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Utama Loops, knots, gauge theories, and quantum gravity

Loops, knots, gauge theories, and quantum gravity

, ,
This text provides a self-contained introduction to applications of loop representations and knot theory in particle physics and quantum gravity. Loop representations (and the related topic of knot theory) are of considerable current interest because they provide a unified arena for the study of the gauge invariant quantization of Yang-Mills theories and gravity, and suggest a promising approach to the eventual unification of the four fundamental forces. This text begins with a review of calculus in loop space and the fundamentals of loop representations. It then goes on to describe loop representations in Maxwell theory, Yang-Mills theories as well as lattice techniques. Applications in quantum gravity are then discussed in detail. Following chapters move on to consider knot theory, the braid algebra and extended loop representations in quantum gravity. A final chapter assesses the current status of the theory and points out possible directions for future research. This self-contained introduction will be of interest to graduate students and researchers in theoretical physics and applied mathematics.
Tahun: 1996
Penerbit: Cambridge University Press
Bahasa: english
Halaman: 334
ISBN 10: 0521473322
ISBN 13: 9780521473323
Series: Cambridge monographs on mathematical physics
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Loops, Knots, Gauge Theories and Quantum Gravity RODOLFO ^GAMBINI Universidad de la Republica, Montevideo JORGE PULLIN Pennsylvania State University CAMBRIDGE

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain © Cambridge University Press 1996 This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1996 First paperback edition 2000 Printed in the United Kingdom at the University Press, Cambridge Typeset in ll/13pt Computer Modern A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication data Gambini, Rodolfo. Loops, knots, gauge theories and quantum gravity / Rodolfo Gambini and Jorge Pullin; [foreword by Abhay Ashtekar]. p. cm. - (Cambridge monographs on mathematical physics) Includes bibliographical references and index. ISBN 0 521 47332 2 (he) 1. Quantum gravity - Mathematics. 2. Loops (Group theory). 3. Knot theory. 4. Gauge fields (Physics). 5. Quantum field theory. I. Pullin, Jorge. II. Title. III. Series. QC178.G25 1996 530.1'43-dc20 95-35159 CIP ISBN 0 521 47332 2 hardback ISBN 0 521 65475 0 paperback

A Gabriela y Martha

Contents Foreword xiii Preface xv 1 Holonomies and the group of loops 1 1.1 Introduction 1 1.2 The group of loops 3 1.3 Infinitesimal generators of the group of loops 7 1.3.1 The loop derivative 8 1.3.2 Properties of the loop derivative 10 1.3.3 Connection derivative 17 1.3.4 Contact and functional derivatives 21 1.4 Representations of the group of loops 25 1.5 Conclusions 27 2 Loop coordinates and the extended group of loops 29 2.1 Introduction 29 2.2 Multitangent fields as description of loops 30 2.3 The extended group of loops 32 2.3.1 The special extended group of loops 33 2.3.2 Generators of the SeL group 36 2.4 Loop coordinates 38 2.4.1 Transverse tensor calculus 38 2.4.2 Freely specifiable loop coordinates 42 2.5 Action .of the differential operators 44 2.6 Diffeomorphism invariants and knots 47 2.7 Conclusions 50 3 The loop representation 52 3.1 Introduction 52 3.2 Hamiltonian formulation of systems with constraints 54 3.2.1 Classical theory 54 3.2.2 Quantum theory 56 3.3 Yang-Mills theories 58

Contents 3.4 3.5 3.6 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 5 5.1 5.2 5.3 5.4 5.5 6 6.1 6.2 6.3 6.4 3.3.1 Canonical formulation 3.3.2 Quantization Wilson loops 3.4.1 The Mandelstam identities 3.4.2 Reconstruction property Loop representation 3.5.1 The loop transform 3.5.2 The non-canonical algebra 3.5.3 Wavefunctions in the loop representation Conclusions Maxwell theory The Abelian group of loops Classical theory Fock quantization Loop representation Bargmann representation 4.5.1 The harmonic oscillator 4.5.2 Maxwell-Bargmann quantization in terms of loops Extended loop representation Conclusions Yang-Mills theories Introduction Equations for the loop average in QCD The loop representation 5.3.1 SUB) Yang-Mills theories 5.3.2 SU(N) Yang-Mills theories Wilson loops and some ideas about confinement Conclusions Lattice techniques Introduction Lattice gauge theories: the ZB) example 6.2.1 Covariant lattice theory 6.2.2 The transfer matrix method 6.2.3 Hamiltonian lattice theory 6.2.4 Loop representation The SUB) theory 6.3.1 Hamiltonian lattice formulation 6.3.2 Loop representation in the lattice 6.3.3 Approximate loop techniques Inclusion of fermions 59 61 63 64 67 72 76 80 85 87 88 89 91 92 96 103 103 105 109 112 113 113 115 118 118 121 124 130 131 131 133 133 136 138 141 144 145 147 150 156

Contents xi 6.5 Conclusions 159 7 Quantum gravity 161 7.1 Introduction 161 7.2 The traditional Hamiltonian formulation 164 7.2.1 Lagrangian formalism 164 7.2.2 The split into space and time 164 7.2.3 Constraints 168 7.2.4 Quantization 169 7.3 The new Hamiltonian formulation 171 7.3.1 Tetradic general relativity 172 7.3.2 The Palatini action 173 7.3.3 The self-dual action 174 7.3.4 The new canonical variables 175 7.4 Quantum gravity in terms of connections 179 7.4.1 Formulation 179 7.4.2 Triads to the right and the Wilson loop 180 7.4.3 Triads to the left and the Chern-Simons form 185 7.5 Conclusions 187 8 The loop representation of quantum gravity 188 8.1 Introduction 188 8.2 Constraints in terms of the T algebra 189 8.3 Constraints via the loop transform 192 8.4 Physical states and regularization 196 8.4.1 Diffeomorphism constraint 196 8.4.2 Hamiltonian constraint: formal calculations 197 8.4.3 Hamiltonian constraint: regularized calculations 201 8.5 Conclusions 208 9 Loop representation: further developments 209 9.1 Introduction 209 9.2 Inclusion of matter: Weyl fermions 209 9.3 Inclusion of matter: Einstein-Maxwell and unification 215 9.4 Kalb-Ramond fields and surfaces 218 9.4.1 The Abelian group of surfaces 218 9.4.2 Kalb-Ramond fields and surface representation 220 9.5 Physical operators and weaves 221 9.5.1 Measuring the geometry of space in terms of loops 222 .9.5.2 Semi-classical states: the weave 228 9.6 2+1 gravity 230 9.7 Conclusions 237

xii Contents 10 Knot theory and physical states of quantum gravity 238 10.1 Introduction 238 10.2 Knot theory 239 10.3 Knot polynomials 242 10.3.1 The Artin braid group 243 10.3.2 Skein relations, ambient and regular isotopies 245 10.3.3 Knot polynomials from representations of the braid group 249 10.3.4 Intersecting knots 251 10.4 Topological field theories and knots 253 10.4.1 Chern-Simons theory and the skein relations of the Jones polynomial 254 10.4.2 Perturbative calculation and explicit expressions for the coefficients 260 10.5 States of quantum gravity in terms of knot polynomials 264 10.5.1 The Kauffman bracket as a solution of the constraints with cosmological constant 264 10.5.2 The Jones polynomial and a state with A = 0 265 10.5.3 The Gauss linking number as the key to the new so- lution 272 10.6 Conclusions 274 11 The extended loop representation of quantum gravity 275 11.1 Introduction 275 11.2 Wavefunctions 277 11.3 The constraints 280 11.3.1 The diffeomorphism constraint 281 11.3.2 The Hamiltonian constraint 283 11.4 Loops as a particular case 285 11.5 Solutions of the constraints 289 11.6 Regularization 291 11.6.1 The smoothness of the extended wavefunctions 292 11.6.2 The regularization of the constraints 294 11.7 Conclusions 300 12 Conclusions, present status and outlook 302 12.1 Gauge theories 302 12.2 Quantum gravity 304 References 309 Index 319

Foreword For about twenty years after its invention, quantum electrodynamics re- mained an isolated success in the sense that the underlying ideas seemed to apply only to the electromagnetic force. In particular, its techniques did not seem to be useful in dealing with weak and strong interactions. These interactions seemed to lie outside the scope of the framework of local quantum field theory and there was a wide-spread belief that the best way to handle them would be via a more general, abstract S-matrix theory. All this changed dramatically with the discovery that non-Abelian gauge theories were renormalizable. Once the power of tlie gauge princi- ple was fully recognized, local quantum field theory returned to the scene and, by now, dominates our thinking. Quantum gauge theories provide not only the most natural but also the only viable candidates we have for the description of electroweak and strong forces. The basic dynamical variables in these theories are represented by non-Abelian connections. Since all the gauge invariant information in a connection is contained in the Wilson loops variables (i.e., traces of holonomies), it is natural to try to bring them to the forefront. This is precisely what is done in the lattice approaches which are the most suc- cessful tools we have to probe the non-perturbative features of quantum gauge theories. In the continuum, there have also been several attempts to formulate the theory in terms of Wilson loops. In the perturbative approach, it is known that Wilson loop "Schwinger functions" are finite to all orders after renormalization. This is a strong indication that they may be also mathematically meaningful in a non-perturbative treatment of the continuum theory. Since these are functions on an appropriate space of loops, one can derive differential equations they satisfy on that loop space. The hope is that once a complete set of equations is obtained, physical "boundary conditions" will lead to unique solutions which in turn will determine the theory. Thus, the space of loops offers a natural arena for the quantum theories of connections.

xiv Foreword In the last few years, this relation between connections and loops has acquired another dimension. In these developments, the emphasis is on the Hamiltonian formulation. It turns out that there is a remarkable mathematical interplay between measures on the spaces of connections and functions on the loop space, which gives rise to a generalization of the Fourier transform, called the loop transform. This transform can be defined rigorously. As a result, quantum states can be regarded either as gauge invariant functions of connections or as suitable functions of loops. The loop picture suggests new strategies for defining operators and provides a number of new insights. Quite surprisingly, it turns out that these insights are especially useful while dealing with a force that one does not, normally, associate with theories of connections: gravity. General relativity is usually thought of as a theory of metrics and, therefore, quite removed from theories of other interactions. One can, however, think of it also as a dynamical the- ory of connections. This idea is not new. Indeed, such a reformulation was obtained already by Einstein and Schrodinger. In their new version, the Levi Civita connection is regarded as the basic variable; metric is a secondary, derived object. The problem was that the equations of the theory became more complicated. It turns out, however, that if one uses chiral connections in place of Levi Civita, the equations actually simplify. With this observation, general relativity moves closer to theories govern- ing other fundamental forces. As in other theories, one can now represent states of quantum gravity as functions of (chiral) connections or, via loop transform, of loops. Thus, the loop representation offers a unified arena for the quantum description of all four fundamental interactions. In the case of general relativity, further structures arise because physical states are required to be invariant under the action of diffeomorphisms. In the loop representation, they depend not on individual loops but also on the (generalized) knot to which the loop belongs. There is thus an unexpected interplay between loops, knots, gauge fields and gravity. This monograph is devoted to this interplay. The authors are eminently qualified to unfold this saga as they are among the leaders in the field. Indeed, many of the developments that I have alluded to are due to them and their close colleagues. They provide not only a comprehensive sum- mary of the entire subject, but, in the last few chapters, also a glimpse of two frontier areas of active research. Graduate students would find this unified treatment of a large subject extremely useful. More advanced researchers would be able to appreciate the fascinating confluence of ideas from particle physics general relativity and contemporary mathematics. University Paxk, 1996 Abhay Ashtekar

