Wilson lines in quantum field theory 2nd edition

We also intend to give an introductory idea of how to implement elementary calculations utilizing Wilson lines within the context of modern quantum field theory, in particular, in Quantu

Igor O Cherednikov, Tom Mertens, Frederik Van der Veken

Wilson Lines in Quantum Field Theory

Mathematical Physics

|

Edited by

Michael Efroimsky, Bethesda, Maryland, USA

Leonard Gamberg, Reading, Pennsylvania, USA

Dmitry Gitman, São Paulo, Brazil

Alexander Lazarian, Madison, Wisconsin, USA

Boris Smirnov, Moscow, Russia

Volume 24

Igor O Cherednikov, Tom Mertens,

Frederik Van der Veken

1211 Geneva Switzerland frederik.van.der.veken@cern.ch

ISBN 978-3-11-065092-1

e-ISBN (PDF) 978-3-11-065169-0

e-ISBN (EPUB) 978-3-11-065103-4

ISSN 2194-3532

Library of Congress Control Number: 2019951784

Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de.

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Cover image: Science Photo Library / Parker, David

Typesetting: VTeX UAB, Lithuania

Printing and binding: CPI books GmbH, Leck

www.degruyter.com

Aristotle held that human intellectual activity or philosophical (in a broad sense)

knowledge can be seen as a threefold research program This program contains physics, the most fundamental branch, which tries to find the right way to deal with

meta-“Being” as such; mathematics, an exact science studying calculable – at least, in principle – abstract objects and formal relations between them; and, finally, physics,

the science working with changeable things and the causes of the changes Therefore,

physics is the science of evolution – in the first place the evolution in time To put it

in more ‘contemporary’ terms, at any energy scale there are things which a physicisthas to accept as being ‘given from above’ and then try to formulate a theory of how

do these things, whatever they are, change Of course, by increasing the energy and,therefore, by improving the resolution of experimental facility, one discovers thatthose things emerge, in fact, as a result of evolution of other things, which shouldnow be considered as ‘given from above’.1

The very possibility that the evolution of material things, whatever they are, can

be studied quantitatively is highly non-trivial First of all, to introduce changes ofsomething, one has to secure the existence of something that does not change Indeed,changes can be observed only with respect to something permanent Kant proposed

that what is permanent in all changes of phenomena is substance Although ena occur in time and time is the substratum, wherein co-existence or succession of

phenom-phenomena can take place, time as such cannot be perceived Relations of time areonly possible on the background of the permanent Given that changes ‘really’ take

place, one derives the necessity of the existence of a representation of time as the stratum and defines it as substance Substance is, therefore, the permanent thing only

sub-with respect to which all time relations of phenomena can be identified

Kant gave then a proof that all changes occur according to the law of the

connec-tion between cause and effect, that is, the law of causality Given that the requirement

of causality is fulfilled, at least locally, we are able to use the language of differential equations to describe quantitatively the physical evolution of things There is, how-

ever, a hierarchy of levels of causality For example, Newton’s theory of gravitation

is causal only if we do not ask how the gravitational force gets transported from one

massive body to another The concept of a field as an omnipresent mediator of all

in-teractions allows us to step up to a higher level of causality The field approach tothe description of the natural forces culminated in the creation in the 20th century of

the quantum field theoretic approach as an (almost) universal framework to study the

physical phenomena at the level of the most elementary constituents of matter

1 It is worth noticing that this scheme is one of the most consistent ways to introduce the concept of

the renormalization group, which is crucial in a quantum field theoretical approach to describe the

three fundamental interactions.

To be more precise, the quantitative picture of the three fundamental

interac-tions is provided by the Standard Model, the quantum field theory of the strong, weak

and electromagnetic forces The aesthetic attractivity and unprecedented predictivepower of this theory is due to the most successful, and nowadays commonly accepted,

way to introduce the interactions by adopting the principle of local (gauge) symmetry.

This principle allows us to make use of the local field functions, which depend onthe choice of the specific gauge and, as such, do not represent any observables, toconstruct a mathematically consistent and phenomenologically useful theory In anygauge field theory we need, therefore, gauge-invariant objects, which are supposed

to be the fundamental ingredients of the Lagrangian of the theory, and which can beconsistently related, at least, in principle, to physical observables

The most straightforward implementation of the idea of a scalar gauge invariantobject is provided by the traced product of field strength tensors

issues throughout the book

2 The terminology and the choice of the signature ± will be explained below.

Preface | VII

The Wilson line (4) is gauge covariant, but, in contrast to the field strength, the

transformation law reads

U γ[y, x] → U(y)U γ[y, x]U†(x), (5)

so that the transformation operators U, U†are defined in different space-time points

For closed paths x = y, so that we speak about the Wilson loop:

