††institutetext: Department of Physics, Brown University,
182 Hope Street, Providence, RI 02912, U.S.A.

# Cosmology meets cohomology

Shounak De<EMAIL_ADDRESS>and Andrzej Pokraka
<EMAIL_ADDRESS>
###### Abstract

The cosmological polytope and bootstrap programs have revealed interesting
connections between positive geometries, modern on-shell methods and bootstrap
principles studied in the amplitudes community with the wavefunction of the
Universe in toy models of FRW cosmologies. To compute these FRW correlators,
one often faces integrals that are too difficult to evaluate by direct
integration. Borrowing from the Feynman integral community, the method of
(canonical) differential equations provides an efficient alternative for
evaluating these integrals. Moreover, we further develop our geometric
understanding of these integrals by describing the associated _relative_
twisted cohomology. Leveraging recent progress in our understanding of
relative twisted cohomology in the Feynman integral community, we give an
algorithm to predict the basis size and simplify the computation of the
differential equations satisfied by FRW correlators.

##  1 Introduction

Over the past decade, an increasing number of concepts from the theory of
hypergoemetric functions and more broadly algebraic geometry have entered
modern methods for studying scattering amplitudes. In particular, the
framework of twisted cohomology (originally formulated to study the linear
relations among hypergoemetric functions) has been used to construct CHY
amplitudes Cachazo:2013gna ; Mizera:2017rqa , provide a geometric framework
for understanding KLT relations in string amplitudes Kawai:1985xq ;
Mizera:2017cqs ; Casali:2019ihm ; Stieberger:2022lss , the double copy
Mizera:2019blq  and more.

In the Feynman integral community, twisted intersection theory
Mastrolia:2018uzb ; Frellesvig:2019kgj ; Mizera:2019gea ; Mizera:2019vvs ;
Frellesvig:2020qot ; Weinzierl:2020xyy ; Chestnov:2022xsy ; Chen2022  provides
an algebraic alternative to integral reduction111Integral reduction is the
process of decomposing an arbitrary integral into a linear combination of
basis (also called master) integrals that only need to be computed once. via
integration-by-parts (IBP) identities Chetyrkin:1981qh ; Gehrmann:1999as ;
Laporta:2000dsw . While these techniques are still being optimized for
applications to physics, they have the potential to circumvent solving large
linear systems of IBPs thereby streamlining one of the most expensive steps in
the pipeline for multi-loop phenomenological calculations.

Recently, the procedure for calculating intersection numbers of Feynman
integrals has been streamlined by using _relative_ twisted cohomology Caron-
Huot:2021xqj ; Caron-Huot:2021iev . By working with relative twisted
cohomology, one can set the regulator in previous works to zero at the start
of the calculation. As a consequence, many intermediate quantities vanish or
simplify, streamlining the overall algorithm.

The last decade has also seen the discovery of rich geometrical structures in
the study of flat-space scattering amplitudes which provide answers to elegant
questions posed purely in the kinematic space. While such insights have
transformed our understanding of the flat-space S-matrix, the situation for
cosmological correlators is still underdeveloped. The lack of such methods for
cosmology and the complexity of even tree-level results222One already
encounters branch cuts at the level of trees in cosmological correlators
whereas their flat-space counterparts are rational functions in the kinematic
invariants. means that the repository of theoretical data is significantly
smaller. The problem is further complicated by the fact that inflationary
correlators live on the late-time boundary of de Sitter, which is space-like.
The understanding of the holographic dual to theories with space-like
boundaries is still in its infancy, while the boundary description for anti-de
Sitter space is well established due to the time-like nature of its boundary.
Having said that, the cosmological polytope and bootstrap programs have taken
important steps towards a boundary description of inflationary correlators.

While de Sitter models the accelerated expansion of the early Universe, the
Friedmann-Robertson–Walker (FRW)333While the most commonly used term is the
FRW metric, it is also sometimes called the FLRW metric after Friedmann,
Lemaître, Robertson and Walker. metric describes a homogeneous, isotropic,
expanding (or contracting) Universe. The FRW spacetime is often considered to
be the standard model of cosmology post inflation. Field theoretic correlation
functions in FRW cosmologies represent quantum fluctuations, which in the
early Universe are thought to have sourced the observed distribution of
matter. Comparing these predictions to current and future cosmological
datasets will help constrain new physics beyond the Standard Model.

Unlike the de Sitter geometry (which has provided a rich playground for
studying inflationary correlators in the literature so far), FRW Universes are
not maximally symmetric spacetimes and have a reduced isometry group. This
reduces the applicability of the cosmological bootstrap program to derive
differential equations for correlators beyond de Sitter space. However, it
turns out that correlators in FRW cosmologies (the main focus of this
work444We will consider FRW cosmologies with a future spacelike boundary where
our boundary observables live.) furnish some of the simplest examples of
relative twisted cohomology.555Mathematically, FRW correlators are integrals
associated to hyperplane arrangements where the coordinate hyperplanes are
twisted and all other divisors are left untwisted. Thus, they provide fertile
testing ground for developing our understanding of relative twisted
cohomologies, which in turn will deepen our understanding of the underlying
physics.

While the standard tool for computing cosmological correlation functions is
perturbation theory, the past decade has seen novel computational methods with
interesting and fruitful connections to scattering amplitudes (e.g., symbology
Hillman:2019wgh ) and mathematics (the combinatorial description in terms of
the cosmological polytope ArkaniHamed2017 ; Benincasa:2019vqr ; Arkani-
Hamed:2018bjr ). Many of these methods manifest the simplicity of the final
answer that is hidden at intermediate steps in perturbation theory. In
particular, the singularity structure of these correlators is particularly
evident in the polytopal picture and encodes principles of unitary (by cutting
rules) as well as causality (Steinmann relations).

Our main goal is two fold. First, we aim to provide efficient methods for
computing the differential equations of FRW wavefunction coefficients.
Deriving differential equations for cosmological correlators was first
considered in the upcoming work nimafriends  (also see the talks ptalk ;
blectures ; ltalk ). We also hope that by formally constructing the cohomology
associated to the integrals appearing in the FRW wavefunction coefficients and
elucidating their mathematical structure will lead to a deeper physical
understanding of these objects. Secondly, our techniques are quite general and
are applicable to many integrals seen in both physics and mathematics. Thus,
we expect this work to be useful beyond the current setting of cosmological
correlators.

While primarily aimed at physicists, we think that much of this work will
interest mathematicians. While many of our claims have been well tested, this
paper lacks formal proofs and it would be interesting to understand these
claims more formally. As far as the authors know, Matsumoto:aa  is the only
paper discussing relative twisted cohomology in the mathematics literature.

The paper is organized as follows. In section 2, we set up the toy model of
conformally coupled scalars in a general FRW background studied in this work
and explain how to uplift flat-space wavefunction coefficients to their FRW
counterparts. In section 3, we study the integral representation of the two-
site/four-point FRW integral in detail. To evaluate this integral, we derive
the differential equation satisfied by a basis of integrals for the
corresponding integral family. We obtain these differential equations using
three different perspectives: twisted cohomology and standard IBPs in section
3.1, the dual (relative) twisted cohomology and dual IBPs in section 3.2, and
intersection numbers in section 3.3. The dual perspective (section 3.2) yields
computational advantages as well as novel geometric understanding and is
central to this work. In section 3.4, we explicitly integrate these
differential equations to obtain the first correction to the de Sitter two-
site/four-point function. In section 4, we provide two more worked examples:
the one-loop two-site/two-point and the tree-level three-site/five-point FRW
integrals. In this section, we illustrate how to use the techniques developed
in section 3 on more complicated examples. We conclude in section 5 with a
discussion of our results and directions for future work.

##  2 Cosmological correlators and the Wavefunction of the Universe

We consider the model of a conformally-coupled scalar field in a general FRW
cosmology with non-conformal polynomial interactions, studied extensively in
ArkaniHamed2017 ; Arkani-Hamed:2018bjr ; Hillman:2019wgh ; Benincasa:2018ssx .
The action for such a theory in a $(d+1)$-dimensional spacetime is

$\displaystyle\mathcal{S}=\int\text{d}^{d}x\,\text{d}\eta\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\xi
R\phi^{2}-\sum_{k\geq 3}\frac{\lambda_{k}}{k!}\phi^{k}\right]~{},$ (1)

with the FRW metric and the constant $\xi$ defined as

$\displaystyle\text{d}s^{2}\equiv
g_{\mu\nu}\text{d}x^{\mu}\text{d}x^{\nu}=a^{2}(\eta)\left[-\text{d}\eta^{2}+\text{d}x_{i}\text{d}x^{i}\right],\quad\xi=\frac{d-1}{4d}~{}.$
(2)

The FRW metric above has been written in comoving coordinates with conformal
time $\eta\in(-\infty,0]$ and the index $i=1,\dots,d$ runs over the spatial
dimensions. $a(\eta)$ denotes the scale factor and the choice of its
representation in various stages of the Universe’s evolution will be crucial
for the class of integrals dealt with in our paper. Importantly, the choice of
$\xi$ sets the scalar to be conformally-coupled thereby making the action
$\mathcal{S}$ conformally equivalent to the following flat-space action

$\displaystyle
S[\phi]=\int\text{d}^{d}x\,\text{d}\eta\left[\frac{1}{2}(\partial\phi)^{2}-\sum_{k\geq
3}\frac{\lambda_{k}(\eta)}{k!}\phi^{k}\right]~{}.$ (3)

More precisely, the conformal transformation

$\displaystyle g_{\mu\nu}\rightarrow a^{2}(\eta)g_{\mu\nu}\
,\quad\phi\rightarrow a^{-\Delta}(\eta)\phi\ ,\quad\Delta=\frac{d-1}{2}~{},$
(4)

allows one to rewrite the original action $\mathcal{S}$ (1) as the action of a
massless scalar field in $(d+1)$-dimensional flat-space (3) with time-
dependent couplings given by

$\displaystyle\lambda_{k}(\eta)\equiv\lambda_{k}[a(\eta)]^{(2-k)\Delta+2}~{}.$
(5)

It is worth stressing that the conformal equivalence between the original
action $\mathcal{S}$ (1) and the flat-space action $S[\phi]$ (3) allows us to
do meaningful computations in a general FRW cosmology (described by the scale
factor $a(\eta)$) simply by doing flat-space perturbation theory (albeit with
time-dependent couplings given by (5)), as we shall see below.

Of central importance, is the wavefunction of the universe $\Psi[\Phi]$.
Formally, this is computed as a path integral by integrating over all bulk
field configurations $\phi(\mathbf{x},\eta)$ with non-vanishing Dirichlet
boundary condition in the future $\phi(\mathbf{x},\eta=0)=\Phi(\mathbf{x})$.
It is standard to work in momentum space where the wavefunction can be
expanded as

$\displaystyle\Psi[\Phi]=\int_{\phi(-\infty(1-i\epsilon))}^{\phi(0)=\Phi}\mathcal{D}\phi\,e^{iS[\phi]}\equiv\exp{\left[i\sum_{n}\frac{1}{n!}\int\prod_{i}\text{d}^{d}\mathbf{k}_{i}\,\Phi(\mathbf{k}_{i})\,\tilde{\psi}_{n}(\\{\mathbf{k}_{i}\\})\right]}~{},$
(6)

and the standard $i\epsilon$ prescription selects the adiabatic/Bunch-
Davies/Hartle-Hawking vacuum at the early-time boundary $\eta\to-\infty$. The
kernels $\tilde{\psi}_{n}$ are called wavefunction coefficients and
$\\{\mathbf{k}_{i}\\}$ denotes the set of all spatial momentum. Due to spatial
translation invariance on the future boundary, the wavefunction coefficients
contain an overall momentum-conserving $\delta$-function. It is useful to
extract this $\delta$-function and compute the “stripped” wavefunction
coefficients $\psi_{n}$:

$\displaystyle\tilde{\psi}_{n}(\\{\mathbf{k}_{i}\\})=\delta^{(d)}\left(\sum_{i=1}^{n}\mathbf{k}^{(i)}\right){\psi}_{n}(\\{\mathbf{k}_{i}\\})~{}.$
(7)

Our main objective is the computation of these stripped wavefunction
coefficients.

Traditionally, the wavefunction coefficients $\psi_{n}$ are obtained
perturbatively, using Feynman diagrammatics, which we review here. We
approximate the path integral (6) by its saddle point
$\Psi[\Phi]\approx\exp{(iS[\Phi_{\textrm{cl}}])}$ where $\Phi_{\textrm{cl}}$
is the boundary profile corresponding to the classical solution to the
equations of motion $\phi_{\textrm{cl}}$. To derive the Feynman rules, we
write the classical solution as

$\displaystyle\phi_{\textrm{cl}}(\mathbf{k},\eta)=\mathcal{K}(E,\eta)\Phi(\mathbf{k})+\int\text{d}\eta^{\prime}\mathcal{G}(E;\eta,\eta^{\prime})\frac{\delta
S_{\textrm{int}}}{\delta\phi(\mathbf{k},\eta^{\prime})}~{}.$ (8)

Substituting the above solution into the action, one can verify that the path
integral evaluates to the expression in (6) by taking functional derivatives
of the path integral with respect to the boundary value $\Phi(\mathbf{k})$
(which acts like a source term). To ensure that the solution (8) satisfies the
correct Dirichlet boundary conditions, one needs to appropriately solve for
the two distinct Green’s functions, the bulk-to-boundary propagator
$\mathcal{K}(E_{v},\eta_{v})$ associated with a vertex $v$ and the bulk-to-
bulk propagator $\mathcal{G}(E_{e},\eta_{v},\eta^{\prime}_{v})$ associated
with an edge $e$ connecting a vertex at time $\eta_{v}$ to another one at time
$\eta^{\prime}_{v}$. Explicitly, they are given by the expressions

$\displaystyle\mathcal{K}(E_{v},\eta_{v})$
$\displaystyle=e^{iE_{v}\eta_{v}}~{},$ (9)
$\displaystyle\mathcal{G}(E_{e},\eta_{v},\eta_{v}^{\prime})$
$\displaystyle=\frac{1}{2E_{e}}\left(e^{{-}iE_{e}(\eta_{v}{-}\eta_{v}^{\prime})}\theta\left(\eta_{v}{-}\eta_{v}^{\prime}\right)+e^{{-}iE_{e}(\eta_{v}^{\prime}{-}\eta_{v})}\theta\left(\eta_{v}^{\prime}{-}\eta_{v}\right)-e^{iE_{e}(\eta_{v}{+}\eta_{v}^{\prime})}\right)~{},$
(10)

where $E_{v}=\sum_{j}|\mathbf{k}_{j}|$ denotes the sum of the energies of $j$
external lines connected to the vertex $v$, and $E_{e}=|\mathbf{k}_{e}|$ is
the energy of an internal line. The bulk-to-boundary propagator,
$\mathcal{K}(E,\eta)$, solves the linearized, homogeneous EOM, oscillates with
positive frequency in the far past, and approaches unity at the late time
boundary as $\eta\to 0$. The bulk-to-bulk propagator,
$\mathcal{G}(E,\eta,\eta^{\prime})$, solves the inhomogeneous wave equation
$\sqrt{-g}(\Box-m^{2})\mathcal{G}(E,\eta,\eta^{\prime})=i\delta(\eta-\eta^{\prime})$,
subject to the boundary condition that it vanishes identically when either of
its time arguments approach zero. The combined boundary conditions on the
Green’s functions guarantee that the path integral in (6) as well as the
classical solution (8) satisfy the correct Dirichlet boundary conditions. The
problem of computing the wavefunction coefficients $\psi_{n}$ is now thus
reduced to an exercise in perturbative diagrammatic expansion using Feynman
rules for the fluctuations of the theory at hand. We note that to compute
these wavefunction contributions, it suffices to implement flat-space
perturbation theory which thereby surpasses the need to introduce complicated
Green’s functions for a general FRW spacetime. As we shall see in the
subsequent sections, for various FRW cosmologies this is facilitated by an
operation involving integration over the external energies that accounts for
the time-dependent couplings (5) of the theory.

###  2.1 FRW cosmologies

In this work, we study the analytic structure of integrals associated to the
wavefunction coefficients in a general FRW Universe, going beyond the
inflationary expansion modelled by a de Sitter Universe. Before presenting the
integrals tackled in this paper, we give a brief description of the FRW
backgrounds describing the Universe’s evolution at its various stages:

#### de Sitter Universe

de Sitter (dS) space models the accelerated expansion of the inflationary
Universe to a good approximation and thus provides a starting point for a
boundary description of inflationary correlators. The isometry group of dS
space has been utilized to exploit the conformal symmetry of cosmological
correlators in the recent cosmological bootstrap program Arkani-Hamed:2018kmz
; Baumann:2022jpr ; Pajer:2020wnj . We recall that the de Sitter line element
in flat slicing is given by

$\displaystyle\text{d}s^{2}=\frac{-\text{d}\eta^{2}+\text{d}x_{i}\text{d}x^{i}}{\eta^{2}}~{},$
(11)

with $\eta\in(-\infty,0]$ and the Hubble scale set to unity. Comparing with
the FRW metric (2), one sees that the scale factor behaves as $a(\eta)=1/\eta$
for a de Sitter Universe. Much of the theoretical data concerning cosmological
correlators come from our understanding of polynomial interactions in dS
space. A particularly well-studied model is the $\lambda_{3}\phi^{3}$ theory
in dS4, which provides a rich framework for understanding the analytic
structure of inflationary correlators ArkaniHamed2015 .

#### Radiation/Matter dominated Universe

For most of its history the Universe was dominated by a single component, as
suggested by the different scalings of the energy density $\rho$ with the
scale factor $a(\eta)$. A phase of radiation domination (RD) where
$\rho\propto a^{-4}$ was followed by an era of matter domination (MD) where
$\rho\propto a^{-3}$. The dependencies of the scale factor on the conformal
time in these phases are obtained by integrating the first Friedmann equation
for a flat, single-component Universe Baumann:2022mni . This leads to the
following FRW solutions for the scale factor

$\displaystyle a(\eta)\propto\begin{cases}\eta\quad(\textrm{RD})\\\
\eta^{2}\quad(\textrm{MD})~{},\end{cases}$ (12)

where the proportionality factors depend on dimensionless density parameters
and will be set to unity for our discussions.

