Intersection theory of the complex quartic Kontsevich model

Finn Bjarne Kohl${}^{\,\text{a, b}}$, Raimar Wulkenhaar${}^{\,\text{b}}$

${}^{\text{a}}$ Institute for Theoretical Physics, Westfälische Wilhelms-
Universität Münster, Wilhelm-Klemm-Straße 9, 48149 Münster, Germany
${}^{\text{b}}$ Mathematical Institute, Westfälische Wilhelms-Universität
Münster,
Einsteinstraße 62, 48149 Münster, Germany

We expand correlation functions of the Langmann-Szabo-Zarembo (LSZ) model in
terms of intersection numbers on the moduli space of complex curves. This
provides an explicit, physically motivated example for the expansion of
correlation functions generated by Chekhov-Eynard-Orantin topological
recursion. To this end, we unify notation as well as different conventions
present in the literature and use a set of moduli of the spectral curve
adapted to the physically motivated model. The presentation focuses on an
illustrative, step-by-step comprehension of the work.

###### Contents

  1. 1 Introduction
  2. 2 Topological recursion
  3. 3 Intersection numbers on moduli spaces
  4. 4 Moduli space of curves and topological recursion
    1. 4.1 Topological recursion in terms of intersection numbers on ${\overline{{\mathcal{M}}}_{g,n}}$
    2. 4.2 Parameters $x_{a,n}$ and $y_{a,n}$
  5. 5 The LSZ-model
  6. 6 Result and Outlook
  7. A Expansion data of the combinatorial limit of the LSZ model with quartic interaction

##  1 Introduction

Topological recursion has been developed by L. Chekhov, B. Eynard, and N.
Orantin formalizing the solution strategy of certain matrix models
Chekhov_2006a ; Chekhov_2006b ; Chekhov_2006c ; Eynard:2007kz . This
technique, which turned out to be a universal structure underlying to various
problems, uses complex analytic tools to recursively compute essential objects
called correlation functions from initial data provided by a spectral curve.
The theory is embedded in a web of mathematical fields connected by deep
theorems since the 90s Belliard21 .
Then M. Kontsevich Kontsevich:1992ti  proved E. Wittens conjecture
Witten:1990hr  relating intersection numbers on the moduli space of complex
curves to a special class of integrable hierarchies. In his seminal work he
interpreted the moduli problem first in terms of enumerative geometry. Then
used complex analytic tools, which can in modern language be formulated in
terms of topological recursion, to apply work of Dijkgraaf, Verlinde and
Verlinde DIJKGRAAF1991435  to prove integrability.
Beyond this prime example, there are various cases where the correlation
functions computed from topological recursion have enumerative geometric
interpretation. Furthermore, it is generally provided that theories obeying
topological recursion correspond to a solution for a class of integrable
hierarchies Eynard:2007kz .
In this work the relation of topological recursion to the moduli space of
complex curves is studied, based on work of B. Eynard Eynard1104 ; Eynard1110
. There he proves the expansion of correlators generated by topological
recursion from regular spectral curves with finitely many ramification points
in terms of intersection numbers on the moduli space of curves. We provide an
illustrative step-by-step example expanding correlators of a physically
motivated theory. By that means we intend not only to give more insight into
the structures involved but also to unify notation as well as conventions.

Quantum field theories provide a proper framework to describe observations in
particle physics since their introduction in the beginning of the past
century. Despite its great success, its formulation is plagued with a number
of issues and remains a topic of active research in physics and mathematics
throughout the decades. Inspired by the ambition to find a quantum theory of
gravity in the beginning of this century the study of quantum field theories
on non-commutative space has received attention. Prominent instances are the
Grosse-Wulkenhaar model a theory of hermitian fields with quartic interaction
GW05 ; Grosse:2012uv  as well as the Langmann-Szabo-Zarembo (LSZ) model
Langmann_2004 .
Recently significant progress has been made by the solution of the LSZ model
with quartic interaction by Branahl and Hock in BH2205 . In their work the
authors are able to interpret Dyson-Schwinger equations for correlators of the
model as abstract loop equations. The analysis of their pole structure allowed
Branahl and Hock to show that the correlators of the LSZ model follow Chekhov-
Eynard-Orantin topological recursion Eynard:2007kz .111A similar analysis is
conjectured HW2103 ; BHW2208  to yield the solvability of the Grosse-
Wulkenhaar model by Borot-Shadrin blobbed topological recursion Borot:2015hna
. In BH2205  the authors provide the information about the spectral curve,
which constitutes the central object of the analysis in this work.

In the following we are going to describe the necessary parts of the theory of
topological recursion and of intersection numbers on the moduli space of
complex curves (cf. Sections 2 and 3). Thereafter, the expansion of
correlation functions generated by topological recursion in terms of
intersection numbers is recalled, following Eynard Eynard1110  (cf. Section
4.1) and in terms of different expansion data (cf. Section 4.2). These
sections should provide a unified notation that can be used in the later
analysis. In Section 5 we give a short reminder on the model that is
investigated in this work and comment on its combinatorial limit. Details and
results of the analysis can be found in Section 6, with the data listed in
Appendix A.

##  2 Topological recursion

In the beginning of the 2000s the study of matrix models culminated in the
introduction of topological recursion in Eynard:2007kz  as a central framework
that computes the correlators of the system recursively. The subsequent work
generalized this framework immensely and yielded numerous applications
throughout mathematics and physics Eynard:2016yaa ; Belliard21 . Proving
topological recursion for a specific problem/model is in general a hard task
and has crucial implications. Importantly, it implies exact solvability and
integrability in terms of some associated integrable hierarchy. Most often it
is revealed that certain physically motivated models correspond to specific
enumerative problems and vice versa. The task of showing Eynard-Orantin
topological recursion for the LSZ model (see Section 5) was completed in
BH2205 . This extends the proof of exact solvability of the original work and,
importantly, establishes similarities to other models such as the Grosse-
Wulkenhaar model.

In the following, the original Eynard-Orantin topological recursion is
reviewed. For details the reader is referred to Eynard:2007kz  or various
later works reviewing and generalizing the framework.
Topological recursion computes an infinite sequence of admissible correlators
$\\{\omega_{g,n}\\}_{g,n}$ indexed by $g\in{\mathbb{N}}$, and
$n\in{\mathbb{N}}^{\times}$, which are meromorphic 1-forms in each of their
$n$ variables and are defined on an $n$-fold product of a Riemann surface
$\Sigma$ with genus $g$. The recursion is conducted in $(2(g-1)+n)$ and the
initial data is provided by a spectral curve ${\mathcal{S}}$. It consists of a
Riemann surface $\Sigma$ as well as two meromorphic functions $x$ and $y$,
which are maps from $\Sigma$ to a base Riemann surface $\Sigma_{0}$ .
The first correlator is, then, given by

$\displaystyle\omega_{0,1}(z_{1})=y(z_{1})\differential{x(z_{1})}\,.$ (2.1)

Note that only regular spectral curves are considered here. This requires the
zeros of $\differential{x}$, the ramification points
$\\{a_{i}\\}_{i=1,\dots\mathfrak{b}}$, to be simple as well as $y$ and
$\differential{y}$ is non-vanishing there. Furthermore, the so-called Bergman
kernel $B$, which is defined to be a normalized, symmetric second-kind
bilinear meromorphic differential, having a double pole on the diagonal and no
other pole, gives the correlator

$\displaystyle\omega_{0,2}(z_{1},z_{2})=B(z_{1},z_{2})\overset{z_{1}\to
z_{2}}{\sim}\frac{\differential{z_{1}}\differential{z_{2}}}{(z_{1}-z_{2})^{2}}+\text{holomorphic}\,.$
(2.2)

At this point all ”unstable” correlators are provided by the spectral curve
${\mathcal{S}}=(\Sigma,x,y,B)$. The recursion (also illustrated in Figure 1)

$\displaystyle\omega_{g,n+1}(z,I)=\sum\nolimits_{i=1}^{\mathfrak{b}}{\underset{q\to
a_{i}}{\text{{Res }}}}$
$\displaystyle\;K_{i}(z,q)\left[\omega_{g-1,n+2}(q,\varsigma_{i}(q),I)\color[rgb]{1,1,1}\sum_{m}^{m}\right.$
$\displaystyle\left.+\sum_{g_{1}+g_{2}=g}\sum_{I_{1}\sqcup
I_{2}=I}^{\prime}\omega_{g_{1},1+\absolutevalue{I_{1}}}(q,I_{1})\;\omega_{g_{2},1+\absolutevalue{I_{2}}}(\varsigma_{i}(q),I_{2})\right]$
(2.3)

computes all higher correlators, which are meromorphic with poles only at the
ramification points with vanishing residue, where $\varsigma_{i}$ is the local
involution of $x$ at the $i$’th marked point.

Figure 1: Illustration of equation (2). According to decompositions in terms
of pairs of pants a Riemann surface of type $(g,n+1)$ can be found from
glueing one of type $(g-1,n+2)$ or two of type
$(g_{1},1+\absolutevalue{I_{1}})$ and $(g_{2},1+\absolutevalue{I_{2}})$. Note
that topological recursion is not associated to a decomposition in terms of
pairs of pants but pairs of pants and cylinders, i.e. surfaces of topology
$(0,2)$ and $(0,3)$.

These correlators obey so-called abstract loop equations Borot:2013lpa ,
however, do not provide the most general solution to the latter Borot:2015hna
. In the recursion above the set $I$ is given by $I=\\{z_{1},\dots,z_{n}\\}$
and in the primed sum terms containing $(0,1)$-type correlators are excluded.
The recursion kernel $K_{i}$ is composed of the initial data as

$\displaystyle
K_{i}(z,q)=\frac{\frac{1}{2}\int_{\varsigma_{i}(q)}^{q}B(z,\bullet)}{\omega_{0,1}(q)-\omega_{0,1}(\varsigma_{i}(q))}\,.$
(2.4)

##  3 Intersection numbers on moduli spaces

Initiated by the work of M. Kontsevich Kontsevich:1992ti , B. Eynard was able
to relate the correlation functions $\omega_{g,n}$ computed by topological
recursion to intersection theory on the moduli space of complex curves
Eynard1104 ; Eynard1110 .
In order to provide a brief reminder (following Eynard1110 ), let
${{\mathcal{M}}_{g,n}}$ be the moduli space of smooth irreducible complex
curves of genus $g$ with $n$ marked points, which is a complex orbifold of
dimension $\dim_{\mathbb{C}}{{\mathcal{M}}_{g,n}}=3(g-1)+n$. As families of
smooth curves may degenerate into non-smooth curves, ${{\mathcal{M}}_{g,n}}$
is not compact. Its well known Deligne-Mumford compactification
${\overline{{\mathcal{M}}}_{g,n}}$, containing also stable nodal curves222The
stability condition can be expressed as the finiteness of the automorphism
group of the respective complex curve and is fulfilled if and only if for
every irreducible component $\chi=2(1-g)-n<0$. In the vicinity of a nodal
singularity the curve is isomorphic to ${\mathbb{C}}{[\\![x,y]\\!]}/(xy)$.,
fixes this.
Intersection numbers on ${\overline{{\mathcal{M}}}_{g,n}}$ are defined as
integrals of cohomology classes. In order to make this precise, let

