# Integrable hierarchy for the resolved conifold

Murad Alim  and  Arpan Saha  Fachbereich Mathematik, Universität Hamburg,
Bundesstr. 55, 20146, Hamburg

###### Abstract.

We provide a direct proof of a conjecture of Brini relating the Gromov–Witten
theory of the resolved conifold to the Ablowitz–Ladik integrable hierarchy at
the level of primaries. We use a functional representation of the
Ablowitz–Ladik hierarchy as well as a difference equation for the
Gromov–Witten potential. We express the solution of the difference equation as
an analytic function which is a specialization of a Tau function put forward
by Bridgeland in the study of wall-crossing phenomena of Donaldson–Thomas
invariants.

##  1\. Introduction

The study of integrable structures as the underlying organizing principle of
Gromov–Witten (GW) theory has been a great source of insights and interactions
between various areas of mathematics such as algebraic and symplectic
geometry, the study of integrable systems, and mathematical physics.

The KdV integrable hierarchy for GW theory of a point was put forward in the
works of Witten [Wit91] and Kontsevich [Kon92], relating the intersection
theory on the Deligne–Mumford compactification of the moduli space of curves
to $2$d topological gravity. For $\mathbb{P}^{1}$, the relation of GW theory
to the Toda integrable hierarchy was studied in [EHY95, EY94, EYY95, EHX97,
Pan00]. In [Pan00], a conjectured difference equation for the GW potential
implied a corresponding difference equation for Hurwitz numbers which was
proved in [Oko00].

A general construction of an integrable hierarchy for a given Gromov–Witten
theory was given by Dubrovin and Zhang in [DZ01], see also [Dub14] for an
overview, yet explicit examples remain sparse, see also [ADK+06] for a
relation between topological string theory and integrable hierarchies. For the
Calabi–Yau threefold geometry given by the resolved conifold, Brini
conjectured in [Bri12] that the equivariant descendent Gromov–Witten potential
of the resolved conifold with anti-diagonal $\mathbb{C}^{\times}$ action
coincides with the logarithm of a tau function of the Ablowitz–Ladik (AL)
hierarchy introduced in [AL75] under a suitable identification of the
variables. In particular, the conjecture implies that the _primary_ GW
potential of the resolved conifold (i. e. without gravitational descendants)
coincides with the restriction of a tau function of the AL hierarchy to the
small phase space.

In [Bri12], Brini provided a proof of the coincidence of the descendent
Gromov–Witten potential with the logarithm of an AL tau function to genus $1$,
and of the primary Gromov–Witten potential with the restriction of the
logarithm of the tau function to the small phase space up to genus $2$. A
proof of the conjecture in the stationary sector to all genera was
subsequently given by Takasaki in [Tak13] using a melting crystal model of
Gromov–Witten theory of the resolved conifold. This was essentially a two-step
proof: First, the descendent potential was identified with a tau function of
the $2$d Toda hierarchy. And then, it was shown that owing to a certain
factorization property which the specific tau function in question possessed,
it could be interpreted as a tau function of the relativistic Toda hierarchy,
a reduction of the $2$d Toda hierarchy which was known to be equivalent to the
Ablowitz–Ladik hierarchy.

In complementary recent developments, Bridgeland put forward in [Bri19] a
Riemann–Hilbert problem associated to wall-crossing of Donaldson–Thomas
invariants. This was applied in [Bri20] to the resolved conifold, putting
forward a Tau function as a solution to the Riemann–Hilbert problem, which in
particular provides the Gromov–Witten potential as an asymptotic expansion.

In this work, we give a proof of the identification of the primary
Gromov–Witten potential for both the stationary and non-stationary sectors
with an AL tau function to all genera. The main ingredients of our proof are
two difference equations. One of these is obtained from a certain functional
representation of the AL hierarchy due to Vekslerchik [Vek98, Vek02], while
the other was proved by one of us in [Ali20] by adapting ideas which appeared
in the study of the exact WKB method in [IKT18]. In particular, by combining
this with the results of Bridgeland in [Bri20], we are able to obtain a
closed-form expression for the tau function, thus providing a new, but perhaps
expected, link between integrable hierarchies of GW theory and Bridgeland’s DT
Riemann–Hilbert problem.

The outline of this paper is as follows: In §2, we recall notions from the
Gromov–Witten theory of the resolved conifold, introduce the difference
equation proved in [Ali20], and construct a solution of it using Bridgeland’s
Tau function [Bri20]. In §3, we recall Vekslerchik’s functional representation
of the Ablowitz–Ladik hierarchy. In §4, we show how to take the dispersionless
limit of the AL hierarchy in this setting and give a version of Dubrovin’s
proof of the correspondence between the equivariant genus zero Gromov–Witten
potential of the resolved conifold and the dispersionless AL hierarchy adapted
to our setting. Finally, in §5, we bring all the ingredients together to
extend a result of Brini.

### Acknowledgements

We would like to thank Andrea Brini and Jörg Teschner for correspondence and
comments on the draft. This work is supported through the DFG Emmy Noether
grant AL 1407/2-1.

##  2\. Gromov–Witten potential and difference equation

Gromov–Witten (GW) theory of a non-singular algebraic variety $X$ is concerned
with the study of integrals over the moduli spaces of maps from Riemann
surfaces into $X$. Let $X$ be a Calabi–Yau threefold. The GW potential of $X$
is the following formal power series:

$F(\lambda,\boldsymbol{t})=\sum_{g\geq
0}\lambda^{2g-2}F^{g}(\boldsymbol{t})=\sum_{g\geq
0}\lambda^{2g-2}\sum_{\beta\in
H_{2}(X,\mathbb{Z})}N^{g}_{\beta}\,q^{\beta}\,,$ (2.1)

where $q^{\beta}:=\exp(2\pi i\langle\beta,\boldsymbol{t}\rangle)$ is a formal
variable living in a suitable completion of the effective cone in the group
ring of the small phase space $H_{2}(X,\mathbb{Z})$, $\lambda$ is a formal
parameter corresponding to the topological string coupling, and
$N^{g}_{\beta}$ are the GW invariants.

The GW potential can be written as:

$F=F_{\beta=0}+\tilde{F}\,,$ (2.2)

where $F_{\beta=0}$ denotes the contribution from constant maps (the
stationary sector) and $\tilde{F}$ the contribution from non-constant maps
(the non-stationary sector). The constant-map contributions at genus 0 and 1
are $\boldsymbol{t}$ dependent and the higher-genus constant-map contributions
take the universal form [FP00]:

$F_{\beta=0}^{g}=\frac{(-1)^{g-1}\,\chi(X)\,B_{2g}\,B_{2g-2}}{4g(2g-2)\,(2g-2)!}\,,\quad
g\geq 2\,,$ (2.3)

where $\chi(X)$ is the topological Euler characteristic of $X$ and $B_{k}$
denotes the $k$-th Bernoulli number.