Given the above dependencies of the scale factor $a(\eta)$, the different FRW
solutions (2) corresponding to a de Sitter (11), flat, radiation or a matter-
dominated Universe (12) can be encapsulated by the following metric
parameterized by $\varepsilon$

$\displaystyle\text{d}s^{2}=\eta^{-2\varepsilon}\left[\frac{-\text{d}\eta^{2}+\text{d}x_{i}\text{d}x^{i}}{\eta^{2}}\right]\begin{cases}\varepsilon=0\quad(\textrm{dS})\\\
\varepsilon=-1\quad(\textrm{flat})\\\ \varepsilon=-2\quad(\textrm{RD})\\\
\varepsilon=-3\quad(\textrm{MD})~{},\end{cases}$ (13)

such that the scale factor is

$\displaystyle a(\eta)=\frac{1}{\eta^{1+\varepsilon}}\,.$ (14)

While the differential equations (DEQ) that we derive in our paper hold for
any value of $\varepsilon$, the solution to them is always presented as a
Laurent series expansion in $\varepsilon$, as we shall soon see. When solving
the DEQ, our analysis will thereby be restricted to cosmologies closely
resembling a dS Universe. The consequence of $\varepsilon\sim 0$ is two fold:
i) in the analysis of inflationary correlators so far, the literature assumes
a perfect dS expansion ($\varepsilon=0$). However, inflationary cosmology
dictates that for inflation to end one has to consider small departures from
dS, often leading to inflation being referrred as a quasi-dS period. ii) Even
if one models inflation as a perfect dS expansion, computing wavefunction
coefficients as a series in $\varepsilon$ could potentially shed crucial
insights into physics beyond inflation, i.e., after the reheating surface.

Mathematically, the parameter $\varepsilon$ corresponds to the exponent of the
“twist” associated to the wavefunction coefficients of an FRW Universe. This
identification, allows us to import the technology of relative twisted
cohomology and intersection theory to cosmological correlators. Such ideas
were first introduced in physics in the context of Feynman integrals. There,
the parameter $\varepsilon$ is the dimensional regularization parameter. In
both cases, this exponent is generally non-integer giving rise to branch cut
singularities on the integration manifold that partially dictates the
singularity structure of the final object.

For the sake of simplicity, we shall focus on a theory of conformally coupled
scalars described by a cubic $\lambda_{3}\phi^{3}$ interaction in a
(1+3)-dimensional FRW Universe described by the action

$\displaystyle\mathcal{S}=\int\text{d}^{3}x\,\text{d}\eta\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{6}R\phi^{2}-\frac{\lambda_{3}}{3!}\phi^{3}\right]~{}.$
(15)

However, it is straightforward to generalize our analysis to theories with
polynomial interactions $\lambda_{k}\phi^{k}$ in an FRW Universe (13) obeying
the condition

$\displaystyle[a(\eta)]^{(2-k)\frac{(d-1)}{2}+2}=\frac{1}{\eta^{1+\varepsilon}}~{},$
(16)

such that the time-dependent couplings (5) are given by
$\lambda_{k}(\eta)=\frac{\lambda_{k}}{\eta^{1+\varepsilon}}$. These time-
dependent couplings are most conveniently treated in (temporal) Fourier space

$\displaystyle\frac{1}{\eta^{1+\varepsilon}}=\frac{(-i)^{1+\varepsilon}}{\Gamma(1+\varepsilon)}\int_{0}^{\infty(1-i\epsilon)}\text{d}x\,x^{\varepsilon}e^{ix\eta}~{}.$
(17)

Before proceeding to furnish the integral representations for FRW correlators,
a few comments are in order regarding the above Fourier-space expression.
First, it is worth noting the difference between the FRW parameter
$\varepsilon$ and the standard $i\epsilon$ regulator ensuring the convergence
of the integral. Secondly, it is crucial to note that the integral converges
only for $\varepsilon>-1$ due to the presence of the gamma functions. This is
seemingly in contradiction with our boundary choices of $\varepsilon$ denoting
the various FRW solutions in (13). However, as we will see in section 3, the
differential equations satisfied by the FRW cosmological correlators can be
solved for any value of the parameter $\varepsilon$. In what follows, we
suppress the overall constants arising from the gamma functions and the
factors of $i$ appearing in (17) to avoid clutter. Given a choice of
$\varepsilon$ for the FRW Universe one lives in, such overall factors and
coupling constants can be reinstated as they are needed.

###  2.2 Cosmological correlators in FRW backgrounds

Equipped with the above conceptual preliminaries, we now turn our attention to
the construction of the wavefunction coefficients in flat-space and their
uplifts to a general FRW cosmology.

We first define the flat-space $n$-site wavefunction coefficient at $L$-loops
$\psi_{n,\textrm{flat}}^{(L)}$ which essentially makes up the energy integrand
of the FRW wavefunction coefficient

$\displaystyle\psi_{n,\textrm{flat}}^{(L)}(X_{v},Y_{e})=\int_{-\infty}^{0}\prod_{v\in\mathcal{V}}\text{d}\eta_{v}\,e^{iX_{v}\eta_{v}}\prod_{e\in\mathcal{E}}\mathcal{G}(Y_{e},\eta_{v},\eta^{\prime}_{v})~{},$
(18)

where $\mathcal{V},\mathcal{E}$ denotes the set of vertices and edges of a
given $n$-site, $L$-loop Feynman graph. We also introduce the notation $X_{v}$
associated with the sum of energies flowing to the boundary from a vertex $v$
and $Y_{e}$ denote the energies of the edges (internal lines) $e$. It is worth
stressing that we follow standard practice (in this field) and do not perform
the integrals over the spatial loop momenta for the case of loop-level
contributions. Such considerations are left for future work. The computation
of the nested time integrals given by the formal expression (18) has been
carried out in the literature using several approaches: i) explicitly
computing the $3^{e}$ bulk time integrals arising from the three-term
structure of the propagator (10), ii) a combinatorial representation to
implement boundary old-fashioned perturbation theory which one would obtain
via the Lippmann-Schwinger kernel ArkaniHamed2017 , iii) triangulations of the
cosmological polytope (and its dual) ArkaniHamed2017 ; Benincasa:2019vqr ;
Arkani-Hamed:2018bjr ; Juhnke-Kubitzke:2023nrj . Using any of the approaches,
one can compute the two- and three-site flat-space wavefunction coefficients
at tree-level to be

$\displaystyle\psi_{2,\textrm{flat}}^{(0)}$
$\displaystyle=\scalebox{1.15}{$\frac{1}{(X_{1}{+}X_{2})(X_{1}{+}Y)(X_{2}{+}Y)}$}~{},$
(19) $\displaystyle\psi_{3,\textrm{flat}}^{(0)}$
$\displaystyle=\scalebox{1.15}{$\frac{X_{1}{+}Y_{1}{+}2X_{2}{+}Y_{2}{+}X_{3}}{(X_{1}{+}X_{2}{+}X_{3})(X_{1}{+}Y_{1})(X_{3}{+}Y_{2})(X_{3}{+}Y_{2})(X_{2}{+}Y_{1}{+}Y_{2})(X_{1}{+}X_{2}{+}Y_{2})(X_{2}{+}X_{3}{+}Y_{1})}$
},$ (20)

while the two-site contribution at one loop is given by

$\displaystyle\psi_{2,\text{flat}}^{(1)}$
$\displaystyle=\scalebox{1.15}{$\frac{2\left(X_{1}+X_{2}+Y_{1}+Y_{2}\right)}{\left(X_{1}+X_{2}\right)\left(X_{1}+X_{2}+2Y_{1}\right)\left(X_{1}+Y_{1}+Y_{2}\right)\left(X_{2}+Y_{1}+Y_{2}\right)\left(X_{1}+X_{2}+2Y_{2}\right)}$
}.$ (21)

To compute wavefunction coefficients relevant to FRW cosmologies, we perform
an uplift by integrating the integrand (18) over the energies related to the
time-dependent coupling constants (17). To account for this extra time-
dependence, we denote the shifted vertex energies as $E_{v}=x_{v}+X_{v}$ while
also tacking on the twist factor $x_{v}^{\varepsilon}$ at each vertex. As a
result, the full FRW wavefunction contributions take the form

$\displaystyle\psi_{n,\textrm{FRW}}^{(L)}(X_{v},Y_{e})=\int_{0}^{\infty}\prod_{v\in\mathcal{V}}\text{d}x_{v}\,x_{v}^{\varepsilon}\,\psi_{n,\textrm{flat}}^{(L)}(x_{v}+X_{v},Y_{e})=\int_{\mathcal{R}}\,\prod_{v\in\mathcal{V}}\text{d}x_{v}\,x_{v}^{\varepsilon}\,\psi_{n,\textrm{flat}}^{(L)}(E_{v},Y_{e})~{},$
(22)

where $\mathcal{R}$ defines the rectangular region $E_{v}>X_{v}$ for each
vertex $v$. In the mathematics literature, the integrals (22) belong to the
class of functions which are often called generalized Euler integrals or
GKZ666Gelfand, Kapranov and Zelevinsky. hypergeometric functions Matsubara-
Heo:2023ylc . As we will see in section 3.4, the FRW wavefunction coefficients
belong to the subset of this function space whose series expansion in
$\varepsilon$ involves only multiple polylogarithms.

As an example of this prescription, consider the uplift of
$\psi^{(0)}_{2,\text{flat}}\to\psi^{(0)}_{2,\text{FRW}}$. Using the two-site
flat-space wavefunction coefficient (19) and the uplift relation (22), one
obtains the explicit realization of the corresponding FRW wavefunction
coefficient given by the integral expression

$\displaystyle\psi_{2,\text{FRW}}^{(0)}(X_{1},X_{2};Y)=\int_{0}^{\infty}\text{d}x_{1}\wedge\text{d}x_{2}\
\frac{(x_{1}x_{2})^{\varepsilon}}{(x_{1}{+}x_{2}{+}X_{1}{+}X_{2})(x_{1}{+}X_{1}{+}Y)(x_{2}{+}X_{2}{+}Y)}~{}.$
(23)

The $\varepsilon\to 0$ limit of such integrals reduces to the well-studied
cases arising in inflationary correlators in dS4 Arkani-Hamed:2018kmz ;
ArkaniHamed2015 . It is worth mentioning that while this prescription for the
uplift of $\psi_{n,\textrm{flat}}\to\psi_{n,\textrm{FRW}}$ is technically true
only for $\varepsilon\neq-1$, the flat-space wavefunction coefficient appears
as the coefficient of the leading log divergence in the $\varepsilon\to-1$
limit. Explicitly setting $\varepsilon\to-1+\mu$ in (23) yields

$\displaystyle\psi^{(0)}_{2,\text{FRW}}(X_{1},X_{2},;Y)\big{|}_{\varepsilon\to-1+\mu}$
$\displaystyle=\int_{0}^{\infty}x_{1}^{-1+\mu}\
\text{d}x_{1}\int_{0}^{\infty}\text{d}x_{2}\ x_{2}^{-1+\mu}\
\psi^{(0)}_{2,\text{flat}}(x_{1}+X_{1},x_{2}+X_{2};Y)$
$\displaystyle=\int_{0}^{\infty}\text{d}x_{1}\ x_{1}^{-1+\mu}\
\frac{\psi^{(0)}_{2,\textrm{flat}}(x_{1}+X_{1},X_{2};Y)}{\mu}+\mathcal{O}(\mu^{0})$
$\displaystyle=\frac{\psi^{(0)}_{2,\textrm{flat}}(X_{1},X_{2};Y)}{\mu^{2}}+\mathcal{O}(\mu^{-1})~{}.$
(24)

Thus, for the flat-space case, the integral (23) returns the original
wavefunction coefficient multiplied by a divergent factor $1/\mu^{2}$. This
factor corresponds to the infinite volume of the energy integrals that we
should not have integrated over.

The main objective of the paper is to illustrate how one can study the
analytic structure of FRW wavefunction coefficients using techniques from
relative twisted cohomology, intersection theory and canonical differential
equations that has already been developed in context to flat-space scattering
amplitudes. In particular, in the subsequent sections, we will derive the
differential equations for the tree-level two-site/four-point and three-
site/five-point and the one-loop two-site/two-point FRW wavefunction
coefficients in a theory comprised of conformally coupled scalars obeying a
$\lambda_{3}\phi^{3}$ interaction (15). In the process, we argue that relative
twisted cohomology provides a particularly efficient and powerful formalism
for studying FRW wavefunction coefficients.

##  3 The tree-level two-site FRW correlator

Figure 1: The Feynman diagram for the tree-level two-site/four-point
wavefunction coefficient in $\lambda_{3}\phi^{3}$ theory.

For the $\lambda_{3}\phi^{3}$ theory of conformally coupled scalars (15), the
two-site flat-space wavefunction coefficient at tree-level is also the four-
point function from the boundary perspective as depicted in figure 1. With the
uplift prescription laid out in section 2.2, the corresponding FRW
wavefunction coefficient has the integral representation

$\displaystyle\psi_{2,\text{FRW}}^{(0)}(X_{1},X_{2};Y)=\int_{0}^{\infty}\text{d}x_{1}\wedge\text{d}x_{2}\
\frac{(x_{1}x_{2})^{\varepsilon}}{(x_{1}{+}x_{2}{+}X_{1}{+}X_{2})(x_{1}{+}X_{1}{+}Y)(x_{2}{+}X_{2}{+}Y)}~{}.$
(25)

While it is possible to integrate the above two-site/four-point integral,
direct integration becomes intractable at higher points. Similar integrals
often appear in the study of dimensionally regulated Feynman integrals where
there is a rich literature describing their evaluation and understanding their
analytic structure (see Eden:1966dnq ; Chetyrkin:1981qh ; Bern:1994cg ;
Gehrmann:1999as ; Laporta:2000dsw ; Britto:2004nc ; Smirnov:2006ry ;
Spradlin:2011wp ; Smirnov:2012gma ; Duhr:2014woa ; Gardi:2022wro ;
Mizera:2021icv ; Mizera:2021fap ; Borinsky:2023jdv ; Matsubara-Heo:2023ylc
for a very incomplete list). In particular, we will use the method of
canonical differential equations Henn:2013pwa ; Henn:2014qga  and some ideas
from intersection theory Mastrolia:2018uzb ; Frellesvig:2019kgj ;
Mizera:2019gea ; Mizera:2019vvs ; Frellesvig:2020qot ; Weinzierl:2020xyy ;
Chestnov:2022xsy ; Caron-Huot:2021xqj ; Caron-Huot:2021iev  to evaluate the
two-site FRW correlator (25).

In order to implement these methods, we have to consider the family of
integrals defined by the singularities of the integrand in (25). This family
is made up of integrals whose integrands are rational differential forms on
the manifold $X$ (to be defined later) multiplied by a universal multi-valued
function, $u$, called the twist

$\displaystyle\psi_{2,\text{FRW}}^{(0)}(X_{1},X_{2};Y;\boldsymbol{\mu},\boldsymbol{\nu})$
$\displaystyle=\int u\
\frac{N(\mathbf{x})}{\prod_{j=1}^{2}T_{j}^{\mu_{j}}\prod_{k=1}^{3}S_{k}^{\nu_{k}}(\mathbf{x})}\text{d}^{2}\mathbf{x}\quad(\nu_{i},\mu_{i}\in\\{0,1,2,\dots\\}),$
(26) $\displaystyle u$
$\displaystyle=\prod_{i=1}^{2}T_{i}^{\varepsilon}\quad\text{
($\varepsilon\notin\mathbb{C}\setminus\mathbb{Z}$ for FRW cosmologies) }.$
(27)

In particular, the integral (25) corresponds to $\mu=\\{0,0\\}$ and
$\nu=\\{1,1,1\\}$ above. While we can have any arbitrary polynomial numerator
$N(\mathbf{x})$, the analytic structure of the integral is (mostly) determined
by the singular surfaces $T_{a}=0$ and $S_{a}=0$. Moreover, the behavior of
the integrand at the singular surfaces $T_{a}=0$ and $S_{a}=0$ is very
different and needs distinct mathematical treatment. The hyperplanes777By a
hyperplane, we mean a codimension-1 surface cut out by a linear equation. We
will use this terminology even for lines in 2-dimensions and planes in
3-dimensions.

$\displaystyle T_{1}=x_{1}\quad\text{and}\quad T_{2}=x_{2}~{},$ (28)

correspond to branch surfaces or _twisted singularities_ of the integrand.
These singularities are said to regulated by $\varepsilon$ and are mild in the
sense that the integral in the neighbourhood of any $T_{a}=0$ has a well
defined analytic continuation. On the other hand, the (hyper-)planes

$\displaystyle S_{1}$ $\displaystyle=x_{1}+X_{1}+Y_{1}~{},$ (29)
$\displaystyle S_{2}$ $\displaystyle=x_{2}+X_{2}+Y_{1}~{},$ (30)
$\displaystyle S_{3}$ $\displaystyle=x_{1}+x_{2}+X_{1}+X_{2}~{}.$ (31)

correspond to poles or _relative singularities_. These singularities are
dangerous and cannot be avoided by analytic continuation.

Having understood the possible locations of singularities, we can define the
integration manifold $X=\mathbb{C}^{2}\setminus(T\cup S)$ where
$T=\cup_{i=1}^{2}\\{T_{i}=0\\}$ are the twisted singular surfaces and
$S=\cup_{i=1}^{3}\\{S_{i}=0\\}$ are the relative singular surfaces. The
analytic structure of this family of integrals is almost entirely determined
by how the surfaces $T_{a}$ and $S_{a}$ intersect amongst themselves.

