The mapping of non-planar diagrams into planar diagrams and the exact solution of QCD

Abstract

In this report I present a method that allows the construction of an exact solution of the low energy (strong coupling) limit of quantum chromodynamics (QCD) by using some homological algebraic and topological tools. If employed properly, this method may give a consistent solution of the confinement problem of QCD as well as new methods of solving strongly coupled problems in condensed matter theory. I employ the universal coefficient theorem in order to construct a method comparable to the quantum field theoretical renormalization group prescription. This allows the mapping of non-planar diagrams into planar diagrams having the Ext and T or constructions as obstructions which have to be taken into account. The main result is the ability to transform the theory behind the topological genus expansion depending on planar diagrams, spheres and toruses of various genera (as is the case in SU (N)-gauge theories and QCD) into a deformed theory depending only on planar diagrams. This is equivalent to the renormalization prescription and the renormalization group methods employed in standard quantum field theories. This can be interpreted as the emergence of effective theories due to the coarse graining of the high energy domain. In the case of the topological expansion the change of the coefficients in (co)homology leads to changes in the observed (co)homology of the topological spaces on which the diagrams are constructed, changes controlled by the Ext or T or groups in the same way in which the charge or mass flows are controlled by the renormalization group equations. These can be seen as a result of a " coarse graining " of the topological invariants used. The general nature of this observation may lead to more practical applications as well: the integration over non-trivial manifolds may be translated into fast integrations over spheres. This may lead to fast imaging procedures that may be useful in the exploration of real time biological phenomena taking place in unaccessible regions.

Full text

The mapping of non-planar diagrams into planar diagrams and the exact solution of
QCD
Andrei Patrascu1
1University College London, Department of Physics and Astronomy, London, WC1E 6BT, UK
In this report I present a method that allows the construction of an exact solution of the low
energy (strong coupling) limit of quantum chromodynamics (QCD) by using some homological
algebraic and topological tools. If employed properly, this method may give a consistent solution of
the confinement problem of QCD as well as new methods of solving strongly coupled problems in
condensed matter theory. I employ the universal coefficient theorem in order to construct a method
comparable to the quantum field theoretical renormalization group prescription. This allows the
mapping of non-planar diagrams into planar diagrams having the Ext and T or constructions as
obstructions which have to be taken into account. The main result is the ability to transform
the theory behind the topological genus expansion depending on planar diagrams, spheres and
toruses of various genera (as is the case in SU (N)-gauge theories and QCD) into a deformed theory
depending only on planar diagrams. This is equivalent to the renormalization prescription and
the renormalization group methods employed in standard quantum field theories. This can be
interpreted as the emergence of effective theories due to the coarse graining of the high energy
domain. In the case of the topological expansion the change of the coefficients in (co)homology
leads to changes in the observed (co)homology of the topological spaces on which the diagrams are
constructed, changes controlled by the Ext or T or groups in the same way in which the charge or
mass flows are controlled by the renormalization group equations. These can be seen as a result
of a ”coarse graining” of the topological invariants used. The general nature of this observation
may lead to more practical applications as well: the integration over non-trivial manifolds may be
translated into fast integrations over spheres. This may lead to fast imaging procedures that may
be useful in the exploration of real time biological phenomena taking place in unaccessible regions.
COVER LETTER
During the end of the 1950’s Alexander Grothendieck observed the importance of the coefficient groups in coho-
mology [1]. Three decades later, he presented his ”Esquisse d’un Programme” [2] to the main french funding body.
His proposal has been rejected. Another three decades later, in the 21st century, his research proposal is considered
one of the most inspiring and important collection of ideas in pure mathematics. His ideas brought together algebraic
topology, geometry, Galois theory, etc. becoming the origin for several new branches of mathematics. Today, less
than one year after his death, Grothendieck is considered one of the most influential mathematicians worldwide. His
ideas were important for the proofs of some of the most remarkable mathematical problems like the Weil Conjectures,
Mordell Conjectures and the solution of Fermat’s last theorem. However, Grothendieck’s work remained a subject
to be discussed only in some restricted circles of pure mathematicians. Very few if any physical applications ever
emerged from his ideas.
