward to the coarse curve $C$. By the universal property satisfied by
basic log curves, the log curve $C/S$ pulls back from a basic log curve
$C^{min}/S^{min}$. The resulting morphism of log structures on $S$ is a simple
morphism of log structures. The tuple of data given by the orders of twisting
at marked points, the prestable curve $C/S$ and the morphism
$\mathcal{M}_{S^{min}}\rightarrow\mathcal{M}_{S}$ gives a family of log
twisted curves. Conversely, every log twisted curve is shown is shown to come
from a unique such orbifold enhancement of the underlying family of stable
curve. Moreover, the canonical log enhancement of a twisted curve satisfies a
similar universal property familiar from the theory of ordinary stable curves,
namely any other log enhancement of the twisted curve $\mathcal{C}\rightarrow
S$ is realized by an appropriate log morphism on the base induced by a unique
morphism of log structures on $S$. As a result, we can think of a twisted log
curve both as in the definition as above, or as the data of a log twisted
curve together with a morphism of log schemes
$(S,\mathcal{M}_{S})\rightarrow(S,\mathcal{M}_{S}^{\prime})$, with
$\mathcal{M}_{S}^{\prime}$ as above.

Twisted log curves arise via tropical considerations in the following way for
us: Suppose that we have a family of prestable log curve $C/S$, and for
simplicity we assume the dual graph over all schematic points is given by
$G_{\tau}$. Tropicalization gives a family of tropical curves
$\Gamma\rightarrow\Sigma(S)=\sigma$, which is induced by a morphism
$\sigma\rightarrow\tau=\mathbb{R}_{\geq 0}^{|E(G_{\tau})|}$. Suppose we have a
lattice refinement $\Gamma^{\prime}\rightarrow\Gamma$ such that for all cones
$\omega$ of $\Gamma^{\prime}$, the map
$\omega_{\mathbb{N}}\rightarrow\sigma_{\mathbb{N}}$ is surjective. Such a
modification is determined by numbers $k_{e}$ for each edge and leg given by
$|coker(\sigma_{e^{\prime},\mathbb{N}}^{gp}\rightarrow\sigma_{e,\mathbb{N}}^{gp})|$
for each edge and leg $e$ refined by an edge/leg $e^{\prime}$ of
$\Gamma^{\prime}$. Since for all $e\in E(G_{\tau})$ and all integral tropical
curves in the family $\Gamma$, the length of the edge $e$ must be divisible by
$k_{e}$, the cone map $\sigma\rightarrow\tau=\mathbb{R}_{\geq
0}^{|E(G_{\tau})|}$ will factor through a simple lattice refinement
$\tau^{\prime}$ of $\tau$ determined by the numbers $k_{e}$. Moreover, the
refinement $\Gamma^{\prime}\rightarrow\Gamma$ is induced by pulling back a
refinement $\Gamma^{\prime}_{\tau^{\prime}}\rightarrow\Gamma_{\tau^{\prime}}$
of the pullback of the universal curve to $\tau^{\prime}$.

Now recall from Lemma 3.1 that the morphism $\sigma\rightarrow\tau^{\prime}$
corresponds to a morphism of Artin fans
$\mathcal{A}_{\sigma}\rightarrow\mathcal{A}_{\tau^{\prime}}$, and by composing
the strict morphism $(S,\mathcal{M}_{S})\rightarrow\mathcal{A}_{\sigma}$ with
this map and pulling back the log structure, we produce a log structure we
call $\mathcal{M}^{tw}$. Similarly, the morphism
$\tau^{\prime}\rightarrow\tau$ induces a simple morphism of log structures
$\mathcal{M}_{S}^{min}\rightarrow\mathcal{M}_{S}^{tw}$ with
$\mathcal{M}_{S}^{min}$ the basic log structure on $S$. Hence we find the data
of a log twisted curve together with a map
$(S,\mathcal{M}_{S})\rightarrow(S,\mathcal{M}_{S}^{tw})$. By the remarks in
the paragraph following Proposition 6.2, this gives by a twisted log curve
$\mathcal{C}$ with coarse curve $C$. Moreover, the morphism to the coarse
curve tropicalizes to $\Gamma^{\prime}\rightarrow\Gamma$.

We now give a lemma describing how puncturings of twisted log curve
$\mathcal{C}$ may be produced from puncturing of the coarse log curve $C$.

######  Lemma 6.3.

Suppose $C\rightarrow S$ is a log curve, $\rho:\mathcal{C}\rightarrow C$ is a
twisted log curve over $C$ and $C^{\circ}\rightarrow C$ is a puncturing of
$C$. Then
$\mathcal{C}^{\circ}:=\mathcal{C}\times_{C}C^{\circ}\rightarrow\mathcal{C}$ is
a puncturing of the twisted log curve $\mathcal{C}$.

###### Proof.

First observe that the map
$\rho^{*}\mathcal{M}_{C}\rightarrow\mathcal{M}_{\mathcal{C}}$ is an integral
morphism of log structures, hence $\mathcal{C}\rightarrow C$ is integral.
Thus, the coherent log structure $\mathcal{C}^{\circ}$ is also fine. To verify
the second condition of Definition 3.2, we note that for $p\in\mathcal{C}$ a
punctured point, and $s\in\mathcal{M}_{\mathcal{C}^{\circ},p}$, we have
$s\notin\mathcal{M}_{\mathcal{C},p}$ if and only if its image
$\overline{s}\in\overline{\mathcal{M}}_{\mathcal{C}^{\circ},p}$ is not
contained in
$\overline{\mathcal{M}}_{\mathcal{C},p}\subset\overline{\mathcal{M}}_{\mathcal{C}^{\circ},p}$.
Since the log structure on $\mathcal{C}^{\circ}$ is fine, we can compute the
stalk $\overline{\mathcal{M}}_{\mathcal{C}^{\circ},p}$ in the category of
integral monoids. Letting
$Q_{p}=\overline{\mathcal{M}}_{C^{\circ},p}\subset\overline{\mathcal{M}}_{S,\pi(p)}\oplus\mathbb{Z}$,
$\overline{\mathcal{M}}_{\mathcal{C},p}=\overline{\mathcal{M}}_{S,\pi(p)}\oplus\mathbb{N}$,
the pushout is given by the smallest submonoid of
$\overline{\mathcal{M}}_{S,\pi(s)}\oplus\mathbb{Z}$ containing
$\overline{\mathcal{M}}_{S,\pi(s)}\oplus\mathbb{N}$ and $i(Q)$, with
$i:\overline{\mathcal{M}}_{C,p}^{gp}\rightarrow\overline{\mathcal{M}}_{\mathcal{C},p}^{gp}$.
In particular, if $\overline{s}\notin\overline{\mathcal{M}}_{\mathcal{C},p}$,
we can factor $\overline{s}$ as
$\overline{s}=\overline{s}_{1}+\overline{s}_{2}$ with
$\overline{s}_{1}\in\overline{\mathcal{M}}_{s,\pi(s)}\oplus\mathbb{N}$ and
$\overline{s}_{2}=i(\overline{t})$ for
$\overline{t}\in\overline{\mathcal{M}}_{C^{\circ},p}\setminus\overline{\mathcal{M}}_{C,p}$.
By picking lifts of $s_{1}\in\mathcal{M}_{\mathcal{C},p}$ and
$t\in\mathcal{M}_{C^{\circ},p}$, we have $s_{1}+s_{2}$ differs from $s$ only
by a unit. In particular, $\alpha_{\mathcal{C}^{\circ}}(s)=0$ if and only if
$\alpha_{C^{\circ}}(t)=0$, which holds since $C^{\circ}$ is a punctured log
curve. Similarly, writing $s=(e_{1},e_{2})$ and $s_{i}=(e_{i1},e_{i2})$, then
$e_{1}$ is equal to $e_{11}+e_{21}$ up to a unit in
$\mathcal{O}_{S,\pi(p)}^{\times}$. Since $\alpha_{S}(e_{21})=0$ as $C^{\circ}$
is a punctured log curve, we have $\alpha_{S}(e_{1})=0$, as required. ∎

