# Co-homology of Differential Forms and Feynman diagrams

Sergio L. Cacciatori 1,2 Maria Conti 1,2 Simone Trevisan 1,2 1 Dipartimento di
Scienza ed Alta Tecnologia, Università degli Studi dell’Insubria;
2 INFN, Sezione di Milano, Via Celoria 16, 20133 Milano, Italy

##  1 Introduction

Feynman diagrams are introduced in the context of quantum interacting field
theory, as a graphical representation of the solution of a system of first
order differential equations, admitting a path-ordered exponential expression.
Usually, the matrix of the system is composed of two terms: one identifying
the solution in absence of interactions, i.e. the free solution; and a second
one, carrying information on the interaction, treated as a perturbation to the
free evolution, and characterized by the strength of the interaction, i.e. the
coupling constant, considered as a small quantity. The perturbative expansion
of the path-ordered exponential, obtained by a series expansion in the
coupling constant, gives rise to the Dyson series, containing an infinite
sequence of iterated integrals, whose iteration number increases with the
perturbative order: at any given order, hence for any given power of the
coupling constant the integrands are formed by an ordered product of functions
\- to better say, distributions -, representing the free evolution (the
propagators), and the insertion of interaction terms (the vertices).

Dyson series can be used to describe the evolution of physical systems whose
dynamics follows Volterra-type model, within quantum as well as classical
physics. Therefore, the predictive power of a theoretical model, aiming at
describing the dynamics of physical systems, on a wide spectrum of physical
scales, being them either so microscopic as colliding elementary particles, or
as macroscopic as coalescing astrophysical binary systems, may depend on our
ability of evaluating Feynman integrals, alias solving systems of differential
equations.

Hamiltonian and Lagrangian carry information about the free and the
interactive dynamics, and the basic rules to build Feynman graphs can
systematically be derived from them. In particular, the interaction between
two elementary entities, experiencing the presence of each other through the
mediation of a third entity, can be described like a scattering event.
Therefore, quantities like the impact parameter, the cross section, the
scattering angle, or the interaction potential turn out to be related to the
scattering amplitude, which ultimately admits a representation in terms of
Feynman graphs.

”Perturbation theory means Feynman diagrams” [1], yet the diagrammatic
approach is not limited to the perturbative regime: ”Perturbation theory is a
very useful device to discover very useful equations and properties that may
hold true even if the perturbation expansion fails” [2].

More modern approaches based on analyticity and unitarity, so called on-shell
and unitarity-based methods [3, 4, 5, 6], make use of the factorization
properties of scattering amplitudes (exposed by using complex variables, to
build suitable combinations of energy and momenta of the interacting objects)
in order to group more efficiently the contributing Feynman diagrams, and
exploit recursive patterns, hard to identify within the pure diagrammatic
approach. In this case, the symmetries, which do not necessarily hold for the
individual diagrams, and which are inherited from the lower-order amplitudes,
yield novel representations of the scattering amplitudes (see f.i. [7, 8]).

Therefore, scattering amplitudes, at any given order in perturbation theory,
can be canonically built out of linear combination of Feynman graphs, and
equivalently out of products (convolutions) of lower order amplitudes.
Independently of the strategy adopted for their generation, the evaluation of
scattering amplitudes beyond the tree-level approximation requires the
evaluation of multivariate Feynman integrals.

Dimensional regularization played a crucial role in the formal mathematical
developments of gauge theories and of Feynman integrals. Exploiting the
analytic continuation in the space-time dimensions $d$ of the interacting
fields, it is possible to modify the number of integration variables in order
to stabilize otherwise ill-defined (mathematically non existing) integrals
emerging in the evaluation of quantities which ultimately have to be compared
with numbers coming from (physically existing) experiments.

Within the dimensional regularization scheme, Feynman integrals, are not
independent functions. They obey relations that can be established at the
integrand level, namely among the integrands related to different graphs,
systematized in the so called integrand decomposition method for scattering
amplitudes [9, 10, 11, 12, 13, 14, 15, 16], as well as relation that hold,
instead, just upon integration. The latter are contiguity relations known as
integration-by-parts (IBP) identities [17], which play a crucial role in the
evaluation of scattering amplitudes beyond the tree-level approximation.
Process by process, IBP identities yield the identification of an elementary
set of integrals, the so-called master integrals (MIs), which can be used as a
basis for the decomposition of multi-loop amplitudes [18]. MIs are special
integrals, as elementary Feynman integrals that admit a graphical
representation (in terms of products of scalar propagators and scalar
interaction vertices). At the same time, IBP relations can be used to derive
differential equations [19, 20, 21, 22, 23, 24, 25, 26], finite difference
equations [27, 28], and dimensional recurrence relations [29, 30] obeyed by
MIs. The solutions of those equations are valuable methods for the evaluation
of MIs, for those cases where their direct integration might turn out to be
prohibitive (see f.i. [31, 32, 33]).

The study of Feynman integrals, the systems of differential equations they
obey, the iterated integral representation of their solution [34, 35, 36] has
been stimulating a vivid interplay and renovated interests between field
theoretical concepts and formal mathematical ideas in Combinatorics, Number
Theory, Differential and Algebraic Geometry, and Topology (see, f.i. [37, 38,
39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]).

The geometric origin of the analytic properties of Feynman integrals finds its
roots in the application of topology to the S-matrix theory [50, 51, 52]. In
more recent studies, co-homology played an important role for identifying
relations among Feynman integrals and to expose deeper properties of
scattering amplitudes [39, 41, 53, 54, 55, 56, 45, 46, 57, 58, 59, 60, 61, 62,
63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73]

In this editorial, we elaborate on the recently understood vector space
structure of Feynman integrals [59, 61, 60, 62, 63, 64, 65, 66] and the role
played by the intersection theory for twisted de Rham (co)-homology to access
it.

As observed in ref. [59], after looking at Feynman integrals as generating a
vector space, one can se intersection numbers of differential forms [74] as a
sort of scalar product over it. From this viewpoint, the intersection products
with a basis of MIs mimic the projection of a vector into a basis. For
example, using intersection projections for 1-forms, applied to integral
representations of the Lauricella $F_{D}$ functions allowed to easily re-
derive continuity relations for such functions and the decomposition in terms
of MIs for those Feynman integrals on maximal cuts that admit a representation
as a one-fold integral [59, 61]. For more general cases, when one has to deal
with multifold integral representations [61, 63, 65], it have been introduced
the multivariate intersection numbers [75, 76, 77, 78, 79, 80, 81, 46, 82].
For the case of meromorphic $n$-forms, an iterative method for the
determination of intersection numbers was proposed in [60] and successively
refined in [63, 65, 66]. The only simple case is for logarithmic (dlog)
differential forms, which bring simple poles only, whose intersection numbers
can be computed by employing the global residue theorem [66].

Within this approach, the number of MIs, proven to be finite [83], is the
dimension of the vector space of Feynman integrals [63], and corresponds to
the dimension of the homology groups [53], or equivalently of the cohomology
group [59, 61, 63, 65], and can be related to topological quantities such as
the number of critical points [53], Euler characteristics [84, 85, 86, 87], as
well as to the dimension of quotient rings of polynomials, for zero
dimensional ideals, in the context of computational algebraic geometry [65].

Another interesting consequence of intersection number is about their
underlying geometrical meaning, which leads to determining linear relations,
equivalent to IBP relations, and quadratic relations for Feynman integrals
[59, 61, 63, 65], colled twisted Riemann periods relations (TRPR) [74] since
represent a twisted version of the well-known bilinear Riemann relations. Some
of such quadratic relations were already noticed by using number-theoretic
methods to Feynman calculus, and given rise to conjectures [54, 55, 56, 88,
89], whose proof has been given only recently [70, 71], while other bilinear
relations, proposed in [90], have yet to be understood in the light of the
TRPR.

As stated in advance, in this work we first concentrate on the basic aspects
leading to the definition of a vector space of Feynman integrals; the we move
onto the mathematical description ( starting from an elementary point of view)
of the instruments needed in order to tackle the geometrization program just
described. Finally, we devote some time to address the problem of the actual
computation of intersection numbers. More precisely, in Section 2 we mostly
describe the Baikov representation of Feynman integrals and its role in
uncovering the underlying cohomological structure, while in Section 3 we
consider the vector structure of Feynman integrals - including bilinear
identities - and also precisely describe how the number of MIs can be
computed. In Section 4 we provide an elementary illustration of cohomologies,
while in Section 5 we highlight the link between cohomology theory and
integration theory. In Section 6 we give an extensive lookout at the advanced
mathematical constructions behind. Finally, in Section 7 we make some explicit
examples of practical techniques adopted to compute intersection numbers. The
three appendices include some technical details.

##  2 Feynman integral representation

We aim at describing the properties of (scalar) Feynman integrals,
representing the most general type of integrals appearing in the evaluation of
Scattering Amplitudes, left over after carrying out the spinor and the Lorentz
algebra (spinor-helicity decomposition, Dirac-Clifford gamma algebra, form
factor decomposition), generically indicated as,

$I_{\nu_{1},\cdots,\nu_{N}}=\int\prod_{i=1}^{L}\frac{d^{D}q_{i}}{\pi^{D/2}}\prod_{a=1}^{N}\frac{1}{D_{a}^{\nu_{a}}}\,.$
(1)

In the classical literature, the evaluation of Feynman integrals is carried
out by direct integration, in position and/or momentum-space representation,
making use of Feynman, or equivalently Schwinger, parameters. More advanced
methods make use of differential equations, as an alternative computational
strategy, which turns out to be very useful whenever the direct integration
becomes prohibitive, for instance, due to the number of the physical scales in
the scattering reaction (number of particles and/or masses of particles). In
this work, we would like to approach the multi-loop Feynman calculus in a
different fashion from the direct integration, making use of novel properties
that emerge when Feynman integrals are cast in suitable parametric
representation, such as the so called Baikov representation [91] (see also
[92] for review). First, we observe that the integration variables involved in
the integral (1) are the usual $L$ loop momenta $q_{i}$, which are not Lorentz
invariants. Baikov representation consists in a change of variables in which
the new integration variables are actually Lorentz invariants: namely, the
independent scalar products one can build using the $L$ loop momenta $q_{i}$
and $E$ independent external momenta $p_{j}$. Using these ideas, one can put
the Feynman Integrals in the following form, called Baikov representation:

$I_{\nu_{1},\cdots,\nu_{M}}=K\int_{\Gamma}B^{\gamma}\prod_{a=1}^{M}\frac{dz_{a}}{z_{a}^{\nu_{a}}}\,.$
(2)

For a proof, see Appendix A.

Before getting more into depth of the meaning behind Eq. (2), an observation
is necessary. By comparing the original integral with Eq. (2), one observes
that the number of integration variables changes from $LD$ to $M$. When we
perform the projection (A.4) of each 4-momentum onto the space generated by
the vectors coming next, it is actually clear that this process cannot
continue indefinitely, as all the vectors are certainly not independent if
they lie in the physical 4D space. The decomposition we describe in Eq. (A.4)
is to be thought in an abstract (sufficiently large) dimension $D$. Since the
final expression is an analytic function of $D$, we get the physically
meaningful result via an analytic continuation down to $D=4$. This discussion
can be also summarised by saying that we are implicitly using dimensional
regularization to make sense of the expression (1), which is obviously
divergent in $D=4$.

The representation in Eq. (2) highlights new properties of the original
integral (1), and allows to study its topological structure as Aomoto-Gel’fand
integral [59]. In fact, extending the integral in Eq. (2) to the complex
space, it takes the form

$I=K\int_{C}u(\vec{z})\phi(\vec{z})\,,$ (3)

where $K$ is constant prefactor, $u=B^{\gamma}$ is a multivalued function such
that $u(\partial C)=0$ and $\phi$ is an $M$-form

$\phi\equiv\hat{\phi}d^{M}z=\frac{dz_{1}\wedge\cdots\wedge
dz_{M}}{z_{1}^{\nu_{1}}\cdots z_{M}^{\nu_{M}}}\,.$ (4)

Because of the Stokes theorem, given a certain $(M-1)$-form $\xi$ the
following identity holds:

$\int_{C}d(u\xi)=\int_{\partial C}u\xi=0\,,$ (5)

as $u\xi$ is integrated along $\partial C$ where $u$ vanishes. Eq. (5) can
also be rewritten as

$\int_{C}d(u\xi)=\int_{C}(du\wedge\xi+ud\xi)=\int_{C}u(\underbrace{d\log
u}_{\omega}\wedge+d)\xi\equiv\int_{C}u\nabla_{\omega}\xi=0.$ (6)

Eq. (6) states that, because of the introduction of the connection
$\nabla_{\omega}=d+\omega\wedge$ where $\omega\equiv d\log u$, it holds

$\int_{C}u\phi=\int_{C}u(\phi+\nabla_{\omega}\xi)\,,$ (7)

as the second term in the right side gives a null contribution. Eq. (7)
identifies an equivalence class, addressed as

${}_{\omega}{\bra{\phi}}:=\,\,\,\phi\sim\phi+\nabla_{\omega}\xi\,.$ (8)

This equivalence class defines twisted cohomologies (twisted because the
derivative involved is not simply $d$ as in the de Rham cohomology but it is
the covariant derivative $\nabla_{\omega}$). Representatives of a class are
called twisted cocycles.

In this fashion, the integral $I$,

$I=\int_{C}u\phi$ (9)

arises as a pairing between the twisted cycle $|C]$ and the twisted cocycle
$\bra{\phi}$. For notation ease, the subscript $\omega$ is understood, and
restored when needed.

Aomoto-Gel’fand integrals admit linear and quadratic relations which can be
used to simplify the evaluation of scattering amplitudes. In particular,
linear relations can be exploited to express any integral as linear
combination of an independent set of functions, called master integrals (MIs).
The decomposition of Feynman integrals in terms of MIs was proposed in [17]
and later systematised in [18], and represents the most common computational
technique for addressing multi-loop calculus nowadays. The novel insights we
elaborate on in this work allow to explore the underpinning vector space
structure obeyed by Aomoto-Gel’fand integrals, in order to investigate the
properties of Feynman integrals making use of co-homological techniques.
Accordingly, the decomposition of any generic integral
$I=\int_{C}u\phi\equiv\bra{\phi}C]$ in terms of a basis of MIs, say $J_{i}$,
can be achieved in a twofold approach:

$\displaystyle
I=\sum_{i=1}^{\nu}c_{i}\int_{C}u\,e_{i}=\sum_{i=1}^{\nu}c_{i}\,J_{i}$ (10)

or

$\displaystyle
I=\sum_{i=1}^{\nu}c_{i}\int_{C_{i}}u\,\phi=\sum_{i=1}^{\nu}c_{i}\,J_{i}\,.$
(11)

The former decomposition involves a basis formed by independent equivalence
classes $\\{e_{i}\\}$ of the underlying twisted cohomology, while the latter,
a basis formed by independent equivalence classes $\\{C_{i}\\}$ of the twisted
homology. Remarkably the dimension of the twisted homology and co-homology
spaces is the same.

Let us finally remark that although Baikov representation turned out to be
useful to uncover the cohomological structure of Feynman integrals [59], there
is no commitment in using it necessarily. Other parametric representations,
such as Feynman-Schwinger [93], $n$-dimensional polar coordinates [94], and
Lee-Pomeransky representation [53], to name a few, can be equivalently used
[64]. In fact, the integral in (1) can be cast in the form [53]

$I_{\nu_{1},\cdots,\nu_{N}}=\frac{\Gamma(D/2)}{\Gamma((L+1)D/2-|n|)\prod_{a}\Gamma(\nu_{a})}\int_{0}^{\infty}\cdots\int_{0}^{\infty}\frac{dz_{a}}{z_{a}^{1-\nu_{a}}}\mathcal{G}^{-D/2}\,,$
(12)

with $|n|=\sum_{a}\nu_{a}$ and $\mathcal{G}=\mathcal{F}+\mathcal{U}$, where
$\mathcal{F}$ and $\mathcal{U}$ are the Symanzik polynomials. The latter are
defined as

$\mathcal{F}=\det{A}C-(A^{\text{adj}})_{ij}B_{i}B_{j}\,,\quad\mathcal{U}=\det{A}\,,$
(13)

with $A$, $B$ and $C$ being the matrices that appear in the decomposition of
the denominators, as

$D_{a}=A_{a,ij}q_{i}q_{j}+2B_{a,ij}q_{i}p_{j}+C_{a}\,,$ (14)

where

$A_{ij}=\sum_{a}z_{a}A_{a,ij}\,,\quad B_{i}=\sum_{a}z_{a}B_{a,ij}p_{j}\,,\quad
C=\sum_{a}z_{a}C_{a}\,.$ (15)

We observe that, although the integral in Eq. (12) has the structure of
Aomoto-Gel’fand integrals Eq. (9), the polynomial $\mathcal{G}$ does not
vanish on the boundary of the integration domain, therefore surface terms can
emerge in in the (co)-homology decomposition. It turns out that these extra
terms can be related to integrals belonging to simpler sectors, i.e. with
fewer denominators than the ones in the original diagram.

##  3 The twisted cohomology vector space

###  3.1 Vector-space structure

To study the co-homology of dimensionally regulated Feynman integrals, we
consider integrals of the form

$I=\int_{\mathcal{C}_{R}}u(\mathbf{z})\,\varphi_{L}(\mathbf{z})=\langle\varphi_{L}|\mathcal{C}_{R}],$
(16)

regarded as a pairing between $\langle\varphi_{L}|$ and the function
$u(\mathbf{z})$, integrated over the contour $|\mathcal{C}_{R}]$. In
particular $u(\mathbf{z})$ is a multivalued function,
$u(\mathbf{z})=\mathcal{B}(\mathbf{z})^{\gamma}$ (or
$u(\mathbf{z})=\prod_{i},\mathcal{B}_{i}(\mathbf{z})^{\gamma_{i}}$), with

$\mathcal{B}(\partial\mathcal{C}_{R})=0\ .$ (17)

The pairing so defined is not strictly speaking a pairing of forms but of
equivalence classes of $n$-forms, so that two differential forms in the same
class differ by covariant derivative-terms whose contribution under
integration over a contour vanishes. Let us see how this works.