In this report I wish to connect several ideas formulated by Grothendieck to some areas of modern research in
physics. The applications are manifold : (co)homologies with coefficients in various sheafs and the universal coefficient
theorem appear to have applications in the topological genus expansion invented by ’t Hooft [3] and applied to
quantum chromodynamics (QCD) and string theory, but also to strongly coupled electronic systems or condensed
matter physics; Anabelian geometry may describe how the fundamental group Gof an algebraic variety Vdetermines
how Vcan be mapped into another geometric object Wunder the assumption that Gis strongly non-commutative.
This may lead us to a possible re-thinking of the renormalization group principle as applied to topological expansions.
Out of this plethora of possibilities I will focus now on one: Is there a general method of finding solutions for theories
in regions where the standard approaches are unavailable? More precisely, is it possible to always map regions that
cannot be reached via standard approaches into regions that can easily be solved? The answer to this question may
have widespread consequences reaching from the genus expansion of string theory to the mystery of the P vs. NP
dilemma [4].
Until now, the dualities which are so helpful in probing otherwise unaccessible regions of various theories, were
discovered more by accident or by educated guesswork. I intend to show in this paper that there exists an underlying
structure to the dualities, a structure that connects them to various symmetries and can provide an algorithm for the
identification of dualities to simpler theories anytime when they are needed.
In order to present here an outline of the proposed idea, let me start with QCD-type theories. QCD is the theory of

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quarks and gluons, two of the fundamental constituents of nuclear matter. It is a non-abelian gauge theory presenting
the well known properties of confinement and asymptotic freedom. Because of these properties, the theory can be
described perturbatively having the coupling constant as an expansion parameter in the domain of high energies.
However, in the region of low energies, the coupling constant becomes large hence unsuitable for a perturbative
expansion. It is in this region where most of the interesting physics occurs because this is the energy domain closest
to our immediate experience. A possible solution was to replace the coupling as a series expansion parameter with the
inverse of the order of the gauge group. In this way if we have a SU (N) gauge theory, the series expansion would be
in 1
N. When Nis sufficiently large we may have a pertinent series expansion. The Feynman diagrams in this theory
arrange themselves on various topological surfaces, each term in this new series expansion being itself a topological
surface of a given genus. I am using in this report the universal coefficient theorem and several results first observed
by Grothendieck in order to show that, with certain corrections to the algebraic structures, it is always possible to
map non-planar diagrams (of higher genus) to lower genus diagrams and finally to planar diagrams. The obvious
obstruction to this operation can be encoded in the universal coefficient theorem.
[1] A. Grothendieck, Inst. des Hautes Etudes Scientifiques, Pub. Math. 29 (29) pag. 95 (1966)
[2] A. Grothendieck, Published in Schneps and Lochak (1997, I), pp.5-48; English transl., ibid., pp. 243-283
[3] G. ’t Hooft, Nucl. Phys. B 72, 3 pag. 461 (1974)
[4] A. Patrascu, arXiv:1403.2067
REPORT ARTICLE
After almost half a century of research quantum chromodynamics (QCD) still remains a mystery. While we can
compute its high energy - small coupling - regions fairly accurately, many unknowns still persist in its low energy region
where standard perturbative approaches in the coupling constant cannot be exploited. The theory itself describes
the behavior of quarks and gluons, two of the most fundamental objects of standard particle physics. Yet, as has
already been noticed by ’t Hooft [1], the theory bears strong similarities with another theory, considered even more
fundamental, a theory that may even prescribe how to talk about quantum gravity and finally how to unify physics in
general : string theory. Indeed, one of the first motivations for string theory was the description of mesons [2], namely
states of quarks and antiquarks together with the color ”bond” between them [3]. This bond has a very peculiar
behavior. While the electromagnetic interaction is mediated via a field that becomes weaker and more diffuse at
larger distances (hence at lower energies), the potential associated to color becomes denser and denser, increasing the
strength of the interaction. This gives to the strong interaction an especially important behavior at low energies. We
can still try to develop QCD in the same way as we did with the other interactions: try to construct the associated
Feynman diagrams, to apply the renormalization prescriptions and to finally solve the problem. In doing so we obtain
a series expansion. This series expansion is meaningful only in the high energy domain. In the low energy domain
something very interesting occurs. It has been noted by ’t Hooft [1] that the gluon lines can be represented using
double arrows instead of simple lines, one arrow for each of the indexes of the field matrix in the adjoint representation
of the gauge group. When doing this, the Feynman diagrams start looking more like rubber bands, having their own
thickness associated to the propagators. They can also be linked together such that in the end they fill a surface of a
specific topology (see supplemental material [4]).