We now proceed to a proof of the representability claim:

######  Proposition 6.4.

Let $\mathscr{M}(\tilde{X}/B,\tilde{\tau})$ be the fibered category defined
over log schemes such that the objects of the groupoid over $S$ are the
following data:

  1. (1)

A twisted log curves $\mathcal{C}\rightarrow S$ over an ordinary log curve
$C\rightarrow S$, in particular a log structure $\mathcal{M}_{S}$ on $S$.

  2. (2)

A punctured log map $C^{\circ}\rightarrow X$ marked by the type $\tau$.

  3. (3)

A commuting diagram of logarithmic morphisms:

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We further require the top horizontal morphism to be representable.

Then the fibered category of $\mathscr{M}_{\tilde{X},\tilde{\tau}}$ over fine
and saturated log schemes is isomorphic to
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})$.

###### Proof.

We recall that $\mathscr{M}_{\tilde{X},\tilde{\tau}}$ is defined as the fiber
product given in Display 6.1. Hence, to prove the isomorphism, we must show
that $\mathscr{M}(\tilde{X}/B,\tilde{\tau})$ satisfies the the same universal
property as $\mathscr{M}_{\tilde{X}/B,\tilde{\tau}}$. Clearly we have a
morphism
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})\rightarrow\mathscr{M}(X/B,\tau)$ by
taking the map associated with $C^{\circ}\rightarrow\mathcal{X}$. By
definition, the tropical type of every twisted log map
$\mathcal{C}^{\circ}\rightarrow\tilde{X}$ associated with a geometric point of
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})$ is a tropical lift of a log stable map
$C\rightarrow X$. Hence, the proof of Proposition $5.4$ of [Joh22a] applies in
this setting to give a strict and idealized strict morphism
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})\rightarrow\mathfrak{M}_{\tilde{\mathcal{X}},\tilde{\tau}}$
which factors the natural morphism
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})\rightarrow\mathfrak{M}(\mathcal{X}/B,\tau)$.

In order to show the resulting square is cartesian, we must first construct a
morphism of stacks
$\mathfrak{M}_{\tilde{\mathcal{X}},\tilde{\tau}}\rightarrow\mathfrak{M}(\tilde{\mathcal{X}}/B,\tilde{\tau})$
which factors
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})\rightarrow\mathfrak{M}(\tilde{\mathcal{X}}/B,\tilde{\tau})$,
with the latter stack defined analogously to
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})$. The construction works essentially
unchanged from Section $5.1$ of [Joh22a], although is simpler in this setting
due to the uniqueness of a tropical lift of a tropical type.

Suppose we have a fine and saturated scheme $S$, together with an $S$-valued
object of $\mathfrak{M}_{\tilde{\mathcal{X}},\tilde{\tau}}$, giving a morphism
$C^{\circ}/S\rightarrow X$. By the same reduction that appears in Section
$5.1$ of [Joh22a], we may assume the map
$S\rightarrow\mathcal{A}_{\mathfrak{M}_{\tilde{X},\tau}}$ factors through an
inclusion of an idealized Artin cone
$\mathcal{A}_{\tau^{\prime},I}\subset\mathcal{A}_{\mathfrak{M}_{\tilde{X},\tau}}$,
hence $\Sigma(S)=\tau^{\prime}$. Associated to $\tau^{\prime}$ is a universal
family of punctured tropical curves
$\tilde{\Gamma}^{\circ}_{\tau^{\prime}}\rightarrow\tilde{X}$ over the cone
$\tau^{\prime}$, and a map of cone complexes
$\tilde{\Gamma}^{\circ}_{\tau^{\prime}}\rightarrow\Gamma_{\tau^{\prime}}$
given cone wise by a lattice refinement. Since $C^{\circ}/S\rightarrow X$ is
induced by a morphism $S\rightarrow\mathfrak{M}_{\tilde{\mathcal{X}},\tau}$,
we have $\Sigma(C^{\circ})\rightarrow\Sigma(X)$ restricted to cones associated
with vertices of $\tau$ factors through the lattice refinement
$\Sigma(\tilde{X})\rightarrow\Sigma(X)$.

In order to lift extend the tropical lift to edges and legs, we must refine
the lattices of the edges and legs of $\Gamma_{\tau}$. To produce this
refinement, note that since the integral points of the maximal cones of
$\Gamma_{\tau^{\prime}}^{\circ}$ naturally embed into the lattice
$\tau_{\mathbb{N}}^{{}^{\prime}gp}\times\mathbb{Z}$, and the refinement
morphism restricts to an isomorphism on the submodule
$\tau_{\mathbb{N}}^{gp^{\prime}}$, the refinement restricted to a given
maximal cone associated with an edge or leg $e$ must be of the form
$\tau_{\mathbb{N}}^{{}^{\prime}gp}\times
r_{e}\mathbb{Z}\subset\tau_{\mathbb{N}}^{{}^{\prime}gp}\times\mathbb{Z}$ for
some natural number $r_{e}\in\mathbb{N}_{\geq 0}$. Hence, this lattice
refinement induces a twisted log curve $\mathcal{C}\rightarrow C$. The
tropical lift $\tilde{\Gamma}\rightarrow\Sigma(\tilde{X})$ of
$\Gamma\rightarrow\Sigma(X)$ guarantees there exists a map
$\tilde{f}:\mathcal{C}^{\circ}\rightarrow\tilde{\mathcal{X}}$ lifting the map
$C^{\circ}\rightarrow\mathcal{X}$.

