Higher dimensional algebra V: 2-groups (2009)
A 2-group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various...
CATEGORIFIED SYMPLECTIC GEOMETRY AND THE STRING LIE 2-ALGEBRA (2009)
Abstract. Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a...
Higher Gauge Theory, Homotopy Theory and n-Categories Lectures at Topics in Homotopy Theory, (2009)
These are rough notes for four lectures on higher gauge theory, aimed at explaining how this theory is related to some classic themes from homotopy theory, such as Eilenberg–Mac Lane spaces. After...
Division algebras and supersymmetry (2009)
Supersymmetry is deeply related to division algebras. Nonabelian Yang--Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or...
Higher-Dimensional Algebra VII: Groupoidification (2009)
Baez, John C., Hoffnung, Alexander E., Walker, Christopher D.
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed...
A Prehistory of n-Categorical Physics (2009)
This paper traces the growing role of categories and n-categories in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts which manifest...
The Algebra of Grand Unified Theories (2009)
The Standard Model of particle physics may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three "grand unified theories": theories that unify...
Physics, Topology, Logic and Computation: A Rosetta Stone (2009)
In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology:...
Categorified Symplectic Geometry and the String Lie 2-Algebra (2009)
Baez, John C., Rogers, Christopher L.
Multisymplectic geometry is a generalization of symplectic geometry suitable for n-dimensional field theories, in which the nondegenerate 2-form of symplectic geometry is replaced by a nondegenerate...
Physics, Topology, Logic and Computation: (2009)
A Rosetta Stone, John C. Baez, Mike Stay
Category theory is a very general formalism, but there is a certain special way that physicists use categories which turns out to have close analogues in topology, logic and computation. A category...
Groupoidification Made Easy (2008)
Baez, John C., Hoffnung, Alexander E., Walker, Christopher D.
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed...
Infinite-Dimensional Representations of 2-Groups (2008)
Baez, John C., Baratin, Aristide, Freidel, Laurent, Wise, Derek K.
A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector...
Categorified Symplectic Geometry and the Classical String (2008)
Baez, John C., Hoffnung, Alexander E., Rogers, Christopher L.
A Lie 2-algebra is a "categorified" version of a Lie algebra: that is, a category equipped with structures analogous those of a Lie algebra, for which the usual laws hold up to isomorphism. In the...
Convenient Categories of Smooth Spaces (2008)
Baez, John C., Hoffnung, Alexander E.
A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold,...
Spin foam models of Riemannian quantum gravity (2008)
John C. Baez, J. Daniel Christensen, Thomas R. Halford, C. Tsang
Abstract. Using numerical calculations, we compare three versions of the Barrett– Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described...
(on leave from the University of California at Riverside) (2008)
Existence and uniqueness questions are treated for quantum fields on × S 1 satisfying the nonlinear Klein-Gordon equation (⊓ ⊔ + m 2)φ + λ:P ′ (φ):v = 0, where P is a given real polynomial,...
Higher-dimensional algebra V: 2-groups, available as math-QA/0307200 (2008)
Abstract. A 2-group is a `categorified ' version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G * G! G has beenreplaced by a functor....
Higher-dimensional algebra V: 2-groups, available as math-QA/0307200 (2008)
Abstract. A 2-group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G × G → G has been replaced by a...
The Classifying Space of a Topological 2-Group (2008)
Baez, John C., Stevenson, Danny
Categorifying the concept of topological group, one obtains the notion of a 'topological 2-group'. This in turn allows a theory of 'principal 2-bundles' generalizing the usual theory of principal...
Infinite-Dimensional Representations of 2-Groups (2008)
Baez, John C., Baratin, Aristide, Freidel, Freidel, Wise, Derek K.
A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector...
statement of the Fundamental Theorem of Hecke Operators Theorem 18. We (2008)
This is a very rough draft, which only attempts to lead up to a precise
Diffeomorphism-Invariant Spin Network States (2007)
We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P ! M is a...
Spin Foam Perturbation Theory (2007)
We study perturbation theory for spin foam models on triangulated manifolds. Starting with any model of this sort, we consider an arbitrary perturbation of the vertex amplitudes, and write the...
From a vector space V equipped with a Yang-Baxter operator R one may form the r-symmetric algebra SRV = TV=hv\Omega
(on leave from the University of California at Riverside) (2007)
Existence and uniqueness questions are treated for quantum fields on \Theta S 1 satisfying the nonlinear Klein-Gordon equation (ut + m 2
Integrability for relativistic spin networks (2007)
Abstract. The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a...