$\displaystyle\pi:\;{\overline{{\mathcal{M}}}_{g,n+1}}\to{\overline{{\mathcal{M}}}_{g,n}}$
(3.1)

be the forgetful morphism, under which the information about the $(n+1)$’st
marked point is lost. Furthermore, denote by $s_{i}$ the $i$’th canonical
section of $\pi$ and by $D_{i}$ the associated divisor, for $1\leq i\leq n$,
and $\omega_{\pi}$ the associated relative dualizing sheaf. Then the classes
of interest in this article include products of $\psi$-classes

$\displaystyle\psi_{i}=c_{1}(s_{i}^{*}(\omega_{\pi}))\,,$ (3.2)

which are differential forms of complex degree one. In the smooth case these
simplify to the first Chern class of co-tangent bundle at the $i$’th marked
point.
Further, the structure is enriched by including

$\displaystyle\kappa_{k}=\pi_{*}c_{1}\left(\omega_{\pi}\otimes{\mathcal{O}}\left(\sum\nolimits_{i}D_{i}\right)\right)^{k+1}\,,$
(3.3)

which are forms333for more details see Mumford1983 ; Arbarello:1994sda ; LX09
of complex degree $k$. Note that

$\displaystyle(\pi_{n+1})_{*}\left(\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\psi_{n+1}^{a_{n+1}+1}\right)=$
$\displaystyle\;\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\kappa_{a_{n+1}}\,,$
(3.4)
$\displaystyle(\pi_{n+1}\pi_{n+2})_{*}\left(\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\psi_{n+1}^{a_{n+1}+1}\psi_{n+2}^{a_{n+2}+1}\right)=$
$\displaystyle\;\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\left(\kappa_{a_{n+1}}\kappa_{a_{n+2}}+\kappa_{a_{n+1}+a_{n+2}}\right)\,,$
(3.5)

and in general

$\displaystyle(\pi_{n+1}\cdots\pi_{n+k})_{*}\left(\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\psi_{n+1}^{a_{n+1}+1}\cdots\psi_{n+k}^{a_{n+k}+1}\right)=$
$\displaystyle\;\psi_{1}^{a_{1}}\cdots\psi_{n}^{a_{n}}\sum_{\tau\in\mathfrak{S}_{k}}\kappa_{\tau}\,,$
(3.6)

which might provide insight into the significance of $\kappa$-classes. In the
above ${\mathfrak{S}}_{k}$ is the symmetric group of order $k$, and
$\kappa_{\tau}=\kappa_{\absolutevalue{\tau_{1}}}\cdots\kappa_{\absolutevalue{\tau_{m}}}$
for the decomposition of $\tau$ into cycles $\tau_{i}$, for $i=1,\dots,m$.
Intersection numbers on ${\overline{{\mathcal{M}}}_{g,n}}$ are then defined as
integrals of products of powers of $\psi$\- and $\kappa$-classes

$\displaystyle{\langle(\dots)\rangle_{g,n}}=\int_{{\overline{{\mathcal{M}}}_{g,n}}}(\dots)\,,$
(3.7)

if and only if the degree of the compound class matches the dimension of the
moduli space (and zero otherwise).
In order to conveniently express the correlation functions associated to
spectral curves with several branchpoints in terms of intersection numbers,
following Eynard1110  it is useful to enrich the moduli space of complex
curves by the information provided by some coloring map $\sigma$. Therefore,
define

$\displaystyle{\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}=\left\\{({\mathcal{C}};p_{1},\dots,p_{n};\sigma)\right\\}\,,$
(3.8)

where $({\mathcal{C}};p_{1},\dots,p_{n})\in{\overline{{\mathcal{M}}}_{g,n}}$
and the coloring map

$\displaystyle\sigma:\;{\mathcal{C}}\backslash\\{\text{nodes}\\}\to\\{1,\dots,{\mathfrak{b}}\\}$
(3.9)

is a continuous map for ${\mathfrak{b}}\in{\mathbb{N}}^{\times}$. Note that
continuity of $\sigma$ implies that it must be constant on each component of
the curve ${\mathcal{C}}$.
In the two examples below ${\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}$
is decomposed into factors of ${\overline{{\mathcal{M}}}_{g,n}}$, providing
some idea of the structure of the enriched moduli space of colored complex
curves. This should stress the fact that the colored moduli space is only an
auxiliary construction abbreviating notation and structuring the calculations.
All computations are equivalently done on (factors of the) ordinary
${\overline{{\mathcal{M}}}_{g,n}}$.

######  Example 3.0.1.

Consider ${\overline{{\mathcal{M}}}_{0,4}^{2}}$. Points in this space are
colored curves of genus zero with four marked points as well as in its
boundary nodal curves of topological type $(0,3)\times(0,3)$. As curves in the
bulk only have one component (on which the color is supposed to be constant)
and ${\mathfrak{b}}=2$, one finds two copies of ${{\mathcal{M}}_{0,4}}$ in
${\overline{{\mathcal{M}}}_{0,4}^{2}}$. Considering all possible colorings of
boundary curves, one finds

$\displaystyle{\overline{{\mathcal{M}}}_{0,4}^{2}}=$
$\displaystyle\,{{\mathcal{M}}_{0,4}}\cup{{\mathcal{M}}_{0,4}}\cup({{\mathcal{M}}_{0,3}}\times{{\mathcal{M}}_{0,3}})^{\cup
12}$ $\displaystyle=$
$\displaystyle\,{\overline{{\mathcal{M}}}_{0,4}}\sqcup{\overline{{\mathcal{M}}}_{0,4}}\sqcup({{\mathcal{M}}_{0,3}}\times{{\mathcal{M}}_{0,3}})^{\sqcup
6}\,.$ (3.10)

Note that curves in ${\overline{{\mathcal{M}}}_{0,4}^{2}}$ of type
$(0,3)\times(0,3)$, which are unichrome, are in the above decomposition in
terms of compact moduli spaces found in ${\overline{{\mathcal{M}}}_{0,4}}$.
This gives rise to the different amount of factors of
$({{\mathcal{M}}_{0,3}}\times{{\mathcal{M}}_{0,3}})$ in the above (cf. Figure
2).

Figure 2: This depicts different colorings of curves of topological type
$(0,4)$ and $(0,3)\times(0,3)$, elements of
${\overline{{\mathcal{M}}}_{0,4}^{2}}$. Curves in the same gray box are in the
same compact component of ${\overline{{\mathcal{M}}}_{0,4}^{2}}$. Note that
the labelling of the marked points is omitted here to avoid cluttering.

######  Example 3.0.2.

As a second example take the decomposition of
${\overline{{\mathcal{M}}}_{1,1}^{2}}$. Curves of type $(1,1)$ in
${\overline{{\mathcal{M}}}_{1,1}^{2}}$ may degenerate into curves of type
$(0,3)$. As all curves have only one component, there are no polychrome
curves. Thus, one finds

$\displaystyle{\overline{{\mathcal{M}}}_{1,1}^{2}}=$
$\displaystyle\;{{\mathcal{M}}_{1,1}}\cup{{\mathcal{M}}_{1,1}}\cup{{\mathcal{M}}_{0,3}}\cup{{\mathcal{M}}_{0,3}}$
$\displaystyle=$
$\displaystyle\;{\overline{{\mathcal{M}}}_{1,1}}\sqcup{\overline{{\mathcal{M}}}_{1,1}}\,.$
(3.11)

An illustration can be found in Figure 3.

Figure 3: This depicts different colorings of curves of topological type
$(1,1)$ and $(0,3)$, elements of ${\overline{{\mathcal{M}}}_{1,1}^{2}}$.
Curves in the same gray box are in the same compact component of
${\overline{{\mathcal{M}}}_{1,1}^{2}}$. Note that the labelling of the marked
points is omitted here to avoid cluttering.

##  4 Moduli space of curves and topological recursion

Building on the work of M. Kontsevich Kontsevich:1992ti , relating symplectic
invariants associated to the Airy spectral curve to intersection numbers of
$\psi$-classes, B. Eynard is able to show the expansion of the correlation
functions generated by topological recursion in terms of $\psi$\- and
$\kappa$-classes. We cite his theorem here as Theorem 4.1, presented in
Subsection 4.1. His proof for spectral curves with one simple branchpoint
relies on deformation arguments (cf. Eynard1104 ). It is extended to spectral
curves with finitely many simple branchpoints by glueing-arguments (cf.
Eynard1110 ).

###  4.1 Topological recursion in terms of intersection numbers on
${\overline{{\mathcal{M}}}_{g,n}}$

######  Theorem 4.1 (Eynard ’11).

Let ${\mathcal{S}}=(\Sigma;x,y;B)$ be a spectral curve with ramification
points $\\{a_{1},\dots,a_{\mathfrak{b}}\\}$ and consider local spectral curves
${\mathcal{S}}_{i}=(\Sigma_{i};x,y;B_{\text{loc.}})$ where
$\Sigma_{i}\subset\Sigma$ is a neighborhood of the ramification point $a_{i}$,
$x$ and $y$ are the restrictions of $x$ and $y$ to $\Sigma_{i}$, and
$B_{\text{loc.}}$ an arbitrary Bergman kernel on $\Sigma_{i}\times\Sigma_{i}$.
Then,

$\displaystyle\omega_{g,n}({\mathcal{S}};z_{1},\dots,z_{n})=2^{\dim_{\mathbb{C}}{{\mathcal{M}}_{g,n}}}\bigintsss_{{\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}}\prod_{m=1}^{k}\hat{\Lambda}_{a_{m}}\prod_{(q_{i},q_{j})\in\\{\text{nodes}\\}}c(q_{i},q_{j})\prod_{i=1}^{n}\hat{B}_{a_{\sigma(p_{i})}}(z_{i};1/\psi_{i})\,.$
(4.1)

While the left-hand-side is described in Section 2, the right-hand-side is
expressed in terms of intersection numbers and local expansion data of the
spectral curve. The different contributions are introduced in the following
sections.

Note that this theorem allows for a slightly more general setup in terms of
the spectral curve. The spectral curve of the LSZ model is given in a global
form and thus $B_{\text{loc.}}=B$. Furthermore, the combinatorial limit of the
LSZ model considered here enforces ${\mathfrak{b}}=2$. The restriction to
${\mathfrak{b}}=2$ on the one hand simplifies the analysis immensely, but
allows on the other hand still to observe phenomena that would not appear in
the trivial case ${\mathfrak{b}}=1$.

####  4.1.1 Local expansion data of spectral curves

In the same manner as the correlation functions are computed in topological
recursion through local residuum calculations, the data required to express
these in terms of intersection numbers is extracted from local expansions of
the spectral curve at the ramification points.
To that end, in the vicinity of a simple ramification point
$a\in\\{a_{i}\\}_{i=1,\dots,{\mathfrak{b}}}$ define the local variable
$\zeta_{a}$ in terms of the function $x$ by

$\displaystyle\zeta_{a}(z)=\sqrt{x(z)-x(a)}$ (4.2)

associated to the two sheets of the complex curve ramifying at $a$. In the
following, the data, $y\differential{x}$ and $B$, defined on the spectral
curve are expanded in $\zeta_{a}$ obtaining expansion data $t_{a,k}$ and
$B_{a,k;a^{\prime},k^{\prime}}$. These encode as moduli the spectral curve
$(\Sigma,y,x,B)$, a B-model geometry444The nomenclature of the ”models” has
been adopted in various works and originates in the string theory literature.
In the context of topological recursion there is the notion of the ”four
sides” of a problem. A nice presentation of this can be found in Belliard21 .
From this point of view what we call A-model here would correspond to the
C-side and the B-model would be the B-side.. B-model moduli are related
through Laplace transformation to the moduli of the A-model, $\hat{t}_{a,k}$
and $\hat{B}_{a,k;a^{\prime},k^{\prime}}$. The transformation is given by a
Laplace transformation in terms of $x$ and the integration path is the
steepest descending contour $\gamma_{a}$ such that
$x(\gamma_{a})=\left[x(a),\infty\right[$.