To see that $\tilde{f}$ is representable, note that this is true if the map
induced on stabilizer groups of geometric points is injective. Thus, let
$q^{\prime}\in\mathcal{C}(\mathds{k})$ be a geometric point contained in
either a nodal or marked section of $\mathcal{C}$, and suppose
$Stab(q^{\prime})\cong\mu_{r}$. In particular, the induced morphism
$m_{q}:q^{\prime}\rightarrow\rho(q^{\prime}):=q$ is a generalized root stack
of the log point $q$, introduced in the last paragraph of Section $3.1$.
Letting $\omega^{\prime}=\Sigma(q^{\prime})$ and $\omega=\Sigma(q)$, this map
is induced by a lattice refinement
$\omega^{{}^{\prime}gp}_{\mathbb{N}}\rightarrow\omega^{gp}_{\mathbb{N}}$,
which with a slight abuse of notation we will refer to as $m_{q}$ as well.
Moreover, we have an induced map $B\mu_{r}\rightarrow\tilde{\mathcal{X}}$
which lifts a map $Spec\text{ }\mathds{k}\rightarrow\mathcal{X}$. Letting
$p=c\circ\tilde{f}(q)\in\mathcal{X}(\mathds{k})$, the morphism on stabilizer
groups $Stab(q)\rightarrow Stab(p)$ induced by $c\circ
f:\mathcal{C}\rightarrow\mathcal{X}$ must be the zero map, hence the morphism
$B\mu_{r}\rightarrow\mathcal{X}$ must factor through a geometric point. Since
$\tilde{\mathcal{X}}$ is a generalized root stack over $\mathcal{X}$
constructed via a fiber diagram in algebraic stacks, the morphism
$B\mu_{r}\rightarrow\tilde{\mathcal{X}}$ must factor through a root stack of
the log point $p$, denoted by $m_{p}:p^{\prime}\rightarrow p$. Letting
$\Sigma(p^{\prime})=\sigma^{\prime}$ and $\Sigma(p)=\sigma$, the morphism
$m_{p}$ is induced by a lattice refinement
$\sigma^{{}^{\prime}gp}_{\mathbb{N}}\rightarrow\sigma^{gp}_{\mathbb{N}}$,
which again with a slight abuse of notation we will refer to as $m_{p}$ as
well.

Since $p^{\prime}\rightarrow\tilde{\mathcal{X}}$ is representable, in order to
show that $\tilde{f}$ is representable, it suffices to show that
$\tilde{f}|_{q^{\prime}}:B\mu_{r}\rightarrow p^{\prime}$ is representable.
Recall that if $m_{p}:p^{\prime}\rightarrow p$ is a generalized root stack
realized as a log étale modification of a log point $p$ with
$\Sigma(p)=\sigma$ and $\Sigma(p^{\prime})=\sigma^{\prime}$ induced by a
lattice refinement $m_{p}$, then
$Stab(p^{\prime})=\sigma^{gp}_{\mathbb{N}}/m_{p}(\sigma^{{}^{\prime}gp}_{\mathbb{N}})$.
Since $m_{p}:p^{\prime}\rightarrow p$ and $m_{q}:q^{\prime}\rightarrow q$ are
log monomorphisms, $f:q\rightarrow p$ determines
$\tilde{f}:q^{\prime}\rightarrow p^{\prime}$, we must have
$\Sigma(f)^{gp}_{\mathbb{N}}((\Sigma(m_{q})(\omega^{{}^{\prime}gp}_{\mathbb{N}}))\subset
m_{p}(\sigma^{{}^{\prime}gp}_{\mathbb{N}})$ and the morphism on stabilizers is
induced by $\omega_{\mathbb{N}}^{gp}\rightarrow\sigma_{\mathbb{N}}^{gp}$. This
induced map is injective only if

$\Sigma(f)^{gp}_{\mathbb{N}}((m_{q}(\omega^{{}^{\prime}gp}_{\mathbb{N}})))=\Sigma(f)^{gp}_{\mathbb{N}}(\omega_{\mathbb{N}}^{gp})\cap
m_{p}(\sigma^{{}^{\prime}gp}_{\mathbb{N}}).$

By the minimal choice of lattice refinement at each leg and edge of $G_{\tau}$
taken to allow a lift of $\tilde{\Gamma}\rightarrow\Gamma$ along the
refinement $\Sigma(\tilde{X})\rightarrow\Sigma(X)$, the representability of
$\tilde{f}$ follows.

In total we find an $S$-valued object of
$\mathfrak{M}(\tilde{\mathcal{X}}/B,\tilde{\tau})$. The proofs of Propositions
$5.7$ and $5.8$ of [Joh22a] apply to show there is a morphism
$\mathfrak{M}_{\tilde{\mathcal{X}},\tau}\rightarrow\mathfrak{M}(\tilde{\mathcal{X}}/B,\tilde{\tau})$
which factors
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})\rightarrow\mathfrak{M}(\tilde{\mathcal{X}}/B,\tilde{\tau})$.
Finally the proof of Proposition $6.1$ of [Joh22a] works to show that
$\mathscr{M}(\tilde{X}/B,\tilde{\tau})$ satisfies the required universal
property.

∎

The morphism
$\mathscr{M}(\tilde{X}/b,\tilde{\tau})\rightarrow\mathfrak{M}_{\tilde{\mathcal{X}},\tilde{\tau}}$
can be equipped with the pulled back obstruction theory from
$\mathscr{M}(X/B,\tau)\rightarrow\mathfrak{M}(\mathcal{X}/B,\tau)$, which may
equivalently be given by the dual of:

$R\pi_{*}f^{*}T_{\tilde{X}/B}^{log}=R\pi_{*}f^{*}c^{*}T_{X/B}^{log}$

The equality holds because the map to the coarse moduli space
$c:\tilde{X}\rightarrow X$ is log étale. Virtual pullback of the fundamental
class of $\mathfrak{M}_{\tilde{\mathcal{X}},\tau}$ is the sense of [Man08]
equips $\mathscr{M}(\tilde{X}/B,\tilde{\tau})$ with a virtual class
$[\mathscr{M}(\tilde{X}/B,\tilde{\tau})]^{vir}\in A_{exp\text{
}dim}(\mathscr{M}(\tilde{X}/B,\tilde{\tau}))$. By the same arguments as in
[Joh22a], the virtual classes are related by the following formula:

######  Corollary 6.5.

Letting $r=|coker(\tilde{\tau}^{gp}\rightarrow\tau^{gp})|$, and
$st:\mathscr{M}(\tilde{X},\tilde{\tau})\rightarrow\mathscr{M}(X/B,\tau)$ the
map which takes a twisted log map to the corresponding map on coarse spaces,
we have the equality:

$st_{*}([\mathscr{M}(\tilde{X}/B,\tilde{\tau})]^{vir})=\frac{1}{r}[\mathscr{M}(X/B,\tau)]^{vir}.$