One perhaps surprising fact is that such families of integrals are spanned by
a finite basis of integrals $\\{I_{a}\\}$ Lee:2013hzt ; Bitoun:2018afx . This
means that the partial derivatives of these integrals with respect to the
kinematic (non-integration) variables can be expressed as a linear combination
of the original basis of integrals

$\displaystyle\partial_{z}I_{a}=B^{(z)}_{ab}I_{b}~{},$ (32)

where $z\in\\{X_{1},X_{2},Y\\}$ for the example at hand. Such systems of
differential equations have a closed solution in terms of a path-ordered
exponential. Then, since we can can choose a basis such that the differential
equation is proportional to $\varepsilon$:
$\underline{\boldsymbol{B}}=\varepsilon\underline{\boldsymbol{A}}$, the path-
ordered exponential

$\displaystyle\mathbf{I}=\mathbf{I}_{0}+\varepsilon\int\underline{\boldsymbol{A}}\cdot\mathbf{I}_{0}+\varepsilon^{2}\int\underline{\boldsymbol{A}}\cdot\int\underline{\boldsymbol{A}}\cdot\mathbf{I}_{0}+\cdots$
(33)

can be truncated at any desired order in $\varepsilon$.888FRW integrals turn
out to be sufficiently simple that it is always possible to chose such a
basis. For each order in $\varepsilon$, sophisticated technology exists in the
amplitudes literature for handling the resulting iterated integrals
bams/1183539443 . Lastly, $\mathbf{I}_{0}$ corresponds to the boundary
conditions of our basis integrals.

Thus, we can integrate by differentiation! Since obtaining
$\underline{\boldsymbol{A}}$ and the boundary conditions $\mathbf{I}_{0}$ are
orders of magnitude easier than direct integration, we pursue this strategy in
the following sections.

Before moving on, we note that the iterated integrals appearing in (33) will
not produce anything more complicated than multiple polylogrithms because the
matrix $\underline{\boldsymbol{A}}$ contains at most simple poles (see
Spradlin:2011wp ; Duhr:2014woa ; Duhr:2019tlz ; Gardi:2022wro  for some of the
mathematical techniques used to simplify such integrals). If one was powerful
enough to sum the series (33), one would recover the corresponding generalized
hypergeometric function. Further, note that $\varepsilon$ carries weight $-1$
so that the expansion of these hypergeometric functions have uniform
transcendental weight.

To aid us in the computation of differential equations, we introduce a few
ideas from intersection theory. In section 3.1, we introduce the twisted
cohomology of FRW integrals, compute the differential equation and obtain the
basis size from geometric quantities. Along the way we will comment on some
computation bottlenecks of the usual twisted cohomology picture. To simplify
the computation of the differential equation as well as improve our geometric
understanding of such integrals, we introduce the vector space dual to FRW
integrals in section 3.2. In particular, the analogous algorithm for computing
the DEQ presented in section 3.1 simplifies and in some cases becomes trivial
in the dual space. We also describe how to compute differential equations of
logarithmic forms using the intersection number (an inner product on the FRW
cohomology) bypassing integration-by-parts in section 3.3. In particular, we
describe how to compute the differential equations without leaving the space
of logarithmic forms. Lastly, in section 3.4, we will integrate the
differential equation found in both sections 3.1 and 3.2 to obtain an explicit
formula for the two-site/four-point FRW correlator.

###  3.1 Differential equations from twisted cohomology

To compute the differential equations, it is useful to change perspective and
consider the _integrands_ as the fundamental objects of interest instead of
the corresponding integrals.999Since we will always be dealing with a basis of
integrals there is no loss of information by considering integrands instead of
integrals. The allowed FRW integrands are any holomorphic rational
differential top-forms $\varphi$ on $X$,101010Holomorphic differential forms
only contain the holomorphic variables of the complex manifold $X$. A top form
has form degree equal to the complex dimension of the manifold
$\text{dim}_{\mathbb{C}}X$. In this example, $\text{dim}_{\mathbb{C}}X=2$.
multiplied by the multi-valued twist

$\displaystyle\text{FRW integrands:}\ u\ \varphi.$ (34)

Mathematically, we denote the space of (holomorphic) differential $p$-forms by
$\Omega^{p}(X)$. Since the twist is universal to all FRW integrands, we can
factor it out and only work with the single-valued forms $\varphi$ by
introducing a covariant derivative: $\text{d}(u\ \varphi)=u\nabla\varphi$
where $\nabla=\text{d}+\omega\wedge$ and $\omega=\text{d}\log u$ is a flat
connection ($\text{d}\omega=0$). The connection $\omega$ is analogous to the
gauge field $A_{\mu}$ of QED and keeps track of the phase of $u$.

Notice that integrands are not uniquely defined. One can always shift an
integrand by a total derivative without changing the resulting integral

$\displaystyle\int u\ \varphi=\int\Big{(}u\ \varphi+\text{d}(u\
\psi)\Big{)}=\int u\Big{(}\varphi+\nabla\psi\Big{)}$ (35)

for any $\psi\in\Omega^{\text{dim}_{\mathbb{C}}X-1}(X)$.111111In the example
at hand, $\text{dim}_{\mathbb{C}}X-1=2-1=1$. Thus, the unique objects are
holomorphic top-forms modulo covariant derivatives. This is is made precise by
the _twisted cohomology group_

$\displaystyle[\varphi]\in
H^{p}(X,\nabla)=\frac{\\{\varphi\in\Omega^{n}(X)|\nabla\varphi=0\\}}{\\{\nabla\psi|\psi\in\Omega^{p-1}(X)\\}}.$
(36)

The numerator corresponds to “covariantly closed” differential forms where the
condition $\nabla\varphi=0$ ensures that the integral of $u\varphi$ does not
depend on the choice of path – only on the end points. The denominator
corresponds to the set of all possible “covariantly exact” forms (covariant
derivatives) that we can add to $\varphi$ without changing the resulting
integral. We will use the freedom to add any $\nabla\psi$ to $\varphi$ to
derive the FRW differential equations. In the Feynman integral literature,
this procedure often goes by the name _integration-by-parts_.

Since twisted cohomology is well studied, we can borrow a key fact from the
mathematical literature: only the top-dimensional twisted cohomology group is
non-trivial.121212The usual proof of these theorems assume some conditions
that we break by letting the $S_{i}$ have integer exponents. Fortunately, in
most cases, this statement remains true even when the $S_{i}$ have integer
exponents. A proof of this fact will appear in a forthcoming publication by
one of the authors. As a consequence, the dimension of the top-dimensional
cohomology (or equivalently the size of the FRW basis) is equal to the Euler
characteristic $\chi(X)$. Moreover, it is known that for hyperplane
arrangements the Euler characteristic is equal to the number of bounded
chambers yoshida1997hypergeometric ; aomoto2011theory .131313When all
singularities are twisted, a more practical way to compute this Euler
characteristic is by counting the critical points of the connection
Lee:2013hzt ; Mizera:2017rqa . While this formula does not hold for FRW
cohomology, it does hold for the dual cohomology. Thus, it is the recommend
way for computing the dual basis size when working in higher dimensions. For
our particular example, there are four bounded chambers (figure 2). This
should be familiar from the positive geometry and cosmological polytope
picture ArkaniHamed2017 ; Arkani-Hamed:2018bjr ; Benincasa:2019vqr ;
Kuhne:2022wze ; Juhnke-Kubitzke:2023nrj ; Albayrak:2023hie . In fact, the
canonical forms associated to the bounded chambers provide a natural
$\varepsilon$-form basis for this family of integrals141414The canonical form
of a chamber bounded by the lines $A,B,C$ is given by the $\text{d}\log$-form
$\text{d}\log\frac{A}{B}\wedge\text{d}\log\frac{B}{C}$ or any permutation of
$A,B$ and $C$. The canonical form for a chamber bounded by more than three
lines is obtained by triangulating the bounded chamber and adding up the
canonical forms associated to the triangulation. This construction generalizes
straightforwardly to higher dimensional spaces.

$\displaystyle\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle=\text{d}\log\frac{S_{1}}{S_{2}}\wedge\text{d}\log\frac{S_{2}}{S_{3}},$
(37)
$\displaystyle\color[rgb]{1,0.39,0.13}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.39,0.13}\pgfsys@color@cmyk@stroke{0}{0.61}{0.87}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle=\text{d}\log\frac{T_{1}}{S_{2}}\wedge\text{d}\log\frac{S_{2}}{S_{3}},$
(38)
$\displaystyle\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}\vartheta_{3}$
$\displaystyle=\text{d}\log\frac{T_{2}}{S_{1}}\wedge\text{d}\log\frac{S_{1}}{S_{3}},$
(39)
$\displaystyle\color[rgb]{1,0.52,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.52,1}\pgfsys@color@cmyk@stroke{0}{0.48}{0}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle=\bigg{(}\frac{x_{1}+X_{1}+X_{2}}{T_{2}S_{1}S_{3}}+\frac{x_{2}+X_{1}+X_{2}}{T_{1}S_{2}S_{3}}-\frac{1}{T_{1}T_{2}}\bigg{)}\text{d}x_{1}\wedge\text{d}x_{2}.$
(40)

The last form has been written out explicitly, since its construction requires
one to triangulate the pink bounded chamber in figure 2 and add up the
canonical forms on each piece of the triangulation. Notice that $u\
\vartheta_{1}$ is simply the integrand of (25) (up to a sign).

Figure 2: Hyperplane arrangement for the two-site/four-point integrand. The
twisted planes are pictured in blue while the untwisted planes are red. For
the FRW-cohomology, it is common to use the canonical forms of the bounded
chambers, which are colored in orange, purple, green and pink (section 3.1).
For the dual-cohomology, the basis is determined by uplifting the basis on
each 1- and 2-boundary since the bulk cohomology is empty. On each 2-boundary,
the cohomology is spanned by the constant function. Each 1-boundary only has a
non-trivial cohomology if it crosses both twisted lines (section 3.2).

Let us now compute the differential equation for $\vartheta_{1}$. Taking the
kinematic covariant derivative of $\vartheta_{1}$ yields

$\displaystyle\dot{\vartheta}_{1}\equiv\nabla_{\text{kin}}\vartheta_{1}$
$\displaystyle=\frac{2\left(2x_{1}+x_{2}+2X_{1}+X_{2}+Y\right)}{S_{1}^{2}S_{2}S_{3}^{2}}\
\text{d}x_{1}\wedge\text{d}x_{2}\wedge\text{d}X_{1}$
$\displaystyle\quad+\frac{2\left(x_{1}+2x_{2}+X_{1}+2X_{2}+Y\right)}{S_{1}S_{2}^{2}S_{3}^{2}}\
\text{d}x_{1}\wedge\text{d}x_{2}\wedge\text{d}X_{2}$
$\displaystyle\quad+\frac{2\left(x_{2}X_{1}+x_{1}X_{2}+x_{1}x_{2}+X_{1}X_{2}-Y_{1}^{2}\right)}{S_{1}^{2}S_{2}^{2}S_{3}}\
\text{d}x_{1}\wedge\text{d}x_{2}\wedge\text{d}Y_{1},$ (41)

where $\nabla_{\text{kin}}=\text{d}_{\text{kin}}+\text{d}_{\text{kin}}\log
u=\text{d}_{\text{kin}}$ and
$\text{d}_{\text{kin}}=\partial_{X_{1}}\text{d}X_{1}+\partial_{X_{2}}\text{d}X_{2}+\partial_{Y}\text{d}Y$
for the current example. Note that $\dot{\vartheta}_{1}$ is not directly
expressible in terms of our basis (37). However, we have the freedom to add
the covariant derivative of any 1-form to (3.1) without changing the value of
the integral. One can check that

$\displaystyle\dot{\vartheta}_{1}+\nabla\text{IBP}$
$\displaystyle=\varepsilon\
\text{d}\log\left(\left(X_{1}+Y\right)\left(X_{2}+Y\right)\right)\wedge\vartheta_{1}$
$\displaystyle+\varepsilon\
\text{d}\log\left(\frac{Y-X_{1}}{X_{1}+Y}\right)\wedge\vartheta_{2}+\varepsilon\
\text{d}\log\left(\frac{X_{2}+Y}{Y-X_{2}}\right)\wedge\vartheta_{3},$ (42)

where

IBP
$\displaystyle=\frac{2(x_{1}+X_{1})\text{d}x_{2}\wedge\text{d}Y-2(x_{2}+X_{2})\text{d}x_{1}\wedge\text{d}Y+2Y(\text{d}x_{1}\wedge\text{d}X_{2}-\text{d}x_{2}\wedge\text{d}X_{1})}{S_{1}S_{2}S_{3}}.$
(43)

The IBP form above is not so difficult to guess or construct for the current
example. However, constructing such IBP forms at the level of the three-
site/five-point graph or higher is much more difficult. This integration-by-
parts step can be simplified and in some cases trivialized by working in the
dual cohomology of section 3.2. While the change to the dual cohomology
requires a new layer of abstraction, the computational and conceptual
simplifications are well worth it!

Equation (3.1) corresponds to the first row of our differential equation.
Computing the other rows of $\underline{\boldsymbol{A}}$ in an analogous
fashion yields

$\displaystyle\underline{\boldsymbol{A}}_{\vartheta}=\varepsilon\
\text{d}{\tiny\begin{pmatrix}\log\left(\left(X_{1}+Y\right)\left(X_{2}+Y\right)\right)&\log\left(\frac{Y-X_{1}}{X_{1}+Y}\right)&\log\left(\frac{X_{2}+Y}{Y-X_{2}}\right)&0\\\
0&\log\left(\left(X_{1}-Y\right)\left(X_{1}+X_{2}\right)\right)&\log\left(\frac{X_{2}+Y}{X_{1}+X_{2}}\right)&\log\left(\frac{X_{2}+Y}{X_{1}+X_{2}}\right)\\\
0&\log\left(\frac{X_{1}+Y}{X_{1}+X_{2}}\right)&\log\left(\left(X_{2}-Y\right)\left(X_{1}+X_{2}\right)\right)&\log\left(\frac{X_{1}+X_{2}}{X_{1}+Y}\right)\\\
0&\log\left(\frac{Y-X_{1}}{X_{1}+Y}\right)&\log\left(\frac{X_{2}+Y}{Y-X_{2}}\right)&\log\left(\left(X_{1}+Y\right)\left(X_{2}+Y\right)\right)\end{pmatrix}},$
(44)

where we have included a subscript $\vartheta$ on $\underline{\boldsymbol{A}}$
to signify the choice of basis. Indeed, such differential equations come with
an inherent gauge redundancy that is the choice of basis. Under a change of
basis $\vartheta_{i}=U_{ij}\varphi_{j}$, the matrix
$\underline{\boldsymbol{A}}$ transforms like a gauge field

$\displaystyle\nabla_{\text{kin}}\boldsymbol{\varphi}=\underline{\boldsymbol{A}}_{\varphi}\cdot\boldsymbol{\varphi},$
(45)

where

$\displaystyle\underline{\boldsymbol{A}}_{\varphi}=\underline{\boldsymbol{U}}^{-1}\cdot\underline{\boldsymbol{A}}_{\vartheta}\cdot\underline{\boldsymbol{U}}-\underline{\boldsymbol{U}}\cdot\text{d}_{\text{kin}}\underline{\boldsymbol{U}}.$
(46)

The equation above gives the rule for how to compare the differential
equations for different choices of bases. Notice that (44) is actually quite
dense. This hints that there may be a better choice of basis that simplifies
the form of $\underline{\boldsymbol{A}}$. In fact, the dual cohomology will
point us to such a basis in section 3.2.

Before moving onto describe the dual cohomology, we note that the matrix
$\underline{\boldsymbol{A}}$ is integrable:
$\text{d}_{\text{kin}}A_{ij}+A_{ik}\wedge A_{kj}=0$. This condition places
strong consistency conditions on $\underline{\boldsymbol{A}}$ and provides an
important check of our calculations. If $\underline{\boldsymbol{A}}$ were not
integrable, the path ordered exponential would depend on the path of
integration in kinematic space.

###  3.2 Differential equations from the dual (relative twisted) cohomology

Since the twisted cohomology is a vector space, there exists a dual vector
space and an associated inner product called the _intersection number_. The
dual cohomology is the _relative twisted_ cohomology
$\check{H}^{p}(X,\nabla)=H^{p}(\check{X},\check{X}\cap S,\check{\nabla})$
where $\check{X}=\mathbb{C}^{2}\setminus T$,
$\check{\nabla}=\text{d}+\check{\omega}\wedge$ is the dual covariant
derivative, $\check{\omega}=\text{d}\log\check{u}$ is the dual connection and
$\check{u}=1/u$ is the dual twist Matsumoto:aa ; Caron-Huot:2021xqj ; Caron-
Huot:2021iev . In this section, we will unpack the ideas underlying relative
twisted cohomology and describe how this simplifies computing differential
equations.

We also opt to avoid delving into the details of how the intersection number
is computed. Instead, we give a simple formula for the intersection of
logarithmic forms in appendix A that covers all cases considered in this work.
However, to provide some motivation for why the dual cohomology takes its
particular form, it is useful to give the definition of the intersection
number in the main text.

The intersection number of a FRW-form $\varphi\in H^{n}(X,\nabla)$ with a dual
form $\check{\varphi}\in H^{n}(\check{X},\check{X}\cap S,\check{\nabla})$ is
given by

$\displaystyle\langle\check{\varphi}|\varphi\rangle\propto\frac{1}{(2\pi
i)^{\text{dim}_{\mathbb{C}}X}}\int_{X}\text{Reg}[\check{\varphi}]\wedge\varphi$
(47)

where the overall sign is conventional and $\text{Reg}[\check{\varphi}]$ is a
regulated version of the dual form.151515We skip the details of the
regularization procedure in this work since the simple formula in appendix A
covers all cases encountered in this work. For details on the regularization
procedure we direct the interested reader to matsumoto1998kforms ;
Mizera:2017rqa ; Mizera:2019gea ; Mizera:2019vvs ; Frellesvig:2019uqt ;
Weinzierl:2020xyy ; Chestnov:2022xsy ; Caron-Huot:2021iev ; Caron-Huot:2021xqj
and the references therein. While this looks like it could be a difficult
integral to perform, the intersection number localizes on the maximal
codimension intersections of the singular surfaces $T_{i}$ and $S_{i}$ (the
point $S_{12}$ in figure 1 for example). Thus, the intersection number is
actually algebraically evaluated as a series of residues. Moreover, it is
simple to compute for logarithmic forms, which encompass all FRW-forms.