##### Times $t_{a,k}$

The times $t_{a,k}$, defined as the expansion of $y(z)$ in terms of
$\zeta_{a}$ as

$\displaystyle y(z)\overset{z\to a}{=}\sum_{k\geq
0}t_{a,k+2}\,\zeta_{a}(z)^{k}\,,$ (4.3)

define the dual times $\hat{t}_{a,k}$ through the Laplace transformation by

$\displaystyle\exp(-\sum_{k\geq
0}\hat{t}_{a,k}u^{-k})=2\sqrt{\frac{u}{\pi}}\int_{z\in\gamma_{a}}e^{-u\,\zeta_{a}(z)^{2}}\differential{y(z)}\,.$
(4.4)

By inserting the expansion of $y$ and performing Gaussian integrals, one can
establish the relation

$\displaystyle\sum_{k\geq 0}\hat{t}_{a,k}u^{-k}=-\log(2\sum_{k\geq
0}t_{a,2k+3}\frac{(2k+1)!!}{2^{k}}u^{-k})\,,$ (4.5)

known as the Schur transform.

##### Coefficients $B_{a,k;a^{\prime},k^{\prime}}$

Analogously, the Bergman-kernel is expanded in terms of the local variables

$\displaystyle B(z,z^{\prime})\overset{\begin{subarray}{c}z\to a\\\
z^{\prime}\to
a^{\prime}\end{subarray}}{=}\left(\frac{\delta_{a,a^{\prime}}}{(\zeta_{a}(z)-\zeta_{a^{\prime}}(z^{\prime}))^{2}}+\sum_{k,k^{\prime}\geq
0}B_{a,k;a^{\prime},k^{\prime}}\,\zeta_{a}(z)^{k}\zeta_{a^{\prime}}(z^{\prime})^{k^{\prime}}\right)\differential{\zeta_{a}(z)}\differential{\zeta_{a^{\prime}}(z^{\prime})}\,.$
(4.6)

The expansion coefficients $B_{a,k;a^{\prime},k^{\prime}}$ determine their
duals defined by

$\displaystyle\sum_{k,k^{\prime}\geq
0}\hat{B}_{a,k;a^{\prime},k^{\prime}}\,u^{-k}v^{-k}=$
$\displaystyle\;\delta_{a,a^{\prime}}\frac{uv}{u+v}+\frac{\sqrt{uv}}{2\pi}\int_{z\in\gamma_{a}}e^{-u\zeta_{a}(z)^{2}}\int_{z^{\prime}\in\gamma_{a^{\prime}}}e^{-v\zeta_{a}(z^{\prime})^{2}}B(z,z^{\prime})$
$\displaystyle=$ $\displaystyle\;\hat{B}_{a,a^{\prime}}(u,v)\,.$ (4.7)

Expansion and evaluation of integrals yields

$\displaystyle\sum_{k,k^{\prime}\geq
0}\hat{B}_{a,k;a^{\prime},k^{\prime}}\,u^{-k}v^{-k}=\frac{1}{2}\sum_{k,k^{\prime}\geq
0}B_{a,2k;a^{\prime},2k^{\prime}}\,\frac{(2k-1)!!}{2^{k}}\frac{(2k^{\prime}-1)!!}{2^{k^{\prime}}}\,u^{-k}v^{-k}\,.$
(4.8)

Remark that from the relations (4.4) and (4.8) it is explicit that the A-model
moduli are determined by only half the B-model moduli.
The expansion data for the spectral curve associated to the LSZ model in the
combinatorial limit with quartic interaction is provided in Appendix A.

####  4.1.2 Integration class

The local expansion data of the spectral curve described in the previous
section provides all the information needed to define the integrand on the
right-hand-side of equation (4.1) in Theorem 4.1.
The three factors of the integrand are due to contributions of elements in
${\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}$ – the bulk of the
different components of a curve, its nodal points as well as the marked points
(see Figure 4). The product of these factors is integrated over the moduli
space of colored complex curves. Comparing to the structure of
${\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}$ explained in Section 3
this amounts to an integration over the ordinary moduli space accompanied with
a sum over colors.

Figure 4: Sketch of an element of ${\overline{{\mathcal{M}}}_{g,n}}$, for
$g=2$ and $n=3$. The different components are colored in different shades of
green, the marked points $p_{i}$ are illustrated in violet and the nodes
$(q_{i},q_{j})$ indicated in red.

##### Bulk contribution

Each component ${\mathcal{C}}_{m}$ of a colored curve
${\mathcal{C}}=(\cup_{m=1}^{k}{\mathcal{C}}_{m})\in{\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}$
contributes a factor

$\displaystyle\hat{\Lambda}_{a}=\exp(\sum_{k}\hat{t}_{a,k}\kappa_{k}+\frac{1}{2}\sum_{\delta\in\partial{{\mathcal{M}}_{g,n}}}\sum_{k,k^{\prime}}\hat{B_{\text{loc.}}}_{a,k;a,k^{\prime}}{\ell_{\delta}}_{*}(\psi^{k}\psi^{\prime
k^{\prime}}))\,,$ (4.9)

where the sum $\sum_{\delta\in\partial{{\mathcal{M}}_{g,n}}}$ runs over the
co-dimension one boundary divisors $\delta$ of ${{\mathcal{M}}_{g,n}}$. Note
that at a boundary divisor the smooth curve degenerates yielding a node. Thus,
a boundary divisor $\delta$ is isomorphic to either
${\overline{{\mathcal{M}}}_{g_{1},n_{1}}}\times{\overline{{\mathcal{M}}}_{g_{2},n_{2}}}$,
with $g_{1}+g_{2}=g$ and $n_{1}+n_{2}=n+2$, or
${\overline{{\mathcal{M}}}_{g-1,n+2}}$. For a fixed $\delta$ the map
$\ell_{\delta}$ denotes the embedding

$\displaystyle\ell_{\delta}:\;\delta\hookrightarrow{\overline{{\mathcal{M}}}_{g,n}}\,.$
(4.10)

By that means, the $\psi$-classes in equation (4.9) are defined on either
${\overline{{\mathcal{M}}}_{g_{1},n_{1}}}\times{\overline{{\mathcal{M}}}_{g_{2},n_{2}}}$
or ${\overline{{\mathcal{M}}}_{g-1,n+2}}$, where they are evaluated at the
marked points, which make up the node on the respective component in
$\partial{\overline{{\mathcal{M}}}_{g,n}}$. To be precise, for some class
$\eta\in H^{\bullet}({\overline{{\mathcal{M}}}_{g,n}})$ one evaluates via the
projection formula

$\displaystyle{\langle\eta\;\ell_{\delta}{}_{*}(\psi^{k}\psi^{\prime
k^{\prime}})\rangle_{{\overline{{\mathcal{M}}}_{g,n}}}}={\langle\ell_{\delta}{}^{*}(\eta)\;\psi^{k}\psi^{\prime
k^{\prime}}\rangle_{\delta}}\,.$ (4.11)

If, further,
$\delta\simeq{\overline{{\mathcal{M}}}_{g_{1},n_{1}}}\times{\overline{{\mathcal{M}}}_{g_{2},n_{2}}}$,
one can decompose
$\ell_{\delta}{}^{*}(\eta)=\sum_{i+j=\dim\delta-(k+k^{\prime})}\rho_{1,i}\wedge\rho_{2,j}$
into components in
$H^{i}({\overline{{\mathcal{M}}}_{g_{1},n_{1}}},{\mathbb{Q}})\otimes
H^{j}({\overline{{\mathcal{M}}}_{g_{2},n_{2}}},{\mathbb{Q}})$, with
$i+j=\dim\delta-(k+k^{\prime})$ via the theorem by Künneth and find

$\displaystyle\eqref{equ:projection-
formula}\;=\sum_{i+j=\dim\delta-(k+k^{\prime})}{\langle\rho_{1,i}\;\psi^{k}\rangle_{{\overline{{\mathcal{M}}}_{g_{1},n_{1}}}}}{\langle\rho_{2,j}\;\psi^{\prime
k^{\prime}}\rangle_{{\overline{{\mathcal{M}}}_{g_{2},n_{2}}}}}\,.$ (4.12)

Note that due to the condition $\text{deg}(\text{integration
class})=\dim(\text{moduli space})$ some terms in this sum will vanish.
It may also be remarked that $\hat{\Lambda}_{a}$ as well as
$\hat{B_{\text{loc.}}}_{a,k;a,k^{\prime}}$ is indexed by only one color. This
is due to the fact that one component of a complex curve can be colored in
just one single color (cf. Section 3).

##### Node contribution

The moduli space of colored complex curves contains components of polychrome
nodal curves. These nodes contribute in the integral in equation (4.1) in
Theorem 4.1 through

$\displaystyle c(q_{i},q_{j})=\sum_{d,d^{\prime}\geq
0}\left(\hat{B}_{a_{\sigma(q_{i})},d;a_{\sigma(q_{j})},d^{\prime}}-\delta_{a_{\sigma(q_{i})},a_{\sigma(q_{j})}}\hat{B_{\text{loc.}}}_{a_{\sigma(q_{i})},d;a_{\sigma(q_{j})},d^{\prime}}\right)\;\ell_{*}\left[\psi(q_{i})^{d}\psi(q_{j})^{d^{\prime}}\right]\,.$
(4.13)

The map $\ell$ here includes the boundary component that the considered nodal
curve corresponds to into ${\overline{{\mathcal{M}}}_{g,n}}$. Its pushforward
embeds the $\psi$-classes, which are evaluated on smooth marked points on a
moduli space isomorphic to the boundary component, into
$H^{\bullet}({\overline{{\mathcal{M}}}_{g,n}},{\mathbb{Q}})$.
Note that if $B=B_{\text{loc.}}$, only not self-connecting nodes give a non-
zero result, that is those with $\sigma(q_{i})\neq\sigma(q_{j})$. This is also
observed when decomposing ${\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}$
into components of colored factors of ${\overline{{\mathcal{M}}}_{g,n}}$. Only
polychrome multi-component curves are due to novel components in
${\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}$, while unichrome curves
are parametrized by one of the ${\mathfrak{b}}$ factors of
${\overline{{\mathcal{M}}}_{g,n}}$ in terms of the disjoint decomposition of
${\overline{{\mathcal{M}}}_{g,n}^{{\mathfrak{b}}}}$ (cf. Example 3.0.1 and
3.0.2).