###  3.2 Dual cohomology groups

Let $\xi_{L}$ be an $(n{-}1)$-differential form and $\mathcal{C}_{R}$ an
integration contour such that (17) holds true. Thus, we can use Stokes theorem
to write

$0=\int_{\partial\mathcal{C}_{R}}u\,\xi_{L}=\int_{\mathcal{C}_{R}}d(u\,\xi_{L})=\int_{\mathcal{C}_{R}}u\left(\frac{du}{u}\wedge+d\right)\xi_{L}=\int_{\mathcal{C}_{R}}u\,\nabla_{\omega}\,\xi_{L}=\langle{\nabla_{\omega}\xi_{L}}|{\cal
C}_{R}]\ ,$ (18)

where

$\nabla_{\omega}=d+\omega\wedge,\qquad\omega=d\log u.$ (19)

As a consequence we immediately get

$\langle{\phi_{L}}|{\cal C}_{R}]=\langle{\phi_{L}}|{\cal
C}_{R}]+\langle{\nabla_{\omega}\xi_{L}}|{\cal
C}_{R}]=\langle{\phi_{L}+\nabla_{\omega}\xi_{L}}|{\cal C}_{R}]\ ,$ (20)

which meaans that the forms $\varphi_{L}$ and
$\varphi_{L}+\nabla_{\omega}\xi_{L}$ give the same result upon integration, as
stated above, and we can consider them as equivalent for the purpose of
computing intersections. This is an equivalence relation and we can say that
the two forms are in the same equivalence class, writing

$\phi_{L}\sim\phi_{L}+\nabla_{\omega}\xi_{L}\ .$ (21)

Of course the equivalence classes of $n$-forms defined by the equivalence
relation (21), that we will denote with $\langle\varphi_{L}|$, define a vector
space, called the _twisted cohomology group_ $H_{\omega}^{n}$.

Analogously, every time two $n$-dimensional contours differ by a boundary,
they have the same boundary so that if (17) holds for one of them it holds for
the other one. Using again the Stokes theorem, it follows that when used as
integration contours for a closed $n$-form $u\phi_{L}$ they give the same
result and, again, can be considered equivalent. We will indicate the
equivalence classes of integration contours so obtained with
$|\mathcal{C}_{R}]$. They define the _twisted homology group_
$H_{n}^{\omega}$, which is indeed an abelian group identifiable with a vector
space after tensorization with $\mathbb{R}$.

Let us consider, now, what happens when using $u^{-1}$ in the integral
definition, instead of $u$. Let us define a _dual_ integral

$\tilde{I}=\int_{\mathcal{C}_{L}}u(\mathbf{z})^{-1}\,\phi_{R}(\mathbf{z})=[{\cal
C}_{L}|\phi_{R}\rangle\ ,$ (22)

as a pairing between the dual twisted cycle $[{\cal C}_{L}|$ and the dual
twisted cocycle $|\phi_{R}\rangle$. It is clear by construction that what we
did before can be repeated in the same way just replacing the covariant
derivative with

$\nabla_{-\omega}=d-\omega\wedge,\qquad\omega=d\log u.$ (23)

Then, we immediately get an equivalence relation for dual twisted cocycles,
analogous to (21),

$\varphi_{R}\;\sim\;\varphi_{R}+\nabla_{-\omega}\xi_{R}$ (24)

which defines equivalence classes denoted by $|\varphi_{R}\rangle$. These
equivalence classes define the _dual_ vector space
$(H^{n}_{\omega})^{\ast}=H^{n}_{-\omega}$.

As mentioned earlier, an equivalence relation among dual integration contours
may be also considered, yielding to the identification of the dual homology
vector space $(H_{n}^{\omega})^{\ast}=H_{n}^{-\omega}$, referred to as the
_dual twisted homology group_ , whose elements are denoted by
$[\mathcal{C}_{L}|$.

###  3.3 Number of Master Integrals

We have shown that we can recognize a vector space structure in Feynman
integrals, so that a given integral may be expressed in some basis of the
twisted cohomology: if one is able to compute the coefficients of the
decomposition, which we discuss in Section 7, the computation will be reduced
to the evaluation of some fixed Master Integrals. All this reasoning would be
useless if the dimension of such vector space turned out to be infinite.
Luckily, the number of the independent MIs, which we refer to as $\nu$, is
known to be actually finite [83]. Moreover, it is known how to compute $\nu$
in practice, as we show in this section.

There are actually many interpretations of $\nu$: here we focus mainly on the
interpretation given by Lee and Pomeransky [53]. Firstly, $\nu$ is the number
of the many independent equivalence classes in the associated cohomology
group. Due to the Poincaré duality between cohomology and homology classes, it
is equivalent to study the dimension of the homology space, which consists in
counting how many non homotopically contractible closed paths exist in the
space of integration contours.

We consider a particularly simple case in order to have a clear understanding
of the usual reasonings followed when counting independent cycles. Once having
derived the associated Baikov representation (2), it is appropriate to perform
a maximal cut (for a more detailed discussion on cuts performed on Feynman
integrals in the context of Baikov representation, we recommend to take a look
at [95] and [96]): out of the $M$ total denominators, we put to 0 the $N$
original ones corresponding to physical propagators. Of course, putting a
propagator on mass shell is the same as performing a residue, which reduces
the level of complexity of the integral. We consider the case where the
integral along the maximal cut reduces to one with a single variable:

$I=\int\frac{dz_{1}}{z_{1}^{\nu_{1}}}B(z_{1})^{\gamma}.$ (25)

We stress that, by taking the maximal cut, all the original physical
propagators were eliminated, so that in Eq. (25) the variable $z_{1}$ is
related to one of the fake propagators introduced in Section 2: hence we can
consider $\nu_{1}<0$ so it does not introduce any additional singularities.
When looking at how many types of non equivalent integration contours one can
build, it is clear that the topology of the space must be taken into account.
Suppose that $B(z_{1})$ has $m$ distinct zeros: the power $\gamma$ introduces
in the $z_{1}$ complex space $m$ cuts starting from each zero point and going
to infinity. Supposing that the integrand is well-behaved at infinity (if this
were not true, then the whole integral $I$ would not converge), we can connect
at infinity the $m$ paths one can draw around the $m$ cuts. The resulting
closed path is actually contractible in a single point, hence only $m-1$ paths
are independent (Figure 1).
Qualitatively, notice that if $m$ is the order of the polynomial $B(z_{1})$,
then $m-1$ is the order of the polynomial $\partial_{z_{1}}B$, hence it is
related to the number of zeros of $\partial_{z_{1}}B$. Getting back to the
notation

$I=\int_{C}u\phi\,,$ (26)

where $u=B^{\gamma}$, it is equivalent to the number of solutions of

$\omega=d\log u=\gamma(\partial_{z_{1}}B/B)dz_{1}=0\,,$ (27)

called the number of proper zeros. Eq. (27) suggests a deep connection between
the number $\nu$ of MIs and the number of critical points of $B$.

Im$(z)$Re$(z)$ Figure 1: Complex plane with $m=5$ cuts (undulate blue curves).
Each cut is encircled by a path going to infinity while never crossing any
cut. Dashed green lines connect at infinity the full green lines and overall
create a closed path which is clearly contractible in 0.

As shown more extensively in [53], this connection is actually much more
general: given an integral of the form (26), in which $\phi$ is a holomorphic
$M$-form and $u$ is a multivalued function such that $u(\partial C)=0$ , then
the number of Master Integrals is

$\nu=\,\,\,\text{number of solutions of the
system}\,\,\,\begin{cases}\omega_{1}=0\\\ \vdots\\\ \omega_{n}=0\end{cases},$
(28)

where

$\omega=d\log u(\vec{z})=\sum_{i=1}^{n}\partial_{z_{i}}\log
u(\vec{z})dz_{i}=\sum_{i=1}^{n}\omega_{i}dz_{i}.$ (29)

Summing up, the number $\nu$ of MIs, which is the dimension of both the
cohomology and homology groups thanks to the Poincaré duality, is equivalent
to the number of proper critical points of $B$, which solve $\omega=0$. We
mention that $\nu$ is also related to another geometrical object: the Euler
characteristic $\chi$ [53][87]. It is found that is linked to
$\chi(P_{\omega})$, where $P_{\omega}$ is a projective variety defined as the
set of poles of $\omega$, through the relation [63]

$\nu=\dim H^{n}_{\pm\omega}=(-1)^{n}\left(n+1-\chi(P_{\omega})\right).$ (30)

While we do not delve into the details of this particular result, we highlight
how, once again, $\nu$ relates the physical problem of solving a Feynman
integral into a geometrical one.

###  3.4 Linear and bilinear identities

The bases of dual twisted cocycles $\\{e_{i}\\}\in H^{n}_{\omega}$ and
$\\{h_{i}\\}\in H^{n}_{-\omega}$, as well as the bases of the dual twisted
cycles $|{\cal C}_{R,i}]\in H_{n}^{\omega}$ and $[{\cal C}_{L,i}|\in
H_{n}^{-\omega}$ can be used to express the identity operator in the
respective vector spaces. In particular, in the cohomology space, the identity
resolution reads as,

$\displaystyle\sum_{i,j=1}^{\nu}|h_{i}\rangle\left(\mathbf{C}^{-1}\right)_{ij}\langle
e_{j}|=\mathbb{I}_{c}$ (31)

where we defined the _metric matrix_

$\mathbf{C}_{ij}=\langle e_{i}|h_{j}\rangle\ ,$ (32)

whose elements are intersection numbers of the twisted basic forms. We can do
the same in the homology space, where the resolution of the identity takes the
form

$\displaystyle\sum_{i,j=1}^{\nu}|\mathcal{C}_{R,i}]\left(\mathbf{H}^{-1}\right)_{ij}[\mathcal{C}_{L,j}|=\mathbb{I}_{h}\,,$
(33)

with the metric matrix now given by

$\mathbf{H}_{ij}=[\mathcal{C}_{L,i}|\mathcal{C}_{R,j}]\ ,$ (34)

in terms of intersection numbers of the basic twisted cycles.

Linear and bilinear relations for Aomoto-Gel’fand-Feynman integrals, as well
as the differential equations and the finite difference equation they obey are
a consequence of the purely algebraic application of the identity operators
defined above: this is the simple observation made in [59], yielding what we
consider as the profound developments reached in the subsequent studies .

In fact, the decomposition of any generic integral
$I=\langle\varphi_{L}|\mathcal{C}_{R}]$ in terms of a set of $\nu$ MIs
$J_{i}=\langle e_{i}|\mathcal{C}_{R}]$ reads as

$I=\sum_{i=1}^{\nu}c_{i}\,J_{i}$ (35)

and can be understood as coming from the underpinning decomposition of
differential forms, as

$\langle\phi_{L}|=\sum_{i=1}^{\nu}c_{i}\,\langle e_{i}|\,,$ (36)

(because the integration cycle is the same for all the integrals of eq. (35)).
Likewise, the decomposition of a _dual_ integral
$\tilde{I}=[\mathcal{C}_{L}|\varphi_{R}\rangle$ in terms of a set of $\nu$
_dual_ MIs $\tilde{J}_{i}=[\mathcal{C}_{L}|h_{i}\rangle$

$\tilde{I}=\sum_{i=1}^{\nu}\tilde{c}_{i}\,\tilde{J}_{i}$ (37)

becomes

$|\phi_{R}\rangle=\sum_{i=1}^{\nu}\tilde{c}_{i}\,|h_{i}\rangle.$ (38)

By applying the identity operator $\mathbb{I}_{c}$ to the forms
$\langle\phi_{L}|$ and $|\phi_{R}\rangle$,

$\displaystyle\langle\phi_{L}|$ $\displaystyle=$
$\displaystyle\langle\phi_{L}|\mathbb{I}_{c}=\sum_{i,j=1}^{\nu}\langle\phi_{L}|h_{j}\rangle\left(\mathbf{C}^{-1}\right)_{ji}\langle
e_{i}|\ ,$ (39) $\displaystyle|\phi_{R}\rangle$ $\displaystyle=$
$\displaystyle\mathbb{I}_{c}|\phi_{R}\rangle=\sum_{i,j=1}^{\nu}|h_{i}\rangle\left(\mathbf{C}^{-1}\right)_{ij}\langle
e_{j}|\phi_{R}\rangle\ ,$ (40)

the coefficients $c_{i}$, and $\tilde{c}_{i}$ of the linear relations in Eqs.
(36), (38) read as

$\displaystyle c_{i}$
$\displaystyle=\sum_{j=1}^{\nu}\,\langle\phi_{L}|h_{j}\rangle\,\left(\mathbf{C}^{-1}\right)_{ji}\,,$
(41) $\displaystyle\tilde{c}_{i}$
$\displaystyle=\sum_{j=1}^{\nu}\,\left(\mathbf{C}^{-1}\right)_{ij}\,\langle
e_{j}|\phi_{R}\rangle\,.$ (42)

The latter two formulas, dubbed _master decomposition formulas_ for twisted
cycles [59, 61], imply that the decomposition of any (dual) Aomoto-Gel’fand-
Feynman integral can be expressed as linear combination of (dual) master
integrals is an algebraic operation, similar to the decomposition/projection
of any vector within a vector space, that can be executed by computing
intersection numbers of twisted de Rham differential forms (cocycles).

Alternatively, by using the identity operator $\mathbb{I}_{h}$ in the homology
space, one obtains the decomposition of (dual) twisted cycles $|{\cal C}_{R}|$
and $|{\cal C}_{L}|$ in terms of (dual) bases,

$\displaystyle|{\cal C}_{R}]=\sum_{i}a_{i}\,|\mathcal{C}_{R,i}]\ ,\qquad{\rm
and}\qquad\left[{\cal
C}_{L}|\right.=\sum_{i}{\tilde{a}}_{i}\,[\mathcal{C}_{L,i}|\ ,$ (43)

where

$\displaystyle
a_{i}=\sum_{j=1}^{\nu}\left(\mathbf{H}^{-1}\right)_{ij}[\mathcal{C}_{L,j}|{\cal
C}_{R}]\ ,\qquad{\rm and}\qquad{\tilde{a}}_{i}=\sum_{i=1}^{\nu}[{\cal
C}_{L}|\mathcal{C}_{R,j}]\left(\mathbf{H}^{-1}\right)_{ji}\ .$ (44)

They imply the decomposition of the (dual) integrals,
$I=\langle\varphi_{L}|\mathcal{C}_{R}]$ and
$\tilde{I}=[\mathcal{C}_{L}|\varphi_{R}\rangle$, in terms of MIs,
$J_{i}^{\prime}$ and ${\tilde{J}}_{i}^{\prime}$,

$\displaystyle
I=\langle\varphi_{L}|\mathcal{C}_{R}]=\sum_{i=1}^{\nu}a_{i}\,J_{i}\
,\qquad{\rm
and}\qquad\tilde{I}=[\mathcal{C}_{L}|\varphi_{R}\rangle=\sum_{i=1}^{\nu}{\tilde{a}}_{i}\,{\tilde{J}}_{i}\
,$ (45)

with

$\displaystyle J_{i}^{\prime}=\langle\varphi_{L}|\mathcal{C}_{R,i}]\
,\qquad{\rm
and}\qquad{\tilde{J}}_{i}^{\prime}=[\mathcal{C}_{L,i}|\varphi_{R}\rangle\ ,$
(46)

where the MIs are characterised by the independent elements of the homology
bases.

In the above formulas, $\mathbf{C}$ and $\mathbf{H}$ are
$(\nu\times\nu)$-matrices of intersection numbers, which, in general, differ
from the identity matrix. For orthonormal intersections of basic cocycles,
$\langle e_{i}|$ and dual-forms $|h_{i}\rangle$, and basic cycles,
$|\mathcal{C}_{R,i}]$ and their dual $[\mathcal{C}_{L,i}|$, they turn into the
unit matrix, hence simplifying the decomposition formulas in Eqs. (41,42,44).
As usual, one can use the orthonormalization Gram-Schmidt procedure in order
to get an orthonormal basis with respect to the intersection product. For
1-forms it is possible to construct orthonormal bases in a direct way starting
from the expression of $\omega$ [61].

We can now get the quadratic identities simply by inserting the identity
operators $\mathbb{I}_{c}$ and $\mathbb{I}_{h}$ in the pairing between the
twisted cocyles or cycles,

$\displaystyle\langle\phi_{L}|\phi_{R}\rangle$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{\nu}\langle\phi_{L}|\mathcal{C}_{R,i}]\left(\mathbf{H}^{-1}\right)_{ij}[\mathcal{C}_{L,j}|\phi_{R}\rangle$
(47) $\displaystyle\left[\mathcal{C}_{L}|\mathcal{C}_{R}\right]$
$\displaystyle=$
$\displaystyle\sum_{i,j=1}^{\nu}[\mathcal{C}_{L}|h_{i}\rangle\left(\mathbf{C}^{-1}\right)_{ij}\langle
e_{j}|\mathcal{C}_{R}]\,.$ (48)

These are the Twisted Riemann’s Period Relations (TRPR) [74], see also Eq.
(157).

For applications of twisted intersection numbers to bilinear relations and to
Gel’fand-Kapranov-Zelevinski systems see [97, 82, 98, 54, 55, 56, 88, 89, 70,
71, 99].

##  4 Pictorial (co)homology

Before providing a geometric interpretation of Feynman-type integrals, we want
to recall some fundamental facts of topology, homology and cohomology, only at
an intuitive level, for those who need to refresh these concepts.

###  4.1 The Euler characteristic

One of the most known topological facts is probably the Euler theorem, [100],
relating the numbers of faces, edges, and vertices in a tessellation of a
compact simply connected region of the plane, like an electricity grid or a
Feynman diagram without external legs: the number $v$ of vertices minus the
number $j$ of edges plus the number $f$ of faces is always equal to 1,

$\displaystyle v-j+f=1,$ (49)

independently on the details of the tessellation. This can be quite easily
understood as follows. The rules of the game are: any face is simply connected
and has a 1-dimensional boundary that is the union of edges and vertices (no
lines or vertices are internal to a face); any edge is a simple line ending at
two vertices (that may coincide). No vertices belong to an internal point of
an edge.
With this rules, given a simply connected region, the simple tessellation we
can do is with just 1 face (covering the whole region), 1 edge (the whole
boundary cut in a point), 1 vertex (the cut point). Therefore, in this case,
$v-j+f=1$. Now, suppose it has given a tessellation. We can modify it by
adding an edge. This can be done in four different ways:

  * •

the new edge starts and ends in already existing (and possibly coincident)
vertices (Fig. 2). In this case the new edge cuts a face in two and the new
tessellation has $v^{\prime}=v$, $j^{\prime}=j+1$, $f^{\prime}=f+1$ so that
$v^{\prime}-j^{\prime}+f^{\prime}=v-j+f$;

Figure 2: The red edge starts and ends at the same vertex, while the blue one
belongs between two vertices.

