# Laplace transform of the $x-y$ symplectic transformation formula in
Topological Recursion

Alexander Hock  Mathematical Institute, University of Oxford, Andrew Wiles
Building, Woodstock Road, OX2 6GG, Oxford, UK email
<EMAIL_ADDRESS>
###### Abstract.

The functional relation coming from the $x-y$ symplectic transformation of
Topological Recursion has a lot of applications, for instance it is the higher
order moment-cumulant relation in free probability or can be used to compute
intersection numbers on the moduli space of complex curves. We derive the
Laplace transform of this functional relation, which has a very nice and
compact form as a formal power series in $\hbar$. We apply the Laplace
transformed formula to the Airy curve and the Lambert curve.

##  1\. Introduction

Topological Recursion (TR) is a universal structure which generates from the
so-called spectral curve a family of multi-differentials $\omega_{g,n}$ on the
spectral curve (Riemann surface). TR occurs to be related to seemingly
different areas of mathematics and mathematical physics. To give an incomplete
list, it is related to volumes of moduli spaces, Hurwitz numbers, intersection
numbers of moduli spaces, Gromov-Witten theory, enumerative combinatorics,
random matrix theory, quantum field theory on noncommutative spaces, free
probability and quantum knot theory [Eyn14].

Knowing properties which hold in general for any spectral curve can give
therefore new insight into the applications of TR. For instance, the multi-
differential $\omega_{g,n}$ are symmetric, but generated via a non-symmetric
formula. This symmetry is in almost all application obvious from the
beginning. Transforming the spectral curve under a symplectic transformation
can leave the $\omega_{g,n}$ invariant. From this one can deduce that certain
models for instance in random matrix theory are equivalent. However, there is
a very specific symplectic transformation the $x-y$ symplectic transformation
which leaves actually the spectral curve invariant but generates completely
different multi-differential. Recently, the relation between these two
different families of multi-differentials were found in its most simple
representation [Hoc22a, ABDB+22]. The result is a functional relation which
has already generalised the higher order moment-cumulant relation in free
probability [BCGF+21b]. Equivalently, this functional relation relates fully
simple and ordinary maps in enumerative combinatorics [BCGF21a, BDBKS21]. Due
to free probability, it can be understood as the qunatised version of a
moment-cumulant relation. The $x-y$ symplectic transformation also reproves
known results for intersection numbers on the moduli space of complex curves
$\overline{\mathcal{M}}_{g,n}$ and might give new algorithms or closed
formulas to compute them.

An other important tool in TR is the Laplace transformation. In [Eyn11a,
Eyn11b], it was shown that the Laplace transform of the $\omega_{g.n}$ has a
direct interpretation in terms of intersection numbers on
$\overline{\mathcal{M}}_{g,n}$. As an application of TR in Gromov-Witten
theory, or more generally, in topological string theory, the Laplace transform
has an interpretation as mirror symmetry. More precisely, the $A$-model and
$B$-model are two different approaches to studying the geometry of Calabi-Yau
manifolds in topological string theory. They are related through mirror
symmetry. Based on observations in [BKMP09], the $A$-model and $B$-model are
related to TR and the mirror map has the interpretation of the Laplace
transform (see also [EMS09, Mul12] for details).

Consequently, it is completely natural to bring together the functional
relation of the $x-y$ symplectic transformation and the Laplace transform.
Actually, we observe that the functional relation behaves very well under the
Laplace transformation. The differential operator in the functional relation
becomes after Laplace transform a multiplication which sums up perfectly in
terms of formal power series. The Laplace transform of $\omega_{g,n}$ and of
its not-necessarily connected sibling is given in Corollary 2.9 and
Proposition 2.6.

The functional relation is not valid in general for spectral curves with
logarithmic singularities. However, we consider the example of the Lambert
curve [BM08], which encodes Hurwitz numbers. After a small transformation of
this curve, we apply successfully the Laplace transform formula of the $x-y$
symplectic transformation to compute Hurwitz numbers. This is the first step
into generalising the $x-y$ symplectic transformation to spectral curves with
logarithmic singularities, which will have application in topological string
theory and quantum knot theory.

### Acknowledgement

This work was supported through the Walter-Benjamin fellowship111“Funded by
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
Project-ID 465029630.

##  2\. Formula for $x-y$ Symplectic Transformation in Topological Recursion

We start to recap the theory of TR and its property under $x-y$ symplectic
transformation in more details. TR is an algorithm which computes recursively
in the negative Euler characteristic $-\chi=2g+n-2$ from some initial data,
the so-called spectral curve, a family of multi-differentials denoted by
$\omega_{g,n}$, which are also commonly called correlators.

More precisely, the spectral curve is the tuple $(\Sigma,x,y,B)$, where
$\Sigma$ is a compact Riemann surface with $x,y:\Sigma\to\mathbb{C}$ are
meromorphic functions with simple and distinct ramification points on
$\Sigma$. The multi-differentials $\omega_{g,n}$ live on $\Sigma^{n}$ with
$\omega_{0,1}(z)=y(z)\,dx(z)$ and $\omega_{0,2}=B$, where $B$ is symmetric
with double pole on the diagonal and no residue, bi-residue 1 and normalised
such that the $A$-periods vanish. In particular for $\Sigma=\mathbb{P}^{1}$
the complex projective line (Riemann sphere), the bilinear differential is
$B(z_{1},z_{2})=\frac{dz_{1}\,dz_{2}}{(z_{2}-z_{2})^{2}}$.

Then for negative Euler characteristic $\chi<0$, all $\omega_{g,n}$ are
defined via [EO07a]

$\displaystyle\omega_{g,n+1}(I,z)$ (2.1)
$\displaystyle:=\sum_{\beta_{i}}\operatorname{Res}\displaylimits_{q\to\beta_{i}}K_{i}(z,q)\bigg{(}\omega_{g-1,n+2}(I,q,\sigma_{i}(q))+\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}g_{1}+g_{2}=g\\\
I_{1}\uplus I_{2}=I\\\
(g_{i},I_{i})\neq(0,\emptyset)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\omega_{g_{1},|I_{1}|+1}(I_{1},q)\omega_{g_{2},|I_{2}|+1}(I_{2},\sigma_{i}(q))\\!\bigg{)}.$

The following notation is used:

  * •

$I=\\{z_{1},\dots,z_{n}\\}$ is a collection of $n$ variables $z_{j}$

  * •

the ramification points $\beta_{i}$ of $x$ are defined by $dx(\beta_{i})=0$

  * •

the local Galois involution $\sigma_{i}\neq\mathrm{id}$ with
$x(q)=x(\sigma_{i}(q))$ is defined in the vicinity of $\beta_{i}$ with fixed
point $\beta_{i}$

  * •

the recursion kernel $K_{i}(z,q)$ is also locally defined in the vicinity of
$\beta_{i}$ by

$\displaystyle
K_{i}(z,q)=\frac{\frac{1}{2}\int^{q}_{\sigma_{i}(q)}B(z,\bullet)}{\omega_{0,1}(q)-\omega_{0,1}(\sigma_{i}(q))}.$

It is not obvious from the definition, but it turns out that all
$\omega_{g,n}$ are symmetric in its variables, and for $\chi<0$ all
$\omega_{g,n}$ have poles just located at the ramification points, with
vanishing residues. In other words, for $\chi<0$ the $\omega_{g,n}$ are exact
1-forms in each variable. We define the primitives

$\displaystyle\Phi_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n})):=\int_{o}^{z_{1}}...\int_{o}^{z_{n}}\omega_{g,n}(z_{1},...,z_{n})$
(2.2)

with base point $o$ which will not play any role. Note that $\Phi_{0,1}$ and
$\Phi_{0,2}$ might not be globally defined on $\Sigma$, since they may have
some branch cuts.

The special case $n=0$ is usually denoted by $\mathcal{F}^{(g)}=\omega_{g,0}$,
the so-called free energy of genus $g$. They can be obtained by
$\Phi_{0,1}(z)$. For negative Euler characteristic $\chi<0$, the free energies
are defined by:

$\displaystyle\sum_{\beta_{i}}\operatorname{Res}_{z\to\beta_{i}}\Phi_{0,1}(z)\omega_{g,1}(z)=(2-2g)\mathcal{F}^{(g)}\qquad
g>1.$ (2.3)

The free energies $\mathcal{F}^{(0)}$ and $\mathcal{F}^{(1)}$ obey a more
involved formula, see [EO07a].

For later purpose, we define also $W_{g,n}$ via

$\displaystyle
W_{g,n}(x_{1}(z_{1}),...,x_{n}(z_{n}))dx_{1}(z_{1})...dx_{n}(z_{n}):=\omega_{g,n}(z_{1},...,z_{n}).$
(2.4)

The genus summation with formal parameter $\hbar$ will be denoted by

$\displaystyle W_{n}(x_{1},...,x_{n}):=$
$\displaystyle\sum_{g=0}^{\infty}\hbar^{2g+n-2}W_{g,n}(x_{1},...,x_{n})$ (2.5)
$\displaystyle\Phi_{n}(x_{1},...,x_{n}):=$
$\displaystyle\sum_{g=0}^{\infty}\hbar^{2g+n-2}\Phi_{g,n}(x_{1},...,x_{n}).$
(2.6)

######  Example 2.1.

For $\Sigma=\mathbb{P}^{1}$ and $x$ unramified, i.e. $x$ has no ramification
point, then all $W_{g,n}=0$ for $\chi=2g+n-2<0$. The correlators with positive
Euler characteristic are

$\displaystyle W_{0,1}(x(z))=$ $\displaystyle y(z),$ $\displaystyle
W_{0,2}(x_{1}(z_{1}),x_{2}(z_{2}))=$
$\displaystyle\frac{1}{x_{1}^{\prime}(z_{1})x^{\prime}_{2}(z_{2})(z_{1}-z_{2})^{2}}.$

###  2.1. $x-y$ Symplectic Transformation

Symplectic transformations play a very important role in the theory of TR. All
transformations which leaves the symplectic form

$\displaystyle|dx\wedge dy|$

invariant are generated by the three transformations:

  * •

$(x,y)\to(x,y+R(x))$, where $R(x)$ is a rational function in $x$

  * •

$(x,y)\to(\frac{ax+b}{cx+d},\frac{(cx+d)^{2}}{ad-bc}y)$ with
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in SL_{2}(\mathbb{C})$

  * •

$(x,y)\to(y,x)$.

It was proved in [EO07b, EO13] that for meromorphic $x,y$ the free energies
are invariant (up to a known normalisation constant) under all symplectic
transformations.

The same is not true for $\omega_{g,n}$! Note however that the first two
symplectic transformations listed above leave indeed $\omega_{g,n}$ for
$\chi<0$ invariant. This happens due to the fact that $x$ and $y$ enter the
recursion in (2.1) just through the recursion kernel and the ramification
points $\beta_{i}$ of $x$ which are both invariant. New inside was achieved
recently on the third symplectic transformation, the so-called $x-y$
symplectic transformation.

In a series of papers [BDBKS20, BDBKS22b, BDBKS22a], Bychkov et al. derived an
involved formula for two different sets of connected moments coming from
topological partition functions. This functional relation between two families
of moments was proved to be the $x-y$ symplectic transformation for a certain
type of matrix models [BDBKS21]. Due to the relation between random matrix
theory and free probability, this functional relation was shown [BCGF+21b] to
give the moment-cumulant relation in free probability. A fairly simple version
of the same functional relation for genus $g=0$ was derived in [Hoc22b] with a
completely different technique via loop insertion operator, and for all genus
in [Hoc22a]. Shortly later, Alexandrov et al. [ABDB+22] have shown that the
simplified version from [Hoc22a] is indeed the symplectic transformation for
any meromorphic $x,y$.

To formulate the functional relation, we define for the spectral curve
$(\Sigma,y,x,B)$ (now with $x$ and $y$ interchanged) the corresponding multi-
differential with $\omega_{g,n}^{\vee}$. More precisely,
$\omega_{0,1}^{\vee}(z)=x(z)dy(z)$,
$\omega_{0,2}^{\vee}(z_{1},z_{2})=B(z_{1},z_{2})$ and all
$\omega_{g,n}^{\vee}$ via (2.1) with interchanged role of $x$ and $y$. This
means $\omega_{g,n}^{\vee}$ has poles just located at the ramification points
of $y$ for $\chi<0$. Similarly, we define in this setting

$\displaystyle
W^{\vee}_{g,n}(y_{1}(z_{1}),...,y_{n}(z_{n}))dy_{1}(z_{1})...dy_{n}(z_{n}):=$
$\displaystyle\omega_{g,n}^{\vee}(z_{1},...,z_{n})$ (2.7) $\displaystyle
W_{n}^{\vee}(y_{1},...,y_{n}):=$
$\displaystyle\sum_{g=0}^{\infty}\hbar^{2g+n-2}W^{\vee}_{g,n}(y_{1},...,y_{n})$
(2.8) $\displaystyle\Phi^{\vee}_{g,n}(y_{1}(z_{1}),...,y_{n}(z_{n})):=$
$\displaystyle\int_{o}^{z_{1}}...\int_{o}^{z_{n}}\omega_{g,n}^{\vee}(z_{1},...,z_{n})$
(2.9) $\displaystyle\Phi_{n}^{\vee}(y_{1},...,y_{n}):=$
$\displaystyle\sum_{g=0}^{\infty}\hbar^{2g+n-2}\Phi_{g,n}^{\vee}(y_{1},...,y_{n}).$
(2.10)

Next, we need the following graphs describing the structure of the functional
relation:

######  Definition 2.2.

Let $\mathcal{G}_{n}$ be the set of connected bicoloured graph $\Gamma$ with
$n$ $\bigcirc$-vertices and $\bullet$-vertices, such that the following holds:

  * -

the $\bigcirc$-vertices are labelled from $1,...,n$

  * -

edges are only connecting $\bullet$-vertices with $\bigcirc$-vertices

  * -

$\bullet$-vertices have valence $\geq 2$.

For a graph $\Gamma\in\mathcal{G}_{n}$, let $r_{i}(\Gamma)$ be the valence of
the $i^{\text{th}}$ $\bigcirc$-vertex.

