# Quantum $K$-invariants and Gopakumar–Vafa invariants I
The quintic threefold

Y.-C. Chou<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Y.-P. Lee
<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute of Mathematics,
Academia Sinica, Taipei 10617, Taiwan, and Department of Mathematics,
University of Utah, Salt Lake City, Utah 84112-0090, U.S.A.

###### Abstract.

We prove a conjecture of Jockers–Mayr and Garoufalidis–Scheidegger, relating
genus zero quantum $K$-invariants and Gopakumar–Vafa invariants on the quintic
threefold.

###### Contents

  1. 0 Introduction
    1. 0.1 Relating enumerative invariants on quintic threefold
    2. 0.2 Contents of the paper
  2. 1 GW, GV, QK and the JMGS Conjecture
    1. 1.1 Gromov–Witten invariants
    2. 1.2 Gopakumar–Vafa invariants
    3. 1.3 Quantum $K$-invariants
    4. 1.4 JMGS conjecture for the quintic
  3. 2 Review of Givental’s framework of quantum $K$-theory
    1. 2.1 The symplectic loop space formalism
    2. 2.2 Fake quantum $K$-theory
    3. 2.3 $J^{K}$-function as a graph sum via Kawasaki’s HRR
    4. 2.4 Reconstruction theorem
  4. 3 Proof of the conjecture
    1. 3.1 Outline of the proof
    2. 3.2 Analytic properties of $J^{K}(0)$ and a degree argument
    3. 3.3 The degree 0 terms
    4. 3.4 Higher degree terms
    5. 3.5 Conclusion of the proof
  5. 4 Some computations of contributions from Kawasaki’s strata
  6. A Alternative proof of Theorem 3.4

##  0\. Introduction

###  0.1. Relating enumerative invariants on quintic threefold

On the Calabi–Yau threefolds (CY3), there are two sets of _integral_
enumerative invariants. The first, called the _Gopakumar–Vafa invariants_
[GV98], was introduced in theoretical physics as “new topological invariants
on _Calabi–Yau threefolds_ ”, counting the “numbers of BPS states”. We refer
the readers to [MT18] and references therein for various mathematical
definitions of these invariants. There is a simple relation between the
Gopakumar–Vafa invariants and the Gromov–Witten invariants (1.1) (1.2),

$\begin{split}&\sum_{g=0}^{\infty}~{}\sum_{\beta\in
H_{2}(M,\mathbb{Z})}{{\mathrm{GW}}}_{g,\beta}q^{\beta}\lambda^{2g-2}\\\
=&\sum_{g=0}^{\infty}~{}\sum_{k=1}^{\infty}~{}\sum_{\beta\in
H_{2}(M,\mathbb{Z})}{{\mathrm{GV}}}_{g,\beta}{\frac{1}{k}}\left(2\sin\left({\frac{k\lambda}{2}}\right)\right)^{2g-2}q^{k\beta},\end{split}$

which all viable mathematical definitions must satisfy. In this paper, we will
use the above (invertible) relation as the definition of the Gopakumar–Vafa
invariants (in terms of Gromov–Witten invariants). The integrality of this ad
hoc definition was proven by E. Ionel and T. Parker [IP18].

The second is the _quantum $K$-invariants_ [Giv00, Lee04], a $K$-theoretic
variant of the (cohomological) Gromov–Witten invariants. Whereas Gromov–Witten
theory produces _rational_ enumerative invariants, quantum $K$-theory,
counting (the alternating sum of) the rank of sheaf cohomology, produces
_integral_ invariants by definition. The close relationship between the
quantum $K$-theory and quantum cohomology was understood since the early phase
of quantum $K$-theory. In fact, many $K$-theoretic results were inspired by
their cohomological counterparts. See, e.g., [GL03, Lee04]. In [GT14] and
subsequent works, A. Givental and his collaborators completely characterized
genus zero quantum $K$-invariants in terms of genus zero Gromov–Witten
invariants.

Therefore, there is a relation between quantum $K$-invariants and
Gopakumar–Vafa invariants for Calabi–Yau threefolds via Gromov–Witten
invariants. In [JM19] and [GS22, Conjecture 1.1], H. Jockers, P. Mayr and S.
Garoufalidis, E. Scheidegger proposed a simple _linear_ relation between these
two sets of integral invariants for _quintic threefold_ at genus zero. They
also provided numerical evidence for this conjecture. The main purpose of this
article is to prove this conjecture as stated in [GS22, Conjecture 1.1], cf.
Conjecture 1.1 in Section 1. According to [GS22, Section 3.3], this also
proves the version proposed by Jockers and Mayr.

We plan to generalize these results to other Calabi–Yau threefolds (Part II)
and to higher genus (Part III) in future works. In particular, some heuristic
relations to the “multiple cover formula” will be discussed in Part II, where
the proof of genus $0$ result for all Calabi–Yau threefolds will use different
arguments.

###  0.2. Contents of the paper

Conjecture 1.1 is phrased in terms of an explicit expression for the _small
$J^{K}$-function_ in quantum $K$-theory, which is a generating function of
$K$-theoretic Gromov-Witten invariants, or quantum $K$-invariants. For a
smooth projective variety $X$, the quantum $K$-invariants are defined as

$\langle\tau_{d_{1}}(a_{1})\dots\tau_{d_{n}}(a_{n})\rangle^{X,{K}}_{g,n,\beta}:=\chi\Big{(}\overline{\mathcal{M}}_{g,n}(X,\beta);\Big{(}\otimes_{i=1}^{n}{\rm
ev}_{i}^{*}(a_{i})L_{i}^{d_{i}}\Big{)}\otimes\mathcal{O}^{{\rm
vir}}\Big{)}\in\mathbb{Z}.$

It gives a deformation of the ordinary $K$-ring $K^{0}(X)$ of $X$, analogous
to the relation between quantum cohomology and ordinary cohomology. We
consider the generating function, the _big $J^{K}$-function_, which determines
the _genus $0$_ quantum $K$-theory of $X$:

$\begin{split}t\rightarrow&J^{{K}}(t,q,Q)\\\
&:=(1-q)+t(q)+\sum_{\alpha}\Phi_{\alpha}\sum_{n,d}\frac{Q^{d}}{n!}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X,{K}}_{0,n+1,d}.\end{split}$

The _small_ $J$-function is a specialization of the big $J$-function to $t=0$.
Jockers–Mayr and Garoufalidis–Scheidegger formulated an explicit expression of
the small $J^{K}$-function in terms of the Gopakumar–Vafa invariants. We
recall the basic definitions and formulations of quantum $K$-theory and
Gopakumar–Vafa invariants in Section 1. The Conjecture 1.1 is formulated in
Section 1.4.

In Section 2, we recall necessary formulations and results from the works of
A. Givental and his collaborators. There are generally two approaches to
compute the genus zero quantum $K$-theory for the quintic threefold. In
[GT14], Givental and Tonita prove that genus 0 $K$-theoretic GW-invariants of
$X$ can be expressed in terms of its cohomological ones. Their framework can
be understood as the _adelic characterization_ of quantum $K$-theory. This is
done by applying Kawasaki’s Hirzebruch–Riemann–Roch formula, expressing Euler
characteristics of vector bundles over a Deligne–Mumford _stack_ (moduli of
stable maps in our case) into the integration over the components of its
inertia stack, called Kawasaki strata. By analyzing the combinatorial
structure of inertia stacks of moduli of stable map, they derive a recursive
relation relating invariants in different strata and then give the
characterization of genus $0$ $K$-theoretic GW-invariants in terms of the
cohomological ones. See Section 2.3 for details.

Another approach is called the _explicit reconstruction theorem_ , which in
principal gives an algorithm to compute all quantum $K$-invariants from an
initial condition. There are various versions of the reconstruction theorem. A
particular case of sufficient initial condition for the quintic threefold is
given by Givental’s $I^{K}$-function. See [LP04], [Giv15b] and [Ton18].

In principal, either the _virtual orbifold Hirzebruch–Riemann–Roch_ or the
_reconstruction theorem_ gives an explicit algorithm to compute the
$J^{K}$-function. In particular, a computation up to degree $7$ is given in
[GS22] using the explicit reconstruction alone. However, the complexity for
explicit computation grows very fast as the degrees increase. In Section 3, we
combine both methods and utilize the _analytic properties_ of the generating
functions to give a proof of Conjecture 1.1.

To streamline the proof, some computations needed to establish the proof are
left to Section 4. In the appendix, an alternative proof of Theorem 3.4 is
given to show that one can usually prove a theorem using either reconstruction
theorem or Riemann–Roch, when the complexity of the former is manageable.

### Acknowledgements

This research is partially supported by the Simons Foundation, the NSTC,
Academia Sinica and University of Utah.

##  1\. GW, GV, QK and the JMGS Conjecture

###  1.1. Gromov–Witten invariants

Let $X$ be a smooth complex projective variety, and
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ be the M. Kontsevich’s moduli space of
$n$-pointed, genus $g$, degree $\beta$ stable maps. Given
$i\in\\{1,\dots,n\\}$, there is an evaluation map

$\begin{split}{\rm ev}_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)&\rightarrow
X\\\ [f:(C;x_{1},\dots,x_{n})]&\mapsto f(x_{i}),\end{split}$

and a line bundle $L_{i}:=x_{i}^{*}w_{\mathcal{C}/\overline{\mathcal{M}}}$ on
$\overline{\mathcal{M}}_{g,n}(X,\beta)$, where
$w_{\mathcal{C}/\overline{\mathcal{M}}}$ is the relative dualizing sheaf of
the universal curve
$\mathcal{C}\rightarrow\overline{\mathcal{M}}_{g,n}(X,\beta)$ and
$x_{i}:\overline{\mathcal{M}}_{g,n}(X,\beta)\rightarrow\mathcal{C}$ is the
$i$-th mark point.

(Cohomological) Gromov-Witten invariants of $X$ are defined to be

$\langle\tau_{d_{1}}(\phi_{1})\dots\tau_{d_{n}}(\phi_{n})\rangle^{X,{H}}_{g,n,\beta}:=\pi^{H}_{*}\left(\cup_{i=1}^{n}{\rm
ev}_{i}^{*}(\phi_{i})c_{1}(L_{i})^{d_{i}}\cap[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{{\rm
vir}}\right)\in\mathbb{Q},$

where

$\pi:\overline{\mathcal{M}}_{g,n}(X,\beta)\to
pt:=\operatorname{Spec}(\mathbb{C})$

is the structural map and $\pi^{H}_{*}$ is the (cohomological) pushforward to
the point. Here $\phi_{1},\dots,\phi_{n}\in H(X)$,
$d_{1},\dots,d_{n}\in\mathbb{Z}_{\geq 0}$, and
$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{{\rm vir}}$ are the (cohomological)
virtual fundamental classes.

In this paper, we will be concerned with only genus zero invariants, which can
be encoded into a formal power series, called genus-0 descendant potential of
$X$:

$F^{{H}}_{0}(t)=\sum_{n\geq 0}\sum_{\beta}\frac{Q^{\beta}}{n!}\langle
t(L),\dots,t(L)\rangle^{X,{H}}_{0,n,\beta}.$

Here the sum is over all curve class $\beta\in H_{2}(X)_{\geq 0}$ and
$Q^{\beta}$ are formal variables, called the Novikov variables, which keep
track of the curve classes. $t(q)$ stands for any polynomial of one variable
with coefficients in $H(X)$. That is $\displaystyle
t(q)=\sum_{k\in\mathbb{Z}_{\geq
0}}\sum_{\alpha=1}^{N}t_{k}^{\alpha}\phi_{\alpha}q^{k}$ with
$\\{\phi_{\alpha}\\}_{\alpha=1}^{N}$ a basis in $H(X)$.

###  1.2. Gopakumar–Vafa invariants

In theoretical physics, R. Gopakumar and C. Vafa in [GV98] introduced new
topological invariants on _Calabi–Yau threefolds_ (CY3) $X$, called
_Gopakumar–Vafa invariants_. These invariants represent the counts of “numbers
of BPS states” on $X$. Unlike the Gromov–Witten invariants, which are defined
for any symplectic manifolds or orbifolds, Gopakumar–Vafa invariants only make
sense for Calabi–Yau threefolds (or variants).

The virtual dimensions for moduli spaces (of stable maps) to CY3 are always
equal to the number of marked points. That is,

$\operatorname{vdim}\overline{\mathcal{M}}_{g,n}(X,\beta)=n.$

By general properties of the moduli spaces, more precisely the string
equation, the divisor equation and dilaton equation, all Gromov–Witten
invariants can be easily reconstructed from

${{\mathrm{GW}}}_{g,\beta}:=\pi^{H}_{*}\left([\overline{\mathcal{M}}_{g,0}(X,\beta)]^{{\rm
vir}}\right)=\int_{[\overline{\mathcal{M}}_{g,0}(X,\beta)]^{{\rm vir}}}1.$

That is, all $n$-pointed invariants with descendants can be recovered from
$0$-pointed counting. In fact, a _closed formula_ was obtained in [FL19] in
terms of generating functions. We will therefore focus on
${{\mathrm{GW}}}_{g,\beta}$.

There have been various attempts at defining Gopakumar–Vafa invariants
mathematically. We refer the readers to [MT18] and references therein. However
the definitions might be, they are expected to be compatible with the dilaton
and divisor equations. The counting of the BPS states is again reduced to
similarly defined ${{\mathrm{GV}}}_{g,\beta}$.

A remarkable relation conjectured in [GV98] between GV and GW can be expressed
in terms of generating functions:

(1.1) $\begin{split}&\sum_{g=0}^{\infty}~{}\sum_{\beta\in
H_{2}(M,\mathbb{Z})}{{\mathrm{GW}}}_{g,\beta}q^{\beta}\lambda^{2g-2}\\\
=&\sum_{g=0}^{\infty}~{}\sum_{k=1}^{\infty}~{}\sum_{\beta\in
H_{2}(M,\mathbb{Z})}{{\mathrm{GV}}}_{g,\beta}{\frac{1}{k}}\left(2\sin\left({\frac{k\lambda}{2}}\right)\right)^{2g-2}q^{k\beta},\end{split}$

In this paper, we are mainly concerned with genus zero invariants on $X$, the
quintic CY3, and the above relation can be written as

(1.2)
$\begin{split}{{\mathrm{GW}}}_{g=0,\beta=d[\operatorname{line}]}=:{{\mathrm{GW}}}_{d}&=\sum_{e|d}\frac{1}{e^{3}}{{\mathrm{GV}}}_{d/e},\\\
{{\mathrm{GV}}}_{g=0,\beta=d[\operatorname{line}]}=:{{\mathrm{GV}}}_{d}&:=\sum_{e|d}\frac{\mu(e)}{e^{3}}{{\mathrm{GW}}}_{d/e},\end{split}$

where we have used the Möbius inversion and $\mu(e)$ is the Möbius function.
For our purpose, we will _use ( 1.2) as the definition of
${{\mathrm{GV}}}_{d}$_. The first 4 terms are listed for readers’ convenience.

$d$ | 1 | 2 | 3 | 4
---|---|---|---|---
${\mathrm{GW}}_{d}$ | 2875 | 4876875/8 | 8564575000/27 | 15517926796875/64
${\mathrm{GV}}_{d}$ | 2875 | 609250 | 317206375 | 242467530000

This _ad hoc_ definition has among other things one difficulty. Namely, the
Gopakumar–Vafa invariants are to be intrinsically _integers_ , while the
Gromov–Witten invariants are generally _rational_ numbers, as the above table
demonstrates. Fortunately, the integrality of this definition has been shown
in [IP18]. There is a variant of the Gromov–Witten theory which also produces
_integral_ invariants, namely, the _quantum $K$-theory_ [Giv00, Lee04]. This
leads to the possibility of relating quantum $K$-invariants with
Gopakumar–Vafa invariants for Calabi–Yau threefolds.