This note is concerned with an equivariant version of the GW potential of the
CY threefold given by the total space of a rank two bundle over the projective
line:

$\mathcal{O}(-1)\oplus\mathcal{O}(-1)\rightarrow\mathbb{P}^{1}\,.$ (2.4)

This corresponds to the resolution of the conifold singularity in
$\mathbb{C}^{4}$ and is known as the resolved conifold. Its small phase space
is spanned by an identity element $\mathrm{id}$ and the Kähler class $\omega$
of the base $\mathbb{P}^{1}$. The associated coordinates shall be denoted
$x:=\langle\mathrm{id},\boldsymbol{t}\rangle$ and
$t:=\langle\omega,\boldsymbol{t}\rangle$ respectively. The (non-equivariant)
GW potential for this geometry was determined in physics [GV98, GV99], and in
mathematics [FP00] with the following outcome for the non-constant maps:

$\tilde{F}^{0}=\mathrm{Li}_{3}(q)\,,\quad\tilde{F}^{g}=\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\mathrm{Li}_{3-2g}(q)\,,\quad
g\geq 1\,,$ (2.5)

where $q:=\exp(2\pi{\mathrm{i}}t)$ and the polylogarithm is defined by:

$\mathrm{Li}_{s}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}\,,\quad
s\in\mathbb{C}\,.$ (2.6)

There is an anti-diagonal $\mathbb{C}^{\times}$ action on
$\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, given by fibrewise
$\mathbb{C}^{\times}$ action on the $\mathcal{O}(-1)$ with opposite
characters. This $\mathbb{C}^{\times}$ action is especially interesting as it
acts trivially on the canonical bundle, meaning that
$\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ is then _equivariantly_ Calabi–Yau.
With respect to this action, one can define an equivariant analogue of
Gromov–Witten theory where the ordinary cohomology ring of the resolved
conifold is replaced by the equivariant cohomology. The equivariant
Gromov–Witten invariants of rank $2$ bundles over curves were computed by
Bryan and Pandharipande in [BP08]. The theory significantly simplifies in the
case of the resolved conifold with anti-diagonal action: It is then just the
contribution from constant maps which is affected. In particular, the genus
zero Gromov–Witten potential becomes:

$F_{\mathrm{ad}}^{0}=\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}}+\tilde{F}^{0},$
(2.7)

where the $\mathrm{ad}$ in the subscript denotes that it is equivariant with
respect to the _anti-diagonal_ action and $\kappa$ is the equivariant
parameter.

The following theorem was proved in [Ali20]:

######  Theorem 1.

[Ali20] The contribution of the non-constant maps $\tilde{F}(\lambda,t)$ to
the GW potential of the resolved conifold satisfies the following difference
equation:

$\tilde{F}(\lambda,t+\check{\lambda})-2\tilde{F}(\lambda,t)+\tilde{F}(\lambda,t-\check{\lambda})=\left(\frac{1}{2\pi}\frac{\partial}{\partial
t}\right)^{2}\,\tilde{F}^{0}(t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,.$
(2.8)

We proceed by determining the solution of the difference equation. The latter
already appears as a building block of the Tau function determined by
Bridgeland in [Bri20] as a solution to a Riemann–Hilbert problem associated to
wall-crossing of DT invariants of the resolved conifold. The special functions
in [Bri20] involve the multiple sine functions which are defined using the
Barnes multiple Gamma functions [Bar04]. For a variable $z\in\mathbb{C}$ and
parameters $\omega_{1},\ldots,\omega_{r}\in\mathbb{C}^{\times}$, these are
defined by:

$\sin_{r}(z\,|\,\omega_{1},\dots,\omega_{r}):=\Gamma_{r}(z\,|\,\omega_{1},\dots,\omega_{r})\cdot\Gamma_{r}\left(\sum_{i=1}^{r}\omega_{i}-z\,|\,\omega_{1},\dots,\omega_{r}\right)^{(-1)^{r}}.$
(2.9)

For further definitions, see e. g. [Bri20, Rui00] and references therein. We
will furthermore need the generalized Bernoulli polynomials, defined by the
generating function:

$\frac{x^{r}\,e^{zx}}{\prod_{i=1}^{r}(e^{\omega_{i}x}-1)}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\,B_{r,n}(z\,|\,\omega_{1},\,\dots,\omega_{r})\,.$
(2.10)

Consider now the function $G(z\,|\,\omega_{1},\omega_{2})$ of [Bri20, Sec.
4.2], defined by:

$G(z\,|\,\omega_{1},\omega_{2}):=\exp\left(\frac{\pi i}{6}\cdot
B_{3,3}(z+\omega_{1}\,|\,\omega_{1},\omega_{1},\omega_{2})\right)\cdot\sin_{3}(z+\omega_{1}\,|\,\omega_{1},\omega_{2},\omega_{3}),$
(2.11)

and define a function111The subscript np of the function stands for non-
perturbative. In [Bri20], it is suggested that the Tau function of the DT
Riemann–Hilbert problem gives a non-perturbative definition of the GW theory
of the resolved conifold. The non-perturbative content of this function is
discussed in a companion paper.:

$F_{\mathrm{np}}(\lambda,t)=\log G(t\,|\,\check{\lambda},1)\,.$ (2.12)

We obtain the following:

######  Proposition 2.

The function $F_{\mathrm{np}}(\lambda,t)$ is a solution of the difference
equation (2.8). Moreover, $F_{\mathrm{np}}$ has an asymptotic expansion as
$\lambda\rightarrow 0$ given by:

$F_{\mathrm{np}}(\lambda,t)\sim\sum_{g=0}^{\infty}\lambda^{2g-2}\tilde{F}^{g}(t)\,,$
(2.13)

where $\tilde{F}^{g}(t)$ are the non-constant parts of the conifold free
energies defined in (2.5). The solution is furthermore unique given the above
asymptotic expansion.

###### Proof.

By [Bri20, Prop.4.2], the function $G(t\,|\,\check{\lambda},1)$ is single
valued for $\check{\lambda}\notin\mathbb{R}_{<0}$ and satisfies the difference
equation:

$\frac{G(t+\check{\lambda}\,|\,\check{\lambda},1)}{G(t\,|\,\check{\lambda},1)}=H(t+\check{\lambda}\,|\,\check{\lambda},1)^{-1}\,,$
(2.14)

where $H$ is denoted by $F$ in [Bri20] and is given by:

$H(t\,|\,\omega_{1},\omega_{2}):=\exp\left(-\frac{\pi i}{2}\cdot
B_{2,2}(t\,|\,\omega_{1},\omega_{2})\right)\cdot\sin_{2}(z\,|\,\omega_{1},\omega_{2})\,.$
(2.15)

This in turn satisfies the difference equation [Bri20, Prop 4.1]:

$\frac{H(t+\omega_{1}\,|\,\omega_{1},\omega_{2})}{H(t\,|\,\omega_{1},\omega_{2})}=\frac{1}{1-x_{2}}\,,\quad
x_{2}=\exp\left(\frac{2\pi it}{\omega_{2}}\right).$ (2.16)

From (2.14) we obtain, by taking the logarithm:

$\begin{split}&\log G(t+\check{\lambda}\,|\,\check{\lambda},1)+\log
G(t-\check{\lambda}\,|\,\check{\lambda},1)-2\log G(t\,|\,\check{\lambda},1)\\\
&=-\left(\log H(t+\check{\lambda}\,|\,\check{\lambda},1)-\log
H(t\,|\,\check{\lambda},1)\right)=\log(1-q)\,,\quad q=\exp(2\pi
it)\,.\end{split}$ (2.17)

