aainstitutetext:  Department of Physics, Brown University, Providence, RI
02912, USA bbinstitutetext:  Department of Physics and Astronomy, Uppsala
University, Box 516, 75120 Uppsala, Sweden

# A double copy from twisted (co)homology at genus one

Rishabh Bhardwaj<EMAIL_ADDRESS>a Andrzej Pokraka
<EMAIL_ADDRESS>a Lecheng Ren<EMAIL_ADDRESS>b Carlos
Rodriguez<EMAIL_ADDRESS>
###### Abstract

We study the twisted (co)homology of a family of genus-one integrals — the so
called Riemann-Wirtinger integrals. These integrals are closely related to
one-loop string amplitudes in chiral splitting where one leaves the loop-
momentum, modulus and all but one puncture un-integrated. While not actual
one-loop string integrals, they share many properties and are simple enough
that the associated twisted (co)homologies have been completely characterized
Goto2022 . Using intersection numbers — an inner product on the vector space
of allowed differential forms — we derive the Gauss-Manin connection for two
bases of the twisted cohomology providing an independent check of Mano2012 .
We also use the intersection index – an inner product on the vector space of
allowed contours – to derive a double-copy formula for complex genus-one
integrals over a torus. Similar to the celebrated KLT formula between open-
and closed-string tree-level amplitudes, the intersection indices form a
genus-one KLT kernel defining bilinears in meromorphic Riemann-Wirtinger
integrals that are equal to their complex counterparts.

††preprint: UUITP–37/23

##  1 Introduction

The double copy is a framework for computing closed-string/gravitational
amplitudes using simpler open-string/gauge theory amplitudes as input. This
framework has revealed important mathematical structures in field theory and
string theory as well as providing an efficient method for computing
scattering amplitudes that would otherwise be intractable. In particular, the
double copy is a bilinear map on open-string/gauge theory amplitudes that
makes the slogan $\text{gravity}=(\text{gauge theory})^{2}$ precise. This is
extremely surprising and useful because gravitational amplitudes are orders of
magnitude harder to compute than their gauge theory analogues.

The first instance of a double copy in the physics literature was discovered
by Kawai-Lewellen-Tye (KLT) Kawai:1985xq . These authors factorized tree-level
closed string scattering amplitudes into bilinears of tree-level open string
amplitudes, their complex conjugates and the so-called KLT kernel. In
parallel, similar relations appeared in the mathematics literature, more
specifically in Aomoto’s work on complex Selberg integrals Aomoto87 . In the
decades since the seminal work Kawai:1985xq , there has been a search for the
analogous statement at genus-one and higher Tourkine:2016bak ; He:2016mzd ;
Ochirov:2017jby ; Hohenegger:2017kqy ; Casali:2019ihm ; Casali:2020knc ;
Stieberger:2021daa ; Edison:2021ebi ; Stieberger:2022lss ; Stieberger:2023nol
.

While its origin may be in string theory, the double copy is perhaps best
understood in the field theory limit where there are several formulations
Adamo:2022dcm . In the field theory limit, the matrix elements of the KLT
kernel can be understood as color ordered amplitudes of bi-adjoint scalar
theory Cachazo:2013iea  or generalizations thereof Mizera:2016jhj ;
Mizera:2017cqs ; Chi:2021mio ; Chen:2022shl ; Chen:2023dcx .

In the scattering equation approach by Cachazo, He and Yuan (CHY)
Cachazo:2013hca ; Cachazo:2013gna ; Cachazo:2013iea , with a loop-level
extension in Geyer:2015jch ; Geyer:2015bja ; Geyer:2016wjx ; Geyer:2018xwu ;
Geyer:2021oox ; Geyer:2022cey , $n$-point color-ordered amplitudes at $L$
loops are represented by an integral over the $(n+2L)$-punctured Riemann
sphere. Here, the integrand is a numerator specific to the scattering process
divided by a product of $n$ differences between the punctures. In particular,
the integrand only ever has simple poles when two punctures collide. Moreover,
this integral representation of the amplitude localizes onto the locus of the
scattering equations turning the integral into a discrete sum. To obtain
gravity amplitudes in this formalism, one simply replaces the denominator of a
color-ordered CHY integrand with the numerator factor corresponding to another
color-ordered CHY integrand.

Additionally, the color-kinematics duality pioneered by Bern, Carrasco, and
Johansson (BCJ) Bern:2008qj ; Bern:2019prr ; Bern:2022wqg ; Adamo:2022dcm  has
been tested and used at loop-level in a wide range of theories and up to four-
loops in $\mathcal{N}\geq 4$ supergravities Bern:2011qn ; Bern:2013uka ;
Bern:2014sna . Schematically, the numerators of tree-level amplitudes or loop-
level integrands are organized into the product of a color factor and a
kinematic factor such that both the color and kinematic factors obey the
Jacobi identity satisfied by the color algebra. Then, the corresponding
gravity amplitude/integrand is obtained by replacing all color factors in the
color-ordered amplitude/integrand with another copy of the kinematic factors,
which can even belong to a different theory.

In the CHY and string theory formulation of the double copy one inevitably
encounters integrals of the generalized Euler-Mellin type. Twisted
(co)homology provides a general mathematical framework for studying such
integrals. In particular, it provides the necessary machinery to construct the
double copy for any family of Euler-Mellin integrals berkesch2014euler ;
Matsubara-Heo:2020lzo . In this language, the double copy of gravitational CHY
and tree-level closed string amplitudes follows from a generalization of the
well-known twisted Riemann bilinear relations hanamura1999hodge  (reviewed in
section 2).

The integrand of a one-loop string amplitude also takes the form of a
generalized Euler-Mellin integral on the torus. Thus, it is tempting to try
using twisted (co)homology to build the corresponding double copy. In fact,
this line of research was initiated in Casali:2019ihm  and Casali:2020knc .
Using twisted (co)homology, the authors derive the monodromy relations of
Tourkine:2016bak ; Hohenegger:2017kqy  for string integrals at genus-one and
higher. While the authors of Casali:2019ihm  comment on the prospect of
deriving the one-loop KLT kernel, the presence of unexpected, unphysical
cycles in the basis of homology and $B$-cycle monodromies that depend on the
other punctures complicate the analysis. The genus-one KLT formulae of
Stieberger:2022lss ; Stieberger:2023nol  derived from contour deformations
call for an understanding in terms of twisted (co)homology.

Generalized Euler-Mellin integrals also make a prominent appearance in the
computation of Feynman integrals/field theory amplitudes. In particular,
twisted (co)homology has found interesting applications to the study of CHY
amplitudes delaCruz:2017zqr ; Mizera:2017rqa ; Mizera:2019gea ;
Mazloumi:2022nvi , cosmological correlators De:2023xue  and Feynman integrals
(in particular the coaction Abreu:2019xep ; Britto:2021prf  and integral
reduction Mastrolia:2018uzb ; Mizera:2019ose ; Frellesvig:2019kgj ;
Frellesvig:2019uqt ; Mizera:2019vvs ; Frellesvig:2020qot ; Chen:2020uyk ;
Caron-Huot:2021xqj ; Caron-Huot:2021iev ; Chen:2022lzr ; Chen:2023kgw ).
Moreover, the study of elliptic Feynman integrals has boomed over the last
decade (see the review Bourjaily:2022bwx  and Giroux:2022wav ;
Frellesvig:2023iwr ; McLeod:2023qdf ; He:2023qld ; Gorges:2023zgv  for some
recent works). Since both the $\alpha^{\prime}$-expansion of one-loop string
integrands (pre $\tau$ and loop-momentum integration) and elliptic Feynman
integrals evaluate to the same class of functions — elliptic multiple
polylogarithms (eMPLs) Broedel:2014vla ; Broedel:2017jdo  — it is interesting
to explore one-loop string integrands and their connection to field theory
objects using tools from twisted (co)homology.

Motivated by the possibility of a KLT kernel for one-loop string integrands
from twisted (co)homology and aiming for a better understanding the function
space of genus-one integrals, we study the family of so-called Riemann-
Wirtinger integrals. This family of integrals is closely related to one-loop
string integrands and serves as a toy model for their twisted (co)homology,
which has been fully characterized mano2009 ; Mano2012 ; ghazouani2016moduli ;
Goto2022 . This allows us to present an explicit double copy formula for
Riemann-Wirtinger integrals with a KLT kernel built out of intersection
indices an inner product on the space of homology!

#### Double copy teaser:

The simplest example of this double copy or genus-one KLT formula is for $n=3$
punctures where the twisted (co)homology is 2-dimensional. Explicitly,

$\displaystyle\int_{M}\text{d}^{2}z_{1}\ e^{2\pi
is_{1A}(z_{1}-\bar{z}_{1})}\left|\frac{\vartheta_{1}(z_{12}|\tau)}{\vartheta_{1}(z_{13}|\tau)}\right|^{2s_{12}}\
\overline{F(z_{1a},\eta|\tau)}\ F(z_{1b},\eta|\tau)$ (1)
$\displaystyle\qquad=\frac{i}{2}\frac{\sin\pi s_{12}}{\sin\pi
s_{1A}}\begin{pmatrix}[\gamma_{2A}|\varphi_{b}\rangle&[\gamma_{23}|\varphi_{b}\rangle\end{pmatrix}\cdot\begin{pmatrix}0&e^{-i\pi(s_{1A}-s_{12})}\\\
-e^{i\pi(s_{1A}-s_{12})}&2i\sin\pi(s_{1A}-s_{12})\end{pmatrix}\cdot\begin{pmatrix}\overline{[\gamma_{2A}|\varphi_{a}\rangle}\\\
\overline{[\gamma_{23}|\varphi_{a}\rangle}\end{pmatrix}$

where

$\displaystyle[\gamma|\varphi_{a}\rangle=\int_{\gamma}\text{d}z_{1}\ e^{2\pi
is_{1A}z_{1}}\left(\frac{\vartheta_{1}(z_{12}|\tau)}{\vartheta_{1}(z_{13}|\tau)}\right)^{s_{12}}F(z_{1a},\eta|\tau)\,,$
(2)

$a,b\in\\{2,3\\}$ and the integration contours are
$\gamma_{2A}=\\{z_{1}\in[z_{2},z_{2}+1]\\}$ and
$\gamma_{23}=\\{z_{1}\in[z_{2},z_{3}]\\}$. We also set $z_{ij}=z_{i}-z_{j}$
and often fix $z_{2}$ as the origin: $z_{2}=0$. In the complex integral of
(1), $M$ is the fundamental parallelogram with the punctures removed, and the
measure is chosen such that $\int_{P}\text{d}^{2}z_{1}=1$. Here,
$\vartheta_{1}(z|\tau)$ is the odd Jacobi-theta function, $F(z,\eta|\tau)$ is
the Kronecker-Eisenstein series and $\\{s_{12},s_{1A}\\}$ are real numbers to
be thought of as Mandelstam invariants and $\eta\in\mathbb{C}$. The genus-one
KLT formula (1) is contingent on the combination

$\displaystyle-\eta+s_{1A}\tau-s_{12}z_{3}=s_{1B}\,.$ (3)

conspiring to a real constant $s_{1B}$ (we also generalize to complex
Mandelstams $s_{\bullet}\in\mathbb{C}$). To our knowledge, this is the first
genus-one KLT formula where the Mandelstam variables are generic and the
modulus can have arbitrary real part: $\mathrm{Re}(\tau)\neq 0$.

Similar double copy formulae will be given for the generalization of the
complex $z_{1}$-integral (1) to additional unintegrated punctures
$z_{4},z_{5},\cdots,z_{n}$. At $n$-points, the KLT kernel is an
$(n-1)\times(n-1)$ matrix whose entries are known rational function of the
monodromies $e^{2\pi is_{\bullet}}$.

#### Outline of paper:

In section 2, we review the formalism of twisted (co)homology and the
associated double copy in the context of tree-level string amplitudes. Then,
in section 3, we review one-loop string integrands and introduce the family of
Riemann-Wirtinger integrals. The twisted cohomology of Riemann-Wirtinger
integrals is constructed in section 4. Using the intersection number — an
inner product on cohomology — we verify the Gauss-Manin connection satisfied
by this family of integrals that was derived through independent means in
Mano2012  (section 4.3). These intersection calculation are interesting in
their own right and have potential application to the integral reduction of
hyperelliptic Feynman integrals Giroux:2022wav . Then, we study the
corresponding leading order solution in the $\alpha^{\prime}$ expansion
(section 4.4). The twisted homology is constructed in section 5. We compute
the intersection indices — an inner product on homology — for a basis of
homology in section 5.2. These intersection indices make up the double copy
kernel. We also provide boundary values for the differential equation of
section 5.3. The double copy of Riemann-Wirtinger integrals is constructed in
section 6 and verified numerically for both real (section 6.2) and complex
(section 6.3) Mandelstams. In 6.4 we write a version of the double copy of
Riemann-Wirtinger integrals that has nice modular properties, which looks like
a closed-integral after integrating out the loop momentum. Finally, in section
7, we discuss the possibility of defining Riemann-Wirtinger integrals where
two punctures are integrated. Understanding this is an important step towards
making this technology applicable to string integrands. We conclude in section
8 with a discussion and future directions.

##  2 Fundamentals of twisted (co)homology and the double copy

In this section, we give an overview of twisted (co)homology in the context of
a simple example: the tree-level four-point open string amplitude (or the
Euler beta function). We slowly build up the twisted (co)homology formalism
ending with the twisted Riemann bilinear relations. Then, we describe how
these relations can be generalized to produce the double copy and express the
tree-level four-point closed string amplitude as a square of the open string
amplitudes.

Historically, twisted (co)homology has been primarily developed to study
properties of hypergeometric functions; see the textbooks aomoto2011theory ;
yoshida2013hypergeometric  or deligne1986monodromy ; Matsumoto1994 ; cho1995 ;
kita_matsumoto_1997 ; Matsumoto1998 ; matsumoto1998kforms ; hanamura1999hodge
; mimachi2002intersection ; Mimachi2003 ; mimachi2004intersection ; Majima2000
; Goto2015 ; Matsumoto:2018aa ; goto2020homology ; Matsubara-Heo:2020lzo ;
MatsubaraHeo2023a  for a very incomplete list of research papers. Fortunately
for physicists, these authors are often concerned with explicit calculations
and many worked examples can be found in their writtings. Interestingly, the
twisted (co)homology version of the double copy was first discovered by
mathematicians studying Selberg integrals that appear in conformal field
theory correlation functions hanamura1999hodge ; mimachi2002intersection ;
Mimachi2003 ; mimachi2004intersection ; Matsubara-Heo:2020lzo . More recently,
the twisted (co)homology double copy was proved from a motivic perspective
Brown:2018omk ; Brown:2019jng . In particular, we want to test whether the
double copy prescription of hanamura1999hodge ; mimachi2002intersection ;
Mimachi2003 ; mimachi2004intersection ; Brown:2018omk ; Brown:2019jng ;
Matsubara-Heo:2020lzo  holds at higher genus and access the possible
application to one-loop closed string amplitudes.

###  2.1 A simple example

To understand the construction of twisted (co)homology and its relation to the
double copy, it is best to have a simple example in mind. To this end,
consider the tree-level massless four-point open string integral (Euler beta
function)

$\displaystyle\mathcal{A}^{\mathrm{open}}(s,t)\propto\int_{0}^{1}\text{d}z\
z^{s-1}(1-z)^{t-1}$ (4)

where we have used the $\text{SL}(2,\mathbb{C})$ gauge symmetry to fix three
of the punctures to $0,1$ and $\infty$ and are not concerned with the overall
constant kinematic factors and the momentum conserving delta function. It
satisfies interesting linear relations such as

$\displaystyle\begin{aligned}
\mathcal{A}^{\mathrm{open}}(s+1,t)=\frac{s}{s+t}\mathcal{A}^{\mathrm{open}}(s,t)\,,\qquad\text{and}\qquad\mathcal{A}^{\mathrm{open}}(s,t+1)=\frac{t}{s+t}\mathcal{A}^{\mathrm{open}}(s,t)\,,\end{aligned}$
(5)

as well as, the quadratic identity

$\displaystyle\mathcal{A}^{\mathrm{open}}(s,t)\mathcal{A}^{\mathrm{open}}(-s,-t)$
$\displaystyle=-\pi\left(\frac{1}{s}+\frac{1}{t}\right)\left(\frac{1}{\tan(\pi
s)}+\frac{1}{\tan(\pi t)}\right).$ (6)

The analogue of (5) in quantum field theory corresponds to the dimension
shifting identities and differential equations satisfied by dimensionally
regulated Feynman integrals. While Feynman integrals also satisfy quadratic
identities like (5) they are not particularly well studied in the literature.
The corresponding closed string amplitude is Mizera:2017cqs

$\displaystyle\mathcal{A}^{\mathrm{closed}}(s,t)\propto\int_{\mathbb{C}}|z|^{2s}|1-z|^{2t}\frac{\text{d}z\wedge\text{d}\bar{z}}{|z|^{2}|1-z|^{2}}=\frac{-\pi(s+t)^{2}\Gamma(s)\Gamma(t)\Gamma(-s-t)}{\Gamma(1-s)\Gamma(1-t)\Gamma(1+s+t)}\,.$
(7)

Again, we have dropped the proportionality factors and gauge fixed three of
the punctures. Of course it is well known that the integral (7) admits a
double copy

$\displaystyle\mathcal{A}^{\mathrm{closed}}(s,t)=\left(\frac{1}{\tan(\pi
s)}+\frac{1}{\tan(\pi
t)}\right)^{-1}\left(\mathcal{A}^{\mathrm{open}}(s,t)\right)^{2}\,.$ (8)

That is, gravity (closed string) amplitudes are the square of gauge (open
string) amplitudes.

###  2.2 Lightning review of twisted (co)homology

Twisted (co)homology provides a systematic formalism for understanding
relations such as (5), (6), (8) and more! Roughly speaking, twisted
(co)homology is the (co)homology theory for integrals whose integrands have
prescribed multiplicative monodromies. More explicitly, the theory of twisted
(co)homology applies to integrals of the form

$\displaystyle\int_{\gamma}u\ \varphi$ (9)

where $u$ is a universal multi-valued function called the twist, $\varphi$ is
a single-valued differential form on the space $M=E\setminus\\{u=0,\infty\\}$
and $\gamma$ is a contour on $M$. In this language, equation (4) becomes

$\displaystyle\mathcal{A}^{\mathrm{open}}(s,t)=\int_{0}^{1}u\ \varphi\,,$ (10)

with

$\displaystyle
u=z^{s}(1-z)^{t},\quad\varphi=\text{d}\log\frac{z}{1-z}\quad\text{and}\quad
M=\mathbb{CP}\setminus\\{0,1,\infty\\}\,.$ (11)

Note that the monodromies of the twist in this example are $e^{2\pi is}$ and
$e^{2\pi it}$ as $z$ winds around the origin and $z=1$ (there is also the
monodromy $e^{-2\pi i(s+t)}$ as $z$ winds about the point $\infty$). The so-
called _dual_ 111The adjective “dual” is a matter of convention and its
significance will become clear once we have introduced the intersection
pairings. local system,

$\displaystyle\check{\mathcal{L}}(s,t)=\mathbb{C}u,$ (12)

keeps track of the branch choices of $u$. For this reason, twisted
(co)homology is also known as (co)homology with coefficients in a local
system.

Importantly, the integral (9) is not well defined until a branch choice for
$u$ is made. Thus, twisted contours must store information associated to the
topological integration contour $\gamma$ as well as a branch choice
$u_{\gamma}$ on this contour. Explicitly, a twisted contour is the tensor
product $\gamma\otimes u_{\gamma}$ where $\otimes u_{\gamma}$ is the
aforementioned coefficient in the dual local system. Returning to our example,
we can now write the open-string four-point amplitude as an integral over a
twisted contour:

$\displaystyle\mathcal{A}^{\mathrm{open}}(s,t)=\int_{(0,1)\otimes
u}\varphi\,.$ (13)

While the differential form $\varphi$ is an ordinary differential form on $M$,
we will also call it a twisted differential form.222Technically, one should
work with $\mathcal{L}$-valued differential forms. However, since the local
system is a trivial line bundle, there exists a global trivialization and we
can work with ordinary complex differential forms without loss of information.