###  1.3. Quantum $K$-invariants

The formulation of _quantum $K$-theory_ is similar to that of quantum
cohomology, i.e., Gromov–Witten theory. The $K$-theoretic Gromov-Witten
invariants, or _quantum $K$-invariants_, of $X$ are defined to be

$\langle\tau_{d_{1}}(\Phi_{1})\dots\tau_{d_{n}}(\Phi_{n})\rangle^{X,{K}}_{g,n,\beta}:=\chi\Big{(}\overline{\mathcal{M}}_{g,n}(X,\beta);\Big{(}\otimes_{i=1}^{n}{\rm
ev}_{i}^{*}(\Phi_{i})L_{i}^{d_{i}}\Big{)}\otimes\mathcal{O}^{{\rm
vir}}\Big{)}\in\mathbb{Z}.$

Here $\Phi_{1},\dots,\Phi_{n}\in K^{0}(X)$, $d_{1},\dots,d_{n}\in\mathbb{Z}$,
and $\mathcal{O}^{{\rm vir}}$ is the virtual structure sheaf on
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ [Lee04]. We encode all genus 0
invariants into a formal power series, called genus-0 descendant potential of
$X$:

$F^{{K}}_{0}(t)=\sum_{n\geq 0}\sum_{\beta}\frac{Q^{\beta}}{n!}\langle
t(L),\dots,t(L)\rangle^{X,{K}}_{0,n,\beta}.$

Here the sum is over all curve class $\beta\in H_{2}(X)_{\geq 0}$. $t(q)$
stands for any Laurent polynomial of one variable $q$ with coefficients in
$K^{0}(X)$

$t(q)=\sum_{k\in\mathbb{Z}}\sum_{\alpha=1}^{N}t_{k}^{\alpha}\Phi_{\alpha}q^{k}$

with $\\{\Phi_{\alpha}\\}_{\alpha=1}^{N}$ a basis in $K^{0}(X)$. We may
rewrite $F^{{K}}_{0}(t)$ as a more transparent form as:

$F^{{K}}_{0}(t)=\sum_{n\geq
0}\sum_{\beta}\frac{Q^{\beta}}{n!}\sum_{\begin{subarray}{c}k_{1},\dots,k_{n}\in\mathbb{Z}\\\
\alpha_{1},\dots,\alpha_{n}\in\\{1,\dots,N\\}\end{subarray}}t_{k_{1}}^{\alpha_{1}}\cdots
t_{k_{n}}^{\alpha_{n}}\langle\tau_{k_{1}}(\Phi_{\alpha_{1}}),\dots,\tau_{k_{n}}(\Phi_{\alpha_{n}})\rangle^{X,{K}}_{0,n,\beta}.$

$K^{0}(X)$ here stands for the topological $K$-theory of complex vector
bundles. For the quintic threefold, the main focus of this paper, Grothendieck
groups of coherent sheaves and complex vector bundles coincide.

###  1.4. JMGS conjecture for the quintic

In [JM19] and [GS22] H. Jockers, P. Mayr and S. Garoufalidis, E. Scheidegger
formulate a conjectural relation between the Gopakumar–Vafa invariants and
quantum $K$-invariants for the _quintic threefold_ $X$. The conjecture is
formulated in terms of _small $J$-functions_, a generating function in quantum
$K$-theory as well as quantum cohomology.

For simplicity, let us start with the small $J^{H}$-function for quantum
cohomology [Giv96] 111The peculiar evaluation at $0$ is a reminder that the
small $J$-function is a restriction of the _big $J$-function_. Cf. Definition
2.2.

$\begin{split}J^{H}(0)&:=-z+\sum_{d\geq
1,\alpha}Q^{d}\langle\frac{\phi_{\alpha}}{-z-\psi}\rangle^{H}_{0,1,d}\phi^{\alpha}\\\
&=-z+\sum_{d\geq
1}Q^{d}\Big{(}-\frac{d{\mathrm{GW}}_{d}}{5z}H^{2}-\frac{2{\mathrm{GW}}_{d}}{5z^{2}}H^{3}\Big{)},\end{split}$

where $H$ is the hyperplane class pulled back from $\mathbb{P}^{4}$. It
follows directly from the divisor and dilaton equations that

$\langle\rangle^{H}_{0,0,d}=:{\mathrm{GW}}_{d},\quad\langle\psi\rangle^{H}_{0,1,d}=-2{\mathrm{GW}}_{d},\quad\langle
H\rangle^{H}_{0,1,d}=(\int_{d}H){\mathrm{GW}}_{d}=d\cdot{\mathrm{GW}}_{d}.$

To phrase the conjecture in $K$-theory, recall that

$K^{0}(X)=\frac{\mathbb{Q}[P]}{(1-P)^{4}},$

where $P=\mathcal{O}(-1)|_{X}$. We fix a basis
$\\{\Phi_{\alpha}\\}_{\alpha=0}^{3}=\\{(1-P)^{\alpha}\\}_{\alpha=0}^{3}$ for
$K^{0}(X)$. The inner product reads:

$(\Phi_{a},\Phi_{b})^{K}:=\chi(X,\Phi_{a}\Phi_{b})=\left(\begin{matrix}0&5&-5&5\\\
5&-5&5&0\\\ -5&5&0&0\\\ 5&0&0&0\end{matrix}\right).$

The dual basis $\\{\Phi^{\alpha}\\}_{\alpha=0}^{3}$ is given by

$\displaystyle\Phi^{0}$ $\displaystyle=\frac{1}{5}(1-P)^{3},$
$\displaystyle\quad\Phi^{1}=\frac{1}{5}((1-P)^{2}+(1-P)^{3}),$
$\displaystyle\Phi^{2}$ $\displaystyle=\frac{1}{5}((1-P)+(1-P)^{2}),$
$\displaystyle\quad\Phi^{3}=\frac{1}{5}(1+(1-P)-(1-P)^{3}).$

The small $J^{K}$-function for quantum $K$-theory is defined similarly.
Jockers–Mayr and Garoufalidis–Scheidegger conjecture that $J^{K}$ can be
expressed as a _linear_ combination of GV invariants.

######  Conjecture 1.1 ([JM19, GS22] ).

$\begin{split}&\frac{1}{1-q}\left[J^{K}(0)\right]:=\frac{1}{1-q}\left[(1-q)+\sum_{\alpha}\sum_{M\geq
1}\Phi_{\alpha}\langle\frac{\Phi^{\alpha}}{1-qL}\rangle^{K}_{0,1,M}Q^{M}\right]\\\
=&1+(1-P)^{2}\sum_{d,r\geq
1}a(d,r,q^{r}){\mathrm{GV}}_{d}Q^{dr}+(1-P)^{3}\sum_{d,r\geq
1}b(d,r,q^{r}){\mathrm{GV}}_{d}Q^{dr},\end{split}$

where

$\begin{split}5a(d,r,q)&=\frac{d(r-1)}{1-q}+\frac{d}{(1-q)^{2}},\\\
5b(d,r,q)&=\frac{rd+r^{2}-d-1}{1-q}+\frac{d+3}{(1-q)^{2}}-\frac{2}{(1-q)^{3}}.\end{split}$

The main purpose of this paper is to prove the above conjecture.

##  2\. Review of Givental’s framework of quantum $K$-theory

In this section, we review relevant results from [GT14, Giv15a, Giv15b,
Giv15c]. In these works and many others, A. Givental and his collaborators
have established groundbreaking works on the structural understanding of
quantum $K$-theory, including a characterization of the genus zero quantum
$K$-theory in terms of quantum cohomology. This is essential for our proof as
${\mathrm{GV}}_{d}$ in this article are actually defined in terms of
${\mathrm{GW}}$ by (1.2).

###  2.1. The symplectic loop space formalism

Let $\Lambda=\mathbb{Q}[\\![Q]\\!]$ be the Novikov ring. Givental’s loop space
for quantum $K$-theory is defined as

$\mathcal{K}:=K^{0}(X)(q)\otimes\Lambda.$

By definition, an element in $\mathcal{K}$ is a $Q$-series with coefficients
rational function of $q$.

######  Remark 2.1.

Let $K:=K^{0}(X)\otimes\Lambda$. An element in $\mathcal{K}$ can also be
understood as a rational function of $q$ with coefficients in $K$ in the
$Q$-adic sense, i.e. modulo any power of the maximal ideal in the Novikov
ring.

$\mathcal{K}$ is quipped with a symplectic form $\Omega$, i.e., the
$\Lambda$-valued non-degenerate anti-symmetric bilinear form:

$\mathcal{K}\ni f,g\rightarrow\Omega(f,g):=\Big{(}{\rm Res}_{q=0}+{\rm
Res}_{q=\infty}\Big{)}(f(q),g(q^{-1}))^{K}\frac{dq}{q},$

where $(\cdot,\cdot)^{K}$ denote the $K$-theoretic intersection pairing on
$K$:

$(a,b)^{K}:=\chi(X,a\otimes b)=\int_{X}{\rm td}(T_{X}){\rm ch}(a){\rm ch}(b).$

$\mathcal{K}$ admits the following Lagrangian polarization with respect to
$\Omega$:

$\begin{split}\mathcal{K}&=\mathcal{K}_{+}\oplus\mathcal{K}_{-}\\\
&:=K[q,q^{-1}]\,\oplus\,\\{f(q)\in\mathcal{K}|f(0)\neq\infty,\
f(\infty)=0\\}.\end{split}$

######  Definition 2.2.

The big $J$-function of $X$ in the $K$-theory is defined as a map
$\mathcal{K}_{+}\rightarrow\mathcal{K}$:

$\begin{split}t\mapsto&J^{{K}}(t):=J^{{K}}(t,q,Q)\\\
&:=(1-q)+t(q)+\sum_{\alpha}\Phi_{\alpha}\sum_{n,d}\frac{Q^{d}}{n!}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X}_{0,n+1,d},\end{split}$

where $\\{\Phi_{\alpha}\\}$ and $\\{\Phi^{\alpha}\\}$ are Poincaré-dual basis
of $K^{0}(X)$ with respect to $(\cdot,\cdot)^{K}$.

######  Conventions 2.3.

In this series of papers, the variables $q$ and $Q$ in $J^{{K}}(t,q,Q)$ will
stay in the background and the $J$-function is often denoted by $J^{{K}}(t)$.

######  Remark 2.4.

The definition of $J$-function contains three ingredients. The first summand,
$1-q$, is called the _dilaton shift_. The second one, $t(q)$, is call the
_input_. These two lie in $\mathcal{K}_{+}$.

We claim that the last summand, the correlator part, lies in
$\mathcal{K}_{-}$. We need to show that it is a rational function of $q$, with
no pole at $q=0$ and with a zero at $q=\infty$. On each particular moduli
stack, there exists a polynomial $P(q)$ such that $P(L^{-1})=0$ and $P(0)\neq
0$ since $L^{-1}$ is invertible. On such a moduli space, we have
$P(q)-P(L^{-1})=F(q,L)(L^{-1}-q)$ for some $F(q,L)$, which is a polynomial in
$q$ with $\deg_{q}F<\deg_{q}P$. Thus,

$\frac{1}{1-qL}=\frac{\frac{P(q)-P(L^{-1})}{P(q)}}{1-qL}=\frac{\frac{P(q)-P(L^{-1})}{1-qL}}{P(q)}=\frac{L^{-1}F(q,L)}{P(q)},$

We conclude that each correlator is a rational function in $q$, has no pole at
$q=0$, vanishes at $q=\infty$ and hence lies in $\mathcal{K}_{-}$.

Note that this last term (in $\mathcal{K}_{-}$) actually has poles only at
roots of unity, cf. Remark 2.12.

Let

$\mathcal{L}^{K}:=\\{(1-q+t(q),J^{K}(t)|_{\mathcal{K}_{-}})\in\mathcal{K}_{+}|t(q)\in\mathcal{K}_{+}\\}$

be the graph of the big $J^{K}$-function in $\mathcal{K}$. We have

######  Proposition 2.5 ([GT14, § 2,§ 3]).

The Lagrangian submanifold $\mathcal{L}^{K}$ has the following properties:

  1. (1)

$\mathcal{L}^{K}$ is cone over $\Lambda$;

  2. (2)

For $f\in\mathcal{L}^{K}$, we have
$T_{f}\mathcal{L}^{K}\cap\mathcal{L}^{K}=(1-q)T_{f}\mathcal{L}^{K}$; Indeed,
$\mathcal{L}^{K}$ is ruled by a finite-parameter family of such spaces.

  3. (3)

Identifying $T^{*}\mathcal{K}_{+}$ with
$\mathcal{K}=\mathcal{K}_{+}\oplus\mathcal{K}_{-}$ as symplectic spaces, we
have

$J^{K}(q)=(1-q)+t(q)+dF_{0}^{{K}}.$

###  2.2. Fake quantum $K$-theory

The fake quantum $K$-invariants of $X$ are defined to be

$\langle\tau_{d_{1}}(a_{1})\dots\tau_{d_{n}}(a_{n})\rangle^{X,{\rm
fake}}_{g,n,d}:=\int_{[\overline{\mathcal{M}}_{g,n}(X,d)]^{{\rm vir}}}{\rm
td}(T_{\overline{\mathcal{M}}_{g,n}(X,d)}^{{\rm vir}}){\rm
ch}(\otimes_{i=1}^{n}{\rm ev}_{i}^{*}(a_{i})L_{i}^{d_{i}}),$

where $[\overline{\mathcal{M}}_{g,n}(X,d)]^{{\rm vir}}$ is the (cohomological)
virtual fundamental class and $[T^{{\rm vir}}]$ the virtual tangent bundle,
the virtual difference of deformation and obstruction.

In the fake quantum $K$-theory, $L_{i}$ are unipotent. We expand
$\displaystyle\frac{1}{1-qL}$ as follows:

$\frac{1}{1-qL}=\sum_{k\geq 0}\frac{q^{k}}{(1-q)^{k+1}}(L-1)^{k}.$

This motivates the following definitions of symplectic loop space and
$J_{X}^{{\rm fake}}$. Let

$\mathcal{K}^{{\rm fake}}:=\\{\text{$Q$ power series with coefficients Laurent
series in $1-q$}\\}.$

$\mathcal{K}^{{\rm fake}}$ is endowed with the symplectic form $\Omega^{{\rm
fake}}$

$\Omega^{{\rm fake}}(f,g):=-{\rm Res}_{q=1}(f(q),g(q^{-1}))^{K}\frac{dq}{q},$

and a corresponding Lagrangian polarization:

$\begin{split}\mathcal{K}^{{\rm fake}}&=\mathcal{K}_{+}^{{\rm
fake}}\oplus\mathcal{K}_{-}^{{\rm fake}}\\\ :&=K[\\![q-1]\\!]\oplus{\rm
span}_{K}\left\\{\frac{q^{k}}{(1-q)^{k+1}}\right\\}_{k\geq 0}.\end{split}$

######  Definition 2.6.

The big $J$-function of $X$ in the fake $K$-theory is defined as a map
$\mathcal{K}_{+}^{{\rm fake}}\rightarrow\mathcal{K}^{{\rm fake}}$:

$t\rightarrow J^{{\rm
fake}}(t):=(1-q)+t(q)+\sum_{\alpha}\Phi_{\alpha}\sum_{n,d}\frac{Q^{d}}{n!}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X,{\rm
fake}}_{0,n+1,d}.$

Same as in Remark 2.4, $J^{{\rm fake}}$ has dilation shift $(1-q)$, input
$t(q)$, and a sum of correlator lying in $\mathcal{K}_{-}^{{\rm fake}}$. The
graph of the $J^{{\rm fake}}$-function is a Lagrangian cone $\mathcal{L}^{{\rm
fake}}\subset\mathcal{K}^{{\rm fake}}$, which is overruled in the sense that
its tangent space $T$ are tangent to $\mathcal{L}^{{\rm fake}}$ along
$(1-q)T$.