U γ≡U γ[x, x] =𝒫exp[±ig ∮ dz μ A μ(z)]

which transforms similarly to the field strength

U γ=U γ[x, x] → U(x)U γ U†(x). (7)

The simplest scalar gauge invariant objects made from Wilson loops are, therefore,

the traced Wilson loops

𝒲γ=Tr U γ

From the mathematical point of view, one can construct a loop space whose

ele-ments are the Wilson loops defined on an infinite set of contours The recast of a tum gauge field theory in loop space is supposed to enable one to utilize the scalargauge-invariant field functionals as the fundamental degrees of freedom, instead ofthe traditional gauge-dependent boson and fermion fields Physical observables arethen supposed to be expressed in terms of the vacuum averages of the products ofWilson loops

quan-𝒲(n)

{γ} = ⟨0| Tr Uγ1Tr U γ2⋅ ⋅ ⋅Tr U γ n|0⟩ (8)The concept of Wilson lines finds an enormously wide range of applications in

a variety of branches of modern quantum field theory, from condensed matter andlattice simulations, to quantum chromodynamics, high-energy effective theories, andgravity However, there exist surprisingly few reviews or textbooks which contain amore or less comprehensive pedagogical introduction into the subject Even the basics

of the Wilson lines theory may put students and non-experts in significant trouble Incontrast to generic quantum field theory, which can be taught with the help of plenty ofexcellent textbooks and lecture courses, the theory of Wilson lines and loops still lackssuch a support The objective of the present book is, therefore, to collect, overview andpresent in the appropriate form the most important results available in literature, withthe aim to familiarize the reader with the theoretical and mathematical foundations

of the concept of Wilson lines and loops We also intend to give an introductory idea

of how to implement elementary calculations utilizing Wilson lines within the context

of modern quantum field theory, in particular, in Quantum Chromodynamics

The target audience of our book consists of graduate and postgraduate studentsworking in various areas of quantum field theory, as well as curious researchers from

other fields Our lettore modello is assumed to have already followed standard

uni-versity courses in advanced quantum mechanics, theoretical mechanics, classicalfields and the basics of quantum field theory, elements of differential geometry, etc.However, we give all necessary information about those subjects to keep with thelogical structure of the exposition Chapters 2, 3, and 4 were written by T Mertens,Chapter 5 by F F Van der Veken Preface, Introduction and general editing are due to

I O Cherednikov In our exposition we used extensively the results, theorems, proofsand definitions, given in many excellent books and original research papers For thesake of uniformity, we usually refrain from citing the original works in the main text

We hope that the dedicated literature guide in Appendix D will do this job better.Besides this, we have benefited from presentations made by our colleagues atconferences and workshops and informal discussions with a number of experts.Unfortunately, it is not possible to mention everybody without the risk of missingmany others who deserve mentioning as well However, we are happy to thank ourcurrent and former collaborators, from whom we have learned a lot: I V Anikin,

E N Antonov, U D’Alesio, A E Dorokhov, E Iancu, A I Karanikas, N I Kochelev,

E A Kuraev, J Lauwers, L N Lipatov, O V Teryaev, F Murgia, N G Stefanis, and

P Taels Our special thanks go to I V Anikin, M Khalo, and P Taels for reading parts

of the manuscript and making valuable critical remarks on its content We greatlyappreciate the inspiring atmosphere created by our colleagues from the ElementaryParticle Physics group in University of Antwerp, where this book was written We aregrateful to M Efroimsky and L Gamberg for their invitation to write this book, and to

the staff of De Gruyter for their professional assistance in the course of the preparation