  * •

the new edge extends from an already existing vertex to a new vertex attached
to an existing edge (Fig. 3). The new edge separates a face into two faces and
the new vertex separates the old edge into two edges. Therefore, the new
tessellation has $v^{\prime}=v+1$, $j^{\prime}=j+2$, $f^{\prime}=f+1$, so that
$v^{\prime}-j^{\prime}+f^{\prime}=v-j+f$;

Figure 3: The blue edge starts and ends at the same new vertex, while the red
one starts from an old vertex and ends in a new one.

  * •

the new edge starts from and ends to a unique new vertex, inserted in an old
edge (Fig. 3). The new edge cuts a face into two and the new vertex cuts the
old edge into two. Therefore, the new tessellation has $v^{\prime}=v+1$,
$j^{\prime}=j+2$, $f^{\prime}=f+1$, so that
$v^{\prime}-j^{\prime}+f^{\prime}=v-j+f$;

  * •

the new edge extends from a new vertex to a different new vertex attached to
two different or the same existing edge (Fig. 4). The new edge separates a
face into two faces. The new vertices separate two edges into two parts each
or a unique edge in three. Therefore, the new tessellation has
$v^{\prime}=v+2$, $j^{\prime}=j+3$, $f^{\prime}=f+1$, so that
$v^{\prime}-j^{\prime}+f^{\prime}=v-j+f$.

Figure 4: The red edge lies between two different new vertices on the same
edge, while the blue one lies between two new vertices belonging on two
different edges.

Therefore, we see that $\chi:=v-j+f$ is invariant, and since any tessellation
can be obtained from the most elementary one by acting with these operations,
we see that $\chi=1$. What happens if in place of a simply connected region we
consider a region with one hole? We can again consider a tessellation for it
and prove that $\chi=v-j+f$ is invariant. This follows from the fact that if
we close the hole by adding the face corresponding to it, we get a simply
connected region with $f^{\prime}=f+1$, $v^{\prime}=v$, $j^{\prime}=j$, so
that

$\displaystyle
1=v^{\prime}-j^{\prime}+f^{\prime}=v-j+f+1,\quad\Rightarrow\quad\chi=v-j+f=0.$
(50)

Changing the topology, the number $\chi$ is changed. From the same reasoning,
we see that if we consider $k$ holes, then

$\displaystyle\chi=1-k.$ (51)

Having understood the mechanism, we can go beyond and see that $\chi$ does not
change if we deform the given region a bit, without changing its topology. For
example, we can deform a simply connected region $S$ to cover part of a
sphere. But if we close this surface to cover the sphere, the topology changes
and what we do to $(v,j,k)$ of the original piece of surface is to add a face
so that for the sphere (Fig. 5)

$\displaystyle\chi_{S^{2}}=\chi_{S}+1=2.$ (52)

$\boldsymbol{\equiv}$ Figure 5: A sphere is obtained adding a face to a disc:
the Euler characteristics increases by 1.

Each time we make a hole on the sphere, we diminish $\chi_{S^{2}}$ by 1. What
does it change if we pass from a sphere to a torus? We can get a torus from a
sphere in the following way. First, we can cut two spherical caps out of the
sphere (say along the arctic and antarctic polar circles). What remains is a
deformed piece of a cylinder and since it is like a sphere with two holes it
has $\chi=0$. The torus is then obtained by gluing two of such pieces along
the circles (Fig. 6). Since both have $\chi=0$, we see that for the torus
$T^{2}$ we have

$\displaystyle\chi_{T^{2}}=0.$ (53)

Figure 6: A torus is obtained by gluing two cylinders along the boundaries.

If we practice $k$ holes on the torus we get $\chi=-k$. If we want to pass to
a surface of genus $2$, we have to practice two holes in the torus surface and
glue the extremities of a piece of cylinder to the holes. The holes diminish
$\chi$ by two, while the cylinder is harmless, so $\chi=-2$ in this case. With
the same construction we see that for a surface $K_{g,k}$ of genus $g$ and $k$
holes on the surface we have

$\displaystyle\chi_{K_{g,k}}=2-2g-k.$ (54)

This topological number is called the Euler characteristic of the surface.

###  4.2 Simplicial (co)homology

The simplest tessellation we can think for a surface is a triangulation. The
surface is then equivalent to the union of triangles, which are equivalent to
two dimensional simplexes (the convex hull generated by three non aligned
points in $\mathbb{R}^{N}$). A triangle has the union of three segments (and
three points) as boundary. Each segment is a one dimensional simplex and each
point a zero dimensional simplex. Two points are the boundary of a 1-simplex.
This way, we see the surface as a collection of simplexes glued together. If
the triangulation is fine enough, we can guarantee that two simplexes of given
dimension meet at most at a simplex of lower dimension (a face). Moreover, all
sub simplexes of any simplex is also in the collection. Such a collection is
called a simplicial complex. Thus, we can see a surface as a simplicial
complex.
Suppose we want to compute the Euler characteristic starting from it. We have
to count the number of faces, edges and vertices. But now, for example, let us
consider a face. It is a filled triangle whose boundary is an empty triangle.
So it has one face, three edges and three vertices. If we squash the triangle
along an edge until it collapses over the other two edges, the face disappears
and we are left with two edges and three vertices (Fig. 7).

$\boldsymbol{a}$$\boldsymbol{b}$$\boldsymbol{c}$$\boldsymbol{a}$$\boldsymbol{b}$$\boldsymbol{c}$$\boldsymbol{a}$$\boldsymbol{b}$$\boldsymbol{c}$
Figure 7: The edge $ab$ is squashed until the triangle reduces to two edges.

But $\chi$ is left invariant! The idea is then to say that the squashed edge
is equivalent to the other two. If we orient the triangle, call it $(abc)$, so
that it has a well defined travelling direction, what we have done can be
reformulated by saying that $(ab)$ can be squashed to $(ac)\cup(cb)$ after
elimination of the face. In practice one realizes this by saying that the
boundary of the 2-simplex $(abc)$ is

$\displaystyle\partial(abc)=(bc)-(ac)+(ab),$ (55)

where the sign respects the orientation, and stating that if a loop is a
boundary, it is trivial: $\partial(abc)\sim 0$, which means
$(ac)\sim(bc)+(ab)$. Also, if we take a union of such triangles, we see that
we can eliminate the common edges (intersections between simplexes) without
changing $\chi$, so to obtaining a simply connected polygon $P$ that is a
boundary of a region. Again, we will get that one of the edges of the polygon
is equivalent to the sum (union) of the remaining ones. It is easy to check
that this can be written as

$\displaystyle P=$ $\displaystyle\sum_{j}T_{j},$ (56) $\displaystyle\partial
P=$ $\displaystyle\sum_{j}\partial T_{j}\sim 0,$ (57)

where $T_{j}$ are the simplexes composing $P$. However, it could happen that
the polygon so obtained is not simply connected but it has a number of holes.
The best we can say this way is that, after collapsing the triangles, the
“more external boundary” of the polygon is equivalent to the sum of boundaries
of the internal holes, se Fig. 8.

Figure 8: All triangle are squashed so the ring is reduced to a closed path
homotopic to a circle.

Strictly speaking, however, these are not really boundaries, since there is
indeed a hole and not a face of which these are boundaries.111here one has to
be careful: if we take a sphere with a hole, its boundary is not a boundary of
the hole, but it is for its complement. In the above example it is not so.
This construction shows that in place of counting all edges, one is interested
in counting how much unions of edges forming closed paths are independent in
the above sense (so unions of closed paths are not boundaries).
The same construction can be done for counting vertices. If two vertices $(a)$
and $(b)$ are the two tips of an edge, then, in place of counting 2 vertices
and 1 edge we can just count 1 vertex and 0 edges, without changing $\chi$. In
this case we can restate this by saying that $(a)\sim(b)$

$\displaystyle\partial(ab)=(b)-(a)$ (58)

and saying that, again, boundaries are trivial: $\partial(ab)\sim 0$,
$(a)\sim(b)$. Notice that if a union $L=\sum_{j}e_{j}$ of edges forms a closed
path than $\partial L=0$. In particular, then, $\partial(\partial P)=0$ for
any polygon $P$. Finally, we notice that if a connected surface has a
boundary, then by collapsing triangles starting from the boundary, one can
always reduce to zero the number of faces, while if it has no boundary (like a
sphere or a torus) than we can reduce the number of faces to one. This is
exactly the number of closed surfaces that are not boundaries (obviously for
dimensional reasons, things could change if we worked, for example, in three
dimensions).
We can summarize such discussion as follows. We call a $j-chain$ the finite
union of $j$-dimensional simplexes, with relative coefficients:

$\displaystyle c=\sum_{a=1}^{N}b_{a}s_{a},\qquad{\rm dim}(s_{a})=j.$ (59)

The sign of $b_{a}$ establishes the orientation and its modulus is the “number
of repetitions”. So, the set of chains is a linear space over $\mathbb{Z}$. We
say that $c$ is closed if $\partial c=0$ and that it is exact if $c=\partial
b$ for $b$ a $(j+1)$-chain. We are then interested in counting the closed
chains that are independent with respect to the relation that two close chains
are equivalent if they differ by an exact chain. This space is called the
$j$-th simplicial homology group

$\displaystyle H_{j}(S,\mathbb{Z}),$ (60)

of the surface $S$. Then, we can define the Betty numbers $b_{j}={\rm
dim}H_{j}$ so that the above reasoning leads us to the result

$\displaystyle\chi_{S}=\sum_{j=0}^{2}(-1)^{j}b_{j}.$ (61)

This result does not change if we change the coefficients allowing $b_{a}$ to
take value in $\mathbb{K}=\mathbb{Q},\mathbb{R},\mathbb{C}$. In this case the
Homology groups become vector spaces. Also, we can consider the cohomology
groups by replacing simplexes $s_{k}$ with continuous maps ($j$-cochain)

$\displaystyle\sigma_{k}:s_{k}\rightarrow\mathbb{K},$ (62)

and the boundary operators $\partial$ with the coboundary operators $\delta$,
sending $j$-cochains in $(j+1)$-cochains, defined by

$\displaystyle\delta(\sigma_{j})(c_{j+1})\equiv\sigma_{j}(\partial c_{j+1}).$
(63)

This defines the cohomology groups $H^{j}(S,\mathbb{K})$. It follows that
$H^{j}(S,\mathbb{K})$ is the linear dual of $H_{j}(S,\mathbb{K})$, so that are
isomorphic as vector spaces.

###  4.3 Morse theory

Another way of understanding the topology of a differentiable manifold $M$ is
to use properties of functions $f:M\rightarrow\mathbb{R}$. Assume these
functions are smooth, with non degenerate isolated critical points. A critical
point is a point $p$ where $df(p)=0$. It is non degenerate if its Hessian is
different from zero. We will not delve here into the details (see [101] for
these), but we want just to show how such function may capture the topological
properties by looking at the explicit example of a torus. Referring to Fig. 9
let consider the map that at $p$ associates the quote $z=f(p)$. The critical
points are the points where the horizontal plane is tangent to the surface. We
see that there are 4 critical points: a maximum, a minimum and two saddle
points.

$\boldsymbol{z}$$\boldsymbol{p_{1}}$$\boldsymbol{p_{2}}$$\boldsymbol{p_{3}}$$\boldsymbol{p_{4}}$bowlribbonpantspantsbowl
Figure 9: A torus is cut in pieces with different topologies. Notice that the
ribbon can be pasted to the bowl or to the pants without change the topology
neither of the bowl or the pants: cutting away pieces not containing critical
points does not change the topology.

Let us see how this function can give us a new way of reconstruct the surfaces
starting from pieces. Starting from above, assume we use the horizontal plane
to cut the surface. First, we will meet the top $p_{1}$ of the surface,
corresponding to the maximum. If we cut a little bit below, we get a small
bowl faced down. Then, let us move below. If we cut before meeting the second
critical point, we cut out a sort of ribbon, whose boundaries are
homologically equivalent. This will happen each time we take two cuts not
containing any critical point in between. Then we move even below until
meeting the saddle point. If we cut a little bit below we get a piece that
looks like a pair of pants. If we go even more below, until passing the second
saddle point $p_{3}$, we get a second pair of pants (with the legs
upside).222If we cut before passing the saddle point we get two cylinders,
which correspond to trivial pieces Going below the point $p_{4}$ we are left
with a second bowl. Forgetting the trivial pieces, we see that we can
reconstruct the surface gluing together the shapes obtained after cutting
horizontally around the critical points. A maximum gives a bowl down, any
saddle point gives a pair of pants and a minimum gives a bowl up. Each of
these shapes is understood by looking at the signature of the Hessian at each
critical point. Starting from above, we have just to look at the principal
ways to go down (Fig 10):

  * •

at $p_{1}$ the Hessian is negative, so it has two principal directions going
down (the eigenvectors). We say that $p_{1}$ has Morse index $2$;

  * •

at $p_{2}$ and $p_{3}$ the Hessian is indefinite, it has only one negative
eigenvalue at which it corresponds a descending direction. We say that $p_{2}$
and $p_{3}$ have Morse index $1$;

  * •

at $p_{4}$ the Hessian is positive. There are not descending directions and we
say that $p_{4}$ has Morse index $0$.

If $m(n)$ is the number of critical points with Morse index $n$, then we can
define the Euler number associated to our surface $S$ ($M$ stays for Morse):

$\displaystyle\chi_{M}(S)=\sum_{n=0}^{2}(-1)^{n}m(n).$ (64)

NONE Figure 10: Principal descending directions from the critical points at
any piece.

Notice that in our example we get $\chi_{M}(S)=0$. If we deform the surface
without changing the topology, like in figure 11, the number of critical
points can change, but $\chi_{M}(S)$ remains unchanged.

$\boldsymbol{0}$$\boldsymbol{2}$$\boldsymbol{2}$$\boldsymbol{1}$$\boldsymbol{0}$$\boldsymbol{1}$$\boldsymbol{1}$$\boldsymbol{1}$
Figure 11: Deforming the torus does not changes the Morse index.

It is a topological invariant and, in this case, it is zero, equal to the
Euler characteristic of the torus. It is possible to prove that $\chi_{M}(S)$
is always equal to the Euler characteristic, in any dimension, for any closed
compact manifold. In the next picture we can see that, indeed, for a sphere
$S^{2}$ we get $\chi_{M}(S^{2})=2$ and for a surface $\Sigma_{g}$ of genus $g$
one has $\chi_{M}(\Sigma_{g})=2-2g$.

###  4.4 Cellular (co)homology

Another way to look at homology (and then cohomology by duality, as usual) is
to look for a cellular decomposition of our surface, [102]. This means that we
have to obtain the surface (up to deformations preserving the topology) by
gluing discs of different dimensions. A disc is meant to be a full ball in
given dimension. So a 0-dimensional disc is a point and a 1-dimensional disc
is a segment. The rule is that one starts from the lower dimensional discs,
then glues the boundaries of the successive dimensional discs to the lower
dimensional structure obtained previously. For example, one can get a sphere
starting from a 0-cell and a 2-cell, gluing the boundary of the two cell to
the point (Fig. 12).

Figure 12: A cellular decomposition of the sphere: a disc is glued to a point.

Or starting from a point, a segment and 2 discs, first gluing the boundaries
of the segments to the point (getting $S^{1}$) and gluing the boundaries of
the two 2-cells to $S^{1}$ (Fig 13).

Figure 13: A different cellular decomposition of the sphere.

These are different constructions of $S^{2}$ gluing discs, and we can think
many others. The point is that if we indicate with $d(n)$ the numbers of cells
of dimension $n$ then we get for the sphere

$\displaystyle\chi_{CW}(S^{2}):=\sum_{n=0}^{2}(-1)^{n}d(n)=2$ (65)

independently on the construction. Notice that, again, it coincides with the
Euler characteristic, and, again, it is not by chance but it is a general
result: the cellular decomposition gives us another way to compute the Euler
characteristic. In Figure 14 it is shown how to get a genus $g$ surface
$\Sigma_{g}$ by gluing a point, $2g$ segments and a 2-disc.

$\boldsymbol{p}$$\boldsymbol{a_{1}}$$\boldsymbol{a_{1}}$$\boldsymbol{a_{2}}$$\boldsymbol{a_{2}}$$\boldsymbol{a_{3}}$$\boldsymbol{a_{g}}$$\boldsymbol{a_{g}}$$\boldsymbol{b_{1}}$$\boldsymbol{b_{2}}$$\boldsymbol{b_{3}}$$\boldsymbol{b_{g}}$$\boldsymbol{b_{1}}$$\boldsymbol{b_{2}}$$\boldsymbol{b_{g}}$$\boldsymbol{p}$$\boldsymbol{a_{1}}$$\boldsymbol{b_{1}}$$\boldsymbol{a_{1}}$$\boldsymbol{b_{g}}$$\boldsymbol{a_{2}}$$\boldsymbol{b_{1}}$$\boldsymbol{a_{g}}$$\boldsymbol{b_{2}}$$\boldsymbol{a_{1}}$$\boldsymbol{a_{2}}$$\boldsymbol{a_{g}}$$\boldsymbol{b_{1}}$$\boldsymbol{b_{2}}$$\boldsymbol{b_{g}}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$$\boldsymbol{p}$
Figure 14: A cellular decomposition of a genus $g$ surface. All the boundaries
of the $2g$ one dimensional cells are glued to a point, giving a one
dimensional skeleton. Then the boundary of the two dimensional disc is glued
to the skeleton, gluing the red points to p and the remaining parts to the
corresponding lines (two segments for each curve) respecting the drawn
orientations.