##### Contribution of marked points

In terms of graphs marked points correspond to half edges. This is reflected
in the way marked points contribute in Theorem 4.1. Each marked point
contributes by a single sided Laplace transformation

$\displaystyle\hat{B}_{a}(z;u)=$
$\displaystyle\;-\sqrt{\frac{u}{\pi}}\int_{z\in\gamma_{a}}e^{-u\;\zeta_{a}(z)^{2}}B(z,z^{\prime})$
$\displaystyle=$ $\displaystyle\;\sum_{d\geq
0}u^{-d}\;\differential{\xi_{a,d}(z)}\,.$ (4.14)

The expansion coefficients $\differential{\xi_{a,d}(z)}$ are one-forms and can
be expanded as

$\displaystyle\differential{\xi_{a,d}}(z)\overset{z\to
a^{\prime}}{=}\delta_{a,a^{\prime}}\;\frac{(2d+1)!!}{2^{d}}\frac{\differential{\zeta_{a}(z)}}{\zeta_{a}(z)^{2d+2}}-\frac{(2d-1)!!}{2^{d}}\sum_{k\geq
0}B_{a,2d;a^{\prime},k}\;\zeta_{a^{\prime}}(z)^{k}\differential{\zeta_{a^{\prime}}(z)}\,.$
(4.15)

Before comparing the left-hand-side and right-hand-side of equation (4.1) in
Theorem 4.1, one needs to express both in terms of the same variables. The
approach, followed here, is to evaluate the right-hand-side explicitly in
terms of $\zeta$ and express the result then in terms of $z$. Therefore it is
useful to express the basis of differential forms in terms of
$\differential{z}$. This is conveniently done using

$\displaystyle\differential{\xi_{a,d}}(z)=-\frac{(2d-1)!!}{2^{d}}\;{\underset{z^{\prime}\to
a}{\text{{Res }}}}B(z^{\prime},z)\;\zeta_{a}(z^{\prime})^{-2d-1}\,.$ (4.16)

In order to prepare the analysis described in Section 6 two examples are
provided. These can already be found in the original work Eynard1110 .

######  Example 4.1.1.

Consider $\omega_{0,4}$ for a spectral curve with ${\mathfrak{b}}=2$ and
$B=B_{\text{loc.}}$. As described in example 3.0.1, the moduli space
${\overline{{\mathcal{M}}}_{0,4}^{2}}$ decomposes into two factors of
${\overline{{\mathcal{M}}}_{0,4}}$ and six factors of
${\overline{{\mathcal{M}}}_{0,3}}\times{\overline{{\mathcal{M}}}_{0,3}}$. Thus
Theorem 4.1 expands to

$\displaystyle 2^{-1}\omega_{0,4}(z_{1},\dots,z_{4})$
$\displaystyle=\bigintsss_{{\overline{{\mathcal{M}}}_{0,4}}}\hat{\Lambda}_{a_{1}}\;\prod\nolimits_{i=1}^{4}\hat{B}_{a_{1}}(z_{i};1/\psi_{i})+\bigintsss_{{\overline{{\mathcal{M}}}_{0,4}}}\hat{\Lambda}_{a_{2}}\;\prod\nolimits_{i=1}^{4}\hat{B}_{a_{2}}(z_{i};1/\psi_{i})$
$\displaystyle\quad+\left(\bigintsss_{{\overline{{\mathcal{M}}}_{0,3}}\times{\overline{{\mathcal{M}}}_{0,3}}}\hat{\Lambda}_{a_{1}}\;\hat{\Lambda}_{a_{2}}\;c(q_{1},q_{2})\;\prod_{i=1}^{2}\hat{B}_{a_{1}}(z_{i};1/\psi_{i})\prod_{j=3}^{4}\hat{B}_{a_{2}}(z_{j};1/\psi_{j})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.\color[rgb]{1,1,1}\bigintsss\color[rgb]{0,0,0}+\text{symm}(z_{1},\dots,z_{4})\right)\,.$
(4.17)

Noting that $\dim{\overline{{\mathcal{M}}}_{0,3}}=0$ one expands the integrand
in the second line according to definitions (4.9), (4.13), and (4.1.2) keeping
only those parts of degree zero. This yields for the integral over
${\overline{{\mathcal{M}}}_{0,3}}\times{\overline{{\mathcal{M}}}_{0,3}}$

$\displaystyle\bigintsss_{{\overline{{\mathcal{M}}}_{0,3}}\times{\overline{{\mathcal{M}}}_{0,3}}}e^{\hat{t}_{a_{1},0}\kappa_{0}}e^{\hat{t}_{a_{2},0}\kappa_{0}}\;\hat{B}_{a_{1},0,a_{2},0}\,\psi(q_{1})^{0}\psi(q_{2})^{0}\;\prod\nolimits_{j=1}^{2}\differential{\xi_{a_{1},0}(z_{j})}\psi_{j}^{0}\prod\nolimits_{j^{\prime}=3}^{4}\differential{\xi_{a_{2},0}(z_{j^{\prime}})}\psi_{j^{\prime}}^{0}$
$\displaystyle=e^{(\hat{t}_{a_{1},0}+\hat{t}_{a_{2},0}){\langle\kappa_{0}\rangle_{0,3}}}\;\hat{B}_{a_{1},0,a_{2},0}\;{\langle
1\rangle_{0,3}}{\langle
1\rangle_{0,3}}\;\differential{\xi_{a_{1},0}(z_{1})}\differential{\xi_{a_{1},0}(z_{2})}\differential{\xi_{a_{2},0}(z_{3})}\differential{\xi_{a_{2},0}(z_{4})}\,.$
(4.18)

As $\dim{\overline{{\mathcal{M}}}_{0,4}}=1$, the integrand of the integral
over ${\overline{{\mathcal{M}}}_{0,4}}$ is expanded to degree one. Note that
with every order of $\hat{B_{\text{loc.}}}$ in the expansion of the bulk
contribution $\hat{\Lambda}$ the dimension of the moduli space that is
actually integrated over is reduced by one (cf. equation (4.11)). This yields

$\displaystyle\bigintsss_{{\overline{{\mathcal{M}}}_{0,4}}}\left(e^{\hat{t}_{i,0}\kappa_{0}}(1+\hat{t}_{i,1}\kappa_{1})\right)\left(1+\frac{1}{2}\sum\nolimits_{\delta\in\partial{\overline{{\mathcal{M}}}_{0,4}}}\hat{B}_{a_{i},0,a_{i},0}\psi^{0}\psi^{\prime
0}\right)\prod\nolimits_{j=1}^{4}\left(\differential{\xi_{a_{i},0}(z_{j})}\psi_{j}^{0}+\differential{\xi_{a_{i},1}(z_{j})}\psi_{j}^{1}\right)$
$\displaystyle=\left(e^{\hat{t}_{a_{i},0}{\langle\kappa_{0}\rangle_{0,4}}}\hat{t}_{a_{i},1}{\langle\kappa_{1}\rangle_{0,4}}+e^{2\hat{t}_{a_{i},0}{\langle\kappa_{0}\rangle_{0,3}}}\frac{6}{2}\hat{B}_{a_{i},0,a_{i},0}\;{\langle
1\rangle_{0,3}}{\langle
1\rangle_{0,3}}\right)\prod\nolimits_{j=1}^{4}\differential{\xi_{a_{i},0}(z_{j})}$
$\displaystyle\qquad+\left(e^{\hat{t}_{a_{i},0}{\langle\kappa_{0}\rangle_{0,4}}}\;{\langle\psi_{1}\rangle_{0,4}}\;\differential{\xi_{a_{i},1}(z_{1})}\differential{\xi_{a_{i},0}(z_{2})}\differential{\xi_{a_{i},0}(z_{3})}\differential{\xi_{a_{i},0}(z_{4})}+\text{symm}(z_{1},\dots,z_{4})\right)\,.$
(4.19)

In conclusion

$\displaystyle 2^{-1}\omega_{0,4}=$
$\displaystyle\,\left(e^{\hat{t}_{a_{1},0}{\langle\kappa_{0}\rangle_{0,4}}}\hat{t}_{a_{1},1}{\langle\kappa_{1}\rangle_{0,4}}+3e^{2\hat{t}_{a_{1},0}{\langle\kappa_{0}\rangle_{0,3}}}\hat{B}_{a_{1},0,a_{1},0}\;{\langle
1\rangle_{0,3}}{\langle
1\rangle_{0,3}}\right)\prod\nolimits_{j=1}^{4}\differential{\xi_{a_{1},0}(z_{j})}$
$\displaystyle\quad+e^{\hat{t}_{a_{1},0}{\langle\kappa_{0}\rangle_{0,4}}}\;{\langle\psi\rangle_{0,4}}\;\sum\nolimits_{i=1}^{4}\left(\differential{\xi_{a_{1},1}(z_{i})}\prod\nolimits_{j\neq
i}^{4}\differential{\xi_{a_{1},0}(z_{j})}\right)$
$\displaystyle+\left(e^{\hat{t}_{a_{2},0}{\langle\kappa_{0}\rangle_{0,4}}}\hat{t}_{a_{2},1}{\langle\kappa_{1}\rangle_{0,4}}+3e^{2\hat{t}_{a_{2},0}{\langle\kappa_{0}\rangle_{0,3}}}\hat{B}_{a_{2},0,a_{2},0}\;{\langle
1\rangle_{0,3}}{\langle
1\rangle_{0,3}}\right)\prod\nolimits_{j=1}^{4}\differential{\xi_{a_{2},0}(z_{j})}$
$\displaystyle\quad+e^{\hat{t}_{a_{2},0}{\langle\kappa_{0}\rangle_{0,4}}}\;{\langle\psi\rangle_{0,4}}\;\sum\nolimits_{i=1}^{4}\left(\differential{\xi_{a_{2},1}(z_{i})}\prod\nolimits_{j\neq
i}^{4}\differential{\xi_{a_{2},0}(z_{j})}\right)$
$\displaystyle+e^{(\hat{t}_{a_{1},0}+\hat{t}_{a_{2},0}){\langle\kappa_{0}\rangle_{0,3}}}\;\hat{B}_{a_{1},0,a_{2},0}\;{\langle
1\rangle_{0,3}}{\langle 1\rangle_{0,3}}\;$
$\displaystyle\qquad\qquad\qquad\quad\times\frac{1}{2}\sum\nolimits_{i=1}^{4}\sum\nolimits_{j\neq
i}^{4}\differential{\xi_{a_{1},0}(z_{i})}\differential{\xi_{a_{1},0}(z_{j})}\prod\nolimits_{k\neq
i,j}^{4}\differential{\xi_{a_{2},0}(z_{k})}$ $\displaystyle=$
$\displaystyle\;\sum_{\sigma\in\\{1,2\\}}\left[e^{2\hat{t}_{a_{\sigma},0}}\left(\hat{t}_{a_{\sigma},1}+3\hat{B}_{a_{\sigma},0,a_{\sigma},0}\;\right)\prod\nolimits_{j=1}^{4}\differential{\xi_{a_{\sigma},0}(z_{j})}\right.$
$\displaystyle\qquad\qquad\qquad\left.+e^{2\hat{t}_{a_{\sigma},0}}\;\sum\nolimits_{i=1}^{4}\left(\differential{\xi_{a_{\sigma},1}(z_{i})}\prod\nolimits_{j\neq
i}^{4}\differential{\xi_{a_{1},0}(z_{j})}\right)\;\right]$
$\displaystyle+e^{(\hat{t}_{a_{1},0}+\hat{t}_{a_{2},0})}\;\hat{B}_{a_{1},0,a_{2},0}\frac{1}{2}\sum\nolimits_{i=1}^{4}\sum\nolimits_{j\neq
i}^{4}\differential{\xi_{a_{1},0}(z_{i})}\differential{\xi_{a_{1},0}(z_{j})}\prod\nolimits_{k\neq
i,j}^{4}\differential{\xi_{a_{2},0}(z_{k})}\,.$ (4.20)

######  Example 4.1.2.

In the same setting as above, consider $\omega_{1,1}$. In Example 3.0.2 it is
laid down that
${\overline{{\mathcal{M}}}_{1,1}^{2}}\simeq{\overline{{\mathcal{M}}}_{1,1}}\sqcup{\overline{{\mathcal{M}}}_{1,1}}$.
Thus from Theorem 4.1 one finds