Even in the twisted setting, there exists a version of Stokes’ theorem
aomoto2011theory ; yoshida2013hypergeometric

$\displaystyle\int_{\gamma\otimes
u_{\gamma}}\nabla_{\omega}\varphi=\int_{\gamma}u_{\gamma}\nabla_{\omega}\varphi=\int_{\gamma}\text{d}(u_{\gamma}\
\varphi)=\int_{\partial\gamma}u_{\gamma}\
\varphi=\int_{\partial_{\omega}(\gamma\otimes u_{\gamma})}\varphi\;.$ (14)

Here, we have introduced the covariant derivative
$\nabla_{\omega}:=\text{d}+\omega\wedge$ where $\omega:=\text{d}\log(u)$ is a
flat connection $\omega\wedge\omega=0$ (this guarantees that the covariant
derivative is nilpotent: $\nabla_{\omega}\circ\nabla_{\omega}=0$). Since the
total derivative of the multi-valued integrand $\text{d}(u\ \varphi)$ is
equivalent to the covariant derivative of the single-valued differential form
$\nabla_{\omega}\varphi$, we can effectively “forget” about $u$ and work only
with single-valued differential forms by replacing the normal exterior
derivative with the covariant derivative.

The boundary of a twisted contour is defined to be the topological boundary of
the contour with each boundary component tensored with the corresponding
branch induced from the original branch choice. This is most easily
illustrated on a one-dimensional contour that has been smoothly triangulated.
Using the standard notation for $m$-simplicies, the (triangulated) path from
point $a$ to $b$ is denoted by $\langle ab\rangle$ and has boundary
$\partial\langle ab\rangle=\langle b\rangle-\langle a\rangle$. Then, the
boundary of the corresponding twisted contour $\langle ab\rangle\otimes
u_{\langle ab\rangle}$ is

$\displaystyle\partial_{\omega}\left(\langle ab\rangle\otimes u_{\langle
ab\rangle}(z)\right):=\langle b\rangle\otimes u_{\langle b\rangle}(b)-\langle
a\rangle\otimes u_{\langle a\rangle}(a)\,,$ (15)

where the branch choices $u_{\langle b\rangle}$ and $u_{\langle b\rangle}$ are
induced from $u_{\langle ab\rangle}$ by restriction. While the generalization
to higher dimensional simplices is straightforward, it will not be needed in
this work. The interested reader is encouraged to consult aomoto2011theory ;
yoshida2013hypergeometric  as well as mimachi2003intersection ;
mimachi2004intersection ; Mizera:2017cqs ; Casali:2019ihm ; Casali:2020knc ;
Duhr:2023bku ; Baune:2023uut .

In our running example, both the twisted contour and twisted differential form
are _closed_ (i.e., their respective image under $\partial_{\omega}$ or
$\nabla_{\omega}$ vanishes). The twisted contour in the definition of the beta
function is closed because $u$ vanishes at the end points

$\displaystyle\partial_{\omega}\left[(0,1)\otimes
u_{(0,1)}(z)\right]=\partial_{\omega}\left[\langle 01\rangle\otimes u_{\langle
01\rangle}(z)\right]=\langle 1\rangle\otimes u_{\langle 1\rangle}(1)-\langle
0\rangle\otimes u_{\langle 0\rangle}(0)=0\,.$ (16)

On the other hand, the twisted differential form in (13) is closed because it
is closed in the usual sense (has no image under d) and has vanishing wedge
product with $\omega$:

$\displaystyle\nabla_{\omega}\left(\text{d}\log\frac{z}{1-z}\right)=\text{d}^{2}\log\frac{z}{1-z}+\omega\wedge\text{d}\log\frac{z}{1-z}=0\,.$
(17)

Closed twisted contours and differential forms are called twisted _cycles_ and
_cocycles_ respectively. Intuitively, cycles are contours with no boundary.
Unfortunately, cocycles do not have a simple intuitive interpretation — they
are simply differential forms that vanish when (covariantly) differentiated.

While twisted cycles and cocycles are the “interesting” twisted
contours/differential forms on $M$, they are not uniquely defined. The
integral of a twisted cocycle ($\varphi$) over a twisted cycle ($\gamma\otimes
u_{\gamma}$) does not change when the twisted cycle (cocycle) is shifted by an
_exact_ twisted contour (differential form). Here, an exact twisted contour is
the image of the boundary operator ($\partial_{\omega}(\delta\otimes
u_{\delta})$) and an exact twisted differential form is the image of the
covariant derivative ($\nabla_{\omega}\psi$). Using Stokes theorem, it is easy
to verify that integrals over twisted (co)cycles are indeed invariant under
such shifts

$\displaystyle\int_{\gamma\otimes
u_{\gamma}}\left(\varphi+\nabla_{\omega}\psi\right)$
$\displaystyle=\int_{\gamma\otimes
u_{\gamma}}\varphi+\int_{\partial_{\omega}(\gamma\otimes
u_{\gamma})}\psi=\int_{\gamma\otimes u_{\gamma}}\varphi\,,$ (18)
$\displaystyle\int_{\gamma\otimes u_{\gamma}+\partial_{\omega}(\delta\otimes
u_{\delta})}\varphi$ $\displaystyle=\int_{\gamma\otimes
u_{\gamma}}\varphi+\int_{\delta\otimes
u_{\delta}}\nabla_{\omega}\varphi=\int_{\gamma\otimes u_{\gamma}}\varphi\,,$
(19)

since $\partial_{\omega}(\gamma\otimes u_{\gamma})=0=\nabla_{\omega}\varphi$
by definition. To remove the above redundancy in defining twisted (co)cycles,
it is useful to define the twisted (co)homology and work with equivalence
classes of (co)cycles.

The $p$-th twisted homology $H_{p}$ group is the space of closed twisted
$p$-contours modulo exact twisted $(p+1)$-contours

$\displaystyle H_{p}(M,\check{\mathcal{L}}_{\omega})$
$\displaystyle=\frac{\ker\left(\partial_{\omega}:C_{p}(M,\check{\mathcal{L}}_{\omega})\to
C_{p-1}(M,\check{\mathcal{L}}_{\omega})\right)}{\text{im}\left(\partial_{\omega}:C_{p+1}(M,\check{\mathcal{L}}_{\omega})\to
C_{p}(M,\check{\mathcal{L}}_{\omega})\right)}\,,$ (20)

where $C_{p}(M,\check{\mathcal{L}}_{\omega})$ is the space of twisted
$p$-contours. Its elements (twisted homology classes) are equivalence classes
of twisted cycles and are denoted by a square bra. Explicitly, given a twisted
$p$-cycle $\gamma\otimes u_{\gamma}\in C_{p}(M,\check{\mathcal{L}}_{\omega})$,
its twisted homology class is

$\displaystyle[\gamma\otimes u_{\gamma}|=\\{\gamma\otimes
u_{\gamma}+\partial_{\omega}(\beta\otimes
u_{\beta})\,|\,\forall\,(\beta\otimes u_{\beta})\in
C_{p+1}(M,\check{\mathcal{L}}_{\omega})\\}\,.$ (21)

When two twisted $p$-cycles, $\gamma\otimes u_{\gamma}$ and
$\gamma^{\prime}\otimes u_{\gamma^{\prime}}$, differ by the boundary of a
twisted $(p+1)$-contour $\beta\otimes u_{\beta}$,

$\displaystyle\gamma\otimes u_{\gamma}\simeq\gamma^{\prime}\otimes
u_{\gamma^{\prime}}\iff\gamma\otimes u_{\gamma}-\gamma^{\prime}\otimes
u_{\gamma^{\prime}}=\partial_{\omega}(\beta\otimes u_{\beta})\,,$ (22)

we say that they are homologous or equivalent in homology.

Similarly, the $p$-th twisted cohomology $H^{p}$ group is the space of closed
twisted $p$-forms modulo exact twisted $(p-1)$-forms

$\displaystyle H^{p}(M,\nabla_{\omega})$
$\displaystyle=\frac{\ker\left(\nabla_{\omega}:\Omega^{p}(M,\nabla_{\omega})\to\Omega^{p+1}(M,\nabla_{\omega})\right)}{\text{im}\left(\nabla_{\omega}:\Omega^{p-1}(M,\nabla_{\omega})\to\Omega^{p}(M,\nabla_{\omega})\right)}\,,$
(23)

where $\Omega^{p}(M,\nabla_{\omega})$ is the space of twisted $p$-forms. Its
elements (twisted cohomology classes) are equivalence classes of twisted
cocycles and are denoted by an angle ket. Explicitly, given a twisted
$p$-cocycle
$\varphi\in\ker(\nabla_{\omega}:\Omega^{p}(M,\nabla_{\omega})\to\Omega^{p+1}(M,\nabla_{\omega}))$,
its twisted cohomology class is

$\displaystyle|\varphi\rangle=\\{\varphi+\nabla_{\omega}\psi\,|\,\forall\,\psi\in\Omega^{p-1}(M,\nabla_{\omega})\\}\,.$
(24)

When two twisted $p$-cocycles, $\varphi$ and $\varphi^{\prime}$, differ by the
covariant derivative of a twisted $(p-1)$-form $\nabla_{\omega}\psi$,

$\displaystyle\varphi\simeq\varphi^{\prime}\iff\varphi-\varphi^{\prime}=\nabla_{\omega}\psi\,,$
(25)

we say that $\varphi$ and $\varphi^{\prime}$ are cohomologous or equivalent in
cohomology. In the physics literature, shifting a differential form by a total
derivative is commonly known as integration-by-parts and is the standard
method for reducing integrals that appear in the calculation of multi-loop
perturbative amplitudes to a minimal basis of so-called master integrals.
Developing fast integration-by-parts computer algorithms and researching
alternative integral reduction methods is currently an active area of research
Klappert:2020nbg ; Barakat:2022qlc ; Smirnov:2023yhb ; Wu:2023upw ;
Heinrich:2023til  since integral reduction is essential to precision
phenomenology. The intersection number, introduced below, is one of the most
promising alternatives to traditional integration-by-parts algorithms being
developed Mastrolia:2018uzb ; Mizera:2019ose ; Frellesvig:2019kgj ;
Frellesvig:2019uqt ; Mizera:2019vvs ; Chen:2020uyk ; Frellesvig:2020qot ;
Caron-Huot:2021xqj ; Caron-Huot:2021iev ; Chen:2022lzr ; Chen:2023kgw .

An important fact about twisted (co)homology is that only the middle degree
(co)homology is non-trivial

$\displaystyle
H_{p}(M,\check{\mathcal{L}}_{\omega})=0=H^{p}(M,\nabla_{\omega})\quad\text{if}\quad
p\neq n\,,$ (26)

where $n$ is the complex dimension of $M$ ($n=\dim_{\mathbb{C}}M$) or
equivalently half the real dimension of $M$
($n=\frac{1}{2}\dim_{\mathbb{R}}M$). The dimensions of the middle (co)homology
groups is given by the Euler characteristic of $M$ aomoto2011theory 333In the
context of Feynman integrals, it is well known that the Euler characteristic
gives the number of independent integrals (dimension of the cohomology)
Bitoun:2018afx .

$\displaystyle\dim H_{n}(M,\check{\mathcal{L}}_{\omega})=\dim
H^{n}(M,\nabla_{\omega})=|\chi(M)|\,.$ (27)

Another important fact is that twisted (co)homology groups are actually vector
spaces (existence of an inner product structure)!

###  2.3 Pairings

Figure 1: Diagram detailing the connections between twisted (co)homology
groups and their duals. The  period pairings (integration) are denoted by the
vertical arrows, while the  intersection pairings are denoted by the
horizontal arrows. Poincaré duality is indicated by the diagonal arrows. We
have also included some subscripts and superscripts for the experts and can be
safely ignored by readers unfamiliar with twisted (co)homology.

Now that we have defined the twisted (co)homology, we can interpret
integration as a pairing between homology and cohomology

$\displaystyle[\bullet|\bullet\rangle:H_{p}(M,\check{\mathcal{L}}_{\omega})\times
H^{p}(M,\nabla_{\omega})\to\mathbb{C}\qquad\text{where}\qquad[\gamma\otimes
u_{\gamma}|\varphi\rangle:=\int_{\gamma\otimes u_{\gamma}}\varphi\,.$ (28)

Formally, this pairing is know as the _period_ pairing. In particular, we can
now think of the four-point open-string amplitude (4) as a twisted period

$\displaystyle\mathcal{A}^{\mathrm{open}}(s,t)=\bigg{[}(0,1)\otimes
z^{a}(1-z)^{b}\ \bigg{|}\ \text{d}\log\frac{z}{1-z}\bigg{\rangle}\,.$ (29)

However, it turns out that we can construct other pairings that involve only
homology or only cohomology: the _intersection index_ 444Here, we are
following the naming convention of Pham hwa1966homology ;
pham2011singularities . Often the intersection index is called the homological
intersection number. We prefer intersection index because it is more succinct.
and _intersection number_. While it may seem unnatural to try and pair cycles
with cycles and cocycles with cocycles, these pairings define inner products
on the vector space of homology and cohomology. Using the intersection
pairings we can project any twisted (co)cycle onto a basis of (co)homology. In
particular, this provides a systematic way of discovering linear relations
between integrals. Moreover, both intersection pairings are essential for
constructing a double copy.

The intersection pairing pairs the twisted (co)homology with an associated
dual twisted (co)homology. The dual twisted (co)homology is simply defined to
be the twisted (co)homology associated to the inverse twist:
$\check{u}:=u^{-1}$:

$\displaystyle[H_{p}(M,\check{\mathcal{L}}_{\omega})]^{\vee}:=H_{p}(M,\mathcal{L}_{\omega})\qquad\text{and}\qquad[H^{p}(M,\nabla_{\omega})]^{\vee}:=H^{p}(M,\check{\nabla}_{\omega})$
(30)

where $\mathcal{L}_{\omega}=\mathbb{C}u^{-1}$ is the local system and
$\check{\nabla}_{\omega}=\nabla_{-\omega}=\text{d}-\omega\wedge$ is the dual
covariant derivative. Dual integrals or dual periods correspond to the
following pairing

$\displaystyle\langle\bullet|\bullet]:H^{p}(M,\check{\nabla}_{\omega})\times
H_{p}(M,\mathcal{L}_{\omega})\to\mathbb{C}\qquad\text{where}\qquad\langle\check{\varphi}|\check{\gamma}\otimes\check{u}]:=\int_{\check{\gamma}\otimes\check{u}_{\check{\gamma}}}\check{\varphi}\,,$
(31)

where $|\check{\gamma}\otimes\check{u}_{\check{\gamma}}]$ and
$\langle\check{\varphi}|$ denote dual homology and cohomology classes. For
example, the integral $\mathcal{A}^{\mathrm{open}}(-s,-t)$ corresponds to the
dual period

$\displaystyle\mathcal{A}^{\mathrm{open}}(-s,-t)=\bigg{\langle}\text{d}\log\frac{z}{1-z}\bigg{|}(0,1)\otimes
z^{-s}(1-z)^{-t}\Bigg{]}\,.$ (32)

From Poincaré duality aomoto2011theory ; pham2011singularities  (see figure
1), the (normal) twisted cohomology is isomorphic to the dual homology
$H^{p}(M,\nabla_{\omega})\cong H_{n-p}(M,\mathcal{L}_{\omega})$ and the
(normal) twisted homology is isomorphic to the dual cohomology
$H_{p}(M,\check{\mathcal{L}}_{\omega})\cong
H^{n-p}(M,\check{\nabla}_{\omega})$. Moreover, the _dual_ local system
$\check{\mathcal{L}}_{\omega}$ corresponds to the functions that are
annihilated by the _dual_ covariant derivative
($\check{\nabla}(\mathbb{C}u)=0$) while the local system
$\mathcal{L}_{\omega}$ corresponds to the functions that are annihilated by
the covariant derivative ($\nabla(\mathbb{C}u^{-1})=0$). This is why the dual
local system appears in the (normal) twisted homology.

Now that the dual (co)homology has been defined, we can introduce the
intersection pairings

$\displaystyle\begin{aligned} :&\ H_{p}(M,\check{\mathcal{L}}_{\omega})\times
H_{n-p}(M,\mathcal{L}_{\omega})\to\mathbb{C}\\\
&\quad\text{by}\quad[\gamma\otimes
u_{\gamma}|\check{\gamma}\otimes\check{u}_{\check{\gamma}}]:=\sum_{x\in\gamma\cap\check{\gamma}}\left(u_{\gamma}\check{u}_{\check{\gamma}}|_{x}\right)\big{[}\mathrm{Reg}[\gamma]|\check{\gamma}\big{]}^{\text{top}}_{x}\,,\end{aligned}$
(33)

and

$\displaystyle\langle\bullet|\bullet\rangle:$ $\displaystyle\
H^{n-p}(M,\check{\nabla}_{\omega})\times
H^{p}(M,\nabla_{\omega})\to\mathbb{C}\quad\text{by}\quad\langle\check{\varphi}|\varphi\rangle:=\int_{M}\mathrm{Reg}[\check{\varphi}]\wedge\varphi\,.$
(34)

In (33) and (34), the dual cycle or dual cocycle must be regularized. In
particular, they must be replaced by a (co)homologous (co)cycle that has
compact support. For cycles, this regularization procedure introduces rational
functions of the monodromies (i.e., $e^{2\pi is}$ and $e^{2\pi it}$). On the
other hand, the regularization of dual cocycles introduces rational functions
of the exponents in the twist (i.e., $s$ and $t$). We will postpone the
explicit recipe for the regularization procedure to sections 4 and 5.

In definition (33), the set $\gamma\cap\check{\gamma}$ corresponds to the set
of all points where $\gamma$ and $\check{\gamma}$ intersect. Here,
$[\bullet|\bullet]^{\text{top}}_{x}$ is the topological intersection number at
$x$ that evaluates to $\pm 1$ depending on the relative orientation of
$\gamma$ and $\check{\gamma}$. Additionally, the factor
$(u_{\gamma}\check{u}_{\check{\gamma}}|_{x})$ simply evaluates to a phase
since $\check{u}=u^{-1}$. Thus, we see that the intersection index (33) counts
the number of times a (regularized) twisted cycle and dual cycle intersects
(with sign) weighted by rational functions of the monodromies. Similarly, the
intersection number (34) counts the number of overlapping singularities
between twisted forms and dual forms weighted by rational functions of the
exponents in the twist. While not obvious without knowing the regularization
procedure, the integral in (34) is simple to compute and always evaluates to a
sequence of residues!

Using the intersection pairings as inner products on (co)homology, we have the
following decompositions of identity

$\displaystyle\mathds{1}=\left|\check{\gamma}_{i}\otimes\check{u}_{\check{\gamma}_{i}}\right]H_{ij}^{-1}\left[\gamma_{j}\otimes
u_{\gamma_{j}}\right|\qquad\text{and}\qquad\mathds{1}=\left|\varphi_{i}\right\rangle
C_{ij}^{-1}\left\langle\check{\varphi}_{j}\right|\,.$ (35)

Here, $H_{ij}=[\gamma_{i}\otimes u_{\gamma_{i}}|\check{\gamma}_{j}\otimes
u_{\check{\gamma}_{j}}]$ is the homology intersection matrix corresponding to
the bases $\\{[\gamma_{i}\otimes u_{\gamma_{i}}|\\}$ and
$\\{|\check{\gamma}_{i}\otimes u_{\check{\gamma}_{i}}]\\}$. Similarly,
$C_{ij}=\langle\check{\varphi}_{i}|\varphi_{j}\rangle$ is the cohomology
intersection matrix corresponding to the bases $\\{|\varphi_{i}\rangle\\}$ and
$\\{\langle\check{\varphi}_{i}|\\}$. Using these formula, any (co)cycle can be
projected onto a basis.