Preface Loops have been used as a tool to study classical and quantum Yang-Mills theory since the work of Mandelstam in the early 1960s. They have led to many insights concerning the non-perturbative dynamics of the theory including the issue of confinement and the lattice formulation. Since the inception of the Asthekar new variables, loop techniques have also found important applications in quantum gravity. Due to the diffeomorphism invariance of the theory they have led to surprising connections with knot theory and topological field theories. The intention in this book is to present several of these results in a com- mon framework and language. In particular it is an attempt to combine ideas developed some time ago in the context of Yang-Mills theories with the recent applications in quantum gravity. It should be emphasized that our treatment of Yang-Mills theories only covers a small part of all results obtained with loops: that which seems of most relevance for applications in gravity. This book should allow people from outside the field to gain access in a pedagogical way to the current state of the art. Moreover, it allows experts within this wide field with heterogeneous backgrounds to learn about specific results outside their main area of expertise and as a reference volume. It should be well suited as an introductory guide for graduate students who want to get started in the subject. Subjects in this area are being developed at two different levels: one more "mathematical", related to constructive quantum field theory and the other more "physical" in which several subtleties are ignored in order to gain rapid insight into the theory. This book will largely concentrate on this latter approach. We will present in some detail the mathematics underlying loops but only at the level needed for a physicist to operate with the resulting formalism. Due to the rapid development of these ideas in recent times we wrote

xvi Preface this book with expediency in mind in order to offer it to the public as soon as possible. We ask readers to forgive any notational and conven- tions errors that might have arisen in this process. In spite of our efforts to deliver the book expediently, during the time it took to write it the field has already evolved. Although we have kept the manuscript updated, some aspects of the presentation may not be completely in line with cur- rent thinking. For instance, the recently pinpointed convergence issues of extended loops and their possible solutions are only briefly mentioned. The rapid development over the last few months of a measure theory in infinite-dimensional spaces and its impact on a rigorous definition of a loop representation is only briefly discussed in chapter 3 and its implica- tions for later chapters are ignored. A similar remark applies to the recent developments concerning the solution of the Mandelstam constraints in terms of spin network states and the possibility of having a basis of in- dependent loop states. We have tried to present a current outlook from our perspective in chapter 12, including very recent references to work in progress. Many people have contributed to make this book possible. We cannot attempt to exhaustively list all the colleagues from which we have bene- fited through discussions and interactions. We like to thank the colleagues with which we have had a closer interaction and whose ideas have cer- tainly left an imprint on this book: Abhay Ashtekar, Bernd Briigmann, Cayetano Di Bartolo, Jorge Griego, Jerzy Lewandowski, Jose Mourao, Carlo Rovelli, Lee Smolin and Antoni Trias. We also wish to thank Ab- hay Ashtekar, John Baez, John Baker, Jose Mourao, Peter Peldan, Carlo Rovelli, Thorsten Schwander, Jose Antonio Zapata and especially Bernd Briigmann and Jerzy Lewandowski for many useful comments about the manuscript. This work was supported in part by grants NSF-INT-94-06269, NSF- PHY-94-06269, NSF-PHY93-96246, NSF-PHY-92-07225, by an Alfred P. Sloan fellowship to JP and by research funds of the University of Utah, The Pennsylvania State University, the Eberly Family research fund, the PSU Office for Minority Faculty Development, Conicyt and PEDECIBA. State College, 1996 Rodolfo Gambini, Jorge Pullin

1 Holonomies and the group of loops 1.1 Introduction In this chapter we will introduce holonomies and some associated con- cepts which will be important in the description of gauge theories to be presented in the following chapters. We will describe the group of loops and its infinitesimal generators, which will turn out to be a fundamental tool in describing gauge theories in the loop language. Connections and the associated concept of parallel transport play a key role in locally invariant field theories like Yang-Mills and general relativity. All the fundamental forces in nature that we know of may be described in terms of such fields. A connection allows us to compare points in neighboring fibers (vectors or group elements depending on the description of the particular theory) in an invariant form. If we know how to parallel transport an object along a curve, we can define the derivative of this object in the direction of the curve. On the other hand, given a notion of covariant derivative, one can immediately introduce a notion of parallel transport along any curve. For an arbitrary closed curve, the result of a parallel transport in general depends on the choice of the curve. To each closed curve 7 in the base manifold with origin at some point o the parallel transport will associate an element H of the Lie group G associated to the fiber bundle. The parallel transported element of the fiber is obtained from the original one by the action of the group element H. The path dependent object HG) is usually called the holonomy. It has been considered in various contexts in physics and given different names. For instance, it is known as the Wu-Yang phase factor in particle physics. Curvature is related to the failure of an element of the fiber to return to its original value when parallel transported along a small closed curve. When evaluated on an infinitesimal closed curve with basepoint o, the

2 1 Holonomies and the group of loops holonomy has the same information as the curvature at o. Knowledge of the holonomy for any closed curve with a base point o allows one, un- der very general hypotheses, to reconstruct the connection at any point of the base manifold up to a gauge transformation. An important fact about holonomies is their invariance under the set of gauge transforma- tions which act trivially at the base point. We will later show that this will imply that the physical configurations of any gauge theory can be faithfully and uniquely (up to transformations at the base point) repre- sented by their holonomies. They can therefore be used to encode all the kinematical information about the theory in question. Since the early 1960s several descriptions of gauge theories in terms of holonomies have been considered. They seem to be particularly well suited to study the non-perturbative features at the quantum level. In recent years interest in the non-local descriptions of gauge theories has been greatly increased by the introduction of a new set of canonical variables that allow one to describe the phase space of general relativity in a manner that resembles an SUB) Yang-Mills theory. In fact, holonomies may well provide a common geometrical framework for all the fundamental forces in nature A generalization of the notion of holonomy may be defined intrinsically without any reference to connections. It will turn out that this point of view has more than a purely mathematical interest and is the origin of im- portant results that are relevant to the physical applications. Holonomies can be viewed as homomorphisms from a group structure defined in terms of equivalence classes of closed curves onto a Lie group G. Each equiva- lence class of closed curves is what we will technically call a loop and the group structure defined by them is called the group of loops. The group of loops is the basic underlying structure of all the non-local formulations of gauge theories in terms of holonomies. In particular, when quantizing the theory, wavefunctions in the "loop representation" are really functions dependent on the elements of the group of loops*. This is the physical reason why it is important to understand the structure of the group of loops, since it is the "arena" where the quantum loop representation takes place. In spite of the fact that the group of loops is not a Lie group, it is pos- sible to define infinitesimal generators for it. When they are represented in the space of functions of loops, they give rise to differential operators in loop space. Some of these operators have appeared in various physical contexts and have been given diverse names such as "area derivative", * In this context the group of loops is usually referred to as "loop space" and we will loosely use this terminology when it does not give rise to ambiguities. Notice that it is not related to the "loop groups" in the main mathematical literature.

1.2 The group of loops 3 "keyboard derivative", "loop derivative". In most of these presentations the group properties of loops were largely ignored and this resulted in var- ious inconsistencies. In the approach we follow in this chapter all these operators arise simply and consistently as representations of the infinites- imal generators of the group of loops. In many presentations, loop space is formulated with parametrized curves. In this context differential operators are usually written in terms of functional derivatives. The group structure of loops is hidden by these formulations and it is easy to overlook it, again leading to inconsistencies. In this book we will deal with unparametrized loops which allow for a cleaner formulation, only resorting to parametrizations for some particu- lar results. This chapter is structured in the following way. In section 1.2 we de- fine the group of loops and discuss its topology and its action on open paths. In section 1.3 we introduce the infinitesimal generators of the group and their differential representation. We also introduce differential oper- ators acting on open paths. In section 1.3.3 we introduce the connection derivative, its relation to the loop derivative and to usual notions of gauge theory. In section 1.3.4 we discuss the contact and functional derivatives in loop space and their relations with diffeomorphisms. In section 1.4 we introduce the idea of representations of the group of loops in a Lie group and we retrieve the classical kinematics of gauge theories. We end with a summary of the ideas developed in this chapter. 1.2 The group of loops We start by considering a set of parametrized curves on a manifold M that are continuous and piecewise smooth. A curve p is a map p : [0, si] U [si, s2] ¦ ¦ ¦ [sn-u 1] -» M A.1) smooth in each closed interval [s^Sj+i] and continuous in the whole do- main. There is a natural composition of parametrized curves. Given two piecewise smooth curves p\ and p2 such that the end point of p\ is the same as the beginning point of p2, we denote by p\ op2 the curve: for a €[0,1/2] ( , fo, ae [1/2,1]. I1-2) The curve traversed in the opposite orientation ("opposite curve") is given by p-1(s):=p(l-s). A.3) In what follows, we will mainly be interested in unparametrized curves. We will therefore define an equivalence relation by identifying the curve

4 1 Holonomies and the group of loops p and p o <f> for all orientation preserving differentiable repaxametrizations <j) : [0,1] —¦ [0,1]. It is important to note that the composition of un- paxametrized curves is well defined and independent of the members of the equivalence classes used in their definition. We will now consider closed curves l,m,..., that is, curves which start and end at the same point o. We denote by Lo the set of all these closed curves. The set Lo is a semi-group under the composition law (/, m) —¦ lom. The identity element ("null curve") is defined to be the constant curve i(s) = o for any s and any parametrization. However, we do not have a group structure, since the opposite curve Z is not a group inverse in the sense that I ol'1 ^ i. Holonomies axe associated with the parallel transport around closed curves. In the case of a trivial bundle the connection is given by a Lie- algebra-valued one form Aa on M. The parallel transport around a closed curve I € Lo is a map from the fiber over o to itself given by the path ordered exponential (for the definition of path ordered exponential see reference [1]), J A.4) In the general case of a principal fiber bundle P(M, G) with group G over M the holonomy map is defined as follows. We choose a point 6 in the fiber over o and by using the connection A we lift the closed curve / in M to a curve I in P such that the beginning point is 1@) =6 A.5) and the end point is given by l(l)=i@)HA(l), A.6) which defines HaA)- The holonomy Ha is an element of the group G and the product denotes the right action of G. The main property of Ha is HA(lom) = HA(l)HA(m). A.7) A change in the choice of the point on the fiber over o replacing 6 for 6' = bg induces the transformation HA(l)=g-1HA(l)g. A.8) In order to transform the set Lo into a group, we need to introduce a further equivalence relation. The rationale for this relation is to try to identify all closed curves leading to the the same holonomy for all smooth connections, since curves with the same holonomy carry the same infor- mation towards building the physical quantities of the theory. The classes of equivalence under this relation axe what we will from now on call loops