Again, we see that

$\displaystyle\chi_{CW}(\Sigma_{g})=\chi_{E}(\Sigma_{g})=\chi_{M}(\Sigma_{g}).$
(66)

###  4.5 De Rham cohomology

Finally, we want just to recall what is probably the most commonly known
cohomology, which is the de Rham cohomology, [103]. On a smooth surface we can
have 0-forms (functions), 1-forms and 2-forms. The external differential maps
$k$-forms into $(k+1)$-forms. A $k$-form $\phi$ is said to be closed if
$d\phi=0$, while it is called exact if $\phi=d\psi$ for $\psi$ a $(k-1)$-form.
An exact form is also closed since $d^{2}=0$. This is an obvious consequence
of the Schwartz’s lemma for double derivatives. So, exact forms generate a
subspace of closed forms. The de Rham cohomology group of degree $k$ is the
space of closed $k$-forms identified up to exact forms. If $\Omega^{k}(S)$ are
the $k$-forms:

$\displaystyle H^{k}_{dR}(S)=\\{\omega\in\Omega^{k}(S)|d\omega=0\\}/\sim,$
(67)

where $\omega_{1}\sim\omega_{2}$ if and only if
$\omega_{1}-\omega_{2}=d\lambda$ for some $\lambda\in\Omega^{k-1}(S)$. It
turns out that, under reasonable hypotheses, $H^{k}_{dR}(S)$ has finite
dimension as a real vector space. Its dimension $b_{k}={\dim}H^{k}_{dR}(S)$ is
called the $k$-th Betty number of $S$ and is a topological invariant. For
$k=0$ the closed forms are the locally constant functions. They are closed but
not exact, so $H^{0}_{dR}(S)$ is the space of locally constant functions. If
$S$ is connected, any constant function is just identified by its value, so
$H^{0}_{dR}(S)=\mathbb{R}$ and $b_{0}=1$. On a bidimensional surface, any
2-form is closed. Suppose that $S$ is connected and closed (i.e. has no
boundary) and that $\omega_{1}\sim\omega_{2}$ are two equivalent 2-forms.
Then, $\omega_{2}-\omega_{1}=d\psi$ for $\psi$ a 1-form, and using Stokes
theorem:

$\displaystyle\int_{S}\omega_{2}-\int_{S}\omega_{1}=\int_{S}d\psi=\int_{\partial
S}\psi=0,$ (68)

so two equivalent forms have the same integral. The vice versa is also true
and we get that for a closed connected surface also $b_{2}=1$. It is difficult
but possible to prove that if the surface has genus $g$, then $b_{1}=2g$.
Since $b_{j}$ are invariants, it follows that also

$\displaystyle\chi_{dR}(S):=\sum_{n=0}^{2}(-1)^{n}b_{n}$ (69)

is an invariant. We see that if $S=\Sigma_{g}$, we get

$\displaystyle\chi_{dR}(\Sigma_{g})=2-2g=\chi_{CW}(\Sigma_{g})=\chi_{E}(\Sigma_{g})=\chi_{M}(\Sigma_{g}).$
(70)

Once more, this is a general result. We can now stop here, we will discuss a
little bit further about the relation of the de Rham cohomology with other
cohomologies in a more general setting (where we will also see a further
cohomology, the singular cohomology).

##  5 Integrals and Cohomologies

While calculus with functions of one real variable gives us the impression of
a strict relation between differential and integral calculus, it is pretty
evident that when passing to more variables, or just to one complex variable,
things change drastically, and the theory of integration looks deeply
different. Indeed, the sensitivity of integration to global questions becomes
quickly manifest. Integration looks more related to (co)homology and Hodge
theory. In this section we want to summarise some facts we probably will have
to consider in a full program of investigation about the geometry underlying
Feynman integrals. Of course, we cannot be exhaustive, but our aim is to be at
least suggestive.
A nice way to see it is to pass through complete elliptic integrals. They are
one dimensional integrals whose structure resembles that of Feynman integrals,
hiding the main characterizations we need, and are also the origin of larger
dimensional integration theory, starting from Poincaré and developed by
Lefschetz and Hodge [104, 105, 106, 107, 108].

###  5.1 Elliptic integrals

Consider the elliptic integrals of first and second kind, [109]:

$\displaystyle K(k)$
$\displaystyle=\int_{0}^{1}\frac{dx}{\sqrt{(1-x^{2})(1-k^{2}x^{2})}}=\frac{1}{2}\int_{-1}^{1}\frac{dx}{\sqrt{(1-x^{2})(1-k^{2}x^{2})}},$
(71) $\displaystyle E(k)$
$\displaystyle=\int_{0}^{1}\frac{\sqrt{1-k^{2}x^{2}}}{\sqrt{1-x^{2}}}\
dx=\frac{1}{2}\int_{-1}^{1}\frac{\sqrt{1-k^{2}x^{2}}}{\sqrt{1-x^{2}}}\ dx.$
(72)

It is known that these integrals cannot be expressed in terms of elementary
functions of $k$. In both cases the last integral extends between two branch
points of the integrands. There are four branch points, at $z_{\pm}=\pm 1$ and
$\tilde{z}_{\pm}=\pm k^{-1}$, $k\neq\pm 1$. As the integrand is a multivalued
function, we can extend it to a single valued function on a Riemann surface.
To this end, we can first extend $x$ to $z$ on the complex plane, with two
cuts, one from $-1$ to $1$, the other from $-k^{-1}$ to $k^{-1}$. Notice that
in both cases the integrand is regular at infinity, so we can think at it as
defined on the Riemann sphere $\mathbb{P}^{1}$ with the two given cuts. If we
try to cross one of the cuts, the function flips the sign and the only way to
keep the function single valued is to assume that we end up to a second copy
of the cut Riemann sphere, where at the doubled point it takes the same value
as on the original sphere, but with the opposite sign. With this standard
construction we end up with a well defined single valued function on a two
dimensional surface obtained gluing the edges of each cut of a sphere to the
edges of the corresponding cut on the second sphere. This is topologically a
torus, that is a Riemann surface of genus two or, from the complex point of
view, an elliptic curve, Fig. 15.

$\boldsymbol{x_{1}}$$\boldsymbol{x_{2}}$$\boldsymbol{x_{4}}$$\boldsymbol{x_{3}}$$\boldsymbol{x_{1}}$$\boldsymbol{x_{2}}$$\boldsymbol{x_{4}}$$\boldsymbol{x_{3}}$$\boldsymbol{x_{2}}$$\boldsymbol{x_{3}}$$\boldsymbol{\mathbb{P}^{1}}$$\boldsymbol{x_{1}}$$\boldsymbol{x_{4}}$$\boldsymbol{\mathbb{P}^{1}}$$\boldsymbol{\mathbb{P}^{2}}$$\boldsymbol{\mathbb{P}^{1}}$$\boldsymbol{\cong}$
Figure 15: Two double cut $\mathbb{P}^{1}$ planes glued along the cuts are
topologically equivalent to a torus.

We see that the integrals can then be thought as along the edge of the cut
where the root is positive. If we go back integrating along the other edge of
the cut, we get the same result. Since the whole cut traveled this way is a
closed path $\gamma$ along the elliptic curve, we can write

$\displaystyle K(k)$
$\displaystyle=\frac{1}{4}\oint_{\gamma}\frac{dx}{\sqrt{(1-x^{2})(1-k^{2}x^{2})}},$
(73) $\displaystyle E(k)$
$\displaystyle=\frac{1}{4}\oint_{\gamma}\frac{\sqrt{1-k^{2}x^{2}}}{\sqrt{1-x^{2}}}\
dx.$ (74)

Here, $\gamma$ is one of the generators of the first homology group of the
elliptic curve. In this form the elliptic integrals looks like a 1-form
integrated over a closed cycle. Since the first homology group of an elliptic
curve is two dimensional, there is a second independent curve, which we can
represent, for example, as the path going from $z_{+}$ to $\tilde{z}_{+}$ on
one of the spheres and coming back from the other sphere. This second path,
say $\beta$, intersects $\gamma$ transversally in a point. After having chosen
an orientation we can define the intersection product $\beta\cdot\gamma=1$,
meaning that the velocities $\dot{\beta}$ and $\dot{\gamma}$ at the
intersection point form a basis for the tangent space correctly oriented. Of
course this means that $\gamma\cdot\beta=-1$. However, in place of considering
the same integral changing $\gamma$ with $\beta$, let us first consider the
other canonical integrals $K(k^{\prime})$ and $E(k^{\prime})$, with
$k^{\prime}=\sqrt{1-k^{2}}$. These integrals are related to the previous ones
by the Legendre’s quadratic relation

$\displaystyle
K(k)E(k^{\prime})+E(k)K(k^{\prime})-K(k)K(k^{\prime})=\frac{\pi}{2}.$ (75)

To understand the meaning of this quadratic relation it is convenient to pass
to a more canonical description of the integrals of first and second kind.
Consider an integral of the form

$\displaystyle I_{1}=\int_{x_{1}}^{x_{2}}\frac{dx}{\sqrt{P(x)}},$ (76)

where $P$ is a fourth order polynomial with simple roots among which there are
$x_{1}$ and $x_{2}$ (not necessarily real). Since the integrand is not single
valued, this integral depends on the specification of the path connecting the
two roots. However, it is clear that we can work exactly as before, so that we
can write333$\gamma$ runs along $[-1,1]$ twice, so a further factor
$\frac{1}{2}$ appears

$\displaystyle I_{1}=\frac{1}{2}\int_{\gamma}\frac{dx}{y},$ (77)

where $\gamma$ is a closed path on the genus one complex line defined by the
equation

$\displaystyle y^{2}=P(x)$ (78)

in $\mathbb{P}^{2}$.444$(x,y)$ are non homogeneous coordinates on
$\mathbb{P}^{2}$ that can be related to homogeneous ones $(z_{0}:z_{1}:z_{2})$
by $x=z_{1}/z_{0}$, $y=z_{2}/z_{0}$ on the patch $z_{0}\neq 0$. Notice that in
the form (76) we are working just on a $\mathbb{P}^{1}$ projected component
$\mathbb{P}^{2}\mapsto\mathbb{P}^{1}$ (this is the reason why the branch
points appear). On $\mathbb{P}^{1}$ there is the action of the Möbius
fractional group $PGL(2,\mathbb{C})$, which allows to move three points in any
desired position. It works as follows. Let $z$ be the inhomogeneous coordinate
on $\mathbb{P}^{1}$, and

$\displaystyle A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in GL(2,\mathbb{C}).$
(79)

Its action on $z$ is thus defined by

$\displaystyle Az=\frac{az+b}{cz+d}.$ (80)

For $\lambda\in\mathbb{C}_{*}$, we se that $\lambda A$ gives the same action
as $A$, so it reduces to an action of the projective group
$PGL(2,\mathbb{C})$. Moving one of the roots at infinity, this way it is
always possible to bring the integral in the form $I_{1}=\chi J_{1}$, where

$\displaystyle J_{1}=\int_{\gamma}\frac{dx}{y},$ (81)

where now the elliptic curve is defined by the Weierstrass normal form555For
example, assume $P(x)=a(x-x_{1})(x-x_{2})(x-x_{3})(x-x_{4})$, and take the
transformation  $\displaystyle x(z)=\frac{x_{1}z-b}{z-d},\qquad
z(x)=\frac{xd-b}{x-x_{1}},$ (82) which sends $x_{1}$ to $\infty$. Then, the
integral (76) takes the form  $\displaystyle
I_{1}=2\sqrt{\frac{x_{1}d-b}{a\prod_{j=2}^{4}(x_{1}-x_{j})}}\int_{z(x_{2})}^{\infty}\frac{dz}{\sqrt{4(z-z(x_{2}))(z-z(x_{3}))(z-z(x_{4}))}}.$
(83) Finally, fixing $b,d$ such that $\sum_{j=2}^{4}z(x_{j})=0$, we get the
Weierstrass normal form.  [110]

$\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}.$ (84)

Another choice is to bring it in the Legendre normal form

$\displaystyle y^{2}=(1-x^{2})(1-k^{2}x^{2}),$ (85)

so relating it to the complete elliptic integral of the first kind. We are
interested in the Weierstrass form now. Associated to it, the elliptic
integrals of the second kind can be written in the form

$\displaystyle J_{2}=\int_{\gamma}\frac{xdx}{y}.$ (86)

Let us now quickly see how $J_{1}$ and $J_{2}$ are related to the theory of
Weierstrass $\wp$ and $\zeta$ elliptic functions. For $\omega_{1}$ and
$\omega_{2}$ two complex numbers such that

$\displaystyle\tau:=\frac{\omega_{2}}{\omega_{1}}$ (87)

has strictly positive imaginary part, one defines the Weierstrass gamma
function $\wp$ as [111]

$\displaystyle\wp(z)=\frac{1}{z^{2}}+\sum_{(m,n)\in\mathbb{Z}^{2}_{0}}\left(\frac{1}{(z-m\omega_{1}-n\omega_{2})^{2}}-\frac{1}{(m\omega_{1}+n\omega_{2})^{2}}\right),$
(88)

where $m,n$ are relative integers with $(m,n)\neq(0,0)$. It is biperiodic,
with periods $\omega_{1}$ and $\omega_{2}$, and satisfies the differential
equation

$\displaystyle(\wp^{\prime})^{2}=4\wp^{3}-g_{2}\wp-g_{3},$ (89)

with

$\displaystyle g_{2}$
$\displaystyle=\sum_{(m,n)\in\mathbb{Z}^{2}_{0}}\frac{60}{(m\omega_{1}+n\omega_{2})^{4}},$
(90) $\displaystyle g_{3}$
$\displaystyle=\sum_{(m,n)\in\mathbb{Z}^{2}_{0}}\frac{140}{(m\omega_{1}+n\omega_{2})^{6}}.$
(91)

The Weierstrass $\zeta$ function is

$\displaystyle\zeta(z)=\frac{1}{z}+\sum_{(m,n)\in\mathbb{Z}^{2}_{0}}\left(\frac{1}{z-m\omega_{1}-n\omega_{2}}+\frac{1}{m\omega_{1}+n\omega_{2}}+\frac{z}{(m\omega_{1}+n\omega_{2})^{2}}\right).$
(92)

It is then clear that $\zeta^{\prime}(z)=-\wp(z)$. However, $\zeta(z)$ is not
biperiodic, but it satisfies the relations

$\displaystyle\zeta(z+\omega_{j})=\zeta(z)+2\eta_{j},\quad\eta_{j}=\zeta(\frac{\omega_{j}}{2}),\quad
j=1,2,3,$ (93)

with $\omega_{3}=\omega_{1}+\omega_{2}$. In this way, the Legendre’s quadratic
relation takes the form

$\displaystyle\eta_{1}\omega_{2}-\omega_{1}\eta_{2}=i\pi,$ (94)

which can be easily proven after integrating $\zeta$ along a fundamental
parallelogram centred in $0$, with fundamental periods as edges [111]. This
means that the rectangle is $(abcd)$, with

$\displaystyle(a,b,c,d)=(-\frac{\omega_{1}+\omega_{2}}{2},\frac{\omega_{1}-\omega_{2}}{2},\frac{\omega_{1}+\omega_{2}}{2},\frac{-\omega_{1}+\omega_{2}}{2}),$
(95)

see figure 16.

$\boldsymbol{\rm{Re}(z)}$$\boldsymbol{\rm{Im}(z)}$$\boldsymbol{\frac{\omega_{1}}{2}}$$\boldsymbol{\frac{\omega_{2}}{2}}$$\boldsymbol{a}$$\boldsymbol{c}$$\boldsymbol{b}$$\boldsymbol{d}$
Figure 16: Fundamental parallelogram.

Now, in the fundamental region, the function $z$ has a unique simple pole,
sited in $z=0$, with residue 1. Therefore, we can write

$\displaystyle 2\pi
i=\int_{(abcd)}\zeta(z)dz=\int_{a}^{b}\zeta(z)dz+\int_{b}^{c}\zeta(z)dz+\int_{c}^{d}\zeta(z)dz+\int_{d}^{a}\zeta(z)dz.$
(96)

On the other hand, we have that

$\displaystyle(d,c)=(a+\omega_{2},b+\omega_{2}),\qquad(b,c)=(a+\omega_{1},d+\omega_{1}),$
(97)

which imply

$\displaystyle\int_{b}^{c}\zeta(z)dz=$
$\displaystyle\int_{a+\omega_{1}}^{d+\omega_{1}}\zeta(z)dz=\int_{a}^{d}\zeta(t+\omega_{1})dt=\int_{a}^{d}(\zeta(t)+2\eta_{1})dt=\int_{a}^{d}\zeta(t)dt+2\eta_{1}(d-a)$
(98) $\displaystyle=$
$\displaystyle\int_{a}^{d}\zeta(t)dt+2\eta_{1}\omega_{2},$ (99)

and

$\displaystyle\int_{c}^{d}\zeta(z)dz=$
$\displaystyle-\int_{d}^{c}\zeta(z)dz=-\int_{a+\omega_{2}}^{b+\omega_{2}}\zeta(z)dz=-\int_{a}^{b}\zeta(t+\omega_{2})dt=-\int_{a}^{b}(\zeta(t)+2\eta_{2})dt$
(100) $\displaystyle=$
$\displaystyle-\int_{a}^{b}\zeta(t)dt-2\eta_{2}(b-a)=-\int_{a}^{b}\zeta(t)dt-2\eta_{2}\omega_{1}.$
(101)

Inserted in (96) we get

$\displaystyle 2\pi i=2\eta_{1}\omega_{2}-2\eta_{2}\omega_{1},$ (102)

which is the assert.
Finally, it is also possible to prove (but we omit the proof) that

$\displaystyle\omega_{1}$
$\displaystyle=J_{1},\qquad\eta_{1}=\frac{1}{2}J_{2},$ (103)
$\displaystyle\omega_{2}$
$\displaystyle=\tilde{J}_{1},\qquad\eta_{2}=\frac{1}{2}\tilde{J}_{2},$ (104)

where $\tilde{J}_{k}$ are obtained replacing $\gamma$ with $\beta$, so we can
further rewrite the Legendre’s relation in the form

$\displaystyle\tilde{J}_{1}J_{2}-\tilde{J}_{2}J_{1}=2\pi i.$ (105)

It is in this form that the Legendre’s relation has its deepest geometrical
meaning.