Let $I\subset\\{1,...,n\\}$ be the set associated to a $\bullet$-vertex, where
$I$ is the set labellings of $\bigcirc$-vertices connected to this
$\bullet$-vertex. Let $\mathcal{I}(\Gamma)$ be the set of all sets $I$ for a
given graph $\Gamma\in\mathcal{G}_{n}$.

The automorphism group $\mathrm{Aut}(\Gamma)$ consists of permutations of
edges which preserve the structure of $\Gamma$ considering the labellings. A
graph $\Gamma\in\mathcal{G}_{n}$ is up to automorphisms completely
characterised by the set $\mathcal{I}(\Gamma)$.

Adapted to the definition and functions above, the functional relation reads
in a very compact form:

######  Theorem 2.3 ([Hoc22a, ABDB+22]).

Let $x,y$ be two meromorphic functions on a compact Riemann surface with
simple distinct ramification points, which generates via TR (2.1) the multi-
differentials $\omega_{g,n}$ and $\omega_{g,n}^{\vee}$ as above. Let
$\Phi_{n},\Phi_{n}^{\vee},W_{n},W_{n}^{\vee}$ be as above defined from
$\omega_{g,n}$ and $\omega_{g,n}^{\vee}$. For $I=\\{i_{1},...,i_{n}\\}$, let

$\displaystyle\hat{\Phi}^{\vee}_{n}(y_{I};\hbar,u_{I}):=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(\varepsilon_{i_{1}},...,\varepsilon_{i_{n}})\in\\{1,-1\\}^{n}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(-1)^{\\#(\varepsilon_{i}=-1)}\Phi^{\vee}_{n}\bigg{(}y_{i_{1}}+\varepsilon_{i_{1}}\frac{\hbar
u_{i_{1}}}{2},...,y_{i_{n}}+\varepsilon_{i_{n}}\frac{\hbar
u_{i_{n}}}{2}\bigg{)}$ (2.11)

and for $I=\\{i,i\\}$ (and genus zero spectral curve) the special case

$\displaystyle\hat{\Phi}^{\vee}_{n}(y_{I};\hbar,u_{I}):=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(\varepsilon_{1},\varepsilon_{2})\in\\{1,-1\\}^{2}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(-1)^{\\#(\varepsilon_{i}=-1)}\bigg{[}\Phi^{\vee}_{2}\bigg{(}y_{i}+\varepsilon_{1}\frac{\hbar
u_{i}}{2},y_{j}+\varepsilon_{2}\frac{\hbar u_{i}}{2}\bigg{)}$ (2.12)
$\displaystyle\qquad\qquad\qquad-\log\bigg{(}y_{i}+\varepsilon_{1}\frac{\hbar
u_{i}}{2}-y_{j}-\varepsilon_{2}\frac{\hbar u_{i}}{2}\bigg{)}\bigg{]}_{j=i}$

(for higher genus spectral curves the logarithm has to be replaces by the
appropriate Theta-function). Let further be $\hat{O}$ a differential operator
acting from the left

$\displaystyle\hat{O}(y_{i}):=$
$\displaystyle\sum_{m}\bigg{(}-\frac{\partial}{\partial{x_{i}}}\bigg{)}^{m}\bigg{(}-\frac{dy_{i}}{dx_{i}}\bigg{)}[u_{i}^{m}]\frac{\exp\bigg{(}\hat{\Phi}^{\vee}_{1}(y_{i};\hbar,u_{i})-x_{i}u_{i}\bigg{)}}{\hbar
u_{i}}.$ (2.13)

Then, the functional relation holds as a formal expansion in $\hbar$

$\displaystyle\boxed{W_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))=\sum_{\Gamma\in\mathcal{G}_{n}}\frac{1}{|\mathrm{Aut}(\Gamma)|}\prod_{i=1}^{n}\hat{O}(y_{i}(z_{i}))\prod_{I\in\mathcal{I}(\Gamma)}\hat{\Phi}^{\vee}_{n}(y_{I}(z_{I});\hbar,u_{I}).}$

###### Proof.

The functional relation stated in the theorem is slightly different from the
one in [Hoc22a, ABDB+22]. First of all, we have changed the role of $x$ and
$y$ comparing to [Hoc22a]. Next, as a formal expansion in $\hbar$, the weight
function $\hat{\Phi}^{\vee}_{n}(y_{I};\hbar,u_{I})$ can be written as

$\displaystyle\hat{\Phi}^{\vee}_{n}(y_{I};\hbar,u_{I})=$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\sum_{(\varepsilon_{i_{1}},...,\varepsilon_{i_{n}})\in\\{1,-1\\}^{n}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(-1)^{\\#(\varepsilon_{i}=-1)}\Phi^{\vee}_{n}\bigg{(}y_{i_{1}}+\varepsilon_{i_{1}}\frac{\hbar
u_{i_{1}}}{2},...,y_{i_{n}}+\varepsilon_{i_{n}}\frac{\hbar
u_{i_{n}}}{2}\bigg{)}$ $\displaystyle=$ $\displaystyle\bigg{(}\prod_{i\in
I}\hbar u_{i}S(\hbar
u_{i}\partial_{x_{i}})\bigg{)}\big{(}W_{n}(x_{I})\bigg{)},$

where $S(u)=\frac{e^{u/2}-e^{-u/2}}{u}$, see [Hoc22a] for more details.
Expanding the lhs and the rhs in $\hbar$ gives for each coefficient the
relation stated in [Hoc22a, ABDB+22]. ∎

Now, we still want to understand more properties of the functional relation of
Theorem 2.3.

######  Example 2.4.

Consider Example 2.1 with $x,y$ interchanged, i.e. $y$ is unramified. All
$W_{g,n}^{\vee}=0$ for $2g+n-2>0$. Therefore, all
$\hat{\Phi}^{\vee}_{n}(y_{I}(z_{I});\hbar,u_{I})=0$ for $n>2$. Let
$\mathcal{G}_{n}^{2}\subset\mathcal{G}_{n}$ be the set of graphs defined in
Definition 2.2 with just 2-valent $\bullet$-vertices, then

$\displaystyle
W_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))=\sum_{\Gamma\in\mathcal{G}_{n}^{2}}\frac{1}{|\mathrm{Aut}(\Gamma)|}\prod_{i=1}^{n}\hat{O}(y_{i}(z_{i}))\prod_{I\in\mathcal{I}(\Gamma)}\hat{\Phi}^{\vee}_{2}(y_{I}(z_{I});\hbar,u_{I}).$

Note that all diagonal
$\hat{\Phi}^{\vee}_{2}(y_{i}(z_{i}),y_{i}(z_{i});\hbar,u_{i},u_{i})$ as
defined in (2.12) can be included in the exponential of the Operator $\hat{O}$
defined in (2.13). Also the other
$\hat{\Phi}^{\vee}_{2}(y_{i}(z_{i}),y_{j}(z_{j});\hbar,u_{i},u_{j})$ with
$i\neq j$ can be collected as an exponential such that multiple
$\bullet$-vertices connecting the same $\bigcirc$-vertices are generated
through expansion of this exponential (this was already discussed in [ABDB+22,
§7]). The symmetry factor becomes redundant. Thus, we have the alternative
form

$\displaystyle
W_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))=\prod_{i=1}^{n}\hat{O}^{2}(y_{i}(z_{i}))\sum_{\Gamma\in\tilde{\mathcal{G}}_{n}^{2}}\prod_{I\in\mathcal{I}(\Gamma)}\bigg{(}e^{\hat{\Phi}^{\vee}_{2}(y_{I}(z_{I});\hbar,u_{I})}-1\bigg{)},$

where
$\tilde{\mathcal{G}}_{n}^{2}\subset\mathcal{G}_{n}^{2}\subset\mathcal{G}_{n}$
is the set of graphs defined in Definition 2.2 with 2-valent
$\bullet$-vertices just connecting two different $\bigcirc$-vertices and at
most one $\bullet$-vertex connects the same $\bigcirc$-vertices. Equivalently,
$\tilde{\mathcal{G}}_{n}^{2}$ is the set of connected labeled graphs with $n$
vertices (A001187). The modified operator is

$\displaystyle\hat{O}^{2}(y_{i}):=$
$\displaystyle\sum_{m}\bigg{(}-\frac{\partial}{\partial{x_{i}}}\bigg{)}^{m}\bigg{(}-\frac{dy_{i}}{dx_{i}}\bigg{)}[u_{i}^{m}]\frac{\exp\bigg{(}\hat{\Phi}^{\vee}_{1}(y_{i};\hbar,u_{i})-x_{i}u_{i}+\frac{1}{2}\hat{\Phi}^{\vee}_{2}(y_{i},y_{i};\hbar,u_{i},u_{i})\bigg{)}}{\hbar
u_{i}}.$

The symmetry factor $\frac{1}{2}$ inside the exponential comes from the
automorphism $\mathrm{Aut}(\Gamma)$ swapping the two edge of a single
$\bullet$-vertex connected to the same $\bigcirc$-vertex. The permutation of
$k$ $\bullet$-vertices connected to the same $\bigcirc$-vertex is $k!$ and
also an automorphism in $\mathrm{Aut}(\Gamma)$, which is collected in the
expansion of the exponentials.

The example shows that terms coming from the graph expansion can be nicely
collected in an exponential. This is not a surprise since the original
derivation came indeed from not-necessarily-connected correlators [BDBKS22a].
Rephrasing these computational steps backwards, a even more compact formula
can be provided for the not-necessarily-connected correlators
$\overset{\circ}{W}_{n}$ defined by

$\displaystyle\overset{\circ}{W}_{n}(x_{I}):=\sum_{\lambda\vdash
I}\prod_{i=1}^{l(\lambda)}W_{|\lambda_{i}|}(x_{\lambda_{i}}),$ (2.14)

where $\lambda\vdash I$ is a set partition of $I=\\{1,...,n\\}$, i.e.
$\lambda=\\{\lambda_{1},...,\lambda_{l(\lambda)}\\}$ of length $l(\lambda)$
and blocks $\lambda_{i}\subset I$. Note that $\overset{\circ}{W}_{n}(x_{I})$
has for all $n>1$ and at each order in $\hbar$ in general poles on the
diagonal on the variables $z_{i},z_{j}$.

######  Corollary 2.5.

The not-necessarily-connected correlators $\overset{\circ}{W}_{n}$ satisfy the
functional relation as formal expansion in $\hbar$

$\displaystyle\overset{\circ}{W}_{n}$
$\displaystyle(x_{I}(z_{I}))=\sum_{m_{1},...,m_{n}}\prod_{i=1}^{n}\bigg{(}-\frac{\partial}{\partial{x_{i}(z_{i})}}\bigg{)}^{m_{i}}\bigg{(}-\frac{dy_{i}(z_{i})}{dx_{i}(z_{i})}\bigg{)}[u_{i}^{m_{i}}]\frac{1}{\hbar
u_{i}}$ (2.15) $\displaystyle\times\exp\bigg{(}\sum_{k\geq
1}\frac{1}{k!}\sum_{i_{1},...,i_{k}=1}^{n}\hat{\Phi}^{\vee}_{k}(y_{i_{1}}(z_{i_{1}}),...,y_{i_{k}}(z_{i_{k}});\hbar,u_{i_{1}},...,u_{i_{k}})$
$\displaystyle\qquad\qquad-\sum_{i=1}^{n}u_{i}x_{i}(z_{i})\bigg{)}.$

###### Proof.

Since $\overset{\circ}{W}_{n}(x_{I})$ includes all not-necessarily-connected
correlators, it is a rather classical result that the exponential generates
these from the connected correlators (very similar to the discussion in
Example 2.4). The automorphisms are split in two groups. The first is
generating the symmetry factors $\frac{1}{k_{i}!}$ permuting a
$\bullet$-vertex connected with $k_{i}$ edges to the $i$-th $\bigcirc$-vertex.
The second permutes the same $\bullet$-vertices with the same number of edge
connecting to the some $\bigcirc$-vertices, which is collected in the
expansion of the exponential. Since all $\hat{\Phi}^{\vee}_{k}$ are symmetric,
we can reorder the summation in the exponential (via multinomial theorem) as

$\displaystyle\sum_{\begin{subarray}{c}k_{1},...,k_{n}\geq 0\\\
k_{1}+...+k_{n}=k>0\end{subarray}}\frac{\hat{\Phi}^{\vee}_{k}(\overbrace{y_{1}(z_{1}),...,y_{1}(z_{1})}^{k_{1}},...,\overbrace{y_{n}(z_{n}),...,y_{n}(z_{n})}^{k_{n}};\hbar,u_{1},...)}{k_{1}!...k_{n}!}$
$\displaystyle=$ $\displaystyle\sum_{k\geq
1}\frac{1}{k!}\sum_{i_{1},...,i_{k}=1}^{n}\hat{\Phi}^{\vee}_{k}(y_{i_{1}}(z_{i_{1}}),...,y_{i_{k}}(z_{i_{k}});\hbar,u_{i_{1}},...,u_{i_{k}}),$

and get the claimed result. ∎

We observe that the not-necessarily-connected correlator
$\overset{\circ}{W}_{n}$, which are generated by the spectral curve $(x,y)$,
are related to the connected correlators $W_{g,n}^{\vee}$ generated by the
spectral curve $(y,x)$ through a differential operator acting on an
exponential. However, the form of the differential operator taking as $m$-th
derivative of the $m$-th order in the $u$ expansion can have a more
transparent explanation. Furthermore, we will also give insight on the factor
$\bigg{(}-\frac{dy_{i}(z_{i})}{dx_{i}(z_{i})}\bigg{)}$ through some formal
observations.