In our example of the beta function, only the first (co)homology is non-
trivial. Moreover, it is a one dimensional vector space since $|\chi(M)|=1$.
Therefore, we can choose the bases of (co)homology and dual (co)homology to be

$\displaystyle
H_{1}(M,\check{\mathcal{L}}_{\omega}){=}\text{Span}\left\\{\left[(0,1)\otimes
z^{s}(1-z)^{t}\right|\right\\}\,,\quad
H^{1}(M,\nabla_{\omega}){=}\text{Span}\left\\{\left|\text{d}\log\frac{z}{1-z}\right\rangle\right\\}\,,$
(36) $\displaystyle
H_{1}(M,\mathcal{L}_{\omega}){=}\text{Span}\left\\{\left|(0,1)\otimes
z^{-s}(1-z)^{-t}\right]\right\\}\,,\quad
H^{1}(M,\check{\nabla}_{\omega}){=}\text{Span}\left\\{\left\langle\text{d}\log\frac{z}{1-z}\right|\right\\}\,.$
(37)

Note that one can choose the same topological cycles and differential forms
for the dual bases since the underlying topology of $M$ is the same. With
these choices, equation (35) becomes

$\displaystyle\mathds{1}=\left|(0,1)\otimes
z^{-s}(1-z)^{-t}\right]H^{-1}\left[(0,1)\otimes z^{s}(1-z)^{t}\right|$ (38)

and

$\displaystyle\mathds{1}=\left|\text{d}\log\frac{z}{1-z}\right\rangle
C^{-1}\left\langle\text{d}\log\frac{z}{1-z}\right|\,,$ (39)

where

$\displaystyle H=\frac{i}{2}\left(\frac{1}{\tan(\pi s)}+\frac{1}{\tan(\pi
t)}\right)\,,\qquad\text{and}\qquad C=(2\pi i)\frac{s+t}{st}\,.$ (40)

Now, we can understand the origin of the first equation of (5) by acting with
the decomposition of identity on $|z\ \text{d}\log\frac{z}{1-z}\rangle$:

$\displaystyle\bigg{|}z\
\text{d}\log\frac{z}{1-z}\bigg{\rangle}=\frac{s}{s+t}\bigg{|}\text{d}\log\frac{z}{1-z}\bigg{\rangle}\,,$
(41)

where $\langle z\
\text{d}\log\frac{z}{1-z}|\text{d}\log\frac{z}{1-z}\rangle=\frac{s}{s+t}C$. We
also remark that the leading term in the $\alpha^{\prime}$ expansion of $H$
corresponds to a four-point bi-adjoint scalar amplitude
($s,t\propto\alpha^{\prime})$.

###  2.4 Quadratic identities and the double copy

Having introduced all the pieces featured in (5) (equations (29), (32) and
(40)), we turn our attention to the quadratic identity (6). This identity
follows from the twisted Riemann bilinear relations (also known as twisted
period relations)

$\displaystyle\underline{\boldsymbol{C}}=\left(\check{\underline{\boldsymbol{P}}}\cdot\underline{\boldsymbol{H}}^{-1}\cdot\underline{\boldsymbol{P}}\right)^{\top}\qquad\text{or}\qquad\underline{\boldsymbol{H}}=\underline{\boldsymbol{P}}\cdot\underline{\boldsymbol{C}}^{-1}\cdot\check{\underline{\boldsymbol{P}}}\,.$
(42)

Here, $\underline{\boldsymbol{P}}$ and $\check{\underline{\boldsymbol{P}}}$
are the twisted period matrices for a given basis choice of the twisted
(co)homology and dual (co)homology,

$\displaystyle P_{ij}=\big{[}\gamma_{i}\otimes
u_{\gamma_{i}}\big{|}\varphi_{j}\big{\rangle}\qquad\text{and}\qquad\check{P}_{ij}=\big{\langle}\check{\varphi}_{i}\big{|}\check{\gamma}_{j}\otimes
u_{\check{\gamma}_{j}}\big{]}\,.$ (43)

Now, while (6) and (42) are not quite what we mean by a double copy they are
only one step away. In the remainder of this section we will describe how to
generalize equation (42) and obtain the double copy (8).

The closed string amplitude (7) already looks like a double copy since its
integrand is simply two copies of the same thing (with one complex conjugated)

$\displaystyle\mathcal{A}^{\mathrm{closed}}(s,t)=\int_{\mathbb{C}}\overline{(u\
\varphi)}\wedge(u\ \varphi)=\int_{\mathbb{C}}|u|^{2}\
\bar{\varphi}\wedge\varphi$ (44)

where $|u|^{2}=|z|^{2s}|1-z|^{2t}$ and $\varphi=\text{d}\log\frac{z}{1-z}$. It
also looks like the intersection number (34) where we use the complex
conjugate of $\varphi$ instead of the regulated dual form. That is, we can
interpret $\mathcal{A}^{\mathrm{closed}}$ as the pairing

$\displaystyle\langle\bar{\bullet}|\bullet\rangle:H^{n}(\bar{M},\nabla_{\bar{\omega}})\times
H^{n}(M,\nabla_{\omega})\to\mathbb{C}\quad\text{by}\quad\langle\bar{\varphi}_{a}|\varphi_{b}\rangle:=\int_{M}|u|^{2}\
\bar{\varphi}_{a}\wedge\varphi_{b}\,.$ (45)

Note that the complex conjugated dual local system has the same monodromies as
the local system for real exponents ($s,t\in\mathbb{R}$). For example,
$\bar{u}\to\overline{e^{2\pi is}}\bar{u}=e^{-2\pi is}\bar{u}$ and $u^{-1}\to
e^{-2\pi is}u^{-1}$ as $z$ winds around the origin. Thus, the local system is
isomorphic to the complex conjugate of the dual local system:

$\displaystyle\mathbb{C}u^{-1}=:\mathcal{L}_{\omega}(s,t)\cong\overline{\check{\mathcal{L}}_{\omega}(s,t)}:=\mathbb{C}\bar{u}$
(46)

for real exponents $s$ and $t$. Replacing dual (co)cycles by complex
conjugated (co)cycles in the Riemann bilinear relations yields the _double
copy_

$\displaystyle\boxed{\langle\bar{\varphi}_{a}|\varphi_{b}\rangle:=\int_{M}|u|^{2}\
\bar{\varphi}_{a}\wedge\varphi_{b}=\left(\
\overline{\underline{\boldsymbol{P}}}\cdot\underline{\boldsymbol{H}}^{-1}\cdot\underline{\boldsymbol{P}}\
\right)_{ba}}\,.$ (47)

Note that since the
$\overline{\check{\mathcal{L}}_{\omega}}\cong\mathcal{L}_{\omega}$, we do not
need to recompute the intersection matrix $\underline{\boldsymbol{H}}$.
Moreover, since the intersection matrix $\underline{\boldsymbol{H}}$ is
meromorphic in the exponents of the twist, we can take these exponents to be
complex. We can check that the above prescription indeed reproduces equation
(8)

$\displaystyle\bar{P}H^{-1}P$ $\displaystyle=\left(\frac{1}{\tan(\pi
s)}+\frac{1}{\tan(\pi
t)}\right)^{-1}\left(\mathcal{A}^{\mathrm{open}}(s,t)\right)^{2}\,,$
$\displaystyle=\frac{-\pi(s+t)^{2}\Gamma(s)\Gamma(t)\Gamma(-s-t)}{\Gamma(1-s)\Gamma(1-t)\Gamma(1+s+t)}\,,$
$\displaystyle=\int_{\mathbb{C}}|z|^{2s}|1-z|^{2t}\frac{\text{d}z\wedge\text{d}\bar{z}}{|z|^{2}|1-z|^{2}}=\mathcal{A}^{\mathrm{closed}}(s,t)\,.$
(48)

Note that the double copy (47) (and (2.4)) are manifestly single-valued. Thus,
the double copy procedure is a mechanism to generate interesting single-valued
integrals.

The main goal of this work is to explore the structure of twisted cohomology
at higher genus and test if the double copy prescription of Brown:2019jng
holds. For one-loop string amplitudes, the KLT kernel has been long sought
after Stieberger:2014hba ; Tourkine:2016bak ; Stieberger:2016xhs ;
Ochirov:2017jby ; Hohenegger:2017kqy ; Casali:2019ihm ; Casali:2020knc ;
Stieberger:2021daa ; Stieberger:2022lss  culminating in Stieberger:2022lss ;
Stieberger:2023nol . Our goal is to asses whether twisted (co)homology is
indeed the correct formalism to compute KLT kernels for higher genus string
amplitudes. Our hope is that twisted (co)homology could simplify the procedure
of determining the KLT kernel and turn the complicated contour deformation
arguments in Stieberger:2022lss ; Stieberger:2023nol  into more linear algebra
like statements.

##  3 One-loop string integrals and the Riemann-Wirtinger integral

In this section, we introduce one-loop string integrals after chiral splitting
(section 3.1) and explain how the Riemann-Wirtinger family of integrals (the
main focus of this work) is related (section 3.2).

###  3.1 One-loop string integrals

In the chiral splitting formalism DHoker:1988pdl , $n$-point one-loop open-
string integrals take the form

$\displaystyle\int\text{d}^{D}\ell\int\text{d}\tau\ e^{\pi
i\alpha^{\prime}\tau\ell^{2}}\int\text{d}^{n}z\ e^{2\pi
i\alpha^{\prime}\ell\cdot\sum_{j=1}^{n}k_{j}z_{j}}\prod_{1\leq
j<k<n}e^{\alpha^{\prime}k_{j}\cdot
k_{k}\log\frac{\vartheta_{1}(z_{j}-z_{k})}{\vartheta_{1}^{\prime}(0)}}f_{\text{string}}(\ell^{\mu},z_{j},\tau)$
$\displaystyle\propto\int\text{d}^{D}\ell\int\text{d}\tau\ e^{\pi
i\alpha^{\prime}\tau\ell^{2}}\int\text{d}^{n}z\ \text{KN}^{\tau}_{n}\
f_{\text{string}}(\ell^{\mu},z_{j},\tau)\,$ (49)

where $D$ is the space-time dimension. Here, $\text{KN}^{\tau}_{n}$ is the
Koba-Nielsen for $n$-punctures is a multi-valued function universal to all
one-loop string integrals

$\displaystyle\text{KN}^{\tau}_{n}=\left(\prod_{i=1}^{n}e^{2\pi
is_{iA}z_{i}}\right)\left(\prod_{1\leq j<k\leq
n}\vartheta_{1}^{s_{jk}}(z_{j}-z_{k})\right)$ (50)

where $\vartheta_{1}$ is the Jacobi theta function and we have hidden most of
the loop-momentum dependence in the $s_{i}$’s

$\displaystyle s_{iA}=\ell\cdot k_{i}\,.$ (51)

The $s_{ij}$’s are the familiar planar Mandelstam invariants
$s_{ij}=s_{ji}=\frac{1}{2}(k_{i}+k_{j})^{2}=k_{i}\cdot k_{j}$ where
$k_{i}^{\mu}$ is the momentum associated to the $i^{\text{th}}$ particle.
Moreover, these variables satisfy the momentum conservation constraint
$\sum_{j\neq i}s_{ij}=0$.

One-loop string integrals are integrals over the moduli space of the
$n$-punctured torus: $\mathcal{M}_{1,n}$. The modulus $\tau$ (ratio of the
$A$\- and $B$-cycle periods) controls the shape of the torus, while the
$z_{i}$ are the position of punctures on the torus. Meromorphic functions on
the torus are called elliptic functions and must be doubly periodic under
integer shifts of the $A$\- and $B$-cycle periods. Another way of saying this
is that elliptic functions transform covariantly under modular transformations
($\text{SL}(2,\mathbb{Z})$ transformation of the periods). In particular, the
modulus can always be mapped to the upper half plane using a modular
transformation. Thus, without loss of generality, we normalize the $A$-cycle
period to one and the $B$-cycle period to $\tau\in\mathbb{H}$.

These periods define a lattice $\Lambda_{\tau}=\mathbb{Z}+\tau\mathbb{Z}$ that
tiles $\mathbb{C}$ and the torus is the corresponding quotient
$E=\mathbb{C}/\Lambda_{\tau}$. We define the fundamental parallelogram to be
$P=\\{z=a+b\tau|a,b\in[0,1]\\}$ and assume all $z_{i}\in P$ (one can always
preform a modular transformation to move a $z_{i}$ outside of the fundamental
parallelogram into $P$). While functions on the torus should be doubly
periodic it is often convenient to work with quasi-periodic functions like the
odd Jacobi-theta function $\vartheta_{1}$. We also use conventions for the
Jacobi-theta function such that

$\displaystyle\vartheta_{1}(z+1|\tau)=-\vartheta_{1}(z|\tau),\qquad\vartheta_{1}(z+\tau|\tau)=-e^{-i\pi(\tau+2z)}\vartheta_{1}(z|\tau).$
(52)

In later sections, we will often omit the explicit dependence on $\tau$ since
we always work at fixed $\tau$.

In this work, we concentrate on the $\text{d}\tau\ \text{d}^{D}\ell$
integrands of string integrals, which are themselves integrals over the
configuration space of the $n$-punctured torus. Under $A$\- and $B$-cycle
shifts of the punctures, the Koba-Nielsen factor becomes

$\displaystyle A\text{-cycle}:$
$\displaystyle\qquad\frac{\text{KN}^{\tau}_{n}|_{z_{i}\to
z_{i}+1}}{\text{KN}^{\tau}_{n}}=\exp\left[2\pi is_{iA}\right],$ (53)
$\displaystyle B\text{-cycle}:$
$\displaystyle\qquad\frac{\text{KN}^{\tau}_{n}|_{z_{i}\to
z_{i}+\tau}}{\text{KN}^{\tau}_{n}}=\exp\left[2\pi
i\left(s_{iA}\tau+\underset{j\neq
i}{\sum_{j=1}^{n}}s_{ij}z_{j}\right)\right],$ (54)

where we have used momentum conservation $\sum_{j\neq i}s_{ij}$ = 0.

The function $f_{\text{string}}$ in (3.1) is a meromorphic and almost periodic
function of punctures $z_{i=1,\dots,n}$, modulus $\tau$ and the loop momentum
$\ell^{\mu}$. At genus-one, products of the Kronecker-Eisenstein function,

$\displaystyle F(z,\eta|\tau)$
$\displaystyle=\frac{\vartheta_{1}^{\prime}(0|\tau)\vartheta_{1}(z+\eta|\tau)}{\vartheta_{1}(z|\tau)\vartheta_{1}(\eta|\tau)},$
(55)

at $z\to z_{i}-z_{j}$ form generating series for one-loop integrands
$f_{\text{string}}$. Here, $\eta$ is treated as a formal parameter and the
coefficients of the $\eta$-expansion of $F$ appear in the one-loop integrands:

$\displaystyle F(z,\eta|\tau)=\sum_{k\geq 0}g^{(k)}(z|\tau)\ \eta^{k-1}.$ (56)

With the exception of $g^{(0)}(z|\tau)=1$, all other Kronecker-Eisenstein
coefficients $g^{(k\geq 1)}(z|\tau)$ are meromorphic functions of $z$ with at
most simple poles. In particular, $g^{(k\geq 1)}(z)\text{d}z$ should be
thought of as the genus-one analogue of $\text{d}\log$-forms. These functions
are the integration kernels that define elliptic multiple polylogarithms
(eMPLs), which are becoming increasingly well studied in the physics
literature (see Duhr:2020gdd ; DHoker:2020hlp ; Bourjaily:2021vyj ;
Kristensson:2021ani ; Duhr:2021fhk ; Hidding:2022vjf ; Giroux:2022wav ;
Frellesvig:2023iwr ; McLeod:2023qdf ; He:2023qld ; Gorges:2023zgv  for an
incomplete survey of the last few years).

Elliptic functions cannot have only a simple pole.555Double periodicity forces
elliptic functions to have a double pole or more than one simple pole.
Therefore, the $g^{(k\geq 1)}$ are not elliptic functions and have non-trivial
$B$-cycle transformations

$\displaystyle A\text{-cycle}:$ $\displaystyle\qquad
g^{(k)}(z+1|\tau)=g^{(k)}(z|\tau),$ (57) $\displaystyle B\text{-cycle}:$
$\displaystyle\qquad g^{(k)}(z+\tau|\tau)=\sum_{j=0}^{k}\frac{(-2\pi
i)^{j}}{j!}g^{(k-j)}(z,\tau).$ (58)

While the $g^{(k\geq 1)}$’s are not elliptic, one can construct doubly
periodic linear combinations. On the other hand, the generating series $F$ has
comparatively simple $B$-cycle monodromies

$\displaystyle A\text{-cycle}:$ $\displaystyle\qquad
F(z+1,\eta|\tau)=F(z|\tau),$ (59) $\displaystyle B\text{-cycle}:$
$\displaystyle\qquad F(z+\tau,\eta|\tau)=F(z,\eta|\tau)e^{-2\pi i\eta}.$ (60)

Thus, it is often more convenient to consider a generating series of one-loop
string integrands built from Kronecker-Eisenstein functions.

Moreover, the Kronecker-Eisenstein functions satisfy a genus-one analogue of
partial fractions called the Fay identity

$\displaystyle F(z_{ik},\zeta_{i})F(z_{jk},\zeta_{j})$
$\displaystyle=F(z_{ik},\zeta_{ij})F(z_{ji},\zeta_{j})+F(z_{jk},\zeta_{ij})F(z_{ij},\zeta_{i})$
(61) $\displaystyle z_{ij}$ $\displaystyle=z_{i}-z_{j}$ (62)
$\displaystyle\zeta_{ij}$ $\displaystyle=\zeta_{i}+\zeta_{j}\,.$ (63)

Since the Fay identity holds order by order in $\eta$, the coefficient
functions $g^{(k)}$ also satisfy partial fraction like identities. However,
these $g^{(k)}$ identities are much more complicated — another reason to
prefer working with the generating functions $F$ over the $g^{(k)}$’s.

In section 6.4, we will also use the non-meromorphic but doubly-periodic
version of the Kronecker Eisenstein series:

$\displaystyle\Omega(z,\eta|\tau)=\exp\big{(}2\pi
i\eta\frac{\mathrm{Im}\,z}{\mathrm{Im}\,\tau}\big{)}F(z,\eta|\tau)\,.$ (64)

Like its meromorphic cousin, the doubly-periodic Kronecker-Eisenstein series
can be $\eta$-expanded

$\displaystyle\Omega(z,\eta|\tau)=\sum_{k\geq 0}f^{(k)}(z|\tau)\
\eta^{k-1}\,,$ (65)

to obtain doubly-periodic coefficients $f^{(k)}$.

###  3.2 The Riemann-Wirtinger integral

The one-loop string integrals (3.1) are closely related to a family of genus-
one integrals — the so-called Riemann-Wirtinger integral — recently studied in
the mathematics literature Goto2022 ; ghazouani2017moduli ;
ghazouani2016moduli ; Mano2012 ; mano2009 . The integrals are essentially the
$\text{d}z_{1}$-part of (3.1)

$\displaystyle\int_{\gamma_{i}}u(z_{1})\ F(z_{1}-z_{j},\eta|\tau)\
\text{d}z_{1},\quad u(z_{1})=e^{2\pi
is_{1A}z_{1}}\prod_{i=2}^{n}\vartheta_{1}^{s_{1i}}(z_{1}-z_{i}),$ (66)

with $n\geq 3$ punctures and an extra condition

$\displaystyle\boxed{s_{1B}=s_{1A}\tau+\sum_{j=2}^{n}s_{1j}z_{j}-\eta=\mathrm{const.}\,,}$
(67)

where $s_{1B}\in\mathbb{C}$ is a generic complex number.666To convert from the
notation of Goto2022  to our notation one makes the replacement
$c_{\infty}\to-s_{1B}$, $c_{i>0}\to s_{1i}$, $c_{0}\to s_{1A}$. Crucially,
this condition ensures that both the $A$\- and $B$-cycle monodromies of the
integrand in (66) are treated symmetrically:

$\displaystyle A\text{-cycle}:$
$\displaystyle\qquad\frac{\text{KN}^{\tau}_{n}\ F(z,\eta|\tau)|_{z_{1}\to
z_{1}+1}}{\text{KN}^{\tau}_{n}\ F(z|\tau)}=e^{2\pi is_{1A}},$ (68)
$\displaystyle B\text{-cycle}:$
$\displaystyle\qquad\frac{\text{KN}^{\tau}_{n}\ F(z,\eta|\tau)|_{z_{1}\to
z_{1}+\tau}}{\text{KN}^{\tau}_{n}\ F(z|\tau)}=e^{2\pi is_{1B}}.$ (69)

The twist also has monodromies as $z_{1}$ loops around another puncture
$z_{j}$

$\displaystyle z_{1}\text{ loop around }z_{j}:$
$\displaystyle\qquad\frac{\text{KN}^{\tau}_{n}|_{\circlearrowleft_{1j}}}{\text{KN}^{\tau}_{n}}=e^{2\pi
is_{1j}}.$ (70)

To understand the linear and quadratic relations among Riemann-Wirtinger
integrals, we construct the corresponding twisted (co)homology. In particular,
we need to understand how the local system is modified at genus-one.