To describe the relation between fake $K$-theory and cohomological theory,
take $H=H^{*}(X)[\\![Q]\\!]$, with Poincaré pairing $(a,b)^{H}=\int_{X}a\cdot
b$. The Givental’s loop space

$\mathcal{H}:=H[\\![z,z^{-1}],$

is endowed with a symplectic form $\Omega^{H}$:

$\Omega(f,g):={\rm Res}_{z=0}(f(-z),g(z))^{H}dz,$

and a corresponding Lagrangian polarization:

$\begin{split}\mathcal{H}&=\mathcal{H}_{+}\oplus\mathcal{H}_{-}\\\
&=H[\\![z]\\!]\oplus z^{-1}H[z^{-1}].\end{split}$

######  Definition 2.7.

The big $J$-function of $X$ in the cohomology theory is defined as a map
$\mathcal{H}_{+}\rightarrow\mathcal{H}$:

$t\rightarrow
J^{H}(t):=-z+t(z)+\sum_{\alpha}\phi_{\alpha}\sum_{n,d}\frac{Q^{d}}{n!}\langle\frac{\phi^{\alpha}}{-z-\psi_{1}},t(\psi_{2}),\dots,t(\psi_{n+1})\rangle^{X,H}_{0,n+1,d},$

where $\\{\phi_{\alpha}\\}$ and $\\{\phi^{\alpha}\\}$ are Poincaré dual basis
in $H(X)$ with respect to $(\cdot,\cdot)^{H}$. We denote the graph of the big
$J^{H}$-function as $\mathcal{L}^{H}$.

Introduce the quantum Chern character by

$\begin{split}{\rm qch}:\mathcal{K}^{{\rm fake}}&\rightarrow\mathcal{H}\\\
\sum_{k}f_{k}(q-1)^{k}&\rightarrow\sqrt{{\rm td}(TX)}\sum_{k}{\rm
ch}(f_{k})(e^{z}-1)^{k}.\end{split}$

It is symplectic, i.e. qch${}^{*}\Omega^{H}=\Omega^{{\rm fake}}$.

######  Theorem 2.8 ([CG07]).

Denote by $\Delta$ the Euler–Maclaurin asymptotic of the infinite product

$\Delta\sim\sqrt{{\rm td}(T_{X}-1)}\prod_{m=1}^{\infty}{\rm
td}((T_{X}-1)\otimes L_{z}^{-m}),$

where $L_{z}$ is the universal line bundle with first chern class $z$. Then
$\mathcal{L}^{{\rm fake}}$ can be obtained by $\mathcal{L}^{H}$ by the
multiplication of $\Delta$:

${\rm qch}(\mathcal{L}^{{\rm fake}})=\Delta\mathcal{L}^{H}.$

###  2.3. $J^{K}$-function as a graph sum via Kawasaki’s HRR

Given a nonsingular projective variety $M$ with a locally free sheaf $E$, the
Hirzebruch-Riemann-Roch formula expresses the Euler characteristic in terms of
a cohomological integration:

$\chi(M;E):=\sum_{i=0}^{\dim M}(-1)^{i}h^{i}(M;E)=\int_{M}{\rm
td}(TM)\cdot{\rm ch}(E).$

A generalization to smooth Deligne-Mumford stacks $\mathcal{M}$ was first
established by T. Kawasaki [Kaw79] in the analytic setting.

Let $I\mathcal{M}=\sqcup_{i}\mathcal{M}_{i}$ be the inertia stack of
$\mathcal{M}$, with $\mathcal{M}_{i}$ connected components. Following
Givental, we will refer to them as _Kawasaki strata_. Kawasaki’s formula reads

$\chi(\mathcal{M},E)=\sum_{i}\frac{1}{m_{i}}\int_{\mathcal{M}_{i}}{\rm
td}(T_{\mathcal{M}_{i}}){\rm ch}\Big{(}\frac{{\rm
Tr}(E|_{\mathcal{M}_{i}})}{{\rm
Tr}(\Lambda^{*}N_{\mathcal{M}_{i}}^{*})}\Big{)},$

where the sum over $i$ runs through all connected components.

As in [GT14], we apply a (virtual) version of the above formula, valid for
quasi-smooth DM stacks, to compute the (big) $J$-function. Kawasaki’s strata
are visualized in Figure 1 as in [GT14, § 7].

dilaton
shiftinputheadarms$(1-q)$+$t(q)$+$\frac{1}{1-qL_{1}}$$\displaystyle\sum_{r\geq
2}\sum_{\zeta_{r}}$$\frac{1}{1-qL_{1}^{1/r}\zeta_{r}}$stemlegstail Figure 1.
Kawasaki strata

Notations of Figure 1 are explained below. A point in the inertia stack is
represented by a stable map with symmetry, which we denote by $g$.

  * •

The black points denote the marked points.

  * •

Given a stable map with symmetry $g$, the action of $g$ fixes the marked
points and acts on $L_{1}$ with an eigenvalue, which we denote $\zeta$. The
strata with $\zeta=1$ are separated from those where $\zeta_{r}\neq 1$ in the
Figure 1. $\displaystyle\sum_{\zeta_{r}}$ denotes the sum over all primitive
$r$-th roots of unity.

  * •

When $\zeta=1$, the _head_ is a stable map with its source curve the maximal
connected subcurve containing the first marked points where $g$ is trivial.
The head has marked points which are either the marked points of the original
curve or nodal points attached to arms.

  * •

An _arm_ is a stable map whose source curve is obtained as a connected
component of the original curve when the head is removed. It has its own
“first marked point”, the nodal point where it is attached to the head.

By definition, an arm can have any symmetry satisfying the following
condition: the eigenvalue of the symmetry at its “first marked point” $\neq
1$, or else it is part of the head.

  * •

When $\zeta=\zeta_{r}\neq 1$, we consider the stem curve defined in the next
paragraph. It is a connected subcurve, containing the first marked point with
$g$ as a symmetry of order $r$. The quotient stable map is called the _stem_.

The stem curve contains the maximal chain of $\mathbb{P}^{1}$ containing the
_first marked point_ and with the same action of the symmetry on each of them.
In other words, a node connects two $\mathbb{P}^{1}$’s should be balanced,
i.e., the eigenvalues of $g$ on the two branches are inverse. Furthermore, it
can have other irreducible components attached on the “side” of the chain
permuted by $g$. The opposite end of the maximal chain of $\mathbb{P}^{1}$’s
is either a node or a marked point. In either case, we refer it as the _“last
marked point”_.

The stem carries two kinds of marked points:

    1. (1)

Ramified/stacky points at the first and last marked points. The input at the
first marked point is ${\displaystyle(1-qL_{1}^{1/r}\zeta_{r})^{-1}}$.

    2. (2)

Unramified/smooth marked points coming from symmetric configurations of
$r$-tuples of nodes on the cover. (These nodes connect to legs; see below.)

  * •

A _tail_ is a stable map whose source curve is obtained as _the_ connected
component of the original curve after removing the stem curve and is attached
the last marked point. Its “first marked point”, the nodal point where it is
attached to the stem, is fixed by the $g$ action with the eigenvalue
$\neq\zeta$, (or else it is part of the stem.)

  * •

_Legs_ are connected components after removing the stem curve and the tail.
They are permuted by the action of $g$.

We write

$\begin{split}J^{{K}}(t)&=(1-q)+t(q)+\sum_{n,d,\alpha}\frac{Q^{d}\Phi_{\alpha}}{n!}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X,{K}}_{0,n+1,d}\\\
&=(1-q)+t(q)+\sum_{n,d,\alpha}\frac{Q^{d}\Phi_{\alpha}}{n!}\sum_{\zeta}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X_{\zeta}}_{0,n+1,d},\end{split}$

where $\sum_{\zeta}$ runs through all roots of unity (including 1),
$X_{\zeta}$ is the Kawasaki strata where $g$ acts on $L_{1}$ with eigenvalue
$\zeta$, and $\langle\rangle^{X_{\zeta}}$ denotes the contribution of
$X_{\zeta}$ in the Kawasaki’s formula. In other words,
$\langle\rangle^{X,{K}}$ represent (true) quantum $K$-invariants while
$\langle\rangle^{X_{\zeta}}$ stand for (collections of) _cohomological_
invariants. Explicit formulas will be reorganized in Proposition 2.10.

Let arm$(L)$, leg${}_{\zeta}(L)$, and tail${}_{\zeta}(L)$ denote the
totalities of the contributions from arm, leg, and tail respectively. They can
be computed as follows:

$\begin{split}{\rm arm}(q)&=\sum_{n,d\neq
0,\alpha}\frac{Q^{d}\Phi_{\alpha}}{n!}\sum_{\zeta^{\prime}\neq
1}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X_{\zeta^{\prime}}}_{0,n+1,d}\\\
{\rm leg}_{\zeta}(q)&=\Psi^{r}({\rm arm}(q)_{t=0})\\\ {\rm
tail}_{\zeta}(q)&=\sum_{n,d\neq
0,\alpha}\frac{Q^{d}\Phi_{\alpha}}{n!}\sum_{\zeta^{\prime}\neq\zeta}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X_{\zeta^{\prime}}}_{0,n+1,d},\end{split}$

where $\zeta$ is the primitive $r$-th roots of unity, $\zeta^{\prime}$ in the
above sums are arbitrary roots of unity and $\Psi^{r}$ are the _Adams
operations_. Recall that Adams operations are additive and multiplicative
endomorphisms of $K$-theory or more generally $\lambda$-rings, acting on line
bundles by $\Psi^{r}(L)=L^{\otimes r}$. Here $\Psi^{r}$ also act on the
Novikov variables by $\Psi^{r}(Q^{d})=Q^{rd}$. In the definition of
leg${}_{\zeta}(q)$, $t=0$ is imposed because legs will be permuted by $g$, an
automorphism, which excludes the appearance of marked points.

The following propositions (2.9, 2.10) justify the above definitions.

######  Proposition 2.9 (Essentially [GT14, Proposition 1]).

$J^{{K}}(t)|_{q=1}=J^{{\rm fake}}(t+{\rm arm}),$

where $(\cdot)|_{q=1}$ is the power series expansion at $q=1$ (of the rational
function).

For $\zeta\neq 1$ a primitive $r$-th roots of unity, the stem space is
isomorphic to the moduli space

$\overline{\mathcal{M}}_{0,n+2,d}^{X}(\zeta):=\overline{\mathcal{M}}_{0,n+2}\left([X/\mathbb{Z}_{r}],d;(g,1,\dots,1,g^{-1})\right).$

Here the group elements $g,1$ signal the twisted sectors in which the marked
points lie. To simplify the notation, we introduce the notation

(2.1)
$\begin{split}&\Big{[}T_{1}(L),T(L),\dots,T(L),T_{n+2}(L)\Big{]}^{X_{\zeta}}_{0,n+2,d}\\\
:=&\int_{[\overline{\mathcal{M}}_{0,n+2,d}^{X}(\zeta)]^{{\rm vir}}}{\rm
td}(T_{\overline{\mathcal{M}}}){\rm ch}\left(\frac{{\rm
ev}_{1}^{*}(T_{1}(L)){\rm ev}_{n+2}^{*}(T_{n+2}(L))\prod_{i=2}^{n+1}{\rm
ev}_{i}^{*}T(L)}{{\rm
Tr}(\Lambda^{*}N^{*}_{\overline{\mathcal{M}}})}\right),\end{split}$

where $[\overline{\mathcal{M}}_{0,n+2,d}^{X}(\zeta)]^{{\rm vir}}$ is the
virtual fundamental class, $T_{\overline{\mathcal{M}}}$ is the (virtual)
tangent bundle to $\overline{\mathcal{M}}_{0,n+2,d}^{X}(\zeta)$, and
$N_{\overline{\mathcal{M}}}$ is the (virtual) normal bundle of
$\overline{\mathcal{M}}_{0,n+2,d}^{X}(\zeta)$ considered as Kawasaki strata in
$\overline{\mathcal{M}}_{0,nr+2}(X,rd)$.

######  Proposition 2.10 ( [GT14, §7] ).

Let $\zeta$ be a primitive $r$-th roots of unity. We have

$\begin{split}&\sum_{n,d,\alpha}\frac{Q^{d}\Phi_{\alpha}}{n!}\langle\frac{\Phi^{\alpha}}{1-qL},t(L),\dots,t(L)\rangle^{X_{\zeta}}_{0,n+1,d}\\\
=&\sum_{n,d,\alpha}\frac{Q^{rd}\Phi_{\alpha}}{n!}\Big{[}\frac{\Phi^{\alpha}}{1-q\zeta
L^{1/r}},{\rm leg}_{\zeta}(L),\dots,{\rm
leg}_{\zeta}(L),\delta_{\zeta}(L^{1/r})\Big{]}^{X_{\zeta}}_{0,n+2,d},\end{split}$

where

$\delta_{\zeta}(q)=(1-\zeta^{-1}q)+t(\zeta^{-1}q)+{\rm
tail}_{\zeta}(\zeta^{-1}q).$

###### Sketch of Proof.

The statement is scattered in [GT14, §7]. A sketch of the proof is given
below.

The factor $\displaystyle\frac{1}{1-q\zeta L^{1/r}}$, coming from the
“ramified” first marked point on a stem curve, with $\zeta$ the eigenvalue of
the $g$-action. The power $1/r$ on $L$ comes from the comparison of the
cotangent line between the marked point on the stem and its $r$-fold cover.

The contributions at unramified marked points can be written in terms of legs,
where we have used the following lemma.

######  Lemma 2.11 ([GT14, §7 Lemma] ).

Let $V$ be a vector bundle, and $g$ the automorphism of $V^{\otimes r}$ acting
by the cyclic permutation of the factors. Then

${\rm Tr}(g|V^{\otimes r})=\Psi^{r}(V).$

The input $\delta(q)$ in the last marked point contains three parts. They come
from the following cases:

  * •

If the last marked point is a regular point on the covering curve, the
infinitesimal translation of that point gives the factor
$1-\zeta^{-1}L^{1/r}$.

  * •

If the last marked point is also a marked point on the covering curve, we keep
the input $t(\zeta^{-1}L^{1/r})$.

  * •

If the last marked point is the node with a tail attached on the cover, the
contribution is tail${}_{\zeta}(\zeta^{-1}L^{1/r})$, since the tail can be any
stable map with eigenvalue at the first marking $\neq\zeta^{-1}$. This is
exactly the definition of tail${}_{\zeta}(q)$

As for the first marked point, the nontrivial action of $g$ creates the
$r$-the roots $\zeta^{-1}$ and $L^{1/r}$. This finish the proof. ∎

######  Remark 2.12.

Since $L^{1/r}$ is unipotent, we do the following expansion:

$\frac{1}{1-q\zeta L^{1/r}}=\sum_{i\geq 0}\frac{(\zeta q)^{i}}{(1-\zeta
q)^{i+1}}(L^{1/r}-1)^{i}.$

This expresses $J^{{K}}|_{\mathcal{K}_{-}}$ as sums of rational functions with
poles only at roots of unity. Cf. Remark 2.4.

###  2.4. Reconstruction theorem

We now specialize to the quintic $X$. There are several reconstruction
theorems that determine parts of the overruled Lagrangian cone
$\mathcal{L}^{{K}}$ from certain initial values. See [LP04, Giv15a, Giv15c,
IMT15].