Preface|V

1 Introduction: What are Wilson lines?|1

2 Prolegomena to the mathematical theory of Wilson lines|6

2.1 Shuffle algebra and the idea of algebraic paths|7

2.1.1 Shuffle algebra: Definition and properties|7

2.1.2 Chen’s algebraic paths|22

2.1.3 Chen iterated integrals|41

2.2 Gauge fields as connections on a principal bundle|48

2.2.1 Principal fiber bundle, sections and associated vector bundle|48

2.2.2 Gauge field as a connection|53

2.2.3 Horizontal lift and parallel transport|59

2.3 Solving matrix differential equations: Chen iterated integrals|61

2.3.1 Derivatives of a matrix function|61

2.3.2 Product integral of a matrix function|63

2.3.3 Continuity of matrix functions|66

2.3.4 Iterated integrals and path ordering|67

2.4 Wilson lines, parallel transport and covariant derivative|70

2.4.1 Parallel transport and Wilson lines|70

2.4.2 Holonomy, curvature and the Ambrose–Singer theorem|71

2.5 Generalization of manifolds and derivatives|76

2.5.1 Manifold: Fréchet derivative and Banach manifold|77

2.5.2 Fréchet manifold|82

3 The group of generalized loops and its Lie algebra|86

3.1 Introduction|86

3.2 The shuffle algebra over Ω = ⋀ M as a Hopf algebra|86

3.3 The group of loops|94

3.4 The group of generalized loops|94

3.5 Generalized loops and the Ambrose–Singer theorem|100

3.6 The Lie algebra of the group of the generalized loops|101

4 Shape variations in the loop space|108

5 Wilson lines in high-energy QCD|137

5.1 Eikonal approximation|137

5.1.1 Wilson line on a linear path|137

5.1.2 Wilson line as an eikonal line|146

5.2 Deep inelastic scattering|148

5.2.1 Kinematics|149

5.2.2 Invitation: the free parton model|150

5.2.3 A more formal approach|152

5.2.4 Parton distribution functions|160

5.2.5 Operator definition for PDFs|163

5.2.6 Gauge invariant operator definition|165

5.2.7 Collinear factorization and evolution of PDFs|169

5.3 Semi-inclusive deep inelastic scattering|176

5.3.1 Conventions and kinematics|176

5.3.2 Structure functions|178

5.3.3 Transverse momentum dependent PDFs|180

5.3.4 Gauge-invariant definition for TMDs|183

A.1 General topology|187

A.2 Topology and basis|188

A.9 Separation properties|207

A.10 Local compactness and compactification|209

A.11 Quotient topology|210

A.12 Fundamental group|212

A.14 Differential calculus|219

A.15 Stokes’ theorem|224

A.16 Algebra: Rings and modules|225

A.17 Algebra: Ideals|228

A.19 Hopf algebra|231

A.20 Topological, C∗-, and Banach algebras|240

A.21 Nuclear multiplicative convex Hausdorff algebras and the Gel’fand

spectrum|241

Contents | XI

B Notations and conventions in quantum field theory|249

B.1 Vectors and tensors|249

B.2 Spinors and gamma matrices|250

B.3 Light-cone coordinates|252

B.4 Fourier transforms and distributions|254

B.5 Feynman rules for QCD|255

C.2.1 Calculating products of fundamental generators|259

C.2.2 Calculating traces in the adjoint representation|262

D Brief literature guide|265

Bibliography|266

Index|269

1 Introduction: What are Wilson lines?

The idea of gauge symmetry suggests that any field theory must be invariant under the

local (i e., depending on space-time points) transformations of field functions

ψ(x) → U(x)ψ(x), ψ(x) → ψ(x)U†(x), (1.1)

where the matrices U(x) belong to the fundamental representation of a given Lie

group In other words, the Lagrangian has to exhibit local symmetry We shall mostly

deal with special unitary groups, SU(N c), which are used in Yang–Mills theories though a number of important results can be obtained by using only the general form

Al-of this gauge transformation, it will be sometimes helpful to use the parameterization1

U(x) = e±igα(x), (1.2)where

α(x) = t a α a (x), t a= λ a

2 ,

and λ a are the generators of the Lie algebra of the group U.

Consider for simplicity the free Lagrangian for a single massless fermion field ψ(x)

gauge invariance, consists in the introduction of the set of gauge fields

1 The coupling constant g can be chosen to have a positive or a negative sign As this is merely a matter

of convention, we leave the choice open and will write ±g.

Thus, the usual derivative has to be replaced by the covariant derivative:

a naive form the bi-local field products and Green’s functions are not gauge invariant:

Δ(y, x) → ψ(y)U†

Therefore, the problem arises of how to find an operator T[y,x] , which transports the field ψ(x) to the point y, so that

T[y,x] ψ(x) → U(y)[T[y,x] ψ(x)]. (1.14)

In this case, we have

ψ(y)T[y,x] ψ(x) → ψ(y)U†(y)U(y)[T[y,x] ψ(x)] = ψ(y)T[y,x] ψ(x), (1.15)

so that the product (1.13) becomes gauge invariant

Consider first the Abelian gauge group U(1) In this case

U(x) = e±igα(x), (1.16)

2 See, in particular, references in the section ‘Gauge invariance in particle physics’, Appendix D.

1 Introduction: What are Wilson lines? | 3

where α(x) is a scalar function Then3

It is instructive to see that the choice of the sign in equation (4) depends on the

pa-rameterization of the symmetry transformation U(x) Taking the line integral for the

integrand 𝜕μ α explicitly, one concludes that in order to save the gauge invariance, the

sign should be chosen as

sign = ±

In what follows we shall always specify the signature conventions we adopt

In the non-Abelian case the situation is more involved The fields at different

space-time points A μ (z) and A μ (z󸀠), equation (1.5), do not commute, so the

exponen-tial of non-commuting functions is ill-defined An infinitesimal version of equation

(1.15) suggests the following equation to the transporter T[y,x]:

is not affected by the choice of path, but, as we will see, the transporter becomes a

functional of the path The path γ is assumed to be parameterized by the coordinate

z ∈ γ

3 Note that the sign in front of 𝜕μ α(x) is independent on the sign choice of g.

4 Any ordering of the field functions is not needed in the classical Abelian case.

depending on the parameter t in such a way that

dz μ = ̇z μ (t)dt, z(0) = x, z(t) = y.