In the following, we review the construction of the local system since there
are new features at genus-one. Specifically, the “total” local system is the
tensor product of two local systems: one from the multi-valuedness of the
twist and one from the quasi-periodicity of the Kronecker-Eisenstein function.
While the local system will be important, much of the underlying mathematical
formalism (see appendix A and references therein) can be ignored by first-time
readers.

We begin with the local system and dual local system associated to the multi-
valuedness of the twist

$\displaystyle\mathcal{L}_{\omega}$
$\displaystyle:=\mathcal{L}_{\omega}(s_{1A},s_{12},\dots,s_{1n})=\mathbb{C}u^{-1}\,,$
(71) $\displaystyle\check{\mathcal{L}}_{\omega}$
$\displaystyle:=\mathcal{L}_{-\omega}(-s_{1A},-s_{12},\dots,-s_{1n})=\mathbb{C}u\,,$
(72) $\displaystyle\omega$ $\displaystyle:=\text{d}\log(u)\,.$ (73)

These local systems are line bundles that keep track of the branch choices of
the twist and dual (inverse) twist. More precisely, the universal covering
$\tilde{M}$ of $M$ is a principle $\pi_{1}(M)$ bundle where $\pi_{1}(M)$ is
the first homotopy group. By exponentiating the monodromies of $u$ we define a
one-dimensional representation of $\pi_{1}(M)$. Then, the local system is the
associated line bundle. More succinctly, the (dual) local system is the
locally constant sheaf of solutions to the equation $\nabla f=0$
($\check{\nabla}f=0$) with the flat777By flat, we mean that
$\omega\wedge\omega=0\implies\nabla^{2}=0$. (dual) covariant derivative
$\nabla=\text{d}+\omega\wedge$ ($\check{\nabla}=\text{d}-\omega\wedge$). In
other words, the (dual) local system is the set of all possible branch choices
for the dual twist $\check{u}=u^{-1}$ (twist $u$). These covariant derivatives
replace the normal exterior derivative in the twisted setting.

So far the genus-one construction mirrors the genus-zero construction in
section 2. However, at genus-one, we have an additional local system
associated to the multi-valued-ness of the Kronecker-Eisensten functions (60).
Following Mano2012 ; Goto2022 , we define a 1-dimensional representation of
the fundamental group $\pi_{1}(E)$ for $\eta\in\mathbb{C}$. We denote this by
$e_{\eta}:\pi_{1}(E)\simeq\Lambda_{\tau}\ni\gamma\mapsto
e_{\eta}(\gamma)\in\mathbb{C}^{*}$ such that $e_{\eta}(1)=1$ and
$e_{\eta}(\tau)=e^{-2\pi i\eta}$. This representation corresponds to the
allowed phases of our integrand modulo the twist (recall equation (59) and
(60)). We denote the rank-1 local system on $E$ determined by the above
representation of $\pi_{1}(E)$ by $\mathcal{L}_{\eta}$.

Thus, the combined local system of the integrand of (66) and its dual are

$\displaystyle\mathcal{L}_{\omega,\eta}(s_{1A},s_{12},\dots,s_{1n},s_{1B})=\mathcal{L}_{\omega}\otimes\mathcal{L}_{\eta}=\mathbb{C}u^{-1}\otimes\mathcal{L}_{\eta}\,,$
(74)
$\displaystyle\check{\mathcal{L}}_{\omega,\eta}=\mathcal{L}_{\omega}(-s_{1A},-s_{12},\dots,-s_{1n},-s_{1B})\otimes\mathcal{L}_{-\eta}=\mathbb{C}u\otimes\mathcal{L}_{-\eta}\,.$
(75)

As usual, these local systems are connected by the involution
$\vee:s_{\bullet}\mapsto-s_{\bullet}$. From this, we construct the twisted
cohomology and homology of the Riemann-Wirtinger integral
($H^{p}(M,\mathcal{L}_{\omega,\eta})$ and
$H_{p}(M,\check{\mathcal{L}}_{\omega,\eta})$) as well as their duals
($H^{p}(M,\check{\mathcal{L}}_{\omega,\eta})$ and
$H_{p}(M,\mathcal{L}_{\omega,\eta})$) in sections 4 and 5.

##  4 Twisted cohomology of the Riemann-Wirtinger integral

In this section, we construct the twisted cohomology of the Riemann-Wirtinger
integral. We define the twisted cohomology for $\eta\neq 0$ in section 4.1 and
give two different bases. Then, in section 4.2, we define the associated dual
twisted cohomology and an inner product called the intersection number. Using
the intersection number, we compute the differential equations satisfied by
the Riemann-Wirtinger integrals in section 4.3. In section 4.4, we solve the
differential equations to order $\mathcal{O}(\alpha^{\prime})$ in terms of
eMPLs for generic boundary values. We postpone a more detailed discussion of
boundary values to section 5 where we introduce a basis of cycles. Lastly, in
section 4.5, we describe the cohomology in the $\eta\to 0$ limit and discuss
the corresponding differential equations.

###  4.1 Basis of cohomology

Recall that the twisted cohomology is essentially the cohomology with respect
to the covariant derivative $\nabla=\text{d}+\omega\wedge$ where
$\omega=\text{d}\log u$ and $\nabla^{2}=0$. This covariant derivative keeps
track of the monodromies generated by the twist $u$. It also means that we can
trade the multi-valued integrand $u\varphi$ for simpler integrands
($\mathcal{L}_{\eta}$-valued differential forms) since
$\text{d}(u\varphi)=u(\nabla\varphi)$. We denote the space of twisted
differential $p$-forms on $M$ by
$\Omega^{p}_{\eta}(M)=\Omega^{p}(M)\otimes\mathcal{L}_{\eta}$.

Because the coefficients $s_{\bullet}$ are generic, the total derivative
always vanishes

$\displaystyle\int\text{d}(u\ \varphi)=\int u(\nabla\varphi)=0.$ (76)

This means that the Riemann-Wirtinger integrand is not unique — we can always
add a total covariant derivative to $\varphi$ without changing the value of
the integral. For this reason, we would like to work with equivalence classes
of forms that are unique and independent of such shifts. We would also like
any representative of such a class to be closed: $\nabla\varphi=0$. This
ensures that there are no boundary terms in the twisted version of Stokes’
theorem. Thus, we are lead to define the twisted cohomology

$\displaystyle
H^{p}(M,\mathcal{L}_{\omega,\eta})=\frac{\text{ker}\left(\nabla:\Omega^{p}_{\eta}(M)\to\Omega^{p+1}_{\eta}(M)\right)}{\text{im}\left(\nabla:\Omega^{p-1}_{\eta}(M)\to\Omega^{p}_{\eta}(M)\right)}$
(77)

Explicitly, the twisted cohomology class represented by $\varphi$ is

$\displaystyle|\varphi\rangle=\\{\varphi+\nabla\psi|\psi\in\Omega^{\text{deg}\varphi-1}_{\eta}(M)\quad\text{and}\quad\nabla\varphi=0\\}\,.$
(78)

Here, the local system appears in the second argument of $H^{p}$ instead of
$\nabla$ because $\nabla$ only knows about the $\mathcal{L}_{\omega}$ half of
the local system ${\mathcal{L}_{\omega,\eta}}$.

For each $p=0,\dots,\text{dim}_{\mathbb{R}}M$, the twisted cohomology is a
finite dimensional vector space. Recall the middle dimensional theorem (26),
which states that only the cohomology for
$p=\frac{1}{2}\text{dim}_{\mathbb{R}}M$ is non-trivial. This translates into
the fact that there are no non-trivial closed 0- or 2-forms

$\displaystyle\text{dim}H^{p=0,2}(M,\mathcal{L}_{\omega,\eta})=0.$ (79)

By a theorem from Mano2012 ; Goto2022 , we also know that the dimension of the
twisted cohomology of 1-forms is

$\displaystyle\text{dim}H^{1}(M,\mathcal{L}_{\omega,\eta})=n-1$ (80)

where $n$ is the number of punctures $z_{i}$.

We close this section by introducing two useful bases for the twisted
cohomology. The first basis contains forms that have at most a simple poles at
one of the punctures $z_{i\geq 2}$

$\displaystyle\varphi_{a}$
$\displaystyle=F(z_{1}-z_{a+1},\eta)\,\text{d}z_{1}\quad\text{for}\quad
a=1,2,\cdots,n-1.$ (81)

The second is a spanning set of forms

$\displaystyle\xi^{(p)}_{1}$ $\displaystyle=\eta
F(z_{1}-z_{p},\eta)\,\text{d}z_{1}$ $\displaystyle\xi^{(p)}_{p}$
$\displaystyle=\partial_{1}F(z_{1}-z_{p},\eta)\,\text{d}z_{1}$ (82)
$\displaystyle\xi^{(p)}_{a\geq 2}$
$\displaystyle=\left[F(z_{1}-z_{a},\eta)-F(z_{1}-z_{p},\eta)\right]\text{d}z_{1}\qquad\text{for}\qquad
a\neq p$

subject to a single relation

$\displaystyle 0\simeq\left[2\pi is_{1A}+s_{1p}\,g^{(1)}(\eta)+\underset{j\neq
p}{\sum_{j=2}^{n}}s_{1j}\left(F(z_{j}-z_{p},\eta)-g^{(1)}(z_{j}-z_{p})\right)\right]|\xi^{(p)}_{1}\rangle$
$\displaystyle-(s_{1,p}-1)\eta|\xi^{(p)}_{p}\rangle+\eta\,\underset{j\neq
p}{\sum_{j=2}^{n}}s_{1j}\,F(z_{j}-z_{p},\eta)\ |\xi^{(p)}_{j}\rangle.$ (83)

This identity follows from the fact that $\nabla\xi^{(p)}_{p}\simeq 0$ in
cohomology (and can be shown using the Fay identity or intersection numbers).
This basis has a well defined $\eta\to 0$ limit and spans the twisted
cohomology in this limit Goto2022  (see section 4.5). In the next section, we
will verify that the intersection matrix of these sets has rank $n-1$
confirming that the set $\\{\varphi_{a}\\}$ is indeed a basis and that the set
$\\{\xi_{a}^{(p)}\\}$ is over-complete.

###  4.2 Intersection numbers: an inner product on cohomology

Recall from section 2.3 that the dual cohomology isomorphic to the twisted
cohomology $H^{p}(M,\mathcal{L}_{\omega,\eta})$ is defined by changing the
sign of all $s_{\bullet}$ and consequently $\eta$

$\displaystyle\left[H^{p}(M,\mathcal{L}_{\omega,\eta})\right]^{\vee}:=H^{p}(M,\check{\mathcal{L}}_{\omega,\eta}).$
(84)

The dual covariant derivative is
$\check{\nabla}=\text{d}+\check{\omega}\wedge$ where
$\check{\omega}=\text{d}\log{u^{-1}}=-\omega$. As a consequence of changing
the signs of the $s_{\bullet}$’s, the dual Riemann-Wirtinger integral comes
with a dual twist $\check{u}=u^{-1}$. We can also think of $\vee$ as a map
from $H^{p}(M,\mathcal{L}_{\omega,\eta})$ to the dual cohomology:
$\check{\varphi}=[\varphi]^{\vee}:=\varphi|_{s_{\bullet}\to-
s_{\bullet},\eta\to-\eta}$. In particular, the set $\\{\check{\varphi}_{a}\\}$
is basis for the dual cohomology and $\\{\check{\xi}^{(p)}_{a}\\}$ spans the
dual cohomology subject to the dualized version of (4.1).

The intersection number pairs the cohomology and its dual to form an inner
product

$\displaystyle\langle\bullet|\bullet\rangle:H^{p}(M,\check{\mathcal{L}}_{\omega,\eta})\times
H^{p}(M,\mathcal{L}_{\omega,\eta})\to\mathbb{C}(s_{\bullet})$ (85)

by

$\displaystyle\langle\check{\varphi}|\varphi\rangle=\int_{M}\text{Reg}[\check{\varphi}]\wedge\varphi$
(86)

where Reg maps the representative $\check{\varphi}$ to a new representative
that has no support in the neighbourhood of each puncture $z_{i\geq
2}$.888Formally, $\mathrm{Reg}$ is a map from the dual twisted cohomology to
the compactly supported dual twisted cohomology
$\mathrm{Reg}:H^{p}(M,\check{\nabla})\to H^{p}_{c}(M,\check{\nabla})$. Note
that our definition for the intersection number differs from Goto2022  by a
sign because we choose to put the dual forms on the left hand side because it
is more natural in the bra-ket notation of quantum mechanics.

Since the Riemann-Wirtinger integral is one-dimensional we only need to know
the map $\mathrm{Reg}$ in this case. Explicitly, for any 1-form
$\check{\varphi}$, its image under $\mathrm{Reg}$ is

$\displaystyle\mathrm{Reg}[\check{\varphi}]$
$\displaystyle:=\check{\varphi}-\sum_{i=2}^{n}\nabla\bigg{(}\Theta(|z_{1i}|<\epsilon)\
\check{\psi}_{i}\bigg{)}$
$\displaystyle=\bigg{(}1{-}\sum_{i=2}^{n}\Theta(|z_{1i}|<\epsilon)\bigg{)}\check{\varphi}-\sum_{i=2}^{n}\check{\psi}_{i}\
\text{d}\Theta(|z_{1i}|<\epsilon),$ (87)

where $\epsilon$ is arbitrarily small enough such that the Heaviside-$\Theta$
functions do not overlap. Here, the 0-form $\check{\psi}_{i}$ is a local
primitive for $\check{\varphi}$ near $z_{1}=z_{i}$:
$\check{\nabla}\check{\psi}_{i}=\check{\varphi}$ for $z_{1}\sim z_{i}$. While
any global primitive is indeed multi-valued, the local primitive
$\check{\psi}_{i}$ is single-valued and can be expressed as a Laurent
polynomial in $z_{1}-z_{i}$. Clearly, $\mathrm{Reg}[\check{\varphi}]$ has no
support for $z_{1}\sim z_{i}$. Moreover, it differs from $\check{\varphi}$ by
a total derivative and is thus equivalent in cohomology.999Forms differing by
a covariant derivative are in the same cohomology class and are said to be
cohomologous. The replacement of $\check{\varphi}$ with
$\text{Reg}[\check{\varphi}]$ does not change the intersection number because
it only depends on the class not the representative. It is essential that we
use the compactly supported version of $\check{\varphi}$ otherwise the
intersection number would not be finite.101010More precisely, for generic
$s_{\bullet}$, Reg is an isomorphism between
$H^{p}(M,\check{\mathcal{L}}_{\omega,\eta})$ and its compactly supported
version $H^{p}_{c}(M,\check{\mathcal{L}}_{\omega,\eta})$. Alternatively, one
could also regulate Riemann-Wirtinger cohomology instead of the dual
cohomology. Moreover, since only the anti-holomorphic part of
$\mathrm{Reg}[\check{\varphi}]$ survives the wedge product in (86), the
intersection number of Riemann-Wirtinger 1-forms is given by the following
simple residue formula

$\displaystyle\langle\check{\varphi}|\varphi\rangle=-\sum_{i=2}^{n}\int_{M}\check{\psi}_{i}\
\text{d}\Theta(|z_{1i}|<\epsilon)\wedge\varphi=-\sum_{i=2}^{n}\text{Res}_{z_{1}=z_{i}}[\check{\psi}_{i}\varphi].$
(88)

One important feature of this formula is that the intersection number vanishes
whenever the forms $\check{\varphi}$ and $\varphi$ do not have overlapping
singularities. Moreover, the intersection numbers satisfy the following
relation

$\displaystyle[\langle\check{\varphi}|\phi\rangle]^{\vee}=-\langle\check{\phi}|\varphi\rangle\,.$
(89)

Thus, one only has to compute the upper/lower triangular part of the
intersection matrix.

In particular, we can already predict that the intersection matrix
$\langle\check{\varphi}_{a}|\varphi_{b}\rangle$ is diagonal. To compute the
proportionality constant, we need to construct the primitives of
$\check{\varphi}_{a}$

$\displaystyle\check{\psi}_{a,i=a}$
$\displaystyle=-\frac{1}{s_{1a}}+\left[\frac{2\pi
is_{1A}}{(s_{1a}{-}1)s_{1a}}-\frac{g^{(1)}(-\eta)}{s_{1a}{-}1}+\frac{\sum_{i\neq
a}s_{1i}\
g^{(1)}(z_{1a})}{(s_{1a}{-}1)s_{1a}}\right]z_{1a}+\mathcal{O}\left(z_{1a}^{2}\right),$
(90) $\displaystyle\check{\psi}_{a,i\neq a}$
$\displaystyle=-\frac{F(z_{ia},-\eta)}{s_{1i}{-}1}z_{1i}+\mathcal{O}\left(z_{1i}^{2}\right),$
(91)

where $\check{\nabla}\check{\psi}_{a,i}=\check{\varphi}_{a}$ near
$z_{1}=z_{i}$. Using (88), we obtain

$\displaystyle(C_{\varphi})_{ab}=\langle\check{\varphi}_{a}|\varphi_{b}\rangle=2\pi
i\frac{\delta_{ab}}{s_{1a}}.$ (92)

In particular, this intersection matrix has full rank verifying that the set
$\\{\varphi_{a}\\}$ forms a basis.

Equations (90) and (91) are also enough to determine almost all of the
intersection numbers for the set $\\{\xi^{(p)}_{a}\\}$:

$\displaystyle\frac{\langle\check{\xi}^{(p)}_{1}|\xi^{(p)}_{a}\rangle}{-2\pi
i}$ $\displaystyle=\begin{cases}\frac{\eta^{2}}{s_{1p}}&b=1,\\\
-\frac{\eta}{s_{1p}}&a=1,b\geq 2\text{ and }b\neq p,\\\ \frac{2i\pi
s_{1A}-s_{1p}\ g^{(1)}(\eta)+\sum_{i=2;i\neq p}^{n}s_{1i}\
g^{(1)}(z_{pi})}{(s_{1p}-1)s_{1p}}&b=p,\end{cases}$ (93)
$\displaystyle\frac{\langle\check{\xi}^{(p)}_{a\geq 2,a\neq
p}|\xi^{(p)}_{b\geq 2}\rangle}{-2\pi i}$
$\displaystyle=\begin{cases}-\frac{s_{1a}+s_{1p}}{s_{1a}s_{1p}}&b=a\\\
-\frac{1}{s_{1p}}&b\neq a,p\\\ \frac{-s_{1p}\ F(z_{ap},\eta)-s_{1p}\
g^{(1)}(\eta)+\sum_{i=2;i\neq p}^{n}s_{1i}\ g^{(1)}(z_{pi})+2i\pi
s_{1A}}{(s_{1p}-1)s_{1p}}&b=p.\end{cases}$ (94)

For $a=b=p$, one needs to go beyond the local primitives (90) and (91)

$\displaystyle\frac{\langle\check{\xi}^{(p)}_{p}|\xi^{(p)}_{p}\rangle}{-2\pi
i}$ $\displaystyle=(2\pi s_{1A})^{2}+2s_{1p}^{2}\
g^{(2)}(\eta)+\sum_{i=2,i\neq p}^{n}s_{1p}s_{1i}\
g^{(1)}(z_{pi})-\left(\sum_{i=2,i\neq p}^{n}s_{1i}\
g^{(1)}(z_{pi})\right)^{2}$ $\displaystyle-4i\pi s_{1A}\sum_{i=2,i\neq
p}^{n}s_{1i}\ g^{(1)}(z_{pi})-2s_{1p}\sum_{i=2,i\neq p}^{n}s_{1i}\
\wp(z_{pi}).$ (95)

All remaining $\langle\xi^{(p)}_{a}|\xi^{(p)}_{b}\rangle$ intersection numbers
can be obtained from the above using the relation
$[\langle\check{\varphi}|\phi\rangle]^{\vee}=-\langle\check{\phi}|\varphi\rangle$.
One can then verify that the $n\times n$ intersection matrix
$\langle\xi^{(p)}_{a=1,\dots,n}|\xi^{(p)}_{b=1,\dots,n}\rangle$ has rank $n-1$
confirming that the set $\\{\xi^{(p)}\\}_{a=1,\dots,n}$ is over-complete. In
particular, the identity (4.1) follows from a straightforward application of
the resolution of identity (35). Choosing $\\{\xi_{a}^{(p)}\\}_{a\neq p}$ and
$\\{\check{\xi}_{a}^{(p)}\\}_{a\neq p}$ for our basis of cohomology and dual
cohomology, we set $C_{ab}=\langle\check{\xi}_{a}^{(p)}|\xi_{b}^{(p)}\rangle$
and apply (35) to $|\xi_{p}^{(p)}\rangle$:

$\displaystyle|\xi_{p}^{(p)}\rangle\simeq\sum_{a,b\neq
p}|\xi_{a}^{(p)}\rangle\ C_{ab}^{-1}\
\langle\check{\xi}_{b}^{(p)}|\xi_{p}^{(p)}\rangle=\frac{\mathrm{RHS}\eqref{eq:xirel}}{\eta(s_{12}-1)}\,.$
(96)

While more complicated than the forms $\varphi_{a}$, the spanning set
$\xi^{(p)}_{a}$ is still useful because it spans the cohomology in the
$\eta\to 0$ limit (section (4.5)).