A convenient choice of the initial values for the reconstruction theorem is
the Givental’s $I^{K}$-function for $X$. First consider [GL03]

$J^{K}_{\mathbb{P}^{4}}(0)=(1-q)\sum_{d\geq
0}\frac{Q^{d}}{\prod_{k=1}^{d}(1-Pq^{k})^{5}},$

where $P=\mathcal{O}(-1)$ and hence $(1-P)^{5}=0$. Then [Giv15b] gives

$I^{K}=\sum_{d\geq
0}I_{d}^{K}Q^{d}=(1-q)\sum_{d=0}^{\infty}Q^{d}\frac{\prod_{k=1}^{5d}(1-P^{5}q^{k})}{\prod_{k=1}^{d}(1-Pq^{k})^{5}}=J^{K}(t^{*}),$

where the input $t^{*}$ can be computed:

$t^{*}(q)=I^{K}|_{\mathcal{K}_{+}}-(1-q).$

######  Theorem 2.13 (Essentially [Giv15c, Theorem 2]).

For $X$ the quintic threefold, we have

(2.2) $\begin{split}&J^{K}(0)=\sum_{d\geq 0}I_{d}^{K}Q^{d}\cdot\\\
&\quad\exp\left(\sum_{k>0}\frac{\sum_{i=0}^{3}\Psi^{k}(\epsilon_{i}(Q))(1-P^{k}q^{kd})^{i}}{k(1-q^{k})}\right)\sum_{i=0}^{3}r_{i}(q,Q)(1-Pq^{d})^{i},\end{split}$

for some uniquely determined $\epsilon_{i}(Q)$ and $r_{i}(q,Q)$, where

(2.3) $\begin{split}\epsilon_{i}(Q)=\sum_{j\geq
1}\epsilon_{ij}Q^{j}\in\mathbb{Q}[\\![Q]\\!],\\\ r_{i}(q,Q)=\sum_{j\geq
0}r_{ij}(q)Q^{j}\in\mathbb{Q}[q][\\![Q]\\!].\end{split}$

###### Sketch of proof.

This is essentially [Giv15c, Theorem 2], with minor improvements due to the
special properties of the quintic. Givental’s reconstruction (2.2) holds for
some $\epsilon_{i}$ and $r_{i}(q)\in\mathbb{Q}[q,q^{-1}][\\![Q]\\!]$. Using
the induction on the degree of $Q$ and the condition

$J^{K}(0)|_{\mathcal{K}_{+}}\left(:=\text{projection to $\mathcal{K}_{+}$ of
$J^{K}(0)$}\right)=1-q,$

$\epsilon_{i}(Q)$ and $r_{i}(q,Q)$ are uniquely determined. More precisely,
given any $M$, the coefficients of $Q^{M}$ of the projections of the left hand
side and of the right hand side of (2.2) to $\mathcal{K}_{+}$ must be equal

(2.4)
$\begin{split}M=0,\quad(1-q)=(1-q)\sum_{i=0}^{3}r_{i0}(1-P)^{i}\,\Rightarrow\,\sum_{i=0}^{3}&r_{i0}(1-P)^{i}=1,\\\
\sum_{i}\left[\epsilon_{iM}+(1-q)r_{iM}(q)\right](1-P)^{i}=\sum_{i}f_{i}(q)(1-P)^{i}\,&\Rightarrow\,\epsilon_{iM}=f_{i}(1)\end{split}$

for some polynomials $f_{i}(q)$ of $q$, which can be written as polynomials of
$\epsilon_{i^{\prime}M^{\prime}}$, $r_{i^{\prime}M^{\prime}}$ and (components
of) $I_{M^{\prime}}^{K}$ for $M^{\prime}<M$. This in particular also implies,
by induction, that _$r_{ik}(q)$ are polynomials of $q$_ (instead of Laurent
polynomials). ∎

##  3\. Proof of the conjecture

###  3.1. Outline of the proof

As explained in the Introduction, our proof involves both the adelic
characterization (via Kawasaki’s HRR theorem) and the explicit reconstruction
theorem. It is quite involved. For readers’ convenience, we break it down in
four steps, and relegate some computations to Section 4 and Appendix A so as
not to break the flow of the proof.

(1). The first step is to establish an expression of the $J^{K}$-function in
Corollary 3.3. That is

$\begin{split}\frac{1}{1-q}J^{K}(0)&=1+\frac{1-P}{5}\sum_{M\geq
1}Q^{M}\sum_{r\leq M}\sum_{\zeta^{r}=1}\Big{(}\frac{a_{M,\zeta}}{1-\zeta
q}\Big{)}\\\ &+\frac{(1-P)^{2}}{5}\sum_{M\geq 1}Q^{M}\sum_{r\leq
M}\sum_{\zeta^{r}=1}\Big{(}\frac{b_{M,\zeta}}{1-\zeta
q}+\frac{c_{M,\zeta}}{(1-\zeta q)^{2}}\Big{)}\\\
&+\frac{(1-P)^{3}}{5}\sum_{M\geq 1}Q^{M}\sum_{r\leq
M}\sum_{\zeta^{r}=1}\left(\frac{{\mathbf{d}}_{M,\zeta}}{1-q\zeta}+\frac{e_{M,\zeta}}{(1-\zeta
q)^{2}}+\frac{f_{M,\zeta}}{(1-\zeta q)^{3}}\right),\end{split}$

where $\sum_{\zeta^{r}=1}$ is the sum over _primitive_ $r$-th roots of unity.
We call $a_{M,\zeta},c_{M,\zeta}$, $f_{M,\zeta}$ the coefficients of degree 0
terms, $b_{M,\zeta}$, $e_{M,\zeta}$ the coefficients of degree 1 terms, and
${\mathbf{d}}_{M,\zeta}$ the degree 2 terms. This step uses mainly the
reconstruction theorem.

(2). The ”degree $0$ terms” $a_{M,\zeta},c_{M,\zeta}$, $f_{M,\zeta}$ are the
easiest to handle. Let $\zeta$ be a primitive $r$-th root of unity. In Theorem
3.4, we show that

$\begin{split}a_{M,\zeta}=0,c_{M,\zeta}=\frac{{\mathrm{GV}}^{(3)}_{M/r}}{M^{2}},f_{M,\zeta}=\frac{-2{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}},\end{split}$

where

${\mathrm{GV}}_{n}^{(\gamma)}:=\sum_{d|n}d^{\gamma}{\mathrm{GV}}_{d}.$

Note that the above invariants are defined to be zero if $M/r$ are _not
integers_. We will give two proofs. The one uses adelic characterization is
given in Subsection 3.3. The one in Appendix A uses only the reconstruction
theorem.

(3). For higher degree terms, we show in Theorem 3.5 that

$\begin{split}b_{rd,\zeta}&={\mathrm{GV}}^{(1)}_{d}+\frac{1}{r^{2}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)},\\\
{\mathbf{d}}_{rd,\zeta}&=rd{\mathrm{GV}}^{(-1)}_{d}+{\mathrm{GV}}^{(1)}_{d}+\frac{1}{r^{2}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)}+\frac{1}{r^{3}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{3}}\Big{)},\\\
e_{rd,\zeta}&=\frac{1}{r^{2}}\Big{(}\frac{{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)}+\frac{1}{r^{3}}\Big{(}\frac{3{\mathrm{GV}}_{d}^{(3)}}{d^{3}}\Big{)}.\end{split}$

This step uses the adelic characterization and induction on Novikov variable.

(4). Combining the above, we have

$\begin{split}&\frac{1}{1-q}J^{K}(0)=1+\\\ &+\frac{(1-P)^{2}}{5}\sum_{r\geq
1}\sum_{\zeta^{r}=1}\sum_{d\geq
1}Q^{rd}\left(\frac{{\mathrm{GV}}_{d}^{(1)}-\frac{1}{r^{2}d^{2}}{\mathrm{GV}}^{(3)}_{d}}{1-\zeta
q}+\frac{\frac{1}{r^{2}d^{2}}{\mathrm{GV}}^{(3)}_{d}}{(1-\zeta
q)^{2}}\right)\\\ &+\frac{(1-P)^{3}}{5}\sum_{r\geq
1}\sum_{\zeta^{r}=1}\sum_{d\geq
1}Q^{rd}\left(\frac{rd{\mathrm{GV}}_{d}^{(-1)}+{\mathrm{GV}}_{d}^{(1)}-\frac{1}{r^{2}d^{2}}{\mathrm{GV}}_{d}^{(3)}-\frac{1}{r^{3}d^{3}}{\mathrm{GV}}_{d}^{(3)}}{1-\zeta
q}\right.\\\ &\hskip
156.49014pt\left.+\frac{\frac{{\mathrm{GV}}_{d}^{(3)}}{r^{2}d^{2}}+\frac{3{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}}{(1-\zeta
q)^{2}}+\frac{\frac{-2{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}}{(1-\zeta
q)^{3}}\right).\end{split}$

A direct computation in Theorem 3.6 shows that this coincides with Conjecture
1.1.

###  3.2. Analytic properties of $J^{K}(0)$ and a degree argument

In this subsection, our main task is to prove Theorem 3.2 below. We first
_partially_ expand $I^{K}$ in the basis $\\{(1-P)^{i}\\}_{i=0,1,2,3}$:

(3.1) $\begin{split}&\frac{1}{1-q}I^{{K}}\\\ =&\sum_{d\geq
0}Q^{d}\frac{\prod_{r=1}^{5d}(1-P^{5}q^{r})}{(\prod_{r=1}^{d}(1-Pq^{r}))^{5}}\\\
=&\sum_{d\geq
0}Q^{d}\frac{\prod_{r=1}^{5d}(1-q^{r}-q^{r}(P^{5}-1))}{(\prod_{r=1}^{d}(1-q^{r}-q^{r}(P-1)))^{5}}\\\
=&\sum_{d\geq
0}Q^{d}\frac{\prod_{r=1}^{5d}(1-q^{r})}{(\prod_{r=1}^{d}(1-q^{r}))^{5}}\cdot\\\
&\cdot\Big{(}1+e_{1}(5d)\frac{1-P^{5}}{P^{5}}+e_{2}(5d)\cdot(\frac{1-P^{5}}{P^{5}})^{2}+e_{3}(5d)(\frac{1-P^{5}}{P^{5}})^{3}\Big{)}\cdot\\\
&\cdot\Big{(}1-5e_{1}(d)\frac{1-P}{P}+(15e_{1}(d)^{2}-5e_{2}(d))(\frac{1-P}{P})^{2}-(35e_{1}(d)^{3}\\\
&\qquad-30e_{1}(d)e_{2}(d))(\frac{1-P}{P})^{3}\Big{)},\end{split}$

where

$e_{i}(k):=\sum_{1\leq j_{1}<\cdots<j_{i}\leq
k}\frac{1}{1-q^{j_{1}}}\cdots\frac{1}{1-q^{j_{i}}}.$

For the third equality, the following equalities are used

$\begin{split}\frac{1}{1-q^{r}-q^{r}(P-1)}&=\frac{1}{1-q^{r}}\sum_{i=0}^{3}\Big{(}\frac{q^{r}}{1-q^{r}}\Big{)}^{i}(P-1)^{i}\\\
&=\frac{1}{1-q^{r}}\sum_{i=0}^{3}\Big{(}\frac{1}{1-q^{r}}-1\Big{)}^{i}(P-1)^{i}\\\
&=\frac{1}{1-q^{r}}\frac{1}{P}\sum_{i=0}^{3}(-1)^{i}\Big{(}\frac{1}{(1-q^{r})}\Big{)}^{i}\Big{(}\frac{1-P}{P}\Big{)}^{i},\end{split}$
$\begin{split}1-q^{r}-q^{r}(P^{5}-1)&=(1-q^{r})\Big{(}1-\frac{q^{r}}{1-q^{r}}(P^{5}-1)\Big{)}\\\
&=(1-q^{r})\Big{(}P^{5}-\frac{P^{5}-1}{1-q^{r}}\Big{)}\\\
&=P^{5}(1-q^{r})\Big{(}1+\Big{(}\frac{1}{1-q^{r}}\Big{)}\Big{(}\frac{1-P^{5}}{P^{5}}\Big{)}\Big{)}.\end{split}$

Note that in order to completely expand in terms of the basis, $1/P$ should
have been rewritten as

$\frac{1}{P}=\frac{1}{1-(1-P)}=1+(1-P)+(1-P)^{2}+(1-P)^{3}.$

However, we will not need the precise form of expansion. (3.1) will suffice
for our purpose below.

Fix a root of unity $\zeta$. Let $f\in\mathcal{K}=K^{0}(X)(q)[\\![Q]\\!]$. We
will expand the $Q$-coefficients $f$ as a Laurent series in $(1-\zeta q)$,
$K^{0}(X)[(1-\zeta q)^{-1},(1-\zeta q)]\\!]$, in terms of the basis
$\\{(1-P)^{i}\\}_{i=0}^{3}$ for $K^{0}(X)$. _Define_ the _degree
$\deg_{q=\zeta^{-1}}$_ as follows:

$\deg_{q=\zeta^{-1}}((1-\zeta
q)^{i}(1-P)^{j}):=i+j,\quad\deg_{q=\zeta^{-1}}(0):=\infty.$

For an inhomogeneous element, we define the degree to be the lowest degree of
the homogeneous parts. We then have

(3.2) $\begin{split}\deg_{q=\zeta^{-1}}(a\cdot
b)&\geq\deg_{q=\zeta^{-1}}(a)+\deg_{q=\zeta^{-1}}(b)\\\
\deg_{q=\zeta^{-1}}(a+b)&\geq\operatorname{min}\\{\deg_{q=\zeta^{-1}}(a),\deg_{q=\zeta^{-1}}(b)\\}.\end{split}$

Indeed, one might wish to consider $\deg_{q=\zeta^{-1}}$ defines a (discrete)
_“semi-valuation”_ , and hence an adic topology, for these localized
functions. Expansion of $f\in\mathcal{K}$ in the ascending order of
$\deg_{q=\zeta^{-1}}$, i.e., the $K$-valued Laurent series in $(1-\zeta q)$,
is denoted as $(f)_{q=\zeta^{-1}}$ and is called the _localization of $f$ at
$q=\zeta^{-1}$_. For $f,g\in\mathcal{K}$, write

$f\equiv_{q=\zeta^{-1}}^{n}g\quad\mbox{(resp.\ $f\equiv_{q=\zeta^{-1}}g$)},$

if $(f)_{q=\zeta^{-1}}$ and $(g)_{q=\zeta^{-1}}$ agree up to degree $\leq n$
(resp. they agree to all degrees).

######  Remark 3.1.

The semi-valuation above can be extended to the entire $\mathcal{K}$ by
extending the totally ordered group from $\mathbb{Z}$ to $\mathbb{Z}^{\oplus
2}$ and define the degree of an element $f$ in $K^{0}(X)[(1-\zeta
q)^{-1},(1-\zeta q)]\\!][\\![Q]\\!]$ as

${\deg_{\zeta^{-1}}}\left((1-\zeta q)^{i}(1-P)^{j}Q^{d}\right)=(d,i+j),$

with the total ordering on $\mathbb{Z}^{\oplus 2}$ being the lexicographic
ordering. We will not use this but stick to the $\deg_{q=\zeta^{-1}}$.

######  Theorem 3.2.

For any roots of unity $\zeta$, the $Q$-coefficients of $[J^{{K}}]_{-}$ are
sums of terms with non-negative degrees with respect to $\deg_{q=\zeta^{-1}}$.

###### Proof.

The reconstruction theorem (2.2) states that

$\begin{split}&J^{{K}}(0)=\\\ &\sum_{d\geq
0}I^{{K}}_{d}Q^{d}\exp\left(\sum_{k>0}\frac{\sum_{i=0}^{3}\Psi^{k}(\epsilon_{i}(Q))(1-P^{k}q^{kd})^{i}}{k(1-q^{k})}\right)\sum_{i=0}^{3}r_{i}(q,Q)(1-Pq^{d})^{i}.\end{split}$

The right hand side of the formula has three parts, and we will show that each
part has $\deg_{q=\zeta^{-1}}\geq 0$. In the first part, we claim that
$I^{K}_{d}$ have $\deg_{q=\zeta^{-1}}\geq 0$. This can be seen from the
expansion of $I^{K}_{d}(q)$ in (3.1) and the paragraphs following it. In
particular, the factor

(3.3)
$(1-q)^{-1}I^{K}_{0d}(q)=\frac{\prod_{r=1}^{5d}(1-q^{r})}{\prod_{r=1}^{d}(1-q^{r})^{5}}$

is actually a polynomial in $q$ and $e_{i}(d)$ has poles of order no greater
than $i$ at $q=\zeta^{-1}$.

On the other hand, the last part of formula,
$\sum_{i=0}^{3}r_{i}(q,Q)(1-Pq^{d})^{i}$, consists of only the polynomial in
$q$, as proven in Theorem 2.13. Its $Q$-coefficients therefore have
$\deg_{q=\zeta^{-1}}\geq 0$.