###  2.2. Laplace transform of the $x-y$ symplectic transformation

Let us look at the Laplace transform of (2.15). The integration path $\gamma$
on the Riemann surface depends on $x(z),y(z)$. We will not be precise about
this path but want rather look at the following formal manipulations:

######  Proposition 2.6.

Assume paths $\gamma_{i}\ni z_{i}$ exist such that
$\overset{\circ}{W}_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ is analytic on
$\gamma_{i}$ and the integrand vanishes fast enough at its boundary values.
Assume further that the Laplace transform of $\overset{\circ}{W}_{n}(x_{I})$
along the paths $\gamma_{i}$ converges for each coefficient in $\hbar$, then
the Laplace transform as a formal expansion in $\hbar$ reads

$\displaystyle\int_{\gamma_{1}}dx_{1}(z_{1})e^{-\mu_{1}x_{1}(z_{1})}...\int_{\gamma_{n}}dx_{n}(z_{n})e^{-\mu_{n}x_{n}(z_{n})}\overset{\circ}{W}_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$
$\displaystyle=$
$\displaystyle\int_{\gamma_{1}}\frac{dy_{1}(z_{1})}{\hbar\mu_{1}}...\int_{\gamma_{n}}\frac{dy_{n}(z_{n})}{\hbar\mu_{n}}\exp\bigg{(}\sum_{k\geq
1}\frac{1}{k!}\sum_{i_{1},...,i_{k}=1}^{n}\hat{\Phi}^{\vee}_{k}(y_{i_{1}}(z_{i_{1}}),...,y_{i_{k}}(z_{i_{k}});\hbar,-\mu_{i_{1}},...,-\mu_{i_{k}})\bigg{)}.$

###### Proof.

Take Corollary 2.5 and multiply it with
$\prod_{i=1}^{n}e^{-\mu_{i}x_{i}(z_{i})}dx_{i}(z_{i})$ and integrate over
$\gamma_{i}$ in each variable $z_{i}$. Due to the assumptions, we can
integrate by parts in each variable $x_{i}(z_{i})$ exactly $m_{i}$ times such
that all boundary terms vanish. This yields

$\displaystyle\int_{\gamma_{1}}dx_{1}(z_{1})e^{-\mu_{1}x_{1}(z_{1})}...\int_{\gamma_{n}}dx_{n}(z_{n})e^{-\mu_{n}x_{n}(z_{n})}\overset{\circ}{W}_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$
$\displaystyle=$
$\displaystyle\int_{\gamma_{1}}dx_{1}(z_{1})e^{-\mu_{1}x_{1}(z_{1})}...\int_{\gamma_{n}}dx_{n}(z_{n})e^{-\mu_{n}x_{n}(z_{n})}\sum_{m_{1},...,m_{n}\geq
0}\prod_{i=1}^{n}\bigg{(}-\mu_{i}\bigg{)}^{m_{i}}\bigg{(}-\frac{dy_{i}(z_{i})}{dx_{i}(z_{i})}\bigg{)}[u_{i}^{m_{i}}]$
$\displaystyle\times\frac{1}{\hbar u_{i}}\exp\bigg{(}\sum_{k\geq
1}\frac{1}{k!}\sum_{i_{1},...,i_{k}=1}^{n}\hat{\Phi}^{\vee}_{k}(y_{i_{1}}(z_{i_{1}}),...,y_{i_{k}}(z_{i_{k}});\hbar,u_{i_{1}},...,u_{i_{k}})-\sum_{i=1}^{n}u_{i}x_{i}(z_{i})\bigg{)}.$

As a formal expression in $u_{i}$, we can just substitute all $u_{i}$ with
$-\mu_{i}$ and ignore the $m_{i}$ summation. The factors
$e^{-\mu_{1}x_{1}(z_{1})}$ are cancelled by
$e^{-\sum_{i=1}^{n}u_{i}x_{i}(z_{i})}$ for $u_{i}=-\mu_{i}$. Changing the
integration variable $dx_{i}(z_{i})\to dy_{i}(z_{i})$ finishes the proof. ∎

The proposition gives the most compact and transparent relation between the
correlators $W_{g,n}$ and $W_{g,n}^{\vee}$ generated via TR with spectral
curve $(x,y)$ and $(y,x)$, respectively. We see explicitly that taking the
$m_{i}$-th derivative of a formal expansion of the $m_{i}$-th coefficient is a
very natural operation for it Laplace transform. The appearance of the factor
$-\frac{dy_{i}(z_{i})}{dx_{i}(z_{i})}$ yields the final change of integration
variables. However, the Laplace transform depends on the different paths
$\gamma_{i}$, and interchanging them $\gamma_{i}\to\gamma_{\sigma(i)}$ under
some permutation gives first of all a different Laplace transform, since
$\overset{\circ}{W}_{n}$ has poles on the diagonal and residues at the
diagonal will not necessarily vanish.

######  Example 2.7.

The leading order in $\hbar$ of Proposition 2.6 is $\hbar^{-n}$. The lhs of
the equation expands into
$\overset{\circ}{W}_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))=\prod_{i=1}^{n}\frac{y_{i}(z_{i})}{\hbar}+\mathcal{O}(\hbar^{-n+1})$,
and for the rhs the argument of the exponential expands at leading order to

$\displaystyle\sum_{i=1}^{n}\hat{\Phi}^{\vee}_{1}(y_{i}(z_{i});\hbar;-\mu_{i})+\mathcal{O}(\hbar)$
$\displaystyle=$
$\displaystyle\frac{1}{\hbar}\sum_{i=1}^{n}\bigg{(}\Phi^{\vee}_{1}\bigg{(}y_{i}(z_{i})-\frac{\hbar\mu_{i}}{2}\bigg{)}-\Phi^{\vee}_{1}\bigg{(}y_{i}(z_{i})+\frac{\hbar\mu_{i}}{2}\bigg{)}\bigg{)}+\mathcal{O}(\hbar)$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\mu_{i}x_{i}(z_{i})+\mathcal{O}(\hbar).$

Putting everything together, the leading order reads

$\displaystyle\int_{\gamma_{1}}dx_{1}(z_{1})e^{-\mu_{1}x_{1}(z_{1})}y_{1}(z_{1})...\int_{\gamma_{n}}dx_{n}(z_{n})e^{-\mu_{n}x_{n}(z_{n})}y_{n}(z_{n})$
$\displaystyle=$
$\displaystyle\int_{\gamma_{1}}\frac{dy_{1}(z_{1})}{\mu_{1}}e^{-\mu_{1}x_{1}(z_{1})}...\int_{\gamma_{n}}\frac{dy_{n}(z_{n})}{\mu_{n}}e^{-\mu_{n}x_{n}(z_{n})}$

which is correct under the assumptions and integration by parts.

######  Example 2.8.

Consider the same situation as in Example 2.1 and 2.4, this is $y$ is
unramified thus all $W_{g,n}^{\vee}=0$ for $2g+n-2>0$. As before, this implies
that all $\hat{\Phi}^{\vee}_{k}=0$ for $k>2$. We conclude for this case from
Proposition 2.6

$\displaystyle\int_{\gamma_{1}}dx_{1}(z_{1})e^{-\mu_{1}x_{1}(z_{1})}...\int_{\gamma_{n}}dx_{n}(z_{n})e^{-\mu_{n}x_{n}(z_{n})}\overset{\circ}{W}_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$
$\displaystyle=$
$\displaystyle\int_{\gamma_{1}}\frac{dy_{1}(z_{1})}{\hbar\mu_{1}}...\int_{\gamma_{n}}\frac{dy_{n}(z_{n})}{\hbar\mu_{n}}\exp\bigg{(}\sum_{i=1}^{n}\hat{\Phi}^{\vee}_{1}(y_{i}(z_{i});\hbar;-\mu_{i})+\frac{1}{2}\sum_{i,j=1}^{n}\hat{\Phi}^{\vee}_{2}(y_{i}(z_{i}),y_{j}(z_{j});\hbar;-\mu_{i},-\mu_{j})\bigg{)}.$

Note that on the diagonal
$\hat{\Phi}^{\vee}_{2}(y_{i}(z_{i}),y_{i}(z_{i});\hbar;\mu_{i},\mu_{i})$, we
have to take the normalised primitives (2.12).

The derivation above is also valid for connected correlators $W_{n}$ with
exactly the same steps. We just state the result:

######  Corollary 2.9.

Assume a paths $\gamma_{i}$ exist such that
$W_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$ is analytic on $\gamma_{i}$ and the
integrand vanishes fast enough at its boundary values. Assume the Laplace
transform of ${W}_{n}(x_{I})$ along the path $\gamma$ converges for each
coefficient in $\hbar$, then the Laplace transform as a formal expansion in
$\hbar$ reads

$\displaystyle\int_{\gamma_{1}}dx_{1}(z_{1})e^{-\mu_{1}x_{1}(z_{1})}...\int_{\gamma_{n}}dx_{n}(z_{n})e^{-\mu_{n}x_{n}(z_{n})}{W}_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$
$\displaystyle=$
$\displaystyle\int_{\gamma_{1}}\frac{dy_{1}(z_{1})}{\hbar\mu_{1}}...\int_{\gamma_{n}}\frac{dy_{n}(z_{n})}{\hbar\mu_{n}}\exp\bigg{(}\sum_{i=1}^{n}\hat{\Phi}^{\vee}_{1}(y_{i}(z_{i});\hbar,-\mu_{i})\bigg{)}\sum_{\Gamma\in\mathcal{G}_{n}}\frac{\prod_{I\in\mathcal{I}(\Gamma)}\hat{\Phi}^{\vee}_{|I|}(y_{I}(z_{I});\hbar,-\mu_{I})}{|\mathrm{Aut}(\Gamma)|}.$

######  Example 2.10.

Continue the series of Example 2.1, 2.4 and 2.8, where $y$ is unramified. The
Laplace transform of the connected correlator reads

$\displaystyle\int_{\gamma_{1}}dx_{1}(z_{1})e^{-\mu_{1}x_{1}(z_{1})}...\int_{\gamma_{n}}dx_{n}(z_{n})e^{-\mu_{n}x_{n}(z_{n})}{W}_{n}(x_{1}(z_{1}),...,x_{n}(z_{n}))$
$\displaystyle=$
$\displaystyle\int_{\gamma_{1}}\frac{dy_{1}(z_{1})}{\hbar\mu_{1}}...\int_{\gamma_{n}}\frac{dy_{n}(z_{n})}{\hbar\mu_{n}}\exp\bigg{(}\sum_{i=1}^{n}\hat{\Phi}^{\vee}_{1}(y_{i}(z_{i});\hbar,-\mu_{i})+\frac{1}{2}\hat{\Phi}^{\vee}_{2}(y_{i}(z_{i}),y_{i}(z_{i});\hbar,-\mu_{i},-\mu_{i})\bigg{)}$
$\displaystyle\qquad\qquad\qquad\times\sum_{\Gamma\in\tilde{\mathcal{G}}^{2}_{n}}\prod_{I\in\mathcal{I}(\Gamma)}\bigg{(}e^{\hat{\Phi}^{\vee}_{2}(y_{I}(z_{I});\hbar,-\mu_{I})}-1\bigg{)},$

where $\tilde{\mathcal{G}}^{2}_{n}\subset\mathcal{G}_{n}$ is the set of graphs
with 2-valent $\bullet$-vertices connecting two different $\bigcirc$-vertices
at most with one $\bullet$-vertex which is nothing than the set of connected
graphs with $n$ labelled vertices (A001187).

######  Remark 2.11.

The definition of the Laplace transform in several variables indicates a
possible dependence on the different paths $\gamma_{i}$. Permuting or
geometrically moving two paths $\gamma_{i},\gamma_{j}$ along each other would
pick a non-trivial residue at the diagonal $z_{i}=z_{j}$. However, it turns
out that after computing all path integrals these residues cancel out. This is
related to the fact that the lhs of the functional relation of Theorem 2.3 has
no pole at the diagonal, even though the rhs is indicating it.

###  2.3. Logarithmic $x,y$

In the previous subsection, logarithmic behaviour of $x$ or $y$ was excluded.
However, the TR itself can also be applied to logarithmic $x$ and $y$.
Actually, a lot of very important examples related for instance to Gromov-
Witten theory have logarithms. To verify that the functional relation does not
hold in general for logarithmic $x,y$, we take the example of the so-called
Lambert curve [BM08, EMS09]. This curve encodes Hurwitz numbers, which will be
explained in more details later. The curve is defined via

$\displaystyle x(z)=-z+\log(z),\qquad y(z)=z.$ (2.16)

The spectral curve is of genus zero such that the bilinear differential is
$\omega_{0,2}(z_{1},z_{2})=\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}}$. Starting
with these two functions, the formula of TR (2.1) generates all $W_{g,n}$ and
also $W^{\vee}_{g,n}$. One might check if the functional relation holds for
some examples.

Since $y$ is unramified, this curve reflects the examples of the previous
subsection, i.e. $W^{\vee}_{g,n}=0$ for all $-\chi<2g+n-2$. A short
computation of (2.1) gives for instance

$\displaystyle
W_{1,1}(x(z))=\frac{1}{x^{\prime}(z)}\operatorname{Res}\displaylimits_{q\to
1}K_{i}(z,q)\omega_{0,2}(q,\sigma_{i}(q))=\frac{z^{2}(z-4)}{24(z-1)^{5}}.$

On the other hand, the $(g,n)=(1,1)$-example of the functional relation is

$\displaystyle W_{1,1}(x(z))=$
$\displaystyle-\frac{dy(z)}{dx(z)}W^{\vee}_{1,1}(y(z))+\frac{1}{2}\frac{d}{dx(z)}\bigg{(}\frac{dy(z)}{dx(z)}\hat{W}^{\vee}_{2,0}(y(z),y(z))\bigg{)}-\frac{1}{24}\frac{d^{3}}{dx(z)^{3}}\bigg{(}\frac{1}{\frac{dy(z)}{dx(z)}}\bigg{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{24}\frac{d^{3}}{dx(z)^{3}}\bigg{(}\frac{1}{\frac{dy(z)}{dx(z)}}\bigg{)}$
$\displaystyle=$ $\displaystyle\frac{-6z^{2}+4z-1}{24z(z-1)^{5}}$

where $W^{\vee}_{1,1}(y(z))=0$ and
$\hat{W}^{\vee}_{2,0}(y_{1},y_{2})=W^{\vee}_{2,0}(y_{1},y_{2})-\frac{1}{(y_{1}-y_{2})^{2}}=0$
vanish such that the last term contributes only. This is a clear discrepancy
to the direct computation.