###  4.3 Differential equations for the $\varphi$-basis

Using intersection numbers, we compute the Gauss-Manin differential equation
satisfied by the Riemann-Wirtinger integrals where we differentiate with
respect to the external un-integrated punctures $z_{i\geq 2}$.

The differential operators in the external punctures must preserve the
monodromy structure of the local system in order for the differential
equations to close. For $f\ \text{d}z_{1}\in\Omega^{1}_{\eta}$, the Gauss-
Manin differential operators that preserve monodromies are Mano2012 :

$\displaystyle\nabla_{\text{ext},a}f\ \text{d}z_{1}$
$\displaystyle:=(\partial_{a}f)\ \text{d}z_{1}+(\omega_{a}f)\
\text{d}z_{1}+s_{1a}(\partial_{\eta}f)\ \text{d}z_{1}\qquad\text{for}\qquad
a=2,3,\dots,n\,,$ (97) $\displaystyle\nabla_{\text{ext},\tau}f\ \text{d}z_{1}$
$\displaystyle:=(\partial_{\tau}f)\ \text{d}z_{1}+(\omega_{\tau}f)\
\text{d}z_{1}+s_{1A}(\partial_{\eta}f)\ \text{d}z_{1}-\frac{1}{2\pi
i}\nabla_{\mathcal{M}/\mathcal{B}}(\partial_{\eta}f)\,.$ (98)

Here, $f$ is a meromorphic function on $M$ with $B$-cycle monodromy $e^{-2\pi
i\eta}$. Understanding the precise form of the above differential operators
requires understanding how varying the punctures and moduli on the total space
$M\to\mathcal{M}\overset{\pi}{\to}\mathcal{B}=\\{(z_{2},\dots,z_{n},\tau)\in\mathbb{C}^{n-1}\times\mathbb{H}|z_{i}\neq
z_{j}\\}$ descends to $M$. For Riemann-Wirtinger integrals, the differential
operator $\nabla_{\mathcal{M}/\mathcal{B}}$ is simply $\nabla$ (see
ghazouani2016moduli  appendix B for the derivation in the $\eta=0$
case)111111See also deligne2006equations  or KatzOda  for the general theory..
It is also important to note that we cannot throw away
$\nabla_{\mathcal{M}/\mathcal{B}}(\partial_{\eta}f)$ even though it is a total
covariant derivative because $\partial_{\eta}f$ does not have the allowed
multiplicative monodromy.

To motivate the form of the Gauss-Manin differentials above, we examine how
the connection on $\mathcal{M}$ transforms under the $B$-cycle. We let
$\tilde{\text{d}}$ be the exterior derivative on $\mathcal{M}$ and set
$\tilde{\omega}=\tilde{\text{d}}\log(u)$. This is the obvious uplift of the
connection on $M$ to $\mathcal{M}$. The $B$-cycle monodromy of
$\tilde{\omega}$ is

$\displaystyle\tilde{\omega}=\tilde{\text{d}}\log(u)$
$\displaystyle\underset{z_{1}\to
z_{1}+\tau}{\longrightarrow}\tilde{\omega}+\tilde{\text{d}}\log\left(e^{2\pi
is_{1A}\tau+2\pi i\sum_{i=2}^{n}s_{1i}z_{i}}\right)$
$\displaystyle=\tilde{\omega}+2\pi i\
\tilde{\text{d}}\left(s_{1A}\tau+\sum_{i=2}^{n}s_{1i}z_{i}\right)=\tilde{\omega}+2\pi
i\ \tilde{\text{d}}\eta,$ (99)

When $\eta=0$, the above connection is doubly periodic. Next, we compute the
$B$-cycle monodromy of the the image of the covariant derivative on
$\mathcal{M}$: $\tilde{\nabla}=\tilde{\text{d}}+\tilde{\omega}\wedge$.
Explicitly,

$\displaystyle\tilde{\nabla}\left(F(z_{1a},\eta)\
\tilde{\text{d}}z_{1}\right)\underset{z_{1}\to
z_{1}+\tau}{\longrightarrow}\left(\tilde{\nabla}+2\pi i\
\tilde{\text{d}}\eta\wedge\right)\left(e^{-2\pi i\eta}F(z_{1a},\eta)\
(\tilde{\text{d}}z_{1}+\tilde{\text{d}}\tau)\right),$ $\displaystyle=e^{-2\pi
i\eta}\bigg{[}(\nabla_{\tau}-\nabla_{1})F(z_{1a},\eta)\
\tilde{\text{d}}\tau\wedge\tilde{\text{d}}z_{1}+\sum_{i=2}^{n}\nabla_{i}F(z_{1a},\eta)\
\tilde{\text{d}}z_{i}\wedge\tilde{\text{d}}z_{1}+\cdots\bigg{]},$ (100)

where the dots above indicate differential forms that are independent of
$\tilde{\text{d}}z_{1}$. By adding $-\frac{1}{2\pi
i}\nabla_{1}\partial_{\eta}$ to the $\text{d}\tau$-part of the covariant
derivative $\tilde{\nabla}$ one obtains a differential form whose
$\tilde{\text{d}}z_{1}$ part has the allowed $B$-cycle monodromy. In fact, the
components of this differential are precisely (97) and (98). Note that we have
made the action of the partial derivatives on $\eta$ explicit in (97) and (98)
but left this implicit in (100).

The differential equation for the family of Riemann-Wirtinger integrals or
Gauss-Manin connection on the base space $\mathcal{B}$ is

$\displaystyle\text{d}_{\text{ext}}[\gamma|\varphi_{a}\rangle=A_{ab}[\gamma|\varphi_{b}\rangle\iff\nabla_{\text{ext}}|\varphi_{a}\rangle=A_{ab}\wedge|\varphi_{b}\rangle$
(101)

where

$\displaystyle
A_{ab}|_{\text{d}z_{i}}=C_{bc}^{-1}\langle\check{\varphi}_{c}|\nabla_{\text{ext},i}\
\varphi_{a}\rangle=-\frac{1}{s_{1b}}\langle\check{\varphi}_{b}|\nabla_{\text{ext},i}\
\varphi_{a}\rangle.$ (102)

Breaking this up into the off-diagonal and diagonal parts, the general formula
for the $\text{d}z_{c}$-components of the differential equation are

$\displaystyle\left(A^{\text{off-diag}}_{c}\right)_{ab}=$
$\displaystyle\sum_{d\neq c}\delta_{bd}\ \delta_{ac}\ s_{1b}\
F(z_{bc},\eta)-\sum_{d\neq c}\delta_{ad}\ \delta_{bc}\ s_{1c}\
F(z_{ca},\eta)\,,$ (103)
$\displaystyle\left(A^{\text{diag}}_{c}\right)_{ab}=\delta_{ab}\Bigg{[}$
$\displaystyle-s_{1c}(1-\delta_{ca})\ g^{(1)}(z_{ac})+\delta_{ac}\bigg{(}2\pi
is_{1A}+\sum_{d\neq c}s_{1d}\ g^{(1)}(z_{cd})\bigg{)}\Bigg{]}\,,$ (104)

where $a,b=2,\dots n$. Similarly, the general formula for the
$\text{d}\tau$-component is

$\displaystyle\left(A^{\text{off-diag}}_{\tau}\right)_{ab}$
$\displaystyle=-\frac{(1-\delta_{ab})}{2\pi i}\bigg{[}s_{1b}\
\partial_{\eta}F(z_{ba},\eta)\bigg{]}\text{d}\tau\,,$ (105)
$\displaystyle\left(A^{\text{diag}}_{\tau}\right)_{ab}$
$\displaystyle=\frac{\delta_{ab}}{2\pi i}\bigg{[}\sum_{c=2}^{n}s_{1c}\
g^{(2)}(z_{ac})-s_{1a}\ \partial_{\eta}F(0,\eta)\bigg{]}\text{d}\tau\,,$ (106)

where

$\displaystyle\partial_{\eta}F(0,\eta)=\frac{\vartheta_{1}^{\prime\prime}(\eta)}{\vartheta_{1}(\eta)}-\left(\frac{\vartheta_{1}^{\prime}(\eta)}{\vartheta_{1}(\eta)}\right)^{2}=-\left(\wp(\eta)+G_{2}(\tau)\right)\,,$
(107)

and
$G_{2}(\tau)=-\frac{\vartheta_{1}^{\prime\prime\prime}(0)}{3\vartheta_{1}^{\prime}(0)}$
is the second Eisenstein series. For example, the $\text{d}z_{2}$\- and
$\text{d}\tau$-components of $\underline{\boldsymbol{A}}$ for $n=4$ are

$\displaystyle\underline{\boldsymbol{A}}_{z_{2}}$
$\displaystyle=\begin{pmatrix}2\pi is_{1A}{+}s_{13}\ g^{(1)}_{23}{+}s_{14}\
g^{(1)}_{24}&s_{13}\ F(z_{32},\eta)&s_{14}\ F(z_{42},\eta)\\\ -s_{12}\
F(z_{23},\eta)&s_{12}\ g^{(1)}_{23}&0\\\ -s_{12}\ F(z_{24},\eta)&0&s_{12}\
g^{(1)}_{24}\end{pmatrix},$ (108)
$\displaystyle\underline{\boldsymbol{A}}_{\tau}$ $\displaystyle=\frac{1}{2\pi
i}\begin{pmatrix}\sum_{c=2}^{4}\frac{s_{1c}}{2}\ g^{(2)}(z_{2c})&0&0\\\
0&\sum_{c=2}^{4}\frac{s_{1c}}{2}\ g^{(2)}(z_{3c})&0\\\
0&0&\sum_{c=2}^{4}\frac{s_{1c}}{2}\ g^{(2)}(z_{4c})\end{pmatrix}$
$\displaystyle\qquad-\frac{1}{2\pi i}\begin{pmatrix}s_{12}\
\partial_{\eta}F(0,\eta)&s_{13}\ \partial_{\eta}F(z_{32},\eta)&s_{14}\
\partial_{\eta}F(z_{42},\eta)\\\ s_{12}\ \partial_{\eta}F(z_{23},\eta)&s_{13}\
\partial_{\eta}F(0,\eta)&s_{14}\ \partial_{\eta}F(z_{43},\eta)\\\ s_{12}\
\partial_{\eta}F(z_{24},\eta)&s_{13}\ \partial_{\eta}F(z_{34},\eta)&s_{14}\
\partial_{\eta}F(0,\eta)\end{pmatrix}.$ (109)

We have also checked that equations (103)-(106) match the differential
equations obtained by different methods in Mano2012 . Moreover, our results
agree with those obtained using standard integration-by-parts and the Fay
identity Kaderli:2022qeu  once the condition (67) is imposed and expand on
this below.

It is interesting that these equations differ from those previously obtained
in the math and physics literature, especially in the context of KZB equations
Felder1995 ; Mafra2019 ; Broedel2020 ; Kaderli:2022qeu . Since the authors of
Mano2012  have already commented on how their differential equations relate to
those in Felder1995 , we focus on the differences in the $z_{j}$-differential
equation of this work and section (3.3.1) of Kaderli:2022qeu . Imposing
momentum conservation on the differential equations of Kaderli:2022qeu  yields
(108) and (4.3) up to one key difference: what we mean by the derivative
$\partial_{z_{i}}$ in our work is actually a partial derivative at constant
$s_{1B}$, $(\partial_{z_{i}})_{s_{1B}}$, while the partial derivative
$\partial_{z_{i}}$ of Kaderli:2022qeu  makes no such distinction. Thus, these
two derivatives are related by:

$\displaystyle(\partial_{z_{i}})_{s_{1B}}$
$\displaystyle=\partial_{z_{i}}\big{|}_{\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Kaderli:2022qeu}{\@@citephrase{(}}{\@@citephrase{)}}}}}+s_{1i}\partial_{\eta}\,,$
(110)

where we one has to exchange the punctures $z_{1}$ and $z_{2}$ in
Kaderli:2022qeu  to convert into the conventions used here. Once we take
equation (110) into account, we can readily recover the $z_{i}-$differential
equation in our work from Kaderli:2022qeu .

As an important cross-check, we show that the above differential equation is
integrable. Writing the connection on kinematic space as
$\underline{\boldsymbol{A}}=\sum_{i=2}^{n}\underline{\boldsymbol{A}}_{i}\text{d}z_{i}\
+\underline{\boldsymbol{A}}_{\tau}\text{d}\tau$, we want to show

$\displaystyle\partial_{a}\underline{\boldsymbol{A}}_{b}-\partial_{b}\underline{\boldsymbol{A}}_{a}+[\underline{\boldsymbol{A}}_{a},\underline{\boldsymbol{A}}_{b}]$
$\displaystyle=0\,,$ (111)
$\displaystyle\partial_{\tau}\underline{\boldsymbol{A}}_{a}-\partial_{a}\underline{\boldsymbol{A}}_{\tau}+[\underline{\boldsymbol{A}}_{\tau},\underline{\boldsymbol{A}}_{a}]$
$\displaystyle=0\,,$ (112)

for all $a,b=2,\dots,n$ and $a\neq b$. To show integrability, we will also
need the help from the following identity

$\displaystyle\partial_{\eta}F_{jk}+(g^{(1)}_{ij}-g^{(1)}_{ik})F_{jk}+F_{ji}\
F_{ik}=0\,,$ (113)

which follows from the $\mathcal{O}(\xi^{0})$ term of (61) with
$(i,j,k)=(2,1,3)$, $\zeta_{i}=\xi$ and $\zeta_{j}=\eta-\xi$. Concentrating on
the integrability of the $\text{d}z_{a}$-components, one finds that

$\displaystyle 0$
$\displaystyle=\bigg{(}\partial_{c}\underline{\boldsymbol{A}}_{e}-\partial_{e}\underline{\boldsymbol{A}}_{c}+[\underline{\boldsymbol{A}}_{c},\underline{\boldsymbol{A}}_{e}]\bigg{)}_{ab}$
(114) $\displaystyle=\left(1-\delta_{ab}\right)\sum_{d\neq
c,e}\Bigg{[}-\left(\delta_{ac}\delta_{be}-\delta_{ae}\delta_{bc}\right)s_{1b}s_{1d}\left(\partial_{\eta}F_{ba}+(g_{db}^{(1)}-g_{da}^{(1)})F_{ba}+F_{da}F_{bd}\right)$
$\displaystyle\hskip
80.00012pt+\left(\delta_{ac}\delta_{bd}-\delta_{bc}\delta_{ad}\right)s_{1b}s_{1e}\left(\partial_{\eta}F_{ba}+(g_{eb}^{(1)}-g_{ea}^{(1)})F_{ba}+F_{be}F_{ea}\right)$
$\displaystyle\hskip
80.00012pt-\left(\delta_{ae}\delta_{bd}-\delta_{be}\delta_{ad}\right)s_{1b}s_{1c}\left(\partial_{\eta}F_{ba}+(g_{cb}^{(1)}-g_{ca}^{(1)})F_{ba}+F_{bc}F_{ca}\right)\Bigg{]}\,,$

where we have introduced the shorthand $F_{ab}=F(z_{ab},\eta)$. The
$\tau$-integrability follows similarly, but requires more complicated Fay
identities.

###  4.4 The $\mathcal{O}(\alpha^{\prime})$ solution to the differential
equations

In this section, we study the $\alpha^{\prime}$ expansions of the differential
equations (103)-(106) with $n=4$ punctures and the resulting solutions. The
purpose of this section is to obtain some zero-th order information about the
$\alpha^{\prime}$-expansion of Riemann-Wirtinger integrals; to identify the
kind of objects that appear and how they relate to known functions in the
elliptic Feynman integral and loop string amplitude literature. At leading
order in the $\alpha^{\prime}$-expansion, the Riemann-Wirtinger integrals have
an analytic structure very similar to the leading $\alpha^{\prime}$-expansion
of the analogous string integrals Kaderli:2022qeu . However, at higher orders,
the two solutions start to diverge.

We start by making the $\alpha^{\prime}$ dependence apparent in the
Mandelstams by setting $s_{ij}=\alpha^{\prime}\tilde{s}_{ij}$, where
$\tilde{s}_{ij}=k_{i}\cdot k_{j}$ are the new $\alpha^{\prime}$ independent
Mandelstams. Next, we expand the Kronecker-Eisenstein functions
$F(z_{ij},\eta)$ in the cohomology basis elements in orders of
$\alpha^{\prime}$ remembering that
$\eta=-s_{1B}+\alpha^{\prime}\left(\tilde{s}_{1A}\tau+\sum_{i=2}^{n}\tilde{s}_{1i}z_{i}\right)$.
While all of the Mandelstam variables have a true factor of $\alpha^{\prime}$,
it is unclear if $s_{1B}$ should also be assigned a power of
$\alpha^{\prime}$. In the following we take the conservative approach and
assume that $s_{1B}$ does not scale with $\alpha^{\prime}$ to ensure that
$\eta\neq 0$ since the structure of the cohomology changes in the $\eta\to 0$
limit.

To sub-leading order in $\alpha^{\prime}$,

$\displaystyle F(z_{ij},\eta)$ $\displaystyle\approx
F(z_{ij},{-}s_{1B})\left[1{+}\alpha^{\prime}\bigg{(}\tilde{s}_{1A}\tau{+}\sum_{i=2}^{n}\tilde{s}_{1i}z_{i}\bigg{)}\left[g^{(1)}(z_{ij}{-}s_{1B}){-}g^{(1)}({-}s_{1B})\right]\right].$
(115)

Similarly, the kinematic connections (103)-(106) have an $\alpha^{\prime}$
expansion

$\underline{\boldsymbol{A}}_{a}=\sum_{\ell=1}^{\infty}(\alpha^{\prime})^{\ell}\
\tilde{\underline{\boldsymbol{A}}}^{(\ell)}_{a}\,,$ (116)

that does not truncate due to the non-uniform dependence of $\eta$ on
$\alpha^{\prime}$. This means that the differential equations are not in
$\alpha^{\prime}$-form complicating the expansion of the path ordered
exponential. Still, one can obtain the $\alpha^{\prime}$ expanded solution
order-by-order where the order $k$ DEQ has contributions from all previous
orders. Explicitly, we have

$\displaystyle\text{d}_{\text{ext}}\mathbf{I}_{\gamma}^{(0)}$
$\displaystyle=0\,,$ (117)
$\displaystyle\text{d}_{\text{ext}}\mathbf{I}_{\gamma}^{(1)}$
$\displaystyle=\sum_{i=2}^{n}\text{d}z_{i}\
\tilde{\underline{\boldsymbol{A}}}^{(1)}_{i}\cdot\mathbf{I}^{(0)}_{\gamma}+\text{d}\tau\
\tilde{\underline{\boldsymbol{A}}}^{(1)}_{\tau}\cdot\mathbf{I}^{(0)}_{\gamma}\,,$
(118) $\displaystyle\text{d}_{\text{ext}}\mathbf{I}_{\gamma}^{(k\geq 1)}$
$\displaystyle=\sum_{i=2}^{n}\text{d}z_{i}\
\sum_{l+m=k}\tilde{\underline{\boldsymbol{A}}}^{(l)}_{i}\cdot\mathbf{I}^{(m)}_{\gamma}+\text{d}\tau\
\sum_{l+m=k}\tilde{\underline{\boldsymbol{A}}}^{(l)}_{\tau}\cdot\mathbf{I}^{(m)}_{\gamma}\,,$
(119)

where we have set $\mathbf{I}_{\gamma}:=\alpha^{\prime}\
[\gamma|\boldsymbol{\varphi}\rangle$. The factor of $\alpha^{\prime}$ in the
definition of $\mathbf{I}_{\gamma}$ is there to ensure that there are no
$\alpha^{\prime}$-poles in the series expansion of $\mathbf{I}_{\gamma}$ for
the contours introduced in section 5.