###  5.2 Riemann’s bilinear relations and intersections

To interpret the above result we pass to a more general situation and consider
an oriented Riemann surface $\Sigma$ of genus $g>0$ (the surface of a donut
with $g$ holes or a sphere with $g$ handles). It is well known, [102], Lemma
3.14, that we can realize it from a $4g$-gone with edges, in clockwise order,
$a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\cdots a_{g}b_{g}a_{g}^{-1}b_{g}^{-1}$, where
$a^{-1}$ means the edge is oriented counterclockwise, gluing each edge with
its “inverse”, according to the orientation. Any $a_{j}$ and any $b_{j}$ is a
closed curve and they satisfy the intersection product relations

$\displaystyle a_{j}\cdot a_{k}=b_{j}\cdot b_{k}=0,\qquad a_{j}\cdot
b_{k}=-b_{k}\cdot a_{j}=\delta_{jk}.$ (106)

These curves are homotopically distinct and are a (canonical) basis for the
first homology group $H_{1}(\Sigma,\mathbb{Z})\simeq\mathbb{Z}^{2g}$. Given a
closed 1-form $\omega$ on $\Sigma$, one defines its periods relative to the
given basis as

$\displaystyle\pi^{a}_{j}(\omega):=\oint_{a_{j}}\omega,\qquad\pi^{b}_{j}(\omega):=\oint_{b_{j}}\omega.$
(107)

Given two closed forms $\omega_{1}$ and $\omega_{2}$, using Stokes’s theorem
and closures, it is not difficult to prove that the following identity is
always true

$\displaystyle\int_{\Sigma}\omega_{1}\wedge\omega_{2}=\sum_{j=1}^{g}\left[\pi^{a}_{j}(\omega_{1})\pi^{b}_{j}(\omega_{2})-\pi^{b}_{j}(\omega_{1})\pi^{a}_{j}(\omega_{2})\right].$
(108)

###### Proof.

Assume that $\Sigma$ is a closed oriented Riemann surface of genus $g$. Now,
let us notice that if we cut $\Sigma$ along the representatives $a_{j}$ and
$b_{j}$ we get the $4g$-gon $\tilde{\Sigma}$ associated to the word
$a_{1}b_{1}a_{1}^{-1}b_{1}^{-1}\cdots a_{j}b_{j}a_{j}^{-1}b_{j}^{-1}\cdots
a_{g}b_{g}a_{g}^{-1}b_{g}^{-1}$.

$\boldsymbol{A}$$\boldsymbol{B}$$\boldsymbol{C}$$\boldsymbol{D}$$\boldsymbol{E}$$\boldsymbol{a_{1}}$$\boldsymbol{b_{1}}$$\boldsymbol{b^{-1}_{g}}$$\boldsymbol{a^{-1}_{1}}$$\boldsymbol{b^{-1}_{1}}$$\boldsymbol{b_{2}}$$\boldsymbol{A}$$\boldsymbol{B}$$\boldsymbol{D}$$\boldsymbol{C}$$\boldsymbol{E}$$\boldsymbol{b_{1}}$$\boldsymbol{b^{-1}_{1}}$
Figure 17: Riemann surfaces from the fundamental $4g$-gon.

This polygon is simply connected so any closed form restricted to it is exact.
In particular, this is true for $\omega_{2}$, which has the form
$\omega_{2}=d\phi_{2}$, for $\phi_{2}$ a well defined function on
$\tilde{\Sigma}$ (but not on $\Sigma$). More precisely we can fix a point $o$
in $\tilde{\Sigma}$ (for example the center). Since the polygon is convex, any
$x\in\tilde{\Sigma}$ is connected to $o$ by a segment and we can write

$\displaystyle\phi_{2}(x)\equiv\int_{o}^{x}\omega_{2},$ (109)

where the integration is intended along the segment. Now, if $\omega_{2}$ is
exact and $\omega_{1}$ is closed (indeed exact too on the simply connected
domain), then also $\omega_{1}\wedge\omega_{2}$ is exact. Indeed, the 1-form
$\Omega=-\phi_{2}\omega_{1}$ has the property that
$d\Omega=\omega_{1}\wedge\omega_{2}$, and is well defined on $\tilde{\Sigma}$.
Finally, since as sets

$\displaystyle\Sigma-\bigcup_{j=1}^{g}(a_{j}\bigcup
b_{j})=\tilde{\Sigma}-\partial\tilde{\Sigma},$ (110)

together with $\bigcup_{j=1}^{g}(a_{j}\bigcup b_{j})$ and
$\partial\tilde{\Sigma}$ are subsets of vanishing measure of $\Sigma$ and
$\tilde{\Sigma}$ respectively, we have

$\displaystyle\int_{\Sigma}\omega_{1}\wedge\omega_{2}$
$\displaystyle=\int_{\Sigma-\bigcup_{j=1}^{g}(a_{j}\bigcup
b_{j})}\omega_{1}\wedge\omega_{2}=\int_{\tilde{\Sigma}-\partial\tilde{\Sigma}}\omega_{1}\wedge\omega_{2}=\int_{\tilde{\Sigma}}\omega_{1}\wedge\omega_{2}=\int_{\tilde{\Sigma}}d\Omega=\int_{\partial\tilde{\Sigma}}\Omega$
(111)
$\displaystyle=-\int_{\partial\tilde{\Sigma}}\phi_{2}\omega_{1}=-\int_{\partial\tilde{\Sigma}}\omega_{1}(x)\int_{o}^{x}\omega_{2},$
(112)

where the main trick has been to rewrite the integral over a region where the
integrand is exact, in order to be able to use the Stokes theorem. Since the
boundary $\partial\tilde{\Sigma}$ is the (oriented) union of the edges of the
polygon, we then have

$\displaystyle\int_{\Sigma}\omega_{1}\wedge\omega_{2}$
$\displaystyle=-\sum_{j=1}^{g}\left(\int_{a_{j}}\omega_{1}(x)\int_{o}^{x}\omega_{2}+\int_{b_{j}}\omega_{1}(x)\int_{o}^{x}\omega_{2}+\int_{a_{j}^{-1}}\omega_{1}(x)\int_{o}^{x}\omega_{2}+\int_{b_{j}^{-1}}\omega_{1}(x)\int_{o}^{x}\omega_{2}\right).$
(113)

Now, consider the integrals

$\displaystyle
I^{a}_{j}\equiv\int_{a_{j}}\omega_{1}(x)\int_{o}^{x}\omega_{2}+\int_{a_{j}^{-1}}\omega_{1}(x)\int_{o}^{x}\omega_{2}.$
(114)

The forms $\omega_{1}$ and $\omega_{2}$ are well defined on $\Sigma$, so they
take the same values on the identified points of $a_{j}$ and $a_{j}^{-1}$. If
$y(x)$ is the point on $a_{j}^{-1}$ which identifies with $x\in a_{j}$ in
gluing $\tilde{\Sigma}$ to form $\Sigma$, then, noticing that $a_{j}^{-1}$ is
oriented in the opposite sense of $a_{j}$, we can write

$\displaystyle
I^{a}_{j}=\int_{a_{j}}\omega_{1}(x)\left(\int_{o}^{x}\omega_{2}-\int_{o}^{y(x)}\omega_{2}\right)=\int_{a_{j}}\omega_{1}(x)\left(\int_{o}^{x}\omega_{2}+\int_{y(x)}^{o}\omega_{2}\right).$
(115)

The sum of the two integrals of $\omega_{2}$ is just equivalent to the
integral along a curve from $y(x)$ to $x$, which is homotopically equivalent
to $b_{j}$ traveled in the opposite orientation (from $a_{j}^{-1}$ to
$a_{j}$), see Fig. 18.

$\boldsymbol{a}$$\boldsymbol{b}$$\boldsymbol{a^{-1}}$$\boldsymbol{b^{-1}}$$\boldsymbol{o}$$\boldsymbol{x}$$\boldsymbol{y(x)}$identified
in $\Sigma$$\tilde{\Sigma}$$\int_{o}^{x}\omega_{2}$$\int_{y(x)}^{o}\omega_{2}$
Figure 18: Construction of the integral $I_{b}$ in case $g=1$.

Therefore,

$\displaystyle\int_{o}^{x}\omega_{2}+\int_{y(x)}^{0}\omega_{2}=-\int_{b_{j}}\omega_{2}=-\pi^{b}_{j}(\omega_{2})$
(116)

is independent from $x$. So, we can write

$\displaystyle
I^{a}_{j}=-\pi^{b}_{j}(\omega_{2})\int_{a_{j}}\omega_{1}(x)=-\pi^{b}_{j}(\omega_{2})\pi^{a}_{j}(\omega_{1}).$
(117)

Let us now see what changes for the remaining integrals

$\displaystyle
I^{b}_{j}\equiv\int_{b_{j}}\omega_{1}(x)\int_{o}^{x}\omega_{2}+\int_{b_{j}^{-1}}\omega_{1}(x)\int_{o}^{x}\omega_{2}.$
(118)

Reasoning as above, if $y(x)$ is the point on $b_{j}^{-1}$ identified with $x$
on $b_{j}$ when constructing $\Sigma$ from $\tilde{\Sigma}$, we can write

$\displaystyle
I^{b}_{j}=\int_{b_{j}}\omega_{1}(x)\left(\int_{o}^{x}\omega_{2}+\int_{y(x)}^{o}\omega_{2}\right).$
(119)

Now, the path $y(x)\to o\to x$ on $\Sigma$ becomes a path homotopic to
$a_{j}^{-1}$ traveled in the opposite direction, so it is homotopic to $a_{j}$
and, therefore

$\displaystyle\int_{o}^{x}\omega_{2}+\int_{y(x)}^{o}\omega_{2}=\pi^{a}_{j}(\omega_{2}).$
(120)

Thus,

$\displaystyle
I^{b}_{j}=\int_{b_{j}}\omega_{1}(x)\pi^{a}_{j}(\omega_{2})=\pi^{a}_{j}(\omega_{2})\pi^{b}_{j}(\omega_{1}).$
(121)

Since

$\displaystyle\int_{\Sigma}\omega_{1}\wedge\omega_{2}=-\sum_{j=1}^{g}(I^{a}_{j}+I^{b}_{j}),$
(122)

we get the assertion. ∎

This wonderful formula, known as Riemann’s bilinear relations, gives (105) as
a very particular case and provides the geometric interpretation we were
looking for. Indeed it leads to the concept of intersection number between
closed forms, or, as we will see in a moment, between cohomology cocycles. The
transversality of the homology curves is now replaced by the fact that in any
given point $\omega_{1}\wedge\omega_{2}$ is non degenerate as a bilinear (or
multilinear) map on the tangent space to $\Sigma$. The directions where
$\omega_{1}$ vanishes define locally a curve. Similarly, $\omega_{2}$ defines
locally a second curve. These two curves meet transversally in the given
point. This curves are essentially the Poincaré duals of the forms and this
naive description shows its role in providing a geometrical interpretation of
the intersection number between forms. When $\Sigma$ is provided by a
Riemannian metric $\boldsymbol{h}$, then, in local coordinates, we can write

$\displaystyle\omega_{1}\wedge\omega_{2}=\omega_{1j}\omega_{2k}dx^{j}\wedge
dx^{k}=\omega_{1j}\omega_{2k}\frac{1}{\sqrt{h}}\varepsilon^{jk}\sqrt{h}dx^{1}\wedge
dx^{2},$ (123)

where $\varepsilon^{jk}$ is the usual Levi-Civita tensor density, and
Einstein’s summation convention is adopted. We set

$\displaystyle w_{2}^{j}\equiv\omega_{2k}\frac{1}{\sqrt{h}}\varepsilon^{jk},$
(124)

the Hodge dual of $\omega_{2}$, and identify $\sqrt{h}dx^{1}\wedge dx^{2}=dV$
as the volume element on $\Sigma$. Finally, since closed forms are locally
exact, in a small enough neighbourhood we can write $\omega_{1}=d\phi_{1}$ for
a potential $\phi_{1}$. Notice that the Poincaré dual of $\omega_{1}$ is thus
locally represented by the level curves $\phi_{1}=const$. In the given
neighbourhood we get

$\displaystyle\omega_{1}\wedge\omega_{2}=\partial_{j}\phi_{1}w_{2}^{j}\
dV=\partial_{j}(\phi_{1}w_{2}^{j})\ dV,$ (125)

where we used that $\omega_{2}$ closed implies $\partial_{j}w_{2}^{j}=0$. If
as a small neighbourhood we take a small rectangle $Q$ having edges $a,b,c,d$
with $a$ and $c$ along level curves for $\phi_{1}$ (with values $\kappa_{a}$
and $\kappa_{c}$), while $b$ and $d$ are orthogonal, then

$\displaystyle\int_{Q}\omega_{1}\wedge\omega_{2}=\int_{Q}\partial_{j}(\phi_{1}w_{2}^{j})\
dV=\kappa_{c}\int_{c}n^{c}_{j}w_{2}^{j}ds+\kappa_{a}\int_{a}n^{a}_{j}w_{2}^{j}ds,$
(126)

$n^{c}_{j}$ and $n^{a}_{j}$ being the external normals to $c$ and $a$
respectively. This integral is thus determined by the flux of the vector field
$w_{2}^{j}$, Hodge dual to $\omega_{2}$, through the equipotential surfaces
(lines) of $\omega_{1}$. Notice that even in an extremal case when the
rectangle can be extended until wrapping the surface so that666notice that
this cannot be the general case, or at least not assuming $b$ and $d$ remain
orthogonal to the equipotential lines $a=c$, then we would get

$\displaystyle\int_{Q}\omega_{1}\wedge\omega_{2}=(\kappa_{c}-\kappa_{a})\int_{c}n^{c}_{j}w_{2}^{j}ds,$
(127)

in general with $\kappa_{c}-\kappa_{a}\neq 0$ despite $c$ and $a$ coincide,
since $\phi_{1}$ is not monodromic and takes a different value after wrapping
$\Sigma$ along a closed loop. Indeed, in this case $(\kappa_{c}-\kappa_{a})$
is nothing but the period of $\omega_{1}$ along the closed path. In general,
it is not possible to extend even $c$ to a closed path, and our naive analysis
then stops here. However, it is clear that its correct formulation is given
exactly by the Riemann’s bilinear formula, which generalizes this vision in
terms of fluxes (of the Hodge dual of $\omega_{2}$) through the Poincaré dual
of $\omega_{1}$.
Now, it is time to be a little bit more precise. If we modify $\omega_{1}$ by
an exact form $df$, its periods don’t change777as well as they don’t change
deforming the path $a_{j}$ and $b_{j}$ continuously. On the other hand, since
$\omega_{2}$ is closed too,

$\displaystyle(\omega_{1}+df)\wedge\omega_{2}=\omega_{1}\wedge\omega_{2}+d(f\omega_{2}),$
(128)

and by Stokes’s theorem and the fact that $\Sigma$ has no boundary, we see
that the exact term does not contribute to the l.h.s. integral. The same holds
if we modify $\omega_{2}$ by an exact form. This means that this formula
depends only on the cohomology classes of $\omega_{1}$ and $\omega_{2}$. From
(128), we also see that if we consider the cohomology class of
$\omega_{1}\wedge\omega_{2}$, then, it depends only on the classes
$[\omega_{1}]$ and $[\omega_{2}]$ and not specifically on the forms. So, this
gives rise to a well defined bilinear antisymmetric map

$\displaystyle\cup:H^{1}(\Sigma)\times H^{1}(\Sigma)\longrightarrow
H^{2}(\Sigma),\
([\omega_{1}],[\omega_{2}])\mapsto[\omega_{1}]\cup[\omega_{2}]:=[\omega_{1}\wedge\omega_{2}],$
(129)

called the cup product. The above formula can then be written as

$\displaystyle[\omega_{1}]\cdot[\omega_{2}]\equiv\int_{\Sigma}[\omega_{1}]\cup[\omega_{2}]=\sum_{j=1}^{g}\left[\pi^{a}_{j}([\omega_{1}])\pi^{b}_{j}([\omega_{2}])-\pi^{b}_{j}([\omega_{1}])\pi^{a}_{j}([\omega_{2}])\right].$
(130)

The l.h.s. formula is called the intersection product of the two classes (or
forms). This is because it is in a sense a dual of the intersection formula in
homology. Indeed, let us consider the dual basis to the canonical one, that is
a set of closed one forms $\omega^{a}_{j}$, $\omega^{b}_{j}$, $j=1,\ldots,g$
such that

$\displaystyle\pi^{a}_{j}(\omega^{a}_{k})$
$\displaystyle=\delta_{jk},\qquad\pi^{a}_{j}(\omega^{b}_{k})=0,$ (131)
$\displaystyle\pi^{b}_{j}(\omega^{a}_{k})$
$\displaystyle=0,\qquad\pi^{b}_{j}(\omega^{b}_{k})=\delta_{jk}.$ (132)

Thus,

$\displaystyle[\omega^{a}_{j}]\cdot[\omega^{a}_{k}]$
$\displaystyle=[\omega^{b}_{j}]\cdot[\omega^{b}_{k}]=0,$ (133)
$\displaystyle[\omega^{a}_{j}]\cdot[\omega^{b}_{k}]$
$\displaystyle=-[\omega^{b}_{k}]\cdot[\omega^{a}_{j}]=\delta_{jk},$ (134)

which has exactly the same form of the intersection products among the
elements of the homology basis. This is our final interpretation of the
Legendre’s bilinear formula (105): it gives the intersection product between
the first kind and the second kind differentials. In general, after selecting
a basis $b_{j}$, $j=1,\ldots,2g$ of $H^{1}(\Sigma)$, the matrix
$\boldsymbol{c}$ with elements

$\displaystyle c_{jk}=b_{j}\cdot b_{k}$ (135)

is invertible. Any other element $[\omega]$ in $H^{1}(\Sigma)$ can be written
as

$\displaystyle[\omega]=\sum_{j=1}^{2g}k^{j}b_{j},$ (136)

with $k_{J}\in\mathbb{C}$. Therefore,

$\displaystyle[\omega]\cdot b_{k}=\sum_{j=1}^{2g}k^{j}c_{jk}$ (137)

so that

$\displaystyle k^{j}=\sum_{k=1}^{2g}[\omega]\cdot b_{k}c^{kj},$ (138)

where $c^{kj}$ are the matrix elements of the inverse matrix
$\boldsymbol{c}^{-1}$. On the other hand, for $u=a,b$, the periods of
$[\omega]$ are

$\displaystyle\pi^{u}_{j}([\omega])=\sum_{k=1}^{2g}k^{k}\pi^{u}_{j}(b_{k})=\sum_{l=1}^{2g}\sum_{k=1}^{2g}([\omega]\cdot
b_{l})c^{lk}\pi^{u}_{j}(b_{k}).$ (139)

Thus, we see that after the intersection product is given, for a path $\gamma$
homologically equal to $\gamma=\sum_{j=1}^{g}(m_{a}^{j}a_{j}+m_{b}^{j}b_{j})$,
$m_{u}^{j}\in\mathbb{Z}$, our integral is given by

$\displaystyle\int_{\gamma}\omega=\sum_{u=a,b}\sum_{j=1}^{g}\sum_{l=1}^{2g}\sum_{k=1}^{2g}m_{u}^{j}([\omega]\cdot
b_{l})c^{lk}\pi^{u}_{j}(b_{k}).$ (140)

So, the only integrals we have really to compute are $\pi^{u}_{j}(b_{k})$,
which we can call the “master integrals.” Notice that this expression
resembles the decomposition formulas (45).
At this point we may wonder how much all this can be generalized to higher
dimensions. To get some insight, let us now present some very general
considerations, without any pretense of completeness, but just with the aim to
outline the main concepts one would need to deepen referring to the
appropriate literature.
Remark: in literature, Riemann’s bilinear relations usually are not exactly
(108), but rather one of its consequences ([102], Th.12.2): if $\omega_{j}$,
$j=1,2$ are two holomorphic forms on a genus $g$ closed oriented Riemann
surface $\Sigma$, and $a_{j},b_{j}$, $j=1,\ldots,g$ a canonical basis of
$H_{1}(\Sigma)$, then the following relations are true:

$\displaystyle\sum_{j=1}^{g}(\pi^{a}_{j}(\omega_{1})\pi^{b}_{j}(\omega_{2})-\pi^{a}_{j}(\omega_{2})\pi^{b}_{j}(\omega_{1}))$
$\displaystyle=0$ (141) $\displaystyle{\rm
Im}(\sum_{j=1}^{g}\overline{\pi^{b}_{j}(\omega_{1})}\pi^{a}_{j}(\omega_{1}))$
$\displaystyle>0,$ (142)

where $\bar{z}$ means the complex conjugate of $z$.

