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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

2

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)

10

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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