######  Remark 6.6.

Battistella, Nabijou and Ranganathan constructed in [BNR24] moduli stacks
$\mathscr{M}^{naive}(\tilde{X},\beta^{\prime})$ and
$\mathfrak{M}^{naive}(\tilde{\mathcal{X}},\beta^{\prime})$ of naive twisted
log maps to root stacks for non-realizable tropical types $\beta^{\prime}$
which specify only (gerby) contact orders of marked points, together with a
perfect obstruction theory for the morphism
$\mathscr{M}^{naive}(\tilde{X},\beta^{\prime})\rightarrow\mathfrak{M}^{naive}(\tilde{\mathcal{X}},\beta^{\prime})$.
Additionally, the forgetful map
$\mathfrak{M}^{naive}(\tilde{\mathcal{X}},\beta^{\prime})\rightarrow\mathfrak{M}^{tw}$
remembering only the domain twisted curve is idealized log étale, with the
target equipped with the trivial idealized log structure. Moreover, assuming
$\beta^{\prime}$ is slope sensitive, the reduction of the stack
$\mathfrak{M}^{naive}(\tilde{\mathcal{X}},\beta^{\prime})_{red}$ is stratified
by components corresponding to realizable tropical types, which recover the
log stacks defined above. With the previous assumption, we will omit the
superscript and call the resulting stacks
$\mathscr{M}(\tilde{X},\beta^{\prime})$ and
$\mathfrak{M}(\tilde{\mathcal{X}},\beta^{\prime})$.

In the following, we note how the above corollary determines how punctured
Gromov-Witten invariants behave under a log étale base change, which may be
useful in studying semistable degenerations of varieties:

Suppose $X\rightarrow\mathbb{A}^{1}$ is a log smooth morphism, with
$\mathbb{A}^{1}$ carrying its usual toric log structure, and consider the log
étale morphism $z^{k}:\mathbb{A}^{1}\rightarrow\mathbb{A}^{1}$ induced by
regular function $z^{k}\in\mathds{k}[z]$. We denote by $B$ and $B^{\prime}$
the codomain and domain of $z^{k}$ respectively to differentiate. We now form
the following fiber square:

${X^{\prime}}$${X}$${B^{\prime}}$${B}$$\scriptstyle{b_{k}}$$\scriptstyle{z^{k}}$

Observe that $X^{\prime}\rightarrow X$ is a log étale morphism between log
schemes. Moreover, the induced morphism
$\Sigma(X^{\prime})\rightarrow\Sigma(X)$ is cone wise given by a lattice
refinement. In particular, $X^{\prime}\rightarrow X$ factors through the root
stack $X^{\prime}\rightarrow X_{k}$ induced by the previous lattice
refinement. We wish to understand the log invariants of $X^{\prime}$ in terms
of the log invariants of $X$. To do so, suppose we have a realizable tropical
type $\tau^{\prime}$ of log stable map to $X^{\prime}$. This induces a
realizable tropical type $\tau$ of log stable map to $X$, and a morphism
$st:\mathscr{M}(X^{\prime},\tau^{\prime})\rightarrow\mathscr{M}(X,\tau)$.
Moreover, by identifying $\Sigma(X^{\prime})$ with $\Sigma(X_{k})$,
$\tau^{\prime}$ also defines a moduli stack
$\mathscr{M}(X_{k},\tau^{\prime})$, and we have a morphism
$i:\mathscr{M}(X^{\prime},\tau^{\prime})\rightarrow\mathscr{M}(X_{k},\tau^{\prime})$.
Analogous morphisms also exist between the relative moduli stacks over the
base $B$. We now state a corollary of the results of this section:

######  Corollary 6.7.

With notation as above and
$r=|coker(\tau^{{}^{\prime}gp}_{\mathbb{N}}\rightarrow\tau^{gp}_{\mathbb{N}})|$,
we have $r\mid k$ and:

$\displaystyle
st_{*}([\mathscr{M}(X^{\prime}/B^{\prime},\tau^{\prime})]^{vir})$
$\displaystyle=\frac{k}{r}[\mathscr{M}(X/B,\tau)]^{vir}.$ $\displaystyle
st_{*}([\mathscr{M}(X^{\prime}/B,\tau^{\prime})]^{vir})$
$\displaystyle=\frac{1}{r}[\mathscr{M}(X/B,\tau)]^{vir}.$ $\displaystyle
st_{*}([\mathscr{M}(X^{\prime},\tau^{\prime})]^{vir})$
$\displaystyle=\frac{k}{r}[\mathscr{M}(X,\tau)]^{vir}.$

###### Proof.

We first prove the claim $r\mid k$. Suppose we have an integral tropical map
$h:\Gamma\rightarrow\Sigma(X)$ of type $\tau$, and let $v\in V(G_{\tau})$ be a
vertex. Then if $(h,\Gamma)\in\tau_{\mathbb{N}}$ is in the image of
$\tau^{\prime}_{\mathbb{N}}\rightarrow\tau_{\mathbb{N}}$, we must have
$\Sigma(\pi)h(v)\in k\mathbb{N}\subset\mathbb{N}$, with
$\Sigma(\pi):\Sigma(X)\rightarrow\mathbb{R}_{\geq 0}$ the tropicalization of
the map $\pi:X\rightarrow B$. On the other hand, if $\Sigma(\pi)h(v)\in
k\mathbb{N}$, then all other integral points of $\Gamma$ must also have image
under $\Sigma(\pi)h$ contained in $k\mathbb{N}$ since $G_{\tau}$ is connected
and all contact orders of bounded edges of $G_{\tau}$ have images under $\pi$
given by integral tangent vectors in $k\mathbb{Z}$. Therefore, $(h,\Gamma)\in
im(\tau^{\prime}_{\mathbb{N}})$ if and only if $\Sigma(\pi)h(v)\in
k\mathbb{N}$. Hence, letting $h_{v}:\tau_{\mathbb{N}}\rightarrow\Sigma(X)$ be
the evaluation map at the vertex $v\in V(G_{\tau})$, the quotient group
$\tau_{\mathbb{N}}^{gp}/\tau^{{}^{\prime}gp}_{\mathbb{N}}$ is isomorphic to
the quotient $im(\Sigma(\pi)h_{v})/(k\mathbb{N}\cap im(\Sigma(\pi)h_{v})$).
Since both groups are cyclic groups, the quotient is cyclic of order dividing
$k$.