After this is done, one obtains a classification of diagrams in terms of the topological genus. I use the supplemental
material for a more in-depth description of these mechanisms. It suffices to say here that a theory containing only
diagrams that in the large Nlimit can be represented on a plane or a sphere can at least in principle be solved exactly.
The next corrections depend on the genus of the torus associated to the surface on which the diagrams reside. An
exact solution of the full theory including these corrections remains unknown.
In a standard quantum field theory a renormalization prescription would make a connection between diagrams
of different orders in the expansion. There, the renormalization procedure is based on a regularization prescription
(a method that allows the identification of the ”problems”, in this case the divergencies) and a renormalization
prescription i.e. a method of canceling systematically the divergencies at each order by employing terms of the same
form as those already existing in the Lagrangian. This leads to corrections in the terms controlling the mass, charge,
etc. defined in the original theory [5].
What if something similar would be possible in the topological expansion? Topology tells us that a torus and a

3
FIG. 1: A sphere and a torus. They appear to be topologically different. This difference is measured by topological
invariants that are defined such that they are the same on surfaces that are topologically identical. Homology and
cohomology are such invariants. However, the invariants are not always ideal. In order to define them correctly one
needs additional information, brought to them by what is known as the ”coefficient structure”. There exist
coefficient groups with enough torsion such that the cohomology group that may make the distinction between the
two becomes incapable of doing so. This makes the integration over the cohomology (cohomology being the only
construction that is useful in diagrammatic calculations) equivalent for the two cases. Representative Feynman
ribbon-graph diagrams are presented below each of the topological objects. Furthermore, this method implies
integration over the two manifolds. One may observe that in medical imagery one also has to integrate various
quantities related to the emission of the probing device (say X rays) over topologically non-trivial manifolds. The
fact that the integration could be mapped from a torus on a sphere in a controlled way would make these devices
significantly faster.
sphere, or two toruses of different genera are not homeomorphic, hence cannot be smoothly transformed into one
another. This is however not the only possibility of relating the two objects. In fact, homological algebra gives various
tools by which we can find what are the obstructions to transformations we would like to perform. These obstructions
may in some cases be eliminated by less obvious changes in the way we look at our theories. This article shows
that there exists a systematic prescription that, used properly, maps the information defined on a torus of any genus
into a sphere. There are however strong obstructions to this mapping that can be eliminated by employing various
coefficient groups in the cohomology associated to the space. Figure 1 shows how obviously different a torus is from
a sphere. In fact we can detect the differences naturally, by looking at them. Mathematically however, we need to
define certain invariant objects that on one side should change when a relevant change in the topology of what we
intend to measure occurs and on the other side, must remain invariant to any other variation that does not change
the topology. Whenever such an invariant is incapable to detect a certain change that would result in a change of

4
FIG. 2: A Klein bottle and a Moebius strip. Two unorientable surfaces that cannot be detected by homology with
arbitrary coefficients. The coefficient groups, in order to make the (co)homology capable of detecting these shapes
must have a certain level of torsion. If one uses real or rational coefficients these surfaces cannot be identified as
such. However, the supplemental information will be encoded by the universal coefficient theorem when one attempts
to change the coefficients in homology or cohomology to a coefficient group with torsion. Then, obstructions in the
exact sequence describing the universal coefficient theorem will appear, in the form of T or resp. Ext groups.
topology, we say it is ”blind” to a specific topological transformation. This makes that invariant of a rather low
quality if our desire is an accurate description of a shape.