Like any vector space, we can construct a resolution of the identity from a
basis for the FRW and dual cohomologies $\\{\varphi_{a}\\}$ and
$\\{\check{\varphi}_{a}\\}$

$\displaystyle\mathds{1}=\sum_{a,b}|\varphi_{a}\rangle\ C_{ab}^{-1}\
\langle\check{\varphi}_{b}|.$ (48)

Here, $C_{ab}=\langle\check{\varphi}_{a}|\varphi_{b}\rangle$ is the
intersection matrix associated to our choice of bases. Using this resolution
of identity, any FRW-form $\varphi$ (and hence integral $\int u\ \varphi$) can
be projected onto the basis $\\{\varphi_{a}\\}$

$\displaystyle|\varphi\rangle=\sum_{a}c_{a}\varphi_{a},\qquad
c_{a}=C_{ab}^{-1}\langle\check{\varphi}_{b}|\varphi\rangle.$ (49)

Roughly speaking, this projection is a form of generalized unitary Bern:1994cg
; Bern:2004ky ; Britto:2004nc ; Anastasiou:2006jv ; Britto:2006fc ;
Ossola:2006us ; Britto:2007tt ; Bourjaily:2017wjl ; Feng:2021spv . It can also
be used instead of the integration-by-parts identities of the previous section
to derive the differential equations (section 3.4).

The reason that the dual cohomology takes a different form from the FRW
cohomology is due to the nature of the regularization map in (47). Since the
twisted singularities $T_{a}$ can be regularized thanks to the fact that
$\varepsilon\notin\mathbb{Z}$ is generic, dual forms can have singularities at
the zero loci of the $T_{i}$. On the other hand, singularities at the zero
loci of the relative singularities $S_{i}$ _cannot_ be regularized. Thus, dual
forms must be regular on $\check{X}\cap S$. This is why we only excise $T$ in
the definition of the dual manifold $\check{X}=\mathbb{C}^{2}\setminus T$.

The last piece of mathematical terminology that we need to explain before
moving to the calculation of the differential equation is what the adjective
“relative” means. It is simply a fancy name for keeping track of boundary
terms that arise form Stokes’ theorem. For the FRW cohomology, we did not
allow boundaries and the total covariant derivative always integrated to zero.
For the dual cohomology, the relative singular surfaces become boundaries and
the integral of a total covariant derivative only vanishes up to boundary
terms. These boundary terms are integrals arising from lower-degree forms that
live on the boundary surfaces.

Mechanically, this boundary information is kept track by the so-called
coboundary symbol $\delta_{J}$. A generic dual form $\check{\varphi}\in
H^{n}(\check{X},\check{X}\cap S,\check{\nabla})$ should be thought of as a
vector with components labeled by each boundary $J$

$\displaystyle\check{\varphi}=\sum_{J}\delta_{J}\left(\check{\phi}_{J}\right).$
(50)

Here, $\delta_{J}$ can be thought of as picking out the component of
$\check{\varphi}$ that corresponds to the boundary $S_{J}=\check{X}\cap_{j\in
J}S_{j}$ and $\check{\phi}_{J}$ is an element of the twisted cohomology of the
boundary indexed by $J$: $H^{n-|J|}(\check{X}\cap
S_{J},\check{\nabla})$.161616For the mathematically inclined, the formal
definition of relative twisted cohomology is the direct sum of the twisted
cohomology on each boundary
$\displaystyle\Omega^{n}(\check{X},S;\check{\nabla})$
$\displaystyle=\Omega^{n}(\check{X};\check{\nabla})\ \bigoplus_{a=1}^{m}\
\Omega^{n-1}(\check{X}\cap S_{a};\check{\nabla}|_{a})\ \underset{a\neq
b}{\bigoplus_{a,b=1}^{m}}\ \Omega^{n-2}(\check{X}\cap S_{a}\cap
S_{b};\check{\nabla}|_{ab})$
$\displaystyle\quad\bigoplus\quad\cdots\underset{a_{i}\neq
a_{j}}{\bigoplus_{a_{1},\dots,a_{n}=1}^{m}}\Omega^{0}(\check{X}\cap
S_{a_{1}}\cap\cdots\cap S_{a_{n}};\check{\nabla}|_{a_{1}\cdots a_{n}})$ where
$m$ is the number of relative surfaces $S_{i}$.  Naturally, the corresponding
dual integrals

$\displaystyle\check{I}_{J}=\int_{\gamma}\check{u}\
\check{\varphi}=\sum_{J}\int_{\gamma}\check{u}\
\delta_{J}\Big{(}\check{\phi}_{J}\Big{)}=\sum_{J}\int_{\gamma|_{J}}\check{u}|_{J}\
\check{\phi}_{J},$ (51)

are simply integrals defined on the boundaries.

In addition to indexing the components of $\check{\varphi}$, the $\delta_{J}$
act like differential forms. They anti-commute with themselves

$\displaystyle\delta_{ij}$
$\displaystyle=\delta_{i}\wedge\delta_{j}=-\delta_{j}\wedge\delta_{i}=-\delta_{ji}$
(52)

and the exterior derivative is given by

$\displaystyle\text{d}\delta_{J}(\check{\phi})$
$\displaystyle=(-1)^{|J|}\delta_{J}\left(\text{d}\check{\phi}\right)+(-1)^{|J|}\sum_{k\notin
J}\delta_{Jk}\left(\check{\phi}|_{k}\right)$ (53)

up to boundary terms. The $\delta_{J}$ also convert the covariant derivative
on bulk space to the covariant derivative on the boundary

$\displaystyle\check{\nabla}\delta_{J}(\check{\phi})=(-1)^{|J|}\delta_{J}\left(\check{\nabla}|_{J}\
\check{\phi}\right)+(-1)^{|J|}\sum_{k\notin
J}\delta_{Jk}\left(\check{\phi}|_{k}\right)$ (54)

where $\check{\nabla}|_{J}=\text{d}+\check{\omega}|_{J}\wedge$. Equations (53)
and (54) will do virtually all the heavy lifting in this formalism. To the
zeroth order approximation, the $\delta_{i}$ can be thought of as being dual
to the $\text{d}\log$-form $\text{d}\log S_{i}$ since the $\delta_{i}$
correspond to residues about $S_{i}=0$ in the intersection number (see
equation (134)). The main job of the $\delta_{J}$ is to take care of the
combinatorics of Stokes’ theorem on a manifold with boundary.

Figure 3: The topological spaces appearing in the direct sum decomposition of
the relative cohomology. Since the twist is sufficiently generic, only the
corresponding middle dimensional cohomology is non-trivial. The first diagram
is a real slice of the bulk space where the coordinate planes (in blue) have
been removed. Since the planes defined by the $T_{i}$ do not form a bounded
chamber, $\dim H^{2}(\check{X},\check{\nabla})=0$. Each 1-boundary
$\check{X}\cap S_{i}$ is topologically a Riemann sphere with either 2- or
3-punctures. Only $X\cap S_{3}$ has enough punctures to have a non-trivial
topology. Thus, $\dim H^{1}(\check{X}\cap S_{3},\check{\nabla})=1$ while $\dim
H^{1}(\check{X}\cap S_{i=1,2},\check{\nabla})=0$. The last diagram represents
the 2-boundaries $\check{X}\cap S_{i}\cap S_{j}$. While there is no twisting,
the $0^{\text{th}}$ cohomology of a point is known to be 1-dimensional: $\dim
H^{0}(\check{X}\cap S_{1}\cap S_{j})=1$.

Returning to the two-site/four-point example, we need to choose a basis for
the dual-cohomology. To do this, we simply write down a basis for the
cohomology on each boundary. First, we look at the bulk twisted cohomology
$H^{2}(\check{X},\check{\nabla})$. The topological space $\check{X}$ is
pictured at the top of figure 3. Clearly there are no bounded chambers cut out
by the twisted singularities so $\text{dim}H^{2}(\check{X},\check{\nabla})=0$.
Next, we look at the cohomologies on all possible 1-boundaries. We find two
distinct cases of 1-boundaries (pictured on the left and right in figure 3).
Both boundaries $S_{1}$ and $S_{2}$ are twice punctured Riemann spheres. From
a well known result aomoto2011theory , the dimension of the twisted cohomology
of a punctured Riemann sphere is $\\#(\text{punctures})-2$ (this is the
analogue of counting bounded chambers in 1-dimension). Thus, both
$H^{1}(S_{1},\check{\nabla}|_{1})$ and $H^{1}(S_{2},\check{\nabla}|_{2})$ are
trivial. On the other hand, the twisted cohomology on $S_{3}$ is 1-dimensional
since $S_{3}$ has 3 punctures. We can choose

$\displaystyle\check{\phi}_{1}=\varepsilon\
\text{d}\log\frac{T_{1}|_{3}}{T_{2}|_{3}}$ (55)

for the basis of $H^{1}(S_{3},\check{\nabla}|_{3})$ and then uplift this into
the full dual cohomology by simply tacking on a $\delta_{3}$

$\displaystyle\check{\varphi}_{1}=\delta_{3}\left(\check{\phi}_{1}\right)=\varepsilon\
\delta_{3}\left(\text{d}\log\frac{T_{1}|_{3}}{T_{2}|_{3}}\right).$ (56)

The 1-boundary form has been normalized by $\varepsilon$ to ensure that the
differential equation is in $\varepsilon$-form.171717Intuitively, the more a
form is localized to a boundary the less transcendental the resulting integral
is. Thus, we normalize the more transcendental 1-boundary form by
$\varepsilon$ so that it has the same transcendentality as the 2-boundary
forms.

Lastly, there are three 2-boundaries: $S_{12},S_{23}$ and $S_{13}$. Each of
these are simply a point (bottom of figure 3). While the corresponding
cohomology $H^{0}(S_{ij})$ is not twisted, it is non-trivial and spanned by
the constant function: 1. Thus, we choose

$\displaystyle\check{\varphi}_{2}=\delta_{12}(1),\qquad\check{\varphi}_{3}=\delta_{23}(1),\qquad\check{\varphi}_{4}=\delta_{13}(1),$
(57)

as the remaining elements for our basis of the dual cohomology. Notice that
the basis size is the same as in section 3.1. In the dual cohomology, the
basis size is computed by adding the dimensions of the twisted cohomology on
each possible boundary!181818Sometimes this can over count the basis size when
the hyperplane arrangement is non-generic and some boundaries need to be
identified. We will see an example of this in section 4.2. For higher
dimensional manifolds, one can no longer visualize and count bounded chambers
and we have to switch to alternative methods for computing the Euler
characteristic of each boundary. A practical way to do this is provided by
footnote 13.

The components of the kinematic connection associated to the 2-boundary forms
are trivial to obtain. Taking the (dual) kinematic covariant derivative and
using equation (54), we find

$\displaystyle\check{\nabla}_{\mathrm{kin}}\check{\varphi}_{2}$
$\displaystyle=\delta_{12}\left(\check{\omega}_{\mathrm{kin}}|_{12}\right)=\check{\omega}_{\mathrm{kin}}|_{12}\wedge\check{\varphi}_{2}\equiv\check{A}_{22}\wedge\check{\varphi}_{2},$
(58) $\displaystyle\check{\nabla}_{\mathrm{kin}}\check{\varphi}_{3}$
$\displaystyle=\delta_{23}\left(\check{\omega}_{\mathrm{kin}}|_{23}\right)=\check{\omega}_{\mathrm{kin}}|_{23}\wedge\check{\varphi}_{3}\equiv\check{A}_{33}\wedge\check{\varphi}_{3},$
(59) $\displaystyle\check{\nabla}_{\mathrm{kin}}\check{\varphi}_{4}$
$\displaystyle=\delta_{13}\left(\check{\omega}_{\mathrm{kin}}|_{13}\right)=\check{\omega}_{\mathrm{kin}}|_{13}\wedge\check{\varphi}_{4}\equiv\check{A}_{44}\wedge\check{\varphi}_{4},$
(60)

where
$\check{\nabla}_{\text{kin}}=\text{d}_{\text{kin}}+\check{\omega}_{\text{kin}}$,
$\check{\omega}_{\text{kin}}=\text{d}_{\text{kin}}\log\check{u}$ and
$\check{\omega}_{\text{kin}}|_{J}=\text{d}_{\text{kin}}\log\check{u}|_{J}$.
Consequently, we see that the dual differential equation has the following
upper triangular form

$\displaystyle\check{\nabla}_{\mathrm{kin}}\check{\varphi}_{a}=\check{A}_{ab}\wedge\check{\varphi}_{b}\quad\text{where}\quad\underline{\boldsymbol{\check{A}}}=\begin{pmatrix}\check{A}_{11}&\check{A}_{12}&\check{A}_{13}&\check{A}_{14}\\\
0&\check{\omega}_{\mathrm{kin}}|_{12}&0&0\\\
0&0&\check{\omega}_{\mathrm{kin}}|_{23}&0\\\
0&0&0&\check{\omega}_{\mathrm{kin}}|_{13}\end{pmatrix}.$ (61)

To compute the remaining components associated to the 1-boundary form, we
perform IBP on the 1-boundary. That is, we look for some kinematic 1-forms
$A_{1i}$ such that

$\displaystyle\check{\nabla}_{\mathrm{kin}}\check{\varphi}_{1}+\check{\nabla}\mathrm{IBP}=\sum_{i=1}^{4}\check{A}_{1i}\wedge\check{\varphi}_{i},$
(62)

for some integration-by-parts form $\mathrm{IBP}$. Since $\check{\varphi}_{1}$
is a form on the boundary $S_{3}$, $\mathrm{IBP}$ should also be a form on
this boundary: $\mathrm{IBP}=\delta_{3}\left(\check{\xi}\right)$ where
$\check{\xi}$ is a 1-form in the kinematic variables but a 0-form in the
integration variables. Inserting this definition of $\mathrm{IBP}$ into (62),
we obtain

$\displaystyle\delta_{3}\left(\check{A}_{11}\wedge\check{\phi}_{1}-\check{\nabla}_{\mathrm{kin}}|_{3}\
\check{\phi}_{1}-\check{\nabla}_{\mathrm{int}}|_{3}\ \check{\xi}\right)$
$\displaystyle\qquad+\delta_{12}\left(\check{A}_{12}\right)+\delta_{23}\left(\check{A}_{13}+\check{\xi}|_{2}\right)+\delta_{13}\left(\check{A}_{14}+\check{\xi}|_{1}\right)=0~{}.$
(63)

Again, we see that the 2-boundary components are fixed rather trivially

$\displaystyle\check{A}_{12}=0~{},\quad\check{A}_{13}=-\check{\xi}|_{2}~{},\quad\check{A}_{14}$
$\displaystyle=-\check{\xi}|_{1}~{}.$ (64)

All that remains is to specify $\check{\xi}$. It is easily verified that
equation (3.2) is satisfied by setting $\check{A}_{11}=-2\varepsilon\
\text{d}\log(X_{1}+X_{2})$ and

$\displaystyle\check{\xi}=-\varepsilon\frac{\text{d}(X_{1}+X_{2})}{T_{2}|_{3}}.$
(65)

Putting everything together, we find the dual differential equation

$\displaystyle\underline{\boldsymbol{\check{A}}}=-\varepsilon\
\text{d}{\tiny\begin{pmatrix}2\log(X_{1}+X_{2})&0&\log\left(\frac{X_{2}+Y}{X_{1}-Y}\right)&\log\left(\frac{X_{2}-Y}{X_{1}+Y}\right)\\\
0&\log(X_{1}+Y)(X_{2}+Y)&0&0\\\ 0&0&\log\left((X_{1}-Y)(X_{2}+Y)\right)&0\\\
0&0&0&\log\left((X_{2}-Y)(X_{1}+Y)\right)\end{pmatrix}}$ (66)

for our basis of the dual-cohomology (55) and (57).

The differential equation for the basis of canonical forms (44) is obtained
from the dual differential equation (66) by a gauge transform

$\displaystyle\underline{\boldsymbol{A}}_{\vartheta}=-\left(\underline{\boldsymbol{C}}^{-1}_{\vartheta}\cdot\underline{\boldsymbol{\check{A}}}\cdot\underline{\boldsymbol{C}}_{\vartheta}\right)^{\top}$
$\displaystyle=\varepsilon\
\text{d}{\tiny\begin{pmatrix}\log\left(\left(X_{1}+Y\right)\left(X_{2}+Y\right)\right)&\log\left(\frac{Y-X_{1}}{X_{1}+Y}\right)&\log\left(\frac{X_{2}+Y}{Y-X_{2}}\right)&0\\\
0&\log\left(\left(X_{1}-Y\right)\left(X_{1}+X_{2}\right)\right)&\log\left(\frac{X_{2}+Y}{X_{1}+X_{2}}\right)&\log\left(\frac{X_{2}+Y}{X_{1}+X_{2}}\right)\\\
0&\log\left(\frac{X_{1}+Y}{X_{1}+X_{2}}\right)&\log\left(\left(X_{2}-Y\right)\left(X_{1}+X_{2}\right)\right)&\log\left(\frac{X_{1}+X_{2}}{X_{1}+Y}\right)\\\
0&\log\left(\frac{Y-X_{1}}{X_{1}+Y}\right)&\log\left(\frac{X_{2}+Y}{Y-X_{2}}\right)&\log\left(\left(X_{1}+Y\right)\left(X_{2}+Y\right)\right)\end{pmatrix}}$
(67)

where

$\displaystyle(C_{\vartheta})_{ab}=\langle\check{\varphi}_{a}|\vartheta_{b}\rangle=\begin{pmatrix}0&-1&1&0\\\
1&0&0&0\\\ 1&1&0&1\\\ -1&0&1&-1\end{pmatrix}$ (68)

is the intersection matrix. The intersection matrix is analogous to the
computation of generalized unitary coefficients in the context of Feynman
integrals. In particular, the simple formula for the intersection number of
$\text{d}\log$-forms (appendix A) make such calculations straightforward!