1.2 The group of loops 5 and we will denote them with Greek letters, to distinguish them from the individual curves which form the equivalence classes. Several definitions of this equivalence relation have been proposed. Each of them sheds some light on the group structure so we will take a minute to consider them in some detail. Definition 1 Let HA:L0-*G A.9) be the holonomy map of a connection A defined on a bundle P(M,G). Two curves l,m € Lo are equivalent [2] [4] / ~ m iff HA(l) = HA(m) A.10) for every bundle P(M, G) and smooth connection A. Definition 2 We start by defining loops which are equivalent to the identity. A closed curve I is called a tree[5] or thin [6] if there exists a homotopy of / to the null curve in which the image of the homotopy is included in the image of I. This kind of curves does not "enclose any area" of M. Two closed curves I, m € Lo are equivalent / ~ m iff I om~1 is thin. Obviously a thin curve is equivalent to the null curve. Definition 3 [7] Given the closed curves / and m and three open curves pi, P2 and q such that l=Pi°P2 A-11) m = p\o qo q~~l ap2 A-12) then / ~ m. There is a fourth definition, due to Chen [7], that requires the use of a set of objects (Chen integrals, which we will call "loop multitangents") that we will define in chapter 2, but we will not discuss it here. It can be shown that definitions 2 and 3 are equivalent. Moreover, it is also immediate to notice that two curves equivalent under defini- tions 2 or 3 are also equivalent under definition 1. The reciprocal is not obvious. Partial results can be found in reference [7] and a complete proof for piecewise analytic curves has been presented by Ashtekar and Lewandowski [40]. With any of these definitions one can show that the composition be- tween loops is well defined and is again a loop. In other words if a = [I]

1 Holonomies and the group of loops O O o Fig. 1.1. Curves p and p' differ by a tree. The composition of a curve and its inverse is a tree. and C = [m] then a o C = [I o m] where by [] we denote the equiva- lence classes. From now on we will denote loops with greek letters, to distinguish them from curves^. Notice that with the equivalence relation defined, it makes sense to define an inverse of a loop. Since the composition of a curve with its opposite yields a tree (see figure 1.1) it is natural, given a loop a, to define its inverse a by a o a — t where t is the set of closed curves equivalent to the null curve (thin loops or trees). a~l is the set of curves opposite to the elements of a. We will denote the set of loops basepointed at o by Co. Under the composition law given by o this set is a non-Abelian group, which is called the group of loops. A well known result [5] is that any homomorphism, Co -* G, A.13) where G is a Lie group, defines a holonomy associated with a "general- ized" connection. By generalized we mean that the connection will not, in general, be a smooth function (for instance it could be distributional or worse). One can, by imposing extra smoothness conditions [6, 4] on the homomorphism, ensure that a differentiable principal fiber bundle and a connection are defined such that H is the holonomy of this connection. Recall that under a homomorphism, the composition law of the group of loops is mapped onto the composition law of the Lie group G, A.14) Notice that in this book we will use the word loop in a very precise sense, denoting the holonomic-equivalent classes of curves. Other equivalences can be considered. The idea of a group of loops has appeared in other unrelated contexts [42]. For this reason some authors have proposed calling the holonomic equivalence classes "hoops" to avoid confusion [3].

1.3 Infinitesimal generators of the group of loops 7 and that inverses are mapped to each other, H{cc~l) = (H(a))-\ A.15) We will come back to this property in section 1.4 when we discuss the infinitesimal generators and their relations to the physical quantities. Prom now on we will routinely use functions of loops, such as the holon- omy that we just introduced. Obviously, not any function of curves qual- ifies as a function of loops. An immediate example of this would be to consider the length of a curve, which takes different values on the different curves that form the equivalence class defining a loop. It is useful to introduce a notion of continuity in loop space, since we will be frequently using functions defined on this space. We will define two loops a and /3 to be close, in the sense that a in a neighborhood Ue(C) if there exist at least two parametrized curves a(s) € a and b(s) 6 C such that a(s) € Ut(b(s)) with the usual topology of curves in the manifold*. With this topology, the group of loops is a topological group. It is convenient for future use to introduce an equivalence relation for open curves similar to the one we introduced for closed curves. We will call the equivalence classes of open curves "paths". Given two open curves p% and <fo from the basepoint to a point x in the manifold, we will define these curves to be equivalent iff p%q~xx0 is a tree§. We will denote paths with Greek letters as we do for loops, but indicating the origin and end points, as in qJ. Given two different paths starting and ending at the same points, it is immediate to see that the composition of one with the opposite of the other is a loop. Analogously one can compose loops with paths to produce new paths with the same end points. Furthermore, the notion of topology introduced for loops can immediately be generalized to paths. However, paths cannot be structured into a group, since it is not possible to compose, in general, two paths to form a new path (the end of one of them has to coincide with the beginning of the other in order to do this). 1.3 Infinitesimal generators of the group of loops We will now consider a representation of the group of loops given by operators acting on continuous functions under the topology introduced in the previous section. We will introduce a set of differential operators Lewandowski [4], elaborating on a suggestion by Barrett [6] has introduced a topology denned in terms of homotopies of loops. The group of loops endowed with this topology is a topological Haussdorff group. From now on we will interchangeably use the notations g* and g° to designate the same object, the curve q traversed from a; too. A similar convention will be adopted for paths.

1 Holonomies and the group of loops n ¦^— o Fig. 1.2. The infinitesimal loop that defines the loop derivative. acting on these functions that are related to the infinitesimal generators of the group of loops, in terms of which one can construct the elements of the group. In later chapters we will show that these operators are related to physical quantities of gauge theories. Although the explicit introduction of the differential operators will be made in a coordinate chart, we will show that the definitions do not depend on the particular chart chosen. A more intrinsic definition, also making use of the properties of the group of loops has been proposed by Tavares [43]. 1.3.1 The loop derivative Given ^G) a continuous, complex-valued function of Co we want to con- sider its variation when the loop 7 is changed by the addition of an in- finitesimal loop Sj basepointed at a point x connected by a path tt% to the basepoint of 7, as shown in figure 1.2. That is, we want to evalu- ate the change in the function when changing its argument from 7 to 71-Q o 67 o n° o 7. In order to do this we will consider a two-parameter family of infinitesimal loops 67 that contain in a particular coordinate chart the curve obtained by traversing the vector ua from xa to xa + eiua, the vector va from xa + e\ua to xa + eiua + €2Va, the vector — ua from xa + e\ua + €2Va to xa + €2Va and the vector — va from xa + e2va back to xa as shown in figure 1.2. We will denote these kinds of curves with the notation' 6u6v6u6v. In order not to clutter the notation we will not distinguish between curves and paths here. We also drop the tj dependence of each path. The path Sii = Fu)~l.

1.3 Infinitesimal generators of the group of loops 9 For a given it and 7 a loop differentiable function depends only on the infinitesimal vectors eiua and €2Va. We will assume it has the following expansion with respect to them, 7) = +±e1e2(uavb + vaub)SabGrZ)*A) ^b 6 A-16) where Q,P,S,A are differential operators on the space of functions If ei or 62 vanishes or if u is collinear with v then 6j is a tree and all the terms of the right-hand side except the first one must vanish. This means that Q = P — S — 0. Since the antisymmetric combination (uavb — vaub) vanishes, A need not be zero. That is, a function is loop differentiable if for any path 7r? and vectors u,v, the effect of an infinitesimal deformation is completely contained in the path dependent antisymmetric operator 6j o < o 7) = A + laab(x)AabGrxo)M1), A.17) where aab{x) = 2eie2(it'a^) is the element of area of the infinitesimal loop Sj. We will call this operator the loop derivative. Notice that we have proved that for an arbitrary function of loop space, one does not have contributions from the terms Q,P,S in the expansion A.16). If one considers functions of curves rather than of loops, these terms will in general be present. As an example, they are present if one considers the function given by the length of the curve. On the other hand, not every function of loop space is differentiable. For instance, we will see when we consider knot invariants — functionals of loops invariant under smooth deformations of the loops — that they are not strictly speaking loop differentiable. The reason for this is that sometimes appending an infinitesimal loop could enable us to change the topology of the knots and therefore to induce finite changes in the values of the functions. Loop derivatives of various kinds were considered by several authors. The idea was introduced by Mandelstam [8]. Later generalizations can be found in the work of Chen [7], Makeenko and Migdal [10, 12], Polyakov [44], Gambini and Trias [13, 14, 15], Blencowe [16] and Brugmann and Pullin [26]. Other references can be found in Loll [17]. The various definitions are not equivalent, and many of them refer to objects that are in reality different from the loop derivative we are defining here. One of the main differences is that in many treatments the infinitesimal loop, instead of being appended at an arbitrary fixed point of the manifold defined by a path ir% as is our case, is appended to a point that lies on the loop. Since one is considering functions of arbitrary loops that means that the point where the derivative acts has to be redefined when

10 1 Holonomies and the group of loops considering its value on a new loop. In other words, the domain of the function that results when applying these kinds of derivatives is not the loop space defined in section 1.2, but the space of loops with a marked point. Makeenko and Migdal [12] noticed this fact and this drove them to call it "keyboard derivative." Notice that this is not the case for the derivative we defined. The result of the application of the loop derivative to a function of a loop is also a function of a loop. For each arbitrary open path there is a different derivative. For these definitions to work it is crucial to have a basepoint, which provides a fixed point for any loop on which to attach the open path that defines the derivative. These considerations are of crucial importance. For instance, we will soon prove that our derivative satisfies Bianchi identities, a fact that cannot be proven for derivatives that act only on points of the loop. The relevance of the group of loops and the path dependence of the loop derivative were first recognized by Gambini and Trias [13, 15]. At the end of section 1.2 we noted that the elements of the group of loops have a natural action on open paths, giving as a result a deformation of the path. We can immediately find an example of this fact in terms of a differential operator defined by simply extending the definition of the loop derivative A.17) to give for open paths n< ° h ° < ° 7?) = A + ^a6(z)Aa6Gr*))vI/G2). A.18) We will take some notational latitude to give the same name to the loop derivative acting on paths and on loops. In all cases the context will uniquely determine to which derivative we are referring. Notice that this extension to open paths is not at all clear for derivatives that depend on a point of the loop as is the case of the "keyboard derivative". 1.3.2 Properties of the loop derivative • Tensor character. By its very definition, A.17), it is immediate to see that the loop derivative has to behave as a tensor under local coordinate transformations containing the end point of the path 7r^ for loop differ- entiable functions. One need just require that the whole expression be invariant and notice that the loop derivative is contracted with the tensor aab. Therefore by quotient law, it must be a tensor. Notice that the loop derivative is really associated with the surface spanned by dua and dvb rather than with the individual infinitesimal vectors, being invariant under vector transformations that preserve the element of area. • Commutation relations. The loop derivatives are non-commutative operators. This, as we will see later, is naturally associated with the fact