We now show the equation
$k[\mathscr{M}(X_{k}/B,\tau^{\prime})]^{vir}=i_{*}([\mathscr{M}(X^{\prime}/B^{\prime},\tau^{\prime})])^{vir}$.
To do so, we will apply the Costello-Herr-Wise pushforward theorem of [Cos06]
and [HW21], in particular we need a cartesian squares relating the moduli
stacks above. Let
$\mathcal{X}^{\prime}:=\mathcal{A}_{X^{\prime}}\times_{\mathcal{A}_{\mathbb{N}}}B^{\prime}$,
and consider the commuting diagram:

${\mathscr{M}(X^{\prime}/B^{\prime},\tau^{\prime})}$${\mathscr{M}(X_{k}/B,\tau^{\prime})}$${\mathfrak{M}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})}$${\mathfrak{M}(\mathcal{X}_{k}/B,\tau^{\prime})}$

The claim that the above diagram is cartesian follows closely from the fact
that $X^{\prime}$ is defined via the following pullback diagram in all log
categories:

${X^{\prime}}$${X_{k}}$${\mathbb{A}^{1}}$${\mathbb{A}^{1}/\mu_{k}}$

In particular, the data of a log stable map $C\rightarrow X^{\prime}$ is the
data of a log stable map $X_{k}$, together with a choice of $k^{th}$ root of
the regular function $C\rightarrow\mathbb{A}^{1}$ induced by $C\rightarrow
X_{k}$. Note in particular that $X^{\prime}\rightarrow X_{k}$ is a strict log
morphism. Moreover, since $B^{\prime}\rightarrow B$ is log étale, we have
$T_{X^{\prime}/B^{\prime}}^{log}=q^{*}T_{X_{k}/B}^{log}$ for
$q:X^{\prime}\rightarrow X_{k}$, hence the obstruction theory for the right
vertical morphism pulls back to the obstruction theory for the left vertical
morphism.

Consider the open substacks
$\mathfrak{M}^{\circ}(\mathcal{X}_{k}/B,\tau^{\prime})$ and
$\mathfrak{M}^{\circ}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})$
consisting of prestable log maps of type $\tau^{\prime}$, and let
$S\rightarrow\mathfrak{M}^{\circ}(\mathcal{X}_{k}/B,\tau^{\prime})$ be a
morphism. We claim that $S$ is in the image of
$\mathfrak{M}^{\circ}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})$. Indeed,
the family of tropical maps $\Gamma\rightarrow\Sigma(X)$ associated to the
morphism $\tau^{\prime}\rightarrow\tau$ consists of tropical curves with edges
and legs having integral lattices that already factor through the sublattice
of interest. Hence, the resulting map $C/S\rightarrow\mathcal{X}$ is the
middle horizontal morphism in Diagram 6.2. Since
$\mathfrak{M}^{\circ}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})\rightarrow\mathfrak{M}^{\circ}(\mathcal{X}_{k}/B,\tau^{\prime})$
is degree $k$ onto its image, coming from the choice of $k^{th}$ root of the
resulting map $C\rightarrow B$ induced by $C\rightarrow\mathcal{X}$, we find
$\mathfrak{M}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})\rightarrow\mathfrak{M}(\mathcal{X}_{k}/B,\tau^{\prime})$
is degree $k$.

In order to apply the Costello-Herr-Wise pushforward theorem, we must also
show that the morphism is proper. First, by replacing the relevant moduli
stacks with open substacks which contain the images of
$\mathscr{M}(X^{\prime}/B^{\prime},\tau^{\prime})$ and
$\mathscr{M}(X_{k}/B,\tau^{\prime})$ where necessary, we may assume the
morphism is finite type. Hence, we may prove properness by proving the
morphism satisfies the valuative criterion. To this end, suppose we have a
valuation ring $R$ with field of fractions $K$, and morphisms $Spec\text{
}K\rightarrow\mathfrak{M}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})$ and
$Spec\text{ }R\rightarrow\mathfrak{M}(\mathcal{X}_{k}/B,\tau^{\prime})$ making
the obvious square commute via some chosen isomorphism of $K$-valued points of
$\mathfrak{M}(\mathcal{X}_{k}/B,\tau^{\prime})$. Note that contained in the
data of the latter morphism is a prestable log curve $C_{R}$ over $Spec\text{
}R$, together with a non-basic map $f_{R}:C_{R}\rightarrow\mathcal{X}$ of type
$\tau$. In particular, we have a regular map $Spec\text{
}R\rightarrow\mathbb{A}^{1}$ which pulls back to the map
$C_{R}\rightarrow\mathbb{A}^{1}$ induced by $f_{R}$.

Letting $r\in R$ be the element corresponding to the former regular map, since
$\mathcal{X}^{\prime}$ is constructed as a fiber product in any log category,
in order to lift $f_{R}$ to a morphism to $\mathcal{X}^{\prime}$, it suffices
to pick a $k^{th}$ root of $r$ in $R$. For existence, we note that $r$ has a
$k^{th}$ root in the field $K$, and since valuations rings are integrally
closed, $r$ has a $k^{th}$ root in $R$. Moreover, the morphism $Spec\text{
}K\rightarrow\mathfrak{M}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})$
furnishes one such root. Thus, by the universal property defining
$\mathcal{X}^{\prime}$, there exists a pair of a map
$C_{R}\rightarrow\mathcal{X}^{\prime}$ and an isomorphism between the two
resulting maps $C_{R}\rightarrow\mathcal{X}$ which is unique up to unique
isomorphism. Moreover, the isomorphism realizing the lift is the unique such
isomorphism which upon restriction to the dense open $Spec\text{
}C_{K}\rightarrow\mathcal{X}$ gives the isomorphism realizing the original
commuting square. It follows from the valuative criterion that
$\mathfrak{M}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})\rightarrow\mathfrak{M}(\mathcal{X}_{k}/B,\tau^{\prime})$
is proper. Hence, the Costello-Herr-Wise pushforward theorem of [Cos06] and
[HW21] allows us to conclude the desired equality of virtual classes. By
Corollary 6.5, we derive the first claimed equality of the corollary.

For the second equality, we observe we have the following cartesian diagram:

${\mathscr{M}(X^{\prime}/B^{\prime},\tau^{\prime})}$${\mathscr{M}(X^{\prime}/B,\tau^{\prime})}$${\mathfrak{M}(\mathcal{X}^{\prime}/B^{\prime},\tau^{\prime})}$${\mathfrak{M}(\mathcal{A}_{X^{\prime}}\times_{\mathcal{A}_{\mathbb{N}}}B/B,\tau^{\prime})}$$\scriptstyle{p}$

As before, the both vertical morphisms are equipped with obstruction theories
which are related by pullback. The bottom morphism is finite of degree $k$,
hence we have
$p_{*}([\mathscr{M}(X^{\prime}/B^{\prime},\tau^{\prime})]^{vir})=k[\mathscr{M}(X^{\prime}/B,\tau^{\prime})]^{vir}$
and the second claimed equality of the corollary follows.