Homologies and cohomologies are in general relatively good topological invariants. They are easily computable and
probe the topology relatively well. However there are well known situations when objects with the same homology
have different homotopical and topological properties (see for example Poincare’s homological sphere [6]). While this
is a way in which (co)homologies may fail in their ability to discern topologies, it is not this the method I wish
to insist upon here. Instead, I remind the reader that homologies and cohomologies, in order to be calculated and
measured, must be defined including a set of coefficients (the coefficient group) [7]. For a detailed description of how
these coefficients enter the definition of the (co)homology I refer to the supplemental material. Enough to say that
they determine the sensitivity of the (co)homology to various features that would otherwise remain invisible. Figure
2 shows another set of two surfaces. Both unorientable and both undetectable by (co)homology as such when an
unadapted coefficient group is employed. In the same way in which homology may be insensitive to unorientability for
various shapes, the cohomology may become insensitive to the presence of a torus instead of a sphere. For example
take the torus T1. Its homology in dimension 1 is H1(T1) = Z⊕Zand the 0-dimensional and 2-dimensional homology
groups are each isomorphic to Z. However, the first cohomology group H1(T1;G) with coefficients in a group Gis
isomorphic to the group of homomorphisms from Z⊕Zto the group G. This group Hom(Z⊕Z,G) is trivial if
Gis a torsion group. If not, it is a direct sum of copies of G⊕G. Hence, the torsion of the coefficient group in
cohomology determines the visibility of a torus as such. Otherwise, the information remains only encoded in the
extension Ext that appears in the universal coefficient theorem used when one has to change the coefficient groups
used in cohomology.
The same is valid for the upper genera tori which may become indiscernible from their lower genus counterparts and
finally from a sphere. It is important to understand the role of the universal coefficient theorem in this construction. It
essentially gives the homological algebraic obstruction to the visibility of a torus from the perspective of a (co)homology
group with a given set of coefficients. The coefficient groups behave like an instrument that, if tuned accordingly,
gives us the desired information about a specific topological property. If however, tuned differently they can mask
some information. This masked information will reveal itself in the Ext and T or groups of the universal coefficient

5
theorem. If we go back to the representation of QCD as a genus expansion, we observe that the obstructions given
by Ext or T or manifest themselves as controlled, order by order deformations of the algebras of the gauge groups.
Using for example SO(N) as a gauge group instead of SU (N), would transform the double arrows of the ribbon-graph
representation of the gluon matrix into an un-oriented double line ribbon-graph. In this case, the gluing of diagrams
would allow also Klein bottles as surfaces and the difference with respect to the oriented SU (N) theory would be
encoded in the form of the T or obstruction in the universal coefficient theorem.
It is important to notice that the integration required to calculate the classes of diagrams summarized by a topo-
logical surface implies a measure of integration that depends only on the cohomology group.
Hence, what we care about is not the chain complex description of the space but, instead, the cohomology of the
surface with a certain set of coefficients and the way it transforms under the universal coefficient theorem. Indeed,
these two aspects encode the full information about the theory [7]. As a conclusion, this article shows that there exist
a vast generalization of the notion of ”renormalization” and ”renormalization group” that may transform the difficult
summation over topological genera into the calculation of mere planar or spherical graphs in QCD together with
controllable deformations of the algebras encoding the higher genus diagrams. This may lead to an exact solution for
the strong coupling regime of QCD and a solution of the confinement problem. Also, the same method can prove to
be of major use in the domain of strongly correlated electrons and in strongly coupled condensed matter systems.
[1] G. ’t Hooft, Nucl. Phys. B 72, 3 pag. 461 (1974)
[2] L. Brink, H. B. Nielsen, Nucl. Phys. B vol. 89, Issue 1 (1975)
[3] M. Gell-Mann, Phys. Rev. 125 (3), pag. 1067 (1962)
[4] A. Patrascu, supplemental material
[5] C. G. Callan, Phys. Rev. D 2 pag. 1541 (1970)
[6] E. Dror, Israel J. of Math. 15 pag. 115 (1973)
[7] A. Grothendieck, Inst. des Hautes Etudes Scientifiques, Pub. Math. 29 (29) pag. 95 (1966)
SUPPLEMENTAL MATERIAL
Quantum chromodynamics is the theory of quarks and gluons. It is a non-abelian gauge theory with the gauge
group SU (3) where the fermions live in the direct representation while the gluons live in the adjoint representation.