Furthermore, the intersection matrix is a realization of the isomorphism
between the FRW- and dual-cohomologies since it has full rank. However, in
some sense, the canonical forms (37) are not the most direct way of
establishing the duality between the FRW- and dual-cohomologies since the
intersection matrix is dense.

Instead, one can directly construct a basis for the FRW-cohomology that is
orthonormal to the basis of dual-cohomology (55) and (57). By choosing
$\text{d}\log$ forms that do not share singularities on the maximal
intersections of the hyperplanes $\\{T_{i},S_{i}\\}$, one guarantees that the
corresponding intersection matrix is diagonal. For example, the following
basis

$\displaystyle\varphi_{1}$
$\displaystyle=-\frac{1}{2}\text{d}\log(S_{3})\wedge\text{d}\log\left(\frac{T_{1}}{T_{2}}\right)~{},$
$\displaystyle\varphi_{2}$ $\displaystyle=\text{d}\log S_{1}\wedge\text{d}\log
S_{2}~{},$ $\displaystyle\varphi_{3}$ $\displaystyle=\text{d}\log
S_{2}\wedge\text{d}\log S_{3}~{},$ $\displaystyle\varphi_{4}$
$\displaystyle=\text{d}\log S_{1}\wedge\text{d}\log S_{3}~{},$ (69)

is orthonormal to the $\check{\varphi}_{a}$

$\displaystyle(C_{\varphi})_{ab}=\langle\check{\varphi}_{a}|\varphi_{b}\rangle=\delta_{ab}~{}.$
(70)

Generally, it is much easier to build a basis in the dual-cohomology that has
a simple DEQ. Thus, the basis $\varphi_{a}$ is extremely useful since the
resulting differential equation is the minus transpose of
$\underline{\boldsymbol{\check{A}}}$

$\displaystyle\underline{\boldsymbol{A}}_{\varphi}=-\left(\underline{\boldsymbol{\check{A}}}\right)^{\top},$
(71)

which is much more sparse than $\underline{\boldsymbol{A}}_{\vartheta}$. Then,
since the integral of physical interest is simply related to the
$\varphi$-basis
${\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>signs simply correspond to the first column of
$\underline{\boldsymbol{C}}_{\vartheta}$. By inserting identity
$\mathds{1}=|\varphi_{a}\rangle\left(C_{\varphi}^{-1}\right)_{ab}\langle\check{\varphi}_{b}|$,
one finds
$|\vartheta_{1}\rangle=|\varphi_{a}\rangle\left(C_{\varphi}^{-1}\right)_{ab}\langle\check{\varphi}_{b}|\vartheta_{1}\rangle=\langle\check{\varphi}_{a}|\vartheta_{1}\rangle|\varphi_{a}\rangle=\left(C_{\vartheta}\right)_{a1}|\varphi_{a}\rangle$
as claimed., one can integrate the simpler DEQ
$\underline{\boldsymbol{A}}_{\varphi}$ instead of
$\underline{\boldsymbol{A}}_{\vartheta}$. Moreover, the $\varphi$-basis never
needs to be explicitly constructed – only its existence is important! One can
obtain any FRW integral of interest by integrating
$\left(-\underline{\boldsymbol{\check{A}}}\right)^{\top}$ (without choosing a
basis for the FRW cohomology) and using the intersection matrix to project out
the correct linear combination.

###  3.3 Trivializing the DEQ computation using intersection theory

As advertised in the introduction, one can compute the differential equations
for a family of relative twisted differential forms using the intersection
number instead of IBPs. Explicitly,

$\displaystyle
A_{ij}=C^{-1}_{jk}\langle\check{\varphi}_{k}|\nabla_{\text{kin}}\varphi_{i}\rangle\,,\qquad\check{A}_{ij}=\langle\check{\nabla}_{\text{kin}}\check{\varphi}_{i}|\varphi_{k}\rangle
C^{-1}_{kj}\,,$ (72)

where $C_{ij}=\langle\check{\varphi}_{i}|\varphi_{j}\rangle$ is the
intersection matrix for a given choice of basis. In general, this usually
requires more sophisticated methods for computing the intersection numbers
than those presented in appendix A since the forms
$\nabla_{\text{kin}}\varphi_{j}$ and
$\check{\nabla}_{\text{kin}}\check{\varphi}_{i}$ no longer have at most simple
poles. However, since our forms are logarithmic, we can use a simple trick
that avoids the introduction of double poles and allows us to compute the
differential equations using only the formula presented in appendix
A!202020The methods outlined here also help streamline the integration-by-
parts algorithm presented in sections 3.1 and 3.2.

The idea is to simply promote the exterior derivative d that acts on the
integration variables to the exterior derivative that acts on both the
integration and external variables:
$\text{d}\to\text{D}=\text{d}+\text{d}_{\text{kin}}$. This promotes our basis
to forms valued on the total space. For example,

$\displaystyle{\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle\to\text{D}\log\frac{S_{1}}{S_{2}}\wedge\text{D}\log\frac{S_{2}}{S_{3}}$
$\displaystyle=\frac{-2Y\
\text{d}x_{1}\wedge\text{d}x_{2}}{(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad+\frac{2(x_{2}+X_{2})\
\text{d}x_{1}\wedge\text{d}Y}{(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad+\frac{2Y\
\text{d}x_{2}\wedge\text{d}X_{1}}{(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad-\frac{2Y\
\text{d}x_{1}\wedge\text{d}X_{2}}{(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad-\frac{2(x_{1}+X_{1})\
\text{d}x_{2}\wedge\text{d}Y}{(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad+\text{kinematic 2-forms}\,.$ (73)

Note that the FRW connection does not change since it is independent of
kinematics $\omega\to\text{D}\log(u)=\text{d}\log(u)$.212121On the other hand,
the dual connection does change since the restriction to boundaries introduces
kinematic dependence.

Now, since the difference between $(\text{D}+\text{D}\log(u)\wedge)$ and
$\nabla_{\text{kin}}$ is a total derivative $\nabla$, we can replace
$\nabla_{\text{kin}}$ in (72) by $(\text{D}+\text{D}\log(u)\wedge)$ without
changing the intersection number.222222The intersection number of any total
covariant derivative $\nabla$ or $\check{\nabla}$ always vanishes. Explicitly,
for
$\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>this amounts to the replacement

$\displaystyle{\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color@cmyk@fill{0.40}{0.80}{0.20}{0}\dot{\vartheta_{1}}}=\nabla_{\text{kin}}{\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle\to(\text{D}+\text{D}\log(u)\wedge)\
{\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle=\frac{-2\varepsilon\ (x_{2}X_{1}+x_{1}X_{2}+2x_{1}x_{2})\
\text{d}x_{1}\wedge\text{d}x_{2}\wedge\text{d}Y}{x_{1}x_{2}(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad+\frac{2\varepsilon\ Y\
\text{d}x_{1}\wedge\text{d}x_{2}\wedge\text{d}X_{1}}{x_{1}(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad+\frac{2\varepsilon\ Y\
\text{d}x_{1}\wedge\text{d}x_{2}\wedge\text{d}X_{2}}{x_{2}(x_{1}+x_{2}+X_{1}+X_{2})(x_{1}+X_{1}+Y)(x_{2}+X_{2}+Y)}$
$\displaystyle\quad+\text{kinematic 2-forms}\,.$ (74)

Notice that the right hand side above only has simple poles! Using the
formulae given in appendix A, one finds the first row of the differential
equation $\underline{\boldsymbol{A}}_{\vartheta}$

$\displaystyle(A_{\vartheta})_{1j}=(C_{\vartheta}^{-1})_{jk}\langle\check{\varphi}_{k}|(\text{D}+\text{D}\log(u)\wedge)\
{\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color@cmyk@fill{0.40}{0.80}{0.20}{0}\vartheta_{1}}\rangle\,.$
(75)

The above formula is attractive since it only involves algebraic (residue)
operations. We know where the intersection number localizes and don’t have to
go through the process of constructing IBP-forms.

Before moving on to actually integrating our differential equation, we note
that the ideas outlined here are applicable to the algorithm for computing the
symbol in Chen:2023kgw . By promoting the external derivative to a derivative
on the total space, the two-step algorithm for computing the intersection
number could potentially be reduced to one-step!

###  3.4 Integrating the differential equations

Having constructed the DEQ system for the two-site/four-point FRW correlator
in subsections 3.1 ($\vartheta$-basis) and 3.2 ($\varphi$-basis), we now turn
our attention to solving them. We implement conventional techniques that have
become widespread in the computation of Feynman integrals over the last decade
Henn:2014qga ; Abreu:2022mfk . Differential equations of the form constructed
in sections 3.1 and 3.2 are known as Pfaffian systems whose solutions are
given by the path-ordered exponential as expressed in equation (33). For small
$\varepsilon$, as is the case for FRW cosmologies, such solutions can be
expanded and truncated to any desired order in $\varepsilon$. In this
subsection, we will reproduce the well known de Sitter result corresponding to
the $\mathcal{O}(\varepsilon^{0})$ term of the FRW wavefunction coefficient.
Then, for the first time, we derive an expression for the
$\mathcal{O}(\varepsilon)$ term of the two-site/four-point FRW wavefunction
coefficient.

We begin by reviewing how to integrate systems of coupled linear DEQs in the
context of the two-site/four-point example. The physical integral
$\vartheta_{1}$ can be obtained by directly integrating the
$\underline{\boldsymbol{A}}_{\vartheta}$ system associated to the canonical
basis, for which $\vartheta_{1}$ is a basis element, or by integrating the
$\underline{\boldsymbol{A}}_{\varphi}$ system and using the fact that
${\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>
In either case, the DEQ is

$\displaystyle\nabla_{\text{kin}}\alpha_{i}=A_{\alpha;ij}\wedge\alpha_{j}~{},$
(76)

where $\alpha=\vartheta,\varphi$ denotes our choice of basis. In terms of
actual integrals instead of differential forms, this DEQ becomes

$\displaystyle\text{d}_{\text{kin}}\,\mathbf{g}_{\alpha}(z,\varepsilon)=\underline{\boldsymbol{A}}_{\alpha}\cdot\mathbf{g}_{\alpha}(z,\varepsilon)~{},$
(77)

where $\mathbf{g}_{\alpha}=\int u\,\alpha$ is the vector of integrals
associated to our choice of basis. Here, $z\in\\{X_{1},X_{2},Y\\}$ refers to
the external scales for the tree-level two-site/four-point example at hand and
$\text{d}_{\text{kin}}=\sum_{\\{z\\}}\partial_{z}\text{d}z=\partial_{X_{1}}\text{d}X_{1}+\partial_{X_{2}}\text{d}X_{2}+\partial_{Y}\text{d}Y$
as before. The $4\times 4$ DEQ matrix $\underline{\boldsymbol{A}}_{\alpha}$
can be expanded as

$\displaystyle\underline{\boldsymbol{A}}_{\alpha}=\varepsilon\sum_{\\{z\\}}\underline{\boldsymbol{\Omega}}_{z}\text{d}z=\varepsilon\left(\underline{\boldsymbol{\Omega}}_{X_{1}}\text{d}X_{1}+\underline{\boldsymbol{\Omega}}_{X_{2}}\text{d}X_{2}+\underline{\boldsymbol{\Omega}}_{Y}\text{d}Y\right)~{},$
(78)

where the entries of the $\underline{\boldsymbol{\Omega}}_{z}$ matrices are
rational one-forms. The solution $\mathbf{g}_{\alpha}(z,\varepsilon)$ is a
Laurent series in $\varepsilon$

$\displaystyle\mathbf{g}_{\alpha}(z,\varepsilon)=\sum_{n=-2}^{\infty}\varepsilon^{n}\
\mathbf{f}_{\alpha}^{(n)}(z)~{},$ (79)

where the minimal value of $n$ is determined by boundary conditions (in this
case, it happens to be $-2$).

For the two-site/four-point example, the boundary conditions are generated by
taking the $Y\to 0$ limit where the integrals can actually be performed and
then expanded in $\varepsilon$. We denote this boundary condition by
$\mathbf{g}^{*}_{\alpha}$

$\displaystyle\mathbf{g}_{\alpha}^{*}(X_{1},X_{2},\varepsilon):=\mathbf{g}_{\alpha}^{*}(X_{1},X_{2},0,\varepsilon)=\sum_{n=-2}^{\infty}\varepsilon^{n}\,\mathbf{f}^{*(n)}_{\alpha}(X_{1},X_{2})~{}.$
(80)

For the $\vartheta$-basis, the boundary condition integrals are
straightforward to compute. On the other hand, the integral associated to
$\varphi_{1}$ is divergent. However, due to the simplicity of the
$\underline{\boldsymbol{A}}_{\varphi}$ DEQ, it is easy to see that the
function $a(\varepsilon)(X_{1}+X_{2})^{2\varepsilon}$ is a solution for the
first component of $\mathbf{g}_{\varphi}$. Expanding $a(\varepsilon)$ as a
series in $\varepsilon$, each coefficient can be fixed by demanding that the
vector of boundary conditions $\mathbf{g}_{\varphi}^{*}$ satisfies the DEQ. In
particular, using
$a(\varepsilon)=-\frac{1}{2\varepsilon^{2}}-\frac{\pi^{2}}{4}+\mathcal{O}(\varepsilon)$
is enough to fix the physical integral to $\mathcal{O}(\varepsilon)$.

Having generated the boundary conditions, we are now in a position to
integrate the DEQ system at hand. Given our Laurent series ansatz (79), the
DEQ system takes the form

$\displaystyle\partial_{z}\,\mathbf{g}_{\alpha}(z,\varepsilon)=\underline{\boldsymbol{\Omega}}_{z}\,\cdot\mathbf{g}_{\alpha}(z,\varepsilon)\implies\begin{cases}\partial_{z}\mathbf{f}^{(-2)}=0\\\
\partial_{z}\mathbf{f}^{(a)}=\underline{\boldsymbol{\Omega}}_{z}\cdot\mathbf{f}^{(a-1)}\quad\forall~{}a>2~{}.\end{cases}$
(81)

Following conventional DEQ techniques, we can now solve order by order in the
twist parameter $\varepsilon$. Thus, we are led to the following solution at
order $\varepsilon=0$ for the two-site/four-point correlator, i.e., the
solution corresponding to the physical chamber denoted by
${\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>
$\displaystyle\psi_{\text{2,dS}}^{(0)}$
$\displaystyle=\text{Li}_{2}\left(\frac{X_{1}-Y}{X_{1}+Y}\right)+\text{Li}_{2}\left(\frac{X_{2}-Y}{X_{2}+Y}\right)-\text{Li}_{2}\left(\frac{X_{1}-Y}{X_{1}+Y}\frac{X_{2}-Y}{X_{2}+Y}\right)-\frac{\pi^{2}}{6}~{}.$
(82)

Up to a sign,232323The sign difference arises from the arbitrariness in the
choice of orientation when constructing a canonical form. the above solution
corresponds to the well-know two-site contribution to the wavefunction
coefficient in a dS Universe ($\varepsilon=0$) ArkaniHamed2015 ; Arkani-
Hamed:2018kmz ; Hillman:2019wgh .

In order to go beyond dS, we solve the differential equations arising at the
next order in $\varepsilon$ (only for the physical component
${\color[rgb]{0.6,0.2,0.8}\definecolor[named]{pgfstrokecolor}{rgb}{0.6,0.2,0.8}\pgfsys@color@cmyk@stroke{0.40}{0.80}{0.20}{0}\pgfsys@color<EMAIL_ADDRESS>leading us to the following two-site/four-point contribution to the
wavefunction coefficient in a FRW Universe in the region $X_{1}>X_{2}>Y$

$\displaystyle\psi_{\text{2,FRW}}^{(0)}=$
$\displaystyle-\text{Li}_{3}\left(\frac{Y+X_{1}}{Y-X_{2}}\right)+\text{Li}_{3}\left(-\frac{X_{1}-Y}{Y+X_{2}}\right)+2\text{Li}_{3}\left(\frac{X_{1}-Y}{X_{1}+X_{2}}\right)-2\text{Li}_{3}\left(\frac{Y+X_{1}}{X_{1}+X_{2}}\right)$
$\displaystyle+2\log\left(X_{1}+X_{2}\right)\text{Li}_{2}\left(-\frac{Y+X_{2}}{X_{1}-Y}\right)-2\log\left(X_{1}+X_{2}\right)\text{Li}_{2}\left(\frac{Y-X_{2}}{Y+X_{1}}\right)$
$\displaystyle+\frac{1}{6}\log^{3}\left(X_{2}-Y\right)-\frac{1}{6}\log^{3}\left(X_{2}+Y\right)-\frac{1}{2}\log\left(X_{1}+Y\right)\log^{2}\left(X_{2}-Y\right)$
$\displaystyle-\frac{1}{2}\log^{2}\left(X_{1}+Y\right)\log\left(X_{2}-Y\right)-\log^{2}\left(X_{1}+X_{2}\right)\log\left(X_{2}-Y\right)$
$\displaystyle+\frac{1}{2}\log\left(X_{1}+Y\right)\log^{2}\left(X_{2}+Y\right)+\log^{2}\left(X_{1}+X_{2}\right)\log\left(X_{2}+Y\right)$
$\displaystyle+\frac{1}{2}\log^{2}\left(X_{1}+Y\right)\log\left(X_{2}+Y\right)+\log\left(X_{1}+X_{2}\right)\log^{2}\left(X_{1}-Y\right)$
$\displaystyle-\log\left(X_{1}+X_{2}\right)\log^{2}\left(X_{1}+Y\right)+2\log\left(X_{1}+X_{2}\right)\log\left(X_{1}+Y\right)\log\left(X_{2}-Y\right)$
$\displaystyle+\frac{1}{6}\pi^{2}\left(\log\left(X_{2}-Y\right)-\log\left(X_{2}+Y\right)\right)$
$\displaystyle-2\log\left(X_{1}+X_{2}\right)\log\left(X_{1}-Y\right)\log\left(X_{2}+Y\right)~{}.$
(83)

It is worth noting a few important features of the above expression. First,
the integral representation of $\psi_{2,\text{FRW}}^{(0)}$ is _not_ manifestly
symmetric in $X_{1}$ and $X_{2}$. We have sacrificed this symmetry for the
simple presentation above. A formula valid in the whole physical region that
is manifestly symmetric has been included in the ancillary files. Second, the
above expression has uniform transcendental weight three. This is expected
since the FRW wavefunction is the integral of the dS wavefunction thereby
increasing the transcendental weight from two to three. In the next section,
we will derive the differential equation for the tree-level three-site/five-
point and one-loop two-site/two-point FRW wavefunction coefficients. While we
leave the integration of these differential equation for future work, the
resulting $\mathcal{O}(\varepsilon)$ terms will have transcendental weight
four and three since the corresponding dS wavefunction coefficients are know
to be weight three and two respectively. Interestingly, loop-diagrams
corresponding to cosmological correlators are not more transcendental than
their tree-level counter parts because the integration over the spatial loop
momentum is not performed.