However, if we look at a symplectic transformation of this curve by
transforming

$\displaystyle y\to\tilde{y}=y+x,$ (2.17)

the validity of the functional relation can be rescued. This means we start
with the curve

$\displaystyle x(z)=-z+\log(z),\qquad\tilde{y}(z)=\log(z).$ (2.18)

Let us denote the corresponding correlators with $\tilde{W}_{g,n}$, which are
equal to $W_{g,n}$ of the curve (2.16) due to symplectic transformation. Thus,
the formula of TR yields obviously again

$\displaystyle\tilde{W}_{1,1}(x(z))=\frac{z^{2}(z-4)}{24(z-1)^{5}}$

due to the invariance of the kernel under this transformation. After $x-y$
symplectic transformation, i.e. looking at the correlators
$\tilde{W}^{\vee}_{g,n}$, we see that $\tilde{W}^{\vee}_{g,n}=0$ for all
$-\chi<2g+n-2$ since $y$ is unramified again. However, the regularised
correlators $\hat{\tilde{W}}^{\vee}_{0,2}$ on the diagonal does not vanish. It
is

$\displaystyle\lim_{z^{\prime}\to
z}\frac{1}{\tilde{y}^{\prime}(z)\tilde{y}^{\prime}(z^{\prime})(z-z^{\prime})^{2}}-\frac{1}{(\tilde{y}(z)-\tilde{y}(z^{\prime}))^{2}}=-\frac{1}{12}.$

Inserting everything into the functional relation for $(g,n)=(1,1)$, we find

$\displaystyle\tilde{W}_{1,1}(x(z))=$
$\displaystyle-\frac{d\tilde{y}(z)}{dx(z)}\tilde{W}^{\vee}_{1,1}(y(z))+\frac{1}{2}\frac{d}{dx(z)}\bigg{(}\frac{d\tilde{y}(z)}{dx(z)}\hat{\tilde{W}}^{\vee}_{2,0}(\tilde{y}(z),\tilde{y}(z))\bigg{)}-\frac{1}{24}\frac{d^{3}}{dx(z)^{3}}\bigg{(}\frac{1}{\frac{d\tilde{y}(z)}{dx(z)}}\bigg{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{24}\frac{d}{dx(z)}\bigg{(}\frac{d\tilde{y}(z)}{dx(z)}\bigg{)}-\frac{1}{24}\frac{d^{3}}{dx(z)^{3}}\bigg{(}\frac{1}{\frac{d\tilde{y}(z)}{dx(z)}}\bigg{)}$
$\displaystyle=$ $\displaystyle\frac{z^{2}(z-4)}{24(z-1)^{5}}$

which coincides with the direct computation from TR.

This example reveals that considering logarithmic $x,y$ the $x-y$ symplectic
transformation does not hold in general. But on the other hand, taking the
correct symplectic transformation (2.17) before, it actually can hold. One may
ask if the Laplace transform of Proposition 2.6 and Corollary (2.9) is still
valid if the functional relation is satisfied including logarithms for $x,y$
as it is for the curve (2.18).

######  Remark 2.12.

For logarithmic behaviour of $x,y$, the contour $\gamma$ may cross the branch
cut of the logarithm. The observation of Example 2.7 gives us an alternative
way making sense of the Laplace transform even for $(g,n)=(0,1)$ including
logarithms for $x,y$. Therefore, we define

$\displaystyle\int_{\gamma}dx(z)e^{-\mu
x(z)}y(z):=\int_{\gamma}\frac{dy(z)}{\mu}e^{-\mu x(z)},$

if $y$ has a logarithm and $\gamma$ crosses the branch cut. This avoids to
split the integration contour at the cut and include boundary terms to
regularise the lhs, which is well-defined and could be done in principle.

##  3\. Application to Intersection Numbers on $\overline{\mathcal{M}}_{g,n}$

This section recalls some examples about the connection between TR and
intersection theory on $\overline{\mathcal{M}}_{g,n}$ and applies the derived
formulas. We refer to [Eyn11a, Eyn11b] for much more information.

Let $\overline{\mathcal{M}}_{g,n}$ be the compactified moduli space of complex
curves of genus $g$ with $n$ labelled points. It is compactified in the sense
Deligne and Mumford [DM69]. $\overline{\mathcal{M}}_{g,n}$ is a complex
orbifold of dimension $d_{g,n}=3g-3+n$. Its points
$(C,p_{1},...,p_{n})\in\overline{\mathcal{M}}_{g,n}$ are isomorphism classes
of a complex curve $C$ with $n$ labelled point denoted by $p_{1},...,p_{n}$.
Let $L_{i}$ be the line bundle over $\overline{\mathcal{M}}_{g,n}$ whose fiber
is the contangent space $TC^{\vee}(p_{i})$ of $C$ at $p_{i}$. The first Chern
class of this line bundle is called the $\psi$-class

$\displaystyle\psi_{i}=c_{1}(L_{i}).$

The $\psi$-classes are the easiest examples for tautological classes in
$\overline{\mathcal{M}}_{g,n}$. One can build intersection numbers by wedging
several $\psi$-classes

$\displaystyle\langle\psi_{1}^{d_{1}}...\psi_{n}^{d_{n}}\rangle_{g,n}=\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}...\psi_{n}^{d_{n}}$
(3.1)

with $d_{g,n}=\sum_{i=1}^{n}d_{i}$, otherwise it is defined to vanish. It was
conjectured by Witten [Wit91] and shortly later proved by Kontsevich [Kon92]
that the generating function (3.1) satisfies the KdV equations, see for
instance [LZ04, Eyn16] for more details.

Kontsevich proved that a hermitian matrix model with external field generates
the intersection numbers (3.1) if one identify the parameters of their
generating function with certain coefficients of the matrix model. After
developing TR, this connection constructed by Kontsevich between matrix models
and intersection theory gave birth to the connection between TR and
intersection theory in general. The associated spectral curve $\psi$-class
intersection numbers is known under the name of Airy curve or Witten-
Kontsevich curve, defined by
$(\mathbb{P}^{1},,x=\frac{z^{2}}{2},y=z,\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$
[EO07a].

We will also discuss more general intersection numbers built from $\psi$\- and
Hodge-classes. Let
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ be the
forgetful morphism, forgetting the last labelled point, and $\omega_{\pi}$ the
relative dualising sheaf. Then, the Hodge-class is defined by

$\displaystyle\Lambda(\alpha)=1+\sum_{k=1}^{g}(-1)^{k}\alpha^{-k}c_{k}(\mathbb{E}),$
(3.2)

where $c_{k}$ is the $k$-th Chern class and $\mathbb{E}$ the Hodge bundle
given by pushforward of the relative dualising sheaf, i.e.
$\mathbb{E}=\pi_{*}(\omega_{\pi})$.

Intersection numbers of a mixture of $\psi$\- and Hodge-classes can be
considered and made considerable interest in the past. Especially in the
context of Hurwitz numbers, a relation between intersection numbers of
$\psi$\- and Hodge-classes and the counting problem of ramified coverings over
$\mathbb{P}^{1}$ with a certain ramification profil at infinity. This relation
is the celebrated ELSV formula [ELSV99]. More precisely, let
$h_{g;k\mu_{1},...,\mu_{n}}$ be the number of the equivalence classes of
topologically nonequivalent ramified coverings $f:C\to\mathbb{P}^{1}$, where
$C$ is compact, connected complex curve of genus $g$ and $f$ has ramification
profil $(\mu_{1},....,\mu_{n})$ over infinity and simple ramification else.
The ELSV formula relates the Hurwitz number $h_{g;\mu_{1},...,\mu_{n}}$ to the
following linear Hodge integral

$\displaystyle
h_{g;\mu_{1},...,\mu_{n}}=\frac{(2g+\mu+n-2)!}{|\mathrm{Aut}(\mu_{1},...,\mu_{n})|}\prod_{i=1}^{n}\frac{\mu_{i}^{\mu_{i}}}{\mu_{i}!}\int_{\overline{\mathcal{M}}_{g,n}}\frac{\Lambda(1)}{\prod_{i=1}^{n}(1-\mu_{i}\psi_{i})},$
(3.3)

where $|\mathrm{Aut}(\mu_{1},...,\mu_{n})|$ is the number of permutations
permuting equal $\mu_{i}$’s and $\mu=\mu_{1}+...+\mu_{n}$.

Bouchard and Marino conjectured [BM08] that these linear Hodge integrals
appearing in the ELSV formula (and therefore also Hurwitz numbers) can
actually be computed by TR with the so-called Lambert curve. This was proved
in [EMS09], where the spectral curve was given by
$(\mathbb{P}^{1}\setminus\mathbb{R}_{-},x=-z+\log(z),y=\log(z),\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$.
Note that we have shifted $y$ in the spectral curve due to the observation in
Sec. 2.3, which does not change the result of [EMS09].

Applying the Laplace transform of Corollary 2.9 to the Airy and/or Lambert
curve gives easy formulas to compute $\psi$\- and Hodge-class intersection
numbers.

###  3.1. Airy curve

For the Airy curve, the explicit relation between the correlators
$\omega_{g,n}$ and the intersection numbers is given by:

######  Theorem 3.1 ([EO07a]).

Let the Airy spectral curve be
$(\mathbb{P}^{1},x(z)=z^{2},y(z)=z,\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$,
then the correlator $\omega_{g,n}$ computed by TR generates the $\psi$-class
integral

$\displaystyle\omega_{g,n}(z_{1},...,z_{n})=\sum_{k_{1}+...+k_{n}=d_{g,n}}\langle\psi_{1}^{k_{1}}...\psi_{n}^{k_{n}}\rangle_{g,n}\prod_{i=1}^{n}\frac{(2k_{i}+1)!!}{z_{i}^{2k_{i}+2}}dz_{i}.$
(3.4)

The first step is two extract intersection numbers from $\omega_{g,n}$ through
a Laplace transform:

######  Lemma 3.2.

Let $\mu_{i}\in\mathbb{N}$ and $2g+n-2>0$, we have for $\omega_{g,n}$
generated by the Airy curve of Theorem 3.1

$\displaystyle\frac{1}{\sqrt{2\pi}^{n}}\int_{i\varepsilon-\infty}^{i\varepsilon+\infty}e^{-\mu_{1}x_{1}(z_{1})...-\mu_{n}x_{n}(z_{n})}\omega_{g,n}(z_{1},...,z_{n})=\bigg{\langle}\prod_{i=1}^{n}\frac{\sqrt{\mu_{i}}}{1-\mu_{i}\psi_{i}}\bigg{\rangle}_{g,n}.$
(3.5)

For $2g+n-2\leq 0$, we get
$\bigg{\langle}\frac{1}{(1-\mu\psi)}\bigg{\rangle}_{0,1}=\frac{1}{\mu^{2}}$
and
$\bigg{\langle}\frac{1}{(1-\mu_{1}\psi_{1})(1-\mu_{2}\psi_{2})}\bigg{\rangle}_{0,2}=\frac{1}{\mu_{1}+\mu_{2}}$.

###### Proof.

For $2g+n-2>0$ and each $\mu_{i}$, we apply the integral to the result of
Theorem 3.1. The integral is computed by a version of the well-known Gaussian
integral

$\displaystyle\frac{1}{\sqrt{2\pi}}\int_{i\varepsilon-\infty}^{i\varepsilon+\infty}dz\frac{e^{-\mu
z^{2}/2}}{z^{2k+2}}=\frac{\mu^{k+1/2}}{(2k+1)!!}.$ (3.6)

The sum over $k_{1}+...+k_{n}=d_{g,n}$ can be extended to a sum over all
$k_{i}\geq 0$, since all the intersection number are defined to vanish unless
the condition $k_{1}+...+k_{n}=d_{g,n}$ is satisfied. For each $k_{i}$
summation, we have a geometric series
$\sum_{k_{i}=0}^{\infty}\mu_{i}^{k_{i}}\psi^{k_{i}}=\frac{1}{1-\mu_{i}\psi_{i}}$.

For $2g+n-2\leq 0$, the computation is also straightforward. The Bergman
kernel is first expanded in a geometric series
$\frac{1}{(z_{1}-z_{2})^{2}}=\frac{1}{z_{1}^{2}}\sum_{n}n(\frac{z_{2}}{z_{1}})^{n-1}$,
then apply the Gaussian integral together with
$\frac{1}{(2k-1)!!(-2k-1)!!}=(-1)^{k}$ and write it again as geometric series
again. ∎

Now, we can apply the formulas developed in Sec. 2.2 to the Airy curve. First
of all note that the Airy curve $x(z)=z^{2},y(z)=z$ is not ramified through
$y$. We can easily stick to the examples discussed in Sec. 2.2 and just have
to compute the following primitives:

$\displaystyle\Phi_{1}^{\vee}(y(z))$ $\displaystyle=\frac{1}{\hbar}\int
x(z)dy(z)=\frac{z^{3}}{6\hbar}$ (3.7)
$\displaystyle\hat{\Phi}^{\vee}_{1}(y(z);\hbar,u)$
$\displaystyle=\Phi_{1}^{\vee}\bigg{(}y(z)+\frac{\hbar
u}{2}\bigg{)}-\Phi_{1}^{\vee}\bigg{(}y(z)-\frac{\hbar
u}{2}\bigg{)}=u\frac{z^{2}}{2}+\frac{\hbar^{2}u^{3}}{24}$ (3.8)
$\displaystyle\Phi_{2}^{\vee}(y_{1}(z_{1}),y_{2}(z_{2}))$
$\displaystyle=\log(z_{1}-z_{2})$ (3.9) $\displaystyle
e^{\hat{\Phi}_{2}^{\vee}(y_{1}(z_{1}),y_{2}(z_{2});\hbar,u_{1},u_{2})}-1$
$\displaystyle=\frac{\hbar^{2}u_{1}u_{2}}{(z_{1}-z_{2})^{2}-\frac{\hbar^{2}}{4}(u_{1}+u_{2})^{2}}.$
(3.10)

Taking Example 2.10 and Lemma 3.2 into account, we conclude:

######  Corollary 3.3.

The $\psi$-class intersection numbers can be computed as a formal expansion in
$\hbar$ via

$\displaystyle\bigg{\langle}\prod_{i=1}^{n}\frac{\sqrt{\mu_{i}}}{1-\mu_{i}\psi_{i}}\bigg{\rangle}_{g,n}$
$\displaystyle=$
$\displaystyle[\hbar^{2g+n-2}]\prod_{i=1}^{n}\frac{e^{-\frac{\hbar^{2}\mu_{i}^{3}}{24}}}{\sqrt{2\pi}}\int_{i\varepsilon-\infty}^{i\varepsilon+\infty}dz_{i}\frac{e^{-\mu_{i}\frac{z_{i}^{2}}{2}}}{\hbar\mu_{i}}\sum_{\Gamma\in\tilde{\mathcal{G}}^{2}_{n}}\prod_{\begin{subarray}{c}I\in\mathcal{I}(\Gamma)\\\
\\{i,j\\}=I\end{subarray}}\frac{\hbar^{2}\mu_{i}\mu_{j}}{(z_{i}-z_{j})^{2}-\frac{\hbar^{2}}{4}(\mu_{i}+\mu_{j})^{2}},$

where $\tilde{\mathcal{G}}^{2}_{n}\subset\mathcal{G}_{n}$ is the subset of
graphs with 2-valent $\bullet$-vertices connecting two different
$\bigcirc$-vertices at most with one $\bullet$-vertex which is nothing than
the set of connected graphs with $n$ labelled vertices (A001187).