For the final equality, we cannot assume $\tau^{\prime}$ defines a tropical
type of map to $X^{\prime}$ relative to the projection $X^{\prime}\rightarrow
B^{\prime}$. We nonetheless have maps
$\mathscr{M}(X^{\prime},\tau^{\prime})\rightarrow B^{\prime}$ and
$\mathscr{M}(X_{k},\tau^{\prime})\rightarrow B$, since every regular function
on a family of proper curves pulls back from the base. Letting
$\mathcal{X}_{k}$ be the Artin fan of $X_{k}$ relative to $Spec\text{
}\mathds{k}$, have the following cartesian diagram:

${\mathscr{M}(X^{\prime},\tau^{\prime})}$${\mathscr{M}(X_{k},\tau^{\prime})}$${\mathfrak{M}(\mathcal{A}_{X^{\prime}},\tau^{\prime})\times
B^{\prime}}$${\mathfrak{M}(\mathcal{X}_{k},\tau^{\prime})\times
B}$$\scriptstyle{i}$

Again, the vertical morphisms are equipped with obstruction theories which are
related by pullback. By similar arguments to proving the first claimed
equality of the corollary, the morphism
$\mathfrak{M}(\mathcal{A}_{X^{\prime}},\tau^{\prime})\rightarrow\mathfrak{M}(\mathcal{X}_{k},\tau^{\prime})$
is pure degree $1$. From this, we deduce the bottom horizontal of the above
cartesian diagram is pure degree $k$, and therefore
$i_{*}([\mathscr{M}(X^{\prime},\tau^{\prime})]^{vir})=k[\mathscr{M}(X_{k},\tau^{\prime})]^{vir}$.
By an application of Corollary 6.5, we derive the third claimed equality of
the corollary.

∎

##  7\. Comparing Invariants

To begin the comparison, we note that given a log smooth pair $(X,D)$, there
exists a log étale modification $(\tilde{X},\tilde{D})\rightarrow(X,D)$ such
that $\tilde{X}$ is smooth and $\tilde{D}\subset\tilde{X}$ is a simple normal
crossings divisor. Moreover, after noting that the proof of Corollary $1.6$ of
[Joh22a] does not use the fact that the invariants in question define an
associative algebra, if the structure constants defining
$R_{(\tilde{X},\tilde{D})}$ satisfy the relations stated in Corollary 1.2,
then so to do the log Gromov-Witten invariants giving the structure constants
for $R^{log}_{(X,D)}$. Moreover, by birational invariance of ordinary log
Gromov-Witten invariants i.e. Corollary 1.3.1 of [AW13], in order to prove
Theorem 1.3, it suffices to prove the theorem for a log étale modification.
Thus, without loss of generality, we will assume throughout that $X$ is smooth
and $D$ has simple normal crossings.

In order to prove Theorem 1.1, the most important case for us to consider will
be when the numerical data defining a type $\beta$ used in defining a given
structure constant is slope sensitive, with output contact order $r$
corresponds to a point on a $1$-dimensional cone of $\Sigma(X)$ i.e., the
contact order corresponds to a divisorial valuation on $\mathds{k}(X)$. In
fact, we will consider the target $X_{r}$, which is a root stack of $X$
induced by a lattice refinement of $\Sigma(X_{r})$ of $\Sigma(X)$ such that
$r$ is a primitive lattice point in $\Sigma(X_{r})$. Let $\beta$ be a tropical
type of punctured log curve, with curve class $\textbf{A}\in NE(X)$ and $l$
input contact orders $p_{1},\ldots,p_{l}$, and an output contact order $-r$
with $r\in B(\mathbb{Z})$ associated with a marked point $x_{out}$. By the
assumption that $\beta$ is a slope sensitive and Remark 6.6, there is a
corresponding tropical lift $\beta^{\prime}$ of $\beta$ to $X_{r}$ and an
analogous moduli stacks $\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime})$.

We now note that we may define the evaluation space $\mathscr{P}(X_{r},r)$
analogously to $\mathscr{P}(X,r)$. Letting $\mathscr{D}_{r}$ be the stratum of
$X_{r}$ associated with $r$, the underlying stack of $\mathscr{P}(X_{r},r)$ is
$\mathscr{D}_{r}\times B\mathbb{G}_{m}$, and the log structure is the sub log
structure of the natural product log structure on the product defined as in
Display 4.1. Finally, after noting that the image of the section of the
universal twisted curve over $\mathscr{M}(X_{r},\beta)$ has trivial inertia,
the universal twisted curve has an underlying stack theoretic section with
fiberwise image the marked point above, and the proof of Proposition $3.3$ of
[GS19] applies to produce evaluation maps
$ev_{x_{out}}:\mathscr{M}(X_{r},\beta)\rightarrow\mathscr{P}(X_{r},r)$ and
$ev_{x_{out}}:\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta)\rightarrow\mathscr{P}(X_{r},r)$.

For any $z\in D_{r}$, the proof of Proposition $3.8$ of [GS19] can also be
used to produce a morphism $z^{\prime}:W\rightarrow\mathscr{P}(X_{r},r)$ which
when composed with the morphism
$\mathscr{P}(X_{r},r)\rightarrow\mathscr{P}(X,r)$ maps to $z$. Moreover, since
$r$ is a primitive point along a $1$-dimensional cone of $\Sigma(X_{r})$, the
morphism $z^{\prime}$ is strict. In particular, the fiber products
$\mathscr{M}(X_{r},\beta^{\prime},z):=\mathscr{M}(X_{r},\beta^{\prime})\times_{\mathscr{P}(X_{r},r)}W$
and
$\mathfrak{M}^{ev}(X_{r},\beta^{\prime},z):=\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime})\times_{\mathscr{P}(X_{r},r)}W$
are independent of the choice of log category in which they are taken. For a
realizable type $\tau$ marked by $\beta$ we also consider the analogous fiber
products $\mathscr{M}(X,\tau,z)$ and
$\mathfrak{M}^{ev}(\mathcal{X}_{\tau},z)$. The morphism
$\mathscr{M}(X_{r},\beta^{\prime},z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime},z)$
can be equipped with a perfect obstruction theory as in [GS19] with respect to
which $\mathscr{M}(X_{r},\beta^{\prime},z)$ virtually is pure dimension $0$.
We define an invariant analogous to those that define the structure constants
of $R_{(X,D)}$ by:

$N_{p,q,r}^{\textbf{A}^{\prime}}=deg[\mathscr{M}(X_{r},\beta^{\prime},z)]^{vir}$

For $\boldsymbol{\tau}$ a realizable decorated tropical type marked by
$\beta$, we also consider the analogous moduli stack
$\mathscr{M}(X,\boldsymbol{\tau},z):=\mathscr{M}(X,\boldsymbol{\tau})\times_{\mathscr{P}(X,r)}W$,
which is equipped with a virtual fundamental class
$[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}$. We can express this invariant in
the display above as an integral on a moduli stack without a point
constraints:

######  Lemma 7.1.

Letting $r=km$ with $m$ a primitive integral point of $\Sigma(X)$ and $k\geq
1$, for $\beta^{\prime}$ a tropical lift of a tropical type $\beta$ which is
slope sensitive, we have the equalities:

$[\mathscr{M}(X_{r},\beta^{\prime},z)]^{vir}=[\mathscr{M}(X_{r},\beta^{\prime})]^{vir}\cap
ev_{x_{out}}^{*}(pt)$

In the above equation, the morphism $ev_{x_{out}}$ is the underlying stack
evaluation maps to the divisor $\mathscr{D}_{r}\subset X_{r}$. Moreover, for
any $\alpha\in A^{*}(\mathscr{M}(X,\beta,z))$, the vanishing of the classes
$\alpha\cap[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}$ for all decorated
tropical types $\boldsymbol{\tau}$ marked by $\beta$ implies
$[\mathscr{M}(X_{r},\beta^{\prime})]^{vir}\cap(ev^{*}_{x_{out}}(pt)\cup\alpha)=0$.