##  4 Further examples

In this section, we provide two further examples illustrating how to construct
cosmologies associated to FRW integrals and the differential equations they
satisfy. In particular, we show that there are no obstructions to extending
the techniques applied in this work to loop-level. To illustrate this point,
we construct the cohomologies for the one-loop two-site/two-point FRW integral
as well as their differential equations in section 4.1. Then, in section 4.2,
we construct the cohomologies and associated differential equations for the
tree-level three-site/five-point FRW wavefunction coefficient. Here, we see a
new feature due to the degeneration of boundaries that will be present in most
higher-site/point examples and explain how to account for this degeneracy.

###  4.1 The one-loop two-site FRW correlator

Figure 4: The Feynman diagram for the one-loop two-site/two-point wavefunction
coefficient in $\lambda_{3}\phi^{3}$ theory and the associated hyperplane
arrangement.

In $\lambda_{3}\phi^{3}$ theory, the flat-space one-loop two-site/two-point
wavefunction coefficient receives contributions from a single graph (see the
left of figure 4) given by equation (19), which we quote here for convenience

$\displaystyle\psi_{2,\text{flat}}^{(1)}$
$\displaystyle=\frac{2\left(X_{1}+X_{2}+Y_{1}+Y_{2}\right)}{\left(X_{1}+X_{2}\right)\left(X_{1}+X_{2}+2Y_{1}\right)\left(X_{1}+Y_{1}+Y_{2}\right)\left(X_{2}+Y_{1}+Y_{2}\right)\left(X_{1}+X_{2}+2Y_{2}\right)}~{}.$
(84)

The FRW uplift prescription (22) yields the associated hyperplane arrangement
for the FRW correlator (see the right of figure 4). As usual, the hyperplane
arrangement is determined by the twisted coordinate hyperplanes

$\displaystyle T_{i}$ $\displaystyle=x_{i}\quad\text{for}\quad i=1,2~{},$ (85)

and the singular/boundary hyperplanes are defined by the linear divisors of
$\psi_{2,\text{flat}}^{(1)}|_{X_{i}\to x_{i}+X_{i}}$

$\displaystyle\begin{matrix}[l]S_{1}=x_{1}+x_{2}+X_{1}+X_{2},&\quad&S_{2}=x_{1}+X_{1}+Y_{1}+Y_{2},\\\
S_{3}=x_{2}+X_{2}+Y_{1}+Y_{2},&\quad&S_{4}=x_{1}+x_{2}+X_{1}+X_{2}+2Y_{1},\\\
S_{5}=x_{1}+x_{2}+X_{1}+X_{2}+2Y_{2}.\end{matrix}$ (86)

Note that this hyperplane arrangement is incredibly similar to the tree-level
case and not any more difficult.

Counting the bounded chambers in figure 4 (right), we see that the FRW
cohomology an its dual are 10-dimensional. A convenient choice for the dual
1-boundary forms is

$\displaystyle\check{\varphi}_{1}=\varepsilon\
\delta_{1}\left(\text{d}\log\frac{T_{1}|_{1}}{T_{2}|_{1}}\right),\qquad\check{\varphi}_{2}=\varepsilon\
\delta_{4}\left(\text{d}\log\frac{T_{1}|_{2}}{T_{2}|_{2}}\right),\qquad\check{\varphi}_{3}=\varepsilon\
\delta_{5}\left(\text{d}\log\frac{T_{1}|_{3}}{T_{2}|_{3}}\right).$ (87)

A similarly convenient choice for the 2-boundary forms is

$\displaystyle\begin{matrix}[l]\check{\varphi}_{4}=\delta_{12}(1),\quad&\check{\varphi}_{5}=\delta_{12}(1),\quad&\check{\varphi}_{6}=\delta_{12}(1),\quad&\check{\varphi}_{7}=\delta_{12}(1),\\\
\check{\varphi}_{8}=\delta_{12}(1),&\check{\varphi}_{9}=\delta_{12}(1),&\check{\varphi}_{10}=\delta_{12}(1).&\end{matrix}$
(88)

A set of FRW forms dual to the 1-boundary forms (87) is

$\displaystyle\begin{matrix}[l]\varphi_{1}=-\frac{1}{2}\ \text{d}\log
S_{1}\wedge\text{d}\log\frac{T_{1}}{T_{2}},&\varphi_{2}=-\frac{1}{2}\
\text{d}\log S_{4}\wedge\text{d}\log\frac{T_{1}}{T_{2}},\\\
\varphi_{3}=-\frac{1}{2}\ \text{d}\log
S_{5}\wedge\text{d}\log\frac{T_{1}}{T_{2}},&\end{matrix}$ (89)

while a set of FRW forms dual to (88) is

$\displaystyle\begin{matrix}[l]\varphi_{4}=\text{d}\log
S_{1}\wedge\text{d}\log S_{2},\qquad&\varphi_{5}=\text{d}\log
S_{1}\wedge\text{d}\log S_{3},\qquad&\varphi_{6}=\text{d}\log
S_{2}\wedge\text{d}\log S_{3},\\\ \varphi_{7}=\text{d}\log
S_{2}\wedge\text{d}\log S_{4},&\varphi_{8}=\text{d}\log
S_{2}\wedge\text{d}\log S_{5},&\varphi_{9}=\text{d}\log
S_{3}\wedge\text{d}\log S_{4},\\\ \varphi_{10}=\text{d}\log
S_{3}\wedge\text{d}\log S_{5}.&&\end{matrix}$ (90)

With these choices for our basis, it is easy to see that only the diagonal
intersection numbers are non-trivial. Moreover, the normalization of
$-\frac{1}{2}$ in (87) ensures that our choice of basis are orthonormal: each
$T_{i}|_{\bullet}$ contributes a $\frac{1}{-\varepsilon}$ to the intersection
number while the point at infinity does not contribute since of the forms (87)
are regular at infinity.

Computing the differential equation using the methods of section 3.2 or 3.3
yields

$\displaystyle\underline{\boldsymbol{A}}=\varepsilon\left(\begin{array}[]{cccccccccc}2a_{1}&0&0&0&0&0&0&0&0&0\\\
0&2a_{2}&0&0&0&0&0&0&0&0\\\ 0&0&2a_{3}&0&0&0&0&0&0&0\\\
a_{5}{-}a_{4}&0&0&a_{4}{+}a_{5}&0&0&0&0&0&0\\\
a_{7}{-}a_{6}&0&0&0&a_{6}{+}a_{7}&0&0&0&0&0\\\
0&0&0&0&0&a_{5}{+}a_{6}&0&0&0&0\\\
0&a_{8}{-}a_{5}&0&0&0&0&a_{5}{+}a_{8}&0&0&0\\\
0&0&a_{10}{-}a_{5}&0&0&0&0&a_{5}{+}a_{10}&0&0\\\
0&a_{6}{-}a_{9}&0&0&0&0&0&0&a_{6}{+}a_{9}&0\\\
0&0&a_{6}{-}a_{11}&0&0&0&0&0&0&a_{6}{+}a_{11}\\\ \end{array}\right)~{}.$ (101)

Here, one should read the $a_{i}$’s above as $\text{d}\log(a_{i})$ where the
$a_{i}$ are the letters

$\displaystyle a_{i=1,\dots,11}=\bigg{\\{}$ $\displaystyle X_{1}+X_{2},\
X_{1}+X_{2}+2Y_{1},\ X_{1}+X_{2}+2Y_{2},\ X_{2}-Y_{1}-Y_{2},\ $ $\displaystyle
X_{1}+Y_{1}+Y_{2},\ X_{2}+Y_{1}+Y_{2},\ X_{1}-Y_{1}-Y_{2},\
X_{2}+Y_{1}-Y_{2},\ $ $\displaystyle X_{1}+Y_{1}-Y_{2},\ X_{2}-Y_{1}+Y_{2},\
X_{1}-Y_{1}+Y_{2}\bigg{\\}}.$ (102)

The known de Sitter limit of this alphabet Hillman:2019wgh  is a subset of the
above FRW alphabet with the new letters being $X_{1}+Y_{1}+Y_{2}$ and
$X_{2}+Y_{1}+Y_{2}$.

After integrating the above differential equation, the physical integral can
be computed by taking the appropriate linear combination. In our chosen basis,

$\displaystyle\psi_{2,\text{FRW}}^{(1)}=\frac{1}{4Y_{1}Y_{2}}\int
u\left(-\varphi_{4}+\varphi_{5}+\varphi_{6}-\varphi_{7}-\varphi_{8}+\varphi_{9}+\varphi_{10}\right).$
(103)

While one can get the FRW IBPs algorithm of section 3.1 to finish in a
reasonable amount of time on a laptop for this example, the dual IBPs of
section 3.2 or the intersection algorithm of section 3.3 becomes noticeably
faster here. The simplicity of dual IBPs is essential for the next example.
There, we could not get a _simple_ implementation of the algorithm in section
3.1 to finish in a reasonable time unless supplemented with knowledge of the
dual differential equation.

Our basis choice is motivated by the geometry of the dual cohomology and
produces a sparse differential equation. While this is desirable, there may be
better choices. Since only the un-twisted hyperplanes appear in the physical
FRW form, it would be interesting if one could find a basis that factorizes
the differential equation (101) into physical and un-physical sectors that can
be solved separately.242424Indeed, it looks like this had been achieved to
some extent in ltalk . While the FRW cohomology of the 3-site/5-point is
26-dimensional, in ltalk  they seem to only need 16 elements to get the DEQ
for the physical integral. We leave such questions to future work.

###  4.2 The tree-level three-site FRW correlator

Figure 5: The Feynman diagram for the three-site/five-point FRW wavefunction
coefficient in $\lambda_{3}\phi^{3}$-theory.

We now turn our attention to the tree-level three-site/five-point FRW
wavefunction coefficient where we will find new features not seen at the level
of the two-site/four-point example but are expected to be present for most
higher-site diagrams. Substituting equation (20) into (22), one obtains

$\displaystyle\psi^{(0)}_{3,\text{FRW}}=\int_{0}^{\infty}\left(\bigwedge_{i=1}^{3}\text{d}x_{i}\right)\left.\left(\prod_{a=1}^{3}T_{a}^{\varepsilon}\right)\psi^{(0)}_{3,\textrm{flat}}\right|_{X_{i}\to
x_{i}+X_{i}}~{},$ (104)

where we duplicate the flat space three-site wavefunction coefficient (20)
below for ease of reading

$\displaystyle\psi_{3,\textrm{flat}}^{(0)}$
$\displaystyle=\scalebox{1.15}{$\frac{X_{1}{+}Y_{1}{+}2X_{2}{+}Y_{2}{+}X_{3}}{(X_{1}{+}X_{2}{+}X_{3})(X_{1}{+}Y_{1})(X_{3}{+}Y_{2})(X_{3}{+}Y_{2})(X_{2}{+}Y_{1}{+}Y_{2})(X_{1}{+}X_{2}{+}Y_{2})(X_{2}{+}X_{3}{+}Y_{1})}$
}.$ (105)

The corresponding hyperplane arrangement is determined by the twisted
coordinate hyperplanes

$\displaystyle T_{i}=x_{i}\quad\text{for}\quad i=1,2,3,$ (106)

and the linear denominators of $\psi^{(0)}_{3,\text{flat}}|_{X_{i}\to
x_{i}+X_{i}}$

$\displaystyle\begin{matrix}[l]S_{1}=x_{1}+x_{2}+x_{3}+X_{1}+X_{2}+X_{3}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}}~{},&\quad&S_{2}=x_{1}+X_{1}+Y_{1}~{},\\\
S_{3}=x_{3}+X_{3}+Y_{2}~{},&&S_{4}=x_{2}+X_{2}+Y_{1}+Y_{2}~{},\\\
S_{5}=x_{1}+x_{2}+X_{1}+X_{2}+Y_{2}~{},&&S_{6}=x_{2}+x_{3}+X_{2}+X_{3}+Y_{1}~{}.\end{matrix}$
(107)

Note that we have added an extra factor of $Y_{3}$ into $S_{1}$ that is not
present in the physical arrangement. This has been done for purely pedagogical
reasons since the physical arrangement ($Y_{3}=0$) is non-generic – there are
4-planes crossing at a single point (see figure 6). Adding a non-zero $Y_{3}$
will help us understand how to think about non-generic arrangements. However,
in the end, one can directly set $Y_{3}=0$ at the beginning of the calculation
as we will explain later.

Figure 6: Plots of the planes $S_{i=1,4,5,6}$. The dark blue lines denote the
intersection of these planes and the red dots mark where these lines
intersect. The left plot corresponds to the unphysical but generic hyperplane
arrangement with $Y_{3}\neq 0$. The right plot is the degeneration of the left
plot in the $Y_{3}\to 0$ limit. The bounded chamber with the red dots as
vertices in the right plot vanishes in the $Y_{3}\to 0$ limit. This
degeneration corresponds to the drop in dimension of the cohomology group.
Figure 7: Boundary stratification of the relative twisted cohomology
associated to the three-site/five-point hyperplane arrangement. The purple
vertices are boundaries with non-trivial cohomology while all the blue
vertices are boundaries with trivial cohomology. Notice that there are some
missing 3-boundaries, $\delta_{126},\delta_{135},\delta_{245},\delta_{346}$,
since $S_{1}\cap S_{2}\cap S_{6}=S_{1}\cap S_{3}\cap S_{5}=S_{2}\cap S_{4}\cap
S_{5}=S_{3}\cap S_{4}\cap S_{6}=\emptyset$.

For the generic arrangement ($Y_{3}\neq 0$), the basis size or dimension of
the cohomology group is 26. The basis size is obtained by adding up the
dimension of each twisted cohomology group on each of the non-maximal
codimension boundaries.252525Recall that the dimension of the top-dimensional
twisted cohomology group is given by the Euler characteristic of the
underlying manifold or equivalently by counting the critical points of the
dual connection $\check{\omega}$ (see footnote 13) or by counting the number
of bounded chambers formed by the twisted hyperplanes. Additionally, (for
generic arrangements) each maximal codimension boundary generates a cohomology
class/basis element. The boundary stratification for the three-site/five-point
arrangement is presented in figure 7. Each node denotes the twisted cohomology
on the given boundary. The blue nodes represent boundaries with trivial
cohomology (zero dimensional) while the purple nodes represent boundaries with
non-trivial cohomology. Example arrangements of the twisted hyperplanes on
some boundaries are illustrated in figure 7. Since the twisted hyperplanes do
not form bounded chambers in the bulk space $\delta_{\\{\\}}$ only the
cohomologies on the boundaries are non-trivial. The 1-boundary $\delta_{6}$ is
also pictured figure 7 and is clearly trivial since there are no bounded
chambers. On the other hand, the cohomology on $\delta_{1}$ is 1-dimensional
since the twisted hyperplanes form a single bounded chamber. For the
cohomologies on 2-boundaries, the dimension is determined by the number of
twisted singularities minus two. Thus, the dimension on boundary $\delta_{12}$
is 1-dimensional while the cohomology on the boundary $\delta_{46}$ is
trivial. For the generic arrangement, the untwisted 3-boundaries are all
distinct and 1-dimensional.

For the physical ($Y_{3}=0$) arrangement, the basis size decreases to 25
because the four hyperplanes $S_{1},S_{4},S_{5}$ and $S_{6}$ all intersect at
a point (see figure 6). Sending $Y_{3}\to 0$ shrinks the bounded chamber
pictured in the left of figure 6 to a point decreasing the basis size. Since
we expect that there will be many more instances of such degeneracies for
higher site FRW integrals, we provide a careful treatment of how the generic
arrangement degenerates to the physical arrangement below.262626For the four-
site graph, there are two topologically distinct Feynman diagrams: a line and
a three-pointed star. For example, the hyperplane arrangement corresponding to
the _generic_ three-point star graph consists of 11 boundary hyperplanes in
generic position and the 3 twisted coordinate hyperplanes as usual. We find
that this generic arrangement of the star graph has a total of 377 elements.
The degenerate/physical arrangement has 12 lines where 4 of the boundary
hyperplanes intersect, 18 points where 5 of the boundary hyperplanes intersect
and 1 point where 8 of the boundary hyperplanes intersect. This is expected to
reduce the number of independent elements by a considerable amount. Again, we
emphasize that one does not need to introduce additional parameters to make
the arrangement generic as long as one knows how to consistently treat the
non-generic/degenerate case directly.