For not-necessarily-connected correlators, we can conclude from Example 2.8
and Lemma 3.2:

######  Corollary 3.4.

The sum over all partitions of $\psi$-class intersection numbers satisfies as
a formal expansion in $\hbar$

$\displaystyle\sum_{\lambda\vdash
I}\prod_{j=1}^{l(\lambda)}\sum_{g_{j}=0}\hbar^{2g_{j}+|\lambda_{j}|-2}\bigg{\langle}\prod_{i=1}^{|\lambda_{j}|}\frac{\sqrt{\mu_{\lambda_{j}^{i}}}}{(1-\mu_{\lambda_{j}^{i}}\psi_{\lambda_{j}^{i}})}\bigg{\rangle}_{g_{j},|\lambda_{j}|}$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\frac{e^{-\frac{\hbar^{2}\mu_{i}^{3}}{24}}}{\sqrt{2\pi}}\int_{i\varepsilon-\infty}^{i\varepsilon+\infty}dz_{i}\frac{e^{-\mu_{i}\frac{z_{i}^{2}}{2}}}{\hbar\mu_{i}}\prod_{i<j}\frac{\hbar^{2}\mu_{i}\mu_{j}}{(z_{i}-z_{j})^{2}-\frac{\hbar^{2}}{4}(\mu_{i}+\mu_{j})^{2}},$

where $\lambda\vdash I$ is a set partition of $I$ with $l(\lambda)$ blocks
$\lambda_{j}\subset I$, i.e. $\lambda=(\lambda_{1},...,\lambda_{l(\lambda)})$.
Each block is written as
$\lambda_{j}=(\lambda_{j}^{1},...,\lambda_{j}^{|\lambda_{j}|})$ of cardinality
$|\lambda_{j}|$ and elements $\lambda_{j}^{i}\in I$.

The result of Corollary 3.3 and 3.4 are not claimed to be new formulas,
because these are equivalent to the one given in [ABDB+22, §7] after applying
the Laplace transform as explained in Sec. 2.2. So, performing the Gaussian
integrals, one usually manipulate the integrand through derivatives to get
formulas like (3.6), we just arrive at the formula directly induced by Theorem
3.1 together with Theorem 2.3 as stated in [ABDB+22, §7]. However, we are
still getting a new perspective on the computation of $\psi$-class
intersection numbers in terms of Gaussian integrals.

We want to emphasise also that the order of integration in Corollary 3.3 and
3.4 does not matter. This is related to the fact that the correlators
$\omega_{g,n}$ have just poles at the ramification points and not at the
diagonal for $2g+n-2>0$. The deep algebraic structure which reveals this
property is not yet understood, but it is present in the background of all
these formulas.

###  3.2. Lambert curve

In this subsection, we apply the formulas of Sec. 2.2 to the Lambert curve.
Note that these formulas are just proved for meromorphic $x,y$. The discussion
of Sec. 2.3 has however shown that for the Lambert curve a specific choice of
$y$ can actually work. Through this subsection, we have assigned an asterisk
to those corollaries which assume that for the Lambert curve of the form of
(2.18) satisfies the $x-y$ symplectic transformation. A lot of checks with
computer algebra have confirmed this assumption.

The precise relation between the correlators $\omega_{g,n}$ and the linear
Hodge integrals (or Hurwitz numbers) is given by:

######  Theorem 3.5 ([EMS09]).

Let the Lambert spectral curve be
$(\mathbb{P}^{1}\setminus\mathbb{R}_{-},x(z)=-z+\log(z),y(z)=\log(z),\frac{dz_{1}\,dz_{2}}{(z_{1}-z_{2})^{2}})$,
then the correlator $\omega_{g,n}$ computed by TR generates the linear Hodge
integrals

$\displaystyle\omega_{g,n}(z_{1},...,z_{n})=\sum_{k_{1},...,k_{n}\geq
0}\prod_{i=1}^{n}\frac{k_{i}^{k_{i}+1}}{k_{i}!}\bigg{\langle}\frac{\Lambda(1)}{\prod_{i=1}^{n}(1-k_{i}\psi_{i})}\bigg{\rangle}_{g,n}e^{k_{i}x_{i}(z_{i})}dx_{i}(z_{i})$
(3.11)

Computing the Laplace transform of (3.11), where $\gamma$ is a contour
encircling the origin, will separate the summands in (3.11):

######  Lemma 3.6.

Let $\mu_{i}\in\mathbb{N}$ and $2g+n-2>0$, we have for $\omega_{g,n}$
generated by the Lambert curve of Theorem 3.5

$\displaystyle\operatorname{Res}_{z_{i}=0}e^{-\mu_{1}x_{1}(z_{1})...-\mu_{n}x_{n}(z_{n})}\omega_{g,n}(z_{1},...,z_{n})=\prod_{i=1}^{n}\frac{\mu_{i}^{\mu_{i}+1}}{\mu_{i}!}\bigg{\langle}\frac{\Lambda(1)}{\prod_{i=1}^{n}(1-\mu_{i}\psi_{i})}\bigg{\rangle}_{g,n}.$
(3.12)

For $2g+n-2\leq 0$, we take that the lhs of (3.12) to be defined by this
equation with
$\bigg{\langle}\frac{\Lambda(1)}{(1-\mu\psi)}\bigg{\rangle}_{0,1}=\frac{1}{\mu^{2}}$
and
$\bigg{\langle}\frac{\Lambda(1)}{(1-\mu_{1}\psi_{1})(1-\mu_{2}\psi_{2})}\bigg{\rangle}_{0,2}=\frac{1}{\mu_{1}+\mu_{2}}$.

###### Proof.

For $2g+n-2>0$ and each $\mu_{i}$, we have the same computation

$\displaystyle\operatorname{Res}_{z=0}dx(z)e^{(k-\mu)x(z)}=\operatorname{Res}_{z=0}\frac{(1-z)dz}{z}\frac{e^{(\mu-k)z}}{z^{\mu-k}}=\delta_{\mu,k}.$

For $2g+n-2<0$, the residue at 0 is not well-defined. Thus, we define the
Laplace transform to match the classical known result. The case $(g,n)=(0,1)$
is related to Remark 2.12. ∎

Now, we can apply the formulas developed in Sec. 2.2 to the Lambert curve.
First of all note that the Lambert curve parametrised by
$x(z)=-z+\log(z),y(z)=\log(z)$, i.e.

$\displaystyle x=-e^{y}+y,$ (3.13)

is not ramified through $y$. We can easily stick to the examples discussed in
Sec. 2.2 and just have to compute the following primitives:

$\displaystyle\Phi_{1}^{\vee}(y(z))$ $\displaystyle=\frac{1}{\hbar}\int
x(z)dy(z)=\frac{-z+\frac{\log(z)^{2}}{2}}{\hbar}=\frac{-e^{y(z)}+\frac{y(z)^{2}}{2}}{\hbar}$
(3.14) $\displaystyle\hat{\Phi}^{\vee}_{1}(y(z);\hbar,u)$
$\displaystyle=\Phi_{1}^{\vee}\bigg{(}y(z)+\frac{\hbar
u}{2}\bigg{)}-\Phi_{1}^{\vee}\bigg{(}y(z)-\frac{\hbar u}{2}\bigg{)}=$ (3.15)
$\displaystyle=-zuS(\hbar u)+\log(z)u$
$\displaystyle\Phi_{2}^{\vee}(y_{1}(z_{1}),y_{2}(z_{2}))$
$\displaystyle=\log(z_{1}-z_{2})$ (3.16)
$\displaystyle\hat{\Phi}_{2}^{\vee}(y_{1}(z_{1}),y_{2}(z_{2});\hbar,u_{1},u_{2})$
$\displaystyle=\log\bigg{(}\frac{(z_{1}e^{\hbar u_{1}/2}-z_{2}e^{\hbar
u_{2}/2})(z_{1}e^{-\hbar u_{1}/2}-z_{2}e^{-\hbar u_{2}/2})}{(z_{1}e^{-\hbar
u_{1}/2}-z_{2}e^{\hbar u_{2}/2})(z_{1}e^{\hbar u_{1}/2}-z_{2}e^{-\hbar
u_{2}/2})}\bigg{)}$ (3.17)
$\displaystyle\hat{\Phi}_{2}^{\vee}(y(z),y(z);\hbar,u,u)$
$\displaystyle=-2\log(S(u\hbar)),$ (3.18)

where $S(u)=\frac{e^{u/2}-e^{-u/2}}{u}$ and $\Phi_{2}^{\vee}$ needed to be
regularised due to (2.12) on the diagonal.

######  Remark 3.7.

Note that we have chosen $y(z)=\log(z)$ instead of $y(z)=z$ as proposed in
[BM08], because Sec. 2.3 has shown that for $y(z)=z$ the functional relation
does not hold. Actually, transforming $y(z)=z\mapsto y(z)+x(z)=\log(z)$, is
the first symplectic transformation of a spectral curve mentioned in the
beginning of Sec. 2.2. Thus, both curves have the same $\omega_{g,n}$’s,
except for $\omega_{0,1}$, and therefore generates the linear Hodge integrals
as stated in Theorem 3.5. The important factor $S(u\hbar)$ appearing for
Hurwitz numbers and linear Hodge integrals (see for instance [FP00]) is
directly produced in (3.15) and (3.18) by our chosen spectral curve (3.13) and
the formula of $x-y$ symplectic transformation.

Taking Example 2.10 and Lemma 3.6 into account, we conclude:

######  *Corollary 3.8.

The linear Hodge integrals are computed as a formal expansion in $\hbar$ via

$\displaystyle\prod_{i=1}^{n}\frac{k_{i}^{k_{i}+1}}{k_{i}!}\bigg{\langle}\frac{\Lambda(1)}{\prod_{i=1}^{n}(1-k_{i}\psi_{i})}\bigg{\rangle}_{g,n}$
$\displaystyle=$
$\displaystyle\operatorname{Res}_{z_{i}=0}[\hbar^{2g+n-2}]\prod_{i=1}^{n}\frac{dz_{i}e^{k_{i}z_{i}S(\hbar
k_{i})}}{z_{i}^{1+k_{i}}\hbar k_{i}S(\hbar k_{i})}$
$\displaystyle\qquad\qquad\times\sum_{\Gamma\in\tilde{\mathcal{G}}^{2}_{n}}\prod_{\begin{subarray}{c}I\in\mathcal{I}(\Gamma)\\\
\\{i,j\\}=I\end{subarray}}\bigg{(}\frac{(z_{i}e^{\hbar k_{i}/2}-z_{j}e^{\hbar
k_{j}/2})(z_{i}e^{-\hbar k_{i}/2}-z_{j}e^{-\hbar k_{j}/2})}{(z_{i}e^{-\hbar
k_{i}/2}-z_{j}e^{\hbar k_{j}/2})(z_{i}e^{\hbar k_{i}/2}-z_{j}e^{-\hbar
k_{j}/2})}-1\bigg{)},$

where $\tilde{\mathcal{G}}^{2}_{n}\subset\mathcal{G}_{n}$ is the subset of
graphs with 2-valent $\bullet$-vertices connecting two different
$\bigcirc$-vertices at most with one $\bullet$-vertex which is nothing than
the set of connected graphs with $n$ labelled vertices (A001187).

Explicit computations of Corollary 3.8 with computer algebra have matched all
the Hurwitz numbers listed in [EMS09, Tab. 2].

For not-necessarily-connected correlators, we can conclude from Example 2.8
and Lemma 3.6:

######  *Corollary 3.9.

The sum over all partitions of linear Hodge integrals satisfies as a formal
expansion in $\hbar$

$\displaystyle\prod_{i=1}^{n}\frac{k_{i}^{k_{i}+1}}{k_{i}!}\sum_{\lambda\vdash
I}\prod_{j=1}^{l(\lambda)}\sum_{g_{j}=0}\hbar^{2g_{j}+|\lambda_{j}|-2}\bigg{\langle}\frac{\Lambda(1)}{\prod_{i=1}^{|\lambda_{j}|}(1-k_{\lambda_{j}^{i}}\psi_{\lambda_{j}^{i}})}\bigg{\rangle}_{g_{j},|\lambda_{j}|}$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\operatorname{Res}_{z_{i}=0}\frac{dz_{i}e^{k_{i}z_{i}S(\hbar
k_{i})}}{z_{i}^{1+k_{i}}\hbar k_{i}S(\hbar
k_{i})}\prod_{i<j}\frac{(z_{i}e^{\hbar k_{i}/2}-z_{j}e^{\hbar
k_{j}/2})(z_{i}e^{-\hbar k_{i}/2}-z_{j}e^{-\hbar k_{j}/2})}{(z_{i}e^{-\hbar
k_{i}/2}-z_{j}e^{\hbar k_{j}/2})(z_{i}e^{\hbar k_{i}/2}-z_{j}e^{-\hbar
k_{j}/2})},$

where $\lambda\vdash I$ is a set partition of $I$ with $l(\lambda)$ blocks
$\lambda_{j}\subset I$, i.e. $\lambda=(\lambda_{1},...,\lambda_{l(\lambda)})$.
Each block is written as
$\lambda_{j}=(\lambda_{j}^{1},...,\lambda_{j}^{|\lambda_{j}|})$ of cardinality
$|\lambda_{j}|$ and elements $\lambda_{j}^{i}\in I$.