For the asymptotic expansion of $F_{\mathrm{np}}$, we use [Bri20, Prop 4.6],
which in particular proves the asymptotic expansion of $G$ as
$\omega_{2}\rightarrow 0$ to be:

$\log G(z\,|\,\omega_{2},\omega_{1})\sim\sum_{k\geq 0}\frac{(k-1)\cdot
B_{k}\cdot\omega_{2}^{k-2}}{k!}\cdot\left(\frac{2\pi
i}{\omega_{1}}\right)^{k-2}\textrm{Li}_{3-k}\left(\exp\left(\frac{2\pi
iz}{\omega_{1}}\right)\right).$ (2.18)

It follows that:

$F_{\mathrm{np}}(\lambda,t)\sim\frac{1}{\lambda^{2}}\,\mathrm{Li}_{3}(q)+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\textrm{Li}_{3-2g}(q)\,,$
(2.19)

by noting that $B_{k}=0$ for $k>1$ odd.

Finally, we prove the uniqueness of this solution given the asymptotic
expansion (2.13). Let $F_{1}$ be a solution to the difference equation (2.8)
satisfying (2.13). Then the difference $f(\lambda,t):=F_{1}(\lambda,t)-\log
G(t\,|\,\check{\lambda},1)$ satisfies the following equation:

$f(\lambda,t+\check{\lambda})-f(\lambda,t)=f(\lambda,t)-f(\lambda,t-\check{\lambda})\,.$
(2.20)

For any fixed value of $\lambda$, the function
$f(\lambda,t+\check{\lambda})-f(\lambda,t)$ is periodic in $t$ with period
$\check{\lambda}$. Thus, it must be of the form:

$f(\lambda,t+\check{\lambda})-f(\lambda,t)=\sum_{k\in\mathbb{Z}}\lambda
d_{k}(\lambda)\exp\left(\frac{2k\pi{\mathrm{i}}t}{\check{\lambda}}\right),$
(2.21)

where $d_{k}(\lambda)$ are (possibly meromorphic) functions of $\lambda$. This
can again be rewritten as:

$\begin{split}&f(\lambda,t+\check{\lambda})-(t+\check{\lambda})\sum_{k\in\mathbb{Z}}d_{k}(\lambda)\exp\left(\frac{2k\pi{\mathrm{i}}(t+\check{\lambda})}{\check{\lambda}}\right)\\\
&=f(\lambda,t)-t\sum_{k\in\mathbb{Z}}d_{k}(\lambda)\exp\left(\frac{2k\pi{\mathrm{i}}t}{\check{\lambda}}\right).\end{split}$
(2.22)

This gives another function which is periodic in $t$ with period
$\check{\lambda}$. Thus, the most general $f(\lambda,t)$ satisfying this
equation is:

$f(\lambda,t)=\sum_{k\in\mathbb{Z}}(c_{k}(\lambda)+td_{k}(\lambda))\exp\left(\frac{2k\pi{\mathrm{i}}t}{\check{\lambda}}\right),$
(2.23)

where $c_{k}(\lambda),d_{k}(\lambda)$ are (possibly meromorphic) functions of
$\lambda$.

Since $F_{1}(\lambda,t)$ and $\log G(t\,|\,\check{\lambda},1)$ both satisfy
(2.13) as $\lambda\rightarrow 0$, the functions $c_{k}(\lambda)$ and
$d_{k}(\lambda)$ are forced to vanish. This implies $F_{1}(\lambda,t)=\log
G(t\,|\,\check{\lambda},1)$ and that the solution is indeed unique. ∎

######  Corollary 3.

The equivariant Gromov–Witten potential of the resolved conifold with anti-
diagonal action is:

$F_{\mathrm{ad}}(\lambda,t)=\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}\check{\lambda}^{2}}+\log
G(t\,|\,\check{\lambda},1)\,.$ (2.24)

##  3\. The Ablowitz–Ladik hierarchy

The AL hierarchy [AL75] is a set of infinitely many non-linear differential-
difference equations in two functions
$a,b\colon\mathbb{Z}\rightarrow\mathbb{C}\llbracket z,\tilde{z}\rrbracket$,
where $z$ and $\tilde{z}$ represent two countably infinite tuples of (complex)
flow parameters $z_{i}$ and $\tilde{z}_{i}$ respectively. The simplest non-
trivial flow equation in the hierarchy is the AL system:

$\begin{split}{\mathrm{i}}\dot{a}&=\phantom{+}(\Lambda(a)+\Lambda^{-1}(a))(1-ab)\,,\\\
{\mathrm{i}}\dot{b}&=-(\Lambda(b)+\Lambda^{-1}(b))(1-ab)\,.\end{split}$ (3.1)

Here the dot denotes derivative with respect to a linear combination of the
flow parameters in the tuples $z$ and $\tilde{z}$, while $\Lambda$ denotes the
shift operator whose action on any function $u$ on $\mathbb{Z}$ is given by

$(\Lambda(u))(n)=u(n+1)\,.$ (3.2)

One way of constructing the full AL hierarchy is by means of three tau
functions $\sigma,\rho,\tau\colon\mathbb{Z}\rightarrow\mathbb{C}\llbracket
z,\tilde{z}\rrbracket$ associated to a solution $(a,b)$ of the AL system.
These are _defined_ by

$a=\frac{\sigma}{\tau}\,,\quad b=\frac{\rho}{\tau}\,,\quad
1-ab=\frac{\Lambda(\tau)\Lambda^{-1}(\tau)}{\tau^{2}}\,.$ (3.3)

Note that the tau functions are not independent but satisfy

$\Lambda(\tau)\Lambda^{-1}(\tau)=\tau^{2}-\sigma\rho\,.$ (3.4)

The AL hierarchy is then equivalent to the following system of Hirota bilinear
equations (cf. Vekslerchik [Vek98, Vek02]):