Below, we give a natural basis associated to the generic arrangement where the
element that drops out when passing to the physical arrangement has been
colored in gray.272727While we have chosen to eliminate the forms associated
with the intersection $S_{1}\cap S_{4}\cap S_{5}$, we could have chosen to
eliminate the form associated to the intersections $S_{1}\cap S_{5}\cap
S_{6}$, $S_{1}\cap S_{4}\cap S_{6}$ or $S_{4}\cap S_{5}\cap S_{6}$ instead.
The only non-trivial 1-boundary cohomology occurs on the boundary $\delta_{1}$
and is generated by the single form

$\displaystyle\check{\varphi}_{1}$ $\displaystyle=\varepsilon^{2}\
\delta_{1}\left(\text{d}\log\left(\frac{T_{1}|_{1}}{T_{3}|_{1}}\right)\wedge\text{d}\log\left(\frac{T_{3}|_{1}}{T_{2}|_{1}}\right)\right).$
(108)

This is nothing but the canonical form associated to the bounded chamber on
the boundary $\delta_{1}$. We have also normalized by $\varepsilon^{2}$ since
we expect the intersection of 2-boundary forms to be proportional to
$1/\varepsilon^{2}$. The 2-boundary cohomologies are generated by the
following 9 forms

$\displaystyle\begin{array}[]{ccc}\check{\varphi}_{2}=\varepsilon\
\delta_{12}\left(\text{d}\log\frac{T_{2}|_{12}}{T_{3}|_{12}}\right),&\check{\varphi}_{3}=\varepsilon\
\delta_{13}\left(\text{d}\log\frac{T_{1}|_{13}}{T_{2}|_{13}}\right),&\check{\varphi}_{4}=\varepsilon\
\delta_{14}\left(\text{d}\log\frac{T_{1}|_{14}}{T_{3}|_{14}}\right),\\\
\check{\varphi}_{5}=\varepsilon\
\delta_{15}\left(\text{d}\log\frac{T_{1}|_{15}}{T_{2}|_{15}}\right),&\check{\varphi}_{6}=\varepsilon\
\delta_{16}\left(\text{d}\log\frac{T_{2}|_{16}}{T_{3}|_{16}}\right),&\check{\varphi}_{7}=\varepsilon\
\delta_{26}\left(\text{d}\log\frac{T_{2}|_{26}}{T_{3}|_{26}}\right),\\\
\check{\varphi}_{8}=\varepsilon\
\delta_{35}\left(\text{d}\log\frac{T_{1}|_{35}}{T_{2}|_{35}}\right),&\check{\varphi}_{9}=\varepsilon\
\delta_{56}\left(\text{d}\log\frac{T_{1}|_{56}}{T_{2}|_{56}}\right),&\check{\varphi}_{10}=\varepsilon\
\delta_{56}\bigg{(}\text{d}\log T_{3}|_{56}\bigg{)}.\end{array}$ (112)

Note that the cohomology on all 2-boundaries is 1-dimensional except for the
$\delta_{56}$ boundary, which is 2-dimensional. The 2-boundary forms are
normalized by a single power of $\varepsilon$ since we expect the 2-boundary
intersection to be proportional to $1/\varepsilon$. Finally, the 3-boundary
cohomologies are generated by the following 16 forms

$\displaystyle\begin{array}[]{cccc}\check{\varphi}_{11}=\delta_{123}\left(1\right),&\check{\varphi}_{12}=\delta_{124}\left(1\right),&\check{\varphi}_{13}=\delta_{125}\left(1\right),&\check{\varphi}_{14}=\delta_{134}\left(1\right),\\\
\check{\varphi}_{15}=\delta_{136}\left(1\right),&{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}\check{\varphi}_{16}=\delta_{145}\left(1\right),}&\check{\varphi}_{17}=\delta_{146}\left(1\right),&\check{\varphi}_{18}=\delta_{156}\left(1\right),\\\
\check{\varphi}_{19}=\delta_{234}\left(1\right),&\check{\varphi}_{20}=\delta_{235}\left(1\right),&\check{\varphi}_{21}=\delta_{236}\left(1\right),&\check{\varphi}_{22}=\delta_{246}\left(1\right),\\\
\check{\varphi}_{23}=\delta_{256}\left(1\right),&\check{\varphi}_{24}=\delta_{345}\left(1\right),&\check{\varphi}_{25}=\delta_{356}\left(1\right),&\check{\varphi}_{26}=\delta_{456}\left(1\right).\end{array}$
(117)

Notice that there are some missing 3-boundaries
($\delta_{126},\delta_{135},\delta_{245},\delta_{346}$) since some triples of
the relative surfaces do not intersect at a point: $S_{1}\cap S_{2}\cap
S_{6}=S_{1}\cap S_{3}\cap S_{5}=S_{2}\cap S_{4}\cap S_{5}=S_{3}\cap S_{4}\cap
S_{6}=\emptyset$.

Next we provide a basis of $\text{d}\log$-forms orthonormal to the above dual
basis. A $\text{d}\log$-form dual to the 1-boundary form (108) is

$\displaystyle\varphi_{1}$ $\displaystyle=\frac{1}{3}\ \text{d}\log
S_{1}\wedge\text{d}\log\left(\frac{T_{1}}{T_{3}}\right)\wedge\text{d}\log\left(\frac{T_{3}}{T_{2}}\right).$
(118)

The normalization of $1/3$ ensures that the intersection of (118) with (108)
is unity. It is a simple exercise to see that this is indeed the correct
normalization. First, it is easy to see that $\varphi_{1}$ only has a non-
trivial overlap with $\check{\varphi}_{1}$. Their intersection
$\langle\check{\varphi}_{1}|\varphi_{1}\rangle$ is proportional to the self-
intersection of the canonical form corresponding to the chamber bounded on
$X\cap S_{1}$. Thus, there are three-intersection points contributing to the
intersection number. Since each $T_{i}$ has the same twist, we expect each
intersection point to contribute with the same factor. Hence, the
normalization of $1/3$.

A set of $\text{d}\log$-forms orthonormal to the 2-boundary forms (112) is

$\displaystyle\begin{array}[]{cc}\varphi_{2}=-\frac{1}{2}\ \text{d}\log
S_{1}\wedge\text{d}\log
S_{2}\wedge\text{d}\log\frac{T_{2}}{T_{3}},&\varphi_{3}=-\frac{1}{2}\
\text{d}\log S_{1}\wedge\text{d}\log
S_{3}\wedge\text{d}\log\frac{T_{1}}{T_{2}},\\\ \varphi_{4}=-\frac{1}{2}\
\text{d}\log S_{1}\wedge\text{d}\log
S_{4}\wedge\text{d}\log\frac{T_{1}}{T_{3}},&\varphi_{5}=-\frac{1}{2}\
\text{d}\log S_{1}\wedge\text{d}\log
S_{5}\wedge\text{d}\log\frac{T_{1}}{T_{2}},\\\ \varphi_{6}=-\frac{1}{2}\
\text{d}\log S_{1}\wedge\text{d}\log
S_{6}\wedge\text{d}\log\frac{T_{2}}{T_{3}},&\varphi_{7}=-\frac{1}{2}\
\text{d}\log S_{2}\wedge\text{d}\log
S_{6}\wedge\text{d}\log\frac{T_{2}}{T_{3}},\\\ \varphi_{8}=-\frac{1}{2}\
\text{d}\log S_{3}\wedge\text{d}\log
S_{5}\wedge\text{d}\log\frac{T_{1}}{T_{2}},&\varphi_{9}=-\frac{1}{2}\
\text{d}\log S_{5}\wedge\text{d}\log
S_{6}\wedge\text{d}\log\frac{T_{1}}{T_{2}},\\\ \varphi_{10}=-\frac{3}{2}\
\text{d}\log S_{5}\wedge\text{d}\log S_{6}\wedge\text{d}\log
T_{3}.&\end{array}$ (124)

Following a similar argument as that below (108), the normalization’s of these
forms are simple to fix. The relative sign difference between the
normalization of (124) and (118) comes from the fact that the dual twist has
exponent $-\varepsilon$ and for each boundary we lose a power of
$-\varepsilon$ in the intersection number. Furthermore, the reason why the
normalization of $\varphi_{10}$ differs from the rest is because it is the
only form where the point at infinity (on $X\cap S_{5}\cap S_{6}$) contributes
to its self-intersection.

Lastly, a set of $\text{d}\log$ forms orthonormal to the 3-boundary forms
(117) is

$\displaystyle\begin{matrix}[l]\varphi_{11}=\text{d}\log
S_{1}\wedge\text{d}\log S_{2}\wedge\text{d}\log
S_{3},\qquad\quad\,&\varphi_{12}=\text{d}\log S_{1}\wedge\text{d}\log
S_{2}\wedge\text{d}\log S_{4},\\\ \varphi_{13}=\text{d}\log
S_{1}\wedge\text{d}\log S_{2}\wedge\text{d}\log
S_{5},&\varphi_{14}=\text{d}\log S_{1}\wedge\text{d}\log
S_{3}\wedge\text{d}\log S_{4},\\\ \varphi_{15}=\text{d}\log
S_{1}\wedge\text{d}\log S_{3}\wedge\text{d}\log
S_{6},&{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color<EMAIL_ADDRESS>S_{1}\wedge\text{d}\log S_{4}\wedge\text{d}\log S_{5},}\\\
\varphi_{17}=\text{d}\log S_{1}\wedge\text{d}\log S_{4}\wedge\text{d}\log
S_{6},&\varphi_{18}=\text{d}\log S_{1}\wedge\text{d}\log
S_{5}\wedge\text{d}\log S_{6}-\varphi_{10},\\\ \varphi_{19}=\text{d}\log
S_{2}\wedge\text{d}\log S_{3}\wedge\text{d}\log
S_{4},&\varphi_{20}=\text{d}\log S_{2}\wedge\text{d}\log
S_{3}\wedge\text{d}\log S_{5},\\\ \varphi_{21}=\text{d}\log
S_{2}\wedge\text{d}\log S_{3}\wedge\text{d}\log
S_{6},&\varphi_{22}=\text{d}\log S_{2}\wedge\text{d}\log
S_{4}\wedge\text{d}\log S_{6},\\\ \varphi_{23}=\text{d}\log
S_{2}\wedge\text{d}\log S_{5}\wedge\text{d}\log
S_{6}-\varphi_{10},&\varphi_{24}=\text{d}\log S_{3}\wedge\text{d}\log
S_{4}\wedge\text{d}\log S_{5},\\\ \varphi_{25}=\text{d}\log
S_{3}\wedge\text{d}\log S_{5}\wedge\text{d}\log
S_{6}-\varphi_{10},&\varphi_{26}=\text{d}\log S_{4}\wedge\text{d}\log
S_{5}\wedge\text{d}\log S_{6}{-}\varphi_{10}.\end{matrix}$ (125)

To have a orthogonal basis, we have to subtract $\varphi_{10}$ from forms with
non-trivial $S_{5}\cap S_{6}$ residues since the intersections
$\langle\check{\varphi}_{10}|\text{d}\log S_{\bullet}\wedge\text{d}\log
S_{5}\wedge\text{d}\log S_{6}\rangle\sim\langle\text{d}\log
T_{3}|_{56}|\text{d}\log S_{\bullet}\rangle\neq 0$ are non-vanishing. This is
because both forms share a singularity at infinity on $X\cap S_{5}\cap S_{6}$
and why it is best to choose $\text{d}\log$-forms built out of ratios of the
twisted hyperplanes when constructing a dual basis.

Computing the dual differential equation using the method outlined in section
3.2 or 3.3 yields

$\displaystyle\boldsymbol{\check{\nabla}}\boldsymbol{\check{\varphi}}$
$\displaystyle=\varepsilon\
\underline{\boldsymbol{\check{A}}}\cdot\boldsymbol{\check{\varphi}},$ (126)
$\displaystyle\underline{\boldsymbol{\check{A}}}$
$\displaystyle=\sum\underline{\boldsymbol{\check{A}}}_{i}\ \text{d}\log
a_{i},$ (127)

where the $a_{i}$ are the letters in table 1 and the
$\underline{\boldsymbol{\check{A}}}_{i}$ are $26\times 26$ $\mathbb{Q}$-valued
matrices. The exact form of these coefficient matrices can be found in the
ancillary file (see figure 8 for the general structure or equation 192 for the
explicit result). Of course, since
$\langle\check{\varphi}_{a}|\varphi_{b}\rangle=\delta_{ab}$, we also have
$\nabla\varphi_{a}=\varepsilon\ A_{ab}\wedge\varphi_{b}$ where
$\underline{\boldsymbol{A}}=-(\check{\underline{\boldsymbol{A}}})^{\top}$.

$X_{1}+X_{2}+X_{3}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ | $X_{1}+Y_{1},$ | $X_{2}+X_{3}-Y_{1}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ | $X_{3}+Y_{2},$
---|---|---|---
$X_{1}+X_{2}-Y_{2}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ |  $X_{2}+Y_{1}+Y_{2},$ | $X_{1}+X_{3}-Y_{1}-Y_{2}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ | $X_{1}+X_{2}+Y_{2},$
${\color[rgb]{0.0600000000000001,0.46,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.0600000000000001,0.46,1}\pgfsys@color@cmyk@stroke{0.94}{0.54}{0}{0}\pgfsys@color@cmyk@fill{0.94}{0.54}{0}{0}X_{3}-Y_{2}}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ | $X_{2}+X_{3}+Y_{1},$ | ${\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color@cmyk@fill{0.91}{0}{0.88}{0.12}X_{1}-Y_{1}}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ | $X_{2}-Y_{1}-Y_{2}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$
$X_{3}-2Y_{1}-Y_{2}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ | $X_{2}-Y_{1}+Y_{2},$ | $X_{1}-Y_{1}-2Y_{2}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}+Y_{3}},$ | $X_{2}+Y_{1}-Y_{2},$
$X_{1}-Y_{1},$ |  $X_{3}-Y_{2},$ | ${\color[rgb]{0.72,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.72,0,0}\pgfsys@color@cmyk@stroke{0}{0.89}{0.94}{0.28}\pgfsys@color@cmyk@fill{0}{0.89}{0.94}{0.28}X_{2}+Y_{1}+Y_{2}}{\color[rgb]{.75,.75,.75}\definecolor[named]{pgfstrokecolor}{rgb}{.75,.75,.75}\pgfsys@color@gray@stroke{.75}\pgfsys@color@gray@fill{.75}-Y_{3}},$ | $X_{3}+2Y_{1}-Y_{2},$
| $X_{1}-Y_{1}+2Y_{2},$ | $X_{1}-X_{3}-Y_{1}+Y_{2}$ |
Table 1: The alphabet for the differential equation $a_{i}$. For the generic
arrangement, there are a total of 22 different letters. None of these letters
are singular in the $Y_{3}\to 0$ limit. The colored pairs of letters
correspond to letters that are to be identified in the $Y_{3}\to 0$ limit.
Thus, the physical DEQ has only 19 letters. The symbol alphabet of the de
Sitter three-site/five-point integral forms a subset of the physical 19
letters as expected ArkaniHamed2017 ; Hillman:2019wgh .

Staring from the generic arrangement we will illustrate how to degenerate
$\underline{\boldsymbol{\check{A}}}$ to the physical arrangement. This will
suggest a way to treat the physical case directly without deforming the
physical arrangement. Both methods have been checked to yield the same
resulting differential equation.

In the limit $Y_{3}\to 0$, the boundaries (or equivalently residues)
$\delta_{145},\delta_{146},\delta_{156}$ and $\delta_{456}$ are not
independent since the following intersection matrix does not have full rank

$\displaystyle\langle\check{\varphi}_{a}|\varphi_{b}\rangle=\begin{pmatrix}\,1&1&0&0\\\
\,1&1&0&0\\\ -1&0&1&0\\\ \,1&0&0&1\end{pmatrix}\quad\text{where}\quad
a,b\in\\{16,17,18,26\\}.$ (128)

Choosing to eliminate $\varphi_{16}$ and $\check{\varphi}_{16}$, yields

$\displaystyle\check{\varphi}_{16}=\check{\varphi}_{17}\implies\delta_{145}=\delta_{146},$
(129) $\displaystyle\varphi_{16}=\varphi_{17}-\varphi_{18}+\varphi_{26},$
(130)

The second equation is an equivalence among $\text{d}\log$-forms that can be
checked directly. On the other hand, the first equation tells us that the
residue operators $\text{Res}_{145}$ and $\text{Res}_{156}$ are equivalent and
that we should identify $\delta_{145}$ and $\delta_{156}$. Since all the
$S_{i}\cap S_{j}\cap S_{k}$ are the same for $i,j,k\in\\{1,4,5,6\\}$ one might
wonder why we do not identify all the $\delta_{ijk}$ with
$i,j,k\in\\{1,4,5,6\\}$. Loosely speaking, the different $\delta_{ijk}$ with
$i,j,k\in\\{1,4,5,6\\}$ represent the “directions” that one can approach the
degenerate boundary. Such degenerate configurations often need careful
treatment by blow-ups to resolve. Thankfully, the hyperplane arrangements
associated to cosmological correlators are sufficiently simple that we can get
by _without_ introducing additional mathematical formalism – one simply finds
the independent residues.

The physical DEQ

$\displaystyle\boldsymbol{\check{\varphi}}_{\text{phys}}$
$\displaystyle=(\check{\varphi}_{1},\dots,\check{\varphi}_{15},\check{\varphi}_{17},\dots,\check{\varphi}_{26})^{\top}~{},$
(131)
$\displaystyle\boldsymbol{\check{\nabla}}\boldsymbol{\check{\varphi}}_{\text{phys}}$
$\displaystyle=\varepsilon\
\underline{\boldsymbol{\check{A}}}_{\text{phys}}\cdot\boldsymbol{\check{\varphi}}_{\text{phys}}~{},$
(132)

is obtained from $\underline{\boldsymbol{\check{A}}}$ by eliminating the
$16^{\text{th}}$ row and adding the $16^{\text{th}}$ column to the
$17^{\text{th}}$ column as instructed by equation (129). For the explicit form
of $\underline{\boldsymbol{A}}_{\text{phys}}$, see the ancillary files or
equation (192). Figure 8 also provides a rough sketch of the structure of
$\underline{\boldsymbol{A}}_{\text{phys}}$ without using up a whole page. Once
again, a natural basis of the dual cohomology has lead to a remarkably sparse
differential equation! As a consistency check, we note that the symbol
alphabet for the dS wavefunction coefficient is a subset of our FRW symbol
alphabet (B), as expected ArkaniHamed2017 ; Hillman:2019wgh .

Figure 8: Matrix plot of the connection
$\underline{\boldsymbol{A}}_{\text{phys}}=-\underline{\boldsymbol{\check{A}}}_{\text{phys}}^{\top}$.
Each colored square on this $25\times 25$ grid represents a non-zero element
of $\underline{\boldsymbol{A}}_{\text{phys}}$. Note that this DEQ is very
sparse!