1.3 Infinitesimal generators of the group of loops 11 Fig. 1.3. The two paths used to compute the commutation relation that they correspond to the generators of a non-Abelian group. Their commutation relations can be computed directly from the geometric prop- erties of the group of loops in the following way. Consider two infinitesimal loops 6r)i, 8r\2 given by 6r)i = -Kg o 6u6v6u6v o n° and 8r\2 — Xo ° 6q6r6q6f o^J A-19) and with area elements af = e1e2(uavb - vaub) and of = e3e4{qarb - raqb). A.20) Then we can derive the following relation: 1 o 7) = A + ± A.21) \af < The first equality follows from the definition of the loop derivative and of the loops 6r)i. To prove the second, one expands keeping only terms of first order in each ej and neglecting those of order ef. We will now define an open path by composing the two paths we have been using X>yo = 6vi°Xyo- A-22) This allows us to rewrite the loop composed by the first three loops in the argument of \I/ in the left-hand side of equation A.21) as, 8-qi o Sq6r6q6r o A.23)

12 1 Holonomies and the group of loops Therefore, o 6rj2 o (fS^i) o (S^y1 07) = A + &bAab(xZ))(l - ^Acd(xg))*G). A.24) And again expanding in es and keeping only the first order in each e* we get A.25) where in the last expression Aaj,Gr^)[AC(i(x^)] represents the action of the first loop derivative only on the path dependence of the second derivative. All this implies [Aab(ir*o),Acd(xyo)} = Acd(x^)[Aa6«)], A.26) from which it is immediate to show that *ab«)[*cdWo)] = -Acd(x^)[Aa6«)]. A.27) These expressions highlight the path dependence of the loop derivative, in the sense that they express the variation of the derivative when the path is varied. We will see at the end of this subsection how these expressions can be naturally interpreted as a group commutator when we prove that the loop derivative is a generator of the group of loops. This commutation relation can be viewed in a different light by consid- ering its integral expression. In order to do this, we will introduce a loop dependent operator U(a) on the space of functions of loops which has the effect of introducing a finite deformation in the argument of the function, U(a)^(j) = *(ao7). A.28) The operator has a naturally denned inverse, U{a)-1 = U{oTl), A.29) and has a natural composition law, U(a)U{l3)^{j) = U{a o /3)*G). A.30) We now consider the action of the loop derivative evaluated along a de- formed path, shown in figure 1.4, on a function of loop, and applying the definition of loop derivative A.17) we get A + \<jabAab{a o <))*G) = *(a o nxo o &y o < o a o 7), A.31) where ?7 is the infinitesimal loop associated with the area element aab. We then use the definition of the operator U A-28) to get 1), A.32)

1.3 Infinitesimal generators of the group of loops 13 Fig. 1.4. The deformed path used to derive the integral expression of the commutation relations from which we can read off the identity, Aab(a o <) = U{a)kab{Kxo)U{a)-\ A.33) which expresses the transformation property of the loop derivative under finite deformations of its path dependence. We will see at the end of this chapter that this expression is the reflection in the language of loops of the gauge covariance of the field tensor in a gauge theory. • Bianchi identities. There is a second set of relations that again can be directly obtained from the geometric properties of the group of loops. One can readily see that they are a reflection of the usual Bianchi identities of Yang-Mills theories. In order to describe them we need to introduce a new differential operator, which we will call the end point derivative or Mandelstam covariant derivative [8], that acts on functions of open paths. Given a function of an open path ^{it*), a local coordinate chart at the point x and a vector in that chart ua, we define the Mandelstam derivative by considering the change in the function when the path is extended from x to x + eu by the infinitesimal path 6u shown in figure 1.5 as *« o 6u) = A + euaDa)^(^). A.34) We denote the new path as 7r^+tu. If one performs a coordinate trans- formation, noting that eua is a vector and applying the quotient law, it is immediate to see that Da transforms as a one-form. Having introduced this operator, we are now ready to derive the Bianchi identity. As usual, the fundamental idea is that "the boundary of a bound- ary vanishes". In the group of loops language, this can be expressed by considering a thin loop i. A representative curve of this loop has the shape of a box with sides 6u, 8v and 6w, connected to the origin by the path -Kg as shown in figure 1.6.

14 1 Holonomies and the group of loops Fig. 1.5. The extended path defining the Mandelstam derivative 7/ x - — X 7« Fig. 1.6. The loop used to derive the Bianchi identity for the loop derivative. The curve 8u represents paths that go from a generic point x to x + e\u and similarly for 8v and 8w with increments e^v and 6311; respectively. Explicitly, i = tt^ o 8u6v6w5v6w6u o ir° o 71^ o 6u6w6u6w o 7r° O7r^ o 6w6u6v6u6v6w o 7r° o 7r^ o 6w6v6w6v o tt° o7To o 8v8w8u8w8u8v o 7r° o 7r^ o 8v8u8v8u o 7r°. A.35) Now, since t is a tree and therefore indistinguishable from the identity loop, we have = #(t o 7) V7. A.36) Noting that the tree is built by six infinitesimal loops (the "faces" of the box shown in figure 1.6), each connected to the origin via the path ir and the "sides" of the box, we can rewrite this identity in terms of loop derivatives as

1.3 Infinitesimal generators of the group of loops 15 x A + eie2ne^Ae/«+f3-))(l + e3e2w<>vhAgh(irx)) x A + e3eiti,VAy «+«w))(l + e2e1vkulAklGrxo))^A).A.37) Collecting the terms of first order in each ej, applying the definition of the Mandelstam derivative and noting that u, v,w are arbitrary we get the final form of the Bianchi identities for the loop derivatives, Da&bc{<) + DbAca(nx) + DcAab(nx) = 0. A.38) We will soon see applications of the above derived identities and their crucial role in the formulation of gauge theories in terms of loops. • The Ricci identity. Consider the action of four Mandelstam covariant derivatives along the vectors u, v on a function of an open path ^(-Kg). Keeping the terms of first order in 6162 in the left-hand side of the next expression we get A + elUaDa){l + e2vbDb)(l - elUcDc)(l - e2vdDd)*(irxo) = av b[Da, Db])^(nx). A.39) The action of the four covariant derivatives is equivalent to appending an infinitesimal loop at the end of the path ir% and therefore can be written in terms of the loop derivative, [Da, Db]^(irx) = Aab(irx)^(nxo). A.40) This expression is the loop analogue of the usual expression of the com- mutator of covariant derivatives in terms of the curvature and we will again see its implications for gauge theories at the end of this chapter. • The loop derivative as a generator of the group of loops. Let us now show that we can, by superposition of loop derivatives, gener- ate any finite loop homotopic to the identity. We need to introduce a parametrization for this proof. Let 7E) be a parametrized curve be- longing to the equivalence class defining the finite loop 7 with s € [0,1]. Consider a one-parameter family of parametrized loops r)(s,t) interpolat- ing smoothly between 7E) and the identity loop, such that rj(s, 0) is in the equivalence class of the identity loop and r)(s, 1) = 7(s)- Consider the curves tjA,s) (= 7E)) and rj(l — e,s). The two curves are drawn in figure 1.7 and differ by an infinitesimal element of area. The whole purpose of our proof will be to cover the infinitesimal area separating the two mentioned curves with a "checkerboard" of infinitesimal closed curves

16 1 Holonomies and the group of loops Fig. 1.7. The construction of a finite loop from the loop derivative. The curves 8r)i are determined by two elements in the family r)(t). such that along each of them one can define a loop derivative. One can therefore express the curve 7E) as A.41) 7E) = lim T](s, 1 — e) o 6m o n—»oo where the 6rji are shown in figure 1.7. Analytically, in terms of differential operators on functions of loops we can write'l ?(,,A))= tffa(l-e)) /•1 +e <p dsrfil — e, s)r/ A — e, s)Aaj,(r/(l — e)*)^f(r/(l — e)), Jo A.42) where r)(?, s) = drj(t,s)/ds and r)'(t,s) = dr)(t,s)/dt It is immediate to proceed from r/(l — e, s) inwards just by repeating the same construction, and so continuing until the final curve is the identity. The end result is = Texv([1dt f dsr,a(t, s)r,'b(t, s)Aab(r,(tyo)y(r,(Q))) A.43) \Jo Jo / where the outer integral is ordered in t (T-ordered). This result is the loop version of the non-Abelian Stokes theorem of gauge theories [18] and it shows that the loop derivative is a generator of loop space, i.e., it allows us to generate any finite loop homotopic to the identity. Notice that the expression for the finite element of the group involves a superposition of an infinite number of generators associated with different ' We drop the * dependence of T) where it is not relevant.

1.3 Infinitesimal generators of the group of loops 17 paths. This is not the usual situation that one encounters in Lie groups, where it is enough to exponentiate one generator to obtain any element of the group. This is another indication of the non-Lie character of the group of loops. It is a direct consequence of the impossibility of denning a non-integer number of powers of 6rj. Finally, identifying the loop derivative as a generator of the group of loops allows us to rewrite in a revealing form the commutation relations of the loop derivatives. Applying the definition of the loop derivative, equation A.26) can be cast in the following way [Aa6«), **,(*»)] = lim —(A^! o xvo) - AcOdfl), A.44) €{—»U (J which is the usual expression of the commutator in terms of a linear combination of elements of the algebra. So we see that the group of loops formally obeys commutation relations similar to those of a Lie group. 1.3.3 Connection derivative In section 1.3.1 we introduced the loop derivative. We saw in the previous section that this operator has several properties resembling those of the curvature or field tensor of a gauge theory. We will now introduce a differ- ential operator with properties similar to those of the connection or vector potential of a gauge theory [14, 15, 4]. This operator appears naturally as an intermediate step in the construction of gauge theories from the group of loops. Although one could formulate a gauge theory completely in terms of the path dependent loop derivative alone, the treatment that we will follow will lead us to a more familiar formulation of gauge theories. Let us consider a covering of the manifold with overlapping coordinate patches. We attach to each coordinate patch V1 a path 7To° going from the origin of the loop to a point j/q in V1. We also introduce a continuous function with support on the points of the chart V1 such that it associates to each point x on the patch a path nXi. Given a vector it at a;, the connection derivative of a continuous function of a loop ^G) will be obtained by considering the deformation of the loop given by the path 7To° o irXi o 6u o n^°+(U o -K°i shown in figure 1.8. The path 6u goes from x to x + eu. We will say that the connection derivative 6a exists and is well denned if the loop dependent function of the deformed loop admits an expansion in terms of eua given by #« o 6u o n°x+(U o 7) = A + eu^aWM-y), A-45) where we have written ir% to denote the path iTo o irXi and similarly for its inverse.