$\displaystyle\tau(z,\tilde{z})\tau(z+{\mathrm{i}}[\zeta],\tilde{z})-\rho(z,\tilde{z})\sigma(z+{\mathrm{i}}[\zeta],\tilde{z})$
$\displaystyle=\phantom{\zeta}\Lambda^{-1}(\tau)(z,\tilde{z})\Lambda(\tau)(z+{\mathrm{i}}[\zeta],\tilde{z})\,,$
(3.5a)
$\displaystyle\tau(z,\tilde{z})\sigma(z+{\mathrm{i}}[\zeta],\tilde{z})-\sigma(z,\tilde{z})\tau(z+{\mathrm{i}}[\zeta],\tilde{z})$
$\displaystyle=\zeta\Lambda^{-1}(\tau)(z,\tilde{z})\Lambda(\sigma)(z+{\mathrm{i}}[\zeta],\tilde{z})\,,$
(3.5b)
$\displaystyle\rho(z,\tilde{z})\tau(z+{\mathrm{i}}[\zeta],\tilde{z})-\tau(z,\tilde{z})\rho(z+{\mathrm{i}}[\zeta],\tilde{z})$
$\displaystyle=\zeta\Lambda^{-1}(\rho)(z,\tilde{z})\Lambda(\tau)(z+{\mathrm{i}}[\zeta],\tilde{z})\,,$
(3.5c)
$\displaystyle\tau(z,\tilde{z})\tau(z,\tilde{z}+{\mathrm{i}}[\zeta])-\rho(z,\tilde{z})\sigma(z,\tilde{z}+{\mathrm{i}}[\zeta])$
$\displaystyle=\phantom{\zeta}\Lambda(\tau)(z,\tilde{z})\Lambda^{-1}(\tau)(z,\tilde{z}+{\mathrm{i}}[\zeta])\,,$
(3.5d)
$\displaystyle\tau(z,\tilde{z})\sigma(z,\tilde{z}+{\mathrm{i}}[\zeta])-\sigma(z,\tilde{z})\tau(z,\tilde{z}+{\mathrm{i}}[\zeta])$
$\displaystyle=\zeta\Lambda(\tau)(z,\tilde{z})\Lambda^{-1}(\sigma)(z,\tilde{z}+{\mathrm{i}}[\zeta])\,,$
(3.5e)
$\displaystyle\rho(z,\tilde{z})\tau(z,\tilde{z}+{\mathrm{i}}[\zeta])-\tau(z,\tilde{z})\rho(z,\tilde{z}+{\mathrm{i}}[\zeta])$
$\displaystyle=\zeta\Lambda(\rho)(z,\tilde{z})\Lambda^{-1}(\tau)(z,\tilde{z}+{\mathrm{i}}[\zeta])\,.$
(3.5f)

In the above, $[\zeta]$ denotes the tuple:

$[\zeta]:=\left(\zeta,\frac{\zeta^{2}}{2},\frac{\zeta^{3}}{3},\ldots\right).$
(3.6)

The flows of the hierarchy are then obtained by expanding the above set of
equations order by order in $\zeta$. In particular, the AL system is obtained
by taking the flow along $\partial_{z_{1}}+\partial_{\tilde{z}_{1}}$.

##  4\. Dispersionless limit

The dispersionless limit of any integrable hierarchy is a limit in which the
contribution of higher-derivative terms (which cause dispersion) are
suppressed. This is achieved by scaling the length and times (i. e. the flow
parameters) by a factor $\check{\lambda}^{-1}$ inversely proportional to a
dispersion parameter $\lambda$, and then taking the limit $\lambda\rightarrow
0$. Expanding the relevant flow equations as Taylor series in $\lambda$ about
$\lambda=0$ then gives the small-dispersion expansion of the hierarchy.

The most straightforward way to do this in the case of the AL hierarchy is to
work at the level of tau functions. That is to say, we introduce tau functions
$\sigma_{\lambda},\rho_{\lambda},\tau_{\lambda}\colon\mathbb{C}\rightarrow\mathbb{C}\llbracket
z,\tilde{z}\rrbracket$ of a continuous complex variable $x$, parametrised by
$\lambda$. (It is not a coincidence that the spatial coordinate is denoted
with the same symbol as the group ring parameter associated to the identity
class in $H_{2}(X,\mathbb{Z})$; we will later see that they can be
identified.) The dependence of the tau functions on $\lambda$ is stipulated to
take the following form:

$\begin{split}\log\sigma_{\lambda}=\frac{\varpi_{\lambda}}{\check{\lambda}^{2}}+\frac{s_{\lambda}}{\check{\lambda}}+r_{\lambda}\,,\quad\log\rho_{\lambda}=\frac{\varpi_{\lambda}}{\check{\lambda}^{2}}-\frac{s_{\lambda}}{\check{\lambda}}+r_{\lambda}\,,\quad\log\tau_{\lambda}&=\frac{\varpi_{\lambda}}{\check{\lambda}^{2}}\,.\end{split}$
(4.1)

Here, $\varpi_{\lambda},s_{\lambda},r_{\lambda}$ are analytic in $\lambda$.
So, they have well-defined limits $\varpi,s,r$ as $\lambda\rightarrow 0$.

We also introduce the operator
$\Lambda_{\lambda}:=\exp(\check{\lambda}\partial_{x})$. The small-dispersion
expansion of the AL hierarchy is then given by making the following
replacements in (3.5):

$\sigma\mapsto\sigma_{\lambda}\,,\quad\rho\mapsto\rho_{\lambda}\,,\quad\tau\mapsto\tau_{\lambda}\,,\quad\Lambda\mapsto\Lambda_{\lambda}\,,\quad[\zeta]\mapsto\check{\lambda}[\zeta]\,.$
(4.2)

In the conjectured correspondence between GW theory and integrable
hierarchies, the dispersion parameter $\lambda$ plays the role of the genus
expansion parameter. In particular, the genus zero GW theory should correspond
to the dispersionless limit of the associated hierarchy. In order to relate
the dispersionless limit of the AL hierarchy to the resolved conifold, we will
first identify a suitable set of dependent variables and describe the
dynamical equations of the hierarchy in terms of them.

######  Proposition 4.

The dispersionless limit of the AL hierarchy is given by the following system
of equations in the functions $v=\partial_{x}s$ and $u=-\log(1-e^{2r})$:

$\displaystyle D^{\zeta}_{z}v$
$\displaystyle={\mathrm{i}}\,\frac{\partial}{\partial
x}\log\left(\frac{1-\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}{2}\right),$ (4.3a) $\displaystyle D^{\zeta}_{z}u$
$\displaystyle={\mathrm{i}}\,\frac{\partial}{\partial
x}\log\left(\frac{1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}{2}\right),$ (4.3b) $\displaystyle D^{\zeta}_{\tilde{z}}v$
$\displaystyle={\mathrm{i}}\,\frac{\partial}{\partial
x}\log\left(\frac{1-\zeta e^{-v}+\sqrt{(1+\zeta e^{-v})^{2}-4\zeta
e^{-v}e^{-u}}}{2}\right),$ (4.3c) $\displaystyle D^{\zeta}_{\tilde{z}}u$
$\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial
x}\log\left(\frac{1+\zeta e^{-v}+\sqrt{(1+\zeta e^{-v})^{2}-4\zeta
e^{-v}e^{-u}}}{2}\right),$ (4.3d)

where $D^{\zeta}_{z}$ and $D^{\zeta}_{\tilde{z}}$ denote the operators:

$D^{\zeta}_{z}=\sum_{j=1}^{\infty}\frac{\zeta^{j}}{j}\frac{\partial}{\partial
z_{j}}\,,\quad
D^{\zeta}_{\tilde{z}}=\sum_{j=1}^{\infty}\frac{\zeta^{j}}{j}\frac{\partial}{\partial\tilde{z}_{j}}\,.$
(4.4)

###### Proof.

Making the replacements (4.2) in (3.5) and dividing equations (3.5a) and
(3.5b) by the respective first terms on the left-hand side, we get:

$\displaystyle
1-\left(\frac{\rho_{\lambda}}{\tau_{\lambda}}\right)(z,\tilde{z})\left(\frac{\sigma_{\lambda}}{\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})$
$\displaystyle\quad=\left(\frac{\Lambda^{-1}_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z,\tilde{z})\left(\frac{\Lambda_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\,,$
(4.5a) $\displaystyle
1-\left(\frac{(\sigma_{\lambda}/\tau_{\lambda})(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{(\sigma_{\lambda}/\tau_{\lambda})(z,\tilde{z})}\right)^{-1}$
$\displaystyle\quad=\zeta\left(\frac{\Lambda^{-1}_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z,\tilde{z})\left(\frac{\Lambda_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\left(\frac{\Lambda_{\lambda}(\sigma_{\lambda}/\tau_{\lambda})}{\sigma_{\lambda}/\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\,.$
(4.5b)

Substituting (4.1) into the above then gives us:

$\displaystyle
1-\exp\left(\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-s_{\lambda}(z,\tilde{z}))}{\check{\lambda}}\right)\exp\left(r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})+r_{\lambda}(z,\tilde{z})\right)$
$\displaystyle\quad=\exp\left(\frac{(\Lambda_{\lambda}^{-1}-1)(\varpi_{\lambda})(z,\tilde{z})+(\Lambda_{\lambda}-1)(\varpi_{\lambda})(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}^{2}}\right),$
(4.6a) $\displaystyle
1-\exp\left(-\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-s_{\lambda}(z,\tilde{z}))}{\check{\lambda}}\right)\exp\left(-r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})+r_{\lambda}(z,\tilde{z}))\right)$
$\displaystyle\quad=\zeta\exp\left(\frac{(\Lambda_{\lambda}^{-1}-1)(\varpi_{\lambda})(z,\tilde{z})+(\Lambda_{\lambda}-1)(\varpi_{\lambda})(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}^{2}}\right)$
$\displaystyle\quad\quad\times\exp\left((\Lambda_{\lambda}-1)\left(\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}}+\pi{\mathrm{i}}t_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\right)\right).$
(4.6b)

For convenience, we now introduce the following notation:

$\begin{split}X(z,\tilde{z})&=\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-s_{\lambda}(z,\tilde{z}))}{\check{\lambda}}\,,\\\
Y(z,\tilde{z})&=\frac{r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-r_{\lambda}(z,\tilde{z})}{\check{\lambda}}\,,\\\
Z(z,\tilde{z})&=(\Lambda_{\lambda}-1)\left(\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}}+r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\right).\end{split}$
(4.7)

Eliminating $\varpi_{\lambda}$ from (4.6a) and (4.6b) gives us a quadratic
equation in $e^{X(z,\tilde{z})}$:

$\zeta
e^{Z(z,\tilde{z})}e^{\check{\lambda}Y(z,\tilde{z})}e^{2r_{\lambda}(z,\tilde{z})}e^{2X(z,\tilde{z})}+\left(1-\zeta
e^{Z(z,\tilde{z})}\right)e^{X(z,\tilde{z})}-e^{-\check{\lambda}Y(z,\tilde{z})}=0\,.$
(4.8)

This has two solutions but only one such that $X(z,\tilde{z})$ has a well-
defined limit as $\zeta\rightarrow 0$:

$X(z,\tilde{z})=\log\left(\frac{-1+\zeta e^{Z(z,\tilde{z})}+\sqrt{(1-\zeta
e^{Z(z,\tilde{z})})^{2}+4\zeta
e^{Z(z,\tilde{z})}e^{2r_{\lambda}(z,\tilde{z})}}}{2\zeta
e^{Z(z,\tilde{z})}e^{\check{\lambda}Y(z,\tilde{z})}e^{2r_{\lambda}(z,\tilde{z})}}\right).$
(4.9)

In the dispersionless limit $\lambda\rightarrow 0$, this becomes:

${\mathrm{i}}D^{\zeta}_{z}s=\log\left(\frac{-1+\zeta e^{v}+\sqrt{(1-\zeta
e^{v})^{2}+4\zeta e^{v}e^{2r}}}{2\zeta e^{v}e^{2r}}\right).$ (4.10)

Differentiating with respect to $x$ on both sides and substituting
$e^{2r}=1-e^{-u}$ then gives us:

$D^{\zeta}_{z}v={\mathrm{i}}\,\frac{\partial}{\partial
x}\log\left(\frac{-1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}{2\zeta e^{v}(1-e^{-u})}\right).$ (4.11)

Equation (4.3a) then follows from the identity:

$\frac{-1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2\zeta
e^{v}(1-e^{-u})}=\frac{2}{1-\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}\,.$ (4.12)

We will now use what we have established so far to prove (4.3b). First, we
note that (4.6a) in the limit $\lambda\rightarrow 0$ becomes:

$1-\exp({\mathrm{i}}D^{\zeta}_{z}s)e^{2r}=\exp\left(\frac{\partial^{2}\varpi}{\partial
x^{2}}+{\mathrm{i}}\frac{\partial D^{\zeta}_{z}\varpi}{\partial x}\right).$
(4.13)

Setting $\zeta=0$ then gives us:

$u=-\log(1-e^{2r})=-\frac{\partial^{2}\varpi}{\partial x^{2}}\,.$ (4.14)

Substituting this back into (4.13), we obtain:

$\frac{\partial D^{\zeta}_{z}\varpi}{\partial
x}=-{\mathrm{i}}\log\left(\frac{1-\exp({\mathrm{i}}D^{\zeta}_{z}s)e^{2r}}{e^{-u}}\right).$
(4.15)

Differentiating with respect to $x$ on both sides and substituting (4.10) and
$e^{2r}=1-e^{-u}$ then gives

$D^{\zeta}_{z}u={\mathrm{i}}\,\frac{\partial}{\partial
x}\log\left(\frac{1+\zeta e^{v}-\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}{2\zeta e^{v}e^{-u}}\right).$ (4.16)

Equation (4.3b) then follows from the identity:

$\frac{1+\zeta e^{v}-\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2\zeta
e^{v}e^{-u}}=\frac{2}{1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}\,.$ (4.17)

A similar argument involving (3.5d) and (3.5e) leads to (4.3c) and (4.3d). ∎

######  Remark 5.

By setting $\zeta=0$ in (4.6a) and taking the logarithm of both sides, we
obtain a difference equation involving shifts in $x$:

$\log(1-\exp(2r_{\lambda}))=\frac{\Lambda_{\lambda}(\varpi_{\lambda})-2\varpi_{\lambda}+\Lambda_{\lambda}^{-1}(\varpi_{\lambda})}{\check{\lambda}^{2}}\,.$
(4.18)

######  Remark 6.

By applying $\zeta\partial_{\zeta}$ to (4.3) we get the following alternative
form for the dispersionless AL hierarchy:

$\displaystyle\Delta^{\zeta}_{z}v$
$\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial x}\left(\frac{1+\zeta
e^{v}}{2\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\right)$
$\displaystyle=-{\mathrm{i}}\,\frac{\partial^{2}}{\partial x\partial
u}\tanh^{-1}\left(\frac{1+\zeta e^{v}}{\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}\right),$ (4.19a) $\displaystyle\Delta^{\zeta}_{z}u$
$\displaystyle=\phantom{+}{\mathrm{i}}\,\frac{\partial}{\partial
x}\left(\frac{1-\zeta e^{v}}{2\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}\right)$
$\displaystyle=-{\mathrm{i}}\,\frac{\partial^{2}}{\partial x\partial
v}\tanh^{-1}\left(\frac{1+\zeta e^{v}}{\sqrt{(1+\zeta e^{v})^{2}-4\zeta
e^{v}e^{-u}}}\right),$ (4.19b) $\displaystyle\Delta^{\zeta}_{\tilde{z}}v$
$\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial x}\left(\frac{1+\zeta
e^{-v}}{2\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right)$
$\displaystyle=-{\mathrm{i}}\frac{\partial^{2}}{\partial x\partial
u}\tanh^{-1}\left(\frac{1+\zeta e^{-v}}{\sqrt{(1+\zeta e^{-v})^{2}-4\zeta
e^{-v}e^{-u}}}\right),$ (4.19c) $\displaystyle\Delta^{\zeta}_{\tilde{z}}u$
$\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial x}\left(\frac{1-\zeta
e^{-v}}{2\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right)$
$\displaystyle=-{\mathrm{i}}\,\frac{\partial^{2}}{\partial x\partial
v}\tanh^{-1}\left(\frac{1+\zeta e^{-v}}{\sqrt{(1+\zeta e^{-v})^{2}-4\zeta
e^{-v}e^{-u}}}\right),$ (4.19d)

where $\Delta^{\zeta}_{z}$ and $\Delta^{\zeta}_{\tilde{z}}$ denote the
operators:

$\Delta^{\zeta}_{z}=\sum_{j=1}^{\infty}\zeta^{j}\frac{\partial}{\partial
z_{j}}\,,\quad\Delta^{\zeta}_{\tilde{z}}=\sum_{j=1}^{\infty}\zeta^{j}\frac{\partial}{\partial\tilde{z}_{j}}\,.$
(4.20)

In particular, this is a dispersionless _Hamiltonian_ hierarchy with
generating functions of Hamiltonian densities:

$\begin{split}h(\zeta)&=-{\mathrm{i}}\tanh^{-1}\left(\frac{1+\zeta
e^{v}}{\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\right),\\\
\tilde{h}(\zeta)&=-{\mathrm{i}}\tanh^{-1}\left(\frac{1+\zeta
e^{-v}}{\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right),\end{split}$
(4.21)

and the following Poisson brackets:

$\\{u(x),u(y)\\}=\\{v(x),v(y)\\}=0\,,\quad\\{u(x),v(y)\\}=\\{v(x),u(y)\\}=\delta^{\prime}(x-y)\,.$
(4.22)

There are a number of interesting aspects of the Hamiltonian formulation of
the dispersionless limit of the AL hierarchy. Firstly, the Hamiltonian
densities depend only on $u$ and $v$ explicitly and not on their spatial
derivatives. Secondly, the coefficients of $\delta^{\prime}(x)$ in the Poisson
brackets form a constant symmetric matrix. A Hamiltonian hierarchy that has
this structure (possibly after a suitable change of dependent variables) is
said to be of _hydrodynamic type_. Such hierarchies were first defined and
studied by Dubrovin and Novikov [DN84].

A key fact concerning Hamiltonian hierarchies of hydrodynamic type is that
there is a natural such hierarchy called the _principal hierarchy_ associated
to any (almost) Frobenius manifold, which conversely may be viewed as a
differential-geometric structure on the initial data of such a hierarchy. The
small phase space $H_{2}(X,\mathbb{Z})$ of a projective variety happens to
carry the structure of a(n almost) Frobenius manifold, so this gives a rather
general construction of a dispersionless integrable hierarchy associated to
the (equivariant) genus zero GW theory of any projective variety. It was noted
by Dubrovin in [Dub08] that the dispersionless limit of the AL hierarchy is
contained within the principal hierarchy associated to the equivariant GW
theory of the resolved conifold with anti-diagonal action (hereafter referred
to as simply the principal hierarchy of the resolved conifold). We will now
present a proof of this result in our notation to ensure that the dependent
variables $u$ and $v$ that we have introduced are indeed the variables of the
principal hierarchy identified by Dubrovin.

######  Proposition 7.

The dispersionless limit of AL hierarchy with the initial conditions
$v=2\pi{\mathrm{i}}x/\kappa$ and $u=-2\pi{\mathrm{i}}t$ is contained in the
principal hierarchy of the resolved conifold with the following identification
on the small phase space:

$(2\pi)^{2}\varpi|_{(z,\tilde{z})=(0,0)}=\left.\left(-\frac{uv^{2}}{2}+\tilde{F}^{0}\right)\right|_{(z,\tilde{z})=(0,0)}.$
(4.23)

###### Proof.

The principal hierarchy associated to a(n almost) Frobenius manifold $V$ is
defined on the formal loop space $L(S^{1},V)$, whose elements are maps from
$S^{1}=\mathbb{R}/\mathbb{Z}$ to $V$. In our case, $V$ is the small phase
space $H_{2}(X,\mathbb{Z})$, while $w^{1}:=v$ and $w^{2}:=u$ can be
interpreted as maps from $S^{1}$ to $V$. (Note that the map $w^{1}$ is
distinguished.) Meanwhile, $x$ is interpreted as a coordinate on the universal
cover $\mathbb{R}$ of $S^{1}$.

A potential $\Phi$ on $V$ encodes a metric $\eta$ and an associative product
$\star$ with identity $\partial_{w^{1}}$ on the tangent bundle $TV$:

$\begin{split}\eta_{\mu\nu}:=\eta\left(\frac{\partial}{\partial
w^{\mu}}\,,\frac{\partial}{\partial
w^{\nu}}\right)&=\frac{\partial^{3}\Phi}{\partial w^{1}\partial
w^{\mu}\partial w^{\nu}}\,,\\\ \eta\left(\frac{\partial}{\partial
w^{\kappa}}\,,\frac{\partial}{\partial w^{\mu}}\star\frac{\partial}{\partial
w^{\nu}}\right)&=\frac{\partial^{3}\Phi}{\partial w^{\kappa}\partial
w^{\mu}\partial w^{\nu}}\,.\end{split}$ (4.24)

We also have a Poisson bracket on $L(S^{1},V)$ in terms of the the inverse
metric $\eta^{\mu\nu}$:

$\\{w^{\mu}(x),w^{\nu}(y)\\}=\eta^{\mu\nu}\delta^{\prime}(x-y)-\eta^{\mu\kappa}\Gamma^{\nu}_{\kappa\gamma}\frac{\partial
w^{\gamma}}{\partial x}\delta(x-y)\,,$ (4.25)

where $\Gamma^{\nu}_{\kappa\gamma}$ are the Christoffel symbols of the metric
$\eta$ and repeated indices are summed over. Comparing this with (4.22), we
immediately obtain:

$\eta_{11}=\eta_{22}=0,\quad\eta_{12}=\eta_{21}=1\,.$ (4.26)

Thus, the most general form that the potential $\Phi$ can take is:

$\Phi=\frac{uv^{2}}{2}+f(u)\,.$ (4.27)

A consequence of the associativity of the product $\star$ is the existence of
a pencil of flat torsion-free connections $\nabla^{\xi}$ parametrised by a
parameter $\xi\in\mathbb{C}$:

$\nabla^{\xi}_{\partial/\partial w^{\mu}}\frac{\partial}{\partial
w^{\nu}}=\xi\,\frac{\partial}{\partial w^{\mu}}\star\frac{\partial}{\partial
w^{\nu}}\,.$ (4.28)

As these connections are flat, they admit local basis of parallel $1$-forms
$\alpha_{1}(\xi)$ and $\alpha_{2}(\xi)$ analytic in $\xi$. And as these
connections are torsion-free, the anti-symmetric parts of
$\nabla^{\xi}\alpha_{1}$ and $\nabla^{\xi}\alpha_{2}$ are
$\mathrm{d}\alpha_{1}$ and $\mathrm{d}\alpha_{2}$ (where $\mathrm{d}$ is the
exterior derivative on $V$). Thus, $1$-forms $\alpha_{1}(\xi)$ and
$\alpha_{2}(\xi)$ are closed, and so by Poincaré’s lemma, we can locally find
functions $g_{1}(\xi)$ and $g_{2}(\xi)$ analytic in $\xi$ such that we have:

$\alpha_{1}=\mathrm{d}g_{1}\,,\quad\alpha_{2}=\mathrm{d}g_{2}\,.$ (4.29)

More explicitly, the functions $g_{1}$ and $g_{2}$ are solutions to the
following system of equations:

$\frac{\partial^{2}g}{\partial v^{2}}=\xi\,\frac{\partial g}{\partial
v}\,,\quad\frac{\partial^{2}g}{\partial v\,\partial u}=\xi\,\frac{\partial
g}{\partial u}\,,\quad\frac{\partial^{2}g}{\partial u^{2}}=\xi
f^{\prime\prime\prime}(u)\,\frac{\partial g}{\partial v}\,.$ (4.30)

The functions $g_{1}(\xi)$ and $g_{2}(\xi)$ are generating functions of
Hamiltonian densities, which generate the principal hierarchy via the Poisson
bracket (4.25). The generating functions $h(\zeta)$ and $\tilde{h}(\zeta)$ are
related to these in a non-trivial way (see Remark 8). We can get around the
need to obtain an explicit relationship between them by eliminating $\xi$ from
the first and third equations in (4.30). This gives us the following equation
that by virtue of its linearity is valid for any generating function of
Hamiltonian densities:

$\frac{\partial^{2}g}{\partial
u^{2}}=f^{\prime\prime\prime}(u)\,\frac{\partial^{2}g}{\partial v^{2}}\,.$
(4.31)

Substituting $g=h$ and $g=\tilde{h}$ as in (4.21) into the above, we obtain:

$f^{\prime\prime\prime}(u)=\frac{1}{e^{u}-1}=\mathrm{Li}_{0}(e^{-u})\,.$
(4.32)

This is solved by
$f(u)=-\tilde{F}^{0}(t)=-\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})$ for
$u=-2\pi{\mathrm{i}}t$. So the dispersionless AL hierarchy is contained in the
principal hierarchy of the resolved conifold. The identification (4.23)
meanwhile follows from (4.14) and the initial condition
$v=2\pi{\mathrm{i}}x/\kappa$. ∎

The exponential $e^{\Phi}$ of the potential $\Phi$ of the (almost) Frobenius
manifold will be henceforth called the _topological tau function_ of the
principal hierarchy.

######  Remark 8.

The first two equations of (4.30) imply that any solution $g$ necessarily has
the form:

$g=A(\xi,u)e^{\xi v}+B(\xi)\,,$ (4.33)

where $A$ is a function independent of $v$ while $B$ is a function independent
of $v$ and $u$. Since we are only interested in the flows of the Hamiltonians,
we can without loss of generality set $B$ to be identically zero. Conversely,
any function $g=A(\xi,u)e^{\xi v}$ which also satisfies (4.31) is a solution
of (4.30). Now consider the Mellin transforms:

$\begin{split}\int_{0}^{\infty}h\zeta^{-1-\xi}\mathrm{d}\zeta&=-{\mathrm{i}}e^{\xi
v}\int_{0}^{\infty}\tanh^{-1}\left(\frac{1+\zeta}{\sqrt{(1+\zeta)^{2}-4\zeta
e^{-u}}}\right)\zeta^{-1-\xi}\mathrm{d}\zeta\\\
&=\phantom{+}{\mathrm{i}}e^{\xi
v}\int^{0}_{\infty}\,\,\tanh^{-1}\left(\frac{1+\zeta^{-1}}{\sqrt{(1+\zeta^{-1})^{2}-4\zeta^{-1}e^{-u}}}\right)\zeta^{1+\xi}\,\frac{\mathrm{d}\zeta}{\zeta^{2}}\\\
&=-{\mathrm{i}}e^{\xi
v}\int_{0}^{\infty}\tanh^{-1}\left(\frac{\zeta+1}{\sqrt{(\zeta+1)^{2}-4\zeta
e^{-u}}}\right)\zeta^{-1+\xi}\mathrm{d}\zeta\,,\\\
\int_{0}^{\infty}\tilde{h}\zeta^{-1+\xi}\mathrm{d}\zeta&=-{\mathrm{i}}e^{\xi
v}\int_{0}^{\infty}\tanh^{-1}\left(\frac{1+\zeta}{\sqrt{(1+\zeta)^{2}-4\zeta
e^{-u}}}\right)\zeta^{-1+\xi}\mathrm{d}\zeta\,.\end{split}$ (4.34)

This is of the form $A(\xi,u)e^{\xi v}$. Moreover, being an integral transform
of $h$ and $\tilde{h}$, it satisfies (4.31) with
$f(u)=-\mathrm{Li}_{3}(e^{-u})$ as well. Hence, it must be a solution $g$ of
(4.30). In other words, it is some constant linear combination of the
Hamiltonian density generating functions $g_{1}$ and $g_{2}$ of the principal
hierarchy of the resolved conifold.

##  5\. Identification of the Tau function

In this section, we will extend the genus zero identification of the
dispersionless AL tau function with the topological tau function of the
principal hierarchy of the resolved conifold to an all-genera small-phase-
space identification of the tau function of the small-dispersion expansion of
the AL hierarchy with Bridgeland’s Tau function. The genus $2$ truncation of
this identification was essentially the content of Theorem 1.4 in [Bri12] and
it was proved by analyzing dispersive perturbations of the principal
hierarchy. This is a computationally challenging task, however the control
over expansion in the dispersive parameter offered by the functional
representation of the AL hierarchies will allow us to sidestep these
difficulties.

Here is the gist of our approach. The solution of Prop. 2 is used to define
initial conditions for the tau function of the AL hierarchy while the other
difference equation in Remark 5 is used to show that these initial conditions
reduce to those of the principal hierarchy of the resolved conifold in the
dispersionless limit.

We now prove our main theorem, which extends Theorem 1.4 in [Bri12] to all
genera.