###  5.3 A twisted version

We want now to consider a situation a little bit more involved. In place of
real or complex valued forms over our Riemann surface $\Sigma_{g}$, let us
consider twisted differential forms taking value in a line bundle
$\mathcal{L}$. This is a bundle whose fibres are copies of
$\mathbb{C}_{*}=\mathbb{C}-\\{0\\}$. To it we can associate a dual line bundle
$\mathcal{L}^{\vee}$, which has the following property: if $\psi$ and
$\psi^{\vee}$ are sections of $\mathcal{L}$ and $\mathcal{L}^{\vee}$,
respectively, then, $\psi(z)\psi^{\vee}(z)\equiv f(z)$ defines a function
$f:\Sigma_{g}\rightarrow\mathbb{C}$. In particular, if $u(z)$ is a section of
$\mathcal{L}^{\vee}$, then $u(z)^{-1}$ is a section of $\mathcal{L}$. With
this in mind, we say that $\phi_{L}$ is a $\mathcal{L}$-valued twisted
$k$-form if for any given section $u$ of $\mathcal{L}^{\vee}$, then
$u\phi_{L}$ is a $\mathbb{C}$-valued $k$-form. We will call $\phi_{L}$ a left
$k$-form, and the space of left $k$-forms is

$\displaystyle\Omega^{k}(\Sigma_{g},\mathcal{L})=\Omega^{k}(\Sigma_{g},\mathbb{C})\otimes\mathcal{L},$
(143)

and we write

$\displaystyle\Omega^{*}_{L}=\bigoplus_{k=0}^{2}\Omega^{k}(\Sigma_{g},\mathcal{L}).$
(144)

We say that $\mathcal{L}$ is flat if it admits a global section. When it
happens, it is clear that also $\mathcal{L}^{\vee}$ is flat. Assume $u$ is a
global section of $\mathcal{L}^{\vee}$: we call it the twist. Given a twist,
on $\Omega^{*}_{L}$ we can put a structure of ring as follows. For
$\phi_{1},\phi_{2}\in\Omega^{*}_{L}$, we define

$\displaystyle\phi_{1}\wedge_{u}\phi_{2}:=u\phi_{1}\wedge\phi_{2},$ (145)

in harmony with the condition
$(u\phi_{1})\wedge(u\phi_{2})=u(\phi_{1}\wedge_{u}\phi_{2})$, required by the
mapping
$\Omega^{k}(\Sigma_{g},\mathcal{L})\rightarrow\Omega^{k}(\Sigma_{g},\mathbb{C})$,
$\phi\mapsto u\phi$. Notice that $\Omega^{0}(\Sigma_{g},\mathcal{L})$ are the
sections of $\mathcal{L}$. $u$ also allows us to introduce a connection on
$\mathcal{L}$ that induces a covariant derivative

$\displaystyle\nabla_{\omega}:\Omega^{k}(\Sigma_{g},\mathcal{L})\longrightarrow\Omega^{k+1}(\Sigma_{g},\mathcal{L}),\quad
i.e.\quad d(u\phi_{L})=u\nabla_{\omega}\phi_{L}.$ (146)

Concretely,

$\displaystyle\nabla_{\omega}\phi\equiv
d+\omega\wedge\phi,\qquad\omega=\frac{du}{u}.$ (147)

It is a flat connection, since it satisfies $\nabla_{\omega}^{2}=0$ as one can
easily check. This allows us to say that $\phi$ is closed if
$\nabla_{\omega}\phi=0$, and exact if $\phi=\nabla_{\omega}\psi$ for $\psi$ a
left form of lower rank. Notice that these definitions imply

$\displaystyle\nabla_{\omega}(\phi_{L}\wedge_{u}\tilde{\phi}_{L})=(\nabla_{\omega}\phi_{L})\wedge_{u}\tilde{\phi}_{L}+(-1)^{k}\phi_{L}\wedge_{u}(\nabla_{\omega}\tilde{\phi}_{L})-\omega\wedge\phi_{L}\wedge_{u}\tilde{\phi}_{L},$
(148)

if $\phi_{L}$ has degree $k$.
In a similar way, we can define the right forms as the twisted forms with
value in $\mathcal{L}^{\vee}$, simply by exchanging
$\mathcal{L}\leftrightarrow\mathcal{L}^{\vee}$, $u\leftrightarrow 1/u$,
$\omega\leftrightarrow-\omega$. A right form will be called $\phi_{R}$, when
necessary to distinguish it from a left form. Of course, a right form is
closed if $\nabla_{-\omega}\phi_{R}=0$, and exact if
$\phi_{R}=\nabla_{-\omega}\psi_{R}$.
A first interesting fact is that the standard wedge product between a left
form and a right form satisfies

$\displaystyle d(\phi_{L}\wedge\phi_{R})$
$\displaystyle=d\phi_{L}\wedge\phi_{R}+(-1)^{k_{L}}\phi_{L}\wedge
d\phi_{R}=(\nabla_{\omega}-\omega\wedge)\phi_{L}\wedge\phi_{R}+(-1)^{k_{L}}\phi_{L}\wedge(\nabla_{-\omega}+\omega\wedge)\phi_{R}$
(149)
$\displaystyle=\nabla_{\omega}\phi_{L}\wedge\phi_{R}+(-1)^{k_{L}}\phi_{L}\wedge\nabla_{-\omega}\phi_{R}-\omega\wedge\phi_{L}\wedge\phi_{R}+\omega\wedge\phi_{L}\wedge\phi_{R}$
(150)
$\displaystyle=\nabla_{\omega}\phi_{L}\wedge\phi_{R}+(-1)^{k_{L}}\phi_{L}\wedge\nabla_{-\omega}\phi_{R},$
(151)

where we used that $\phi_{L}\wedge\omega=(-1)^{k_{L}}\omega\wedge\phi_{L}$. In
particular, if $\phi_{L}$ and $\phi_{R}$ are covariantly closed, then,
$\phi_{L}\wedge\phi_{R}$ is closed in the canonical sense. Now, suppose
$\phi_{L}$ and $\phi_{R}$ are 1-forms, well defined over the Riemann surface
$\Sigma_{g}$. As in the previous sections, we can cut it to
$\tilde{\Sigma}_{g}$, the fundamental $4g$-gone, which is simply connected
and, indeed, star shaped with respect to an internal point $o$. Therefore, for
any $z\in\tilde{\Sigma}_{g}$ we can take the segment $[o,z]$ and consider the
function

$\displaystyle\psi_{L}(z)=e^{-\int_{o}^{z}\omega(t)}\int_{o}^{z}e^{\int_{o}^{t}\omega(s)}\phi_{L}(s).$
(152)

It is well defined on $\tilde{\Sigma}_{g}$ and satisfies

$\displaystyle\nabla_{\omega}\psi_{L}(z)=\phi_{L}(z).$ (153)

The formula (151) then shows that

$\displaystyle d(\psi_{L}\phi_{R})=\phi_{L}\wedge\phi_{R},$ (154)

so $\phi_{L}\wedge\phi_{R}$ is exact on $\tilde{\Sigma}_{g}$. Proceeding as in
the proof of (108), we can write

$\displaystyle\int_{\Sigma_{g}}\phi_{L}\wedge\phi_{R}$
$\displaystyle=\int_{\tilde{\Sigma}_{g}}\phi_{L}\wedge\phi_{R}=\int_{\tilde{\Sigma}_{g}}d(\psi_{L}\phi_{R})=\int_{\partial\tilde{\Sigma}_{g}}\psi_{L}\phi_{R}=\int_{\partial\tilde{\Sigma}_{g}}e^{-\int_{o}^{z}\omega(t)}\int_{o}^{z}e^{\int_{o}^{t}\omega(s)}\phi_{L}(t)\phi_{R}(z),$
(155)

and following the same exact procedure, we finally get [74]

$\displaystyle\int_{\Sigma_{g}}\phi_{L}\wedge\phi_{R}=\sum_{j=1}^{g}$
$\displaystyle\left[\left(\int_{a_{j}}e^{\int_{o}^{t}\omega(s)}\phi_{L}(t)\right)\left(\int_{b_{j}}e^{-\int_{o}^{t}\omega(s)}\phi_{R}(t)\right)\right.$
(156)
$\displaystyle\left.-\left(\int_{b_{j}}e^{\int_{o}^{t}\omega(s)}\phi_{L}(t)\right)\left(\int_{a_{j}}e^{-\int_{o}^{t}\omega(s)}\phi_{R}(t)\right)\right],$
(157)

which is the twisted version of (108).

###  5.4 Perversities and thimbles

Now, we are to the point! While we apparently related our integrals just to
the cohomology of the Riemann surface defined by the polynomial, the situation
is more involved in the general case of a Feynman integral. The point is that
the section $u$ is represented by the Baikov polynomial $\mathcal{B}$ as
$u=\mathcal{B}^{\gamma}$. We assume $\gamma\in\mathbb{R}-\mathbb{Z}$. This
section has zeros (or singular points if $\gamma<0$) in the zeros of
$\mathcal{B}$, so it doesn’t define a global section of a line bundle over the
whole domain of the polynomial, but just in the complement of its zeros. Also,
if the integration domain goes towards infinity, it must be chosen so that the
integral converges (so along paths where $\mathcal{B}$ goes to $0$ or to
$\infty$, according if $\gamma$ is positive or negative). This looks like to
say that the form becomes trivial around infinity, to make sense to the
integral. This leads to think that the right (co)homology to be considered is
the one of $\mathbb{C}^{n}-Z$ relative to infinity, where $Z$ is the zero
locus of the polynomial. This is indeed the result of Francis Pham, who proved
that the right homology is the relative homology $H_{n}(\mathbb{C}^{n}-Z,B)$,
$B$ being a relative neighbourhood of infinity:
$B=\\{x\in\mathbb{C}^{n}||\mathcal{B}(x)|>N\\}$ where $N$ is “large enough”,
[112], [113].
The point is that one has to manage the homological tools when singularities
are present. Let us consider as an illustrative example the integral

$\displaystyle\int_{\Gamma}\frac{dz}{(z^{3}+z)^{\alpha}},$ (158)

$\alpha\notin\mathbb{Z}$. The integral has to converge, so $\Gamma$ has to
avoid the zeros of the polynomial, but can reach the infinity, if
$\alpha>\frac{1}{3}$ and the opposite if $\alpha<\frac{1}{3}$. The zeros of
the polynomials are branch points and we have to fix branch cuts to fix the
integrand. Moreover, the map $\mathcal{B}:\mathbb{C}\rightarrow\mathbb{C}$ has
two critical points, which are the points where the gradient vanishes. Let us
briefly see why this could be interesting. If we want to have intuition on the
cohomology to be used, we can start thinking that it is not expected to depend
on the exact value of $\alpha$. Therefore, after rewriting the integral in the
form

$\displaystyle\int_{\Gamma}e^{-\alpha\log(z^{3}+z)}dz,$ (159)

we can estimate it for very large $\alpha$. This can be done by means of the
saddle point method, so that the integral is dominated by the saddle points of
the exponent, which are indeed the critical points of $\mathcal{B}$. Depending
on $\alpha$, one has then to follow the steepest descent or ascent lines, in
such a way that the integrand goes to zero. For example, if we write
$\mathcal{B}(\zeta_{+})=|\mathcal{B}(\zeta_{+})|e^{i\phi_{+}}$, where
$\zeta_{+}$ is a critical point, then

$\displaystyle\mathcal{B}(\zeta_{+})^{-\alpha}=e^{-\alpha\log|\mathcal{B}(\zeta_{+})|-i\alpha\phi_{+}},$
(160)

we see that the path passing from the saddle point is conveniently chosen so
that the phase is constant so to avoid oscillations. Thus, the best choice are
the paths having the constant phase, fixed by the critical point. Then, one
chooses the direction such to move from the critical value to infinity or to
zero, depending on the case if the path is allowed to reach infinity or zero
(in our example, depending if $\alpha>1/3$ or $\alpha<1/3$). Notice that,
having fixed the phase, choosing a direction to move away from the critical
point, the absolute value of the integrand moves monotonically toward $0$ or
$\infty$. Also, since we have to move along these paths, we have to choose the
cuts so that they do not coincide with (part of) the constant phase
integration paths. Now that we have got the hint, let us go back to the
explicit case in some detail.
The branch points are $z_{0}=0,\ z_{\pm}=\pm i$. The critical points solve
$3z^{2}+1=0$, so that they are $\zeta_{\pm}=\pm\frac{i}{\sqrt{3}}$. If we
coordinatize the target space of $\mathcal{B}$ with $t$, then the images of
the branch points and the critical points in the target space are three:

$\displaystyle
t_{0}\equiv\mathcal{B}(z_{i})=0,\qquad\tau_{\pm}=\mathcal{B}(\zeta_{\pm})=\pm\frac{2i}{3\sqrt{3}}.$
(161)

The phases determined by $\tau_{\pm}$ are thus $\phi_{\pm}=\pm\frac{\pi}{2}$.
Therefore, we choose the cuts with phase $0$. Writing $z=x+iy$, this means

$\displaystyle\mathcal{B}(z)=x^{3}-3y^{2}x+x+i(-y^{3}+3x^{2}y+y),$ (162)

so that the cuts are defined by $-y^{3}+3x^{2}y+y=0$, $x^{3}-3y^{2}x+x>0$.
These are

$\displaystyle y$ $\displaystyle=0,\qquad x>0;$ (163) $\displaystyle y$
$\displaystyle=\sqrt{3x^{2}+1},\qquad x<0;$ (164) $\displaystyle y$
$\displaystyle=-\sqrt{3x^{2}+1},\qquad x<0,$ (165)

see figure figure 19. In a similar way, we can determine the descent/ascent
paths from the critical points.

$z$-plane$t$-plane$\boldsymbol{\tau_{+}}$$\boldsymbol{\tau_{-}}$$\boldsymbol{t_{0}}$$\boldsymbol{\phi_{+}}$$\boldsymbol{\phi_{-}}$$\boldsymbol{z_{+}}$$\boldsymbol{z_{-}}$$\boldsymbol{z_{0}}$$\boldsymbol{\zeta_{+}}$$\boldsymbol{\zeta_{-}}$$\boldsymbol{\Gamma_{+}^{d}}$$\boldsymbol{\Gamma_{+}^{a}}$$\boldsymbol{\Gamma_{-}^{d}}$$\boldsymbol{\Gamma_{-}^{a}}$
Figure 19: Cuts, ascent paths and descent paths.