It is a very accurate theory that can be analyzed perturbatively in the high energy regime [1] where the first obvious
expansion parameter (the coupling) can be considered as small [2]. Its results in that domain are accurate enough to
validate it as the correct theory of strong interactions [3]. However, due to the property of confinement (which is not
systematically proved but for which we have strong experimental evidence [4]) the perturbative approach becomes
less and less reliable at low energy [5]. This is because there is a strong contribution towards the coupling constant
from low energy corrections [6]. These make the perturbative approach break down at energies comparable to our
immediate observations [7] i.e. nuclear physics, predictions for the masses of protons, neutrons, hadrons, etc. This
is why non-perturbative methods are being employed at that level. Today we rely in this area mostly on lattice
approaches [8] which are notoriously computationally intensive [9]. There are however different other methods. One
method tries to connect regions unaccessible to a perturbative approach in QCD to regions of other theories that
can easily be solved either via perturbative methods [10] or via methods originating from, say, general relativity [11].
These methods are spin-offs from the study of dualities [12] and are impressively promising today. Another, somehow
related method is the large N expansion [13]. This appeared due to the search for alternative expansion parameters
when the most obvious choice (the coupling constant) became unreliable. For a simplified alternative model (the
Gross-Neveu Lagrangian [14])
L=¯
ψi /
∂ψ +g2
2N(¯
ψψ)2(1)
the 1/N expansion parameter appears naturally in the interaction term. This can be understood with an analogy
to simple quantum mechanics [15] : a state |ψiwhich can be written as a sum of Northonormal states with equal
amplitudes
|ψi=α(|1i+|2i+|3i+... +|Ni) (2)

6
has as normalization condition
N|α|2= 1 (3)
and hence ( ¯
ψψ)2has as coefficient 1/N. The limit N→ ∞ then gives the desired approximation. QCD is more com-
plicated than this Gross-Neveu model. In principle, QCD is a SU (3) theory with gluons in the adjoint representation
of the gauge group. In order to define an accessible expansion parameter we will consider 3 →N→ ∞. In this
way we will write SU (N) instead of SU (3) for the gauge group, the case N= 3 being a restriction to the physical
situation. There will be a covariant derivative
Dµ=∂µ+ig
√NAµ(4)
where the gauge field will be of the form Aµ=AA
µTAwith the TAmatrices normalized according to
T rT ATB=1
2δAB (5)
The indices Aand Brefer to the adjoint representation of the gauge group. The coupling constant is considered to be
g/√Ninstead of g in order to obtain a sensible and non-trivial large Nlimit [16]. One can define the field strength
then as
Fµν =∂µAν−∂νAµ+ig
√N[Aµ, Aν] (6)
the lagrangean being
L=−1
2T rFµν Fµν +
NF
X
k=1
¯
ψk(i/
D−mk)ψk(7)
The large Nlimit can be taken with the number of flavors NFfixed. The quark propagator has the form
< ψa(x)¯
ψb(y)>=δabS(x−y) (8)
and is represented in the Feynman diagrams as a single line. The gluon propagator is
< AA
µ(x)AB
ν(y)>=δAB Dµν (x−y) (9)
where Aand Bare the indexes of the adjoint representation. A gluon can be seen as a N×Nmatrix with two indices,
(Aµ)a
b=AA
µ(TA)a
band the propagator becomes
< Aa
µb(x)Ac
νd (y)>=Dµν (x−y)(1
2δa
dδc
b−1
2Nδa
bδc
d) (10)
One can redraw the gluon lines from the standard Feynman diagrams in a way that explicitly shows these two indices
on each line [17]. In this way one arrives at something called a ”ribbon graph” where each gluon line is now represented
as a double arrow (in the case of an SU (N) theory; an SO(N) theory for example would have only double lines, with
no orientation). Putting together various processes one arrives at a situation where these double arrow graphs fill
a topological space. Indeed, they can be classified with respect to the homotopy class of the topological space they
can fill: sphere or planar diagrams, non-planar diagrams of genus 1 (a single hole torus) or of genus 2 (a double hole
torus) etc [18]. In QCD planar diagrams with quark lines are represented as planar diagrams with the quark line as
the boundary or as spheres with holes represented by the quark loops. These appear with order Nin the expansion.