18 1 Holonomies and the group of loops Fig. 1.8. The path that defines the connection derivative The definition of the connection derivative can be immediately extended to act on functions of open paths, in a similar way to that used for the loop derivative. We will take some notational latitude to give the same name to the connection derivative acting on functions of paths or loops. In all cases the context will uniquely determine to which derivative we are referring. Notice that the deformation introduced in order to define the connec- tion derivative could have been generated by application of successive loop derivatives, as seen in the non-Abelian Stokes theorem. This implies that any function that is loop differentiable should be connection differ- entiable, and that there is a natural relation between the two derivatives. We will now prove the converse relation between the connection and loop derivatives. It will be quite reminiscent of the well known relation be- tween the connection and the curvature (or vector potential and field) in a gauge theory. In order to do this, let us start by considering the following identity in loop space, ?7 = iTg o SuSvSuSv o tt° = ITg O 8U O TT°+eu O TTg+eU O 6v o 6U O <+e2X+e2" ° SO O <, A.46) corresponding to the path shown in figure 1.9. Notice that the first defi- nition is written in such a way that it has the structure of the paths we have used to define loop derivatives, whereas the second has the form of the paths used to define connection derivatives. Specifically, it implies the following identity between differential operators, 6 elUa6a(x))(l + e2v%(x + ei«)) + e2v))(l - e2vd6d(x + e2«))*G). A-47)

1.3 Infinitesimal generators of the group of loops 19 Fig. 1.9. The path that defines the relation between connection derivatives and loop derivatives. We now expand to first order in eiC2 and get Aa6«) = da6b(x) - db6a(x) + [Sa(x),6b(x)]. A.48) Notice that we have obtained the loop derivative for the path irx given by 7i"o° o irXi. This path is uniquely prescribed by the function defining the J'o connection derivative. The loop derivative defined by A-48) automatically satisfies the Bianchi identities due to the fact that it is obtained by a construction in loop space that is totally similar to that used to derive the identities themselves. The idea of introducing the connection derivative was to provide us in the language of loops with a notion of connection or vector potential similar to that of gauge theories. However, the connection in a gauge theory is gauge dependent. How does this dependence manifest itself in the language of loops? It does so through the choice of prescription of path used to compute the connection derivative. We will now study this in some detail. Let us consider a connection derivative at a point x and two prescrip- tions for choosing the path from the origin to x, given by the continuous functions nx — f(x) and Xo = 9(x)i ^ shown in figure 1.10. We again consider two equivalent paths in the group of loops, Xo ° Su o x°x+eu = Xxoon°onxoo8uo n°x+eu o <+™ o X°+?U. A-49) We also introduce a point dependent operator U{x) constructed from the loop dependent deformation operator U^) defined in A.28) and the loop

20 1 Holonomies and the group of loops Fig. 1.10. The path dependence of the connection derivative. associated with the point x, Xo ° nxi by U(x) = U{x% ° 7r?)- This gives the identity between operators: = U(x)(l + eua6^>{x))U~l(x + eu)*G). A.50) Prom which we can immediately compute the change in the connection derivative due to a change in the prescription of the path, and we see that it is totally analogous to the transformation law for a gauge connection under changes of gauge. The usual relation between connections and holonomies in a local chart in a gauge theory can also be written in this language. This relation is just an expression of the fact that the infinitesimal generators associated with connections allow us to construct finite loops. As shown in figure 1.11 one uses infinitesimal increments generated by the connection derivative to build a loop. The expression of this fact in loop space is 7 = lim ?71 o ?72 o • • • o S'-fn, A-52) n—*oo where 6ji = ^o'^i^Xi+eui^ an(^ *n^s relation connects the deformation operator with the connection derivative, C/G) = lim A + (x2 — ^l^aOsi)) k-*oo x(l + (#3 — a;2)a<^a(^2)) •••(]. + (x\ — Xk)aSa(x)). A.53) We can rewrite the above expression as the following path ordered exponential, / r \ A.54) This again is reminiscent of the familiar expression for gauge theories, which yields the holonomy in terms of the path ordered exponential of a

1.3 Infinitesimal generators of the group of loops 21 Fig. 1.11. How to generate a finite loop using the infinitesimal generators. connection. 1.3.4 Contact and functional derivatives The differential operators that we have introduced up to now axe char- acteristic of the group structure of the group of loops and find no direct analogues in the space of parametrized curves. It is worthwhile analyz- ing whether there is any relation between these operators and the usual functional derivative 6/6p(s) acting on functionals of parametrized curves \I/[p(s)]. The operator in loop space that allows us to make this connec- tion is the contact derivative Ca{x), which plays an important role in diffeomorphism invariant theories, as we will see later. We define the contact derivative of a function of loops in terms of the expression Ca(xM-y) = I dybS(x - i/)Aa6(-y«f)*G), A.55) where 7^ is the portion of the loop 7 going from the basepoint to y. This operator was first introduced in the chiral formulation of Yang-Mills theories in the loop representation by Gambini and Trias [13]. It can be considered as the projection of the loop derivative on the tangent to the loop 7, Xa(x,j)=<fdyaS(x-y). A.56)

22 1 Holonomies and the group of loops This expression involving the tangent to the loop will play a role in the ideas of chapter 2. An important property of the contact derivative is that it is the gen- erator of diffeomorphisms on functions of loops. Given the infinitesimal diffeomorphism x—>x'a = xa + eua, A.57) the expression for ^G^), where j'e is the loop obtained by "dragging along" 7 with the diffeomorphism A.57), is given by 6) = (l+eJd3xua(x)Ca(xfj *G) ) A.58) j To prove that Ca actually is a generator of diffeomorphisms, we will show that it satisfies the corresponding algebra, d3xNa(x)Ca(x), I d3yMa(y)Ca(y)j = J dzxCfkNa{x)Ca{x), A.59) where Na, Ma are arbitrary vector fields on the three-manifold. We will start with an auxiliary calculation that will prove useful in what follows. We will evaluate the action of the loop derivative on a contact derivative. To this end we construct the following expression, which holds due to the very definition of loop derivative, ^ I dyaS(y - ar)Aa6 dya8{y - x)Aab{{S-fz ° l)y0)*{hz ° 7)- A-60) z °7 In this expression, 6jz is the infinitesimal loop added to 7 through a path from the origin up to the point z. That is, we evaluate the action of an infinitesimal deformation of area ocd acting on the contact derivative. We now expand the right-hand side of A.60), partitioning the domain of integration into the portions after and before the action of the deformation and use the definition of the loop derivative to expand ^(^7 o 7), f Jo +{ua6(z - z)Ao6G*) + va6(z + u - x)Aab(-yz0+u) -ua6(z + u + v- x)Aab(jz0+u+v) - va6(z + u- x)Aab(-yz0+u+v+a)} dyaS(y - x)Aab{{8lz o 7)g)(l + i^AcdG^))*G). A.61)

1.3 Infinitesimal generators of the group of loops 23 The last term in this expression can be rewritten as dy*6{x - y)@{y - z){\ + \acd A A.62) where @(y — z) is a Heaviside function that orders points along the loop, i.e., it is 1 if z precedes y and zero otherwise. We will be able to combine the zeroth order contribution of this term with the first term in A.61). It should be noticed that the first loop derivative does not act on everything to its right but only on the path inside the second loop derivative 7*f, a fact that as before we denote by enclosing it in brackets. We now consider the expansion of the terms containing the infinitesimally shifted loop. They can be expressed with the use of the Mandelstam derivative, Aa6Go+") = A + ucDc)Aab(jz0), A.63) Aa6Go+1x+") = A + vdDd){\ + ucDc)Aab(Y0), A-64) 6{z + u - x) = A + uada)S(z - x). A.65) We now expand A.61) again / dya6(y - aOA^Xl + \°cdAcdG*))\I>G) +{uaS(z - x)Aab(-yz0) + va(l + u*deN(z - x)(l + nd?>d)Aa6Goz) -na(l + vcDcN(z - x)(l + vdDd)AabAz) - vaS(z - x)Aab(jz)} dya6(y-x)@(y-z)acd l A.66) Of the terms in braces, it can be readily seen that only contributions proportional to uavb are present, neglecting terms of higher order. The other terms combine to give the original expression. We can finally read off the contribution of the loop derivative, dya6(x - 2(d[c6(z - x)Ad]bAz0) + 6{z - + <f dyaS(y - zN(y - x)AC(i + I dya6(y - x) Aa6G?) AcdG^)*G). A.67) Ji With this calculation in hand, it is straightforward to compute the successive action of two diffeomorphisms,

24 1 Holonomies and the group of loops C{N) C{M) *G)= /d3wNd(w) I dzc6(w - z)Acd{^0) x J d3xMb(x) j dya6(y - ar)Aa6G»)*G). A-68) Expanding this expression we get six terms f d3w Jd3xNd{w)Mb{x) x{<f dzc6(w - z)dc6(z - x)Adb(>f0)*(>y) Ji - <j> dz°S(w - z)dd6(z - x t dz°S{w - z)8{z - x)L - j dzc6(w - zN(z - x)DdAcbAz)<HA) + 1 dzc I dya6(w - z)S(y - x)S(y - z)l r r + <b dzc <f> dyaS(w - z)S(y - x)AabM)AcdHz)^H)\. A.69) Ji Ji We should now subtract the same terms with the replacement N +-* M. Since the calculation is tedious but straightforward we describe in words how the terms combine. The fifth and sixth terms, when combined with the similar terms coming from the substitution N +-* M cancel taking into account the commutation relations for the loop derivatives A.26). The first and third terms combined with the first of the substitution N +-* M form a total derivative. The fourth term, combined with the third and fourth of the substitution N <-> M cancel due to the Bianchi identities of the loop derivatives. Finally, the second terms combine to produce exactly C(C^M), which is the correct result of the calculation. We end by pointing out the relation between the contact derivative and the usual functional derivative. This can be immediately recognized by noticing that we can write J d3xua(x)Ca(x) in terms of parametrized curves by d3xua(x)Ca(x) = <p dsua{p(s))c . ,, A.70) Jo bpa(s) where p(s) is one of the parametrized curves in the equivalence class of the loop 7. The way to see this is to notice that if 7 is the equivalence

1.4 Representations of the group of loops 25 class of curves \pa{s)}, then #G') = *[pa(s) +eua(p(s))]. Then, -p(s))-^, A.71) which relates the two derivatives. It should be noticed that these two derivatives only agree when acting on functions of loops. The expression on the right of A.70) can act on functions of parametrized curves, on which the contact derivative is not defined. 1.4 Representations of the group of loops In previous sections we derived several relations between generators of the group of loops. These relations were independent of any particular representation. We will now study their form in the context of a particular representation in terms of a given gauge group. We will see that from them emerges the kinematical structure of gauge theories. Gauge theories arise as representations (homomorphisms H.) of the group of loops onto some gauge group G, H:?0^G, A.72) i.e., 7 —> tf G), A.73) such that #G1)^G2) = #G1 072). Let us assume we are considering a specific Lie group, for instance SU{N), with N2 - 1 generators Xi such that TrXi = 0 and [Xi,Xj] = CijXk, A.74) where ClJ are the structure constants of the group in question. We will assume that the representation is loop differentiable**. This will enable us to obtain the usual local objects associated with the gauge theory (curvature and connection) from the loop language. Let us compute the action of the connection derivative in this repre- sentation. We use the same prescriptions as in the previous section A + eua?a(z))#G) = H(nx0 o Su o n°x+eu o 7) = H(nx o Su o <+6Jtf G). A.75) Since the loop irx°SuoTr°+eu is close to the identity loop (with the topology of loop space) and since H is a continuous, differentiable representation, H(ttx o Su o Tr°+eu) = 1 + ieuaAa(x), A.76) ** If it is not loop differentiable, instead of dealing with holonomies we will have "generalized" holonomies, which are not derived from a smooth connection.