## References

  * [ABDB+22] A. Alexandrov, B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin.  A universal formula for the $x-y$ swap in topological recursion.  12 2022, 2212.00320.
  * [BCGF21a] G. Borot, S. Charbonnier, and E. Garcia-Failde.  Topological recursion for fully simple maps from ciliated maps.  6 2021, 2106.09002.
  * [BCGF+21b] G. Borot, S. Charbonnier, E. Garcia-Failde, F. Leid, and S. Shadrin.  Analytic theory of higher order free cumulants.  12 2021, 2112.12184.
  * [BDBKS20] B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin.  Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type.  12 2020, 2012.14723.
  * [BDBKS21] B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin.  Generalised ordinary vs fully simple duality for $n$-point functions and a proof of the Borot–Garcia-Failde conjecture.  6 2021, 2106.08368.
  * [BDBKS22a] B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin.  Explicit closed algebraic formulas for Orlov–Scherbin $n$-point functions.  Journal de l’École polytechnique — Mathématiques, 9:1121–1158, 2022, 2008.13123.  doi:10.5802/jep.202.
  * [BDBKS22b] B. Bychkov, P. Dunin-Barkowski, M. Kazarian, and S. Shadrin.  Symplectic duality for topological recursion.  6 2022, 2206.14792.
  * [BKMP09] V. Bouchard, A. Klemm, M. Marino, and S. Pasquetti.  Remodeling the B-model.  Commun. Math. Phys., 287:117–178, 2009, 0709.1453.  doi:10.1007/s00220-008-0620-4.
  * [BM08] V. Bouchard and M. Marino.  Hurwitz numbers, matrix models and enumerative geometry.  Proc. Symp. Pure Math., 78:263–283, 2008, 0709.1458.  doi:10.1090/pspum/078/2483754.
  * [DM69] P. Deligne and D. Mumford.  The irreducibility of the space of curves of given genus.  Inst. Hautes Études Sci. Publ. Math., (36):75–109, 1969.  URL http://www.numdam.org/item?id=PMIHES_1969__36__75_0.
  * [ELSV99] T. Ekedahl, S. Lando, M. Shapiro, and A. Vainshtein.  On hurwitz numbers and hodge integrals.  Comptes Rendus de l’Académie des Sciences - Series I - Mathematics, 328(12):1175–1180, 1999.  doi:https://doi.org/10.1016/S0764-4442(99)80435-2.
  * [EMS09] B. Eynard, M. Mulase, and B. Safnuk.  The laplace transform of the cut-and-join equation and the bouchard-marino conjecture on hurwitz numbers.  Publications of The Research Institute for Mathematical Sciences, 47:629–670, 2009.
  * [EO07a] B. Eynard and N. Orantin.  Invariants of algebraic curves and topological expansion.  Commun. Num. Theor. Phys., 1:347–452, 2007, math-ph/0702045.  doi:10.4310/CNTP.2007.v1.n2.a4.
  * [EO07b] B. Eynard and N. Orantin.  Topological expansion of mixed correlations in the Hermitian 2-matrix model and $x$-$y$ symmetry of the $F_{g}$ algebraic invariants.  J. Phys. A: Math. Theor., 41(1):015203, 2007, 0705.0958.  doi:10.1088/1751-8113/41/1/015203.
  * [EO13] B. Eynard and N. Orantin.  About the x-y symmetry of the $F_{g}$ algebraic invariants.  11 2013, 1311.4993.
  * [Eyn11a] B. Eynard.  Intersection numbers of spectral curves.  4 2011, 1104.0176.
  * [Eyn11b] B. Eynard.  Invariants of spectral curves and intersection theory of moduli spaces of complex curves.  Communications in Number Theory and Physics, 8:541–588, 2011, 1110.2949.  doi:10.4310/CNTP.2014.v8.n3.a4.
  * [Eyn14] B. Eynard.  A short overview of the ”Topological recursion”.  12 2014, 1412.3286.
  * [Eyn16] B. Eynard.  Counting Surfaces, volume 70 of Prog. Math. Phys. Birkhäuser/ Springer, 2016.  doi:10.1007/978-3-7643-8797-6.
  * [FP00] C. Faber and R. Pandharipande.  Hodge integrals and Gromov-Witten theory.  Inventiones mathematicae, 139(1):173–199, 2000, math/9810173.  doi:10.1007/s002229900028.
  * [Hoc22a] A. Hock.  A simple formula for the $x$-$y$ symplectic transformation in topological recursion.  11 2022, 2211.08917.
  * [Hoc22b] A. Hock.  On the $x$-$y$ Symmetry of Correlators in Topological Recursion via Loop Insertion Operator.  1 2022, 2201.05357.
  * [Kon92] M. Kontsevich.  Intersection theory on the moduli space of curves and the matrix Airy function.  Commun. Math. Phys., 147:1–23, 1992.  doi:10.1007/BF02099526.
  * [LZ04] S. K. Lando and A. K. Zvonkin.  Graphs on surfaces and their applications, volume 141 of Encyclopaedia of Mathematical Sciences.  Springer-Verlag, Berlin, 2004.  doi:10.1007/978-3-540-38361-1.  With an appendix by Don B. Zagier.
  * [Mul12] M. Mulase.  The Laplace Transform, Mirror Symmetry, and the Topological Recursion of Eynard–Orantin.  In 31st Workshop on Geometric Methods in Physics, pages 127–142, 10 2012, 1210.2106.  doi:10.1007/978-3-0348-0645-9_11.
  * [Wit91] E. 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.  doi:10.4310/SDG.1990.v1.n1.a5.


# DFT+DMFT study of dopant effect in a heavy fermion compound CeCoIn5

Hong Chul Choi<EMAIL_ADDRESS>Theoretical Division, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Center for Correlated Electron
Systems, Institute for Basic Science (IBS), Seoul 08826, Korea  Eric D. Bauer
Materials Physics and Application Division, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545, USA  Filip Ronning  Institute for Materials
Science, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Jian-Xin Zhu<EMAIL_ADDRESS>Theoretical Division, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Center for Integrated
Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico
87545, USA

###### Abstract

We study the dopant-induced inhomogeneity effect on the electronic properties
of heavy fermion CeCoIn5 using a combined approach of density functional
theory (DFT) and dynamical mean-field theory (DMFT). The inhomogeneity of the
hybridization between Ce-$4f$ and conduction electrons is introduced to impose
the inequivalent Ce atoms with respect to the dopant. From the DFT to the
DFT+DMFT results, we demonstrate a variation of the hybridization strength
depending on the hole or electron doping. A drastic asymmetric mass
renormalization could be reproduced in the DFT+DMFT calculation. Finally, the
calculated coherence temperature reflects the different development of the
heavy quasiparticle states, depending on the dopant.

##  I Introduction

Heavy fermion systems (HFSs)Fulde (2012); Fulde and Thalmeier (2006);
Pfleiderer (2009) have shown unconventional superconductivity, Kondo effect,
valence fluctuations, magnetism and exotic coexisting phases. The renormalized
quasiparticle kinetic energies observed in the heavy fermion originates from
the Kondo effect. The strength of the Kondo effect is determined by the
hybridization function ($\Delta$) between localized and conduction electrons,
and the density of state ($N_{F}$) at the Fermi level ($E_{F}$). Strong
hybridization could suppress the renormalization effect over the competition
with the Coulomb interaction. The wavefunction of the $4f$-electrons is deeply
distributed inside the atomic radius close to the atomic wavefunction.
However, the energy gain through the Kondo effect could drive a crossover from
a localized moment into a highly renormalized itinerant quasiparticle state
near $E_{F}$. In addition, the atomic multiplet of Ce-4$f$ valence states
provide an additional incoherent feature, being identified as lower and upper
Hubbard bands. The three peak spectral function $A(k,\omega)$ represents the
characteristic dual nature of the strongly correlated $f$-electron system.

The Ce-based heavy fermion compounds are one prototypical family of HFSs.
CeIn3 shows an antiferromagnetic ground state with $T_{N}$=10 K, where
superconductivity with $T_{c}$=0.21 K emerges under a critical pressure of 2.6
GPa. Mathur et al. (1998); Knebel et al. (2001) CeMIn5 (M=Co, Rh, Ir)Petrovic
et al. (2001a); Hegger et al. (2000); Petrovic et al. (2001b) is synthesized
by inserting MIn2 layers between CeIn3 layers. In these compounds, CeCoIn5 and
CeIrIn5 are unconventional superconductors at $T_{c}$=2.3 K and 0.4 K,
respectively, indicative of Ce-4$f$ electrons being delocalized, whereas
CeRhIn5 is an antiferromagnet ($T_{N}$=3.8 K) with localized 4$f$ electrons at
ambient pressure. The substitution of Cd Pham et al. (2006); Nicklas et al.
(2007); Urbano et al. (2007); Rusz et al. (2008); Booth et al. (2009); Gofryk
et al. (2012); Sakai et al. (2017); Chen et al. (2019), Ru Ou et al. (2013) or
Hg Booth et al. (2009); Gofryk et al. (2012) (hole doping) and Sn Bauer et al.
(2005a, b, 2006); Daniel et al. (2005); Rusz et al. (2008); Booth et al.
(2009); Gofryk et al. (2012); Sakai et al. (2017), Zn Yokoyama et al. (2014);
Park et al. (2020) or Pt Gofryk et al. (2012) (electron doping) has been
reported to either promote antiferromagnetic (AFM) or metallic phases. In
comparison to In ($4d^{10}5p^{1}$), Cd and Sn have the valence states of
$4d^{10}$ and $4d^{10}5p^{2}$, respectively. The one less (more) electron
would be expected to reduce (enhance) the hybridization. Experimentally,
angle-resolved photoemission spectroscopy has indeed demonstrated the
suppression of the band hybridization in Cd-doped CeCoIn5. Chen et al. (2019)
The electron or hole doping affects the ground state of CeCoIn5
asymmetrically. Only one percent Cd substitution causes the AFM ground state
with a local moment on the Ce sites ($\sim 0.7\;\mu_{B}$/Ce); while the Sn
substitution leads to a complete disappearance of superconductivity for a
critical concentration $\sim$ $\sim$ 3.6$\%$ Sn. In addition, the extended
X-ray absorption fine structure measurements has suggested the preference of
dopant atoms to the Ce-In plane, Daniel et al. (2005); Booth et al. (2009)
which means an inhomogeneous distribution of dopants.

Density functional theory (DFT) has proven to be very effective in helping to
unravel the effects of doping in the 115s. For example, DFT could help
identify various local atomic environments observed by NMR. Rusz et al.
(2008); Sakai et al. (2015) DFT calculations also helped to uncover the local
increase (decrease) in the hybridization caused by Sn (Cd) dopants. Gofryk et
al. (2012) Furthermore, the open-core (setting $4f$ states inside the core
state) or the conventional DFT (setting $4f$ states as valence states)
provides a hint of whether $4f$ states might be localized or delocalized.Rusz
et al. (2008) Therefore, DFT proved itself as a powerful tool to incorporate
material-specific information, that is, the structure and atomic configuration
of the real compound. On the other hand, it is known to underestimate the
effect of electronic correlations in heavy fermion systems. This inadequacy
can be remedied by dynamical mean-field theory (DMFT), which has successfully
captured the correlation-induced crossover between localization and
delocalization in strongly correlated systems. Via a strong band
renormalization, the correlation effect on the electronic structure and
optical properties were well reproduced by DFT+DMFT calculations.Shim et al.
(2007); Choi et al. (2012); Nomoto and Ikeda (2014) Using the virtual crysal
approximation, a DFT+DMFT study reproduced the general trends of the doping
effect.Seo et al. (2015) However, the impact of the dopant-induced
inhomogeneity in real compounds has not been addressed.

Here we revisit the effect of doping on CeCoIn5 to take into account the
inhomogeneity with respect to the distance from a dopant to a Ce atom. We
break the equivalence of Ce atoms by placing the dopant close to a specific Ce
atom. The electronic structure is investigated by DFT and DFT+DMFT
calculations. The DFT results give a hint about the different behavior of hole
versus electron substitutions, while its limitation is also shown. The small
change in the hybridization in the DFT quasiparticle could turn into a totally
different phase through the DMFT impurity solver. The inhomogeneity of the
self-energy and the hybridization functions is shown in the DFT+DMFT results.
The different evolution of of the heavy quasiparticle band induced by the
dopant will be discussed with the calculated coherence temperature.

The paper is organized as follows: Section II explains the computational
method. In Sec. III we address calculated crystal structures, the DFT and
DFT+DMFT results. Also, the calculated coherence temperature will be provided
here. Section IV presents a summary and concluding remarks.

##  II Computational details

The charge self-consistent version of DFT+DMFT, Kotliar et al. (2006) as
implemented in Ref. Haule et al., 2010, is based on the full-potential
linearized augmented plane-wave (FP-LAPW) band method. Blaha et al. (2001) The
correlated $4f$-electrons are treated dynamically by the DMFT local self-
energy ($\Sigma(\omega)$), while all other delocalized $spd$ electrons are
treated on the DFT level. The charge and spin fluctuations considered in DMFT
enable the describtion of the Kondo effect correctly. $\Sigma(\omega)$ is
calculated from the corresponding impurity problem, in which full atomic
interaction matrix is taken into account ($F^{0}$= 5.0 eV, $F^{2}$=
8.10693641618 eV $F^{4}$= 5.41543352601 eV and, $F^{6}$= 4.0048265896). Cowan
(1981) A temperature of 5 meV is used in the calculations. To solve the
impurity problem, we used the one-crossing approximation of the local self-
energies. Kotliar et al. (2006) RKmax is 7.0 and 3000 k-point mesh are used
through the calculation. The spin-orbit coupling is always included in the
calculations. The total charge density and the self-energy through the
DFT+DMFT iterations are converged within 10-5 and 10-4, respectively.