###### Proof.

Throughout, we abuse notation and always let $z$ refer to the morphism from
the point constrained moduli space to the unconstrained moduli space. Recall
that the underlying stack of the point constrained moduli stacks are ordinary
fiber products. As a result, the claimed formula now follows if we show the
following equality:

(7.1)
$z_{*}([\mathscr{M}(X_{r},\beta^{\prime})\times_{X_{r}}z]^{vir})=[\mathscr{M}(X_{r},\beta^{\prime})]^{vir}\cap
ev_{x_{out}}^{*}(pt).$

To see this claim, we let
$f:\mathscr{M}(X_{r},\beta^{\prime},z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime},z)$,
and note that we have the following triple of morphisms which comes equipped
with a compatible triple of obstruction theories:

$\mathscr{M}(X_{r},\beta^{\prime},z)\rightarrow\mathscr{M}(X_{r},\beta^{\prime})\rightarrow\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime}).$

The obstruction theory for the left most arrow is pulled back from that of
$\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime},z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime})$,
which itself is pulled back from the natural obstruction theory for
$z:Spec\text{ }\mathds{k}\rightarrow\mathscr{D}_{r}$. In particular, by
Theorem $4.9$ of [Man08], we have:

$[\mathscr{M}(X_{r},\beta^{\prime})\times_{X_{r}}z]^{vir}=f^{!}([\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime},z)])=f^{!}(z^{*}([\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime})]))=z^{*}([\mathscr{M}(X_{r},\beta^{\prime})]^{vir}).$

By pushing forward the equality along $z$, and the equality
$z_{*}z^{*}(A)=A\cap ev_{x_{out}}^{*}(pt)$ for any $A\in
A_{*}(\mathscr{M}(X,\beta^{\prime}))$, we conclude Equation 7.1.

To see the final claim, note that by Corollary 6.5 we have

$ip_{*}([\mathscr{M}(X_{r},\boldsymbol{\tau}^{\prime})]^{vir})\cap
ev_{x_{out}}(pt)=[\mathscr{M}(X,\boldsymbol{\tau})]^{vir}\cap
ev_{x_{out}}(pt),$

for some $i>0$. Moreover, letting
$s:\mathfrak{M}^{ev}(\mathcal{X},\tau,z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X},\tau)\times_{\mathscr{P}(X,r)}W$
be the map between the fs and coherent fiber products, then since
$W\rightarrow\mathscr{P}(X,r)$ is an integral morphism, $s$ is finite and
surjective. Hence, there exists $l>0$ such that:

$z_{*}s_{*}[\mathfrak{M}^{ev}(\mathcal{X},\tau,z)]=l[\mathfrak{M}^{ev}(\mathcal{X},\tau)]\cap
ev_{x_{out}}^{*}(pt)=ilp_{*}([\mathfrak{M}^{ev}(\mathcal{X}_{r},\tau^{\prime})])\cap
ev^{*}_{x_{out}}(pt)$

In particular, by taking virtual pullback of the above equation and capping
with the class $\alpha$, we find
$\alpha\cap[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}=0$ only if
$[\mathscr{M}(X_{r},\tau^{\prime})]^{vir}\cap(ev_{x_{out}}^{*}(pt)\cup\alpha)=0$.
Finally, by Corollary 4.4, which also holds in the setting of punctured maps
to a root stack, the latter vanishing for all types $\tau^{\prime}$ implies
[$\mathscr{M}(X_{r},\beta^{\prime})]^{vir}\cap(ev_{x_{out}}^{*}(pt)\cup\alpha)=0$,
as required. ∎

We now remark that the log étale invariance of the numbers
$N_{p,q,r}^{\textbf{A}}$ demonstrated in the proof [Joh22a] Corollary $1.6$
generalizes to the context of root stacks:

######  Proposition 7.2.

Letting $\beta^{\prime}$ be a slope sensitive tropical type which pushes
forward to the tropical type $\beta$ associated to a structure constant
$N_{p,q,r}^{\textbf{A}}$, then we have the equality:

$N_{p,q,r}^{\textbf{A}}=N_{p,q,r}^{\textbf{A}^{\prime}}$

###### Proof.

In light of Proposition 6.4, which is the analogue of Theorem $1.2$ [Joh22a]
in the setting of a root stack modification, the claim follows essentially
from the proof of Corollary $1.6$ of [Joh22a]. More precisely, letting
$\mathfrak{M}^{ev}(\mathcal{X},\tau,z)=\mathfrak{M}^{ev}(\mathcal{X},\tau)\times_{\mathscr{P}(X,r)}W$
and
$i_{\tau}:\mathfrak{M}^{ev}(\mathcal{X},\tau,z)\rightarrow\mathfrak{M}^{ev}(\mathcal{X},\beta,z)$,
recall from Section $10$ of [Joh22a] that we have a decomposition:

$[\mathfrak{M}^{ev}(\mathcal{X},\beta,z)]=\sum\frac{m_{\tau}}{|Aut(\tau)|}i_{\tau*}([\mathfrak{M}(\mathcal{X},\tau,z)_{red}]).$

The multiplicity $m_{\tau}$ above is
$|coker(ev_{\tau}^{*,gp}:\mathbb{Z}\rightarrow\mathbb{Z})|$, where $\tau$ is a
minimal tropical type marked by $\beta$, necessarily satisfying $dim\text{
}\tau=1$, and
$ev_{\tau}:\tau_{\mathbb{N}}\times_{\boldsymbol{\sigma}(v)_{\mathbb{N}}}\mathbb{N}\rightarrow\mathbb{N}$
is the evaluation morphism at the vertex $v$ containing the output leg of
slope $-r$. By taking the virtual pullback and degree of the above equation,
we have a decomposition
$N_{p,q,r}^{\textbf{A}}=\sum_{\boldsymbol{\tau}}N_{\boldsymbol{\tau}}$, with
$N_{\boldsymbol{\tau}}$ defined as the degree of the contribution to
$[\mathscr{M}(X,\beta,z)]^{vir}$ from decorated type $\boldsymbol{\tau}$.

To produce the analogous decomposition of $N_{p,q,r}^{\textbf{A}^{\prime}}$,
the main ingredient missing in order to apply the argument given in Section
$10$ of [Joh22a] is
$\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime},z)\rightarrow W$ being log
smooth. After recalling from remark 6.6 that
$\mathfrak{M}(\mathcal{X}_{r},\beta^{\prime})\rightarrow\mathfrak{M}^{tw}$ is
idealized log étale, and that $\mathfrak{M}^{tw}$ is log smooth as follows
from [Ols07] Theorem $1.10$, slight modifications made to the proofs of
Propositions $3.14$, Theorem $3.16$, and Lemma $3.17$ of [GS19] show that
$\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime},z)\rightarrow W$ is log
smooth.