Diagrams with no quark lines (gluon lines only) form closed spheres and appear to be of order N2. Some diagrams
however cannot be drawn in a plane with only quark lines at the boundaries or on a sphere with no boundaries. In
that case, these diagrams behave as powers of 1
Nand are represented on toruses of various genera [19]. See figure 1 for
a graphical representation. In a more general sense, increasing the size of the group to SU (N) where N→ ∞ makes
the theory simplify considerably. This is remarkable because one would not expect from a theory with an infinite
number of degrees of freedom to manifest itself in a simpler way than a theory with a finite, and rather small number
of degrees of freedom. It is possible to prove that in some theories, terms in the series expansion in 1
Nare related to
homotopy classes of diagrams. The fact that diagrams become less relevant in the large Nlimit is classified by the
topological space they can fill.

7
FIG. 3: (a) ribbon graphs on a planar topological space (or sphere), (b) ribbon graphs on a toroidal topological
space [18]
The series expansion in this limit is not the classical expansion in the coupling constant but instead in 1
Nand the
diagrams arrange themselves such that each term of this expansion corresponds to a set of diagrams on a topological
space characterized by a given genus.
In order to see the way in which the situation simplifies one should go back to what is known as the ’t Hooft model.
There, one considers a theory similar to QCD but on only two spacetime dimensions and in the large Nlimit. The
theory is confining and there are several simplifications. What is important however, is that in this simplified model
all the diagrams appearing in the theory can be represented on the surface of a plane or on a sphere. This is not a
general fact of QCD. Due to this fact, specific to the case of 2 dimensional QCD in the ’t Hooft model, we may solve
this theory exactly. Complete planarity makes a theory exactly solvable.
It is therefore important to consider the topological properties of the space formed by the ribbon graph diagrams
and particularly to see in what cases non-planar diagrams behave like planar diagrams. One observes that each color
index loop forms a polygon (simplified, it encompasses a 2-simplex) that can be glued together with another index
loop in order to form a surface. The topology of a space can be determined at various homological ”resolutions” with
various homological-algebraic tools. Here ”resolution” refers to the visibility of a certain topological feature. It has
nothing to do with the scale at which one looks and with the analogy to a ”magnifying glass”. Maybe the best analogy
would be a detector sensible to certain topological properties if suitably tuned. In general, what the physicist wishes
is to integrate over such a topological space in order to obtain the answers encoded in the diagrams defined on it. For
this, one needs an integration measure. This measure is in general sensitive to the (co)homology of the space. The
(co)homologies are invariants of the topological space of a certain accuracy (strength). They are in general defined in
terms of chain complexes [20]. The chain complexes are defined starting from the q-simplexes ∆q[22]
∆q={(t0, t1, ..., tq)∈Rq+1|Xti= 1, ti≤0∀i}(11)
together with face maps
fq
m: ∆q−1→∆q(12)
defined as
(t0, t1, ..., tq−1)→(t0, ..., tm−1,0, tm, ..., tq−1) (13)
This abstract construction must be mapped into a realistic space X. In order to do this a continuous map is required
(see fig. 2)
σ: ∆q→X(14)
Considering this, any space can be constructed as a chain
{X}=
l
X
i=1
riσi(15)

8
FIG. 4: Mapping a simplex on an arbitrary manifold [22]
where {ri}is the set of coefficients belonging in general to a ring R. The space Xas seen via the basis formed from
the q-simplexes defined above is denoted Sq(X;R). One defines a boundary map as
∂:Sq(X;R)→Sq−1(X;R) (16)
such that
∂(σ) =
q
X
m=0
(−1)mσ◦fq
m(17)
One can extend the above definition by introducing the covariant functor S∗(−;R). This means that given a continuous
map
f:X→Y(18)
this will induce a homomorphism
f∗:S∗(X;R)→S∗(Y;R) (19)
with the definition
f∗(σ) = f◦σ(20)
Then, the complex (S∗(X;R), ∂) is called the simplicial chain complex of the space Xwith coefficients in R. The
homology of this chain complex with coefficients in Ris then
Hq(X;R) = ker ∂
Im ∂ (21)
where ker represents the kernel of the considered map and Im represents its image.