The operator𝒜γ (t) in the r h s of equation (1.20) is given by

𝒜γ (t) = ±i gA μ [z(t)] ̇z μ (t). (1.21)

It is easy to see that (1.18) solves equation (1.20) in the classical Abelian case

Integrating equation (1.20) from 0 to t yields an integral equation instead of a

T(0) = T[x,x] = T(0) (1.24)Then, for the first non-trivial term in the expansion (1.23) we obtain

1 Introduction: What are Wilson lines? | 5

where the path-ordering operator reads

In this part of the book we give the necessary conceptual thesaurus and overview themain steps towards the construction of the mathematical theory of Wilson lines andloops To be more precise, a goal of this exposition is to demonstrate that gauge the-

ories can be consistently formulated in the principal fiber bundle setting, where the

gauge fields (or potentials) are identified with pullbacks of sections of a connection

one-form in the gauge bundle The gauge potentials give rise to a parallel transport equation in the gauge bundle that can be solved by using product integrals As a result,

we shall show that the solution of the parallel transport equation can be presented as

a Wilson line We shall also discuss its relation to the standard covariant derivative ingauge theories

Then, an alternative way to construct a gauge theory will be discussed, which is

based on the use of the holonomies in the gauge bundle instead of the gauge potentials.

This possibility is based on the Ambrose–Singer theorem, which claims that the entire

gauge invariant content of a gauge theory is included in the holonomies However, the

issues of overcompleteness, reparameterization invariance, and additional algebraicconstraints, coming from the matrix representation of the Lie algebra associated withthe gauge group, impede the straightforward application of the standard loop spaceapproach to gauge field theories An interesting solution to these problems arises if

one extends this setting to the so-called generalized loops, first proposed by Chen and

further studied by Tavares (for references, see section ‘Algebraic paths’ in Literature

Guide D) within the framework of the generalized loop space (GLS) approach Our

ex-position is based mostly on the original works by these authors

Aiming towards the appropriate formulation of the generalized loop space work and having in mind the demonstration of its relation to Wilson lines and loops,

frame-we start with an introductory discussion of the most relevant algebraic concepts Then

we make use of these concepts to construct Chen’s algebraic d-paths, and,

conse-quently, the generalized loop space We end the chapter with a discussion on the ferential operators which can be defined in generalized loop space Assuming thatgauge field theories can be recast within the GLS framework, and given that, to this

dif-end, a suitable action could be found, one can use relevant differential operators to

generate the variations of the generalized degrees of freedom in the GLS, and hence,

to construct the appropriate equations of motion in the GLS Let us mention that the

ambitious program of reformulation of gauge theories in the GLS setting has neverbeen fully accomplished and thus remains a challenge

Note that we give complete definitions of the notions, formulations of theoremsand their proofs only when we find it necessary for the consistency of the exposition.For an extended list of definitions and some helpful theorems and statements we refer

to Appendix A

2.1 Shuffle algebra and the idea of algebraic paths | 7

2.1 Shuffle algebra and the idea of algebraic paths

2.1.1 Shuffle algebra: Definition and properties

2.1.1.1 Algebraic preliminaries

For our purposes it is sufficient to describe an n-dimensional manifold as a topological

space, wherein a neighborhood to each point is equivalent (strictly speaking,

homeo-morphic) to the n-dimensional Euclidean space The fundamental geometrical object

in a manifold we will be concerned about is a path One has a natural intuitive idea of what a path or a loop in a manifold is Mathematically one usually defines a path γ in

a manifold M as the map

γ : [0, 1] → M, t 󳨃→ γ(t).

For closed paths, which are called loops, one just adds the extra condition that theinitial and final points of the path coincide

γ(0) = γ(1) ∈ M.