In the middle $\exp$ factor, we first claim $\epsilon_{0}(Q)=0$. Assuming the
claim, we rewrite the $\exp$ factor as follows

$\begin{split}&\exp\left(\sum_{k>0}\frac{\sum_{i=0}^{3}\Psi^{k}(\epsilon_{i}(Q))(1-P^{k}q^{kd})^{i}}{k(1-q^{k})}\right)\\\
=&\exp\left(\sum_{k>0}\frac{\sum_{i=1}^{3}\Psi^{k}(\epsilon_{i}(Q))(1-P^{k}q^{kd})^{i}}{k(1-q^{k})}\right)\\\
=&\exp\left(\sum_{k>0}\frac{\sum_{i=1}^{3}\Psi^{k}(\epsilon_{i}(Q))(1-q^{kd}-q^{kd}(P^{k}-1))^{i}}{k(1-q^{k})}\right).\end{split}$

One see clearly that the exponent has $\deg_{q=\zeta^{-1}}\geq 0$. We now
return to the proof of claim. By definition $\epsilon_{00}=0$, and assume that
$\epsilon_{0M^{\prime}}=0$ for all $M^{\prime}<M$. By (3.3), $I^{K}_{od}$ is a
polynomial of $q$ and is divisible by $1-q$. Following the reconstruction
theorem and in particular (2.4), we have

$\epsilon_{0M}=-\sum_{d_{1}+d_{2}=M}I^{K}_{0d_{1}}(q)r_{0d_{2}}(q).$

Note that the LHS is a constant in $q$ while the RHS always contains a factor
$I^{K}_{0d_{1}}(q)$, a polynomial in $q$ divisible by $1-q$. We have
therefore, $e_{0M}=\mathrm{RHS}(q=1)=0$.

The theorem follows by the semi-valuation property (3.2). ∎

Theorem 3.2 has the following important consequence on the analytic property
of $\displaystyle{(1-q)^{-1}}J^{{K}}(0)$.

######  Corollary 3.3.

The small $J^{K}$-function has the following expansion in $Q$ and $q$

$\begin{split}\frac{1}{1-q}J^{K}(0)&=1+\frac{1-P}{5}\sum_{M\geq
1}Q^{M}\sum_{r\leq M}\sum_{\zeta^{r}=1}\Big{(}\frac{a_{M,\zeta}}{1-\zeta
q}\Big{)}\\\ &+\frac{(1-P)^{2}}{5}\sum_{M\geq 1}Q^{M}\sum_{r\leq
M}\sum_{\zeta^{r}=1}\Big{(}\frac{b_{M,\zeta}}{1-\zeta
q}+\frac{c_{M,\zeta}}{(1-\zeta q)^{2}}\Big{)}\\\
&+\frac{(1-P)^{3}}{5}\sum_{M\geq 1}Q^{M}\sum_{r\leq
M}\sum_{\zeta^{r}=1}\left(\frac{{\mathbf{d}}_{M,\zeta}}{1-q\zeta}+\frac{e_{M,\zeta}}{(1-\zeta
q)^{2}}+\frac{f_{M,\zeta}}{(1-\zeta q)^{3}}\right),\end{split}$

where $\displaystyle\sum_{\zeta^{r}=1}$ is the sum over all primitive $r$-th
roots of unities.

###### Proof.

By Remark 2.12, the partial fraction expansion of

${(1-q)^{-1}}J^{{K}}(t,q,Q)\in\mathcal{K}=\mathcal{K}_{+}\oplus K_{-}$

consists of the $\mathcal{K}_{+}$-projection, a Laurent polynomial in $q$, and
the $\mathcal{K}_{-}$ projection, a rational function in $q$ with poles only
at roots of unity and vanishing at $\infty$. We know that
${(1-q)^{-1}}J^{{K}}(t=0)|_{\mathcal{K}_{+}}=1$. Therefore, the RHS must have
the specified form, provided that we can prove the coefficients of $Q^{M}$
have poles only at $r$-th roots of unity for $r\leq M$. This can be seen from
the reconstruction theorem as follows.

As in the proof of Theorem 3.2, we employ the reconstruction theorem (2.2) and
analyze the three factors $I^{K}$, the exponential, and the polynomial
$r_{i}(q,Q)$ separately. In (3.1) of $I^{K}_{d}$, the factor $e_{i}(5d)$ in
the coefficient of $Q^{d}$ can be cancelled by the numerator
${\prod_{r=1}^{5d}(1-q^{r})}$ of the overall factor. The remaining denominator
has poles only at $r$-th roots of unity with $r\leq d$. For the exp factor,
$\epsilon_{i}(Q=0)=0$ for all $i$ and hence any expressions with poles at
roots of unity of order $k$ appear with $Q^{jk}$ for $j\geq 1$. Finally, the
third factor $r_{i}$ are polynomial in $q$ and have no poles. This finishes
the proof. ∎

In the above expression, we observe that the terms involving
$a_{M,\zeta},c_{M,\zeta}$, and $f_{M,\zeta}$ have $\deg_{q=\zeta^{-1}}=0$;
those involving $b_{M,\zeta}$ and $e_{M,\zeta}$ have $\deg_{q=\zeta^{-1}}=1$;
those involving ${\mathbf{d}}_{M,\zeta}$ have $\deg_{q=\zeta^{-1}}=2$.

###  3.3. The degree 0 terms

Let $\zeta$ be a primitive $r$-th roots of unity. In this subsection, we
compute $a_{M,\zeta},c_{M,\zeta}$, and $f_{M,\zeta}$.

######  Theorem 3.4.

Let $d:=M/r$. If $M/r$ are not integers, we have

$a_{M,\zeta}=c_{M,\zeta}=f_{M,\zeta}=0.$

Otherwise, we have

$a_{M,\zeta}=0,\quad
c_{M,\zeta}=\frac{{\mathrm{GV}}_{d}^{(3)}}{r^{2}d^{2}},\quad
f_{M,\zeta}=\frac{-2{\mathrm{GV}}_{d}^{(3)}}{r^{3}d^{3}}.$

###### Proof.

The following expansion will be used.

(3.4)
$\begin{split}\frac{1}{1-qL}=&\frac{1-(\zeta^{-1}L-1)+(\zeta^{-1}L-1)^{2}}{1-\zeta
q}\\\ &\quad+\frac{(\zeta^{-1}L-1)-2(\zeta^{-1}L-1)^{2}}{(1-\zeta
q)^{2}}+\frac{(\zeta^{-1}L-1)^{2}}{(1-\zeta q)^{3}},\\\
\frac{1}{1-q}=&\frac{1}{1-\zeta^{-1}}-\frac{\zeta^{-1}}{(1-\zeta^{-1})^{2}}(1-\zeta
q)\\\ &\quad+\frac{\zeta^{-2}}{(1-\zeta^{-1})^{3}}(1-\zeta q)^{2}+O(1-\zeta
q)^{3}.\end{split}$

Let ${\rm Coeff}(f(x);x^{d})$ be the coefficient of $x^{d}$ in $f(x)$. By
definition,

$\begin{split}a_{rd,\zeta}&=5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{K}(0);Q^{rd}\frac{1-P}{1-\zeta q}\Big{)}\\\
&={\rm
Coeff}\Big{(}\frac{1}{1-q}Q^{rd}(1-P)\langle\frac{(1-P)^{2}+(1-P)^{3}}{1-qL}\rangle^{X_{\zeta}}_{0,1,rd};Q^{rd}\frac{1-P}{1-\zeta
q}\Big{)}\\\ &={\rm Coeff}\Big{(}\frac{1}{1-q}\langle\frac{(1-P)^{2}}{1-\zeta
q}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{1-\zeta q}\Big{)}\\\ &=0.\end{split}$

In the last equality, Lemma 4.3 is used. Similarly for $c_{rd,\zeta}$

$\begin{split}c_{rd,\zeta}&=5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{K}(0);Q^{rd}\frac{(1-P)^{2}}{(1-\zeta
q)^{2}}\Big{)}\\\ &={\rm
Coeff}\Big{(}\frac{1}{1-q}Q^{rd}(1-P)^{2}\langle\frac{(1-P)+(1-P)^{2}}{1-qL}\rangle^{X_{\zeta}}_{0,1,rd};Q^{rd}\frac{(1-P)^{2}}{(1-\zeta
q)^{2}}\Big{)}\\\ &={\rm
Coeff}\Big{(}\frac{1}{1-q}\langle\frac{(\zeta^{-1}L-1)(1-P)}{(1-\zeta
q)^{2}}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{(1-\zeta q)^{2}}\Big{)}\\\
&=\frac{{\mathrm{GV}}^{(3)}_{d}}{r^{2}d^{2}}.\end{split}$

For the last equality, we use Lemma 4.4. Note that the factor $(1-\zeta^{-1})$
will be cancelled by the constant term of $(1-q)^{-1}$. For $f_{rd,\zeta}$

$\begin{split}f_{rd,\zeta}&=5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{K}(0);Q^{rd}\frac{(1-P)^{3}}{(1-\zeta
q)^{3}}\Big{)}\\\ &={\rm
Coeff}\Big{(}\frac{1}{1-q}Q^{rd}(1-P)^{3}\langle\frac{1+(1-P)-(1-P)^{3}}{1-qL}\rangle^{X_{\zeta}}_{0,1,rd};Q^{rd}\frac{(1-P)^{3}}{(1-\zeta
q)^{3}}\Big{)}\\\ &={\rm
Coeff}\Big{(}\frac{1}{1-q}\langle\frac{(\zeta^{-1}L-1)^{2}}{(1-\zeta
q)^{3}}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{(1-\zeta q)^{3}}\Big{)}\\\
&=\frac{-2{\mathrm{GV}}_{d}^{(3)}}{r^{3}d^{3}}.\end{split}$

Lemma 4.5 is used in the last equality.

Lastly, when $M/r$ are not integers, we observe that

$\langle(1-P)^{2}\rangle^{X_{\zeta}}_{0,1,M}=\langle(\zeta^{-1}L-1)(1-P)\rangle^{X_{\zeta}}_{0,1,M}=\langle(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,M}=0$

by Lemma 4.3, Lemma 4.4, Lemma 4.5, and the convention that
${\mathrm{GW}}_{M/r}=0$ if $M/r$ are not integers. Geometrically, this
vanishing reflects the fact that the appearance of $r$-th roots of unity in
$q$-expressions comes from the $mr$-fold covers (for $m\in\mathbb{Z}_{\geq
1}$) in the _stem_ contribution, where the degrees of curve classes are
divisible by $mr$. This completes the proof. ∎

###  3.4. Higher degree terms

Higher degree terms can be computed by induction on Novikov variable.

######  Theorem 3.5.

$\begin{split}&\frac{1}{1-q}J^{{K}}(0)=1+\frac{1}{5}\sum_{r\geq
1}\sum_{\zeta:{\rm ord}(\zeta)=r}(1-P)^{2}\sum_{d\geq
1}Q^{rd}\left(\frac{b_{rd,\zeta}}{1-\zeta
q}+\frac{\frac{1}{r^{2}d^{2}}{\mathrm{GV}}^{(3)}_{d}}{(1-\zeta
q)^{2}}\right)\\\ &+\frac{1}{5}\sum_{r\geq 1}\sum_{\zeta:{\rm
ord}(\zeta)=r}(1-P)^{3}\sum_{d\geq
1}Q^{rd}\left(\frac{{\mathbf{d}}_{rd,\zeta}}{1-\zeta
q}+\frac{e_{rd,\zeta}}{(1-\zeta
q)^{2}}+\frac{\frac{-2{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}}{(1-\zeta
q)^{3}}\right),\end{split}$

where

$\begin{split}b_{rd,\zeta}&={\mathrm{GV}}^{(1)}_{d}+\frac{1}{r^{2}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)},\\\
e_{rd,\zeta}&=\frac{1}{r^{2}}\Big{(}\frac{{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)}+\frac{1}{r^{3}}\Big{(}\frac{3{\mathrm{GV}}_{d}^{(3)}}{d^{3}}\Big{)},\\\
{\mathbf{d}}_{rd,\zeta}&=rd{\mathrm{GV}}^{(-1)}_{d}+{\mathrm{GV}}^{(1)}_{d}+\frac{1}{r^{2}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)}+\frac{1}{r^{3}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{3}}\Big{)}.\end{split}$

###### Proof.

It is a purely algebraic computation assuming Lemma 4.3 to Lemma 4.8, where we
use induction on Novikov variable. For the initial ($Q^{1}$) term, it involves
only fake theory and can be computed using equation 4.1.

The expansions in (3.4) imply that

$\begin{split}b_{rd,\zeta}&=5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{{K}}(0);Q^{rd}\frac{(1-P)^{2}}{1-\zeta
q}\Big{)}\\\ &={\rm
Coeff}\Big{(}\frac{1}{1-q}Q^{rd}(1-P)^{2}\langle\frac{(1-P)+(1-P)^{2}}{1-qL}\rangle^{X_{\zeta}}_{0,1,rd};Q^{rd}\frac{(1-P)^{2}}{1-\zeta
q}\Big{)}\\\ &={\rm
Coeff}\Big{(}\frac{1}{1-q}\langle\frac{(1-P)+(1-P)^{2}-(1-P)(\zeta^{-1}L-1)}{1-\zeta
q}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{1-\zeta q}\Big{)}\\\ &\qquad+{\rm
Coeff}\Big{(}\frac{1}{1-q}\langle\frac{(1-P)(\zeta^{-1}L-1)}{(1-\zeta
q)^{2}}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{1-\zeta q}\Big{)}\\\
&=\frac{1}{1-\zeta^{-1}}\langle(1-P)+(1-P)^{2}-(1-P)(\zeta^{-1}L-1)\rangle^{X_{\zeta}}_{0,1,rd}\\\
&\qquad-\frac{\zeta^{-1}}{(1-\zeta^{-1})^{2}}\langle(1-P)(\zeta^{-1}L-1)\rangle^{X_{\zeta}}_{0,1,rd}\\\
&={\mathrm{GV}}^{(1)}_{d}+\frac{1}{r^{2}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)}.\end{split}$

In the last equality, Lemmas 4.3, 4.4, 4.5, and Lemma 4.7 are used.

For $e_{rd,\zeta}$, we have

$\begin{split}&e_{rd,\zeta}=5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{{K}}(0);Q^{rd}\frac{(1-P)^{3}}{(1-\zeta
q})^{2}\Big{)}\\\ =&{\rm
Coeff}\Big{(}\frac{Q^{rd}(1-P)^{3}}{1-q}\langle\frac{1+(1-P)-(1-P)^{3}}{1-qL}\rangle^{X_{\zeta}}_{0,1,rd};Q^{rd}\frac{(1-P)^{3}}{(1-\zeta
q)^{2}}\Big{)}\\\ =&{\rm
Coeff}\Big{(}\langle\frac{(\zeta^{-1}L-1)+(1-P)(\zeta^{-1}L-1)-2(\zeta^{-1}L-1)^{2}}{(1-q)(1-\zeta
q)^{2}}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{(1-\zeta q)^{2}}\Big{)}\\\
&\qquad+{\rm Coeff}\Big{(}\langle\frac{(\zeta^{-1}L-1)^{2}}{(1-\zeta
q)^{3}}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{(1-\zeta q)^{2}}\Big{)}\\\
=&\frac{1}{1-\zeta^{-1}}\langle(\zeta^{-1}L-1)+(1-P)(\zeta^{-1}L-1)-2(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,rd}\\\
&\qquad-\frac{\zeta^{-1}}{(1-\zeta^{-1})^{2}}\langle(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,rd}\\\
=&\frac{1}{r^{2}}\Big{(}\frac{{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)}+\frac{1}{r^{3}}\Big{(}\frac{3{\mathrm{GV}}_{d}^{(3)}}{d^{3}}\Big{)}.\end{split}$

In the last equality, Lemmas 4.4, 4.5, and 4.6 are used.