Hence the homology groups depend on the coefficient rings Rused to define them. For simplicity one can also
restrict the rings to groups. I showed in ref. [20] following [21] that the universal coefficient theorems can express
the (co)homology groups of a space with a certain coefficient group in terms of (co)homology groups of the same
space with a different coefficient group. I also showed in [20] following [21] that some information visible when a
certain group is used becomes invisible when another group is used. The main idea is that the information about
the homotopy class of a function may be accessible when a certain coefficient group is used, while not visible from
the (co)homological perspective when another coefficient group is used. However, the universal coefficient theorem,
written as
0→Ext(Hn−1(C), G)→Hn(C;G)→H om(Hn(C), G)→0 (22)
for cohomology or as
0→Hn(C)⊗G→Hn(C;G)→T or(Hn−1(C), G)→0 (23)

9
for homology gives us the extra information related to the homotopy classes, just that in this case encoded in the
”homological obstruction” given by the Ext respectively T or groups. Here H∗(C) is the *-th dimension homology of
the chain complex C,H∗(C;G) is the *-th dimensional cohomology of the chain complex Cmeasured with coefficients
in G,Hom(H∗(C), G) is the group of all homomorphisms from H∗(C) to the coefficient group G,H∗(C;G) is the *-th
dimensional homology group with coefficients in the group Gand the Ext and T or functors are here the extensions
and the torsions of the respective homologies. These behave as obstructions to the exactness of the short sequence
where they would be absent.
Translated in terms of 1
Nexpansions, this would mean that, when using a certain coefficient group, the integration
measure may ”see” the non-planar graphs as planar while the formal differences between the two types can be found
only in the form of Ext and T or groups and modified group operations. In this way, one can relate theories containing
non-planar diagrams, considered hard to solve today, to theories containing only planar diagrams and homological-
algebraic corrections to some composition rules. These corrections will differ for each topological genus they originate
from. This would make many theories exactly solvable if the above mentioned corrections are correctly understood. In
some sense, this amounts, loosely speaking, to a renormalization procedure: the ”non-solvability” due to non-planar
contributions is eliminated, maintaining the relevant, computable ”non-planar” contributions only in the form of
modifications of group laws as specified by the Ext and/or T or groups. Applications to QCD should be obvious:
exact calculations, new dualities, a new systematic approach to non-perturbative QCD. In the context of string theory
this would allow to probe regions beyond the perturbative string expansion in a systematic way in the same way in
which the renormalization group prescription in renormalizable quantum field theories allows us to go beyond the
strict perturbative regime.
In a sense, until now, the lack of strength of topological invariants (their inability to discern some topological spaces)
was certainly not seen as a desirable property. However, several physical and finally natural phenomena can also not
make the distinction between some topological spaces defined in terms of chain complexes. For example, naturally
a measure of an integral over a topological space is related more to the cohomology of the space than to its actual
mathematically perfectly described shape. Natural phenomena are also defined in terms of integrations over spaces
with certain measures. This reminds us of the coarse graining in the problems analyzed via renormalization groups.
However, in this case it is not the large scale that hides features but the topological invariants and other homological
tools that we naturally use. In this sense I see this idea as a generalization of ”renormalization group approaches”.
The renormalization group transformations now become changes in the group structures used in (co)homology. The
regularization step becomes the identification of the problems originating from the non-planar nature of the corrections
and the translation of these into the language of derived functors (Ext and T or). The standard example of how a
function that looks homotopic to a constant in the cohomology with a set of coefficients, is in fact homotopically
non-trivial when analyzed with another set of coefficients, is presented in what follows.