To explore the properties of these functions, we solve the DEQs to the first
non-trivial order in $\alpha^{\prime}$. Here, the leading order approximation
of the connection matrices is

$\displaystyle\left(\tilde{A}^{(1)}_{\tau}\right)_{ab}=\frac{\delta_{ab}}{2\pi
i}\bigg{[}\sum_{c=2}^{n}s_{1c}\ g^{(2)}(z_{ac})+s_{1a}\
\partial_{s_{1B}}F(0,-s_{1B})\bigg{]}$ $\displaystyle\hskip
60.00009pt+\frac{(1-\delta_{ab})}{2\pi i}\bigg{[}s_{1b}\
\partial_{s_{1B}}F(z_{ba},-s_{1B})\bigg{]}\,,$ (120)
$\displaystyle\left(\tilde{A}_{c=2,\dots,4}^{(1)}\right)_{ab}=\Bigg{[}\sum_{d\neq
c}\delta_{bd}\ \delta_{ac}\ \tilde{s}_{1b}\ F(z_{bc},-s_{1B})-\sum_{d\neq
c}\delta_{ad}\ \delta_{bc}\ \tilde{s}_{1c}\ F(z_{ca},-s_{1B})\Bigg{]}$
$\displaystyle\hskip
60.00009pt+\delta_{ab}\Bigg{[}-\tilde{s}_{1c}(1-\delta_{ca})\
g^{(1)}(z_{ac})+\delta_{ac}\bigg{(}2\pi i\tilde{s}_{1A}+\sum_{d\neq
c}\tilde{s}_{1d}\ g^{(1)}(z_{cd})\bigg{)}\Bigg{]}\,.$ (121)

We also set
$\mathbf{I}_{\gamma}(z_{i=2,\dots,n}^{*},\tau^{*})=\sum_{\ell=0}^{\infty}\left(\alpha^{\prime}\right)^{\ell}\mathbf{c}^{(\ell)}_{\gamma}$
as the boundary value. While the boundary values for a basis of contours is
given in section 5.3, we will keep $\mathbf{c}^{(\ell)}_{\gamma}$ generic in
this section to avoid subtleties with the limit $\tau\to i\infty$ where the
torus degenerates into a nodal sphere.

Obviously, the first term in the solution is simply the leading term of the
boundary value: $\mathbf{I}_{\gamma}^{(0)}=\mathbf{c}^{(0)}_{\gamma}$. Next,
we integrate in $\tau$ to find $\mathbf{I}_{\gamma}^{(1)}(z_{i\geq
2}^{*},\tau)=\int_{\tau^{*}}^{\tau}\text{d}\tau^{\prime}\
\tilde{\underline{\boldsymbol{A}}}^{(1)}_{\tau}(z_{i\geq
2}^{*},\tau^{\prime})\cdot\mathbf{I}_{\gamma}^{(0)}(z_{i\geq
2}^{*},\tau^{\prime})$. Explicitly,

$\displaystyle{I}_{\gamma,a}^{(1)}(z_{i\geq 2}^{*},\tau)$
$\displaystyle=c_{\gamma,a}^{(0)}\sum_{b=2}^{n}s_{1b}\log\left[\frac{\vartheta_{1}^{\prime}(z_{ab}^{*},\tau)}{\vartheta_{1}^{\prime}(z_{ab}^{*},\tau^{*})}\frac{\eta_{\text{\tiny
De}}(\tau^{*})}{\eta_{\text{\tiny
De}}(\tau)}\right]+\sum_{b=2}^{n}c_{\gamma,b}^{(0)}s_{1b}\
\mathcal{F}(z_{ba}^{*},\tau)+c_{\gamma,a}^{(1)}\,,$ (122)

where $\eta_{\text{\tiny De}}(\tau)$ is the Dedekind eta-function121212The
subscript on the Dedekind eta-function is to differentiate from the variable
$\eta=s_{1A}\tau+\sum_{i=2}^{n}s_{1i}z_{i}$.,

$\displaystyle\mathcal{F}(z,\tau)$
$\displaystyle=\int_{\tau^{*}}^{\tau}\text{d}\tau^{\prime}\
\partial_{s_{1B}}F(z,-s_{1B}|\tau^{\prime})=\int_{\tau^{*}}^{\tau}\text{d}\tau^{\prime}\
\left[\frac{1}{s_{1B}^{2}}-\sum_{k=2}^{\infty}(k-1)g^{(k)}(z,\tau^{\prime})(-s_{1B})^{k-2}\right]\,,$
$\displaystyle=\frac{\tau-\tau^{*}}{s_{1B}^{2}}-\sum_{k=2}^{\infty}(-s_{1B})^{k-2}(2\pi
i)^{k-1}\left[\Omega^{(k-1)}(z_{ba}^{*},\tau)-\Omega^{(k-1)}(z_{ba}^{*},\tau^{*})\right]\,,$
(123)

and $\Omega^{(k)}(z,\tau)$ are the known primitives of the $g^{(k)}(z,\tau)$
Wilhelm:2022wow

$\displaystyle g^{(k)}(z,\tau)=(2\pi
i)^{k-1}\partial_{z}\Omega^{(k)}(z,\tau)=\frac{(2\pi
i)^{k-1}}{k-1}\partial_{\tau}\Omega^{(k-1)}(z,\tau)\,.$ (124)

The primitives $\Omega^{(k)}$ are the symbol letters of elliptic multiple
polylogarithms (eMPLs). In fact, the $\Omega^{(k)}$ are actually depth-1 eMPLs
themselves. It is also important to note that $\mathcal{F}(0,\tau)$ is well-
defined since $\partial_{s_{1B}}F(0,-s_{1B})$ appears in the differential
equations. To see this, observe that the primitives are finite at the origin
since the $g^{(k\geq 2)}(0,\tau)<\infty$ are finite at the origin. Moreover,
note that the above expression for $\mathcal{F}$ only makes sense if the
difference $\tau-\tau^{*}$ is finite otherwise one has to apply the tangential
base point regularization of brown2014multiple  and set
$\int^{\tau}_{i\infty}\text{d}\tau^{\prime}=\tau$. However, we find boundary
values in section 5.3, which ensure that the $\tau$-integral is finite without
applying tangential base point regularization.

Once a boundary value is known at finite $\tau$ or when the $\tau$ integration
converges, the $z_{i}$-integrals are readily evaluated in terms of familiar
eMPLs

$\displaystyle{I}_{\gamma,a}^{(1)}(z_{i\geq 2},\tau)$
$\displaystyle=\left(2\pi i\tilde{s}_{1A}\
\tilde{\Gamma}\left(\begin{smallmatrix}0\\\
0\end{smallmatrix};z_{a}{-}z_{a}^{*}|\tau\right)+\sum_{b\neq a}\tilde{s}_{1b}\
\left[\tilde{\Gamma}\left(\begin{smallmatrix}1\\\
0\end{smallmatrix};z_{ab}|\tau\right)-\tilde{\Gamma}\left(\begin{smallmatrix}1\\\
0\end{smallmatrix};z_{ab}^{*}|\tau\right)\right]\right)c^{(0)}_{\gamma,a}$
$\displaystyle~{}~{}~{}+\sum_{b\neq a}\tilde{s}_{1b}\ c^{(0)}_{\gamma,b}\
\left[\Phi\left(z_{ab};s_{1B}|\tau\right)-\Phi\left(z_{ab}^{*};s_{1B}|\tau\right)\right]+{I}_{\gamma,a}^{(1)}(z_{i\geq
2}^{*},\tau)\,,$ (125)

where we define the $z$-integral of the Kronecker-Eisenstein function to be

$\Phi\left(z_{ij};s_{1B}|\tau\right):=\sum_{k\geq
0}s_{1B}^{k-1}\tilde{\Gamma}\left(\begin{smallmatrix}k\\\
0\end{smallmatrix};z_{ij}|\tau\right)~{},$ (126)

and $\tilde{\Gamma}\left(\begin{smallmatrix}0\\\
0\end{smallmatrix};z_{a}{-}z_{a}^{*}|\tau\right)=z_{a}{-}z_{a}^{*}$. Note that
this expression may only be well defined after shuffle regularization
Broedel:2019tlz .

Unsurprisingly, the first $\alpha^{\prime}$ correction,
$\mathbf{I}^{(1)}_{\gamma}$, has a functional form very similar to that found
in Kaderli:2022qeu  where there was no constraint on $\eta$. However, at order
$\mathcal{O}(\alpha^{\prime 2})$ we start to see integrals with powers of
$\tilde{s}_{1A}\tau{+}\sum_{i=2}^{n}\tilde{s}_{1i}z_{i}$ in the integrand and
terms with different transcendentality start to mix. Thus, we expect that only
the leading term $\mathbf{I}^{(1)}_{\gamma}$ is actually representative of the
analogous string integral.

Unfortunately, twisted (co)homology seems to force the condition (67) on us.
To get closer to real string integrals, we must find a way to understand the
(co)homology for unconstrained $\eta$ (67). However, at the level of the
differential equation, we can always shift the $\eta$-derivatives similarly to
(110) to define differential operators that are unconstrained by (67).

###  4.5 The $\eta\to 0$ limit

The $\eta$-dependent differential forms considered in section 4 are useful
because they can be thought of as generating integrals where one performs a
formal series expansion in $\eta$. On the other hand, true string integrals do
not depend on the parameter $\eta$. For this reason, it is interesting to
examine the $\eta\to 0$ limit of the Riemann-Wirtinger integrals. However,
unlike true string integrands, the Riemann-Wirtinger integrand must not have
$B$-cycle monodromies.131313The integral over the loop-momentum guarantees the
double periodicity of string integrals. Therefore, it is possible to have
string integrands that are not doubly periodic. Anything that breaks this
condition is not part of our cohomology.

It turns out that the spanning set $\\{\xi^{(p)}_{a}\\}$ of
$H^{1}(M,{\mathcal{L}_{\omega,\eta}})$ (equation (82)) has a smooth $\eta\to
0$ limit that also spans the cohomology when $\eta=0$:
$H^{1}(M,\mathcal{L}_{\omega,\eta=0})$ Goto2022 . Explicitly, we let
$\zeta^{(p)}=\lim_{\eta\to 0}\xi^{(p)}$ and find

$\displaystyle\zeta^{(p)}_{1}$ $\displaystyle=\text{d}z_{1},$
$\displaystyle\zeta^{(p)}_{p}$
$\displaystyle=\partial_{z_{1}}g^{(1)}(z_{1p})\,\text{d}z_{1},$ (127)
$\displaystyle\zeta^{(p)}_{a\geq 2}$
$\displaystyle=\left[g^{(1)}(z_{1a})-g^{(1)}(z_{1p})\right]\,\text{d}z_{1}\quad\text{for}\quad
a\neq p.$

Each element of this set has no $B$-cycle monodromy. Also note that, to have a
basis, the element $\zeta^{(p)}$ with a double pole must be included. This
spanning set is also subject to the relation

$\displaystyle 2\pi is_{1A}\ |\zeta^{(p)}_{1}\rangle+\underset{j\neq
p}{\sum_{j=2}^{n}}s_{1j}\,\ |\zeta^{(p)}_{j}\rangle\simeq 0$ (128)

that follows from the $\eta\to 0$ limit of (4.1). Moreover, in this limit, the
defining equation for $\eta$ places restrictions on the punctures in the
problem

$\displaystyle 0=s_{1A}\tau+\sum_{i=2}^{n}s_{1i}z_{i}-s_{1B}.$ (129)

Importantly, for the differential equation, this means that the differentials
satisfy the relation

$\displaystyle 0=s_{1A}\text{d}\tau+\sum_{i=2}^{n}s_{1i}\text{d}z_{i}\ .$
(130)

We can compute the intersection numbers directly in the $\eta\neq 0$ limit
case using (88) as before. Of course, this yields the same result as taking
the $\eta\to 0$ limit of (93)-(4.2). For example, at $n=4$ with $p=4$, the
explicit form of the intersection matrix is

$\displaystyle\frac{\langle\zeta^{(4)}_{a}|\zeta^{(4)}_{b}\rangle}{2\pi
i}=\scalebox{0.9}{$\begin{pmatrix}0&0&0&\frac{1}{s_{14}-1}\\\
0&\frac{s_{13}}{s_{12}s_{14}}&-\frac{1}{s_{14}}&\frac{2i\pi
s_{1}+s_{13}\left[g^{(1)}(z_{24})-g^{(1)}(z_{34})\right]}{(s_{14}-1)s_{14}}\\\
0&-\frac{1}{s_{14}}&\frac{s_{12}}{s_{13}s_{14}}&\frac{2i\pi
s_{1}-s_{12}\left[g^{(1)}(z_{24})-g^{(1)}(z_{34})\right]}{(s_{14}-1)s_{14}}\\\
\frac{1}{s_{14}+1}&\frac{2i\pi
s_{1}+s_{13}\left[g^{(1)}(z_{24})-g^{(1)}(z_{34})\right]}{s_{14}(s_{14}+1)}&\frac{2i\pi
s_{1}-s_{12}\left[g^{(1)}(z_{24})-g^{(1)}(z_{34})\right]}{s_{14}(s_{14}+1)}&\frac{\langle\zeta^{(4)}_{4}|\zeta^{(4)}_{4}\rangle}{2\pi
i}\end{pmatrix}$}\,,$ (131)

where

$\displaystyle\frac{\langle\zeta^{(4)}_{4}|\zeta^{(4)}_{4}\rangle}{2\pi i}$
$\displaystyle=\frac{1}{(s_{14}-1)s_{14}(s_{14}+1)}\bigg{[}-4\pi^{2}s_{1}^{2}-2s_{14}\left(s_{12}-s_{14}\right)G_{2}(\tau)$
$\displaystyle\qquad+4i\pi s_{1}\left(s_{12}\ g^{(1)}(z_{24})+s_{13}\
g^{(1)}(z_{34})\right)+s_{12}s_{13}\left(g^{(1)}(z_{24})-g^{(1)}(z_{34})\right)^{2}$
$\displaystyle\qquad+2s_{14}\left(s_{12}\ g^{(2)}(z_{24})+s_{13}\
g^{(2)}(z_{34})\right)\bigg{]}\,.$ (132)

Choosing $\zeta^{(p)}_{p}$ and any set of $n-2$ other $\zeta^{(p)}_{a\neq p}$
forms a basis of the cohomology $H^{1}(M,\mathcal{L}_{\omega,\eta=0})$. After
we choosing a basis, the DEQs can be computed in the usual way. For example,
choosing $\zeta^{(p)}_{a>1}$ for our basis and choosing to eliminate
$\text{d}z_{4}$ using the $\eta=0$ condition, we find the following
$\text{d}z_{2}$ component of the differential equation

$\displaystyle\underline{\boldsymbol{A}}_{\zeta}|_{\text{d}z_{2}}=\sum_{i=1}^{9}\underline{\boldsymbol{\Xi}}_{i}X_{i}$
(133)

where the $X_{i}$ are monomials in the $g$-functions (including the constant
function)

$\displaystyle X_{i}\in\bigg{\\{}$ $\displaystyle
1,g^{(1)}\left(z_{23}\right),g^{(1)}\left(z_{34}\right),g^{(1)}\left(z_{24}\right),g^{(2)}\left(z_{34}\right),g^{(2)}\left(z_{24}\right),$
$\displaystyle\qquad\left(g^{(1)}\left(z_{34}\right)\right)^{2},g^{(1)}\left(z_{24}\right)g^{\text{(1)}}\left(z_{34}\right),\left(g^{(1)}\left(z_{24}\right)\right)^{2}\bigg{\\}}\,,$
(134)

and the coefficient matrices are

$\displaystyle\begin{aligned}
&\underline{\boldsymbol{\Xi}}_{1}=\left(\begin{array}[]{ccc}-\frac{2i\pi
s_{1}s_{13}\left(s_{12}-s_{14}\right)}{s_{12}s_{14}^{2}}&\frac{2i\pi
s_{1}s_{12}}{s_{14}^{2}}&\frac{s_{13}G_{2}(\tau)}{s_{14}}-\frac{4\pi^{2}s_{1}^{2}s_{12}}{\left(1-s_{14}\right)s_{14}^{2}}\\\
\frac{2i\pi s_{1}s_{12}}{s_{14}^{2}}&\frac{2i\pi
s_{1}s_{12}}{s_{14}^{2}}&-\frac{s_{12}G_{2}(\tau)}{s_{14}}-\frac{4\pi^{2}s_{1}^{2}s_{12}}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&0&0\\\ \end{array}\right),\\\
&\underline{\boldsymbol{\Xi}}_{2}=\left(\begin{array}[]{ccc}-\frac{s_{13}}{s_{12}}&1&0\\\
1&-\frac{s_{12}}{s_{13}}&0\\\ 0&0&0\\\
\end{array}\right),\qquad\underline{\boldsymbol{\Xi}}_{3}=\left(\begin{array}[]{ccc}-\frac{s_{12}s_{13}}{s_{14}^{2}}&\frac{s_{12}^{2}}{s_{14}^{2}}&-\frac{4i\pi
s_{1}s_{12}s_{13}}{\left(1-s_{14}\right)s_{14}^{2}}\\\
\frac{s_{12}^{2}}{s_{14}^{2}}&-\frac{s_{12}^{3}}{s_{13}s_{14}^{2}}&\frac{2i\pi
s_{1}s_{12}\left(s_{12}-s_{13}\right)}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&0&0\\\ \end{array}\right),\\\
&\underline{\boldsymbol{\Xi}}_{4}=\left(\begin{array}[]{ccc}\frac{s_{13}^{3}}{s_{12}s_{14}^{2}}&-\frac{s_{13}^{2}}{s_{14}^{2}}&\frac{4i\pi
s_{1}s_{12}s_{13}}{\left(1-s_{14}\right)s_{14}^{2}}\\\
-\frac{s_{13}^{2}}{s_{14}^{2}}&\frac{s_{12}s_{13}}{s_{14}^{2}}&-\frac{2i\pi
s_{1}s_{12}\left(s_{12}-s_{13}\right)}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&-\frac{2i\pi s_{1}s_{12}}{\left(-s_{14}-1\right)s_{14}}&0\\\
\end{array}\right),\qquad\underline{\boldsymbol{\Xi}}_{5}=\left(\begin{array}[]{ccc}0&0&\frac{2s_{12}s_{13}}{s_{14}^{2}}\\\
0&0&-\frac{2s_{12}^{2}}{s_{14}^{2}}\\\ 0&\frac{s_{12}}{s_{14}+1}&0\\\
\end{array}\right),\\\
&\underline{\boldsymbol{\Xi}}_{6}=\left(\begin{array}[]{ccc}0&0&\frac{2s_{13}^{2}}{s_{14}^{2}}\\\
0&0&-\frac{2s_{12}s_{13}}{s_{14}^{2}}\\\ 0&\frac{2s_{12}}{s_{14}+1}&0\\\
\end{array}\right),\qquad\underline{\boldsymbol{\Xi}}_{7}=\left(\begin{array}[]{ccc}0&0&-\frac{s_{12}\left(s_{12}+1\right)s_{13}}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&0&\frac{s_{12}^{2}\left(s_{12}+1\right)}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&0&0\\\ \end{array}\right),\\\
&\underline{\boldsymbol{\Xi}}_{8}=\left(\begin{array}[]{ccc}0&0&-\frac{2s_{12}s_{13}^{2}}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&0&\frac{2s_{12}^{2}s_{13}}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&\frac{s_{12}^{2}}{\left(s_{14}+1\right)s_{14}}&0\\\
\end{array}\right),\qquad\underline{\boldsymbol{\Xi}}_{9}=\left(\begin{array}[]{ccc}0&0&-\frac{s_{13}^{2}\left(s_{13}+1\right)}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&0&\frac{s_{12}s_{13}\left(s_{13}+1\right)}{\left(1-s_{14}\right)s_{14}^{2}}\\\
0&\frac{s_{12}s_{13}}{\left(s_{14}+1\right)s_{14}}&0\\\
\end{array}\right).\end{aligned}$ (135)

It is interesting to note that the $\eta=0$ differential equations are much
more complicated than their $\eta\neq 0$ cousins. Most of the complexity comes
from having to use a form with a double pole in our basis. This translates
into double poles in the differential equation. That is, the double poles
coming from the monomials
$\left(g^{(1)}\left(z_{34}\right)\right)^{2},g^{(1)}\left(z_{24}\right)g^{\text{(1)}}\left(z_{34}\right)$
and $\left(g^{(1)}\left(z_{24}\right)\right)^{2}$ are not spurious.
Connections with higher order poles appear at higher genus and it is possible
to make sense of iterated integrals with such a connection
enriquez2023analogues ; enriquez2023elliptic .