$\displaystyle
2^{-1}\omega_{1,1}(z_{1})=\bigintsss_{{\overline{{\mathcal{M}}}_{1,1}}}\hat{\Lambda}_{a_{1}}\;\hat{B}_{a_{1}}(z_{1};1/\psi_{1})+\bigintsss_{{\overline{{\mathcal{M}}}_{1,1}}}\hat{\Lambda}_{a_{2}}\;\hat{B}_{a_{2}}(z_{2};1/\psi_{2})\,.$
(4.21)

Expanding the integrand and keeping only contributions up to degree one,
yields for both summands in the above $(\sigma=1,2)$

$\displaystyle\bigintsss_{{\overline{{\mathcal{M}}}_{1,1}}}\left(e^{\hat{t}_{a_{\sigma},0}\kappa_{0}}(1+\hat{t}_{i,1}\kappa_{1})\right)\left(1+\frac{1}{2}\sum\nolimits_{\delta\in\partial{\overline{{\mathcal{M}}}_{0,4}}}\hat{B}_{a_{\sigma},0,a_{\sigma},0}\psi^{0}\psi^{\prime
0}\right)(\differential{\xi_{a_{\sigma},0}(z_{1})}\psi_{1}^{0}+\differential{\xi_{a_{\sigma},1}(z_{1})}\psi_{1}^{1})$
$\displaystyle=\left(e^{\hat{t}_{a_{\sigma},0}{\langle\kappa_{0}\rangle_{1,1}}}\hat{t}_{a_{\sigma},1}{\langle\kappa_{1}\rangle_{1,1}}+\frac{1}{2}e^{\hat{t}_{a_{\sigma},0}{\langle\kappa_{0}\rangle_{0,3}}}\hat{B}_{a_{\sigma},0,a_{\sigma},0}\;{\langle
1\rangle_{0,3}}\right)\differential{\xi_{a_{\sigma},0}(z_{1})}$
$\displaystyle\qquad\qquad\qquad+e^{\hat{t}_{a_{\sigma},0}{\langle\kappa_{0}\rangle_{1,1}}}\;{\langle\psi_{1}\rangle_{1,1}}\;\differential{\xi_{a_{\sigma},1}(z_{1})}$
$\displaystyle=e^{\hat{t}_{a_{\sigma},0}}\left(\frac{\hat{t}_{a_{\sigma},1}}{24}+\frac{\hat{B}_{a_{\sigma},0,a_{\sigma},0}}{2}\right)\differential{\xi_{a_{\sigma},0}(z_{1})}+e^{\hat{t}_{a_{\sigma},0}}\;\frac{1}{24}\;\differential{\xi_{a_{\sigma},1}(z_{1})}\,.$
(4.22)

###  4.2 Parameters $x_{a,n}$ and $y_{a,n}$

In order to compare the expressions obtained in the previous section to
results obtained from topological recursion for physical theories it is
convenient to transform to a different equivalent set of moduli of the
spectral curve. In the section above, the A-side moduli $t_{a,k}$ and
$B_{a,k,a^{\prime},k^{\prime}}$ are used to expand the correlators. In the
following, we transform to

$\displaystyle
x_{a,n}=\left.\frac{{\partial_{z}}^{n+2}x(z)}{{\partial_{z}}^{2}x(z)}\right|_{z=a}\,,$
$\displaystyle
y_{a,n}=\left.\frac{{\partial_{z}}^{n+1}y(z)}{\partial_{z}y(z)}\right|_{z=a}\,,$
(4.23)

for $n\in{\mathbb{N}}^{\times}$, supplemented with
$x_{a,0}=\left.{\partial_{z}}^{2}x(z)\right|_{z=a}$ and
$y_{a,0}=\left.\partial_{z}y(z)\right|_{z=a}$. These are well defined, as the
spectral curve is required to be regular. However, the definitions can be
generalized to a larger class of spectral curves.

##### Times $t_{a,k}$

In order to obtain the times $t_{a,k}$ in terms of the parameters $x_{a,n}$
and $y_{a,n}$, one compares the expansion of $y$ containing $x_{a,n}$ and
$t_{a,k}$ to that containing $y_{a,n}$ as

$\displaystyle y(\\{x_{a,n},t_{a,k}\\};z)=y(\\{y_{a,n}\\};z)\,.$ (4.24)

The left-hand-side is obtained by expanding first, as explained in equation
(4.3), in terms of $\zeta_{a}(z)$, and then in terms of $z$ at the
ramification point

$\displaystyle y(\\{x_{a,n},t_{a,k}\\};z)\overset{z\to a}{=}\sum_{k\geq
0}t_{a,k+2}\,\zeta_{a}(z)^{k}\overset{z\to a}{=}\sum_{k\geq
0}t_{a,k+2}\sum_{l\geq k}{\mathfrak{z}}_{a,k,l}(\\{x_{a,n}\\})(z-a)^{l}\,.$
(4.25)

There ${\mathfrak{z}}_{a,k,l}(\\{x_{a,n}\\})$ is the coefficient of
$(z-a)^{l}$ in the expansion of $(\zeta_{a})^{k}=(x(z)-x(a))^{k/2}$, which
depends on the parameters $\\{x_{a,n}\\}$.
The right-hand-side is simply

$\displaystyle y(\\{y_{a,n}\\};z)\overset{z\to
a}{=}y(a)+y_{a,0}(z-a)+y_{a,0}\sum_{n\geq
1}y_{a,n}\frac{(z-a)^{n+1}}{(n+1)!}\,.$ (4.26)

A comparison of coefficients in equations (4.25) and (4.26) yields relations
between the parameters $t_{a,k}$ and $x_{a,n}$ as well as $y_{a,n}$. The
expressions for the first few $t_{a,k}$ can be found in Appendix A.

##### Coefficients $B_{a,k;a^{\prime},k^{\prime}}$

A similar analysis that gave the times $t_{a,k}$ in terms of the new set of
moduli of the spectral curve, also gives the expansion coefficients
$B_{a,k;a^{\prime},k^{\prime}}$ in terms of $x_{a,n}$ and $y_{a,n}$. In
particular, one obtains the set
$\\{B_{a,k;a^{\prime},k^{\prime}}\\}_{k,k^{\prime}\in{\mathbb{N}}}$ by
expanding the Bergman kernel in the local coordinate
$\zeta_{a}=\sqrt{x(z)-x(a)}$. Again, expanding the appearing powers of
$\zeta_{a}(z)$ in $z$ gives

$\displaystyle B(z,z^{\prime})$ $\displaystyle\overset{\begin{subarray}{c}z\to
a\\\ z^{\prime}\to
a^{\prime}\end{subarray}}{=}\left[\frac{\delta_{a,a^{\prime}}}{(\zeta_{a}(z)-\zeta_{a^{\prime}}(z^{\prime}))^{2}}+\sum_{k,k^{\prime}\geq
0}B_{a,k;a^{\prime},k^{\prime}}\,\zeta_{a}(z)^{k}\zeta_{a^{\prime}}(z^{\prime})^{k^{\prime}}\right]\differential{\zeta_{a}(z)}\differential{\zeta_{a^{\prime}}(z^{\prime})}$
(4.27) $\displaystyle\overset{\begin{subarray}{c}z\to a\\\ z^{\prime}\to
a^{\prime}\end{subarray}}{=}\differential{z}\differential{z^{\prime}}\left(\sum_{n\geq
0}\sum_{n=0}^{m}(n+1)(n-m+1){\mathfrak{z}}_{a,1,n+1}{\mathfrak{z}}_{a^{\prime},1,n-m+1}(z-a)^{n}(z^{\prime}-a^{\prime})^{n-m}\right)$
$\displaystyle\qquad\times\left[\frac{\delta_{a,a^{\prime}}}{\left[\sum_{l\geq
1}{\mathfrak{z}}_{a,1,l}\left((z-a)^{l}-(z^{\prime}-a)^{l}\right)\right]^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\sum_{k,k^{\prime}\geq
0}B_{a,k;a^{\prime},k^{\prime}}\sum_{l\geq
0}\sum_{r=0}^{l}{\mathfrak{z}}_{a,k,r}{\mathfrak{z}}_{a^{\prime},k^{\prime},l-r}(z-a)^{r}(z^{\prime}-a^{\prime})^{l-r}\right]\,,$
(4.28)

with ${\mathfrak{z}}_{a,k,l}={\mathfrak{z}}_{a,k,l}(\\{x_{a,n}\\})$ as above.
This expression is compared with the explicit direct expansion of
$B(z,z^{\prime})$ for $z\to a,z^{\prime}\to a^{\prime}$, yielding relations
between $\\{B_{a,k;a^{\prime},k^{\prime}}\\}$ and
$\\{x_{A,n}\\}_{A\in\\{a,a^{\prime}\\}}$. In Appendix A these expressions can
be found for the lowest order coefficients.

Note that in Theorem 4.1 the B-side moduli are used to expand the correlators
in terms of intersection numbers on the moduli space of complex curves.
However, there are explicit relations between A- and B-side moduli given in
equations (4.5) and (4.8).

In the following, the expressions obtained in Examples 4.1.1 and 4.1.2 are
rewritten in terms of the new set of moduli of the spectral curve.

######  Example 4.2.1.

In the discussion of $\omega_{0,4}$ in Example 4.1.1 its expansion in terms of
intersection numbers on the moduli space of complex curves was re-derived.
Using the relation of $t_{a,k}$ and $B_{a,k;a^{\prime},k^{\prime}}$ to
$x_{a,n}$ and $y_{a,n}$ established above and explicitly written for the first
few parameters in Appendix A, the expansion given in equation (4.1.1) is
rewritten to obtain

$\displaystyle 2^{-1}\omega_{0,4}=$
$\displaystyle\sum_{\sigma\in\\{1,2\\}}\frac{1}{48x_{a_{\sigma},0}^{2}y_{a_{\sigma},0}^{2}}\left[\frac{(3-5{\langle\kappa_{1}\rangle_{0,4}}+20{\langle\psi\rangle_{0,4}})x_{a_{\sigma},1}^{2}-3(1-{\langle\kappa_{1}\rangle_{0,4}}+4{\langle\psi\rangle_{0,4}})x_{a_{\sigma},2}}{(z_{1}-a_{\sigma})^{2}(z_{2}-a_{\sigma})^{2}(z_{3}-a_{\sigma})^{2}(z_{4}-a_{\sigma})^{2}}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+\frac{12{\langle\kappa_{1}\rangle_{0,4}}x_{a_{\sigma},1}y_{a_{\sigma},1}-12{\langle\kappa_{1}\rangle_{0,4}}y_{a_{\sigma},2}}{(z_{1}-a_{\sigma})^{2}(z_{2}-a_{\sigma})^{2}(z_{3}-a_{\sigma})^{2}(z_{4}-a_{\sigma})^{2}}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad-24{\langle\psi\rangle_{0,4}}x_{a_{\sigma},1}\sum\nolimits_{i=1}^{4}\left(\frac{1}{(z_{i}-a_{\sigma})^{3}}\prod\nolimits_{j\neq
i}^{4}\frac{1}{(z_{j}-a_{\sigma})^{2}}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad+72{\langle\psi\rangle_{0,4}}\sum\nolimits_{i=1}^{4}\left(\frac{1}{(z_{i}-a_{\sigma})^{4}}\prod\nolimits_{j\neq
i}^{4}\frac{1}{(z_{j}-a_{\sigma})^{2}}\right)\right]$
$\displaystyle+\frac{1/2}{(a_{1}-a_{2})^{2}x_{a_{1},0}x_{a_{2},0}y_{a_{1},0}y_{a_{2},0}}\left(\frac{1}{(z_{1}-a_{1})^{2}(z_{2}-a_{1})^{2}(z_{3}-a_{2})^{2}(z_{4}-a_{2})^{2}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\left.\color[rgb]{1,1,1}\frac{1}{(a_{1})^{2}}\color[rgb]{0,0,0}+\text{symm}(z_{1},\dots,z_{4})\right)\,.$
(4.29)