Finally, for ${\mathbf{d}}_{rd,\zeta}$

$\begin{split}&\quad{\mathbf{d}}_{rd,\zeta}=5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{{K}}(0);Q^{rd}\frac{(1-P)^{3}}{1-\zeta
q}\Big{)}\\\ &={\rm
Coeff}\Big{(}\frac{1}{1-q}Q^{rd}(1-P)^{3}\langle\frac{1+(1-P)-(1-P)^{3}}{1-qL}\rangle^{X_{\zeta}}_{0,1,rd};Q^{rd}\frac{(1-P)^{3}}{1-\zeta
q}\Big{)}\\\ &={\rm
Coeff}\Big{(}\langle\frac{[1-(\zeta^{-1}L-1)+(\zeta^{-1}L-1)^{2}]+(1-P)[1-(\zeta^{-1}L-1)]}{(1-q)(1-\zeta
q)}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{1-\zeta q}\Big{)}\\\ &+{\rm
Coeff}\Big{(}\langle\frac{(\zeta^{-1}L-1)+(1-P)(\zeta^{-1}L-1)-2(\zeta^{-1}L-1)^{2}}{(1-q)(1-\zeta
q)^{2}}\\\ &\qquad\qquad+\frac{(\zeta^{-1}L-1)^{2}}{(1-q)(1-\zeta
q)^{3}}\rangle^{X_{\zeta}}_{0,1,rd};\frac{1}{1-\zeta q}\Big{)}\\\
&=\frac{1}{1-\zeta^{-1}}\langle
1+(1-P)-(\zeta^{-1}L-1)-(1-P)(\zeta^{-1}L-1)+(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,rd}\\\
&-\frac{\zeta^{-1}}{(1-\zeta^{-1})^{2}}\langle(\zeta^{-1}L-1)+(1-P)(\zeta^{-1}L-1)-2(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,rd}\\\
&-\frac{\zeta^{-2}}{(1-\zeta^{-1})^{3}}\langle(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,rd}\\\
&=rd{\mathrm{GV}}^{(-1)}_{d}+{\mathrm{GV}}^{(1)}_{d}+\frac{1}{r^{2}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{2}}\Big{)}+\frac{1}{r^{3}}\Big{(}\frac{-{\mathrm{GV}}_{d}^{(3)}}{d^{3}}\Big{)}.\end{split}$

For the last equality, Lemmas 4.4, 4.5, 4.6, 4.7, and 4.8 are used. ∎

###  3.5. Conclusion of the proof

We have shown that

(3.5) $\begin{split}&\frac{1}{1-q}J^{K}(0)\\\
=&1+\frac{(1-P)^{2}}{5}\sum_{r\geq 1}\sum_{\zeta:{\rm
ord}(\zeta)=r}\sum_{d\geq
1}Q^{rd}\left(\frac{{\mathrm{GV}}_{d}^{(1)}-\frac{1}{r^{2}d^{2}}{\mathrm{GV}}^{(3)}_{d}}{1-\zeta
q}+\frac{\frac{1}{r^{2}d^{2}}{\mathrm{GV}}^{(3)}_{d}}{(1-\zeta
q)^{2}}\right)\\\ &+\frac{(1-P)^{3}}{5}\sum_{r\geq 1}\sum_{\zeta:{\rm
ord}(\zeta)=r}\sum_{d\geq 1}Q^{rd}\cdot\\\
&\cdot\left(\frac{rd{\mathrm{GV}}_{d}^{(-1)}+{\mathrm{GV}}_{d}^{(1)}-\frac{{\mathrm{GV}}_{d}^{(3)}}{r^{2}d^{2}}-\frac{{\mathrm{GV}}_{d}^{(3)}}{r^{3}d^{3}}}{1-\zeta
q}+\frac{\frac{{\mathrm{GV}}_{d}^{(3)}}{r^{2}d^{2}}+\frac{3{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}}{(1-\zeta
q)^{2}}+\frac{\frac{-2{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}}{(1-\zeta
q)^{3}}\right).\end{split}$

######  Theorem 3.6.

Equation 3.5 is equivalent of Conjecture 1.1.

###### Proof.

We use the following expansions

$\begin{split}\frac{1}{1-q^{M}}&\equiv_{q=\zeta^{-1}}\frac{1}{M(1-\zeta
q)-\binom{M}{2}(1-\zeta q)^{2}+\binom{M}{3}(1-\zeta q)^{3}-\dots}\\\
&=\frac{1}{M(1-\zeta q)}+\frac{M-1}{2M}+\frac{M^{2}-1}{12M}(1-\zeta
q)+O((1-\zeta q)^{2})\\\
\frac{1}{(1-q^{M})^{2}}&\equiv_{q=\zeta^{-1}}\frac{1}{M^{2}(1-\zeta
q)^{2}}+\frac{M-1}{M^{2}}\frac{1}{1-\zeta q}+\mbox{ regular terms}\\\
\frac{1}{(1-q^{M})^{3}}&\equiv_{q=\zeta^{-1}}\frac{1}{M^{3}(1-\zeta
q)^{3}}+\frac{3(M-1)}{2M^{3}(1-\zeta q)^{2}}+\frac{2M^{2}-3M+1}{2M^{3}(1-\zeta
q)}\\\ &\qquad+\mbox{ regular terms}.\end{split}$

Start with the coefficient of $(1-P)^{2}Q^{M}$ in the expression of Conjecture
1.1. Let $\zeta$ be any $r$-th roots of unity.

$\begin{split}&5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{{K}}(0);(1-P)^{2}Q^{M}\Big{)}\\\
&=\sum_{k|M}\left(\frac{\frac{M}{k}(k-1)}{1-q^{k}}+\frac{\frac{M}{k}}{(1-q^{k})^{2}}\right){\mathrm{GV}}_{M/k}\\\
&\equiv_{q=\zeta^{-1}}\sum_{k|\frac{M}{r}}\left(\frac{\frac{M}{rk}(rk-1)}{1-q^{rk}}+\frac{\frac{M}{rk}}{(1-q^{rk})^{2}}\right){\mathrm{GV}}_{M/rk}+\mbox{
regular terms}\\\
&\equiv_{q=\zeta^{-1}}\sum_{k|\frac{M}{r}}\left(\frac{\frac{M}{rk}(rk-1)}{rk(1-\zeta_{r}q)}+\frac{\frac{M}{rk}(rk-1)}{(rk)^{2}(1-\zeta_{r}q)}+\frac{\frac{M}{rk}}{((rk)^{2}(1-\zeta_{r}q)^{2}}\right){\mathrm{GV}}_{M/rk}\\\
&\qquad+\mbox{ regular terms}\\\
&\equiv_{q=\zeta^{-1}}\left(\frac{{\mathrm{GV}}_{M/r}^{(1)}-\frac{1}{M^{2}}{\mathrm{GV}}_{M/r}^{(3)}}{1-\zeta
q}+\frac{\frac{1}{M^{2}}{\mathrm{GV}}^{(3)}_{M/r}}{(1-\zeta
q)^{2}}\right)+\mbox{ regular terms}.\end{split}$

This coincides with the equation 3.5 by taking $M=rd$.

Similarly for the coefficient of $(1-P)^{3}Q^{M}$

$\begin{split}&5\,{\rm
Coeff}\Big{(}\frac{1}{1-q}J^{{K}}(0);(1-P)^{3}Q^{M}\Big{)}\\\
&=\sum_{k|M}\left(\frac{M+k^{2}-\frac{M}{k}-1}{1-q^{k}}+\frac{\frac{M}{k}+3}{(1-q^{k})^{2}}-\frac{2}{(1-q^{k})^{3}}\right){\mathrm{GV}}_{M/k}\\\
&\equiv_{q=\zeta^{-1}}\sum_{k|\frac{M}{r}}\left(\frac{M+(rk)^{2}-\frac{M}{rk}-1}{1-q^{rk}}+\frac{\frac{M}{rk}+3}{(1-q^{rk})^{2}}-\frac{2}{(1-q^{rk})^{3}}\right){\mathrm{GV}}_{M/rk}\\\
&\qquad+\mbox{ regular terms}\\\
&\equiv_{q=\zeta^{-1}}\sum_{k|\frac{M}{r}}\left(\frac{{\mathrm{GV}}_{M/r}}{(1-\zeta_{r}q)}\Big{(}(rk)+\frac{M}{rk}+\frac{-M-1}{(rk)^{3}}\right.\Big{)}\\\
&\qquad\left.+\frac{{\mathrm{GV}}_{M/r}}{(1-\zeta_{r}q)^{2}}\Big{(}\frac{M+3}{(rk)^{3}}\Big{)}+\frac{{\mathrm{GV}}_{M/r}}{(1-\zeta_{r}q)^{3}}\Big{(}\frac{-2}{(rk)^{3}}\Big{)}\right)+\mbox{
regular terms}\\\
&\equiv_{q=\zeta^{-1}}\sum_{k|\frac{M}{r}}\left(\frac{M{\mathrm{GV}}_{M/r}^{(-1)}+{\mathrm{GV}}_{M/r}^{(1)}-\frac{1}{M^{2}}{\mathrm{GV}}_{M/r}^{(3)}-\frac{1}{M^{3}}{\mathrm{GV}}_{M/r}^{(3)}}{1-\zeta_{r}q}\right.\\\
&\qquad\left.+\frac{\frac{{\mathrm{GV}}_{M/r}^{(3)}}{M^{2}}+\frac{3{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}}{(1-\zeta_{r}q)^{2}}+\frac{\frac{-2{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}}{(1-\zeta_{r}q)^{3}}\right)+\mbox{
regular terms}.\end{split}$

This coincides with the equation 3.5 by taking $M=rd$. We finish the proof.

∎

##  4\. Some computations of contributions from Kawasaki’s strata

In this section, we prove a few identities needed in the proofs in Section 3.

We start with the fake theory:

(4.1) $\begin{split}\frac{1}{1-q}J^{{\rm
fake}}(0)=1&+\frac{1}{5}(1-P)^{2}\sum_{d}Q^{d}\frac{d{\mathrm{GW}}_{d}}{(1-q)^{2}}\\\
&+\frac{1}{5}(1-P)^{3}\sum_{d}Q^{d}\left(\frac{(3+d){\mathrm{GW}}_{d}}{(1-q)^{2}}-\frac{2{\mathrm{GW}}_{d}}{(1-q)^{3}}\right),\end{split}$

see Theorem 2.8 and reference therein for more detail.

For the stem theory, let $\zeta$ be a primitive $r$-th roots of unity. We
prove the following statements about the invariants on Kawasaki’s strata with
$r\geq 2$ in the following order. Same formulas hold true when $r=\zeta=1$.

$\langle(1-P)^{2}\rangle^{X_{\zeta}}_{0,1,M}=0.$
$\langle(\zeta^{-1}L-1)(1-P)\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})\Big{(}\frac{{\mathrm{GV}}^{(3)}_{M/r}}{M^{2}}\Big{)}.$
$\langle(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})\Big{(}\frac{-2{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}.$
$\langle\zeta^{-1}L-1\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})\Big{(}\frac{-{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}-\zeta^{-1}\Big{(}\frac{2{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}.$
$\langle
1-P\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1}){\mathrm{GV}}^{(1)}_{M/r}+\zeta^{-1}\Big{(}\frac{{\mathrm{GV}}^{(3)}_{M/r}}{M^{2}}\Big{)}.$
$\langle
1\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})M{\mathrm{GV}}^{(-1)}_{M/r}+\zeta^{-1}\Big{(}\frac{{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}.$

Here ${\mathrm{GV}}^{(\gamma)}_{d}:=\sum_{k|d}k^{\gamma}{\mathrm{GV}}_{k}$. If
$(M/r)$ are not integers, ${\mathrm{GV}}^{(\gamma)}_{M/r}$ are set to 0 by
convention.

Recall the notation in (2.1)

$\begin{split}&\Big{[}T_{1}(L),T(L),\dots,T(L),T_{n+2}(L)\Big{]}^{X_{\zeta}}_{0,n+2,d}\\\
:=&\int_{[\overline{\mathcal{M}}_{0,n+2,d}^{X}(\zeta)]^{{\rm vir}}}{\rm
td}(T_{\overline{\mathcal{M}}}){\rm ch}\left(\frac{{\rm
ev}_{1}^{*}(T_{1}(L)){\rm ev}_{n+2}^{*}(T_{n+2}(L))\prod_{i=2}^{n+1}{\rm
ev}_{i}^{*}T(L)}{{\rm
Tr}(\Lambda^{*}N^{*}_{\overline{\mathcal{M}}})}\right).\end{split}$

This can be interpreted as _twisted_ (cohomological) GW invariants with the
twisting class

${\rm td}(T_{\overline{\mathcal{M}}}){\rm ch}\Big{(}\frac{1}{{\rm
Tr}(\Lambda^{*}N^{*}_{\overline{\mathcal{M}}})}\Big{)}.$

Such twisting classes come from the deformation theory of the moduli of stable
maps and consist of three parts. See e.g., [GT14, Section 8].

  1. type $A$.

$\displaystyle{\rm td}(\pi_{*}^{K}{\rm ev}^{*}(TX))\prod_{k=1}^{r-1}{\rm
td}_{\zeta^{k}}(\pi_{*}^{K}{\rm ev}^{*}(T_{X}\otimes\mathbb{C}_{\zeta^{k}}))$,
where $\pi:\mathcal{C}\rightarrow\overline{\mathcal{M}}$ and ${\rm
ev}:\mathcal{C}\rightarrow X/\mathbb{Z}_{r}$ form the universal (orbifold)
stable map diagram. $\mathbb{C}_{\zeta^{k}}$ is the line bundle over
$B\mathbb{Z}_{r}$ with $g$ acts by $\zeta^{k}$, and for any line bundle $l$,
define the invertible multiplicative characteristic classes

${\rm td}(l):=\frac{c_{1}(l)}{1-e^{-c_{1}(l)}},\quad{\rm
td}_{\lambda}(l):=\frac{1}{1-\lambda e^{-c_{1}(l)}}$

  2. type $B$.

$\displaystyle{\rm td}(\pi_{*}^{K}(-L^{-1}))\prod_{k=1}^{r-1}{\rm
td}_{\zeta^{k}}(\pi_{*}^{K}(-L^{-1}\otimes{\rm
ev}^{*}(\mathbb{C}_{\zeta^{k}})))$, where $L=L_{n+3}$ is the universal
cotangent line bundle of

$\mathcal{C}\cong\overline{\mathcal{M}}_{0,n+3}^{X/\mathbb{Z}_{r},d}(g,1,\dots,1,g^{-1},1).$

  3. type $C$.

$\displaystyle{\rm td}^{\vee}(-\pi_{*}^{K}i_{*}\mathcal{O}_{Z_{g}}){\rm
td}^{\vee}(-\pi_{*}^{K}i_{*}\mathcal{O}_{Z_{1}})\prod_{i=1}^{k-1}{\rm
td}^{\vee}_{\zeta^{k}}(-\pi_{*}^{K}i_{*}\mathcal{O}_{Z_{1}})$, where $Z_{1}$
stands for unramified nodal locus, and $Z_{g}$ stands for ramified one with
$i:Z\rightarrow\mathcal{C}$ the embedding of nodal locus. For any line bundle
$l$,

${\rm td}^{\vee}(l)=\frac{-c_{1}(l)}{1-e^{c_{1}(l)}},\quad{\rm
td}^{\vee}_{\lambda}(l)=\frac{1}{1-\lambda e^{c_{1}(l)}}.$

We start with the following two observations.

######  Lemma 4.1.

${\rm td}(T_{\overline{\mathcal{M}}}){\rm ch}\Big{(}\frac{1}{{\rm
Tr}(\Lambda^{*}N^{*}_{\overline{\mathcal{M}}})}\Big{)}=:r^{-n}+T_{0,n+2,d}(\zeta),$

where $T_{0,n+2,d}(\zeta)\in
H^{>0}(\overline{\mathcal{M}}^{X}_{0,n+2,d}(\zeta))$, with $0$, $n$, $d$, and
$\zeta$ inherited from
$\overline{\mathcal{M}}=\overline{\mathcal{M}}^{X}_{0,n+2,d}(\zeta)$.