The single-site DMFT solver for the $4f$ state in each unique Ce cite are
treated separately. The $4f$ state for the unique Ce sites will be denoted as
Ce1 or Ce2. The inter-site correlation beyond the single-site DMFT solver is
included through the hybridization function ($\Delta$). The self-consistent
equation between the local Green’s function
($\mathrm{G_{loc}}=\frac{1}{\omega-\epsilon_{imp}-\Sigma-\Delta}$,
$\epsilon_{imp}$: impurity energy level, $\Sigma$: the self-energy,$\Delta$:
the hybridization function between impurity and bath) and the lattice green
function computed by the DFT eigenvalue ($H_{\mathbf{k}}^{DFT}$) and $E_{F}$,
is defined through the projection operator ($P$) from the lattice to the
impurity and the embedding operator ($E$) vice versa. Using
$\hat{P}_{1,{\mathbf{k}}}$, $\hat{P}_{2,{\mathbf{k}}}$,
$\hat{E}_{1,{\mathbf{k}}}$, and $\hat{E}_{2,{\mathbf{k}}}$ (1:Ce1, 2:Ce2), the
self-consistent equations with the diagonal basis are defined as follows:

$\frac{1}{\omega-\epsilon_{1,imp}-\Sigma_{1}-\Delta_{1}}=\\\
\sum_{\mathbf{k}}\hat{P}_{1,{\mathbf{k}}}[(\omega+E_{F}-H_{\mathbf{k}}^{DFT}-\hat{E}_{1,{\mathbf{k}}}\Sigma_{1})^{-1}],$
(1)

and

$\frac{1}{\omega-\epsilon_{2,imp}-\Sigma_{2}-\Delta_{2}}=\\\
\sum_{\mathbf{k}}\hat{P}_{2,{\mathbf{k}}}[(\omega+E_{F}-H_{\mathbf{k}}^{DFT}-\hat{E}_{2,{\mathbf{k}}}\Sigma_{2})^{-1}].$
(2)

Here ,$\epsilon_{i,imp}$, $\Sigma_{i}$, and $\Delta_{i}$ mean the impurity
energy level, self-energy, and hybridization function, respectively. $i$
represents the different impurity (Ce). The long-range correlation effect is
taken into account through $\Delta_{i}$.

Figure 1: (color online) (a)The schematic crystal structuresMomma and Izumi
(2011) of the 1x1x2 supercell of CeCoIn5 The different colors represent the
different atoms. Indium (In) atoms have the magenta (In1) and purple (In0)
colors. For the doped case, the dopant atom replaces In1 to give the
concentration of 10$\%$-doped structure. Additionally, the two-types of Ce
atoms are distinguished to Ce1 (close to the dopant) and Ce2 (away from the
dopant). (b) The atomic configurations of Cd, In, Sn, and Hg are compared. (c)
The primitive unit cell of CeCoIn5 is provided.  Figure 2: (color online) The
density of states for (a) Ce1-4$f$, (b) Ce2 -4$f$, and (c) Co-3$d$ states
calculated in four different structures with DFT. The labels represent CeCoIn5
(pristine), CeCo(In0.9Cd0.1)5 (Cd-doped CeCoIn5), CeCo(In0.9Hg0.1)5 (Hg-doped
CeCoIn5), and CeCo(In0.9Sn0.1)5 (Sn-doped CeCoIn5). The red vertical line
represents the Fermi level. The blue vertical line is provided as a guideline
to compare the shift of the bands with respect to the dopant.

##  III Numerical Results

Figure 1(a) presents the supercell structure of CeCoIn5 made from the original
primitive unit cell as shown Fig. 1(c). We adopt the minimum size of the 1x1x2
supercell of CeCoIn5, which has two Ce, two Co and ten In atoms for our
calculations. Ce(1) at (0,0,0) and Ce(2) at (0,0,0.5) are equivalent in the
pristine structure. In the primitive cell, In atoms are distinguished as the
ones (In(1)) in the Ce plane and the other four In atoms (In(2)) in between
the Co plane and the Ce-In plane. For simplicity, we choose the dopant atom
(Cd,Sn or Hg) to occur at the In(1) position of the Ce(1) plane. This is
consistent with extended X-ray absorption fine structure measurement, which
indicate that these dopant atoms preferentially substitute on the In(1) site.
These substitutions force Ce(1) and Ce(2) to become inequivalent. In this
work, we examine the electronic structure of CeCoIn5 (pristine),
CeCo(In0.9Cd0.1)5 (Cd-doped CeCoIn5), CeCo(In0.9Hg0.1)5 (Hg-doped CeCoIn5),
and CeCo(In0.9Sn0.1)5 (Sn-doped CeCoIn5).

First we report the conventional DFT calculations of the doped CeCoIn5. Figure
2 shows the density of states (DOS) of (a) Ce(1)-$4f$ states, (b) Ce(2)-$4f$
states, and (c) Co-$3d$ states. We note that the DOS for the two Co atoms are
identical due to the inversion symmetry. Depending on the dopant, the Ce-$4f$
bands are shifted with respect to the pristine case (red line). The two peaks
in Fig. 2(a-b) correspond $j$=5/2 (left peak) and $j$= 7/2 (right peak) states
split by spin-orbit coupling ($\sim$ 0.3 eV). This value is inherited into the
impurity solver in the DFT+DMFT calculation. Hereafter we focus on the $j$=5/2
states, which mainly contribute the states around $E_{F}$.

As Cd and Hg (Sn) have one less (more) electron than In (See Fig. 1(b)), the
shift of the chemical potential with respect to Ce-$4f$ states would be
lowered (increased) in the Cd- and Hg-doped (Sn-doped) cases. The trend in
Fig. 2(a-b) for Ce(1)-$4f$ and Ce(2)-$4f$ in the pristine and doped cases does
not follow this simple intuition. In comparison to the $j$=5/2 peak of the
pristine Ce-$4f$ states, the peak for Cd- and Hg-doped (Sn-doped) cases is
shifted toward (away from) the Fermi energy level. Also, the DOS at $E_{F}$
are increased in Cd- and Hg-doped CeCoIn5 systems, whereas that of the Sn-
doped CeCoIn5 is decreased. Those results indicate the effect of doping on the
$4f$ states could not be described well by the DFT method. On the other hand,
the inhomogeneous feature between Ce(1) and Ce(2) sites could be identified.

We also explore the doping effects on the 3$d$-bands of transition-metal Co.
As shown in Fig. 2(c), the band center of Co-3$d$ states is shifted toward
$E_{F}$ in the Cd- and Hg-doped CeCoIn5 cases, while that of the Sn-doped
CeCoIn5 is moved away from $E_{F}$. Such behavior is in agreement with a rigid
band picture.

Figure 3: (color online) The imaginary part of the local Green’s functions for
(a) Ce1 and (b) Ce2 in different structures with DFT+DMFT.The labels represent
CeCoIn5 (Pristine), CeCo(In0.9Cd0.1)5, CeCo(In0.9Hg0.1)5, and
CeCo(In0.9Sn0.1)5.

In the remainder, we will discuss how correlations influence the electron- and
hole-doped cases. Through the DFT+DMFT calculations, Ce(1)-$4f$ and Ce(2)-$4f$
states are treated as correlated impurity states, denoted by Ce1 and Ce2. The
DOS ($-\mathrm{Im}\mathrm{G_{loc}}$) of Ce1 and Ce2 are provided in Fig. 3. In
CeCoIn5, the peak of $j$=5/2 states grows as temperature lowers. Fig. 3 is a
snapshot in the middle of the development of the quasiparticle peak. In
comparison to Fig. 2 (a,b), the width of the peaks are highly reduced and
pushed towards $E_{F}$. The DOS at $E_{F}$ of the Ce1 in the Sn-doped case is
much larger than others. Also no spin-orbit side band at $-0.3$ eV could be
observed. The spin-orbit side band is attributed to the transition between
$j$=5/2 and $j$=7/2 states. The Sn dopant due to the increased hybridizations
pushes the Ce1 away from the Kondo limit. The enhanced DOS of the Ce1 is
associated with its closer proximity to the Fermi liquid in CeCo(In0.9Sn0.1)5.
In case of the hole-doped calculations (Cd or Hg), Ce1 shows a reduction of
the DOS relative to the pristine case. This opposition behavior is consistent
with stabilizing a magnetic state upon hole doping.

We examine the $4f$ electron occupancy ($n_{f}$) for the pristine and doped
cases. As summarized in Table 1, the calculated $n_{f}$ in Sn-doped case is
larger than the others. This trend looks contradicting to the intuitive view
of valence change from $4f^{1}$ ($\mathrm{Ce}^{3+}$) to $f^{0}$
($\mathrm{Ce}^{4+}$) under strong hybridization. Therefore, within a simple
hybridization picture, one would expect that the hole (electron) doping would
drive the valence of Ce states toward $f^{1}$ ($f^{0}$). Here we bring up a
new insight: As shown in Fig. 4, although the total number of Ce-4$f$ electron
occupation is larger in the Sn-doped case, under the electron doping, the
probability of $f^{1}$ configuration decreases while simultaneously the
probabilities of $f^{0}$ and $f^{2}$ configurations increase. We thus conclude
that the calculated $n_{f}$ measures both the Ce-4$f$ electron count for both
low energy and high energy configuration states. The overall differences of
$n_{f}$ are less than 0.1 electron per Ce atom. However, the correlation
effect enables the fine tuned electron number to bring up the drastic change
of its electronic structure.

Table 1: Ce-$4f$ electron occupancy, $n_{f}$, and probabilities of $4f^{0}$, $4f^{1}$, and $f^{2}$ for Ce1 in the DFT+DMFT calculations. nf (electron/Ce atom) | Ce1 | Ce2 | P($4f^{0}$) | P($4f^{1}$) | P($4f^{2}$)
---|---|---|---|---|---
CeCoIn5 | 1.0432 | 1.0432 | 0.0341 | 0.9126 | 0.0532
CeCo(In0.9Cd0.1)5 | 1.0299 | 1.0329 | 0.0297 | 0.9267 | 0.0434
CeCo(In0.9Hg0.1)5 | 1.0214 | 1.0362 | 0.028 | 0.9224 | 0.0495
CeCo(In0.9Sn0.1)5 | 1.0822 | 1.0452 | 0.064 | 0.8711 | 0.0646
Figure 4: (color online) The calculated probabilities of valence states for $n_{f}$=0 (blue), 1 (green), and 2 (red) of Ce1 in the pristine and doped cases.  Figure 5: (color online) The imaginary part of the hybridization functions for (a) Ce1 and (b) Ce2 in different structures with DFT+DMFT. The labels represent CeCoIn5 (Pristine), CeCo(In0.9Cd0.1)5, CeCo(In0.9Hg0.1)5, and CeCo(In0.9Sn0.1)5.  Table 2: Summary of the hybridization strength $-\mathrm{Im}\Delta(0)$. $-\mathrm{Im}\Delta(0)$ (eV) | Ce1 | Ce2
---|---|---
CeCoIn5 | 0.08999 | 0.08999
CeCo(In0.9Cd0.1)5 | 0.07032 | 0.07345
CeCo(In0.9Hg0.1)5 | 0.06837 | 0.08064
CeCo(In0.9Sn0.1)5 | 0.16292 | 0.0982

We investigate the calculated hybridization function ($-\mathrm{Im}\Delta$ )
for Ce1 and Ce2 as shown Fig. 5(a,b). Compared to Fig. 2(a-b), the DFT+DMFT
calculation shows the hybridization strength increases with the number of
valence electrons. In energy between $-0.1$ eV and 0.1 eV, the dopant-
dependent hybridization functions are well illustrated with respect to the red
lines (pristine). The imaginary part of the hybridization
($-\mathrm{Im}\Delta(\omega=0)$) at EF of Ce1 in Sn-doped CeCoIn5 is enhanced
significantly over the other cases (see Table 2). This enhancement of
hybridization is reduced to be comparable to the pristine case for Ce2. This
could be understood in terms of the relative distances between the Ce atoms
and the Sn atom. The substitution of a hole (Cd, Hg) decreases
$-\mathrm{Im}\Delta(\omega=0)$ around $E_{F}$. Both Cd and Hg dopants give
rise to similar decreases of the $-\mathrm{Im}\Delta(\omega=0)$ on both Ce1
and Ce2 with respect to the pristine cases. The behavior shown in
$-\mathrm{Im}{\Delta}$ and $-\mathrm{Im}\mathrm{G_{loc}}$ could reveal how the
hybridization is changed in each case. The reduction of the hybridization
favors a local-moment ground state over the Fermi liquid phase. We conclude
that Sn-doped ( Cd- or Hg-doped) CeCoIn5 would be in proximity to a Fermi
liquid state (a magnetic ground state).

Figure 6: (color online) The imaginary part of the self-energy for (a) Ce1 and
(b) Ce2 in different structures with DFT+DMFT. The labels represent CeCoIn5
(Pristine), CeCo(In0.9Cd0.1)5, CeCo(In0.9Hg0.1)5, and CeCo(In0.9Sn0.1)5.

To elaborate more insight into the renormalization of the quasiparticle state,
we will examine the imaginary ($\mathrm{Im}$) and real ($\mathrm{Re}$) parts
of the calculated self-energy ($\Sigma$). Figure 6 shows the imaginary part of
the self-energy ($\mathrm{Im}\Sigma$),whose inverse at $E_{F}$ represents the
inverse of the life-time of quasiparticle state. $\mathrm{Im}\Sigma(\omega=0)$
of Ce1 in the Sn-doped case is almost zero at $T$=0.005 eV ($\sim$ 65 K). The
parabolic behavior of $\mathrm{Im}\Sigma$ around $\omega=0$ of Ce1 in Sn-doped
CeCoIn5 is a typical Fermi liquid characteristic. The Fermi liquid phase
represents the presence of the coherent quasiparticle states. The finite
values of of $\mathrm{Im}\Sigma$ ($\omega=0$) indicate the deviation from the
Fermi liquid phase in the other cases. This quasiparticle state of the Ce1 in
CeCo(In0.9Sn0.1)5 would be consistent with the largely enhanced hybridization
function.