######  Example 4.2.2.

In the same way as in the previous example, here, the expansion of
$\omega_{1,1}$ in terms of intersection numbers on the moduli space of complex
curves (see equation (4.22)) is rewritten using $\\{x_{a,n}\\}$ and
$\\{y_{a,n}\\}$. Using the data presented in Appendix A one finds

$\displaystyle 2^{-1}\omega_{1,1}=$
$\displaystyle\;\sum_{\sigma\in\\{1,2\\}}\frac{1}{x_{a_{\sigma},0}y_{a_{\sigma},0}}\left[\frac{(-1+10({\langle\kappa_{1}\rangle_{1,1}}-{\langle\psi\rangle_{1,1}}))x_{a,1}^{2}+(1-6({\langle\kappa_{1}\rangle_{1,1}}-{\langle\psi\rangle_{1,1}}))x_{a,2}}{48(z_{1}-a_{\sigma})^{2}}\right.$
$\displaystyle\;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad-\frac{24{\langle\kappa_{1}\rangle_{1,1}}x_{a,1}y_{a,1}-24{\langle\kappa_{1}\rangle_{1,1}}y_{a,2}}{48(z_{1}-a_{\sigma})^{2}}$
$\displaystyle\qquad\qquad\qquad\qquad\left.+\frac{{\langle\psi\rangle_{1,1}}x_{a,1}}{(z_{1}-a_{\sigma})^{3}}-\frac{3{\langle\psi\rangle_{1,1}}}{(z_{1}-a_{\sigma})^{4}}\right]\,.$
(4.30)

##  5 The LSZ-model

An approach to the study of quantum field theories on non-commutative space is
provided by the expansion of the quantum field in terms of a matrix base
obeying the Moyal algebra. Then properties can be studied in the finite
dimensional approximation and limiting procedures to recover the quantum field
theory limit have to be carried out carefully. The partition function of the
LSZ model in the finite dimensional approximation of complex matrices of size
$N$ is given by

$\displaystyle{\mathcal{Z}}=\int_{M_{N}}\differential{\Phi}\differential{\Phi^{\dagger}}\;\exp(-N\tr[E\,\Phi\Phi^{\dagger}+\tilde{E}\,\Phi^{\dagger}\Phi+\frac{\lambda}{2}\Phi^{\dagger}\Phi\Phi^{\dagger}\Phi])\,.$
(5.1)

In the above, $E$ and $\tilde{E}$ are distinct hermitian matrices of size $N$
with positive distinct eigenvalues $\\{E_{1},\dots,E_{N}\\}$ and
$\\{\tilde{E}_{1},\dots,\tilde{E}_{N}\\}$. Note that these can be assumed to
be diagonal due to the invariance of the partition function under unitary
transformations. The coupling strength is given by $\lambda\in{\mathbb{C}}$.
The above partition function can also be considered a complex analogue of the
quartic Kontsevich model. In contrast to the present LSZ model, the more
constrained setting of the hermitian model results in a more complicated
solution structure.
In the original work Langmann_2004  the authors explicitly computed some
correlation functions in the case of $\tilde{E}=0$ and proved integrability in
the sense of Toda. Recently, in BH2205  this result has been generalized to
all external fields and topological recursion provides the machinery to
compute arbitrary high correlation functions. Furthermore, it was specified
that the partition function (5.1) is in fact a $\tau$-function of the
Kadomtsev-Petviashvili hierarchy (for $N<\infty$), a subclass of the Toda
hierarchy. It was shown that the model is entirely described by the spectral
curve ${\mathcal{S}}_{\text{LSZ}}=({\mathbb{P}}^{1},x_{\tilde{d}},y_{d},B)$
with

$\displaystyle
x_{\tilde{d}}(z)=z-\frac{\lambda}{N}\sum_{k=1}^{\tilde{d}}\frac{\tilde{r}_{k}}{y^{\prime}(\tilde{{\varepsilon}}_{k})(z-\tilde{{\varepsilon}}_{k})}\,,$
$\displaystyle
y_{d}(z)=-z+\frac{\lambda}{N}\sum_{k=1}^{d}\frac{r_{k}}{x^{\prime}({\varepsilon}_{k})(z-{\varepsilon}_{k})}\,,$
(5.2) $\displaystyle
B(z_{1},z_{2})=\frac{\differential{z_{1}}\differential{z_{2}}}{(z_{1}-z_{2})^{2}}\,.$
(5.3)

In the above the points ${\varepsilon}_{k}$ and $\tilde{{\varepsilon}}_{k}$
are defined by $e_{k}=x({\varepsilon}_{k})$ and
$\tilde{e}_{k}=x(\tilde{{\varepsilon}}_{k})$, respectively, where
$\\{e_{k}\\}_{k=1,\dots,d}$ and $\\{\tilde{e}_{k}\\}_{k=1,\dots,\tilde{d}}$
are pairwise distinct eigenvalues of $E$ and $\tilde{E}$ with multiplicities
$r_{k}$ and $\tilde{r}_{k}$ summing up to $N$.
In this work we restrict the analysis to the case of $d=\tilde{d}=1$. Then,
the above functions $x$ and $y$ can be reduced to

$\displaystyle
x(z)=z+\frac{\gamma^{2}}{y^{\prime}(\tilde{{\varepsilon}})(z-\tilde{{\varepsilon}})}\,,$
$\displaystyle
y(z)=-z+\frac{\gamma^{2}}{x^{\prime}({\varepsilon})(-z+{\varepsilon})}\,,$
(5.4)

with ${\varepsilon}={\varepsilon}_{1}$ and
$\tilde{{\varepsilon}}=\tilde{{\varepsilon}}_{1}$ as well as
$\gamma^{2}=-\lambda r_{1}/N=-\lambda\tilde{r}_{1}/N=-\lambda$. Note that also
the implicit definitions of ${\varepsilon}$ and $\tilde{{\varepsilon}}$ in
terms of $e$ and $e$ as well as the derivative of $y$ in the definition of $x$
and vice versa can be made explicit.
This setting is often referred to as the combinatorial limit, as it has been
subject of studies already in the 1990s in the field of combinatorics and
enumeration. Notable works include MORRIS1991703 ; ZJZ03 . In particular, this
model (with arbitrary polynomial interaction) is known to count bipartite
maps555For a nice introduction to maps and graphs the reader might be referred
to Eynard:2016yaa .. These are ordinary maps of even face degree with black or
white colored vertices and no monochromatic edge. An edge as well as its
marking plus the orientation of an edge creates a rooted bipartite graph and a
boundary of even length $2l_{k}$ following the face to the right of the rooted
edge. As long as they do not correspond to the same boundary, also multiple
edges can be marked. Denoting by ${\mathcal{T}}^{(g)}_{2l_{1},\dots,2l_{n}}$
the generating function of bipartite maps with natural embedding into Riemann
surfaces of genus $g$ and $n$ boundaries of length $2l_{1},\dots,2l_{n}$, one
can relate the counting results to the correlators of topological recursion
through

$\displaystyle{\mathcal{T}}^{(g)}_{2l_{1},\dots,2l_{n}}=(-1)^{n}{\underset{z_{1},\dots,z_{n}\to\infty}{\text{{Res
}}}}x(z_{1})^{l_{1}}\cdots
x(z_{n})^{l_{n}}\omega_{g,n}({\mathcal{S}}_{\text{LSZ}};z_{1},\dots,z_{n})$
$\displaystyle\Leftrightarrow\qquad\omega_{g,n}({\mathcal{S}}_{\text{LSZ}};z_{1},\dots,z_{n})=\sum_{l_{1},\dots,l_{n}=1}^{\infty}\frac{{\mathcal{T}}^{(g)}_{2l_{1},\dots,2l_{n}}}{x(z_{1})^{l_{1}+1}\cdots
x(z_{n})^{l_{n}+1}}\differential{x(z_{1})}\cdots\differential{x(z_{n})}\,.$
(5.5)

Remark that comparing this model (with quartic interaction) to its hermitian
counterpart, that is the quartic Kontsevich model, the latter counts ordinary
rooted quadrangulations while the quartic LSZ model counts bipartite ones.
This, again, stresses the relation of both models, which is to be exploited in
the further understanding of the quartic Kontsevich model in higher genus.

##  6 Result and Outlook

The aim of this work is to verify the agreement of correlation functions
calculated directly from topological recursion (Section 2) and expressed in
terms of intersection numbers on the moduli space of curves (Section 4.1),
which is provided by Eynard1110  as a theorem, in the concrete case of the LSZ
model in the combinatorial limit.
Therefore, the correlation functions $\omega_{0,4}$ and $\omega_{1,1}$ are
considered here. These are recursively calculated by computer algebra.
Furthermore, the expansion data listed in Appendix A is used to specify
Section 4.1 to the LSZ model in the combinatorial limit with quartic
interaction, as explained in Section 4.2. Subtracting one expression from the
other and evaluating the (higher) residues at the poles, which are only at the
ramification points of the spectral curve, verifies that both expressions give
the same, i.e. for all $a_{i}\in\\{\text{ram.pts.}\\}$,
$i\in\\{1,2,\dots,n\\}$, and all $\alpha\in{\mathbb{N}}^{n}$

$\displaystyle[(\vec{z}-\vec{a})^{-\alpha}]\left(\omega_{g,n}^{\text{TR}}(\vec{z})-\omega_{g,n}^{\text{IN-
TR}}(\vec{z})\right)=0\,,$ (6.1)

for $(g,n)\in\\{(0,4),(1,1)\\}$. Note that, beyond $g=1$, the expressions
$\omega^{\text{IN-TR}}$ get lengthy and clutter the analysis – beyond $n=4$ in
genus zero, the pole structure gets more and more involved.
By the study of these first examples, the general technique of comparing the
two expressions in equation (6.1) is shown, and different conventions in the
literature are clarified. This will enable further studies of the intersection
theory encoded in physical models such as in the quartic Kontsevich model.
This model is conjectured to be governed by a consistent extension of
topological recursion, called blobbed topological recursion Borot:2015hna .
The correlation functions generated by this extension have contributions that
are holomorphic at the ramification points of the spectral curve.
Nevertheless, there is a theory (provided in Section 3 of Borot:2015hna ) of
how to expand these in terms of intersection numbers on the moduli space of
curves. Furthermore, there is evidence that the genus zero sector of the
quartic Kontsevich model can be extracted from the LSZ-model. To utilize the
explicit expressions found in here for the analysis of (the planar sector of
the) quartic Kontsevich model will be subject of future work.

## Acknowledgments

This work is funded by the Deutsche Forschungsgemeinschaft through the
Research Training Group 2149 ”GRK 2149: Strong and Weak Interactions – from
Hadrons to Dark Matter” as well as supported666funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy EXC 2044 –390685587, Mathematics Münster:
Dynamics–Geometry–Structure by the Cluster of Excellence Mathematics Münster.