26 1 Holonomies and the group of loops where Aa{x) is an element of the algebra of the group, in our example of SU{N). That is, Aa(x) = A\Xl. Therefore, we see that through the action of the connection derivative, 6a(x)H(j) = iAa(x)H(j). A.77) Following similar steps one obtains the action of the loop derivative, Aa6«)tf G) = iFab(x)H(j), A.78) where Fab is an algebra-valued antisymmetric tensor field. Remember that 7Tq is only fixed by the prescription. Changing the prescription for ir% is the way to change the gauge. Suppose we change the prescription by tt* —> tt'* = -k'xo ott^ott;!;. We then simply use equation A.33) that in terms of the field reads F'ab{x) = H(x)Fab(x)H(x)-\ A.79) where H(x) is a shorthand for H{k'xo o n°). Prom equation A.48) we immediately get the usual relation defining the curvature in terms of the potential, Fab{x) = daAb{x) - dbAa{x) + i[Aa, Ab]. A.80) Gauge transformations in terms of the connection are immediate from equation A.51), Aa(x)' = H{x)Aa{x)H{x)-1 - iH{x)daH{x)-1. A.81) Let us act with the deformation operator U{rj) introduced in section 1.3.3 on the representation H(j), U{n)H(-y) = H(r) o 7) = HirfiHi-y). A.82) Now, applying formula A.54), = Pexp (Jdya8a{y)} H(j), A.83) and substituting equation A.77) and comparing terms we get H(v) = P exp (i j> dyaAa{yfj , A.84) which shows that the representation we are considering is given in terms of the usual expression for the holonomy of the connection Aa. Up to now, open paths have not played any relevant physical role. We will now show that open paths are naturally related to the inclusion of material fields coupled to gauge theories. We will consider matter fields that transform under the fundamental representation of the gauge group considered in our example, SU(N).

1.5 Conclusions 27 We will describe the matter field at the point x through a path depen- dent object *&(ir%). The natural extension of the representation introduced at the beginning of this section to the case of open paths is to consider the composition of an open path and a loop, defined by ttGO7?) = #G)?«). A-85) As in other cases, the role of the path choice will be to fix a gauge choice. A local description in a fixed gauge is obtained by fixing a family of paths, each of which are associated with each point in the manifold. The functions ^ will now become functions of points labeled by the fixed prescription tt used to determine the paths tyW (x). The prescription is given through a continuous function from the points of the manifold into the paths f(x) = nx. Notice that if we change the prescription for the path nx —> n'x = tt'x on°ong we get A.86) The Mandelstam derivative Da behaves as the usual covariant derivative of a gauge theory. Consider its action on a function of an open path, A + euaDa)V(TTx0) = *« o 6u) = *« o 8u o n°x+eu o nx+iU) = A + eua6a{x)){l + eubdb)V{nx), A.87) expanding in e and keeping terms of first order, we get W x). A.88) Using the relation between the function of a deformed path and the holon- omy A.85), we get \x). A.89) The usual form of the Ricci identity, [Da,Da] = iFab, A.90) can be obtained directly from the above expression or by considering the representation of equation A.40). 1.5 Conclusions We have seen how gauge theories arise as representations of the group of loops. All the usual kinematical concepts of gauge theories are reflections of properties of the group of loops. It is important to realize that the identities and properties that we proved in this chapter for the loop and connection derivative do not de- pend on any choice of gauge group to represent the group of loops. In this

28 1 Holonomies and the group of loops sense one can think of the corresponding generators of the group as asso- ciated with an "abstract" curvature and connection. It is only when one considers a particular representation of the group of loops in terms of a gauge group that these quantities adopt the usual meaning of connections and curvatures in gauge theories. In the next chapter we will introduce more techniques that will put us in a better position to deal with loops in the context of quantum theories in the loop representation.

Loop coordinates and the extended group of loops 2.1 Introduction Continuing with the idea of describing gauge theories in terms of loops, we will now introduce a set of techniques that will aid us in the descrip- tion of loops themselves. The idea is to represent loops with a set of ob- jects that are more amenable to the development of analytical techniques. The advantages of this are many: whereas there is limited experience in dealing with functions of loops, there is a significant machinery to deal with analytic functions. They even present advantages for treatment with computer algebra. Surprisingly, we will see that the end result goes quite beyond our expectations. The quantities we originally introduced to describe loops immediately reveal themselves as having great potential to replace loops altogether from the formulation and go beyond, allowing the development of a reformulation of gauge theories that is entirely new. This formulation introduces new perspectives with respect to the loop formulation that have not been fully developed yet, though we will see in later chapters some applications to gauge theories and gravitation. The plan for the chapter is as follows: in section 2.2 we will start by introducing a set of tensorial objects that embody all the information that is needed from a loop to construct the holonomy and therefore to reconstruct any quantity of physical relevance for a gauge theory. In sec- tion 2.3 we will show how the group of loops is a subgroup of a Lie group with an associated Lie algebra, the extended loop group. The generators of this Lie group will turn out to be coordinates in the extended loop space, which we discuss in section 2.4. In section 2.5 we will study how the differential operators introduced in the previous chapter act on the loop coordinates. In particular we will study the action of the genera- tor of diffeomorphisms. In section 2.6 we will discuss how to construct

30 2 Loop coordinates and the extended group of loops diffeomorphism invariant quantities in terms of loop coordinates and, in particular, knot invariants. In the conclusion we will discuss the differ- ences and similarities between the group structures we have introduced and the usual Lie groups. The subject of this chapter has been discussed in detail in reference [20], the reader is referred to it for a more technical approach. 2.2 Multitangent fields as description of loops As we discussed in the previous chapter, all the gauge invariant informa- tion present in a gauge field can be retrieved from the holonomy. There- fore the only information we really need to know from loops is that used in the definition of the holonomy, HA{i) = P exp [i j Aa dya^j . B.1) We can write this definition more explicitly as n=l B-2) where the loop dependent objects X are given by r rVn ri/2 f dy? / dy^1 ... dy^S(xn -yn)... S(Xl - Vl) = J7 Jo Jo xn - yn)... S(xi - yiH7(o, yi,...,Un) B-3) and O7(o,yi,... ,yn) is a generalized Heaviside function that orders the points along the contour starting at the origin of the loop, i.e., 7/1- / ! ifo < 2/1 <y2<---yn along the loop ..., yn) - | These relations define the X objects of "rank" n. We shall call them the multitangents of the loop 7. By writing the holonomy in the non-standard form B.2) we have been able to isolate all the loop dependent information in the multitangents of the loop. No more information from the loop is needed in order to com- pute the holonomy than that present in the multitangents of all orders. In what follows, it will be convenient to introduce the notation 1-a»*»G) =Xai-a»(xl,...,xn,'Y) , B.5)

2.2 Multitangent fields as description of loops' 31 with m = (aiXi), which is more suggestive of the role played by the x variables under diffeomorphisms. The X objects transform as multivector densities (they behave as a vector density at the point X{ on the index aj) under the subgroup of coordinate transformations that leaves the base point o fixed. In other words if xa _¦ x'a = Da(x) B.6) then B.7) where J is the Jacobian of the transformation. The Xs are not really "coordinates" in the sense that they are not independent. They are constrained by algebraic and differential relations. The algebraic constraints stem from relations satisfied by the general- ized Heaviside function, 67@,2/1,2/2,3/3) + 67@,2/2,2/1,2/3) + 67@,2/2,2/3,2/1) = 67(o,2/2,2/3), 67@,2/1) = 1, 67(o, yi,y2) + 67@,2/2, yi) = 1, B.8) which imply the following kind of relations among the Xs, B 9) And in general, y^^'() B.10) where the sum goes over all the permutations of the \i variables which preserve the ordering of the /ii,..., /i* and the fik+i, ¦ ¦ ¦ ,fJ-n among them- selves. We have introduced the notation of underlined indices to symbolize the permutation for future use. The differential constraint ensures that the holonomy has the correct transformation properties under gauge transformations, and can be read- ily derived from equation B.2). It is given by va\X\...aiXi ...anxn _ dxfX F(xi-xi-1)-6(xi-xi+i))XaiXl-ai-lXi-iai+lXi+1-anXn. B.11) In this expression, both xq and a;n+i represent the base point of the loop. An important property of the differential constraint is that any multi- tensor density j)a\xi...anxn t]iat satisfies it can be put into equation B.2) and the resulting object is a gauge covariant quantity. When restricted to

32 2 Loop coordinates and the extended group of loops the multitangents of a loop, the resulting object is the holonomy. It is this property that exhibits the relevance of this formulation. In it, loops are only a particular case. One can, in general, deal with arbitrary multiten- sor densities and construct gauge invariant objects, for instance by taking the trace. The multitensor densities need not have the same distributional character as the multitangents associated with a loop. Their divergence structure is dictated by the differential constraint, which requires its so- lutions to be distributional. This will have important consequences later. We will call the space of all multitensors that satisfy the differential con- straints Vo. With this construction in hand, one could go further and forget loops and holonomies altogether. Since one can represent any gauge covariant object using the Ds, one could represent a gauge theory entirely in terms of Ds. This has not been done up to present for non-Abelian theories in a complete fashion (nor for gravity), but it can be easily worked out for an Abelian theory, as we will do in chapter 4. When one allows arbitrary multitensors in B.2) the convergence of the series is not guaranteed. There is no easy way to prescribe multitensors such that the series converges, so we will assume from now on that we work only with multitensors such that the series converges. Even this requirement is not enough to produce an object with a gauge invariant trace. The differential constraint B.11) only ensures that if one performs a gauge transformation on the trace of the holonomy of a multitensor the resulting series has terms that cancel in pairs. For this to imply gauge invariance, it has to happen that [222] 4.1 • • • A,k[A, AUA,k+1... A,nX^ •••'* B.12) goes to zero as N —> oo. A is the parameter of the gauge transformation and is therefore an arbitrary function. Notice that the vanishing of B.12) is not guaranteed by the convergence of the holonomy alone. The question of selecting an appropriate set of multitensors in a precise way in order to ensure convergence of these expressions is at present not settled, see reference [21]. 2.3 The extended group of loops When we introduced the group of loops in the previous chapter, we no- ticed that no one-parameter subgroup existed (since one could only define integer powers of the generators) and therefore it did not form a Lie group. In this section we will introduce a Lie group, the "extended loop group". The group of loops will be a subgroup of it. This construction is of in-