######  Theorem 9.

There exists a solution to the small-dispersion expansion of the AL hierarchy
satisfying the following conditions:

  1. (1)

The tau function $\tau_{\lambda}$ of the solution reduces at leading order in
$\lambda$ to the topological tau function of the principal hierarchy of the
resolved conifold as $\lambda\rightarrow 0$,

  2. (2)

The restriction of the tau function $\tau_{\lambda}$ of the solution to the
small phase space coincides with the exponential of the equivariant GW
potential of the resolved conifold with anti-diagonal action, namely:

$\exp(F_{\mathrm{ad}}(\lambda,t))=\exp\bigg{(}\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}\check{\lambda}^{2}}\bigg{)}G(t\,|\,\check{\lambda},1)\,,$
(5.1)

to all orders in $\lambda$.

###### Proof.

We impose the following initial conditions on
$v_{\lambda}:=\check{\lambda}^{-1}(\Lambda_{\lambda}-1)s_{\lambda}$ and
$\varpi_{\lambda}$:

$v_{\lambda}|_{(z,\tilde{z})=(0,0)}=\frac{2\pi{\mathrm{i}}x}{\kappa}\,,\quad(2\pi)^{2}\varpi_{\lambda}|_{(z,\tilde{z})=(0,0)}=\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}}+\check{\lambda}^{2}\log
G(t\,|\,\check{\lambda},1)\,.$ (5.2)

This entails condition (2). Moreover, the difference equation in Remark 5
implies the following initial condition for
$u_{\lambda}:=-\log(1-\exp(2r_{\lambda}))$:

$u_{\lambda}|_{(z,\tilde{z})=(0,0)}=-2\pi{\mathrm{i}}t\,.$ (5.3)

The initial conditions for $u_{\lambda}$ and $v_{\lambda}$ reduce to those for
$u$ and $v$ in the principal hierarchy of the resolved conifold in the limit
$\lambda\rightarrow 0$. Since the full solution to the principal hierarchy is
fixed by the initial conditions, condition (1) holds as well. ∎

## References

  * [ADK+06] Mina Aganagic, Robbert Dijkgraaf, Albrecht Klemm, Marcos Mariño, and Cumrun Vafa.  Topological strings and integrable hierarchies.  Comm. Math. Phys., 261(2):451–516, 2006.
  * [AL75] Mark J. Ablowitz and John F. Ladik.  Nonlinear differential-difference equations.  Journal of Mathematical Physics, 16(3):598–603, March 1975.
  * [Ali20] Murad Alim.  Difference equation for the Gromov–Witten potential of the resolved conifold.  arXiv preprint arXiv:2011.12759, 2020.
  * [Bar04] Ernest W. Barnes.  On the theory of multiple gamma function.  Trans. Cambridge Phil. Soc., 19:374–425, 1904.
  * [BP08] Jim Bryan and Rahul Pandharipande.  The Local Gromov–Witten theory of curves.  J. Am. Math. Soc., 21:101–136, 2008.
  * [Bri12] Andrea Brini.  The local Gromov–Witten theory of $\mathbb{C}\mathbb{P}^{1}$ and integrable hierarchies.  Comm. Math. Phys., 313(3):571–605, 2012.
  * [Bri19] Tom Bridgeland.  Riemann–Hilbert problems from Donaldson-Thomas theory.  Invent. Math., 216(1):69–124, 2019.
  * [Bri20] Tom Bridgeland.  Riemann–Hilbert problems for the resolved conifold.  J. Differential Geom., 115(3):395–435, 2020.
  * [DN84] Boris A. Dubrovin and Sergei P. Novikov.  On Poisson brackets of hydrodynamic type.  Dokl. Akad. Nauk SSSR, pages 294–297, 1984.
  * [Dub08] Boris A. Dubrovin.  On universality of critical behaviour in Hamiltonian PDEs.  arXiv e-prints, page arXiv:0804.3790, April 2008.
  * [Dub14] Boris A. Dubrovin.  Gromov–Witten invariants and integrable hierarchies of topological type.  In Topology, geometry, integrable systems, and mathematical physics, volume 234 of Amer. Math. Soc. Transl. Ser. 2, pages 141–171. Amer. Math. Soc., Providence, RI, 2014.
  * [DZ01] Boris A. Dubrovin and Youjin Zhang.  Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants.  8 2001.
  * [EHX97] Tohru Eguchi, Kentaro Hori, and Chuan-Sheng Xiong.  Quantum cohomology and Virasoro algebra.  Phys. Lett. B, 402(1-2):71–80, 1997.
  * [EHY95] Tohru Eguchi, Kentaro Hori, and Sung-Kil Yang.  Topological $\sigma$ models and large-$N$ matrix integral.  Internat. J. Modern Phys. A, 10(29):4203–4224, 1995.
  * [EY94] Tohru Eguchi and Sung-Kil Yang.  The topological ${\bf C}{\rm P}^{1}$ model and the large-$N$ matrix integral.  Modern Phys. Lett. A, 9(31):2893–2902, 1994.
  * [EYY95] Tohru Eguchi, Yasuhiko Yamada, and Sung-Kil Yang.  On the genus expansion in the topological string theory.  Rev. Math. Phys., 7(3):279–309, 1995.
  * [FP00] Carel F. Faber and Rahul Pandharipande.  Hodge integrals and Gromov–Witten theory.  Invent. Math., 139(1):173–199, 2000.
  * [GV98] Rajesh Gopakumar and Cumrun Vafa.  M theory and topological strings. 1.  9 1998.
  * [GV99] Rajesh Gopakumar and Cumrun Vafa.  On the gauge theory/geometry correspondence.  Adv. Theor. Math. Phys., 3(5):1415–1443, 1999.
  * [IKT18] Kohei Iwaki, Tatsuya Koike, and Yumiko Takei.  Voros coefficients for the hypergeometric differential equations and Eynard–Orantin’s topological recursion – Part I: For the Weber equation.  arXiv preprint arXiv:1805.10945, 2018.
  * [Kon92] Maxim Kontsevich.  Intersection theory on the moduli space of curves and the matrix Airy function.  Comm. Math. Phys., 147(1):1–23, 1992.
  * [Oko00] Andrei Okounkov.  Toda equations for Hurwitz numbers.  Math. Res. Lett., 7(4):447–453, 2000.
  * [Pan00] Rahul Pandharipande.  The Toda equations and the Gromov–Witten theory of the Riemann sphere.  Lett. Math. Phys., 53(1):59–74, 2000.
  * [Rui00] Simon N. M. Ruijsenaars.  On Barnes’ multiple zeta and gamma functions.  Adv. Math., 156(1):107–132, 2000.
  * [Tak13] Kanehisa Takasaki.  A modified melting crystal model and the Ablowitz–Ladik hierarchy.  Journal of Physics A: Mathematical and Theoretical, 46(24):245202, May 2013.
  * [Vek98] Vadym E. Vekslerchik.  Functional representation of the Ablowitz–Ladik hierarchy.  Journal of Physics A: Mathematical and General, 31(3):1087–1099, Jan 1998.
  * [Vek02] Vadym E. Vekslerchik.  Functional representation of the Ablowitz–Ladik hierarchy. II.  Journal of Nonlinear Mathematical Physics, 9(2):157–180, 2002.
  * [Wit91] Edward Witten.  Two-dimensional gravity and intersection theory on moduli space.  In Surveys in differential geometry (Cambridge, MA, 1990), pages 243–310. Lehigh Univ., Bethlehem, PA, 1991.