The straightforward idea of paths and loops can be generalized to the so-called

algebraic d-paths, where the d-paths are constructed as algebraic objects possessing

certain (desirable) properties The resulting algebraic structure can then be supplied

with a topology, turning it into a topological algebra The topology is used to complete

the algebraic properties with analytic ones, allowing the introduction of the necessarydifferential operators.1

Several algebraic structures must be introduced before we begin the main sion Without going too deep into details, we define a ring as a set wherein two binary

discus-operations of multiplication and addition are defined Putting it another way, a ring is

an Abelian group (with addition being the group operation) supplied with an extra eration (multiplication) If the second operation is commutative, the ring is also calledcommutative The set of integers provides one of the simplest examples of a commu-tative ring Otherwise we speak of noncommutative rings The set of square matrices

op-is an example of a noncommutative ring

Having introduced the notion of a ring, we are able to introduce another algebraic

structure, namely a field, which is defined as a commutative ring where division by

a nonzero element is allowed It is evident that nonzero elements of a field make up

an Abelian group under multiplication For example, the set of real numbers forms afield

1 Most of the material in this section is based on the original works by Chen (see Literature Guide),

where the proofs to a number of the stated theorems can also be found For the sake of brevity we skip those proofs which do not bring more insight than needed into the subject of the book.

One can then construct a vector space over a field In this case, the elements of

the vector space are called vectors, while the elements of the field are scalars, and twobinary operations (addition of two vectors and multiplication of a vector by a scalar)acting within the vector space should be defined One easily captures the idea of avector space by thinking of the usual Euclidean vectors of velocities or forces

The notion of a module over a ring generalizes the concept of a vector space over

a field: now scalars only have to form a ring, not necessary a field For example, any

Abelian group is a module over the ring of integers In what follows ‘K-module’ stands for a module over a ring K.

2.1.1.2 Shuffle algebra

The generalization of the concept of paths in a manifold calls for the introduction of

the notion of a shuffle algebra The shuffle algebra is an algebra based on the shuffle product, which in its turn is defined via (k, l)-shuffles Let us start with the definitions

of these shuffles

Definition 2.1 ((k, l)-shuffle) A (k, l)-shuffle is a permutation P of the k + l letters, such that

P(1) < ⋅ ⋅ ⋅ < P(k) and P(k + 1) < ⋅ ⋅ ⋅ < P(k + l).

Exercise 2.2 How can one explain a (k, l)-shuffle using a deck of cards?

Using these (k, l)-shuffles we can introduce the shuffle multiplication, symbolically

represented by the symbol ∙

Let us consider a set of arbitrary objects Z i

Definition 2.3 (Shuffle multiplication) Using the notations

exterior algebra ⋀ M over the manifold M, and n k the exterior algebra degree of the factor Z k.

Several examples will be instructive to make the shuffle multiplication clear

2.1 Shuffle algebra and the idea of algebraic paths | 9

Example 2.4 (Shuffle multiplication).

– The situation with two objects is evident:

If we consider the objects Z to be one-forms ω (or linear functionals) defined

on some manifold and compare their shuffle products with the usual antisymmetricwedge products, then shuffle product can be treated as a symmetric counterpart to

the wedge product.

Let now M be a manifold and

Ω = ⋀ M =⋀1 M

be the set of one-forms on M We interpret Ω as a K-module, where for the moment we assume that K is a general ring of scalars with a multiplicative unity Introducing the shuffle product on a K-module Ω defines the shuffle K-algebra.2

Definition 2.5 (Shuffle K-algebra) Consider a K-module Ω and the regular tensor algebra over K

based on Ω, denoted by T(Ω) Then T r(Ω) represents the degree r components of the algebra It is easy to see that

T0(Ω) = K.

Replacing the tensor product by the shuffle multiplication generates a new algebra called the shuffle

K-algebra Sh(Ω) based on the K-module Ω.

In this algebra the shuffle product plays a role of the algebra multiplication m, so that

one can write

It is now possible to extend the algebraic structure based on the shuffle product

by introducing the K-linear maps ϵ, Δ.

Definition 2.6 (Co-unit and co-multiplication).

– Co-unit ϵ ∈ Alg(Sh(Ω), K) is defined by

{ ϵ(1Sh) = 1K for n = 0

ϵ(ω1⋅ ⋅ ⋅ω n) =0 for n > 0. (2.3)

– Co-multiplication Δ : Sh(Ω) → Sh(Ω) ⊗ Sh(Ω) acts as

{ { { { {

Δ(1) = 1 for n = 0 Δ(ω1⋅ ⋅ ⋅ω n) =

n

∑

i=0(ω1⋅ ⋅ ⋅ω i) ⊗ (ω i+1⋅ ⋅ ⋅ω n) for n > 0. (2.4)

The map Δ can be considered as a K-module morphism and is also an associative

co-multiplication since

(1 ⊗ Δ)Δ = (Δ ⊗ 1)Δ

Exercise 2.7 Prove the above statement.

The co-multiplication Δ and co-unit ϵ introduces a co-algebra structure on the shuffle

algebra, so that it becomes a bi-algebra We can go a step further and show that the

shuffle algebra is also a Hopf algebra.3For that reason we introduce the notion of an

2.1 Shuffle algebra and the idea of algebraic paths | 11

and the unit map

Theorem 2.9 (Sh(Ω) is a Hopf algebra) The shuffle algebra Sh(Ω) is a Hopf K-algebra

provided that its co-multiplication Δ, co-unit ϵ and antipode J are defined as in equations (2.3), (2.4), and (2.5).

Keeping in mind the algebraic structure of the shuffle algebra discussed above,

we can go on with the study of the algebra homomorphisms4Alg(Sh(Ω), K).

Definition 2.10 (Group multiplication on Alg(Sh(Ω), K)) Consider the algebra homomorphisms

Figure 2.1: Multiplication of algebra morphisms.

The multiplication of algebra morphisms is depicted in Figure 2.1

It is easy to observe that:

Proposition 2.11 The multiplication in Definition 2.10, defined on the algebra

mor-phisms

Alg(Sh(Ω), K), turns it into a group.

We can now study the properties of the algebra homomorphisms Alg(Sh(Ω),Sh(Ω)) To this end, let us define an algebra morphism which might look a bit strange

at the moment, but will turn out to be valuable when considering the group structure

of algebraic paths and loops

Definition 2.12 (L-operator) For

α ∈ Alg(Sh(Ω), K)

define

̃L α= (α ⊗ 1)Δ ∈ Alg(Sh(Ω), Sh(Ω)) (2.8) and

2.1 Shuffle algebra and the idea of algebraic paths | 13

and

(̃L α1α2, ̂L α1α2) = (̃L α2L̃α1, ̂L α2L̂α1) (2.10)The proof of the above statement is straightforward:

shuf-consistent formalism for variations of these paths and loops, we need the operations

of differentiation to be well-defined The appropriate introduction of these ations requires some basic knowledge of category theory We give now a brief intro-

differenti-duction to category theory, restricting ourselves only to those concepts which will beexplicitly used in our discussion

Define first the concept of a category

Definition 2.14 (Category) Category C includes:

1 a class ob(C) of objects a i;

2 a class Hom(C) of morphisms F lwhich can be interpreted as maps between the objects A

mor-phism has a unique source object a i and target object a j:

F l:a i→a j;

3 a binary operation called the composition of morphisms, such that

Hom(a1,a2) ×Hom(a2,a3) →Hom(a1,a3), which exhibit the properties of

1 identity: there exists a unity object 1 ∈ ob(C), such that

1a i=a i 1 = a i;

2 associativity: if

F i:a i→a i+1, then

F3 ⋅ (F2 ⋅F1 ) = (F3 ⋅F2 ) ⋅F1

It is easy to show that there exists a unique identity map.

We also need maps between categories which are captured by the notion of a functor.

Definition 2.15 (Functor) Let C1and C2be two categories A functor F from C1to C2is a map with the following properties:

1 The mapping rule: for each X1∈C1, there exists X2∈C2, such that

1 for the unity 1C1in C1

Hom(a1,a2) →Hom(F(a1),F(a2)) (2.13)

is surjective (injective, bijective).

2.1 Shuffle algebra and the idea of algebraic paths | 15

There exist some special types of functors, of which we only mention the forgetful tor, since we shall deal with it in further discussions.

func-Definition 2.16 (Forgetful functor) Suppose two categories C1and C2are given, and the object X ∈ C1can be regarded as an object of C2by ignoring certain mathematical structures of X Then a functor

U : C1→C2

which ‘forgets’ about any mathematical structure is called a forgetful functor.

Now we are ready to introduce differentiations Let us begin with the notion of a K-module differentiation.

Definition 2.17 (K-module differentiation) Consider K-modules U and a U-module Ω Let F1,F2∈ U.

A differentiation d is a morphism

d : U → Ω

which obeys the rule

d(F1F2 ) =F2 (dF1 ) +F1 (dF2 ) (2.14)

Extending to K-algebras, K-module differentiations form a category:

Definition 2.18 (Category of K-algebra differentiations) Consider K-algebras U and U󸀠 and the entiations

̂ϕ ∈ Hom k(Ω, Ω 󸀠 ), such that

d󸀠̃ϕ = ̂ϕD

and if

F ∈ U, w ∈ Ω

then

̂ϕ(Fw) = (̃ϕF) ̂ϕw.

In what follows, 𝒟 stands for the category of differentiations of unitary commutative K-algebras with

the category morphisms defined above.

The next category of differentiations we shall introduce is the category of pointed ferentiations.

dif-Definition 2.19 (Category of pointed differentiations) Consider a pair (d, p) with the operation

d : U → Ω

being a differentiation, and

p ∈ Alg(U, K).

Such a pair is said to be a pointed differentiation The corresponding category 𝒫𝒟 can be introduced,

such that the morphisms Diff(D, p : D󸀠p󸀠 ) in 𝒫𝒟

(d, p) → (d󸀠 ,p󸀠 ) are given by pairs

(̃ϕ, ̂ϕ) ∈ Diff(d, d󸀠 ), such that

p = p󸀠̃ϕ The category morphisms then define equivalences of differentiations.

Anticipating a forthcoming discussion, we notice that this is the above-mentionedproperty of morphisms

p = p󸀠̃ϕ which guarantees the uniqueness of the initial point of a generalized path.

A subcategory of this pointed differentiation is generated if one imposes the

con-straint of surjectivity on the K-module differentiation.

Definition 2.20 (Surjective pointed differentiation) We call a pointed differentiation (d, p) surjective

if d is surjective.

2.1 Shuffle algebra and the idea of algebraic paths | 17

The last subcategory of pointed differentiations we need to define is the category of splitting pointed differentiations To define these differentiations we also need to in- troduce the notion of a short exact sequence In general, ker F stands for a kernel of a

map

F : A1→A2,

that is, a subset of A1which maps under F into the zero of A2 In other words, the image

of the kernel F is the zero in A2

In what follows,𝒮𝒫𝒟stands for the subcategory of splitting surjective differentiations

of the category𝒫𝒟

Consider an application of the above differentiations to the specific case of shuffle

algebras Applying the K-module differentiation d from Definition 2.17 with

Similarly we can consider the surjective differentiations:

Definition 2.23 (Surjective shuffle module differentiation) Suppose we have the K-module

To see that δ is a differentiation, it is instructive to first consider an example of a shuffle

product of the tensor products

Example 2.24 (Shuffle product of tensor products) Consider

u1,u2∈T1(Ω)

2.1 Shuffle algebra and the idea of algebraic paths | 19

For splitting pointed differentiations we have the following lemma:

Lemma 2.26 (Splitting Pointed Differentiation Homomorphisms) Suppose we have

a splitting pointed differentiation (d, p), (Definition 2.22) A commutative diagram of K-module morphisms is shown in Figure 2.2 (the double line between the K’s indicates that their values are equal) Assuming that

̃

θ(1) = 1

Figure 2.2: Splitting pointed

differen-tial.

and, for all u1∈Sh(Ω), u2∈Ω,

generates a homomorphism between the tensor algebras T(Ω) and T(Ω󸀠) Denote this

homomorphism by T(θ) This algebra morphism is shuffle product preserving, such that we can write it as Sh(θ).

The following special functor can be defined:

Definition 2.27 (Covariant functor to 𝒮𝒫𝒟) Let ΔF denote the covariant functor (Definition 2.15) to

the category of splitting pointed differentiations (Definition 2.22) on the category of K-modules, which

exhibits the properties

ΔF(Ω) = (δ, ϵ) = (δ(Ω), ϵ(Ω))

and for θ ∈ Hom(Ω, Ω󸀠 )

ΔF(θ) = (Sh(θ), Sh(θ) ⊗ θ).

A diagrammatic representation of this definition is given in Figure 2.3

Figure 2.3: Covariant functor to the category of 𝒮𝒫𝒟.

A theorem holds which states that the morphism (̃θ, ̂θ) is a unique homomorphism in

the category of splitting pointed differentiations:

2.1 Shuffle algebra and the idea of algebraic paths | 21

Theorem 2.28 (Uniqueness) Suppose we have K-module Ω and a splitting surjective

pointed differentiation (d󸀠,p󸀠), where

(Defini-(̃ϕ, ̂ϕ) ∈ Diff(d, p; d󸀠,p󸀠)

the morphism ϕ of K-modules.

Continuing along the same line and taking into account that Sh(Ω) ⊗ Sh(Ω) ⊗ Ω is

a Sh(Ω) ⊗ Sh(Ω)-module, and that

ϵ ⊗ ϵ ∈ Alg(Sh(Ω)) ⊗ Sh(Ω), K),

we have the following properties:

1 The morphism of K-modules

1 ⊗ δ : Sh(Ω) ⊗ Sh(Ω) → Sh(Ω) ⊗ Sh(Ω) ⊗ Ω

is a differentiation

2 The pair

(1 ⊗ δ, ϵ ⊗ ϵ)

is a splitting surjective pointed differentiation

We end the discussion on shuffle algebras and their differentiations by stating the

fol-lowing property of the L-operator, defined in equation (2.12), with respect to the

cate-gory of differentiations defined on the shuffle algebras:

Proposition 2.29 (̃L α, ̂L α)is an equivalence in the category of differentiations𝒟 That is,

L α δ = δ̃L α

2.1.2 Chen’s algebraic paths

In this part we introduce the d-paths and d-loops, generalizing the intuitive notion of

paths and loops, as it was originally proposed by Chen

Figure 2.4: Path diagram.