Using the stratification of
$\mathfrak{M}(\mathcal{X}_{r},\beta^{\prime})_{red}$ by the stacks
$\mathfrak{M}(\mathcal{X}_{r},\tau^{\prime})$ remarked upon in Remark 6.6, the
argument from Section $10$ of [Joh22a] now applies to produce the analogous
decompositions:

$[\mathfrak{M}^{ev}(\mathcal{X}_{r},\beta^{\prime},z)]=\sum_{\boldsymbol{\tau}^{\prime}}\frac{m_{\tau^{\prime}}}{|Aut(\tau^{\prime})|}i_{\tau^{{}^{\prime}*}}([\mathfrak{M}(\mathcal{X}_{r},\tau^{\prime},z)_{red}]).$

Virtual pullback of the above equation to
$A_{*}(\mathscr{M}(X,\beta^{\prime},z^{\prime}))$ produces a decomposition
$N_{p,q,r}^{\textbf{A}^{\prime}}=\sum_{\boldsymbol{\tau}^{\prime}}N_{\boldsymbol{\tau}^{\prime}}$
as before. The main claim now reduces to showing
$N_{\boldsymbol{\tau}}=N_{\boldsymbol{\tau}^{\prime}}$, which is the analogue
of Proposition $10.1$ of [Joh22a] in the root stack setting. Using the fact
that the log point constraint $z:W\rightarrow\mathscr{P}(X,r)$ uniquely
determines a point constraint $W\rightarrow\mathscr{P}(X_{r},r)$ together with
the the expression of $\mathscr{M}(X_{r},\boldsymbol{\tau}^{\prime})$ as a
fiber product implied by Proposition 6.4, the proof of Proposition $10.1$ of
loc. cit. applies in the present context to prove the desired equality.

∎

######  Corollary 7.3.

Letting $z:\mathscr{M}(X,\beta,z)\rightarrow\mathscr{M}(X,\beta)$ be the
projection from the fiber product and $ev:\mathscr{M}(X,\beta)\rightarrow
D_{r}$ the evaluation map, we have:

$z_{*}([\mathscr{M}(X,\beta,z)]^{vir})=kp_{*}([\mathscr{M}(X_{r},\beta^{\prime})]^{vir})\cap
ev^{*}(pt).$

###### Proof.

Note that by the projection formula, we may compute the right hand side of the
desired equation by pulling back the class $ev^{*}(pt)$ along the morphism $p$
and capping the resulting class with
$[\mathscr{M}(X_{r},\beta^{\prime})]^{vir}$. This pullback can also be
expressed by pulling back the class of a point along
$\mathscr{D}_{r}\rightarrow D_{r}$ followed by pulling back along the
evaluation map. The resulting class is $\frac{1}{k}ev^{*}([pt^{\prime}])$,
where $[pt^{\prime}]\in A^{dim\text{ }X}(X_{r})$ is the class associated with
a closed immersion $B\mu_{r}\rightarrow X_{r}$. Since the left hand side of
the desired equation is equal to the integral on
$\mathscr{M}(X_{r},\beta^{\prime})$ with insertion $ev^{*}([pt])$ by
Proposition 7.2, the desired equation follows. ∎

######  Corollary 7.4.

Let $\beta$ be a tropical type introduced at the start of this section with
$r=km$ for $m\in B(\mathbb{Z})$ a primitive integral point in a
$1$-dimensional cone of $\Sigma(X)$ and total curve class $\textbf{A}\in
NE(X)$, and suppose $X$ is smooth and slope sensitive with respect to
$\mathscr{M}(X,\beta)$ in the sense of [BNR22]. Then for any $a\geq 0$,
$\psi_{x_{out}}^{a}[\mathscr{M}(X,\boldsymbol{\tau},z)]^{vir}=0$ for all
decorated realizable types $\boldsymbol{\tau}$ marked by $\beta$ only if
$\langle\prod_{i}\psi_{x_{i}}^{a_{i}}[\alpha_{i}],\psi_{x_{out}}^{a}[pt]\rangle_{\textbf{A}}=0$
for all choices of $a_{i}\geq 0$ and insertions $\alpha_{i}\in
H^{*}(D_{p_{i}})$. Moreover, we have
$N_{p_{1},p_{2},r}^{\textbf{A}}=N_{p_{1},p_{2},r}^{orb\textbf{A}}$.

###### Proof.

Let $D=D_{1}+\cdots+D_{p}$ be a decomposition of $D$ into irreducible
components with $D_{1}=D_{r}$. Note by the smoothness assumption that $D_{i}$
is a Cartier divisor for all $i$. For $\textbf{b}=(b_{1},\ldots,b_{p})$, let
$X_{\textbf{b}}$ denote the root stack of $X$ given by taking the $b_{i}^{th}$
root stack of the divisor $D_{i}$ for all $i$. Note that by the notation
introduced in display 4.2, for $b_{i}\gg 0$, we have:

(7.2)
$\langle\prod_{i}\psi_{x_{i}}^{a_{i}}[\alpha_{i}],\psi_{x_{out}}^{a}[pt]\rangle_{\textbf{A}}=b_{1}^{m_{i,-}}\prod_{i>1}b_{i}^{m_{i,-}}\int_{[\mathscr{M}^{orb}(X_{\textbf{b}},\beta)]^{vir}}\psi_{x_{out}}^{a}ev_{x_{out}}^{*}([pt])\prod_{i=1}^{l}\psi_{x_{i}}^{a_{i}}ev_{x_{i}}^{*}([\alpha_{i}]).$

Alternatively, since multiples of $k$ are cofinal in the poset of the natural
numbers with the divisibility relation, it suffices to consider the inverse
system over root stacks of the pair $(X,D)$ with a rooting parameter for
$D_{1}$ divisible by $k$. Hence, letting $X_{r}$ be the log étale modification
of $X$ in which we take $k^{th}$ root of the the Cartier divisor $D_{1}$, we
then have for $b_{i}\gg 0$:

(7.3)
$\langle\prod_{i}\psi_{x_{i}}^{a_{i}}[\alpha_{i}],\psi_{x_{out}}^{a}[pt]\rangle_{\textbf{A}}=(kb_{1})^{m_{1,-}}\prod_{i>1}b_{i}^{m_{i,-}}\int_{[\mathscr{M}^{orb}((X_{r})_{\textbf{b}},\beta)]^{vir}}\psi_{x_{out}}^{a}ev_{x_{out}}^{*}([pt])\prod_{i=1}^{l}\psi_{x_{i}}^{a_{i}}\alpha_{i}.$

On the other hand, using [BNR24] Theorem $B$ for the first equality, which
holds after replacing $\mathscr{M}(X,\beta)$ with $\mathscr{M}(X_{r},\beta)$,
and Equation 7.3 for the second equality, we have: