In memory of Bumsig Kim

# Gopakumar–Vafa invariants $=$ quantum $K$-invariants
on Calabi–Yau threefolds

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.

The main purpose of this article is to discuss a project relating
Gopakumar–Vafa invariants to quantum $K$-invariants on Calabi–Yau threefolds.
Results in genus zero, including recent and forthcoming works, are reported.

##  1\. Introduction

This is an expository article based on Part I [1] and Part II [2] of our
ongoing project about quantum $K$-theory of Calabi–Yau threefolds (CY3). In
this project, we hope to explore various special properties of _quantum
$K$-theory on Calabi–Yau threefolds_, which have not yet been extensively
studied in the mathematical literature. We chose to start with its relation
with the Gopakumar–Vafa invariants partly because we feel that the
_integrality_ of quantum $K$-theory has not received much attention so far.

Gromov–Witten theory of Calabi–Yau threefolds enjoys various beautiful
properties, among them the _Kodaira–Spencer theory of gravity_ and
_holomorphic anomaly equation_ of Bershadsky, Cecotti, Ooguri and Vafa, the
appearance of (quasi-)modular forms etc.. Some of these remarkable phenomena
should have counterparts in $K$-theory. We hope to explore these properties in
the context of quantum $K$-theory in the future.

###  1.1. Gopakumar–Vafa invariants and Quantum $K$-theory

Among the _integral_ (virtual) enumerative invariants on Calabi–Yau threefolds
(CY3), two will be the foci of this paper. The first, called the
_Gopakumar–Vafa invariants_ (GV) [10, 11], was introduced in theoretical
physics as “new topological invariants on _Calabi–Yau threefolds_ ”, counting
the “numbers of BPS states”. There have been various attempts at giving these
BPS invariants rigorous mathematical definitions. At the moment, there are
still unresolved issues of these definitions. We refer the readers to [17] and
references therein for mathematical definitions of various degrees of
generalities of these invariants. Gopakumar and Vafa also argue that these BPS
invariants and the Gromov–Witten invariants are intimately related by the
following formula

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

which all viable mathematical definitions must satisfy. In fact, this formula
is ultimately the most important test for any geometric definition. The
detailed definitions and discussions can be found in Section 2.2. In this
paper, we will use the above (invertible) relation as the _definition_ of the
Gopakumar–Vafa invariants (in terms of Gromov–Witten invariants). As defined,
it is not at all obvious these invariants are integral. The integrality of
this ad hoc definition was proven by E. Ionel and T. Parker using symplectic
techniques in a remarkable work [12] .

The second is the _quantum $K$-invaraints_ (QK) [5, 15], a $K$-theoretic
variant of the (cohomological) Gromov–Witten invariants (GW). The definition
and references for quantum $K$-theory will be recalled in Section 2.3. Whereas
Gromov–Witten invariants are _rational_ numbers, quantum $K$-invariants,
counting alternating sums of ranks of sheaf cohomology (Euler characteristic),
produces _integral_ invariants by definition. Therefore, any relations
established between the quantum $K$-invariants and Gopakumar–Vafa invariants
will automatically produce the _integrality for the GV_.

###  1.2. GV $=$ QK on CY3?

The question is: Are there any relations between the Gopakumar–Vafa invariants
and quantum $K$-invariants on Calabi–Yau threefolds? We think this is the
case. In fact, we think that one should be able to _define_ the Gopakumar–Vafa
invariants (GV) in terms of the quantum $K$-invariants, similar to (1.1).
While this definition would still be _ad hoc_ , it would have the benefit of
being integral by definition. It would also give an alternative proof of
Ionel–Parker’s integrality theorem within the algebraic category.

Why do we believe such relationship exist? On the conceptual level, there is a
clear link between GV and QK through GW. On the one hand, as pointed out
above, Gopakumar–Vafa invariants and Gromov–Witten invariants are
reconstructible from each other by the Gopakumar–Vafa formula (1.1). On the
other hand, Givental and his collaborators has furnished a clear link between
QK and GW via a virtual orbifold Hirzebruch–Riemann–Roch theorem [9, 7].

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. Many
formulas and results were first guessed based on their cohomological
counterparts. See, e.g., [8, 15]. In [9] and subsequent works [7], A. Givental
and his collaborators completely characterized genus zero quantum $K$-theory
in terms of quantum cohomology. Subsequently, Givental generalizes these
results to permutation equivariant setting and to higher genera [7].

Therefore, there is a relation between quantum $K$-invariants and
Gopakumar–Vafa invariants for Calabi–Yau threefolds via Gromov–Witten
invariants. In this series of papers, we seek to explore these links and to
“codify” the relations between QK and GV. Our starting point is the conjecture
by H. Jockers, P. Mayr [13] and S. Garoufalidis, E. Scheidegger [4] in genus
zero. We will formulate a precise version of the conjecture for all Calabi–Yau
threefolds in genus zero based on their works.

The Gopakumar–Vafa formula (1.1), together with results in Section 3.1, can be
interpreted as the following statement: _in genus zero_ , the collection of
all Gopakumar–Vafa invariants of a fixed Calabi–Yau threefold contains exactly
the same information as the collection of all Gromov–Witten invariants. We
hope that in the future to study the higher genus counterpart. We will offer
some heuristic arguments why this might be possible if higher genus
computations can be done for $(-1,-1)$ curves, the so-called multiple cover
formula.

### Acknowledgements

We wish to thank A. Givental, R. Pandharipande, H. Tseng and Y. Wen for
discussions about this work. The research is partially supported by the Simons
Foundation, the NSTC, University of Utah and Academia Sinica.

##  2\. GW, GV and QK

###  2.1. 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

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

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 $\phi_{1},\dots,\phi_{n}\in H(X)$,
$k_{1},\dots,k_{n}\in\mathbb{Z}_{\geq 0}$, and
$[\overline{\mathcal{M}}_{g,n}(X,\beta)]^{{\rm vir}}$ are the (cohomological)
virtual fundamental classes.

The genus $g$ Gromov–Witten invariants of $X$ can be encoded in a generating
function, called genus-$g$ descendant potential

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

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

$t(q)=\sum_{k\in\mathbb{Z}_{\geq
0}}\sum_{\alpha=1}^{N}t_{k}^{\alpha}\phi_{\alpha}q^{k}$

with $\\{\phi_{\alpha}\\}_{\alpha=1}^{N}$ a basis of $H(X)$.

###  2.2. Gopakumar–Vafa invariants

In theoretical physics, R. Gopakumar and C. Vafa in [10, 11] introduced new
topological invariants on _Calabi–Yau threefolds_ (CY3) $X$, which are now
commonly 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 their 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 the dilaton equation, all Gromov–Witten
invariants can be easily reconstructed from $0$-pointed invariants

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

In fact, a _closed formula_ was obtained in [3] in terms of generating
functions. We will therefore focus on $0$-pointed invariants
${{\mathrm{GW}}}_{g,\beta}$.

There have been various attempts at defining Gopakumar–Vafa invariants
mathematically. We refer the readers to [17] and references therein. If one is
interested in defining Gopakumar–Vafa invariants to allow for insertions, the
proposed generalizations are expected to be compatible with the dilaton and
divisor equations. Therefore, the counting of the BPS states is again reduced
to similarly defined ${{\mathrm{GV}}}_{g,\beta}$.

A remarkable relation between GV and GW in [10, 11] can be expressed in terms
of generating functions:

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

We note that this formula gives an invertible relation between GV and GW,
filtered by genus. Namely, one can obtain all
$\\{{\mathrm{GV}}_{g,\beta}\\}_{g\leq g_{0},\beta}$ from
$\\{{\mathrm{GW}}_{g,\beta}\\}_{g\leq g_{0},\beta}$ and vice versa. In this
paper, we use this formula to _define_ the Gopakumar–Vafa invariants.

######  Example 2.1.

For the quintic CY3, the above relation can be written as

(2.2)
$\begin{split}{{\mathrm{GW}}}_{g=0,\beta=d[\operatorname{line}]}=:{{\mathrm{GW}}}_{d}&=\sum_{e|d}\frac{1}{e^{3}}{{\mathrm{GV}}}_{d/e},\\\
{{\mathrm{GV}}}_{g=0,\beta=d[\operatorname{line}]}=:{{\mathrm{GV}}}_{d}&:=\sum_{e|d}\frac{\mu(e)}{e^{3}}{{\mathrm{GW}}}_{d/e},\end{split}$

where we have used the Möbius inversion and $\mu(e)$ is the Möbius function.
For our purpose, we will _use ( 2.2) 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 [12].

There is a variant of the Gromov–Witten theory which also produces _integral_
invariants, namely, the _quantum $K$-theory_ [5, 15]. This leads to the
possibility of relating quantum $K$-invariants with Gopakumar–Vafa invariants
for Calabi–Yau threefolds.

###  2.3. Quantum $K$-invariants

The formulation of _quantum $K$-theory_ is similar to that of quantum
cohomology, i.e., Gromov–Witten theory. The _quantum $K$-invariants_, or
$K$-theoretic Gromov-Witten 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
$\overline{\mathcal{M}}_{g,n}(X,\beta)$ [15]. As in Gromov–Witten theory, all
genus $g$ invariants can be packed into a formal power series, called
genus-$g$ descendant potential of $X$:

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

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 $F^{{K}}_{g}(t)$ as

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

##  3\. Quantum $K$-theory on Calabi–Yau threefolds

###  3.1. GW on CY3

Before we proceed to quantum $K$-theory for the Calabi–Yau threefolds, we
first discuss some relevant statements in Gromov–Witten theory. In the
following, we use $\deg_{\mathbb{C}}$ for the Chow degree, i.e., one half of
the usual degree in cohomology.

######  Proposition 3.1.

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

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

where $\mathrm{tw}$ denotes cohomological GW invariants with twistings.

###### Proof.

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

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

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. ∎

######  Proposition 3.2.

All descendant Gromov–Witten invariants on a Calabi–Yau threefold $X$ can be
reconstructed from $0$-pointed invariants
$\\{\langle\cdot\rangle^{H}_{g,0,\beta}\\}_{g,\beta}$.

###### Proof.

This follows from a combination of virtual dimension counts, the string
equation, dilaton equation and divisor equation. A closed formula is available
in [3, Proposition 1.6]. ∎

###  3.2. QK on CY3

In quantum $K$-theory, things are somewhat different, mostly due to the fact
that $K$-theory is more sensitive to the stack structures. For example, let
$G$ be a finite group acting on $X$ and $\pi:X\to[X/G]$ the $G$-torsor. In
cohomology or Chow,

$H([X/G],\mathbb{Q})=H(X,\mathbb{Q})$

and

$\int_{[X/G]}\alpha=\frac{1}{|G|}\int_{X}\pi^{*}\alpha.$

However, in $K$-theory,

$K([X/G])=K_{G}(X)$

is the $G$-equivariant $K$-theory, which is much richer than $K(X)$.
Furthermore, the pushforward of a vector bundle to a point for $X$ is the
Euler characteristic of the bundle, while the same operation for $[X/G]$
extracts the $G$-invariant part of the sheaf cohomologies, which is much more
intricate.

Nevertheless, one can apply the virtual Hirzebruch–Riemann–Roch theorem on
Deligne–Mumford stacks, [14, 18] which we briefly recall.

Let $M$ be a quasi-smooth DM stack, i.e., a virtual orbifold,

$IM=\sqcup_{i}M_{i}$

its inertia stack and $N_{i}^{{\rm vir}}$ the virtual normal bundle. Let
$\lambda_{-1}(N_{i}^{{\rm vir}})^{*}$ be the “$K$-theoretic Euler class” of
$(N_{i}^{{\rm vir}})^{*}$. For example, if $(N_{i}^{{\rm vir}})^{*}$
decomposes into a direct sum of line bundles $L_{\alpha}$, then

$\lambda_{-1}(N_{i}^{{\rm vir}})^{*}=\prod_{\alpha}(1-L_{\alpha}).$

Let $g_{i}$ be the generic automorphism element of $M_{i}$. For every vector
bundle $E$ on $M_{i}$, $g_{i}$ acts on $E$. Let $E=\oplus_{j}E_{j}$ be an
eigenbundle decomposition with eigenvalues $\epsilon_{j}$ on $E_{j}$. Denote

$\operatorname{Tr}(E):=\oplus_{j}\epsilon_{j}E_{j}.$

Let $F$ be a vector bundle on $M$, and $\tau(F)\in H(IM)$ defined by

$\tau(F)|_{M_{i}}:=\frac{\operatorname{ch}(\operatorname{Tr}F)}{\operatorname{ch}\operatorname{Tr}(\lambda_{-1}(N_{i}^{{\rm
vir}})^{*})}Td(T^{{\rm vir}}_{M_{i}}).$

The virtual HRR states that

(3.2) $\chi(M,F)=\sum_{i}\frac{1}{m_{i}}\int_{[M_{i}^{{\rm
vir}}]}\tau(F)|_{M_{i}},$

where $m_{i}$ is the order of the generic automorphism associated to $M_{i}$.
We now prove a $K$-theoretic version of Proposition3.1.

######  Proposition 3.3 ([2]).

Let $E_{1}\in K(X)$ be any element such that
$\deg_{\mathbb{C}}\operatorname{ch}(E_{1})\geq 2$. Then

$\langle\tau_{k_{1}}(E_{1}),\ldots,\tau_{k_{n}}(E_{n})\rangle^{K}_{g,n,\beta}=0.$

###### Proof.

The proof is a combination of the virtual orbifold Hirzebruch–Riemann–Roch
theorem (3.2) and the arguments in the proof of Proposition 3.1, including the
virtual dimension count and the projection formula. Here two key facts are
used. First, ${\rm ev}^{*}E_{1}$ is pulled back from a scheme and

(3.3) $\operatorname{Tr}({\rm ev}^{*}E_{1})={\rm ev}^{*}E_{1}.$

Second, the virtual dimension of any non-identity component of the intertial
stack is no greater than the virtual dimension of the identity component. More
precisely,

$\begin{split}&\chi\left(\overline{\mathcal{M}}_{g,n}(X,\beta),\prod_{j=1}^{n}L_{j}^{\otimes
k_{j}}{\rm ev}_{j}^{*}E_{j}\right)\\\
=&\sum_{i}\frac{1}{m_{i}}\int_{[(M_{n})_{i}^{{\rm
vir}}]}\tau(\prod_{j=1}^{n}L_{j}^{\otimes k_{j}}{\rm ev}_{j}^{*}E_{j})\\\
=&\sum_{i}\frac{1}{m_{i}}\int_{[(M_{1})_{i}]^{{\rm
vir}}}\operatorname{ch}({\rm
ev}_{1}^{*}E_{1})(\pi_{1})_{*}\left(\tau(L_{1}^{k_{1}}\prod_{j=2}^{n}L_{j}^{k_{j}}{\rm
ev}_{j}^{*}E_{j})\right)\\\ =&0,\end{split}$

where we have used the fact that the virtual dimensions of all $(M_{1})_{i}$
are less than or equal to $1$ and $\deg_{\mathbb{C}}\operatorname{ch}({\rm
ev}_{1}^{*}E_{1})\geq 2$. ∎

At this moment, we do not have a general result in $K$-theory corresponding to
Proposition 3.2. Nevertheless, in genus zero there is a reconstruction theorem
from the set of all one-pointed descendants (i.e., including the cotangent
line bundles) to all quantum $K$-invariants with descendants, in the spirit of
the reconstruction theorem in [16]. The small $J$-function is a generating
function of one-pointed descendants

$J(q,Q)\ :=(1-q)+\sum_{\alpha}\sum_{\beta\neq
0}\Phi_{\alpha}\langle\frac{\Phi^{\alpha}}{1-qL}\rangle_{0,1,\beta}Q^{\beta}$

where $\\{\Phi_{\alpha},\Phi^{\alpha}\\}$ are dual classes with respect to the
$K$-theoretic Poincaré pairing

$(\Phi_{\alpha},\Phi^{\alpha^{\prime}}):=\chi(X,\Phi_{\alpha}\otimes\Phi^{\alpha^{\prime}})=\delta_{\alpha}^{\alpha^{\prime}}.$

######  Proposition 3.4 ([2]).

The small $J$-function in quantum $K$-theory determines all genus zero
(descendant) quantum $K$-invariants for Calabi–Yau threefolds.

###### Proof.

This can be shown by a combination the string equation in the quantum
$K$-theory [15], a reconstruction theorem in quantum $K$-theory [15, (22)
(23)], a version of Riemann–Roch for virtually smooth stacks, together with
virtual dimension counts.

By the string equation, if any insertion is $1=\mathcal{O}$, then it can be
reduced to $K$-invariants of fewer points. One can assume that there is no
insertion of $1$ by induction. We can therefore assume that the insertions at
all marked point look like

$({\rm ev}_{i}^{*}(E_{i})-r_{i}1)$

where $E_{i}$ are vector bundles of rank $r_{i}$. The Chern character of the
$i$-th insertion starts at $\deg_{\mathbb{C}}\geq 1$, otherwise the insertion
is $1$, contradictory to the assumption.

One can now apply the Hirzebruch–Riemann–Roch for the virtually smooth stacks.
Since the $K$-classes pulled back by evaluations are acted trivially by
automorphisms, we conclude that only $\prod_{i}c_{1}({\rm ev}_{i}^{*}(E_{i}))$
contribute to stacky HRR by virtual dimension counting. Therefore, the quantum
$K$-invariants remain unchanged if we replace $({\rm
ev}_{i}^{*}(E_{i})-r_{i}1)$ by the corresponding _line bundles_ $({\rm
ev}_{i}^{*}(\bigwedge^{\operatorname{top}}E_{i})-1)$. That is, one can assume
that the insertions are all linear combinations of line bundles.

Then the reconstruction theorem in quantum $K$-theory in [16] applies: any
descendant quantum $K$-invariants of insertions by line bundles can be
reconstructed from the small $J$-function in the quantum $K$-theory. More
precisely, all cotangent lines can be moved to the first marked points by
induction. Assuming that there are no cotangent line bundles at any but the
first marked point, we may further assume that there are no $1$’s at all other
points after applying the string equation. Therefore the quantum
$K$-invariants look like

${\rm ev}_{1}^{*}(E)\otimes L_{1}^{k}\otimes\prod_{j=2}^{n}({\rm
ev}_{j}^{*}(E_{j})-r_{j}1),$

where $E$ can be arbitrary, e.g., $1$. By Proposition 3.3, ${\rm
ev}_{1}^{*}(E)$ can only contribute through its $\operatorname{ch}_{0}$ or
$\operatorname{ch}_{1}$ to the virtual HRR formula. $\operatorname{ch}_{0}$
can be absorbed by the string equation and

$c_{1}({\rm
ev}_{i}^{*}(E_{i})-r_{i}1)=c_{1}(\bigwedge^{\operatorname{top}}E_{i})-1).$

Thus, vector bundle can be replaced by line bundles without changing the
quantum $K$-invariants. The reconstruction theorem in [16] applies. ∎

##  4\. Multiple cover formula and the JMGS conjecture

###  4.1. GV $=$ QK in genus zero

In [13] and [4] 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. We generalize their conjecture to
all Calabi–Yau threefolds, which should be implicitly in the original proposal
by Jockers and Mayr.

We fix the following notation:

$\\{\Phi_{\alpha}\\}_{\alpha=1}^{N}=\bigsqcup_{i=0}^{3}\\{\Phi_{ij}\\}_{j=1}^{n_{i}},$

where $\\{{\rm ch}(\Phi_{ij})\\}_{j=1}^{n_{i}}$ forms a basis in $H^{2i}(X)$.
In particular,

$\\{\Phi_{0j}\\}_{j=1}^{n_{0}}=\\{\Phi_{01}=\mathcal{O}\\}.$

Let $\\{\Phi^{ij}\\}$ be the dual basis of $\\{\Phi_{ij}\\}$ with respect to
the K-theoretic Poincaré pairing:

$(\Phi_{a},\Phi_{b})^{K}:=\chi(X,\Phi_{a}\Phi_{b}).$

The following fact will be used later.

######  Lemma 4.1.

For $i=0$ and $1$,

${\rm ch}(\Phi^{ij})\in H^{3-i}(X).$

For $i=2$ and $3$,

${\rm ch}(\Phi^{ij})\in H^{\geq 3-i}(X).$

###### Proof.

$\Phi^{ij}$ can be written as:

$\Phi^{ij}:={\rm ch}^{-1}\Big{(}{\rm td}(TX)^{-1}PD({\rm
ch}(\Phi_{ij}))\Big{)},$

where PD denotes the Poincaré dual. The lemma follows from the definition of
$\Phi_{ij}$ and that ${\rm td}(TX)^{-1}\in 1+H^{\geq 2}(X)$. ∎

We now generalize the JMGS conjecture to general Calabi–Yau threefolds.

######  Conjecture 4.2 (cf. [13, 4]).

$\begin{split}\frac{1}{1-q}\left[J(q,Q)\right]:=&\frac{1}{1-q}\left[(1-q)+\sum_{\alpha}\sum_{\beta\neq
0}\Phi_{\alpha}\langle\frac{\Phi^{\alpha}}{1-qL}\rangle_{0,1,\beta}Q^{\beta}\right]\\\
=&1+\sum_{\vec{d}\in
H_{2}(X,\mathbb{Z})}\sum_{r=1}^{\infty}\bigg{[}\sum_{j=1}^{n_{1}}\Phi^{1j}\Big{(}\int_{\vec{d}}{\rm
ch}(\Phi_{1j})\Big{)}\,a(r,q^{r})\,{\mathrm{GV}}_{\vec{d}}\,Q^{r\vec{d}}\\\
&\qquad\qquad\qquad+\Phi^{01}b(r,q^{r})\,{\mathrm{GV}}_{\vec{d}}\,Q^{r\vec{d}}\bigg{]},\end{split}$

where

(4.1)
$\begin{split}a(r,q^{r})&=\frac{(r-1)}{1-q^{r}}+\frac{1}{(1-q^{r})^{2}},\\\
b(r,q^{r})&=\frac{r^{2}-1}{1-q^{r}}+\frac{3}{(1-q^{r})^{2}}-\frac{2}{(1-q^{r})^{3}}.\end{split}$

The main result of Part II is a proof of this conjecture.

######  Theorem 4.3 ([2]).

Conjecture 4.2 holds.

######  Remark 4.4.

By Proposition 3.4, Theorem 4.3 implies that the quantum $K$-theory and
Gopakumar–Vafa theory are equivalent for all Calabi–Yau threefolds in genus
zero.

###  4.2. Multiple cover formula

######  Lemma 4.5.

For the total space $X_{-1,-1}$ of $\mathcal{O}(-1)\oplus\mathcal{O}(-1)\to
P^{1}$, the Gopakumar–Vafa invariants $\operatorname{GV}_{0,1}=1$ (genus zero
and degree $1$) and $\operatorname{GV}_{g,d}=0$ otherwise

###### Proof.

This follows from the Gopakumar–Vafa equation (2.1), and the results in
Gromov–Witten theory: for $g=0$ the Voisin–Aspinwall–Morrison formula

$\operatorname{GW}_{0,0,d}=\frac{1}{d^{3}},$

for $g=1$, the BCOV and Graber–Pandharipande formula

$\operatorname{GW}_{1,0,d}=\frac{1}{12d},$

and for $g\geq 2$ the Faber–Pandharipande formula

$\operatorname{GW}_{g,0,d}=\frac{|B_{2g}|d^{2g-3}}{2g\cdot(2g-2)!}.$

∎

In order to consider the small $J$-function of $X_{-1,-1}$ in quantum
$K$-theory, we consider its compatification by the “infinity divisor”. Let

$Y_{-1,-1}:=P_{P^{1}}(\mathcal{O}(-1)\oplus\mathcal{O}(-1)\oplus\mathcal{O}).$

Let $P=\pi^{*}\mathcal{O}(-1)$ with $\pi:Y_{-1,-1}\rightarrow P^{1}$, and
$t=\mathcal{O}(D_{\infty})$ with $D_{\infty}\subset Y_{-1,-1}$, the infinity
divisor. We denote $Q^{r}:=Q^{r\ell}$, where

$\ell:=[P^{1}]\xhookrightarrow{0}Y_{-1,-1}$

is the line class in the “zero section” of the projective bundle. In the
following, we consider the _specialized_ small $I$-function and $J$-function
of $Y_{-1,-1}$ only for the curve classes in the zero section, i.e., multiples
of $\ell$.

######  Lemma 4.6.

The small $I$-function for $Y_{-1,-1}$, with curve classes in the zero
section, is

$I^{Y_{-1,-1}}(q,Q)=(1-q)\left[1+\sum_{r=1}^{\infty}Q^{r}\frac{(1-Pt)^{2}\prod_{m=1}^{r-1}(1-Ptq^{m})^{2}}{(Pt)^{2r}q^{r(r-1)}\prod_{m=1}^{r}(1-Pq^{m})^{2}}\right].$

###### Proof.

This lemma follows from the computations of the $I$-functions for toric
manifolds via fixed point localization by A. Givental and collaborators. See
[9, 7]. Their formula for small $I$-function gives

$I^{Y_{-1,-1}}(q,Q)=(1-q)\left[1+\sum_{r=1}^{\infty}Q^{r}\frac{\prod_{m={-r+1}}^{0}(1-P^{-1}t^{-1}q^{m})^{2}}{\prod_{m=1}^{r}(1-Pq^{m})^{2}}\right].$

A simple manipulation gives the above presentation. ∎

We note that in this case the small $I$-function includes a factor of
$(1-Pt)^{2}$, the $K$-theoretic normal bundle of $P^{1}$ embedded in
$Y_{-1,-1}$. One may think of this as an $I$-function of a toric completion
$P_{P^{1}}(\mathcal{O}(-1)\oplus\mathcal{O}(-1)\oplus\mathcal{O})$, with curve
class in the base $P^{1}$. This small $I$-function is different from the small
$J$-function, as it has poles at $q=0$. (However, it satisfies
$\lim_{q\to\infty}I^{X_{-1,-1}}(q)=0$.) Using this $I$-function one can obtain
the small $J$-function via the “ _generalized_ mirror transform” (also known
as the explicit reconstruction, or Birkhoff factorization).

######  Proposition 4.7.

The small $J$-function for $Y_{-1,-1}$, with curve classes in the zero
section, is

(4.2)
$\begin{split}\frac{1}{1-q}&J^{Y_{-1,-1}}(q,Q)=1+(1-Pt)^{2}\Big{(}1+(1-P)\Big{)}\sum_{r\geq
1}\,Q^{r}a(r,q^{r})\\\ &+(1-Pt)^{2}(1-P)\sum_{r\geq
1}Q^{r}\,b(r,q^{r}),\end{split}$

where $a(r,q^{r}),b(r,q^{r})$ are defined in (4.1).

###### Proof.

The $K$-ring of $Y_{-1,-1}$ has the following presentation

$K(Y_{-1,-1})=\frac{\mathbb{Z}[P,t]}{\left((1-P)^{2},(1-Pt)^{2}(1-t)\right)}.$

We use the relations of $K(Y_{-1,-1})$ to rewrite $I^{Y_{-1,-1}}$ as follows.
Since $(1-Pt)^{2}(1-t)=0$, we have

(4.3) $\begin{split}&(1-Pt)^{2}(1-Pq^{m}t)=(1-Pt)^{2}(1-Pq^{m}[1-(1-t)])\\\
=&(1-Pt)^{2}(1-Pq^{m}).\end{split}$

Therefore,

$\begin{split}&I^{Y_{-1,-1}}-(1-q)\\\
=&(1-q)\sum_{r=1}^{\infty}Q^{r}\frac{(1-Pt)^{2}}{(Pt)^{2r}q^{r(r-1)}(1-Pq^{r})^{2}}\\\
=&(1-q)\sum_{r\geq
1}Q^{r}\left[\Big{(}\sum_{i=1}^{r-1}\frac{i}{q^{r(r-i)}}\Big{)}(1-Pt)^{2}+\Big{(}\sum_{i=1}^{r-1}\frac{i(2r-i+1)}{q^{r(r-i)}}\Big{)}(1-Pt)^{3}\right]\\\
&+(1-q)\Big{(}(1-Pt)^{2}+(1-Pt)^{3}\Big{)}\sum_{r\geq
1}Q^{r}\Big{(}\frac{r-1}{1-q^{r}}+\frac{1}{(1-q^{r})^{2}}\Big{)}\\\
&+(1-q)(1-Pt)^{3}\sum_{r\geq
1}Q^{r}\Big{(}\frac{r^{2}-1}{1-q^{r}}+\frac{3}{(1-q^{r})^{2}}-\frac{2}{(1-q^{r})^{3}}\Big{)}.\end{split}$

In the first equality, the factor $\prod_{m=1}^{r-1}(1-Ptq^{m})^{2}$ in the
numerator and $\prod_{m=1}^{r-1}(1-Pq^{m})^{2}$ in the denominator cancel each
other due to the presence of $(1-Pt)^{2}$ and (4.3). The second equality
follows from an explicit computation.

The first line after the second equal sign lies in $\mathcal{K}_{+}$ and the
rest lies in $\mathcal{K}_{-}$. Consider the reconstruction theorem [6,
Theorem 2]

$\begin{split}&J^{X_{-1,-1}}(q,Q)=\sum_{d\geq 0}I_{d}^{X_{-1,-1}}Q^{d}\cdot\\\
&\quad\exp\left(\Big{(}\sum_{k>0}\frac{\Psi^{k}}{k}\Big{)}\Big{(}\frac{\delta(Q)(1-Pq^{d})+\sum_{i=0}^{3}\epsilon_{i}(Q)(1-Ptq^{d})^{i}}{(1-q)}\Big{)}\right)\cdot\\\
&\qquad\left(s(q,Q)(1-Pq^{d})+\sum_{i=0}^{3}r_{i}(q,Q)(1-Ptq^{d})^{i}\right),\end{split}$

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

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

A direct computation by induction on the degree of the Novikov variable shows
that

$r_{0}(q,Q)=1,\quad\epsilon_{1}(Q)=\epsilon_{2}(Q)=\epsilon_{3}(Q)=\delta(Q)=r_{1}(q,Q)=s(q,Q)=0.$

and that $r_{2}(q,Q)$ and $r_{3}(q,Q)$ will not change the $\mathcal{K}_{-}$
part. This concludes the proof. ∎

######  Corollary 4.8 (Multiple cover formula [2]).

The small $J$-function for $X_{-1,-1}$ is

$\begin{split}\frac{1}{1-q}&J^{X_{-1,-1}}(q)=1+\Big{(}1+(1-P)\Big{)}\sum_{r\geq
1}\,Q^{r}a(r,q^{r})\\\ &+(1-P)\sum_{r\geq 1}Q^{r}\,b(r,q^{r}),\end{split}$

where $a(r,q^{r}),b(r,q^{r})$ are defined in (4.1).

###### Proof.

Since the zero section $P^{1}$ has normal bundle
$\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, the quantum $K$-invariants of $r\ell$
in $X_{-1,-1}$ are exactly the same as those in $Y_{-1,-1}$. The only
difference in $J$-functions comes from different bases of the $K$-groups and
the Poincaré pairing. The net result is the removal of the factor $(1-Pt)^{2}$
from the specialized $J^{Y_{-1,-1}}(q,Q)$ for non-zero degree terms. ∎

###  4.3. Virtual Clemens’ conjecture

We now give a heuristic derivation of Conjecture 4.2 and a heuristic
interpretation of the relationship between Gopakumar–Vafa invariants and
quantum $K$-invariants at genus zero by a multiple cover formula. This has
served to guide us in our search for the current formulation of Conjecture
4.2, even though the actually proof follows a completely different approach.
Of course, the original formulations of Jockers–Mayr [13] and
Garoufalidis–Scheidegger [4] have been enormous help.

Assume that we are given an “ideal” Calabi–Yau threefold $X$ satisfying a
“virtual Clemens’ conjecture”. That is, there are, up to deformations,
finitely many isolated rational curves $\\{C_{i}\\}$. Furthermore, they are
all smooth $(-1,-1)$ curves.

By Lemma 4.5, each isolated $(-1,-1)$-curve (in any degree $\vec{d}$)
contributes $1$ to the Gopakumar–Vafa invariants, independently of $\vec{d}$.
Therefore, there are ${\mathrm{GV}}_{0,\vec{d}}$ isolated $(-1,-1)$-curves in
degree $\vec{d}$. For each of these isolated curves, quantum $K$-theory allows
multiple $r$-covers of the isolated $(-1,-1)$-curve. The coefficients
$a(r,q^{r})$ and $b(r,q^{r})$ of the $r$-covers come from the $J$-function of
$X_{-1,-1}$. The only addition is the factor of $\int_{\vec{d}}D_{j}$, where
the divisor $D_{j}=\operatorname{ch}(\Phi^{1j})$ comes from the divisor axiom.

In summary, the “virtual Clemens conjecture” implies that GV $=$ QK for all
Calabi–Yau threefolds in genus zero via the multiple cover contributions.

This line of thoughts lead us to believe that, in order to generalize this to
higher genera, the most important ingredient is the higher genus multiple
cover formula in quantum $K$-theory. It is entirely possible that the higher
genus multiple cover formulas will serve as universal coefficients, similar to
the genus zero case. We intend to pursue this in future works.

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