Through $\zeta_{\pm}$ the paths $\Gamma_{\pm}$ are defined by the phase
$\phi_{\pm}=\pm\frac{\pi}{2}$, which correspond to $x^{3}-3y^{2}x+x=0$,
$\pm(-y^{3}+3x^{2}y+y)>0$. These give for $\Gamma_{+}$ the paths

$\displaystyle x$ $\displaystyle=0,\qquad 0<y<1,$ $\displaystyle({\rm descent\
path,}\ \Gamma^{d}_{+});$ (166) $\displaystyle y$
$\displaystyle=\sqrt{\frac{x^{2}+1}{3}},$ $\displaystyle({\rm ascent\ path,}\
\Gamma^{a}_{+}),$ (167)

for $\Gamma_{-}$ the paths

$\displaystyle x$ $\displaystyle=0,\qquad-1<y<0,$ $\displaystyle({\rm descent\
path,}\ \Gamma^{d}_{-});$ (168) $\displaystyle y$
$\displaystyle=-\sqrt{\frac{x^{2}+1}{3}},$ $\displaystyle({\rm ascent\ path,}\
\Gamma^{a}_{-}).$ (169)

These are curves from infinity to infinity around the cuts (ascent paths), or
from a zero to another one (descent paths). They are called Lefschetz
thimbles, for a reason that will become clear soon. Now we want to understand
a little better what we have done just inspired by saddle point methods. To
this hand, let us consider $\mathcal{B}$ as a map defining a fibration over
$\mathbb{C}$. The fibre over $t$ is generically given by a set
$\mathcal{B}^{-1}(t)$ of 3 points. The exceptions are the critical points
where the polynomial has double roots, so the fibre is made just by 2 points.
For the generic fibre at $t$ (which is zero dimensional) the cohomology is
$H^{0}(\mathcal{B}^{-1}(t),\mathbb{Z})=\mathbb{Z}^{3}$, and is invariant when
$t$ varies, so we are in trouble if we move to a critical point. Therefore, to
manage this difficulty, it is convenient to restrict this “fibration” of
cohomology groups to
$\mathbb{C}_{\tau}\equiv\mathbb{C}-\\{\tau_{+},\tau_{-}\\}$ and look at what
happens if we move around the critical values. To see this, notice that if
$\lambda:=e^{\frac{2\pi i}{3}}$ is a primitive third root of $1$, then,

$\displaystyle\mathcal{B}^{-1}(t)=\begin{pmatrix}\sqrt[3]{\frac{t}{2}+\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}+\sqrt[3]{\frac{t}{2}-\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}\\\
\lambda\sqrt[3]{\frac{t}{2}+\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}+\lambda^{2}\sqrt[3]{\frac{t}{2}-\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}\\\
\lambda^{2}\sqrt[3]{\frac{t}{2}+\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}+\lambda\sqrt[3]{\frac{t}{2}-\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}\end{pmatrix},$
(170)

from which we can easily see what happens to its (co)homology when we move on
a loop around the critical points: since $\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}$
vanishes at $t=\tau_{\pm}$, it changes sign in moving along the loop, and we
see that the upper component is invariant, while the lowest components are
interchanged. So, on $V_{t}\equiv
H^{0}(\mathcal{B}^{-1}(t),\mathbb{Z})=\mathbb{Z}^{3}$, the monodromy action
(the action of moving around the points) around $\tau_{\pm}$ is given by the
matrices

$\displaystyle M_{\pm}=\begin{pmatrix}1&0&0\\\ 0&0&1\\\ 0&1&0\end{pmatrix}.$
(171)

The badness of the fibration is measured by these (here coincident) monodromy
matrices. The point is that $V_{t}$ has fixed points given by the eigenspaces
of $M_{\pm}$ corresponding to the eigenvalue 1. These define the spaces of
invariants

$\displaystyle V^{M_{\pm}}\simeq\mathbb{Z}\ \begin{pmatrix}1\\\ 0\\\
0\end{pmatrix}\oplus\mathbb{Z}\begin{pmatrix}0\\\ 1\\\ 1\end{pmatrix}.$ (172)

Their complement are the spaces of coinvariants

$\displaystyle V_{M_{\pm}}\simeq\mathbb{Z}\begin{pmatrix}0\\\ 1\\\
-1\end{pmatrix}.$ (173)

On them the monodromies act as minus the identity. The strategy for capturing
the right cohomology, is to restrict the fibration of cohomologies, keeping
only the coinvariant parts and the information of the action of the
monodromies. Notice that these are irreducible non trivial representations of
$\pi_{1}(\mathbb{C}_{\tau})$, the first homotopy group (generated by the loops
around the critical values). This “almost fibration” with specification of
monodromy constrained structures at singular points substantially describes
what is called a perverse sheaf. The family of coinvariant (co)homology
elements can be then interpreted as follows. In the homology of
$\mathcal{B}^{-1}(t)$ they can be thought as

$\displaystyle
0\cdot\left(\sqrt[3]{\frac{t}{2}+\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}+\sqrt[3]{\frac{t}{2}-\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}\right)+p\cdot\left(\lambda\sqrt[3]{\frac{t}{2}+\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}+\lambda^{2}\sqrt[3]{\frac{t}{2}-\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}\right)$
(174)
$\displaystyle-p\cdot\left(\lambda^{2}\sqrt[3]{\frac{t}{2}+\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}+\lambda\sqrt[3]{\frac{t}{2}-\sqrt{\frac{t^{2}}{4}+\frac{1}{27}}}\right)$
(175)

with $p\in\mathbb{Z}$. The generator is for $p=1$. Then when $t$ varies from a
singular point to infinity (or to 0), we see that these points collapse at the
critical point, collapsing to zero also as an element of
$H_{0}(\mathcal{B}^{-1}(t),\mathbb{Z})$, since we get the same point with
total coefficient $1-1=0$.888If we considered the invariants, the result at
the point would be that the points meet with total coefficient $1+1=2$. These
correspond to the thimbles, and, are called vanishing cycles. Notice that the
pair of points are $0$ dimensional cycles. For a polynomial in $n$ variables,
one gets that the preimages of $t$ are generically $\mathcal{B}^{-}1(t)\simeq
TS^{n-1}$, the total space of the tangent bundle of a sphere, which are
homologically equivalent to spheres $S^{n-1}$. The (perverse) cohomology so
generated, is represented by union of spheres when $t$ varies, for example,
from a critical value to $\infty$ along a line $\gamma$. The result is
homotopic to a cylinder $S^{n-1}\times\gamma$, whose face at the end
corresponding to the critical value collapses to a point. This may be thought
to have the shape of a thimble.
Of course, a rigorous version of this construction in a general setting
requires the use of more sophisticated mathematical tools.

##  6 Some general constructions

In this section we want to discuss some relations between Feynman integrals,
homologies and cohomologies. Even if we will avoid entering the details, we
will illustrate the main ideas on how to deal with singularities by passing to
simplicial (co)homologies, while our original problem remains the one of
computing integrals of differential forms. Therefore, we will start by
recalling some relations among the main (co)homologies we will need to
consider in the successive subsections. We will not really provide notions but
rather only rough ideas and some literature. However, we hope to make this
section at least readable.

###  6.1 On cohomologies and de Rham Theorem

In general, for a given space, there can be several associated homologies
depending on the structures one is considering. We are of course interested in
the de Rham cohomology, which is well defined on smooth manifolds as the
quotient of closed forms with exact forms. But there are other (co)homologies
that are of interest for concrete calculations. For the definitions reported
below and any further lecture we refer to [102], [114].

####  6.1.1 Singular (co)homology

One of the most general is singular homology. A $k$-simplex $\Delta_{k}$ is
the smallest convex subset of $\mathbb{R}^{N}$ (N¿k) generated by $k+1$ points
in general position (so it has dimension $k$). If $x_{j}$, $j=0,\ldots,k$ are
the generating points, it is convenient to use the notation
$\Delta_{k}=\\{x_{0},\ldots,x_{k}\\}$. Then, the $j$-th face of the simplex is
the $(k-1)$-simplex
$\Delta_{k-1}^{j}=\\{x_{0}\ldots,\hat{x}_{j},\ldots,x_{n}\\}$, where the $hat$
means omission of the point. If $X$ is a topological space, then a singular
$k$-simplex in $X$ is a pair $(\Delta_{k},\phi)$, where
$\phi:\Delta_{k}\rightarrow X$ is a continuous map. A singular $k$-chain
$c_{k}$ is a formal finite combination of singular $k$-simplexes
$c_{k}=\sum_{\alpha}a_{\alpha}(\Delta_{k,\alpha},\phi_{\alpha})$,
$a_{\alpha}\in\mathbb{K}$, where $\mathbb{K}$ is an abelian group, for us
$\mathbb{K}=\mathbb{Z},\mathbb{Q},\mathbb{R},\mathbb{C}$. They form the space
$C^{sing}_{k}(X)$ of singular chains. The boundary of a singular simplex
$(\Delta_{k},\phi)$ is the singular $(k-1)$-chain

$\displaystyle\partial(\Delta_{k},\phi)=\sum_{j=0}^{k}(-1)^{j}(\Delta_{k-1}^{j},\phi|_{\Delta_{k-1}^{j}}).$
(176)

This defines by $\mathbb{K}$-linear extension, a sequence of
$\mathbb{K}$-linear maps

$\displaystyle\partial_{k}:C^{sing}_{k}(X)\longrightarrow
C^{sing}_{k-1}(X),\quad c_{k}\mapsto\partial c_{k}.$ (177)

It follows that $\partial_{k-1}\circ\partial_{k}=0$ and one can define the
singular homology groups

$\displaystyle H^{sing}_{k}(X,\mathbb{K})=\frac{{\rm ker}(\partial_{k})}{{\rm
Im}(\partial_{k+1})}.$ (178)

Similarly, we can define a singular $k$-cochain as a linear map

$\displaystyle\psi^{k}:C^{sing}_{k}(X)\longrightarrow\mathbb{K},$ (179)

that form the $\mathbb{K}$-linear space $C_{sing}^{k}(X)$, on which acts the
coboundary operator

$\displaystyle\delta^{k}:C_{sing}^{k}(X)\longrightarrow
C_{sing}^{k+1}(X),\quad i.e.\quad\
\delta^{k}\psi^{k}(c_{k+1})\equiv\psi^{k}(\partial_{k+1}c_{k+1}),\ \forall
c_{k+1}\in C^{sing}_{k+1}(X).$ (180)

From these, one defines the singular cohomology groups

$\displaystyle H_{sing}^{k}(X,\mathbb{K})=\frac{{\rm ker}(\delta^{k})}{{\rm
Im}(\delta^{k-1})}.$ (181)

Remark: replacing simplexes with hypercubes, one gets the singular cubic
(co)homology. However, this gives a non normalized cohomology [102] (also
called a generalized cohomology), that is a cohomology that does not satisfy
the condition $H_{k}(p)=0$ for $k>0$ and $p$ a point.

####  6.1.2 Simplicial (co)homology

This is less general and is first defined for simplicial complexes. A
simplicial complex $\Xi$ is a collection of simplexes such that if a simplex
is in the complex, then all its faces are also in, and, moreover, two
simplexes in the collection can intersect in one and only one common sub-face
of a given dimension. It is called finite, if composed by a finite number of
simplexes, while it is locally finite if for any point there is a small
neighbourhood intersecting just a finite number of simplexes. Given a
simplicial complex $\Xi$, one defines its simplicial $k$-chains as the formal
combinations $c_{k}=\sum_{\alpha}a_{\alpha}\Delta_{k,\alpha}$,
$a_{\alpha}\in\mathbb{K}$, which define the space $C_{k}^{simp}(\Xi)$.
Defining the boundary

$\displaystyle\partial\Delta_{k}=\sum_{j=0}^{k}(-1)^{j}\Delta_{k-1}^{j},$
(182)

we get linear maps

$\displaystyle\partial^{s}_{k}:C^{simp}_{k}(\Xi)\longrightarrow
C^{simp}_{k-1}(\Xi),\quad c_{k}\mapsto\partial c_{k},$ (183)

which satisfy $\partial_{k-1}\circ\partial_{k}=0$ and allow to define the
simplicial homology groups

$\displaystyle H^{simp}_{k}(\Xi,\mathbb{K})=\frac{{\rm
ker}(\partial^{s}_{k})}{{\rm Im}(\partial^{s}_{k+1})}.$ (184)

Similarly, we can define a simplicial $k$-cochain as a linear map

$\displaystyle\xi^{k}:C^{simp}_{k}(\Xi)\longrightarrow\mathbb{K},$ (185)

that form the $\mathbb{K}$-linear space $C_{simp}^{k}(\Xi)$, on which acts the
coboundary operator

$\displaystyle\delta_{s}^{k}:C_{simp}^{k}(\Xi)\longrightarrow
C_{simp}^{k+1}(\Xi),\quad i.e.\quad\
\delta_{s}^{k}\xi^{k}(c_{k+1})\equiv\xi^{k}(\partial_{k+1}c_{k+1}),\ \forall
c_{k+1}\in C^{simp}_{k+1}(\Xi).$ (186)

From these, one defines the simplicial cohomology groups

$\displaystyle H_{simp}^{k}(\Xi,\mathbb{K})=\frac{{\rm
ker}(\delta_{s}^{k})}{{\rm Im}(\delta_{s}^{k-1})}.$ (187)

This construction can then be extended to the case of a topological space $X$
that admits triangulations. If a given triangulation $T$ can be homotopically
deformed to become homeomorphic to a simplicial complex $\Xi_{T}$, then we can
define the simplicial homology groups of $X$ as

$\displaystyle
H^{simp}_{k}(X,\mathbb{K})=H^{simp}_{k}(\Xi_{T},\mathbb{K}),\qquad
H_{simp}^{k}(X,\mathbb{K})=H_{simp}^{k}(\Xi_{T},\mathbb{K}).$ (188)

These are well defined, since it can be proved that the resulting groups are
independent on the triangulation $T$. An important result is that the
following isomorphisms (for spaces admitting simplicial homology) hold true:

$\displaystyle H^{sing}_{k}(X,\mathbb{K})\simeq
H^{simp}_{k}(X,\mathbb{K}),\qquad H_{sing}^{k}(X,\mathbb{K})\simeq
H_{simp}^{k}(X,\mathbb{K}).$ (189)

For a proof, see [102], [114].

####  6.1.3 de Rham theorem

Now we can state the following important result ([115, 116, 117, 118, 102,
114]):

de Rham Theorem. If $X$ is a smooth manifold and
$\mathbb{K}=\mathbb{R},\mathbb{C}$, then

$\displaystyle H^{deRham}_{k}(X,\mathbb{K})$ $\displaystyle\simeq
H^{sing}_{k}(X,\mathbb{K})\simeq H^{simp}_{k}(X,\mathbb{K}),$ (190)
$\displaystyle H_{deRham}^{k}(X,\mathbb{K})$ $\displaystyle\simeq
H_{sing}^{k}(X,\mathbb{K})\simeq H_{simp}^{k}(X,\mathbb{K}).$ (191)

An important remark for us is that the same theorem can be extended by
weakening the hypothesis and assuming that $X$ admits singularities and more
in general a (Withney) stratification or the structure of a stratifold. See
[119] and references therein.
This result allows us to move to simplicial (co)homology to simplify abstract
constructions as we did in section 5. But it is also helpful for simplifying
certain computations like the one of Betti number and the Euler
characteristic. Here we recall that the Betti numbers of $X$ are the numbers

$\displaystyle b_{k}={\rm dim}H_{deRham}^{k}(X,\mathbb{K}),$ (192)

while the Euler characteristic of $X$ is the number

$\displaystyle\chi(X)=\sum_{j=0}^{N}(-1)^{j}b_{j},$ (193)

where $N={\rm dim}X$.

Finally, we remark that further cohomologies can be introduced, like for
example relative (co)homologies $H_{k}(X,B,\mathbb{K})$, cellular
(co)homologies, Čech and sheaf (see [102]), or generalized (co)homologies,
like $K$-theories or (co)bordisms. We demand any further reading to the
standard literature.

###  6.2 Cup products and intersections on smooth manifolds

In the example of closed one forms on a compact closed Riemann surface
$\Sigma$, we have seen that the cup product $\cup:H^{1}(\Sigma)\times
H^{1}(\Sigma)\rightarrow H^{2}(\Sigma)$ can be interpreted as the intersection
product between 1-forms, in a sense, dual to the intersection product among
homological curves (1-cycles), probably more intuitive to be figured out. Now,
we want to understand how much this notion can be generalized. To this hand we
will refer to the nice lecture in [120].
Let $\mathcal{M}$ be an $m$-dimensional oriented compact smooth manifold.
Also, we assume $S_{1}$ and $S_{2}$ are two smooth submanifolds of dimensions
$m_{1}$ and $m_{2}$ respectively. We say that they have transversal
intersection if $N=S_{1}\cap S_{2}\neq\emptyset$, and for any $x\in N$ the
union of the embeddings of $\iota_{j}:T_{x}S_{j}\hookrightarrow T_{x}M$ is the
whole $T_{x}M$. In particular, this implies that $m_{1}+m_{2}-m=m_{N}\geq 0$,
which is the dimension of the intersection. We have to take into account the
orientation. We say that the intersection is oriented on a given connected
component $\bar{N}$ of $N$ if it is possible to choose a basis
$\boldsymbol{v}_{1},\ldots\boldsymbol{v}_{m_{N}}\in T_{x}N$, $x\in\bar{N}$,
such that:

  * •

we can complete it to an oriented basis
$\boldsymbol{w}_{1},\ldots,\boldsymbol{w}_{m_{1}-m_{N}},\boldsymbol{v}_{1},\ldots\boldsymbol{v}_{m_{N}}\in
T_{x}S_{1}$ according to the orientation of $S_{1}$;

  * •

we can complete it to an oriented basis
$\boldsymbol{v}_{1},\ldots\boldsymbol{v}_{m_{N}},\boldsymbol{z}_{1},\ldots,\boldsymbol{z}_{m_{2}-m_{N}}\in
T_{x}S_{2}$ according to the orientation of $S_{2}$;

  * •

$\boldsymbol{w}_{1},\ldots\boldsymbol{w}_{m_{1}-m_{N}},\boldsymbol{v}_{1},\ldots\boldsymbol{v}_{m_{N}},\boldsymbol{z}_{1},\ldots,\boldsymbol{z}_{m_{2}-m_{N}}$
is an oriented basis for $T_{x}M$ according to the orientation of $M$.

See Fig. 20 for a pictorial representation.

$\boldsymbol{N=S_{1}\cap
S_{2}}$$\boldsymbol{S_{1}}$$\boldsymbol{S_{2}}$1212123$\boldsymbol{M}$ Figure
20: Intersection between two oriented surfaces $S_{1}$ and $S_{2}$ in a three
dimensional oriented manifold $M$.

It is clear that it depends only on the connected component and not on the
specific point. So, $\boldsymbol{v}_{1},\ldots\boldsymbol{v}_{m_{N}}$ defines
the orientation of the intersection. We can consider the homology class
$[S_{1}\cap S_{2}]$ associated to $N$, in the sense of simplicial
homology.999so, $[N]$ is defined by any given simplicial decomposition of $N$.
It can be shown that it depends only on the homology classes of $S_{j}$ so it
gives a map $H_{m_{1}}(M,\mathbb{Z})\times H_{m_{2}}(M,\mathbb{Z})\rightarrow
H_{m_{1}+m_{2}-m}(M,\mathbb{Z})$, each time $m_{1}+m_{2}-m\geq 0$.
In particular, when $m_{1}+m_{2}=m$, $N$ is just the union of a set of
isolated points. This set is finite since $M$ is compact, and defines an
element of $H_{0}(M,\mathbb{Z})\simeq\mathbb{Z}$ (since $M$ is connected).
Thus we can associate to the intersection product a number representing such
element. This can be done as follows. If $\sigma(x)$ is the sign associated to
$x\in N$, we set

$\displaystyle S_{1}\cdot S_{2}:=\sum_{x\in N}\sigma(x).$ (194)

It is well defined, since $N$ is finite, and it results to depend on the
homology classes only. So, it shows the connection with the intersection
product we met in the case of a Riemann surface.
However, let us stay general for the moment. There is another interesting map
to be considered in simplicial (co)homology in order to go ahead: the cap
product

$\displaystyle\frown:H^{j}(M,\mathbb{Z})\times
H_{m}(M,\mathbb{Z})\longrightarrow
H_{m-j}(M,\mathbb{Z}),\quad(\mu,u)\mapsto\mu\frown u,$ (195)

defined in such the way that for any given $\nu\in H^{m-j}(M,\mathbb{Z})$, we
must have $\nu(u\frown\mu)\equiv(\nu\cup\mu)(u)$. Then, let $[M]\in
H_{m}(M,\mathbb{Z})$ be the class generated my $M$, also called the
fundamental class. Fixing $u=[M]$, we get a map

$\displaystyle\star:H^{j}(M,\mathbb{Z})\longrightarrow
H_{m-j}(M,\mathbb{Z}),\quad\mu\mapsto\star\mu:=\mu\frown[M].$ (196)

Then, this theorem holds ([121], Th. 1.1.3):

######  Theorem 1 (Poincaré duality).

If $M$ is a connected, closed and oriented topological manifold, then
$\star:H^{j}(M,\mathbb{Z})\longrightarrow H_{m-j}(M,\mathbb{Z})$ is an
isomorphism for all integers $j$.