The $\eta=0$ twisted (co)homology was previously studied in
ghazouani2016moduli . Here, the authors study the moduli space of flat tori
and elliptic hypergeometric functions using algebro-geometric techniques
similar to those used in this paper. In particular, they compute the $n=2$
differential equations (see their appendix B) and construct the double copy in
order to obtain an explicit expression for the so-called Veech map. In both
cases, we find agreement.

##  5 Twisted homology of the Riemann-Wirtinger integral

After reviewing the construction of the twisted homology for Riemann-Wirtinger
integrals, we give a basis for homology (section 5.1). Then, in section 5.2,
we compute the intersection matrix that becomes the double copy kernel in
section 6. In section 5.3, we describe how to take the $\tau\to i\infty$ limit
of the Riemann Wirtinger integral and obtain boundary values for the
differential equations of section 4.3. We also give an alternative
prescription to compute boundary values that works at finite $\tau$ for all
but the $A$\- and $B$-cycles.

###  5.1 Basis of homology

As reviewed in section 2, twisted or loaded contours are represented by the
tensor product of a “normal” topological contour with a branch choice. That
is, given a topological contour $\tilde{\gamma}$ on the manifold $M$, we
“load” it with a branch choice of the twist $u_{\tilde{\gamma}}$ (section of
${\check{\mathcal{L}}_{\omega,\eta}}$) to form a twisted contour $\gamma$

$\displaystyle\gamma=\tilde{\gamma}\otimes u_{\tilde{\gamma}}\,.$ (136)

Technically, the contour should also be loaded with a branch choice
$e_{\tilde{\gamma}}$ on $E$ (section of $\check{\mathcal{L}}_{\eta}$).
However, since we can regard the Kronecker-Eisenstein function $F(z,\eta)$ as
a global meromorphic section of $\check{\mathcal{L}}_{\eta}$, such a loading
will not play an important role in the computation of intersection indices.

The twisted homology is defined in the usual way

$H_{m}(M,{\check{\mathcal{L}}_{\omega,\eta}})=\frac{\text{ker}\left(\partial_{\omega,\eta}:C_{m}(M,{\check{\mathcal{L}}_{\omega,\eta}})\to
C_{m-1}(M,{\check{\mathcal{L}}_{\omega,\eta}})\right)}{\text{im}\left(\partial_{\omega,\eta}:C_{m+1}(M,{\check{\mathcal{L}}_{\omega,\eta}})\to
C_{m}(M,{\check{\mathcal{L}}_{\omega,\eta}})\right)}$ (137)

where $C_{m}(M,{\check{\mathcal{L}}_{\omega,\eta}})$ is the space of twisted
$m$-contours. Elements of the twisted homology are equivalence classes of
closed twisted cycles (twisted contours with no boundary) modulo exact cycles
(twisted contours that are boundaries) and denoted by the square bra.
Explicitly, the homology class of an $m$-cycle $\gamma\otimes
u_{\gamma}\in\ker(\partial_{\omega,\eta}:C_{m}(,\check{\mathcal{L}}_{\omega})\to
C_{m-1}(,\check{\mathcal{L}}_{\omega}))$ is

$\displaystyle[\gamma\otimes u_{\gamma}|=\\{\gamma\otimes
u_{\gamma}+\partial_{\omega}(\beta\otimes u_{\beta})\,|\,(\beta\otimes
u_{\beta})\in C_{m+1}(,\check{\mathcal{L}}_{\omega})\\}\,.$ (138)

That is, a representative of a twisted homology class, $\gamma\otimes
u_{\gamma}$, is a twisted contour that has no boundary and is not a boundary
itself. We also define the dual twisted homology by exchanging the dual local
system $\check{\mathcal{L}}_{\omega,\eta}$ in (137) for the local system
$\mathcal{L}_{\omega,\eta}$

$[H_{m}(M,{\check{\mathcal{L}}_{\omega,\eta}})]^{\vee}:=H_{m}(M,{\mathcal{L}_{\omega,\eta}})=\frac{\text{ker}\left(\check{\partial}_{\omega,\eta}:C_{m}(M,{\mathcal{L}_{\omega,\eta}})\to
C_{m-1}(M,{\mathcal{L}_{\omega,\eta}})\right)}{\text{im}\left(\check{\partial}_{\omega,\eta}:C_{m+1}(M,{\mathcal{L}_{\omega,\eta}})\to
C_{m}(M,{\mathcal{L}_{\omega,\eta}})\right)}$ (139)

where
${\mathcal{L}_{\omega,\eta}}=[{\check{\mathcal{L}}_{\omega,\eta}}]^{\vee}={\check{\mathcal{L}}_{\omega,\eta}}(-s_{1A},-s_{12},\dots,-s_{1n},-s_{1B})=\mathbb{C}u^{-1}$,
$C_{m}(M,{\mathcal{L}_{\omega,\eta}})$ is the space of dual twisted
$m$-contours and $\check{\partial}_{\omega,\eta}=\partial_{-\omega,-\eta}$ is
the dual boundary operator. Representatives of dual homology classes are
topological cycles loaded with a branch choice of the inverse twist:
$\check{\gamma}=\check{\gamma}^{\prime}\otimes
u_{\check{\gamma}^{\prime}}^{-1}$. Given a representative of the dual
homology, its class is denoted by a square ket
$|\check{\gamma}^{\prime}\otimes u_{\check{\gamma}^{\prime}}^{-1}]$.

Since only the middle dimensional twisted cohomology is non-trivial, $\dim
H_{0}=\dim H_{2}=0$. We also know from Mano2012 ; Goto2022  that $\dim
H_{1}=n-1$ where $n$ is the number of punctures. Of course, this counting is
the same for the cohomology (section 4).

Figure 2:  An over-complete set of cycles in blue and an over-complete set of
dual cycles in  green for $n=6$ punctures. The (dual) cycles depicted by solid
lines form a basis and the dashed  red line denotes our choice of  branch cut.
All of the cycles start at the point $z_{2}$ while all of the dual cycles
start at the point $z_{n}=z_{6}$. The solid $n-1=5$ (dual) cycles define a
basis for the (dual) twisted homology. The dashed cycles are naturally defined
but linearly depend on the cycles denoted by solid lines. The contours drawn
here are technically cycles because the end points do not belong to $M$. Such
cycles are known as locally finite or Borel-Moore cycles in the mathematics
literature.

The remaining task is to provide a basis for the twisted homology and its
dual. A particularly convenient choice of topological cycles is depicted in
figure 2 with cycles in blue and dual cycles in  green. All of the cycles
start at the puncture $z_{2}$ while the dual cycles start at $z_{n}$ (where
$n=6$ in figure 2). These topological cycles are then loaded with a branch of
$u$ ($\check{u}$) to form twisted (dual) cycles where the branch cut is
denoted by the dashed red line in figure 2. We will always choose branches
such that on the interval $\Delta_{k}=[z_{k},z_{k+1}]_{k=2,\dots,n-1}$

$\displaystyle\arg\left[\frac{z_{1}-z_{j}}{z_{1}-z_{i}}\right]=\pi\quad\text{
for }2\leq i\leq k<j\leq n\,.$ (140)

Furthermore, note that the cycles in figure 2 intersect only one dual cycle.
This is enough to guarantee that the corresponding intersection matrix is
diagonal.

We have actually drawn an over complete set of $n$ (dual) cycles in figure 2
since we know that $\dim H_{1}=n-1$. To find a linear relation amongst these
cycles, start with a closed circular contour that does not encircle any
singularities. Then, expand this contour without crossing the branch cuts
until the contour runs along the edge of the fundamental parallelogram and the
branch cut. Rewriting the deformed contour in terms of the set
$\\{\gamma_{2A},\gamma_{2B}\\}\cup\\{\gamma_{2j}\\}_{j=3,\dots,n}$ yields the
relation Goto2022

$\displaystyle(e^{2\pi is_{1A}}-1)[\gamma_{2B}|+(1-e^{2\pi
is_{1B}})[\gamma_{2A}|-\sum_{j=3}^{n}e^{-2\pi
i(s_{12}+\cdots+s_{1j})}(1-e^{2\pi is_{1j}})[\gamma_{2j}|\simeq 0,$ (141)

(a similar identity exists for the set of dual cycles
$\\{\check{\gamma}_{nA},\check{\gamma}_{nB}\\}\cup\\{\check{\gamma}_{nj}\\}_{j=2,\dots,n-1}$).
We will often choose the cycles drawn in solid lines as our basis for the
(dual) homology. Moreover, we will also rediscover the relation (141) using
the intersection indices in the next section.

In string theory context, relations like (141) are often called monodromy
relations because one projects a contour with one color ordering onto another
with different color ordering, which corresponds to the winding or exchanging
of punctures. Such identities relate various color orderings for open string
amplitudes. The tree-level gauge theory equivalent of (141) are the so-called
Kleiss-Kuijf and _BCJ_ relations. The Kleiss-Kuijf follow from the leading
$\alpha^{\prime}$ expansion of the string monodromy relations while the BCJ
relations follow from the subleading order in $\alpha^{\prime}$.141414Note
that the BCJ relations are _linear_ relations among color-ordered amplitudes
and should not be confused with color-kinematics duality.

###  5.2 The intersection index: an inner product on homology

In this section, we regularize the cycles defined in the previous section in
order to compute the intersection matrix that becomes the double copy kernel
in section 6. The intersection indices for Riemann-Wirtinger integrals were
first derived in ghazouani2016moduli  for $\eta=0$ and more recently revisited
in Goto2022  for $\eta\neq 0$.

As reviewed in section 2, the intersection index (33) is a pairing between the
twisted homology with local system ${\check{\mathcal{L}}_{\omega,\eta}}$ and
the twisted homology with local system ${\mathcal{L}_{\omega,\eta}}$

$\displaystyle[\bullet|\bullet]:$ $\displaystyle\
H_{p}(M,\check{\mathcal{L}}_{\omega})\times
H_{n-p}(M,\mathcal{L}_{\omega})\to\mathbb{C}\,,$ (142)

given by

$\displaystyle[\gamma\otimes
u_{\gamma}|\check{\gamma}\otimes\check{u}_{\check{\gamma}}]:=\sum_{x\in\gamma\cap\check{\gamma}}\left(u_{\gamma}\check{u}_{\check{\gamma}}|_{x}\right)\big{[}\mathrm{Reg}[\gamma]|\check{\gamma}\big{]}^{\text{top}}_{x}\,,$
(143)

where the set $\gamma\cap\check{\gamma}$ corresponds to the set of all points
where $\gamma$ and $\check{\gamma}$ intersect and the factor
$(u_{\gamma}\check{u}_{\check{\gamma}})|_{x}$, evaluates to a phase. Here,
$[\bullet|\bullet]^{\text{top}}_{x}$ is the topological intersection index at
$x$ that evaluates to $\pm 1$ depending on the relative orientation of
$\gamma$ and $\check{\gamma}$. Following the conventions of Mizera:2017cqs ;
Mimachi_2003 , we define

$\displaystyle\begin{gathered}\includegraphics[scale={0.5}]{figs/topological_intersection_numbers.png}\end{gathered}\,.$
(145)

The only remaining piece is to specify the map $\mathrm{Reg}$.

Figure 3:  The over-complete set of cycles (dual cycles) subject to (141) in
the compact homology (dual homology). Those depicted by solid lines form a
basis. In particular, this choice has an orthogonal intersection matrix since
each cycle only intersects one dual cycles!

In order for the definition (143) to make sense, at least one contour should
have compact support. Thus, $\mathrm{Reg}$ is a map from the normal151515To be
precise, the homology described in section 5.1 is called locally finite or
Borel-Moore hwa1966homology . twisted homology described in section 5.1 to the
compactly supported twisted homology

$\displaystyle\mathrm{Reg}:H_{m}(M,{\check{\mathcal{L}}_{\omega,\eta}})\to
H_{m}^{c}(M,{\check{\mathcal{L}}_{\omega,\eta}})\,.$ (146)

When all of the exponents $s_{\bullet}$ are generic, this map is an
isomorphism and we can use it to compute intersections involving non-compact
cycles.

Compact cycles cannot have endpoints like in figure 2 even if the endpoints do
not belong to the space (as in our case). Thus, compact cycles must wind
around the punctures instead of ending at the punctures. This necessarily
means that the compact cycles will cross a branch cut. To be a twisted cycle
with compact support, it must have no boundary. Because such a cycle crosses
branch cuts there will be relative phases that prevent
$\partial_{\omega,\eta}$ from vanishing. Only by breaking up the compact cycle
into pieces and combining these pieces with particular phases do we obtain a
genuine compact twisted cycle.

This procedure is best illustrated through examples. Consider the cycles of
figure 3 in blue. Clearly, these are compact. The compact cycles corresponding
to the non-compact $\gamma_{2j=3\dots,n,A,B}$ cycles have been broken up into
three pieces: one encircling the puncture $z_{2}$, one encircling the puncture
$z_{j}$ and one connecting each of these circles below the branch cut. When
$j=A$ or $j=B$, the compact cycle encircles $z_{2}{+}1$ or $z_{2}{+}\tau$. To
understand how to combine the contours $m_{j},S_{j}$ and $\ell_{2j}$ into a
genuine compact twisted cycle, we enumerate their twisted boundaries:

$\displaystyle\partial_{\omega,\eta}m_{j}$
$\displaystyle=\begin{cases}P^{(2)}_{j+1}-P^{(2)}_{j}&\text{for}\
i\in\\{1,2,3\\}\,,\\\ e^{2\pi is_{12}}P^{(2)}_{1}-P^{(2)}_{0}&\text{for}\
j=0\,,\end{cases}$ (147) $\displaystyle\partial_{\omega,\eta}S_{j}$
$\displaystyle=(e^{2\pi is_{1j}}-1)P^{(j)}_{0}\quad\text{for}\
j\in\\{3,\dots,n\\},$ $\displaystyle\partial_{\omega,\eta}\ell_{2j}$
$\displaystyle=\begin{cases}P_{0}^{(j)}-P^{(2)}_{0}&\text{for}\
i\in\\{1,2,3\\}\,,\\\ e^{2\pi is_{A}}P^{(2)}_{2}-P^{(2)}_{0}&\text{for}\
j=A\,,\\\ e^{-2\pi is_{B}}P^{(2)}_{3}-P^{(2)}_{1}&\text{for}\
j=B\,.\end{cases}$

Using (147), the reader can verify that the regulated contours

$\displaystyle\mathrm{Reg}[\gamma_{2A}]$ $\displaystyle:=\frac{m_{0}+e^{2\pi
is_{12}}(m_{1}+m_{2}+m_{3})}{e^{2\pi is_{12}}-1}{+}\ell_{2A}{-}e^{2\pi
is_{1A}}\frac{(m_{2}+m_{3}+m_{0})+e^{2\pi is_{1A}}m_{1}}{e^{2\pi is_{12}}-1}$
(148) $\displaystyle\mathrm{Reg}[\gamma_{2B}]$
$\displaystyle:=\frac{m_{1}+m_{2}+m_{3}+m_{0}}{e^{2\pi
is_{12}}-1}+\ell_{2B}-e^{-2\pi
is_{1B}}\frac{(m_{3}+m_{0})+(m_{1}+m_{2})e^{2\pi is_{12}}}{e^{2\pi
is_{12}}-1}$ (149) $\displaystyle\mathrm{Reg}[\gamma_{2j}]$
$\displaystyle:=\frac{m_{0}+e^{2\pi is_{12}}(m_{1}+m_{2}+m_{3})}{e^{2\pi
is_{12}}-1}+\ell_{2j}-\frac{S_{j}}{e^{2\pi is_{1j}-1}}\qquad\text{for}\quad
j=3,\dots,n$ (150)

have no twisted boundary. In practice, this regularisation procedure is
equivalent to an analytic continuation by using a Pochhammer contour
Mizera:2017cqs . The regularization of these cycles almost identical to the
genus-zero case.