2.3 The extended group of loops 33 terest in itself, since it is clear that it is a great advantage to have at our disposal all the machinery of Lie groups to analyze loops. Among other results, by identifying the free parameters of the algebra associated with the extended loop group we will be able to solve automatically the homo- geneous part of the differential and algebraic constraints B.10), B.11) of section 2.2. With some additional construction, we will have a definition for the portion of the multitensor density fields that is unconstrained, i.e., that we can freely specify. They can therefore genuinely be called "coor- dinates" and contain as a subspace the "loop coordinates" or coordinates on loop space. We will elaborate more on this concept in section 2.4. Now we will proceed to construct the extended loop group. 2.3.1 The special extended group of loops Let us start by considering arbitrary* multitensor densities similar to those introduced in section 2.2 and define a quantity E by E = (E,E/Xl,...,E'xl-'Xn,...) = (E,E), B.13) where E is a real number and E^1'"''1*" (for any n ^ 0) is an arbitrary multivector density field. It can be readily checked that the set of these quantities has the structure of a vector space (denoted as ?) with the usual composition laws of addition and multiplication. We will now introduce a product law in ? in the following way: given two vectors Ei and E2, we define Ei x E2 as the vector with components Ei x E2 = {E1E2, EiE2 + EiE2 + Ei x E2), B.14) where E\ x E2 is given by n-l (Ei x j^)-"" = Yl E?-k E%i+1-»n . B.15) t=l For any value of n, the rank n component of the x-product of elements of ? can be expressed as n x E2)-"" = Y,E\l'"'li E%i+1-"n B.16) *=o with the convention * In this chapter we will always discuss real multitensor fields. It is obvious that the formal- ism is unchanged if one allows complex fields. In some applications they seem to play an important role, as we will see in section 3.4.2 (see also [19]).

34 2 Loop coordinates and the extended group of loops The product law is associative and distributive with respect to the addition of vectors. It has a null element (the null vector) and an identity element, given by I = (l,0, ...,0, ...) . B.18) An inverse element exists for all vectors with non-vanishing zeroth rank component. It is given by oo E = E~l\ + ^(-lJVE-'-^E - Ely, B.19) such that E x E = E x E = I . B.20) When evaluating the components of E it should be noticed that the sum involved in B.19) is actually finite due to the fact that (E — El) is a vector with its zeroth rank component equal to zero. Therefore, [Ex.!. xEf1-^ = [(E - EiyY1*1 = 0 if n<i . B.21) The set of all vectors with non-vanishing zeroth rank component (notice the role of E~l in equation B.20)) forms a group with the x-product as composition law. The x-product law has an interesting property when restricted to mul- titangents. In this case it just corresponds to the composition law of loops, XGi)xXG2)=XGlo72), B.22) where X(j) = A,X^1G),... ,Xixl-iXn('y),...). Therefore we see that the product law that gave rise to the group of loops is the same product law we are generalizing to the case of arbitrary multitensor fields. The x-product law can also represent more general compositions than those of two loops sharing a common basepoint, such as the composition of an open path with a loop at its end, assuming a generalization of the definition of multitangents to open paths. After all this construction, let us now make contact with the group of loops. First, let us restrict attention to multitensors (not necessarily as- sociated with a loop) that satisfy the constraints B.10), B.11). Consider the set of vectors X E ? that have their zeroth rank component equal to one, X = A, X). The set X is closed under the x-product law. If Xi G X and X2 G X, it is clear from the definition of the group product that Xi x X2 satisfies the differential constraint. One can also demonstrate that Xi x X2 satisfies the algebraic constraint. In a similar way one can show that the inverse X given by B.19) satisfies the constraints if X does. A detailed proof

2.3 The extended group of loops 35 of these properties can be seen in the appendices of reference [20]. These results show that the group structure under the x-composition law is preserved by the imposition of the algebraic and differential constraints. We call X the Special-extended Loop group (SeL group)t. Note that the zeroth rank component of E plays a role analogous to the determinant in a group of matrices. For this reason we introduce the name Special when selecting E — 1. The group of loops is a subgroup of the SeL group since XG) G X and the composition law of the group of loops o is mapped via B.22) to the x-product. An important question at this point is: is the group SeL just a fancy rewriting of the group of loops, or is it actually a more general structure? We will show that SeL is actually larger than the group of loops by direct construction. Consider the group element Xm = Xx ™. xX. Note that if X gives the multitangent field of certain loop 7, Xm would be the multitangent field of the loop 7 swept itself m times. Applying the binomial expansion we get, m Xm = [I + (X - I)]m = I + Y^ ( 7 )(x - I)* ¦ B-23) The extension of B.23) to real values of m is straightforward, being defined as (i ) B.24) with A real. We usually call this object the analytic extension of X. Note that for A = —1 we recover the expression of the inverse of X. Also in this case, due to B.21) the analytic extension is well defined for all elements of X. One can prove that if X is constrained by the differential and algebraic identities, its analytic extension also satisfies the constraints (again see the appendices of [20]). So, the analytic extension of any X is in X. Moreover, we have XA x X" = XA+" . B.25) We conclude that the set {XA/A G R and X a given element of X} de- fines an Abelian one-parameter subgroup of the X group. For non-integer values of A, the Ath power of a multitangent is not a multitangent. This fact explicitly shows that there exist in X other elements besides the loop coordinates. Tavares [43] has also considered this group. His "shuffle product" is associated with the algebraic constraint in our terminology.

36 2 Loop coordinates and the extended group of loops Matrix representations of the SeL group can be generated through a natural extension of the holonomy. The extended holonomy associated with a non-Abelian connection Aax = Aa(x) is defined as Ha(X.) = A • X, where A = {l,iAalXl,---,inAanXn,...), B.26) X = (l,XaiX1,.. .,xaiX1~a"x",...), B.27) and the dot acts like a generalized Einstein convention including contrac- tions of the discrete indices aj and integrals over the three-manifold in the continuous indices X{. We have HA(X1)HA(X2) = fe=0 j=k = EiJ A^-H ( E Xr^X^-A = JfA(Xi x X2), B.28) 3=0 \k=0 J where convention B.17) has been applied over all the indices. The cor- respondence X —> if^(X) gives a representation of the SeL group into a particular gauge group. In the case of the X group and the connections A belonging to the algebra of a unitary group, Ha(X) is an element of the given unitary group. If one considers multitensors that do not satisfy the algebraic constraint, one still has a group and can construct a representa- tion by considering As that belong to a unitary gauge algebra. However, the corresponding representation will give a holonomy that is not an ele- ment of the gauge group. It will, in general, be an element of the general linear group of the same dimension as the gauge group. This highlights the role of the algebraic constraint in this formalism. The differential con- straint imposed on X ensures that Ha(X) is a gauge covariant quantity provided that the expressions involved in the proofs converge (see chapter 12 for some subtleties on this issue). We have shown that the analytic extension of any element of the SeL group defines a one-parameter subgroup. By studying its properties one can find the algebra associated with the SeL group. 2.3.2 Generators of the SeL group Consider the one-parameter subgroup {XA} and suppose that we increase A by an infinitesimal amount. We can write dXA xX + dX = XA x XdA = XA + -— dX , B.29) and taking A = 0 we get XdA = I + FdX, B.30)

2.3 The extended group of loops 37 where oo / -I \i—1 Introducing B.30) in B.29) we obtain the following differential equation for the elements of {XA} = XA x F = F x XA . B.32) dX This equation can be iteratively integrated to give n \k rX rXi f\n = 1+ ) T7 ' / 1 / 2 • • • / **"n+l ¦**¦ j~. k\ Jo Jo Jo B.33) The process actually stops for any finite rank n component (Fn+1 = Fx ^t.1 xF = 0 in this case). Therefore oo \k XA = I + ^ ^-F* = exp(AF) . B.34) We conclude that the vector F given by B.31) is the generator of the one-parameter subgroup {XA}. It is evident that the generator satisfies the differential constraint. We shall now prove the following fundamental property: F satisfies the homogeneous algebraic constraint (i.e., the sum over permutations defined in equation B.10) vanishes). In other words, the generator of the one-parameter subgroup {XA} is the algebraic free part of X. We know that ("V"A \fli...flktJik+l "'t*n / "Y~A \^l*--^fc t Y^ \tjtk+l-"tjtn /O QK\ -A. I — I -A- J \ -A. I • I titOO) Differentiating with respect to A and evaluating for A = 0 we get d_ d~X As 1 < k < n, we conclude pHi^Hkftk+i-^ = o , l < jfc < n . B.37) Reciprocally, one can demonstrate that the exponential of any alge- braically free quantity produces an object that satisfies the algebraic con- straint. It is important to stress that these results allow us to obtain the general solution for the algebraic constraint (equation B.34) with A = 1

38 2 Loop coordinates and the extended group of loops and its inverse B.31) give the relationship between an object that satisfies the algebraic constraint and its algebraic-free part). The set of all Fs that satisfy the differential constraint and the homo- geneous algebraic constraint forms a vector space T. One can define a bilinear operation on T in the following way, [Fi, F2] = Fi x F2 - F2 x Fi for any Flt F2 G T. B.38) This operation is closed on T. The vector space T together with the bracket operation B.38) defines the Lie algebra associated with the SeL group. 2.4 Loop coordinates The quantities X that we introduced in section 2.2 are not freely speci- fiable. That is, in order to be able to construct a gauge covariant object via equation B.1), the Xs had to satisfy the differential and algebraic constraints B.10), B.11). That they are not freely specifiable is a natural thing, since they are elements of a group. That is why it was impor- tant to find the associated algebra, since its free parameters give us a chance to separate the part of the multitangents that we can freely spec- ify. In the previous section we saw how to construct the set of objects T. These objects had the advantage of being constrained not by the algebraic constraint, but by the homogeneous algebraic constraint. This latter con- straint is very easily solvable, simply by requiring some symmetries on the Fs, given by equation B.37). In terms of the Fs one immediately is able to compute a solution to both the differential and algebraic constraints making use of equation B.34), X = exp(F). B.39) However, the Fs are far from freely prescribable since they are con- strained by the differential constraint. The main intention of this section is to give a prescription for generating the Fs (and through them the Xs) from freely specifiable quantities. In order to do this we will need to in- troduce some technology to deal with transverse tensors. This technology will also be useful for dealing with knot invariants. 2.4-1 Transverse tensor calculus First of all notice that the notion of transversality (divergence equal to zero) is well defined for vector densities, since their divergence can be com- puted without introducing an external metric. For instance; statements