Equivalently, one could compute the physical DEQ without introducing $Y_{3}$
by making the identification $\delta_{145}=\delta_{156}$ at the start of the
calculation. To avoid missing potential boundary elements of the dual
cohomology, we recommend that one assumes that all boundaries are independent
and then check for relations among them. Complications from non-generic
arrangements are unavoidable and present in both the dual and FRW
cohomologies.

Once the $\underline{\boldsymbol{A}}_{\text{phys}}$ system has been
integrated, the three-site/five-point FRW integral (104) can be recovered from
the decomposition of its integrand

$\displaystyle\psi^{(0)}_{3,\text{FRW}}=\frac{1}{4Y_{1}Y_{2}}\int u\
\left(\varphi_{11}{-}\varphi_{13}{+}\varphi_{15}{-}\varphi_{18}{+}\varphi_{19}{-}\varphi_{20}{-}\varphi_{21}{+}\varphi_{22}{-}\varphi_{24}{+}\varphi_{26}\right).$
(133)

Since the dS wavefunction coefficient $\psi^{(0)}_{3,\text{dS}}$ has
transcendental weight three, the leading FRW ($\mathcal{O}(\varepsilon)$) term
must have transcendental weight four.

##  5 Discussions and future directions

The main goal of this work is to elucidate how (relative) twisted cohomology
and intersection theory can be used to study the analytic structure of FRW
integrals. In particular, we construct the differential equations obeyed by
families of FRW integrals. The techniques presented here provide a geometric
framework for understanding Euler/generalized hypergeometric integrals arising
from hyperplane arrangements.

Families of FRW integrals form a particularly simple subset of such integrals
where not all singularities are twisted. The un-twisted singularities
correspond to boundaries in the associated dual cohomology leading to a simple
geometric picture where dual forms are supported only on the boundaries. By
working in the dual space, we gain both conceptual and computational
advantages. In particular, integration-by-parts is much simpler in the dual
cohomology since all forms are restricted to boundaries.

After defining the proper dual cohomology, we can use the intersection number
– an inner product on the vector space of FRW integrals – to decompose any FRW
integral into a chosen basis. In particular, the intersection number can be
used to derive differential equations for a family of FRW integrals bypassing
traditional integration-by-parts algorithms. Intersection numbers have several
advantages over integration-by-parts algorithms. Most importantly, the
intersection number is an algebraic procedure given by a sequence of residues
localizing onto the maximal codimension intersection points of the FRW
hyperplane arrangement. The use of the intersection number for studying FRW
integrals is efficient and practical since the intersection number of
$\text{d}\log$-forms is easy to compute.

For various examples (tree-level two-site/four-point and three-site/five-point
integrals as well as the one-loop two-site/two-point integral), we have
constructed explicit representatives for a basis of the associated (relative)
twisted cohomology. Then, using dual integration-by-parts identities or
intersection numbers, we derived the corresponding differential equations. For
the tree-level two-site/four-point example, we have explicitly integrated the
differential equation to $\mathcal{O}(\varepsilon)$ and obtained the first
correction (83) to the known de Sitter result (82).

As already mentioned a few times throughout the text, finding/choosing a
“good” basis is often artisanal. Motivated by the geometry of the dual
cohomology, our choices for a good basis yielded simple intersection matrices
and sparse differential equations. Yet, from ltalk  it seems that one may be
able to choose a basis where the differential equation decomposes into
physical and un-physical sectors that can be integrated separately. It would
be interesting to further study the physical and mathematical principles
behind this basis choice and subsequent factorization of the differential
equations.

It would also be interesting to understand the cohomology in the de Sitter
limit $\varepsilon\to 0$ where the twist disappears. Such an understanding
would also be useful in the context of $\mathcal{N}=4$ SYM amplitudes where
one could hope to understand soft and collinear singularities cohomologically.
Similarly, it would be interesting to understand the un-twisted cohomology of
the $m=1$ amplituhedron, which has been identified as the complex of bounded
faces of a cyclic hyperplane arrangement in Karp:2016uax .

Another interesting question is whether the tools of intersection theory can
be utilised to help understand how the constraints of bulk unitarity are
encoded in cosmological correlators in a general FRW Universe (with a
boundary). It has been recently pointed out in Goodhew:2020hob ;
Goodhew:2021oqg ; Melville:2021lst ; Baumann:2021fxj  that unitarity imposes
specific constraints that inflationary correlators have to satisfy, closely
resembling the optical theorem in flat space. These systematic cutting rules
that cosmological correlators obey rely on very general properties of the
Green’s functions used to compute the wavefunction Baumann:2021fxj . As a
result, there lies no conceptual obstruction to generalizing these cutting
rules to FRW wavefunction coefficients. However, with the Green’s functions
being rather complicated in a given FRW spacetime, such generalizations could
be computationally cumbersome. Since intersection theory has also enhanced our
understanding of $d$-dimensional generalized unitarity in multi-loop Feynman
integrals Caron-Huot:2021iev ; Caron-Huot:2021xqj , it would be interesting to
see if such techniques simplify some aspects of the cosmological cutting rules
in a general FRW Universe, both at the level of trees and loops.

In this paper, we considered correlation functions of a simplistic, toy-model
FRW theory comprising solely of conformally coupled scalars with a cubic
interaction. In such a scenario, our new result for the leading $\varepsilon$
correction to the four-point function in dS (equation 83) could pave the way
for understanding distinctive signatures during inflation and beyond. Like the
analysis conducted in ArkaniHamed2015  where the four-point function in dS was
utilised to understand distinctive signatures of primordial non-Gaussianities
and also as a probe to predict new particles during inflation, the four-point
function for $\varepsilon\neq 0$ (equation 83) is perhaps a first step towards
understanding physics in an inflationary regime described by a quasi-dS period
or physics beyond the reheating surface. Moreover, correlators of conformally-
coupled scalars act as seeds from which correlators of massless/massive fields
with higher-spin exchanges can be deduced by acting with differential
operators Baumann:2019oyu . Thus, our analysis serves as an initial step
towards analysing physically relevant theories in FRW cosmologies and
facilitating concrete cosmological predictions.

## Acknowledgments

AP would like to thank Sebastian Mizera for pointing out the relevance of
hyperplane arrangements in the cosmological context, Nima Arkani-Hamed and
Aaron Hillman for stimulating discussions, as well as the Institute for
Advanced Study for its hospitality. SD and AP would also like to thank
Heliudson de Oliveira Bernardo, Claudia Fevola and Henrik Munch for useful
discussions. SD and AP are especially grateful to Anastasia Volovich and
Marcus Spradlin for valuable feedback throughout the project and help in
simplifying equation (83). This work was supported in part by the US
Department of Energy under contract DESC0010010 Task F (SD) and by Simons
Investigator Award $\\#376208$ (AP).

##  Appendix A Simple formulae for the intersection number of logarithmic
forms

Fortunately, FRW-forms and their duals are always logarithmic so that the
needed intersection numbers are easy to compute and purely combinatorial. To
avoid getting bogged down by unnecessary mathematical formalism, we provide
simple formulas to compute logarithmic intersection numbers without proof. The
interested reader is encouraged to consult matsumoto1998kforms ;
Mizera:2017rqa ; Mizera:2019gea ; Mizera:2019vvs ; Frellesvig:2019uqt ;
Weinzierl:2020xyy ; Chestnov:2022xsy ; Caron-Huot:2021iev ; Caron-Huot:2021xqj
and the references therein for more details.

To compute the intersection number, one first takes care of any $\delta_{I}$’s
via

$\displaystyle\bigg{\langle}\delta_{I}(\check{\phi}_{I})\bigg{|}\varphi\bigg{\rangle}=\bigg{\langle}\check{\phi}_{I}\bigg{|}\text{Res}_{I}\left[\frac{u}{u|_{I}}\varphi\right]\bigg{\rangle}\underset{\text{if
$\varphi$
$\text{d}\log$}}{=}\bigg{\langle}\check{\phi}_{I}\bigg{|}\text{Res}_{I}\left[\varphi\right]\bigg{\rangle}\equiv\langle\check{\phi}_{I}|\phi_{I}\rangle~{},$
(134)

where $\phi_{I}$ is a short had for $\text{Res}_{I}[\varphi]$. Since FRW-forms
have only simple poles on $S$, the ratio $u/u|_{I}$ is not needed and can be
set to unity in the second equality above. It is also important to note that
the residue is anti-symmetric with respect to the ordering of $I$. In our
conventions, we define $\text{Res}_{I}$ by
$\text{Res}_{I}[\bullet]=\text{Res}_{i_{|I|}}\cdots\text{Res}_{i_{2}}\text{Res}_{i_{1}}[\bullet]$.

After using up all the $\delta_{J}$’s, the remaining intersection number is an
intersection number on the submanifold $X_{I}=X\cap S_{I}$. We provide a
simple formula for its computation that localizes to the maximal codimension
intersection points of the twisted surfaces. The algorithm presented here was
first developed in matsumoto1998kforms  and we aim to provide a user friendly
summary of this work.

The first step is to identify maximal intersection points of the twisted
hyperplanes with themselves and the hyperplane at infinity ($H_{\infty}$):
$\mathbb{CP}^{n-|I|}\setminus(T\cap S_{I})\cup H_{\infty}$. Let us call this
set $\text{Int}_{I}$. For each intersection point, choose _normal_ coordinates
$\boldsymbol{z}^{*}$ such that the origin $\boldsymbol{0}$ corresponds to the
intersection point. Then, the intersection number of logarithmic forms is
given by

$\displaystyle\bigg{\langle}\check{\phi}_{I}\bigg{|}\text{Res}_{I}\left[\varphi\right]\bigg{\rangle}$
$\displaystyle=\sum_{\boldsymbol{z}^{*}\in\text{Int}_{I}}\frac{\text{Res}_{\boldsymbol{z}^{*}=\boldsymbol{0}}\Big{[}\check{\phi}_{I}\Big{]}}{\prod_{i=1}^{n-|I|}(\check{\alpha}_{i})}\text{Res}_{\boldsymbol{z}^{*}=\boldsymbol{0}}\Big{[}\text{Res}_{I}\left[\varphi\right]\Big{]}~{},$
(135)

where, near $\boldsymbol{z}^{*}=\boldsymbol{0}$,
$\check{\omega}_{I}|_{\boldsymbol{z}^{*}}=\sum_{i=1}^{n-|I|}\check{\alpha}_{i}\
\text{d}\log z^{*}_{i}+\mathcal{O}(\boldsymbol{z}^{*})$. For those familiar
with intersection theory, the first term in the product above is a stand-in
for the leading term of the maximal primitive for $\check{\phi}_{I}$.

This formula makes it obvious that the intersection number vanishes if
$\check{\phi}_{I}$ and $\text{Res}_{I}\left[\varphi\right]$ do not share a
singularity at any $\boldsymbol{z}^{*}\in\text{Int}_{I}$. For FRW integrals,
the $\check{\alpha}_{i}$ are simply multiples of $\varepsilon$ (intersection
points at infinity can introduce non-trivial multiples of $\varepsilon$).

###  A.1 Tree-level two-site example

As a simple example, we compute the intersection number
$\langle\check{\varphi}_{1}|\vartheta_{2}\rangle$. Using the $\delta_{3}$ in
$\check{\varphi}_{1}$, we find

$\displaystyle\langle\check{\varphi}_{1}|\vartheta_{2}\rangle$
$\displaystyle=\bigg{\langle}\varepsilon\
\text{d}\log\frac{T_{1}|_{3}}{T_{2}|_{3}}\bigg{|}\text{Res}_{3}\bigg{[}\text{d}\log\frac{T_{1}}{S_{2}}\wedge\text{d}\log\frac{S_{2}}{S_{3}}\bigg{]}\bigg{\rangle}=\varepsilon\
\bigg{\langle}\text{d}\log\frac{T_{1}|_{3}}{T_{2}|_{3}}\bigg{|}\text{d}\log\frac{T_{1}|_{3}}{S_{2}|_{3}}\bigg{\rangle}~{}.$
(136)

Supposing that we eliminated $x_{2}$ when taking the $\delta_{3}$ residue,
there are three twisted intersection points in $x_{1}$:
$\text{Int}=\\{0,-(X_{1}+X_{2}),\infty\\}$. The left and right forms in the
intersection number share singularities only at $x_{1}=0$. Thus, the
intersection number evaluates to

$\displaystyle\langle\check{\varphi}_{1}|\vartheta_{2}\rangle=\varepsilon\
\frac{\text{Res}_{x_{1}=0}\Big{[}\text{d}\log\frac{T_{1}|_{3}}{T_{2}|_{3}}\Big{]}}{-\varepsilon}\text{Res}_{x_{1}=0}\left[\text{d}\log\frac{T_{1}|_{3}}{S_{2}|_{3}}\right]=-1~{},$
(137)

since $\check{\omega}\sim-\varepsilon\ \text{d}\log z^{*}+\mathcal{O}(z^{*})$
near $z^{*}=x_{1}=0$.

###  A.2 Tree-level three-site example

In this section, we work through a more involved example: the intersection
number $\langle\check{\varphi}_{1}|\varphi_{1}\rangle$ in the context of the
tree-level three-site example. Taking care of the $\delta_{1}$ in
$\check{\varphi}_{1}$, yields

$\displaystyle\langle\check{\varphi}_{1}|\varphi_{1}\rangle$
$\displaystyle=\frac{\varepsilon^{2}}{3}\bigg{\langle}\text{d}\log\left(\frac{T_{1}|_{1}}{T_{3}|_{1}}\right)\wedge\text{d}\log\left(\frac{T_{3}|_{1}}{T_{2}|_{1}}\right)\bigg{|}\text{d}\log\left(\frac{T_{1}|_{1}}{T_{3}|_{1}}\right)\wedge\text{d}\log\left(\frac{T_{3}|_{1}}{T_{2}|_{1}}\right)\bigg{\rangle}\,.$
(138)

On $X\cap S_{1}$ there are three finite intersection points: $T_{i}|_{1}\cap
T_{j}|_{1}$ for $i,j\in\\{1,2,3\\}$ and $i\neq j$. Additionally, there are
possible intersection points with the plane at infinity $H_{\infty}$. Thus,
the set of intersection points are

$\displaystyle\text{Int}=\\{T_{1}|_{1}\cap T_{2}|_{1},\ T_{1}|_{1}\cap
T_{3}|_{1},\ T_{2}|_{1}\cap T_{3}|_{1},\ T_{2}|_{1}\cap H_{\infty},\
T_{1}|_{1}\cap H_{\infty},\ T_{3}|_{1}\cap H_{\infty}\\}~{}.$ (139)

Note that the intersection point $T_{3}|_{1}\cap H_{\infty}$ appears in
multiple charts of the $\mathbb{CP}^{2}$ and should only be counted once.

For the example at hand, only the finite intersection points will contribute
since the forms are non-singular on the hyperplanes at infinity. Then, for
each of these intersection points, we use the corresponding hyperplanes as the
normal coordinates $\boldsymbol{z}^{*}=\\{T_{i}|_{1},T_{j}|_{1}\\}$. In these
coordinates, the dual connection becomes $\check{\omega}|_{1}=-\varepsilon\
\text{d}\log z_{1}^{*}-\varepsilon\ \text{d}\log z_{2}^{*}+\mathcal{O}(1)$
near each of the finite intersection points. Using (135), the intersection
(138) becomes

$\displaystyle\langle\check{\varphi}_{1}|\varphi_{1}\rangle=\frac{\varepsilon^{3}}{3}\left(\frac{3}{(-\varepsilon)^{2}}\right)=1~{}.$
(140)

The factor of 3 simply comes from the fact that each of the three finite
intersection points contribute the same factor of $1/(-\varepsilon)^{2}$ to
the intersection number.

##  Appendix B Explicit presentation of the three-site/five-point differential
equation

In this appendix, we record the physical differential equation
$\underline{\boldsymbol{A}}_{\text{phys}}=-(\check{\underline{\boldsymbol{A}}}_{\text{phys}})^{\top}$.
Explicitly, the physical alphabet is

$\displaystyle a_{i=1,\dots,19}=\big{\\{}$ $\displaystyle
X_{1}{+}X_{2}{+}X_{3},\ X_{1}{-}Y_{1},\ X_{2}{+}X_{3}{-}Y_{1},\
X_{1}{+}Y_{1},\ X_{2}{+}X_{3}{+}Y_{1},\ X_{1}{-}Y_{1}{-}2Y_{2},\ $
$\displaystyle X_{1}{+}X_{2}{-}Y_{2},\ X_{3}{-}Y_{2},\
X_{3}{-}2Y_{1}{-}Y_{2},\ X_{2}{-}Y_{1}{-}Y_{2},\
X_{1}{+}X_{3}{-}Y_{1}{-}Y_{2},\ $ $\displaystyle X_{2}{+}Y_{1}{-}Y_{2},\
X_{3}{+}2Y_{1}{-}Y_{2},\ X_{1}{+}X_{2}{+}Y_{2},\ X_{3}{+}Y_{2},\
X_{2}{-}Y_{1}{+}Y_{2},\ $ $\displaystyle X_{1}{-}X_{3}{-}Y_{1}{+}Y_{2},\
X_{2}{+}Y_{1}{+}Y_{2},\ X_{1}{-}Y_{1}{+}2Y_{2}\big{\\}}~{}.$ (141)

Though a slight abuse of notation, we denote $\text{d}\log(a_{i})$ as $a_{i}$
below in order to get $\underline{\boldsymbol{A}}_{\text{phys}}$ to fit onto
the page

(192)

## References

  * (1) F. Cachazo, S. He and E. Y. Yuan, _Scattering equations and Kawai-Lewellen-Tye orthogonality_ , _Phys. Rev. D_ 90 (2014) 065001, [1306.6575].
  * (2) S. Mizera, _Scattering Amplitudes from Intersection Theory_ , _Phys. Rev. Lett._ 120 (2018) 141602, [1711.00469].