Now we shall discuss the properties of the maps shown in Figure 2.4 and the way they

generate the d-paths.

The entire structure is built starting from a given pointed differentiation (d, p), which is mapped to the pointed differentiation (δ, ϵ) by the equivalence of differenti- ations we introduced in Definition 2.19 The δ in the figure is the differentiation intro- duced in Definition 2.23, and ϵ is the co-unit from the co-algebra structure on Sh(Ω) Anticipating that the notion of an ideal will play an important role, let us give the

definition of an ideal and review some of its properties related to kernels of maps

Definition 2.30 (Ring ideal) Consider a ring K Its subset

A ⊂K

2.1 Shuffle algebra and the idea of algebraic paths | 23

is called an ideal if

1 A is a subgroup of K under the addition;

2 for a ∈ A and k ∈ K one has

We can also consider sums and products of ideals.

Definition 2.31 Consider two ideals in K, A1and A2 Then, for a1∈ a1, a2∈ a2, the set

{a1+a2}

is an ideal written as

A 1 + A 2 Similarly, the set

Proposition 2.32 The kernel of a homomorphism

F : A1→A2

is an ideal in A1.

To prove this powerful statement we first need the following theorem:

Theorem 2.33 (Kernel is a subring) Consider two rings K1and K2with the binary ations {+1,2, ∘1,2} Suppose that we have a ring homomorphism

oper-Φ : K1→K2.

Then the kernel of Φ is a subring in K1.

Proof A ring homomorphism of addition is a group homomorphism and the kernel is

a subgroup

ker Φ ≤ K1,where ≤ denotes subgroup Let now

x1,x2∈ker Φ,then

Φ (x1∘1x2) =Φ (x1) ∘2Φ (x2) =0K2∘20K2 =0K2.Therefore,

x1∘1x2∈ker Φ

so that the conditions for a subring are fulfilled

Now we are in a position to show that the kernel of a homomorphism is indeed anideal

Theorem 2.34 (Kernel is an ideal) Let K1,2 be again rings with the corresponding nary operations Consider a ring homomorphism

bi-Φ : K1→K2

Then the kernel of Φ is an ideal in K1.

Proof By Theorem 2.33, ker Φ is a subring of K1 Consider x1∈ker Φ, such that

Φ (x1) =0K2

2.1 Shuffle algebra and the idea of algebraic paths | 25

Suppose now that x2∈K1 Then, given that x1∈ker Φ,

Φ (x2∘1x1) =Φ (x2) ∘2Φ (x1)

=Φ (x2) ∘20K2 =0K2.Taking into account that (now obviously)

Φ (x1∘1x2) ,the theorem is proven

With the aid of ideals we can introduce d-closed differentiations.

Definition 2.35 (d-closed differentiation) Consider a differentiation

This definition calls for a more detailed explanation We take U to be a K-module, so

that U supplied with addition (U, +) is an Abelian group, and we can use elements of

K as scalars in the multiplication with elements of U Expressing this multiplication

as a map, we can write

K × U → U.

We take similarly (Ω, +) to be an Abelian group as a U-module, such that the elements

of U now act as scalars The differentiation d then makes the ideal J a subset of Ω, but

also turns it into a U-(sub)module

U×dJ → Ω.

The term ‘closed’ refers then to the fact that

JΩ ⊂ dJ in Ω, where the elements of J are multiplied by the elements of Ω.

As it was discussed before, kernels of homomorphisms generate ideals The

proposition below explains how a d-closed ideal is related to the kernel of a

ho-momorphism between pointed differentiations

Proposition 2.36 Consider the pointed differentiations (d, p) and (d󸀠,p󸀠), of which

(d, p) is surjective and (d󸀠,p󸀠)splitting Hence, if

2.1.2.2 Chen iterated integrals as extension of line integrals

Given that the concept of ideals is introduced and their relation to kernels of morphisms is clarified, we are in a position to study an ideal in the shuffle algebra To

homo-this end, we define an extension of the notion of line integrals, so-called Chen iterated integrals.

Definition 2.37 (Chen iterated integrals) Consider a line integral along the path γ(t) from the point

Definition 2.38 (Chen iterated integrals without coordinates) Consider a smooth n-dimensional

manifold M, the set of piecewise-smooth paths 𝒫ℳ

γ : I → M

2.1 Shuffle algebra and the idea of algebraic paths | 27

t

∫

0

ω1(t1)dt1] ]

Proposition 2.39 (Chen iterated integrals preserve multiplication) Consider again a

piecewise linear path γ in the manifold M γ and be the set of one-forms Ω on M If we interpret γ as the map