##  Appendix A Expansion data of the combinatorial limit of the LSZ model with
quartic interaction

In the following, the expansion data of the spectral curve associated to the
complex quartic Kontsevich model are provided. Remark that for some models it
is possible to give the expansion coefficients $\hat{t}_{a,k}$ and
$\hat{B}_{a,k;a^{\prime},k^{\prime}}$ for all $k$ and $k^{\prime}$ in closed
form777Instructive examples can for example be found in Eynard1104 .. This is
not possible here, as the integrals in equations (4.4) and (4.1.1) for this
model do not evaluate into some known functions. Therefore, the coefficients
$\hat{t}_{a,k}$ and $\hat{B}_{a,k;a^{\prime},k^{\prime}}$ are expressed in
terms of $\\{x_{a,n}\\}$ and $\\{y_{a,n}\\}$ defined in Section 4.2. A list of
the first coefficients is provided in Tables A.1–A.4.
Recall, the spectral curve contains the information about the two functions

$\displaystyle x(z)=z+\frac{\gamma_{y}^{2}}{z-\tilde{{\varepsilon}}}$
$\displaystyle y(z)=-z+\frac{\gamma_{x}^{2}}{-z+{\varepsilon}}\,,$ (A.1)

with $\gamma_{y}^{2}=-\lambda/y^{\prime}(\tilde{{\varepsilon}})$ and
$\gamma_{x}^{2}=-\lambda/x^{\prime}({\varepsilon})$. The ramification points
of this spectral curve defined by $\differential{x(a)}=0$ are at

$\displaystyle a_{\pm}=\tilde{{\varepsilon}}\pm\gamma_{y}\,.$ (A.2)

Using this information the moduli of the spectral curve $x_{a,n}$ and
$y_{a,n}$ can be specified. One finds

$\displaystyle x_{a_{\pm},n>0}=\frac{(n+2)!}{2(\mp\gamma_{y})^{n}}\,,$
$\displaystyle
y_{a_{\pm},n>0}=\frac{(n+1)!}{\left({\varepsilon}-\tilde{{\varepsilon}}\mp\gamma_{y}\right)^{n}-\gamma_{x}^{-2}\left({\varepsilon}-\tilde{{\varepsilon}}\mp\gamma_{y}\right)^{n+2}}\,,$
(A.3)

as well as

$\displaystyle x_{a_{\pm},0}=\pm 2\gamma_{y}^{-1}\,,$ $\displaystyle
y_{a_{\pm},0}=-1+\gamma_{x}^{2}({\varepsilon}-\tilde{{\varepsilon}}\mp\gamma_{y})^{-2}\,.$
(A.4)