The most important consequence of this theorem, at least as what regards the
applications we are interested in, is the following theorem ([120], Th. 1.1):

######  Theorem 2.

Assume $M$ is a compact, connected, closed smooth manifold and let $S_{j}$
smooth submanifolds of dimension $m_{j}$, $j=1,2$ that (up to an homological
deformation) have transversal intersection. Then, it holds

$\displaystyle\star[S_{1}]\cup\star[S_{2}]=\star[S_{1}\cap S_{2}].$ (197)

For a proof, see [120]. This theorem thus states that cup and cap products are
dual under the Poincaré map. So, it provides a remarkable generalization of
the results discussed for Riemann surfaces. We described it in the case of
simplicial homology, but it clearly works as well if we tensorize with
$\mathbb{K}=\mathbb{R},\mathbb{C}$, and then we can use the isomorphism
$H_{simp}^{*}(M,\mathbb{K})\simeq H_{dR}^{*}(M,\mathbb{K})$ to get it for the
case of the de Rham cohomology (where the cup product is induced by the wedge
product), which better matches with the previous section.
However, we are not yet happy enough, since we had to make use of smoothness,
but, in considering the general integrals we are interested in, we generically
have to consider manifolds with singularities, which may be not homologous to
smooth manifolds, so we need to explore further generalizations.

###  6.3 Lefschetz theorems, Hodge-Riemann bilinear relations and perversities

We want to go beyond in our tour on intersection theory by looking at finer
structures. We will follow mainly [122], indicating some further lectures
explicitly when necessary.
First, let us go back to the case of Riemann surfaces. They are complex curves
of complex dimension 1, and smooth real surfaces of real dimension 2. They can
be embedded in a projective space, for example the elliptic curves can be
embedded in $\mathbb{P}^{2}$ as the zero locus of a quartic or cubic
polynomial, such as

$\displaystyle y^{2}=P_{4}(x),$ (198)

in local non homogeneous coordinates. If in the above equation in
$\mathbb{P}^{2}$ we replace $P_{4}$ with a polynomial $P_{d}$ of degree $d=2g$
or $d=2g-1$, we get an hyperelliptic complex curve of genus $g$. They are
therefore algebraic varieties, that are zeros of polynomial functions in some
real or complex space. When an algebraic variety can be embedded in a
projective space $\mathbb{P}^{N}$ for some $N$, then it is called a projective
variety. Also, we can notice that the one forms on a Riemann surface $\Sigma$,
in local complex coordinates $z$ and $\bar{z}$, have the general form

$\displaystyle\omega=f(z,\bar{z})dz+g(z,\bar{z})d\bar{z},$ (199)

with differential

$\displaystyle d\omega=(\partial_{z}g-\partial_{\bar{z}}f)dz\wedge d\bar{z}.$
(200)

Any two form is

$\displaystyle\Omega=\lambda(z,\bar{z})dz\wedge d\bar{z},$ (201)

and it is closed, $d\Omega=0$ for dimensional reasons. It is cohomologically
trivial if it has the form (200). $\Sigma$ is also a Kähler manifold, which
means the following. It is always possible to put on it a Hermitian metric

$\displaystyle G=g_{z\bar{z}}dz\otimes d\bar{z},$ (202)

that defines at any point $p$ of $\Sigma$ a sesquilinear map

$\displaystyle G_{p}:T_{p}\Sigma\times T_{p}\Sigma\longrightarrow\mathbb{C},$
(203)

such that

$\displaystyle G_{p}(\alpha\xi,\zeta)=G(\xi,\bar{\alpha}\zeta)$ (204)

for any $\alpha\in\mathbb{C}$. Its real part defines a Riemannian metric on
$\Sigma$, while its imaginary part defines a two form

$\displaystyle K=-\frac{i}{2}g_{z\bar{z}}dz\wedge d\bar{z},$ (205)

which is a closed form, called the Kähler form. The same construction can be
made in higher dimensions, with the only difference that in general the Kähler
form is not automatically closed. When it is, then the complex manifold is
said to be a Kähler manifold. It is possible to prove that for a Kähler
manifold $M$, in a local coordinate patch $U$ it is always possible to find a
real valued function $\mathcal{K}:U\rightarrow\mathbb{R}$, called the Kähler
potential, such that

$\displaystyle g_{z\bar{z}}=\partial_{z}\partial_{\bar{z}}\mathcal{K}.$ (206)

An example are the complex projective manifolds $\mathbb{P}^{N}$, that, in non
homogeneous coordinates $z_{1},\ldots,z_{N}$, admit the Kähler potential

$\displaystyle\mathcal{K}_{FS}=\log(1+\sum_{j=1}^{N}|z_{j}|^{2}),$ (207)

generating the Fubini-Study Hermitian metric

$\displaystyle
G_{FS}=\frac{1}{(1+\sum_{j=1}^{N}|z_{j}|^{2})^{2}}\left[(1+\sum_{j=1}^{N}|z_{j}|^{2})\sum_{k=1}^{N}dz_{k}\otimes
d\bar{z}_{k}-\sum_{j,k=1}^{N}z_{j}\bar{z}_{k}dz_{j}\otimes
d\bar{z}_{k}\right].$ (208)

Any projective manifold is a Kähler manifold, since it inherits a Kähler
structure after restriction of the Fubini-Study of the ambient space
$\mathbb{P}^{N}$ in which it is embedded.
The cohomology of a Kähler manifold of complex dimension $m$ has a nice
structure, called a Hodge structure. For any $k=0,1,\ldots,2m$, the cohomology
group $H^{k}(M,\mathbb{C})$ admits a direct decomposition

$\displaystyle H^{k}(M,\mathbb{C})=\bigoplus_{p+q=k}H^{p,q}(M,\mathbb{C}),$
(209)

where $p$ and $q$ run in $\\{0,\ldots,m\\}$, and the elements of
$H^{p,q}(M,\mathbb{C})$ are represented by closed forms of the type

$\displaystyle\omega_{p,q}=\sum_{j_{1},\ldots,j_{p},k_{1},\ldots,k_{q}}\omega_{j_{1}\ldots
j_{p}k_{1}\ldots k_{q}}dz^{j_{1}}\wedge\cdots\wedge dz^{j_{p}}\wedge
d\bar{z}^{k_{1}}\wedge\cdots d\bar{z}^{k_{q}}.$ (210)

For example, for connected Riemann surfaces of genus $g$ we have

$\displaystyle H^{0}(\Sigma,\mathbb{C})$ $\displaystyle\simeq\mathbb{C},$
(211) $\displaystyle H^{1}(\Sigma,\mathbb{C})$
$\displaystyle=H^{1,0}(\Sigma,\mathbb{C})\oplus
H^{0,1}(\Sigma,\mathbb{C})\simeq\mathbb{C}^{g}\oplus\mathbb{C}^{g},$ (212)
$\displaystyle H^{2}(\Sigma,\mathbb{C})$
$\displaystyle=H^{2,0}(\Sigma,\mathbb{C})\oplus
H^{1,1}(\Sigma,\mathbb{C})\oplus H^{0,2}(\Sigma,\mathbb{C})\simeq
0\oplus\mathbb{C}\oplus 0.$ (213)

In particular, $H^{1,0}(\Sigma,\mathbb{C})$ is generated by $g$ holomorphic
forms, which for an hyperelliptic curve in $\mathbb{P}^{2}$

$\displaystyle y^{2}=P(z),$ (214)

are

$\displaystyle\omega_{j}=\frac{z^{j}dz}{y},\quad j=0,\ldots,g-1.$ (215)

$H^{2}(\Sigma,\mathbb{C})$ is generated by the Kähler form, which is also real
valued, $K\in H^{1,1}(M,\mathbb{C})\cap H^{2}(M,\mathbb{R})$.
Now, unfortunately, it is time to become more abstract. Given a cohomology
group of any kind, let us think at it as a finitely generated abelian group
$H$ over $\mathbb{Z}$, and, following [122], let use the notation
$H_{\mathbb{K}}=H\otimes_{\mathbb{Z}}\mathbb{K}$,
$\mathbb{K}=\mathbb{Q},\mathbb{R},\mathbb{C}$. We say that $H$ carries a pure
Hodge structure of weight $k$ if there is a decomposition of abelian groups

$\displaystyle H_{\mathbb{C}}=\bigoplus_{p+q=k}H^{p,q},\quad
i.e.\quad\overline{H^{p,q}}=H^{q,p}.$ (216)

An equivalent formulation is to say that $H$ admits a decreasing Hodge
Filtration, that is a family of abelian subgroups $F^{p}$ of $H_{\mathbb{C}}$
with the properties

$\displaystyle H_{\mathbb{C}}$ $\displaystyle=F^{0}\supset
F^{1}\supset\cdots\supset F^{p}\supset F^{p+1}\supset\cdots,$ (217)
$\displaystyle F^{p}\cap\overline{F^{k+1-p}}$ $\displaystyle=0,$ (218)
$\displaystyle F^{p}\oplus\overline{F^{k-p}}$ $\displaystyle=H_{\mathbb{C}}.$
(219)

Indeed, the equivalence is read by the relations

$\displaystyle H^{p,q}$ $\displaystyle=F^{p}\cap\overline{F^{q}},$ (220)
$\displaystyle F^{p}$ $\displaystyle=\bigoplus_{j\geq p}H^{j,k-j}.$ (221)

On $H_{\mathbb{C}}$ there is a real action of $S^{1}\subset\mathbb{C}_{*}$
given by $\rho(s)a=s^{p-q}a$ for $a\in H^{p,q}$, and extended by $\mathbb{R}$
linearity. In particular, $\rho(i)=w$ is called the Weyl map. A polarization
on the pure Hodge structure is a real bilinear form $Q$ on $H_{\mathbb{R}}$,
which is invariant under the action of $S^{1}$ and such that

$\displaystyle B(a,b):=Q(a,wb)$ (222)

is symmetric and positive definite. After extending $Q$ over $H_{\mathbb{C}}$,
one gets that $i^{2k+p-q}Q(a,\bar{a})>0$ for a non vanishing $a\in H^{p,q}$.
Now, consider a non singular algebraic projective variety of dimension $m$,
$X\subset\mathbb{P}^{N}$ (for instance a projective manifold). Let $H$ be a
generic hyperplane in $\mathbb{P}^{N}$. Then, $H\cap X$ defines an element of
$H_{n-2}(X,\mathbb{Z})$, whose Poincaré dual is $\eta\in H^{2}(X,\mathbb{Z})$,
see [121]. We then define the Lefschetz map

$\displaystyle L:H^{k}(X,\mathbb{Q})\longrightarrow
H^{k+2}(X,\mathbb{Q}),\quad a\mapsto L(a)=\eta\cup a.$ (223)

The degree $k$ primitive vector spaces are the subspaces $P^{k}\subseteq
H^{k}(X,\mathbb{Q})$ defined by $P^{k}:={\rm ker}L^{m-k+1}$. Finally, on
$H^{k}(X,\mathbb{Q})$ let us define the quadratic form

$\displaystyle
Q^{(k)}_{L}([\omega],[\omega^{\prime}]):=(-1)^{\frac{k(k+1)}{2}}\int_{X}\eta^{m-k}\wedge\omega\wedge\omega^{\prime}.$
(224)

Then, $H^{k}(X,\mathbb{Z})$ is a pure Hodge structure of weight $k$ and
$P^{k}$ is a rational hodge structure of weight $k$. Then we have the
following important properties ([122], Th.5.2.1).

######  Theorem 3 (Hard Lefschetz Theorem).

With the above notations, the maps

$\displaystyle L^{k}:H^{m-k}(X,\mathbb{Q})\longrightarrow
H^{m+k}(X,\mathbb{Q})$ (225)

are isomorphisms.

######  Theorem 4 (Primitive Lefschetz decomposition).

For $0\leq k\leq m$ one has the decomposition

$\displaystyle H^{k}(X,\mathbb{Q})=\bigoplus_{j\geq 0}L^{j}P^{k-2j}.$ (226)

Moreover, for any given $j$, $V^{j}\equiv L^{j}P^{k-2j}$ is a pure Hodge
substructure of weight $k$, and for any $j\neq j^{\prime}$, $V^{j}$ and
$V^{j^{\prime}}$ are $Q^{(k)}_{L}$-orthogonal.

######  Theorem 5 (Hodge-Riemann bilinear relations).

For any $0\leq k\leq m$, $Q^{(k)}_{L}$ is a polarization for the pure Hodge
structure $P^{k}_{\mathbb{R}}$. In particular, after complexification,

$\displaystyle
i^{p-q}Q^{(k)}_{L}([\omega],[\bar{\omega}])>0,\qquad\forall[\omega]\in
P^{k}\cap H^{p,q}(X,\mathbb{C}),$ (227)

so it is strictly positive definite.

This result seems to give us a very broad generalization of the properties we
met for integrals along one dimensional cycles on a Riemann surface. But,
again, it is not enough, since we had to require for $X$ to be a nonsingular
variety. The simplest example is $\mathbb{C}_{*}=\mathbb{C}\backslash\\{0\\}$.
Its first cohomology group is obviously one dimensional, and this fact is
incompatible with the existence of a pure Hodge structure. Indeed, assuming
$H^{2k+1}$ has a pure Hodge structure, then the relation
$H^{p,q}=\overline{H^{q,p}}$ implies

$\displaystyle{\rm dim}(H^{2k+1})=2\sum_{j=0}^{k}{\rm dim}(H^{j,2k+1-j})\in
2\mathbb{N},$ (228)

so it must be even dimensional.
Since we have to work with general Feynman integrals, we cannot avoid to work
with singular varieties, so we need to overcome this difficulty. Luckily, this
has been done along the years, passing from Weyl, Serre, Grothendieck, and
finally solved by Deligne [123], [124], [125], who proved the existence of a
mixed Hodge structure on the $H^{k}(X,\mathbb{Q})$ when $X$ is an arbitrary
algebraic complex variety. This means that, given $X$ of complex dimension $m$
and $0\leq k\leq 2m$, it is always possible to find two filtrations:

  * •

a decreasing filtration $F^{*}$, called the Hodge filtration, such that

$\displaystyle H^{k}(X,\mathbb{Q})=F^{0}\supseteq
F^{1}\supseteq\cdots\supseteq F^{m}\supseteq F^{m+1}=\\{0\\};$ (229)

  * •

an increasing filtration $W_{*}$, called the Weight filtration, such that

$\displaystyle\\{0\\}=W_{-1}\subseteq W_{0}\subseteq\cdots\subseteq
W_{2k-1}\subseteq W_{2k}=H^{k}(X,\mathbb{Q});$ (230)

which have the following property. On the complexification of each graded part
of $W_{*}$, $F^{*}$ induces a decreasing filtration that endows the pieces

$\displaystyle gr_{j}^{W}:=W_{j}\otimes\mathbb{C}/W_{j-1}\otimes\mathbb{C}$
(231)

with a pure Hodge structure of weight $j$. We do not intend to delve further
into these concepts, and demand the reader to [123] for details.
At this point, in order to define intersection theory, one has essentially to
tackle the problem on how to define intersections of cycles that may contain
singularities or pass through a singular region. To get an idea, let us follow
the nice exposition in [121], Chp. 1. Let us consider a two dimensional
pinched torus $W$, which looks something like a würstel bent over and glued at
the tips just in one point $p$. It is clear that on this surface, a loop
around the hole is a non trivial generator for $H_{1}(W)$ (but pass through
the singularity), while a loop $L$ wounding the “würstel” shrinks down to a
point when moved to the gluing point $p$ (the singularity), Fig. 21.

$\boldsymbol{p}$ Figure 21: A singular torus

So, $H^{1}(W)$ is one dimensional and the pure Hodge structure is lost. A
relevant observation is that if we think at the singular point $p$ as a zero
dimensional representative of $H_{0}(W)$, it meets the loop $L$ non
transversally. This situation repeats for higher dimensional varieties, where
singularities can be much worse. A strategy is to construct more refined
(co)homologies but starting from the simplicial ones and restricting the
allowed (co)chains. One idea is first to generate a homology starting from a
simplicial one where only triangulations giving locally finite chains are
allowed (so that any point admits a small boundary that intersects only a
finite number of cells of the chain). This construction defines the Borel-
Moore homology groups $H^{BM}_{k}(X,\mathbb{Z})$, [121]. Tensoring with
$\mathbb{C}$ gives the isomorphism $H_{k}^{BM}(X,\mathbb{C})\simeq
H^{k}_{c}(X,\mathbb{C})$ of the compactly supported cohomology, whose cochains
(or closed $k$-forms) have compact support. Assume now that singularities
define an increasing filtration of $X$, that is a sequence of subvarieties

$\displaystyle\emptyset=X_{-1}\subseteq X_{0}\subseteq X_{1}\ldots\subseteq
X_{2m-2}=X_{2m-1}\subseteq X_{2m}=X,$ (232)

where the $X_{j}$, $j<2m$ are the real $j$-dimensional singular sub loci,
called the strata, of which the largest one has at least real codimension 2,
and $X\backslash X_{2m-2}$ is required to be dense in $X$. With this in mind,
we can refine further the homology by allowing only triangulations giving
locally finite $k$-chains whose support intersects all strata transversally.
We call $H^{tr}_{k}(X)$ the obtained homology group. A theorem by McCrory
([121], Th.2.3.2, [126]) then provides the isomorphism $H^{tr}_{k}(X)\simeq
H^{2m-k}(X,\mathbb{Z})$. Assuming that (in a suitable sense, see [121]) $X$ is
oriented, we can use cap product with the fundamental class $[X]$ to map
$H^{2m-k}(X,\mathbb{Z})$ into $H_{k}^{BM}(X,\mathbb{Z})\simeq H^{lf}_{k}(X)$,
the last term being the locally finite simplicial homology. However, since $X$
is singular, the Poincaré map is no more an isomorphism. Therefore,
$H^{lf}_{k}(X)$ and $H^{tr}_{k}(X)$ are not isomorphic, the reason essentially
being that, in general, the locally finite cycles cannot be deformed so to
have transversal intersection with all strata. For example, this is what
happened in our example of the “glued würstel”. Therefore, one needs to do a
final refinement, in order to get a correct intersection theory together with
a good Poincaré duality. This is obtained by restricting which chains can meet