Lemma 1 (The Universal Coefficient Theorem) [21]
If Cis a chain complex of free abelian groups, then there are natural short exact sequences
0→Hn(C)⊗G→Hn(C;G)→T or(Hn−1(C), G)→0 (24)
∀n,G, and these sequences split. Here T or(Hn−1(C), G) is the torsion group associated to the homology. In this way
homology with arbitrary coefficients can be described in terms of homology with the “universal” coefficient group Z[
This lemma is also valid for cohomology groups. Moreover, it is a property of algebraic topology independent of
the existence of an underlying manifold structure for the spaces or groups on which it may be applied. For a proof in
both the homology and the cohomology cases see reference [21]. The following example shows how the choice of the
coefficient group can affect the correct identification of the homotopy type of a function.
Example 2 (Homotopy and coefficient group) [21]
Take a Moore space M(Zm, n) obtained from Snby attaching a cell en+1 by a map of degree m. The quotient
map f:X→X/Sn=Sn+1 induces trivial homomorphisms on the reduced homology with Zcoefficients since the
nonzero reduced homology groups of Xand Sn+1 occur in different dimensions. But with Zmcoefficients the situation
changes, as we can see considering the long exact sequence of the pair (X, Sn), which contains the segment
0 = ˜
Hn+1(Sn;Zm)→˜
Hn+1(X;Zm)f∗
−→ ˜
Hn+1(X/Sn;Zm) (25)
Exactness requires that f∗is injective, hence non-zero since ˜
Hn+1(X;Zm) is Zm, the cellular boundary map
Hn+1(Xn+1 , Xn;Zm)→Hn(Xn, Xn−1;Zm) (26)
being exactly
Zm
m
−→ Zm(27)

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One can see that a map f:X→Ycan have induced maps f∗that are trivial for homology with Zcoefficients but not
so for homology with Zmcoefficients for suitably chosen m. This means that homology with Zmcoefficients can tell
us that fis not homotopic to a constant map, information that would remain invisible if one used only Z-coefficients.
Let me be more accurate and translate this into notions related to integration. This has important consequences
in the way we calculate the terms in the topological expansion of QCD but also in practical calculations of integrals
over topologically non-trivial manifolds in general. In principle the homology groups, Hk(C) relate to the shape of
the manifold. The cohomology groups, Hk(C) relate to the differential forms defined over the manifold. Hence, if
there is a manifold Mcharacterized by a sequence of homology groups, then, one can define the integral
ZM
ω(28)
characterized by the differential form ωand by the manifold M. In QCD the differential form may encode the
integration over all internal bands of a genus term. Integration can be seen as the pairing
Hk(M, R)×Hk(M , R)→R(29)
such that
([M],[ω]) →ZM
ω(30)
where this pairing is constructed with real coefficients and this coefficient structure characterizes also the measure of
integration and implicitly the differential form ω. Here, [M] represents a class in homology and [ω] represents a class
in cohomology. The pairing above however is an isomorphism (one-to-one relation) only when this particular choice
of coefficients is made. For other coefficients this pairing may fail to be an isomorphism. The correction is encoded
in a term controlled by the Ext and T or groups
Hk(M, G)×Hk(M , G)→G(31)
where the map becomes
([M],[ω]) →Z{M}
ω⊕ZExt(Hn−1(C),G)
η(32)
Here, the first integral is over the surface {M}visible when the coefficient structure Gis used and the correction
appears as an integral of another differential form over the extension group constructed from the homology with general
integer coefficients over a lower dimension. Here I simply used the universal coefficient theorem in cohomology. The
non-trivial topology however is not visible from the lower dimension hence the simplification.
In this way I show that properties defined on some more complex topological objects may be acceptably described
on simpler topological objects if controlled changes in the groups used to describe them are being employed.
I wish to underline again, not only the fact that this method brings obvious simplifications for the calculation of the
topological genus expansion in QCD but also in any field where the integration over topologically non-trivial objects
is required and time is an important factor. I give as an example here only the possible biological applications of
this method. In many circumstance it is vital for the integration over a complex topologically non-trivial object to
be performed with the speed and accuracy of a simple integration on a planar surface. In those cases, mapping a
torus-type problem onto a planar problem may prove useful.
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