2.4 Loop coordinates 39 such as daxEaxby = O B.40) are well defined for an object like E which is a vector density on the index a at the point x. Let us introduce the notion of transverse and longitudinal projectors in the multivector density space. In order to do this, it is convenient to endow the space of transverse vector densities of rank one with a natural metric structure. Given two transverse fields Vax and Wax one can define their inner product [22], g(V,W) = daVa = daWa = 0, B.41) where A^ is a "potential" defined in the following way. Construct a two-form Wab = eabcWc. This two-form is curl-free, d[cWab], due to the transversality of Wa. Then one can define the one-form ("potential") A^f by du,A^( = Wab- This one-form is defined up to the addition of a gradient. This will force us to give ad-hoc prescriptions when dealing with expressions in terms of A^f. However, the inner product B.41) is well defined in a prescription independent way since the addition of a gradient to A^ only contributes a total divergence term. The inner product introduced by B.41) gives rise to a covariant metric on the space of transverse vectors, g(V,W)=gOaxbyVaxWb», B.42) which can be explicitly written, for instance, in the transverse ( non- covariant) prescription, = 0 B.43) as I XC _ yC 9oaxby = ~^~ Cafcc 1 To • B.44) y 4?r I x — y \A Notice that due to the use of a non-covariant prescription the final ob- ject has both coordinate and background metric dependence. <?o is a well known object in knot theory, where it plays the role of the kernel of the Gauss knot invariant, as we will see in section 2.6. It is the expression in a particular prescription of the covariant metric in the space of transverse vector densities defined by B.41). Notice that in what follows we will not need to specify a background metric unless we want to give a specific prescription. In general, the covariant metric is defined up to gradients

40 2 Loop coordinates and the extended group of loops that change according to the prescription chosen, 9axby = 90 axby + Paxy,b + Pbyx,a • B.45) Transverse and longitudinal projectors may easily be written without the use of a background metric in terms of g and its inverse in the trans- verse space, gaxby = €abcdc6(x-y). B.46) We define the quantities 6t and 8l (the transverse and longitudinal Dirac deltas) as c ax ~ax cz rt ic\ a1\ <>T by =9 9czby B.47) and SL^by = 6°*by ~ *T°V B.48) where 8axby = SabS(x — y). It is straightforward to check that they have the desired projection properties, = St^p f>LPu = 0 . By using the explicit form of the covariant metric one can prove that haxby = <l>axy,b, B-49) where ~rxy = S(x - y) ¦ B.50) The ambiguity in the definition of the metric induces an ambiguity in the decomposition into transverse and longitudinal parts. Each function <j> that satisfies B.50) determines a particular prescription of the decom- position. It is important to note that the transverse density fields and in particular the contravariant metric B.46) are prescription independent. In the particular case in which we choose the transverse metric to be go we have Id 1 cax 6 by A transverse projector acting on the vector space ? of multitensor den- sities can be immediately introduced through the matrix &r, defined in

2.4 Loop coordinates 41 components as <5r-/i'V..^ = Km *r*% • • • *r"%B • B.53) Given any multivector density E one can construct a multivector den- sity Ey that is transverse or in other words that satisfies the homogeneous part of the differential constraint B.11) by, ET = 8T ¦ E . B.54) The set of all E^s forms a linear vector space &?. The definition of Ey is not unique, it depends on the prescription used in the definition of the projector. Since 8? a projector, relation B.54) is obviously not invertible in gen- eral. However, it turns out that it can be inverted on a subspace of ? given by ?&, the multitensor densities that satisfy the differential con- straint B.11). In order to do this, let us start by evaluating Eft"*" =8»\x...^VnEv?-v", B.55) making use of identity B.48) and the differential constraint and recalling that the first rank component of E is transverse, we then get ED = a ¦ ET. B.56) The soldering quantities <T only depend on the function (j> which char- acterizes the choice of decomposition in transverse and longitudinal parts, frui ^ ? if m = n x- nuu = "j Qft^ <T**>->Ul..J,m , if m < n 0 , if m > n with naixi...anxn — \^ raixi--aj-ixj-i ( i^Xj ,a,jXj \ . aj+ixj+i...anxn Vcij/i.-cn-ij/n-l = 2^Ocl«"cJ-i%-l [Vyj - (Pyi-i)°TCjyi...cn-1yn-i ¦ 3=1 B.58) Again, this definition is not unique and will be prescription dependent. However, starting from a given E/j one can construct an Ey and then uniquely reconstruct the original Ep by applying <T. A crucial property is that the quantities <T satisfy the differential con- straint in their upper indices, as can be checked from their definition. That is, given an arbitrary transverse multitensor density Ey, one can construct a solution of the differential constraint by applying equation B.56).

42 2 Loop coordinates and the extended group of loops The quantities <T have definite transversality properties 6T-(T = 6T, B.59) a-6T = a. B.60) Notice that due to these properties we can relax the requirement to construct a solution to the differential constraint, i.e., given an arbitrary multitensor E, the quantity <T-E is a solution of the differential constraint. Under a change of the prescription (j>\x —> ^2% we §e* a ^["h] satisfying <7[&] = (Tfa] • <T[<h\. B.61) The operations 8? and <T define an isomorphism between vector spaces, ?d the space of multitensors that solve the differential constraint and ?t via, ET = 6T ¦ ED, B.62) ED = (T- ET. B.63) The vector product can be introduced in the vector space ?d and, due to the isomorphism, it is simply given by Em x E/,2 = G ¦ (En x Era). B.64) This last property will have useful applications in section 2.6 where we construct diffeomorphism invariants. We are now ready to combine this construction with the ideas of the last section to define the loop coordinates. 2.4-2 Freely specifiable loop coordinates We saw in section 2.3.2 that one could generate a solution to the differ- ential and algebraic constraints X by considering X = exp(F) B.65) but for this to hold F had to satisfy the differential constraint and the homogeneous algebraic constraint. Let us now consider an arbitrary transverse multitensor Ey. Applying the results of the last subsection, we notice that the quantity <T ¦ Ey satisfies the differential constraint. Unfortunately, it does not satisfy the homogeneous algebraic constraint (if it did, we would be done, since it would be an element of T). We will remedy this situation now. We define a new matrix, given by v\...vm — v v\—vm ^__ lit B.66)

2.4 Loop coordinates 43 where «-"'V..*» = Km #% • • • &*Un . B.67) The matrix tl has the following important property: it satisfies the homogeneous algebraic constraint in the upper indices. This fact imme- diately shows that 0, is a projector. Given an arbitrary vector E, ft ¦ E is an algebraic-free object. In particular we have F = Q • F. Let us now introduce the following set of vectors ^...Vra = @,^...O B.68) with S = (a ¦ ft) , B.69) which written explicitly in components is Vl...Um- These vectors combine the action of <7, which converted an arbitrary multitensor into a solution of the differential constraint, and at, which projects into the space of solutions of the homogeneous algebraic con- straint. That is, given an arbitrary multivector density E, projecting it with S one obtains an element of T. Simply by exponentiating this ele- ment, as we saw in section 2.3.2, we obtain a solution of the differential and algebraic constraint. That is, we just consider, X = exp(<S • E), B.71) and the Es are unconstrained! Notice that expression B.71) is the usual relation between elements of a Lie group (X), a basis of generators S and their free parameters (E). Expression B.71) does not really depend on the portion of the Es that does not satisfy the homogeneous algebraic and differential constraints since the contraction with the «Ss is independent of that portion. There- fore, one will usually concentrate on the set of transverse vectors Y that satisfy the homogeneous algebraic constraint, and we will call this set 3^, y-. Y = 6T-Y and yHL^&^+i-*. = o , 1 < k < n . B.72) The situation is totally analogous, for instance, to that of the Lorentz group. In that case the generators are antisymmetric matrices and there- fore one usually works with free parameters that are antisymmetric ma- trices in spite of the fact that any kind of matrix would do. It is just that one can only code relevant information in its antisymmetric part. Simi- larly here, any arbitrary multitensor E would work as a free parameter,

44 2 Loop coordinates and the extended group of loops but only information coded in the portion that satisfies the homogeneous constraints will be relevant for constructing the Xs via equation B.71), X = exp(<S • Y). B.73) The elements of y are immediately related to those of T by Y = &r ¦ F. B.74) When referring to multitangents rather than arbitrary multitensors we can therefore call the objects Y "loop coordinates" or coordinates in loop space. Abusing the terminology a bit we will also refer to them in this way when we talk about arbitrary multitensor densities not necessarily associated with loops. Since they are solutions to the homogeneous algebraic and differential constraint, the «Ss are elements of T and therefore they form a basis for the algebra as we suggested above. Details of their construction, the proof that they satisfy the algebra and the determination of the structure constants of the SeL can be seen in reference [20]. 2.5 Action of the differential operators In the previous chapter we introduced a series of differential operators that represented the infinitesimal generators of the group of loops. The loop coordinates provide us with an explicit representation in terms of which we can explore the action of the differential operators. We will not discuss in detail the action of all the differential operators, since as we saw, they are related to each other. We will only concentrate on the action of the loop derivative and of the contact derivative. The former can be used as the starting point to compute any other derivative. The latter is related to diffeomorphism invariance and therefore deserves a detailed treatment. Let us therefore start by computing the action of the loop derivative on a multitangent field. By the definition of the loop derivative A.17), A + ^aabAab(Trz0))XaiXl-anXn(j) = XaiXl-a"Xn(irz0 o SuSvSuSv o < o 7), B.75) and recalling the relation between the x-product and the composition law B.22), we can write JlnXnfa Q SuSvSUSv O 7T° O 7) = «) x XzFu6v6u6v) x X«) x XG))OlXl-°»x». B.76) Notice that ~KzFu6v6u6v) is a multitangent basepointed at z, which is in line with the fact that it is composed with an open path that ends at z.

2.5 Action of the differential operators 45 We therefore need to evaluate ~X.(8u8v8u8v) applying the definition of the multitangents B.3). We can do this order by order. We will only make explicit the calculation of the first order, Xz(8u8v8u8v)aiXl = eiuai8(xi - z) + e2vai8(xl + exu - z) -€iuai8(xi + eiu + e2v - z) - e2vai8(xi + e2v - z), B.77) we now expand the Dirac