###### Proof.

Note that $N_{\overline{\mathcal{M}}}$ has virtual dimension $(r-1)n$ since
$\overline{\mathcal{M}}$ is considered as Kawasaki strata in
$\overline{\mathcal{M}}^{X}_{0,nr+2,d}$.

Only the twisting classes on normal bundle give nontrivial constant. It is of
the following form:

$\prod_{k=1}^{r-1}{\rm td}_{\zeta^{k}}\Big{(}\pi^{K}_{*}(E\otimes{\rm
ev}^{*}(\mathbb{C}_{\zeta^{k}}))\Big{)},$

where $E$ is a virtual bundle of rank $n$. The constant term is given by

$\Big{(}\prod_{k=1}^{r-1}\frac{1}{1-\zeta^{k}}\Big{)}^{n}=r^{-n}.$

∎

######  Lemma 4.2.

For any Calabi-Yau threefolds $X$, if $\deg_{\mathbb{C}}\phi_{1}\geq 2$ then

$\langle\tau_{k_{1}}(\phi_{1}),\dots,\tau_{k_{n}}(\phi_{n})\rangle_{g,n,\beta\neq
0}^{\mathrm{tw}}=0,$

where $\mathrm{tw}$ denotes cohomological GW invariants with twistings by any
combinations of the three types defined above.

###### Proof.

This well-known fact has many different proofs. All proofs use the fact that
the virtual dimension of $\overline{\mathcal{M}}_{g,n}(X,\beta)$ is equal to
$n$.

Let

$\pi_{1}:\overline{\mathcal{M}}_{g,n}(X,\beta)\to\overline{\mathcal{M}}_{g,1}(X,\beta)$

be the forgetful map forgetting the last $n$ marked points and $T\in
H^{*}(\overline{\mathcal{M}}_{g,n}(X,\beta))$ be the twisting class. By
projection formula

$\begin{split}\int_{[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{{\rm vir}}}T\
&\prod_{i=1}^{n+1}\Big{(}\psi_{i}^{k_{i}}{\rm ev}_{i}^{*}\phi_{i}\Big{)}\\\
&=\int_{[\overline{\mathcal{M}}_{g,1}(X,\beta)]^{{\rm vir}}}({\rm
ev}_{1}^{*}\phi_{1})\ (\pi_{1})_{*}\left(T\
\psi_{1}^{k_{1}}\prod_{i=2}^{n+1}\psi_{i}^{k_{i}}{\rm
ev}_{i}^{*}\phi_{i}\right).\end{split}$

Since $[\overline{\mathcal{M}}_{g,1}(X,\beta)]^{{\rm vir}}$ has virtual
dimension 1, while $\deg_{\mathbb{C}}(\phi_{i})\geq 2$. This completes the
proof. ∎

######  Lemma 4.3.

$\langle(1-P)^{2}\rangle^{X_{\zeta}}_{0,1,M}=0.$

###### Proof.

It is a consequence of Lemma 4.2 since $\deg_{\mathbb{C}}{\rm ch}(1-P)^{2}\geq
2$. ∎

The following expression of leg and tail contributions will be used.

(4.2) $\begin{split}{\rm
leg}_{\zeta}(q)&=\Psi^{r}\Big{(}[J^{{K}}_{q=1}]_{+}-(1-q)\Big{)}\\\
&=\Psi^{r}\Big{(}(1-P)^{2}(\cdots)+(1-P)^{3}(\cdots)\Big{)}\\\
\delta_{\zeta}(q)&=(1-\zeta^{-1}q)+{\rm tail}_{\zeta}(\zeta^{-1}q)\\\
&=(1-\zeta^{-1}q)+(1-P)^{2}(\cdots)+(1-P)^{3}\cdots.\end{split}$

######  Lemma 4.4.

$\langle(\zeta^{-1}L-1)(1-P)\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})\Big{(}\frac{{\mathrm{GV}}^{(3)}_{M/r}}{M^{2}}\Big{)}.$

###### Proof.

Proposition 2.10 with input $t=0$ gives:

$\begin{split}&\sum_{d}Q^{d}\langle(\zeta^{-1}L-1)(1-P)\rangle^{X_{\zeta}}_{0,1,d}\\\
&\hskip 28.45274pt=\sum_{n,d}\frac{Q^{rd}}{n!}\Big{[}(L^{1/r}-1)(1-P),{\rm
leg}_{\zeta}(L),\dots,{\rm
leg}_{\zeta}(L),\delta_{\zeta}(L^{1/r})\Big{]}^{X_{\zeta}}_{0,2+n.d}.\end{split}$

We expand ${\rm leg}_{\zeta}(q)$, and $\delta_{\zeta}(q)$ as in equation
(4.2). The invariant vanishes unless the dimension of moduli space and the
degree of classes match. We have:

$\begin{split}&\sum_{d}Q^{d}\langle(\zeta^{-1}L-1)(1-P)\rangle^{X_{\zeta}}_{0,1,d}\\\
=&\sum_{d}Q^{rd}\Big{[}(L^{1/r}-1)(1-P),1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,2,d}\\\
=&(1-\zeta^{-1})\sum_{d}Q^{rd}\int_{[\overline{\mathcal{M}}_{0,2,d}^{X}(\zeta)]^{{\rm
vir}}}{\rm ch}\Big{(}(L_{1}^{1/r}-1){\rm ev}_{1}^{*}(1-P))\Big{)}\\\
=&\frac{(1-\zeta^{-1})}{r}\sum_{d}Q^{rd}\int_{[\overline{\mathcal{M}}_{0,2,d}^{X}(\zeta)]^{{\rm
vir}}}c_{1}(L_{1}){\rm ev}_{1}^{*}(H)\\\
=&\frac{(1-\zeta^{-1})}{r^{2}}\sum_{d}Q^{rd}(d{\mathrm{GW}}_{d})=\frac{(1-\zeta^{-1})}{r^{2}}\sum_{d}Q^{rd}\Big{(}\frac{{\mathrm{GV}}^{(3)}_{d}}{d^{2}}\Big{)}.\end{split}$

The extra power of $r$ in the denominator comes from the virtual fundamental
class. Compare the coefficient of $Q^{M}$ gives the result. ∎

######  Lemma 4.5.

$\langle(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})\Big{(}\frac{-2{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}.$

###### Proof.

The computation is similar to Lemma 4.4. We have:

$\begin{split}&\sum_{d}Q^{d}\langle(\zeta^{-1}L-1)^{2}\rangle^{X_{\zeta}}_{0,1,d}\\\
&=\sum_{d}Q^{rd}\Big{[}(L^{1/r}-1)^{2},1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,2,d}\\\
&=(1-\zeta^{-1})\sum_{d}Q^{rd}\int_{[\overline{\mathcal{M}}_{0,2,d}^{X}(\zeta)]^{{\rm
vir}}}{\rm ch}\Big{(}(L_{1}^{1/r}-1)^{2}\Big{)}\\\
&=\frac{(1-\zeta^{-1})}{r^{2}}\sum_{d}Q^{rd}\int_{[\overline{\mathcal{M}}_{0,2,d}^{X}(\zeta)]^{{\rm
vir}}}c_{1}(L_{1})^{2}\\\
&=\frac{(1-\zeta^{-1})}{r^{3}}\sum_{d}Q^{rd}(-2{\mathrm{GW}}_{d})=\frac{(1-\zeta^{-1})}{r^{3}}\sum_{d}Q^{rd}\Big{(}\frac{-2{\mathrm{GV}}^{(3)}_{d}}{d^{3}}\Big{)}.\end{split}$

Comparing the coefficient of $Q^{M}$ gives the result. ∎

######  Lemma 4.6.

$\langle\zeta^{-1}L-1\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})\Big{(}\frac{-{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}-\zeta^{-1}\Big{(}\frac{2{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}.$

###### Proof.

Proposition 2.10 with input $t=0$ gives:

$\begin{split}&\sum_{d}Q^{d}\langle\zeta^{-1}L-1\rangle^{X_{\zeta}}_{0,1,d}\\\
=&\sum_{n,d}\frac{Q^{rd}}{n!}\Big{[}L^{1/r}-1,{\rm leg}_{\zeta}(L),\dots,{\rm
leg}_{\zeta}(L),\delta_{\zeta}(L^{1/r})\Big{]}^{X_{\zeta}}_{0,2+n,d}\\\
=&\sum_{d}Q^{rd}\Big{[}L^{1/r}-1,1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,2,d}+\Big{[}L^{1/r}-1,{\rm
leg}_{\zeta}(L),1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
=&\sum_{d}Q^{rd}\int_{[\overline{\mathcal{M}}_{0,2,d}^{X}(\zeta)]^{{\rm
vir}}}\Big{(}\frac{c_{1}(L_{1})}{r}+\frac{c_{1}(L_{1})^{2}}{2r^{2}}\Big{)}\cdot\Big{(}(1-\zeta^{-1})-\zeta^{-1}\frac{c_{1}(L_{2})}{r}\Big{)}\\\
=&\sum_{d}Q^{rd}\Big{[}(1-\zeta^{-1})\Big{(}\frac{-{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}\Big{)}-\zeta^{-1}\Big{(}\frac{2{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}\Big{)}\Big{]}.\end{split}$

In the third equality, the computation reduce to 2-pointed and untwisted case.
We use the following observations. The twisting class of types A gives no
contribution since $X$ is Calabi-Yau threefold. The twisting class of types B
gives no contribution since $L$ is trivial on
$\overline{\mathcal{M}}^{X}_{0,2,d}(\zeta)$. Finally, the twisting class of
type C and the term with ${\rm leg}_{\zeta}(L)$ give no contribution when
$d\neq 0$ by Lemma 4.2. When $d=0$, the invariant reduces to the Poincaré
pairing on $X$ and equals zero.

Compare the coefficient of $Q^{M}$ gives the result.

∎

######  Lemma 4.7.

$\langle
1-P\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1}){\mathrm{GV}}^{(1)}_{M/r}+\zeta^{-1}\Big{(}\frac{{\mathrm{GV}}^{(3)}_{M/r}}{M^{2}}\Big{)}.$

###### Proof.

The computation is similar to Lemma 4.6 with nonzero contributions when $d=0$.

Proposition 2.10 with input $t=0$ gives:

$\begin{split}&\sum_{d}Q^{d}\langle 1-P\rangle^{X_{\zeta}}_{0,1,d}\\\
&=\sum_{n,d}\frac{Q^{rd}}{n!}\Big{[}1-P,{\rm leg}_{\zeta}(L),\dots,{\rm
leg}_{\zeta}(L),\delta_{\zeta}(L^{1/r})\Big{]}^{X_{\zeta}}_{0,2+n,d}\\\
&=\sum_{d}Q^{rd}\Big{[}1-P,1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,2,d}+\Big{[}1-P,{\rm
leg}_{\zeta}(L),1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
&=\sum_{d}Q^{rd}\zeta^{-1}\Big{(}\frac{{\mathrm{GV}}^{(3)}_{d}}{r^{2}d^{2}}\Big{)}+\Big{[}1-P,{\rm
leg}_{\zeta}(L),1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
&+\Big{[}1-P,\Psi^{r}\Big{(}\frac{(1-P)^{2}}{5}\sum_{k}Q^{k}\langle
1-P\rangle^{{\rm
fake}}_{0,1,k}\Big{)},1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}\end{split}$

The last term corresponds to the contribution of the twisting of unramified
locus in type $C$. It can be computed as follows:

$\begin{split}&\Big{[}1-P,\Psi^{r}\Big{(}\frac{(1-P)^{2}}{5}\sum_{k}Q^{k}\langle
1-P\rangle^{{\rm
fake}}_{0,1,k}\Big{)},1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
&=\Big{[}1-P,\frac{(1-P^{r})^{2}}{5}\sum_{k}Q^{rk}(k{\mathrm{GW}}_{k}),1-\zeta^{-1}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
&=(1-\zeta^{-1})\sum_{k}Q^{rk}(k{\mathrm{GW}}_{k})=(1-\zeta^{-1})\sum_{k}Q^{rk}\Big{(}\frac{1}{k^{2}}{\mathrm{GV}}^{(3)}_{k}\Big{)}.\end{split}$

Note that the constant $r^{2}$ from $(1-P^{r})^{2}$ is cancelled by the
constant in Lemma 4.1 and the Poincaré pairing on $X/\mathbb{Z}_{r}$.

Assume the expression of $J^{K}$ in Theorem 3.5 up to $Q^{M-1}$, a direct
computation shows that:

$\begin{split}{\rm
leg}_{\zeta}(q)&=\Psi^{r}\Big{(}[J^{K}_{q=1}]_{+}-(1-q)\Big{)}\\\
&=\frac{(1-P^{r})^{2}}{5}\sum_{d<M}Q^{rd}\Big{(}{\mathrm{GV}}_{d}^{1}-\frac{1}{d^{2}}{\mathrm{GV}}^{(3)}_{d}\Big{)}\\\
&+\frac{(1-P^{r})^{3}}{5}\sum_{d<M}Q^{rd}\Big{(}d{\mathrm{GV}}_{d}^{(-1)}+{\mathrm{GV}}_{d}^{(1)}-\frac{{\mathrm{GV}}_{d}^{(3)}}{d^{2}}-\frac{{\mathrm{GV}}_{d}^{(3)}}{d^{3}}\Big{)}\\\
&\hskip 256.0748pt\mbox{(mod $Q^{M+1}$)}\end{split}$

We recover the coefficient of $Q^{M}$ on ${\rm leg}_{\zeta}(q)$ since $r\geq
2$. Now we have

$\begin{split}&{\rm Coeff}\Big{(}\Big{[}1-P,{\rm
leg}_{\zeta}(L),1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0};Q^{M}\Big{)}\\\
&\hskip
60.0pt=(1-\zeta^{-1})\Big{(}{\mathrm{GV}}^{(1)}_{M/r}-\frac{r^{2}}{M^{2}}{\mathrm{GV}}^{(3)}_{M/r}\Big{)}.\end{split}$

Compare the coefficient of $Q^{M}$ gives the result. ∎

######  Lemma 4.8.

$\langle
1\rangle^{X_{\zeta}}_{0,1,M}=(1-\zeta^{-1})M{\mathrm{GV}}^{(-1)}_{M/r}+\zeta^{-1}\Big{(}\frac{{\mathrm{GV}}^{(3)}_{M/r}}{M^{3}}\Big{)}.$

###### Proof.

The computation is similar to Lemma 4.7. Proposition 2.10 with input $t=0$
gives:

$\begin{split}&\sum_{d}Q^{d}\langle 1\rangle^{X_{\zeta}}_{0,1,d}\\\
&=\sum_{n,d}\frac{Q^{rd}}{n!}\Big{[}1,{\rm leg}_{\zeta}(L),\dots,{\rm
leg}_{\zeta}(L),\delta_{\zeta}(L^{1/r})\Big{]}^{X_{\zeta}}_{0,2+n,d}\\\
&=\zeta^{-1}\Big{(}\frac{{\mathrm{GV}}^{(3)}_{d}}{r^{3}d^{3}}\Big{)}+\Big{[}1,{\rm
leg}_{\zeta}(L),1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
&+\Big{[}1,\Psi^{r}\Big{(}\frac{(1-P)^{3}}{5}\sum_{k}Q^{k}\langle
1\rangle^{{\rm
fake}}_{0,1,k}\Big{)},1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}.\end{split}$

Note that

$\begin{split}&\Big{[}1,\Psi^{r}\Big{(}\frac{(1-P)^{3}}{5}\sum_{k}Q^{k}\langle
1\rangle^{{\rm
fake}}_{0,1,k}\Big{)},1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
&=\Big{[}1,\frac{(1-P^{r})^{3}}{5}\sum_{k}Q^{rk}{\mathrm{GW}}_{k},1-\zeta^{-1}\Big{]}^{X_{\zeta}}_{0,3,0}\\\
&=r(1-\zeta^{-1})\sum_{k}Q^{rk}({\mathrm{GW}}_{k})=r(1-\zeta^{-1})\sum_{k}Q^{rk}\Big{(}\frac{1}{k^{3}}{\mathrm{GV}}^{(3)}_{k}\Big{)},\end{split}$

and that

$\begin{split}&{\rm Coeff}\Big{(}\Big{[}1,{\rm
leg}_{\zeta}(L),1-\zeta^{-1}L^{1/r}\Big{]}^{X_{\zeta}}_{0,3,0};Q^{M}\Big{)}\\\
&\hskip
60.0pt=r(1-\zeta^{-1})\Big{(}\frac{M{\mathrm{GV}}^{(-1)}_{M/r}}{r}-\frac{r^{3}}{M^{3}}{\mathrm{GV}}^{(3)}_{M/r}\Big{)},\end{split}$

by assuming the expression of $J^{K}$ in Theorem 3.5 up to $Q^{M-1}$.