Since the intersection matrix is a basis dependent object, there is no unique
best choice. Therefore, we will provide the intersection matrix for two
choices of dual cycles in this section. Choosing the dual basis in figure 2
and 3, yields the following diagonal intersection matrix

$\displaystyle H^{(2n)}_{ab}$
$\displaystyle:=[\gamma_{2a}|\check{\gamma}_{nb}]$ (151)
$\displaystyle=\mathrm{diag}\scalebox{0.98}{$\left(\frac{e^{2\pi
i(s_{12}+s_{13})}}{1-e^{2\pi is_{13}}},\frac{e^{2\pi
i(s_{12}+s_{13}+s_{14})}}{1-e^{2\pi is_{14}}},\cdots,\frac{e^{2\pi
i(s_{12}+s_{13}+\dots+s_{1,n-1})}}{1-e^{2\pi is_{1,n-1}}},e^{-2\pi
i(s_{1n}+s_{1B})},e^{-2\pi is_{1A}}\right)$},$

where $a=3,\dots,n-1,A,B$ and $b=3,\dots,n-1,B,A$. Note that we have to invert
the ordering of the $A$\- and $B$-cycles in the dual basis to make the
intersection matrix diagonal. This form of the intersection matrix is most
useful in the decomposition of identity (35) since
$\underline{\boldsymbol{H}}^{(2n)}$ is easy to invert. For example, we can
prove (141) by applying (35) to $[\gamma_{2n}|$

$\displaystyle[\gamma_{2n}|$
$\displaystyle\simeq\sum_{a=3}^{n-1}\frac{[\gamma_{2n}|\check{\gamma}_{na}]}{[\gamma_{2a}|\check{\gamma}_{na}]}[\gamma_{2a}|+\frac{[\gamma_{2n}|\check{\gamma}_{nB}]}{[\gamma_{2A}|\check{\gamma}_{nB}]}[\gamma_{2A}|+\frac{[\gamma_{2n}|\check{\gamma}_{nA}]}{[\gamma_{2B}|\check{\gamma}_{nA}]}[\gamma_{2B}|=\frac{-\mathrm{LHS}\eqref{eq:homology_cycle_overcomplete_basis}}{1-e^{2\pi
is_{1n}}}\,.$ (152)

Here, we have used (151) and

$\displaystyle[\gamma_{2n}|\check{\gamma}_{nb}]_{b=3,\dots,n-1,B,A}$
$\displaystyle=\frac{-1}{1-e^{2i\pi s_{1n}}}\left\\{1,\dots,1,e^{2\pi
is_{1n}}(1-e^{-2\pi is_{1B}}),(1-e^{2\pi is_{1A}})\right\\}\,.$ (153)

While the dual basis $\\{\check{\gamma}_{nb}\\}_{b=3,\dots,n-1,B,A}$ is
convenient for many computations, it may or may not be the best for the double
copy because it requires evaluating two sets of twisted periods:
$[\gamma_{2a}|\varphi_{b}\rangle$ and
$\langle\check{\varphi}_{a}|\check{\gamma}_{nb}]$. On the other hand, if we
use the basis of topological cycles $\gamma_{2a}$ for the dual basis as well
(i.e., use
$\check{\gamma}_{2b}=[\gamma_{2b}]^{\vee}=\gamma_{2b}|_{s_{\bullet}\to-
s_{\bullet}}$ as our dual basis), we could easily obtain the dual periods from
the “known” integrals $[\gamma_{2a}|\varphi_{b}\rangle$. The price one pays is
that the intersection matrix becomes incredibly dense and hard to invert.
Explicitly, we set

$\displaystyle H^{(22)}_{ab}:=[\gamma_{2a}|\check{\gamma}_{2b}]\,,$ (154)

for $a=3,\dots,n-1,A,B$ and $b=3,\dots,n-1,A,B$ where Goto2022

$\displaystyle\begin{aligned} &=\begin{cases}\frac{e^{2\pi is_{1j}}}{1-e^{2\pi
is_{1j}}}&\text{for}~{}j<k\\\ \frac{1}{1-e^{2\pi is_{1j}}}&~{}~{}~{}~{}j>k,\\\
\end{cases}\qquad\qquad~{}[\gamma_{2j}|\check{\gamma}_{2j}]=\frac{1-e^{2\pi
i(s_{12}+s_{1j})}}{(1-e^{2\pi is_{12}})(1-e^{2\pi is_{1j}})},\\\
[\gamma_{2j}|\check{\gamma}_{2A}]&=\frac{e^{2\pi is_{12}}(1-e^{-2\pi
is_{1A}})}{1-e^{2\pi
is_{12}}},\qquad\qquad~{}~{}[\gamma_{2A}|\check{\gamma}_{2j}]=\frac{1-e^{2\pi
is_{1A}}}{1-e^{2\pi is_{12}}},\\\
[\gamma_{2j}|\check{\gamma}_{2B}]&=\frac{e^{2\pi is_{12}}(1-e^{2\pi
is_{1B}})}{1-e^{2\pi
is_{12}}},\qquad\qquad~{}~{}~{}~{}[\gamma_{2B}|\check{\gamma}_{2j}]=\frac{1-e^{-2\pi
is_{1B}}}{1-e^{2\pi is_{12}}},\\\
[\gamma_{2A}|\check{\gamma}_{2A}]&=-\frac{(e^{2\pi is_{1A}}-1)(e^{2\pi
is_{1A}}-e^{2\pi is_{12}})}{e^{2\pi is_{1A}}(1-e^{2\pi is_{12}})},\\\
[\gamma_{2B}|\check{\gamma}_{2B}]&=-\frac{(e^{2\pi is_{1B}}-1)(e^{2\pi
is_{1B}}-e^{2\pi is_{12}})}{e^{2\pi is_{1B}}(1-e^{2\pi is_{12}})},\\\
[\gamma_{2A}|\check{\gamma}_{2B}]&=\frac{e^{2\pi is_{12}}-e^{2\pi
i(s_{1A}+s_{12})}-e^{2\pi i(s_{1B}+s_{12})}+e^{2\pi
i(s_{1B}+s_{1A})}}{1-e^{2\pi is_{12}}},\\\
[\gamma_{2B}|\check{\gamma}_{2A}]&=\frac{1-e^{-2\pi is_{1B}}-e^{-2\pi
is_{1A}}+e^{2\pi i(s_{12}-s_{1B}-s_{1A})}}{1-e^{2\pi is_{12}}}.\end{aligned}$
(155)

Pedagogical computations of some of these intersection indices can be found in
appendix B.

Even though it is not obvious that $\underline{\boldsymbol{H}}^{(22)}$ is
invertible, its determinant is easily computed

$\det(\underline{\boldsymbol{H}}^{(22)})=\frac{1-e^{-2\pi
is_{1n}}}{\prod_{m=2}^{n-1}(1-e^{2\pi is_{1m}})}\neq 0\,,$ (156)

and non-vanishing. Similarly, the determinant of
$\underline{\boldsymbol{H}}^{(2n)}$ is also non-vanishing. However, notice
that these determinants are finite only when
$s_{12},\dots,s_{1n-1}\notin\mathbb{Z}$. Thus, the bases of twisted cycles
introduced in this section are only valid when the Mandelstams are generic:
$s_{12},\dots,s_{1n}\notin\mathbb{Z}$. Also note that when $s_{1A}=s_{1B}$

$\displaystyle\begin{pmatrix}[\gamma_{2A}|\check{\gamma}_{2A}]&[\gamma_{2A}|\check{\gamma}_{2B}]\\\
[\gamma_{2B}|\check{\gamma}_{2A}]&[\gamma_{2B}|\check{\gamma}_{2B}]\end{pmatrix}=\begin{pmatrix}\phantom{-}0&1\\\
-1&0\end{pmatrix}$ (157)

and we can identify $\gamma_{2A}$ and $\gamma_{2B}$ with the usual basis of
the non-twisted homology $H_{1}(E,\mathbb{Z})$.

Before closing this section, we note that intersection indices can be used to
compute the monodromy matrices associated to $z_{j}$ $A$\- and $B$-cycles as
well as encircling $z_{k\geq 2}$ with $z_{j\geq 2}$ (i.e., the circuit matrix
$M_{jk}$) Goto2022 . While we do not explicitly need these matrices in this
work, it is worth stressing that their computation is rather straightforward
from the intersection point of view. Moreover, these matrices may be the key
to understanding Riemann-Witinger integrals when more than one puncture is
integrated (see section 7 for some speculation on how this may work)! While
the derivation of the circuit matrices is almost identical to the analogous
genus-zero case, something new and interesting happens when computing the
$z_{j\geq 2}$ $A$\- and $B$-cycle monodromy matrices. In addition to a phase,
the $z_{j\geq 2}$ $A$\- and $B$-cycles induce a change in the local system
where

$\displaystyle z_{j\geq 2}\ A\text{-cycle}:s_{1B}\mapsto s_{1B}-s_{1j}\,,$
(158) $\displaystyle z_{j\geq 2}\ B\text{-cycle}:s_{1A}\mapsto
s_{1A}+s_{1j}\,.$ (159)

The corresponding monodromy matrices realize a linear map from the homology
with $s_{1A}$ and $s_{1B}$ to the homology with $s_{1B}-s_{1j}$ or
$s_{1A}+s_{1j}$. For more details, see Goto2022 .

###  5.3 Boundary values

In this section, we show that the $\tau\to i\infty$ limit of the Riemann-
Wirtinger family corresponds to generalized Lauricella-D hypergeometric
functions. This degeneration to a nodal sphere yields one method for computing
the boundary values for the differential equations derived in section 4.3. For
$n=3$, this degeneration can be found in mano2009 . However, we fix a typo
there and provide an $n$-point generalization that has been numerically
verified. Finally, using a neat trick, we also provide boundary conditions at
finite $\tau$ for all but the $A$\- and $B$-cycles.

To understand how to compute this degenerate limit, we need to know the form
of the integrand in the $\tau\to i\infty$ limit and how the cycles on the
torus map to cycles on the nodal sphere. Using

$\displaystyle\frac{\vartheta_{1}(z_{1a}+\eta_{0}+s_{1A}\tau)}{\vartheta_{1}(\eta_{0}+s_{1A}\tau)}$
$\displaystyle\ \underset{\tau\to i\infty}{\longrightarrow}e^{\pi
iz_{1a}}\quad\text{for}\quad 0<\mathrm{Re}(s_{1A})<1\,,$ (160)
$\displaystyle\frac{\vartheta_{1}^{\prime}(0)}{\vartheta_{1}(z_{1a})}$
$\displaystyle\ \underset{\tau\to i\infty}{\longrightarrow}\frac{1}{\sin(\pi
z_{1a})}\,,$ (161)

we find that the $\tau\to i\infty$ limit of the Riemann-Wirtinger integrand
becomes

$\displaystyle u\ F(z_{1a},\eta)\ \text{d}z_{1}$ $\displaystyle\
\underset{\tau\to i\infty}{\longrightarrow}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}\,,$ (162)

where $w_{i}=e^{2\pi iz_{i}}$ are the new coordinates on the sphere and

$\displaystyle
u_{\mathrm{tree}}=w_{1}^{s_{1A}}\left(\prod_{i=2}^{n}w_{i}^{-\frac{s_{1i}}{2}}w_{1i}^{s_{1i}}\right)\,,$
(163)

is the analogous twist for tree-level string integrals. Above,
$\eta_{0}=\sum_{i=2}^{n}s_{1i}z_{i}-s_{1B}$ is the $\tau$-independent part of
$\eta$ and drops out of all quantities in the limit. Equations (162) and (163)
are the $n$-point generalization of the analogous formulas in mano2009  where
the factors of $w_{i}^{-\frac{s_{1A}}{2}}$ are missing. Like mano2009 , we
assume that $0<\mathrm{Re}(s_{1A})<1$ since this leads to convergent
integrals. Just from the integrand, it is easy to see that the $\tau\to
i\infty$ limit results in a linear combination of generalized Lauricella-D
hypergeometric functions (see figure 4).

Figure 4: The branch choice for the tree-level twist (Koba-Nielsen) factor
given by (163). The branch choice in the fundamental domain of the torus maps
to the red dotted line connecting the marked points on the rational nodal
curve parameterized by $w=e^{2\pi iz}$. The $A$\- and $B$-cycle
discontinuities are captured by the branch cut from $0$ to $\infty$ denoted by
the purple dashed line.

The contours $\gamma_{2j}$ are the easiest to describe since the path from
$z_{2}$ to $z_{j}$ on the torus maps to an analogous path from $w_{2}$ to
$w_{j}$ with no phase factors. Gauge fixing $z_{2}=0$, we find

$\displaystyle[\gamma_{2j}|\varphi_{k}\rangle_{i\infty}:=\lim_{\tau\to
i\infty}[\gamma_{2j}|\varphi_{a}\rangle=\int_{w_{2}=1}^{w_{j}}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}\,.$ (164)

On the other hand, the $A$\- and $B$-cycle integrals become

$\displaystyle[\gamma_{2A}|\varphi_{k}\rangle_{i\infty}$
$\displaystyle:=\lim_{\tau\to
i\infty}[\gamma_{2j}|\varphi_{a}\rangle=(1-e^{2\pi
is_{1A}})\int_{w_{2}=1}^{\infty}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}\,,$ (165)
$\displaystyle[\gamma_{2B}|\varphi_{k}\rangle_{i\infty}$
$\displaystyle:=\lim_{\tau\to
i\infty}[\gamma_{2j}|\varphi_{a}\rangle=\int_{w_{2}=1}^{0}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}-e^{2\pi
is_{1B}}\int_{w_{2}=1}^{\infty}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}\,.$ (166)

Note that the $A$\- and $B$-cycle degenerations come with explicit factors of
the $A$\- and $B$-cycle monodromies even though the twist $u_{\text{tree}}$
knows nothing about $s_{1B}$.

The phase factor and contour on the nodal sphere for the $A$-cycle integral
(165) is fairly straightforward to derive. The path from $z_{1}=0$ to
$z_{1}=1$ on the torus maps to the contour from $\text{arg}(w_{1})=0$ to
$\text{arg}(w_{1})=2\pi$ with $|w_{1}|=1$ on the nodal sphere. Moreover, the
corresponding punctures are all inside the unit circle since
$\mathrm{Im}\,z_{i}\geq 0$

$\displaystyle[\gamma_{2A}|=$
$\displaystyle\includegraphics[align=c,scale={0.4}]{figs/deformationA0-eps-
converted-to.pdf}\ \underset{\tau\to i\infty}{\longrightarrow}\
\includegraphics[align=c,scale={0.4}]{figs/deformationA1-eps-converted-
to.pdf}\,.$ (167)

We can then deform the contour to be: $\infty+i0^{-}\to 1\to\infty+i0^{+}$

$\displaystyle[\gamma_{2A}|$ $\displaystyle\underset{\tau\to
i\infty}{\longrightarrow}\
\includegraphics[align=c,scale={0.39}]{figs/deformationA2-eps-converted-
to.pdf}=(1-e^{2\pi
is_{1A}})\left(\includegraphics[align=c,scale={0.39}]{figs/deformationA3-eps-
converted-to.pdf}\right)\,,$ (168)

where the phase comes from from crossing the purple branch cut. Equation (168)
is precisely the contour in the boundary value (165)!

On the other hand, the degeneration of the $B$-cycle is more subtle since the
$\tau\to i\infty$ limit of a cycle only makes sense for paths whose imaginary
part is much smaller than $|\tau|$. Thus, we cannot directly map the $B$-cycle
to the nodal sphere. Instead, we define a new path that includes the $B$-cycle
and can be deformed to one with imaginary part much smaller than $|\tau|$. To
this end, we consider the following linear combination:

$\displaystyle[\gamma_{2B}|+[\gamma_{2A}|_{z_{1}\to
z_{1}+\tau}-[\gamma_{2B}|_{z_{1}\to z_{1}+1}$ $\displaystyle=(1-e^{2\pi
is_{1A}})[\gamma_{2B}|+e^{2\pi is_{1B}}[\gamma_{2A}|\,,$ (169)
$\displaystyle=\includegraphics[align=c,scale={0.4}]{figs/deformationB01-eps-
converted-to.pdf}$
$\displaystyle=\quad\includegraphics[align=c,scale={0.4}]{figs/deformationB02-eps-
converted-to.pdf}\,.$

Then, the above path maps to the nodal sphere as follows

$\displaystyle[\gamma_{2B}|+[\gamma_{2A}|_{z_{1}\to
z_{1}+\tau}-[\gamma_{2B}|_{z_{1}\to z_{1}+1}\ \underset{\tau\to
i\infty}{\longrightarrow}\
\includegraphics[align=c,scale={0.4}]{figs/deformationB1-eps-converted-
to.pdf}\,.$ (170)

Accounting for the phase from crossing the branch cut, we find

$\displaystyle[\gamma_{2B}|+[\gamma_{2A}|_{z_{1}\to
z_{1}+\tau}-[\gamma_{2B}|_{z_{1}\to z_{1}+1}\underset{\tau\to
i\infty}{\longrightarrow}\ =$ $\displaystyle(1-e^{2\pi
is_{1A}})\left(\includegraphics[align=c,scale={0.35}]{figs/deformationB2-eps-
converted-to.pdf}\right)$ (171)

Combining (169) and (171) and inserting (168), we obtain the contour in (166).

-eps-converted-to.pdf

As a sanity check, we show that these degenerate integrals satisfy the over-
completeness relation (141). Taking the $\tau\to i\infty$ limit of (141)
yields

$\displaystyle\lim_{\tau\to
i\infty}\eqref{eq:homology_cycle_overcomplete_basis}$
$\displaystyle=(1-e^{2\pi is_{1A}})\int_{0}^{\infty}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}-\sum_{j=3}^{n}e^{-2\pi
i(s_{12}+\cdots+s_{1j})}(1-e^{2\pi
is_{1j}})[\gamma_{2j}|\varphi_{a}\rangle_{i\infty}\,,$
$\displaystyle=\int_{\mathcal{C}_{\infty}}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}-\sum_{j=3}^{n}e^{-2\pi
i(s_{12}+\cdots+s_{1j})}(1-e^{2\pi
is_{1j}})[\gamma_{2j}|\varphi_{a}\rangle_{i\infty}\,,$ (172)

where all of the dependence on the $B$-cycle monodromy has dropped out and we
have identified the first term with the contour $\mathcal{C}_{\infty}$ in
figure 5. Next, we note that

$\displaystyle\sum_{j=3}^{n}e^{-2\pi i(s_{12}+\cdots+s_{1j})}(1-e^{2\pi
is_{1j}})[\gamma_{2j}|\varphi_{a}\rangle_{i\infty}=\sum_{j=3}^{n}\left(1-e^{-2\pi
i\sum_{k=2}^{j-1}s_{1k}}\right)\int_{w_{j-1}}^{w_{j}}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}\,$
$\displaystyle\qquad\qquad=\sum_{j=3}^{n}\left(1-e^{2\pi
i\sum_{k=j}^{n}s_{1k}}\right)\int_{w_{j-1}}^{w_{j}}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}\,=-\int_{\mathcal{C}_{n}}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}\,.$ (173)

Putting the last two equations together, yields

$\displaystyle\lim_{\tau\to
i\infty}\eqref{eq:homology_cycle_overcomplete_basis}=\int_{\mathcal{C}_{\infty}+\mathcal{C}_{n}}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}=\int_{\mathcal{C}_{0}}u_{\mathrm{tree}}\
\frac{\text{d}w_{1}}{w_{1a}}=0$ (174)

since the contour $\mathcal{C}_{0}=\mathcal{C}_{\infty}+\mathcal{C}_{n}$ in
figure 5 is clearly contractible with no singularities inside and thus the
corresponding integral vanishes due to Cauchy.

Figure 5: Contour deformation argument for the over-completness of cycles on
the sphere. Via Cauchy’s theorem, the integral over the contour
$\mathcal{C}_{0}$ vanishes. Moreover, $\mathcal{C}_{0}$ can be deformed into
the contours $\mathcal{C}_{\infty}+\mathcal{C}_{n}$, which can further be
decomposed into the contours of figure 4. Thus, one finds a linear relation
between the contours of figure 4.

The drawback to equations (164-166) is that one has to be careful when
propagating the boundary value at $\tau=i\infty$ to finite $\tau$ since
equation (122) is naively divergent at $\tau=i\infty$. Fortunately, we can
avoid the complication of regulating the $\tau$-integration for the
$\gamma_{2j}$ cycles because we can find a boundary value at finite $\tau$ by
taking the $z_{i}$ to a special configuration!

By taking all of the punctures to the origin, the $\tau$-dependence drops out.
However, this limit must be taken carefully using regularized objects to get a
finite answer. In the end, we find the following prescription for the boundary
value where all punctures are sent to the origin

$\displaystyle[\gamma_{2i}|\varphi_{a}\rangle_{0}$
$\displaystyle:=\left(\prod_{k\neq 2,i}\lim_{z_{k}\to 0}(-2\pi
iz_{k})^{-s_{1k}}\right)\lim_{z_{i}\to 0}(-2\pi
iz_{i})^{-s_{12}-s_{1i}}\int_{\gamma_{2i}}\text{d}z_{1}\ u\
F(z_{1a},\eta|\tau)\,.$ (175)

In the above, we set $z_{1}=xz_{i}$ and the integration region is $x\in[0,1]$
in $x$-space runs from $0$ to $1$. It is also important to note that the limit
$z_{i}\to 0$ is taken before all others.

One can think of the prefactors in (175) as defining a regularized limit
consistent with the shuffle regularisation of eMPLs Broedel2020 ;
Broedel:2019tlz . The eMPL of weight and length one makes an appearance in the
twist because it is simply related to the log of the Jacobi theta function and
its regularization is well understood. The unregulated eMPL is

$\tilde{\Gamma}\left(\begin{smallmatrix}1\\\
0\end{smallmatrix};z|\tau\right)=\lim_{\epsilon\to
0}\int_{\epsilon}^{z}\text{d}w~{}g^{(1)}(w;\tau)=\Omega^{(1)}(z,\tau)-\lim_{\epsilon\to
0}\Omega^{(1)}(\epsilon,\tau)\,,$ (176)

where $\Omega^{(1)}(z,\tau)=\log{\left(\vartheta_{1}(z|\tau)/\eta_{\text{\tiny
De}}(\tau)\right)}$ (recall from section 4.4 that $\eta_{\text{\tiny De}}$ is
the Dedekind eta function and $\Omega^{(1)}$ is an elliptic symbol letter
Wilhelm:2022wow ). As one can clearly see, this is divergent for all $z_{1}$