Figure 7: (color online) The real part of the self-energy for (a) Ce1 and (b) Ce2 in different structures with DFT+DMFT. The labels represent CeCoIn5 (Pristine), CeCo(In0.9Cd0.1)5, CeCo(In0.9Hg0.1)5, and CeCo(In0.9Sn0.1)5.  Table 3: Summary of the quasiparticle weight ($Z$) at the Fermi energy $E_{F}$. $Z$ ($1/(1-\frac{\partial\mathrm{Re}\Sigma}{\partial\omega})$ at EF) | Ce1-$4f$ | Ce2-$4f$
---|---|---
CeCoIn5 | 0.0124 | 0.0124
CeCo(In0.9Cd0.1)5 | 0.0110 | 0.0110
CeCo(In0.9Hg0.1)5 | 0.0121 | 0.0121
CeCo(In0.9Sn0.1)5 | 0.1100 | 0.0139

The renormalization of the band is strongly associated with the mass
enhancement under correlation. The slope of $\mathrm{Re}\Sigma$ at $E_{F}$ is
used to represent how large correlation effect reduces the bandwidth of a
quasiparticle state. Figure 7 provides the variation of $\mathrm{Re}\Sigma$
from $-0.4$ eV to 0.4 eV. The slopes of $\mathrm{Re}\Sigma$ at $E_{F}$ seem to
be similar to the red lines of the pristine case except for that of Ce1 in Sn-
doped case. The quasiparticle weight $Z$ is computed using
$1/(1-\frac{\partial\mathrm{Re}\Sigma}{\partial\omega})$ at $E_{F}$, and is
given in Table 3. The $\mathrm{Re}\Sigma(\omega=0)$ of the Ce1 shows a very
different behavior in comparison to the other cases. More strikingly, the
quasiparticle weight $Z$ (0.11) of the Ce1 in the Sn-doped CeCoIn5 is nearly
one order of magnitude larger than that for both the pristine and hole-doped
cases. The larger quasiparticle weight represents the larger overlap of the
correlate and non-interacting wavefunction at $E_{F}$. The weight for the Ce2
is around 0.0139, which is slightly larger than the other cases (see Fig.
7(b)). The inhomogeneity due to the Sn dopant is consistently observed in the
DFT+DMFT calculations. On the other hand, the quasiparticle weight for the
hole-doped CeCoIn5 is 0.011 (Cd-doped), and 0.012 (Hg-doped), respectively.
This is also consistent with the observation that the bandwidths of the
pristine and the hole-doped CeCoIn5 compounds are similarly renormalized (see
Fig. 3(a)). The inhomogeneity due to the hole dopant is relatively suppressed
in the DFT+DMFT calculations.

Table 4: Calculation of coherence temperature, $T_{coh}=-\pi Z\mathrm{Im}\Delta(0)/4$. $T_{coh}$ (eV) | Ce1-$4f$ | Ce2-$4f$
---|---|---
CeCoIn5 | 0.00087 | 0.00087
CeCo(In0.9Cd0.1)5 | 0.000607 | 0.000634
CeCo(In0.9Hg0.1)5 | 0.000649 | 0.000766
CeCo(In0.9Sn0.1)5 | 0.0140 | 0.001071

Using the calculated $Z$ and $\mathrm{Im}\Delta$, the coherence temperatures
($T_{coh}$) could be predicted using the empirical formula ($T_{coh}=-\pi
Z\mathrm{Im}\Delta(0)/4$). Zhu et al. (2015) The calculated $T_{coh}$ is
summarized in Table 4. The largest $T_{coh}$ is 0.0140 eV for Ce1 in
CeCo(In0.9Sn0.1)5, which is one order of magnitude larger than the used
temperature ($T$=0.005 eV). However, The $T_{coh}$ for Ce2 in
CeCo(In0.9Sn0.1)5 is reduced to 0.001071 eV, which is still higher than all
other coherence temperatures in the hole-doped compounds. These results
suggest the substantial enhancement effect of the Sn dopant on the Kondo
coherence, which pushes the system to be more close to the Fermi liquid phase,
and a robust nonequivalence of the Kondo coherence between Ce1 and Ce2 through
the DFT+DMFT iterations. The hole-doped case show the decrease in both the
$-\mathrm{Im}\Delta$ and $Z$ at $E_{F}$. Therefore, $T_{coh}$ in the hole-
doped cases is lower than those of the pristine case. Unlike the electron-
doped case, Ce1 and Ce2 simultaneously become more incoherent at $T$=0.005 eV.

##  IV Concluding Remarks and Summary

Since dopants prefer to be substituted in the Ce plane, Booth et al. (2009)
our study here has considered the spatial inhomogeneity in the doped CeCoIn5
system. The much enhanced hybridization $\Delta$ and the non Ce-4$f$ density
of states $N_{F}$ of Ce1 near Sn can drive the system deep into the Fermi
liquid state. We have also found through the two single-site DMFT impurity
solvers that the 4$f$-electron behavior is dramatically different on Ce1 and
Ce2. Furthermore, the current study has shown a short-ranged electron doping
effect. On the contrary, in the case of hole-doped CeCoIn5, Ce-$4f$ state
would be more incoherent than in the pristine. We further note that our study
has been limited to the paramagnetic solutions and as such the presence of a
G-type antiferromagnetic configuration ($q=(\pi,\pi,\pi)$), as reported in the
Cd-doped CeCoIn5, Nicklas et al. (2007) is beyond the scope of the present
work. However, the dopant induced incoherence obtained here, indicating that
the state moves away from the Fermi liquid phase, is consistent with the
experiment. It is worthwhile to note that while the Sn doping gives a stronger
inhomogeneity effect on the local electronic structure, the NMR experiment has
observed Sakai et al. (2015) that the Cd doping causes a stronger
inhomogeneity effect on the magnetic fluctuations.

In summary, we have investigated the inhomogeneous hole and electron doping
effect on CeCoIn5 within the DFT+DMFT framework. We have revealed a clear
variation of the hybridization function as well as the highly renormalized
bands of Ce-$4f$ states. In addition, the calculated $T_{coh}$ has provided a
good insight into whether Ce-$4f$ states in a doped CeCoIn5 system would be
coherent or incoherent. Specifically, we have discovered a highly
inhomogeneous influence of Sn substitution on Ce-4$f$ electron states. This
prediction should be accessible to scanning tunneling microscopy experiments
on a (100)- or (010)-oriented surface of a Sn-doped CeCoIn5.

###### Acknowledgements.

Hongchul Choi thanks Soobeom Seo for illuminating discussions. This work was
carried out under the auspices of the U.S. Department of Energy (DOE) National
Nuclear Security Administration (NNSA) under Contract No. 89233218CNA000001.
It was supported by LANL LDRD Program (H.C.C.), the DOE BES ”Quantum
Fluctuations in Narrow Band Systems” project (E.D.B, F.R), and NNSA Advanced
Simulation and Computing Program (J.-X.Z.). Additional support has been
provided in part by the Center for Integrated Nanotechnologies, a DOE BES user
facility, in partnership with the LANL Institutional Computing Program for
computational resources. H.C.C. was also supported by Institute for Basic
Science in Korea (IBS-R009-D1).

## References

  * Fulde (2012) P. Fulde, _Electron correlations in molecules and solids_ , vol. 100 (Springer Science & Business Media, 2012).
  * Fulde and Thalmeier (2006) P. Fulde and P. Thalmeier, _G. zwicknagl solid state physics vol. 60_ (2006).
  * Pfleiderer (2009) C. Pfleiderer, Rev. Mod. Phys. 81, 1551 (2009).
  * Mathur et al. (1998) N. Mathur, F. Grosche, S. Julian, I. Walker, D. Freye, R. Haselwimmer, and G. Lonzarich, Nature 394, 39 (1998).
  * Knebel et al. (2001) G. Knebel, D. Braithwaite, P. Canfield, G. Lapertot, and J. Flouquet, Phys. Rev. B 65, 024425 (2001).
  * Petrovic et al. (2001a) C. Petrovic, P. Pagliuso, M. Hundley, R. Movshovich, J. Sarrao, J. Thompson, Z. Fisk, and P. Monthoux, J. Phys.: Condens. Matter 13, L337 (2001a).
  * Hegger et al. (2000) H. Hegger, C. Petrovic, E. Moshopoulou, M. Hundley, J. Sarrao, Z. Fisk, and J. Thompson, Phys. Rev. Lett. 84, 4986 (2000).
  * Petrovic et al. (2001b) C. Petrovic, R. Movshovich, M. Jaime, P. Pagliuso, M. Hundley, J. Sarrao, Z. Fisk, and J. Thompson, Europhs. Lett. 53, 354 (2001b).
  * Pham et al. (2006) L. Pham, T. Park, S. Maquilon, J. Thompson, and Z. Fisk, Phys. Rev. Lett. 97, 056404 (2006).
  * Nicklas et al. (2007) M. Nicklas, O. Stockert, T. Park, K. Habicht, K. Kiefer, L. Pham, J. Thompson, Z. Fisk, and F. Steglich, Phys. Rev. B 76, 052401 (2007).
  * Urbano et al. (2007) R. Urbano, B.-L. Young, N. Curro, J. Thompson, L. Pham, and Z. Fisk, Phys. Rev. Lett. 99, 146402 (2007).
  * Rusz et al. (2008) J. Rusz, P. M. Oppeneer, N. J. Curro, R. R. Urbano, B.-L. Young, S. Lebegue, P. G. Pagliuso, L. D. Pham, E. D. Bauer, J. L. Sarrao, et al., Phys. Rev. B 77, 245124 (2008).
  * Booth et al. (2009) C. H. Booth, E. D. Bauer, A. D. Bianchi, F. Ronning, J. D. Thompson, J. L. Sarrao, J. Y. Cho, J. Y. Chan, C. Capan, and Z. Fisk, Phys. Rev. B 79, 144519 (2009).
  * Gofryk et al. (2012) K. Gofryk, F. Ronning, J.-X. Zhu, M. Ou, P. Tobash, S. Stoyko, X. Lu, A. Mar, T. Park, E. Bauer, et al., Phys. Rev. Lett. 109, 186402 (2012).
  * Sakai et al. (2017) H. Sakai, F. Ronning, T. Hattori, Y. Tokunaga, S. Kambe, J.-X. Zhu, N. Wakeham, H. Yasuoka, E. D. Bauer, and J. D. Thompson, J. Phys.: Conf. Ser. 807, 032001 (2017).
  * Chen et al. (2019) Q. Chen, F. Ronning, E. Bauer, C. Wen, Y. Huang, and D. Feng, Phys. Rev. B 100, 235148 (2019).
  * Ou et al. (2013) M. Ou, K. Gofryk, R. Baumbach, S. Stoyko, J. Thompson, J. Lawrence, E. Bauer, F. Ronning, A. Mar, and Y. Chen, Phys. Rev. B 88, 195134 (2013).
  * Bauer et al. (2005a) E. Bauer, C. Capan, F. Ronning, R. Movshovich, J. Thompson, and J. Sarrao, Phys. Rev. Lett. 94, 047001 (2005a).
  * Bauer et al. (2005b) E. Bauer, N. Moreno, D. Mixson, J. L. Sarrao, J. Thompson, M. F. Hundley, R. Movshovich, and P. Pagliuso, Physica B 359, 35 (2005b).
  * Bauer et al. (2006) E. Bauer, F. Ronning, C. Capan, M. Graf, D. Vandervelde, H. Yuan, M. Salamon, D. Mixson, N. Moreno, S. Brown, et al., Phys. Rev. B 73, 245109 (2006).
  * Daniel et al. (2005) M. Daniel, E. Bauer, S.-W. Han, C. Booth, A. Cornelius, P. Pagliuso, and J. Sarrao, Phys. Rev. Lett. 95, 016406 (2005).
  * Yokoyama et al. (2014) M. Yokoyama, K. Fujimura, S. Ishikawa, M. Kimura, T. Hasegawa, I. Kawasaki, K. Tenya, Y. Kono, and T. Sakakibara, J. Phys. Soc. Japan 83, 033706 (2014).
  * Park et al. (2020) T. B. Park, S. Shin, S. Lee, S. Seo, H. Jang, J. Kim, H. Lee, H. Wang, H. Lee, and T. Park, Phys. Rev. Mater. 4, 084801 (2020).
  * Sakai et al. (2015) H. Sakai, F. Ronning, J.-X. Zhu, N. Wakeham, H. Yasuoka, Y. Tokunaga, S. Kambe, E. Bauer, and J. Thompson, Phys. Rev. B 92, 121105 (2015).
  * Shim et al. (2007) J. Shim, K. Haule, and G. Kotliar, Science 318, 1615 (2007).
  * Choi et al. (2012) H. C. Choi, B. Min, J. Shim, K. Haule, and G. Kotliar, Phys. Rev. Lett. 108, 016402 (2012).
  * Nomoto and Ikeda (2014) T. Nomoto and H. Ikeda, Phys. Rev. B 90, 125147 (2014).
  * Seo et al. (2015) S. Seo, E. Park, E. Bauer, F. Ronning, J. Kim, J.-H. Shim, J. Thompson, and T. Park, Nat. Commun. 6, 6433 (2015).
  * Kotliar et al. (2006) G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. Marianetti, Rev. Mod. Phys. 78, 865 (2006).
  * Haule et al. (2010) K. Haule, C.-H. Yee, and K. Kim, Phys. Rev. B 81, 195107 (2010).
  * Blaha et al. (2001) P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka, and J. Luitz, Schwarz, Vienna University of Technology, Austria (2001).
  * Cowan (1981) R. D. Cowan, _The theory of atomic structure and spectra_ , 3 (Univ of California Press, 1981).
  * Momma and Izumi (2011) K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011).
  * Zhu et al. (2015) J.-X. Zhu, R. Albers, K. Haule, and J. M. Wills, Phys. Rev. B 91, 165126 (2015).