Compare the coefficient of $Q^{M}$ gives the result. ∎

##  Appendix A Alternative proof of Theorem 3.4

In this Appendix, we give another proof of Theorem 3.4 using only
reconstruction theorem.

We start with the localization at $q=1$, which will relate to the
reconstruction theorem in cohomological theory.

######  Theorem A.1.

Degree $0$ terms with $\zeta=1$ are given as follows:

$\begin{split}a_{M,1}&=0,\\\ c_{M,1}&=-2GW_{M},\\\
f_{M,1}&=MGW_{M}.\end{split}$

###### Proof.

We localize the reconstruction theorem at $q=1$:

$\begin{split}&\frac{1}{1-q}J^{K}(0)\\\ \equiv_{q=1}^{0}&\sum_{d\geq
0}\Big{(}\frac{1}{1-q}I^{K}_{d}\Big{)}_{q=1}Q^{d}\exp\left(\sum_{k>0}\frac{\sum_{i=1}^{3}\Psi^{k}(\epsilon_{i}(Q))(1-P^{k}q^{kd})^{i}}{k(1-q^{k})}\right)_{q=1}\\\
&\qquad\qquad\cdot\left(\sum_{i=0}^{3}r_{i}(q,Q)(1-Pq^{d})^{i}\right)_{q=1}.\end{split}$

We analyze the $I^{K}$ term, the $\exp$ term and the polynomial term at the
end individually.

######  Lemma A.2.

$\Big{(}\frac{1}{1-q}I^{{K}}\Big{)}_{q=1}\equiv_{q=1}^{0}\Big{(}\frac{1}{-z}I^{H}(0)\Big{)}\Big{|}_{z=-(1-q),H=1-P}$

###### Proof.

Since we consider only the degree $0$ part, it suffices to compute the
following terms in the expansion of $I^{{K}}$:

$\begin{split}\left(\frac{\prod_{r=1}^{5d}(1-q^{r})}{(\prod_{r=1}^{d}(1-q^{r}))^{5}}\right)_{q=1}&\equiv_{q=1}^{0}\frac{(5d)!}{(d!)^{5}}\\\
\Big{(}e_{i}(k)(1-P)^{i}\Big{)}_{q=1}&\equiv_{q=1}^{0}s_{i}(k)\Big{(}\frac{1-P}{1-q}\Big{)}^{i},\end{split}$

where

$s_{i}(k)=\sum_{1\leq j_{1}<\dots<j_{i}\leq
k}\frac{1}{j_{1}}\dots\frac{1}{j_{k}}.$

After replacing all terms in the expansion of $I^{{K}}$ with their degree 0
part in the above computation and the substitution $(1-q)=-z$ and $1-P=H$, we
get

$\begin{split}&\frac{1}{-z}\sum_{d\geq
0}Q^{d}\frac{(5d)!}{(d!)^{5}}\Big{(}1+s_{1}(5d)(\frac{5H}{-z})+e_{2}(5d)(\frac{5H}{-z})^{2}+e_{3}(5d)(\frac{5H}{-z})^{3}\Big{)}\\\
&\Big{(}1-5e_{1}(d)(\frac{H}{-z})+(15e_{1}(d)^{2}-5e_{2}(d))(\frac{H}{-z})^{2}-(35e_{1}(d)^{3}-30e_{1}(d)e_{2}(d))(\frac{H}{-z})^{3}\Big{)},\end{split}$

which is exactly the expansion of $\displaystyle\frac{1}{-z}I^{H}(0)$. This
proves the lemma. ∎

######  Lemma A.3.

$\begin{split}&\exp\left(\sum_{k>0}\frac{\sum_{i=1}^{3}\Psi^{k}(\epsilon_{i}(Q))(1-P^{k}q^{kd})^{i}}{k(1-q^{k})}\right)_{q=1}\equiv_{q=1}^{0}\\\
&\qquad\exp\left(\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))(1-P)}{k(1-q)}\right)\exp\left(\sum_{k>0}-\frac{d\Psi^{k}(T_{1}(Q))}{k}\right)\end{split}$

###### Proof.

For each $k$, we observe that

$\begin{split}\frac{1-P^{k}q^{kd}}{k(1-q^{k})}&\equiv_{q=1}^{0}\frac{(1-P^{k})}{k(1-q^{k})}+\frac{1-q^{kd}}{k(1-q^{k})}\equiv_{q=1}^{0}\frac{(1-P)}{k(1-q)}+\frac{d}{k}\\\
\frac{(1-P^{k}q^{kd})^{i}}{k(1-q^{k})}&\equiv_{q=1}^{0}0,\mbox{ for
$i>1$}.\end{split}$

The lemma follows. ∎

######  Lemma A.4.

$\sum_{i=0}^{3}r_{i}(q,Q)(1-Pq^{d})^{i}\equiv_{q=1}^{0}r_{0}(1,Q)$

###### Proof.

For the degree 0 term of the polynomial, we simply put $q=1$ and $P=1$. ∎

Now we have

$\begin{split}&\frac{1}{1-q}J^{K}(0)\\\
\equiv_{q=1}^{0}&\left(\frac{1}{-z}\sum_{d\geq
0}I^{H}_{d}Q^{d}\exp\Big{(}-\tau(Q)\frac{H}{z}\Big{)}\exp\Big{(}d\tau(Q)\Big{)}\Big{(}c(Q)\Big{)}\right)\Big{|}_{z=-(1-q),H=1-P},\end{split}$

where

$\tau(Q)=-\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))}{k},\quad c(Q)=r_{0}(1,Q).$

The above expression coincides with the reconstruction theorem in
cohomological theory:

$\begin{split}\frac{1}{-z}J^{H}(0)&=\frac{1}{-z}\sum_{d\geq
0}I^{H}_{d}Q^{d}\exp\Big{(}-\tau(Q)\frac{H}{z}\Big{)}\exp\Big{(}d\tau(Q)\Big{)}\Big{(}c(Q)\Big{)}\\\
&=1+\sum_{d\geq
0}Q^{d}\Big{(}\frac{-2{\mathrm{GW}}_{d}H^{2}}{(-z)^{2}}+\frac{d{\mathrm{GW}}_{d}H^{3}}{(-z)^{3}}\Big{)}.\end{split}$

We conclude that the degree 0 term in the localization at $q=1$ coincides with
cohomological computation after the substitution $1-q=-z$ and $1-P=H$.

∎

Let $\zeta$ be a primitive $r$-th roots of unity. We generalize the above
computation to all degree 0 terms.

######  Theorem A.5 (=Theorem 3.4).

All degree 0 terms are given as follows:

$\begin{split}a_{M,\zeta}&=0,\\\ c_{M,\zeta}&=\frac{-2GW_{M/r}}{r^{2}},\\\
f_{M,\zeta}&=\frac{\frac{M}{r}GW_{M/r}}{r^{3}}.\end{split}$

All invariants are defined to be zero if $M/r$ is rational.

###### Proof.

Using the reconstruction theorem again but localize at $q=\zeta^{-1}$, we have

$\begin{split}&\frac{1}{1-q}J^{{K}}(0)\\\
&\equiv_{q=\zeta^{-1}}^{0}\Big{[}\sum_{i\geq
0}I^{{K}}_{ri}(q)Q^{ri}\exp\Big{(}\sum_{k>0}\frac{i\Psi^{rk}(T_{1}(Q))}{k}\Big{)}\Big{]}\exp\Big{(}\sum_{k>0}\frac{\Psi^{rk}(T_{1}(Q))(1-P)}{rk(1-\zeta_{r}q)}\Big{)}\\\
&\left[\sum_{j=0}^{r-1}\pi_{+}(I_{j}^{{K}})(\zeta^{-1})\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))}{k}\sum_{l=0}^{j-1}q^{kl}\Big{)}\Big{(}\sum_{i=0}^{3}r_{i}(\zeta^{-1},Q)(1-\zeta^{-j})^{i}\Big{)}Q^{j}\right].\end{split}$

For the above expression, we use the following observations:

  * •

For $m=ri+j$ with $0\leq j<r$, we have

$\begin{split}I^{{K}}_{m}(q)&\equiv_{q=\zeta^{-1}}^{0}I^{{K}}_{rk}(q)\cdot\pi_{+}(I^{{K}}_{j})(q)\\\
&\equiv_{q=\zeta^{-1}}^{0}I^{{K}}_{rk}(q)\cdot\pi_{+}(I^{{K}}_{j})(\zeta_{r}^{-1}),\end{split}$

where $\pi_{+}:\mathcal{K}\rightarrow\mathcal{K}_{+}$ is the projection map.

  * •

With the same notation, we have

$\begin{split}&\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))(1-P^{k}q^{km})}{k(1-q^{k})}\Big{)}\\\
&\equiv_{q=\zeta^{-1}}^{0}\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))(1-q^{km})}{k(1-q^{k})}\Big{)}\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))\cdot
kq^{km}(1-P)}{k(1-q^{k})}\Big{)}\\\
&\equiv_{q=\zeta^{-1}}^{0}\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))}{k}\sum_{l=0}^{m-1}q^{kl}\Big{)}\exp\Big{(}\sum_{k>0}\frac{\Psi^{rk}(T_{1}(Q))(1-P)}{rk(1-\zeta_{r}q)}\Big{)}\\\
&\equiv_{q=\zeta^{-1}}^{0}\exp\Big{(}\sum_{k>0}\frac{i\Psi^{rk}(T_{1}(Q))}{k}\Big{)}\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))}{k}\sum_{l=0}^{j-1}q^{kl}\Big{)}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\exp\Big{(}\sum_{k>0}\frac{\Psi^{rk}(T_{1}(Q))(1-P)}{rk(1-\zeta_{r}q)}\Big{)}.\end{split}$

Our expression of $\displaystyle\frac{1}{1-q}J^{K}(0)$ fits into the following
lemma:

######  Lemma A.6.

Let $f,g\in\mathcal{K}$, and $h,h^{\prime}\in K[q]$. Assume that

$f\cdot g=1\ ({\rm mod}Q),\quad f\cdot g\cdot h|_{\mathcal{K}_{+}}=1,$

and that

$\Psi^{r}((f)_{q=1})(\zeta q)\cdot\Psi^{r}((g)_{q=1})(\zeta q)\cdot
h^{\prime}(q)\Big{|}_{\mathcal{K}_{+}}\equiv_{q=\zeta^{-1},0}1.$

Then we have

  1. (1)

$h^{\prime}(q)=\Psi^{r}(h)(\zeta q)$.

  2. (2)

$\begin{split}\Psi^{r}((f)_{q=1})(\zeta q)&\Psi^{r}((g)_{q=1})(\zeta
q)h^{\prime}(q)\Big{|}_{\mathcal{K}_{-}}\\\ &=\Psi^{r}((f)_{q=1})(\zeta
q)\Psi^{r}((g)_{q=1})(\zeta
q)\Big{|}_{\mathcal{K}_{-}}\cdot\Psi^{r}(h(1)).\end{split}$

###### Proof.

$h^{\prime}$ is uniquely determined by induction on $Q$ since we assume that
$h^{\prime}\in K[q]$. It suffices to prove that

$\Psi^{r}(h)(\zeta q)=h^{\prime}(q)=\Psi^{r}((h)_{q=1})(\zeta q),$

which follows from $(h)_{q=1}(q)=h(q)$ since $h\in K[q]$.

For the second statement, notice that

$\Psi^{r}(h)(\zeta q)\equiv_{q=\zeta^{-1}}^{0}\Psi^{r}(h(1)).$

∎

We claim that:

$\begin{split}&\sum_{j=0}^{r-1}\pi_{+}(I_{j}^{K})(\zeta^{-1})\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))}{k}\sum_{l=0}^{j-1}q^{kl}\Big{)}\Big{(}\sum_{i=0}^{3}r_{i}(\zeta^{-1},Q)(1-\zeta^{-j})^{i}\Big{)}Q^{j}\\\
=&\Psi^{r}(r_{0}(1,Q)).\end{split}$

We prove the claim by taking $h(q)=r_{0}(q,Q)$ in Lemma A.6. To apply the
lemma, we need the following two observations:

  * •

$\frac{1}{1-q}I^{K}(q)\equiv_{q=\zeta^{-1}}^{0}\Psi^{r}\Big{(}(\frac{1}{1-q}I^{K}(q))_{q=1}\Big{)}(\zeta
q)_{1-P=>\frac{1-P}{r}}.$

The description at the end means that replacing $1-P$ by $\frac{1-P}{r}$. The
results follows from

$\begin{split}\frac{\prod_{k=1}^{5rd}(1-q^{k})}{\prod_{k=1}^{rd}(1-q^{k})^{5}}\equiv_{q=\zeta^{-1}}^{0}\frac{(5d)!}{(d!)^{5}},\quad\frac{1}{1-q^{rd}}\equiv_{q=\zeta_{r}^{-1}}^{0}\frac{1}{rd(1-\zeta
q)}.\end{split}$

  * •

$\begin{split}&\exp\Big{(}\sum_{k>0}\frac{\Psi^{rk}(T_{1}(Q))(1-P)}{rk(1-\zeta_{r}q)}\Big{)}\\\
\equiv_{q=\zeta^{-1}}^{0}&\Psi^{r}\Big{(}\exp\Big{(}\sum_{k>0}\frac{\Psi^{k}(T_{1}(Q))(1-P)}{k(1-q)}\Big{)}\Big{)}_{\begin{subarray}{c}q=>\zeta
q\\\ 1-P=>\frac{1-P}{r}\end{subarray}}.\end{split}$

The description at the end means that replacing $q$ by $\zeta q$ and $1-P$ by
$\frac{1-P}{r}$.

Taking $\displaystyle f=\frac{1}{1-q}I^{K}(q)$ and $\displaystyle
g=\exp\Big{(}\sum_{k>0}\frac{\Psi^{rk}(T_{1}(Q))(1-P)}{rk(1-\zeta q)}\Big{)}$
in the lemma, we conclude the claim. Changing the variables from $1-P$ to
$\frac{1-P}{r}$ does not change any arguments and conclusions in the lemma.

Finally, we get

$\begin{split}\frac{1}{1-q}J^{K}(0)&\equiv_{q=\zeta^{-1}}^{0}\Psi^{r}\Big{(}(\frac{1}{1-q}J^{K}(0))_{q=1}\Big{)}_{\begin{subarray}{c}q=>\zeta
q\\\ 1-P=>\frac{1-P}{r}\end{subarray}}\\\
&=1+\frac{(1-P)^{2}}{r^{2}}\sum_{d\geq
1}Q^{rd}\Big{(}\frac{-2{\mathrm{GW}}_{d}}{(1-\zeta
q)^{2}}\Big{)}+\frac{(1-P)^{3}}{r^{3}}\sum_{d\geq
1}Q^{rd}\Big{(}\frac{d{\mathrm{GW}}_{d}}{(1-\zeta q)^{3}}\Big{)}.\end{split}$

This proves the theorem. ∎

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