Table A.1: This lists the first expansion coefficients or times $t_{a,k}$ of $y(z)$ at the ramification point $z=a$ expressed in terms of the parameters $x_{a,n}$ and $y_{a,n}$. $k$ | $t_{a,k}$
---|---
3 | $\begin{subarray}{c}\frac{y_{a,0}}{x_{a,0}^{1/2}}\sqrt{2}\end{subarray}$
4 | $\begin{subarray}{c}\frac{y_{a,0}}{3x_{a,0}}\left(-x_{a,1}+3y_{a,1}\right)\end{subarray}$
5 | $\begin{subarray}{c}\frac{y_{a,0}}{18\sqrt{2}}\left(-12x_{a,1}y_{a,1}+5x_{a,1}^{2}-3x_{a,2}+12y_{a,2}\right)\end{subarray}$
6 | $\begin{subarray}{c}\frac{y_{a,0}}{270x_{a,0}^{3/2}}\left(90x_{a,1}^{2}y_{a,1}-90x_{a,1}y_{a,2}-45x_{a,2}y_{a,1}-40x_{a,1}^{3}+45x_{a,2}x_{a,1}-9x_{a,3}+45y_{a,3}\right)\end{subarray}$
7 | $\begin{subarray}{c}\frac{y_{a,0}}{2160\sqrt{2}x_{a,0}^{2}}\left(-840x_{a,1}^{3}y_{a,1}+840x_{a,1}^{2}y_{a,2}+840x_{a,2}x_{a,1}y_{a,1}-480x_{a,1}y_{a,3}-144x_{a,3}y_{a,1}-360x_{a,2}y_{a,2}+385x_{a,1}^{4}\right.\\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.-630x_{a,2}x_{a,1}^{2}+168x_{a,3}x_{a,1}+105x_{a,2}^{2}-24x_{a,4}+144y_{a,4}\right)\end{subarray}$
8 | $\begin{subarray}{c}\frac{y_{a,0}}{17010x_{a,0}^{5/2}}\left(4200x_{a,1}^{4}y_{a,1}-4200x_{a,1}^{3}y_{a,2}-6300x_{a,2}x_{a,1}^{2}y_{a,1}+2520x_{a,1}^{2}y_{a,3}+1512x_{a,3}x_{a,1}y_{a,1}+3780x_{a,2}x_{a,1}y_{a,2}\right.\\\ \left.-945x_{a,1}y_{a,4}+945x_{a,2}^{2}y_{a,1}-189x_{a,4}y_{a,1}-567x_{a,3}y_{a,2}-945x_{a,2}y_{a,3}-1960x_{a,1}^{5}+4200x_{a,2}x_{a,1}^{3}\right.\\\ \qquad\qquad\qquad\qquad\left.-1260x_{a,3}x_{a,1}^{2}-1575x_{a,2}^{2}x_{a,1}+252x_{a,4}x_{a,1}+378x_{a,2}x_{a,3}-27x_{a,5}+189y_{a,5}\right)\end{subarray}$
9 | $\begin{subarray}{c}\frac{y_{a,0}}{2721600\sqrt{2}x_{a,0}^{3}}\left(-900900x_{a,1}^{5}y_{a,1}+900900x_{a,1}^{4}y_{a,2}+1801800x_{a,2}x_{a,1}^{3}y_{a,1}-554400x_{a,1}^{3}y_{a,3}-498960x_{a,3}x_{a,1}^{2}y_{a,1}\right.\\\ \left.\qquad\qquad\quad-1247400x_{a,2}x_{a,1}^{2}y_{a,2}+226800x_{a,1}^{2}y_{a,4}-623700x_{a,2}^{2}x_{a,1}y_{a,1}+90720x_{a,4}x_{a,1}y_{a,1}+272160x_{a,3}x_{a,1}y_{a,2}\right.\\\ \left.\qquad+453600x_{a,2}x_{a,1}y_{a,3}-60480x_{a,1}y_{a,5}+136080x_{a,2}x_{a,3}y_{a,1}-8640x_{a,5}y_{a,1}+170100x_{a,2}^{2}y_{a,2}\right.\\\ \left.\qquad\qquad-30240x_{a,4}y_{a,2}-60480x_{a,3}y_{a,3}-75600x_{a,2}y_{a,4}+425425x_{a,1}^{6}-1126125x_{a,2}x_{a,1}^{4}+360360x_{a,3}x_{a,1}^{3}\right.\\\ \left.+675675x_{a,2}^{2}x_{a,1}^{2}-83160x_{a,4}x_{a,1}^{2}-249480x_{a,2}x_{a,3}x_{a,1}+12960x_{a,5}x_{a,1}-51975x_{a,2}^{3}\right.\\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\left.+13608x_{a,3}^{2}+22680x_{a,2}x_{a,4}-1080x_{a,6}+8640y_{a,6}\right)\end{subarray}$
Table A.2: This lists the first expansion coefficients or dual times $\hat{t}_{a,k}$ of the Laplace transform of $\differential{y(z)}$ at the ramification point $z=a$ in terms of the parameters $x_{a,n}$ and $y_{a,n}$. These coefficients are dual to the times $t_{a,k}$ (see Table (A.1)) and their relation is provided in the second column. $k$ | $\hat{t}_{a,k}$
---|---
$0$ | $\log\frac{1}{2t_{a,3}}$ | $\begin{subarray}{c}\log\left(\frac{\sqrt{x_{a,0}}}{2\sqrt{2}y_{a,0}}\right)\end{subarray}$
$1$ | $-\frac{3t_{a,5}}{2t_{a,3}}$ | $\begin{subarray}{c}\frac{1}{24x_{a,0}}\left(12x_{a,1}y_{a,1}-5x_{a,1}^{2}+3x_{a,2}-12y_{a,2}\right)\end{subarray}$
$2$ | $\begin{subarray}{c}\frac{9t_{a,5}^{2}}{8t_{a,3}^{2}}\\\ -\frac{15t_{a,7}}{4t_{a,3}}\end{subarray}$ | $\begin{subarray}{c}\frac{1}{48x_{a,0}^{2}}\left(30x_{a,1}^{3}y_{a,1}+6x_{a,1}^{2}y_{a,1}^{2}-30x_{a,1}^{2}y_{a,2}-32x_{a,2}x_{a,1}y_{a,1}-12x_{a,1}y_{a,1}y_{a,2}+20x_{a,1}y_{a,3}\right.\\\ \qquad\qquad\left.+6x_{a,3}y_{a,1}+12x_{a,2}y_{a,2}-15x_{a,1}^{4}+25x_{a,2}x_{a,1}^{2}-7x_{a,3}x_{a,1}-4x_{a,2}^{2}+x_{a,4}+6y_{a,2}^{2}-6y_{a,4}\right)\end{subarray}$
$3$ | $\begin{subarray}{c}-\frac{9t_{a,5}^{3}}{8t_{a,3}^{3}}\\\ +\frac{45t_{a,7}t_{a,5}}{8t_{a,3}^{2}}\\\ -\frac{105t_{a,9}}{8t_{a,3}}\end{subarray}$ | $\begin{subarray}{c}\frac{1}{5760x_{a,0}^{3}}\left(10800x_{a,1}^{5}y_{a,1}+1800x_{a,1}^{4}y_{a,1}^{2}-10800x_{a,1}^{4}y_{a,2}+240x_{a,1}^{3}y_{a,1}^{3}-22200x_{a,2}x_{a,1}^{3}y_{a,1}\right.\\\ \qquad\qquad\left.-3600x_{a,1}^{3}y_{a,1}y_{a,2}+7200x_{a,1}^{3}y_{a,3}-1920x_{a,2}x_{a,1}^{2}y_{a,1}^{2}+1800x_{a,1}^{2}y_{a,2}^{2}\right.\\\ \qquad\qquad\left.+6360x_{a,3}x_{a,1}^{2}y_{a,1}-720x_{a,1}^{2}y_{a,1}^{2}y_{a,2}+15000x_{a,2}x_{a,1}^{2}y_{a,2}+1200x_{a,1}^{2}y_{a,1}y_{a,3}\right.\\\ \qquad\qquad\left.-3000x_{a,1}^{2}y_{a,4}+360x_{a,3}x_{a,1}y_{a,1}^{2}+720x_{a,1}y_{a,1}y_{a,2}^{2}+7920x_{a,2}^{2}x_{a,1}y_{a,1}\right.\\\ \qquad\qquad\left.-1200x_{a,4}x_{a,1}y_{a,1}-3360x_{a,3}x_{a,1}y_{a,2}+2640x_{a,2}x_{a,1}y_{a,1}y_{a,2}-6000x_{a,2}x_{a,1}y_{a,3}\right.\\\ \qquad\qquad\left.-1200x_{a,1}y_{a,2}y_{a,3}-360x_{a,1}y_{a,1}y_{a,4}+840x_{a,1}y_{a,5}-720x_{a,2}y_{a,2}^{2}-1800x_{a,2}x_{a,3}y_{a,1}\right.\\\ \qquad\qquad\left.+120x_{a,5}y_{a,1}-1920x_{a,2}^{2}y_{a,2}+360x_{a,4}y_{a,2}-360x_{a,3}y_{a,1}y_{a,2}+840x_{a,3}y_{a,3}\right.\\\ \qquad\qquad\left.+960x_{a,2}y_{a,4}-5525x_{a,1}^{6}+14775x_{a,2}x_{a,1}^{4}-4830x_{a,3}x_{a,1}^{3}-8900x_{a,2}^{2}x_{a,1}^{2}+1130x_{a,4}x_{a,1}^{2}\right.\\\ \qquad\qquad\left.+3360x_{a,2}x_{a,3}x_{a,1}-180x_{a,5}x_{a,1}+660x_{a,2}^{3}-189x_{a,3}^{2}-300x_{a,2}x_{a,4}+15x_{a,6}-240y_{a,2}^{3}\right.\\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\left.+360y_{a,2}y_{a,4}-120y_{a,6}\right)\end{subarray}$
Table A.3: This lists the expansion coefficients $B_{a,k;a,k^{\prime}}$ of $B(z,z^{\prime})$ at the ramification point $(z,z^{\prime})=(a,a)$ in the local variables $\zeta_{a}(z)=\sqrt{x(z)-x(a)}$. Note that the dual coefficients $\hat{B}_{a,2k;a,2k^{\prime}}$ can be found be equation (4.8) and differ only by a numerical factor. k k’ |  0 |  1 | 2
---|---|---|---
0 |  $\frac{1}{12x_{a,0}}\left(x_{a,1}^{2}-x_{a,2}\right)$ |  $\begin{subarray}{c}\frac{1}{135x_{a,0}^{3/2}}\left(-10\sqrt{2}x_{a,1}^{3}\right.\\\ \qquad\left.+15\sqrt{2}x_{a,2}x_{a,1}-\frac{9x_{a,3}}{\sqrt{2}}\right)\end{subarray}$ | $\begin{subarray}{c}\frac{1}{1440x_{a,0}^{2}}\left(175x_{a,1}^{4}\right.\\\ \qquad\left.-350x_{a,2}x_{a,1}^{2}+120x_{a,3}x_{a,1}\right.\\\ \qquad\qquad\left.+75x_{a,2}^{2}-24x_{a,4}\right)\\\ \;\end{subarray}$
1 |  $\begin{subarray}{c}\frac{1}{135x_{a,0}^{3/2}}\left(-10\sqrt{2}x_{a,1}^{3}\right.\qquad\\\ \left.+15\sqrt{2}x_{a,2}x_{a,1}-\frac{9x_{a,3}}{\sqrt{2}}\right)\end{subarray}$ |  $\begin{subarray}{c}\frac{1}{270x_{a,0}^{2}}\left(40x_{a,1}^{4}\right.\\\ \qquad\left.-80x_{a,2}x_{a,1}^{2}+30x_{a,3}x_{a,1}\right.\\\ \qquad\qquad\left.+15x_{a,2}^{2}-6x_{a,4}\right)\end{subarray}$ | $\begin{subarray}{c}\frac{1}{3780x_{a,0}^{5/2}}\left(-490\sqrt{2}x_{a,1}^{5}\right.\qquad\qquad\\\ \qquad\left.+1225\sqrt{2}x_{a,2}x_{a,1}^{3}-\frac{945x_{a,3}x_{a,1}^{2}}{\sqrt{2}}\right.\\\ \qquad\left.-525\sqrt{2}x_{a,2}^{2}x_{a,1}+126\sqrt{2}x_{a,4}x_{a,1}\right.\\\ \qquad\qquad\left.+\frac{315x_{a,2}x_{a,3}}{\sqrt{2}}-18\sqrt{2}x_{a,5}\right)\\\ \;\end{subarray}$
2 |  $\begin{subarray}{c}\frac{1}{1440x_{a,0}^{2}}\left(175x_{a,1}^{4}\right.\qquad\qquad\\\ \left.-350x_{a,2}x_{a,1}^{2}+120x_{a,3}x_{a,1}\right.\\\ \left.+75x_{a,2}^{2}-24x_{a,4}\right)\end{subarray}$ |  $\begin{subarray}{c}\frac{1}{3780x_{a,0}^{5/2}}\left(-490\sqrt{2}x_{a,1}^{5}\right.\qquad\qquad\\\ \qquad\left.+1225\sqrt{2}x_{a,2}x_{a,1}^{3}-\frac{945x_{a,3}x_{a,1}^{2}}{\sqrt{2}}\right.\\\ \qquad\left.-525\sqrt{2}x_{a,2}^{2}x_{a,1}+126\sqrt{2}x_{a,4}x_{a,1}\right.\\\ \qquad\qquad\left.+\frac{315x_{a,2}x_{a,3}}{\sqrt{2}}-18\sqrt{2}x_{a,5}\right)\end{subarray}$ | $\begin{subarray}{c}\frac{1}{181440x_{a,0}^{5/2}}\left(42875x_{a,1}^{6}\right.\qquad\\\ \qquad\left.-128625x_{a,2}x_{a,1}^{4}+50400x_{a,3}x_{a,1}^{3}\right.\\\ \qquad\left.+86625x_{a,2}^{2}x_{a,1}^{2}-15120x_{a,4}x_{a,1}^{2}\right.\\\ \qquad\left.-37800x_{a,2}x_{a,3}x_{a,1}+3024x_{a,5}x_{a,1}\right.\\\ \qquad\left.-7875x_{a,2}^{3}+2268x_{a,3}^{2}+4536x_{a,2}x_{a,4}\right.\\\ \qquad\qquad\qquad\qquad\left.-324x_{a,6}\right)\end{subarray}$
Table A.4: This lists the expansion coefficients $B_{a_{1},k;a_{2},k^{\prime}}$ of $B(z,z^{\prime})$ at the ramification point $(z,z^{\prime})=(a_{1},a_{2})$ in the local variables $\zeta_{a_{i}}(z)=\sqrt{x(z)-x(a_{i})}$ omitting a factor $\Delta^{-(k+k^{\prime}+2)}$. Here $\Delta=a_{1}-a_{2}$. Note that the dual coefficients $\hat{B}_{a_{1},k;a_{2},k^{\prime}}$ can be found from equation (4.8) and differ only by a numerical factor. k k’ |  0 |  1 | 2
---|---|---|---
0 |  $\frac{2}{\sqrt{x_{a_{1},0}x_{a_{2},0}}}$ |  $\begin{subarray}{c}-\frac{2\sqrt{2}}{3\sqrt{x_{a_{1},0}}x_{a_{2},0}}\left(\Delta x_{a_{2},1}\right.\\\ \qquad\qquad\qquad\left.-6\right)\end{subarray}$ | $\begin{subarray}{c}\frac{1}{6\sqrt{x_{a_{1},0}}x_{a_{2},0}^{3/2}}\left(5\Delta^{2}x_{a_{2},1}^{2}\right.\qquad\;\\\ \left.-3\Delta^{2}x_{a_{2},2}-24\Delta x_{a_{2},1}\right.\\\ \qquad\qquad\left.+72\right)\end{subarray}$
1 |  $\begin{subarray}{c}-\frac{2\sqrt{2}}{3x_{a_{1},0}\sqrt{x_{a_{2},0}}}\left(\Delta x_{a_{1},1}\right.\\\ \qquad\qquad\qquad\left.+6\right)\end{subarray}$ |  $\begin{subarray}{c}\frac{1}{9x_{a_{1},0}x_{a_{2},0}}\left(4\Delta^{2}x_{a_{1},1}x_{a_{2},1}\right.\qquad\\\ \left.-24\Delta x_{a_{1},1}+24\Delta x_{a_{2},1}\right.\\\ \qquad\qquad\qquad\left.-216\right)\end{subarray}$ | $\begin{subarray}{c}\frac{1}{9\sqrt{2}x_{a_{1},0}x_{a_{2},0}^{3/2}}\left(-5\Delta^{3}x_{a_{1},1}x_{a_{2},1}^{2}\right.\qquad\\\ \quad\left.+3\Delta^{3}x_{a_{1},1}x_{a_{2},2}-30\Delta^{2}x_{a_{2},1}^{2}\right.\\\ \quad\left.+24\Delta^{2}x_{a_{1},1}x_{a_{2},1}+18\Delta^{2}x_{a_{2},2}\right.\\\ \quad\left.-72\Delta x_{a_{1},1}+216\Delta x_{a_{2},1}\right.\\\ \qquad\qquad\qquad\left.-864\right)\end{subarray}$
2 |  $\begin{subarray}{c}\frac{1}{6x_{a_{1},0}^{3/2}\sqrt{x_{a_{2},0}}}\left(5\Delta^{2}x_{a_{1},1}^{2}\right.\qquad\;\\\ \left.-3\Delta^{2}x_{a_{1},2}+24\Delta x_{a_{1},1}\right.\\\ \qquad\qquad\left.+72\right)\end{subarray}$ |  $\begin{subarray}{c}\frac{1}{9\sqrt{2}x_{a_{1},0}^{3/2}x_{a_{2},0}}\left(-5\Delta^{3}x_{a_{1},1}^{2}x_{a_{2},1}\right.\\\ \quad\left.+3\Delta^{3}x_{a_{1},2}x_{a_{2},1}+30\Delta^{2}x_{a_{1},1}^{2}\right.\\\ \quad\left.-18\Delta^{2}x_{a_{1},2}-24\Delta^{2}x_{a_{1},1}x_{a_{2},1}\right.\\\ \quad\left.+216\Delta x_{a_{1},1}-72\Delta x_{a_{2},1}\right.\\\ \qquad\qquad\qquad\left.+864\right)\end{subarray}$ | $\begin{subarray}{c}\frac{1}{72x_{a_{1},0}^{3/2}x_{a_{2},0}^{3/2}}\left(25\Delta^{4}x_{a_{1},1}^{2}x_{a_{2},1}^{2}\right.\qquad\quad\\\ \quad\left.-15\Delta^{4}x_{a_{1},2}x_{a_{2},1}^{2}-15\Delta^{4}x_{a_{1},1}^{2}x_{a_{2},2}\right.\\\ \quad\left.+9\Delta^{4}x_{a_{1},2}x_{a_{2},2}+120\Delta^{3}x_{a_{1},1}x_{a_{2},1}^{2}\right.\\\ \quad\left.-120\Delta^{3}x_{a_{1},1}^{2}x_{a_{2},1}+72\Delta^{3}x_{a_{1},2}x_{a_{2},1}\right.\\\ \quad\left.-72\Delta^{3}x_{a_{1},1}x_{a_{2},2}+360\Delta^{2}x_{a_{1},1}^{2}\right.\\\ \quad\left.+360\Delta^{2}x_{a_{2},1}^{2}-216\Delta^{2}x_{a_{1},2}\right.\\\ \quad\left.-864\Delta^{2}x_{a_{1},1}x_{a_{2},1}-216\Delta^{2}x_{a_{2},2}\right.\\\ \quad\left.+3456\Delta x_{a_{1},1}-3456\Delta x_{a_{2},1}\right.\\\ \qquad\qquad\qquad\left.+17280\right)\end{subarray}$

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