Lectures on n-Categories and Cohomology (2006)
Baez, John C., Shulman, Michael
The goal of these talks was to explain how cohomology and other tools of algebraic topology are seen through the lens of n-category theory. Special topics include nonabelian cohomology, Postnikov...
Quantization of strings and branes coupled to BF theory (2006)
Baez, John C., Perez, Alejandro
BF theory is a topological theory that can be seen as a natural generalization of 3-dimensional gravity to arbitrary dimensions. Here we show that the coupling to point particles that is natural in...
Exotic Statistics for Strings in 4d BF Theory (2006)
Baez, John C., Wise, Derek K., Crans, Alissa S.
After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the...
Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings)...
Calabi-Yau Manifolds and the Standard Model (2005)
For any subgroup G of O(n), define a "G-manifold" to be an n-dimensional Riemannian manifold whose holonomy group is contained in G. Then a G-manifold where G is the Standard Model gauge group is...
From Loop Groups to 2-Groups (2005)
Baez, John C., Crans, Alissa S., Stevenson, Danny, Schreiber, Urs
We describe an interesting relation between Lie 2-algebras, the Kac-Moody central extensions of loop groups, and the group String(n). A Lie 2-algebra is a categorified version of a Lie algebra where...
Quantum Quandaries: a Category-Theoretic Perspective (2004)
General relativity may seem very different from quantum theory, but work on quantum gravity has revealed a deep analogy between the two. General relativity makes heavy use of the category nCob, whose...
Higher-dimensional algebra VI: Lie 2-algebras. Theory and Applications of Categories (2004)
, *] : L*L! L satisfying the Jacobi identity up to a completely antisymmetric trilinearnatural transformation called the `Jacobiator', which in turn must satisfy a certain law of its own. This...
Higher-dimensional algebra VI: Lie 2-algebras. Theory and Applications of Categories (2004)
Abstract. The theory of Lie algebras can be categorified starting from a new notion of ‘2-vector space’, which we define as an internal category in Vect. There is a 2category 2Vect having these...
Higher-dimensional algebra VI: Lie 2-algebras, The-ory and Applications of Categories 12 (2004)
The theory of Lie algebras can be categorified starting from a new notion of ‘2-vector space’, which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector...
Higher-Dimensional Algebra VI: Lie 2-Algebras (2003)
Baez, John C., Crans, Alissa S.
The theory of Lie algebras can be categorified starting from a new notion of "2-vector space", which we define as an internal category in Vect. There is a 2-category 2Vect having these 2-vector...
Higher-Dimensional Algebra V: 2-Groups (2003)
Baez, John C., Lauda, Aaron D.
A 2-group is a "categorified" version of a group, in which the underlying set G has been replaced by a category and the multiplication map has been replaced by a functor. Various versions of this...
Asymptotics of 10j symbols (2002)
Baez, John C., Christensen, J. Daniel, Egan, Greg
The Riemannian 10j symbols are spin networks that assign an amplitude to each 4-simplex in the Barrett-Crane model of Riemannian quantum gravity. This amplitude is a function of the areas of the 10...
Higher Yang-Mills Theory (2002)
Electromagnetism can be generalized to Yang-Mills theory by replacing the group U(1)$ by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form...
Spin Foam Models of Riemannian Quantum Gravity (2002)
Baez, John C., Christensen, J. Daniel, Halford, Thomas R., Tsang, David C.
Using numerical calculations, we compare three versions of the Barrett-Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De Pietri,...
Uncertainty in Measurements of Distance (2002)
Ng and van Dam have argued that quantum theory and general relativity give a lower bound of L^{1/3} L_P^{2/3} on the uncertainty of any distance, where L is the distance to be measured and L_P is the...
Abstract. The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of...
Positivity of spin foam amplitudes (2002)
John C. Baez, J. Daniel Christensen
Abstract. The amplitude for a spin foam in the Barrett{Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j...
Positivity of spin foam amplitudes (2002)
John C. Baez, J. Daniel Christensen
Abstract. The amplitude for a spin foam in the Barrett{Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j...
Uncertainty in Measurements of Distance (2002)
Ng and van Dam have argued that quantum theory and general relativity give a lower bound P on the uncertainty of any distance, where ` is the distance to be measured and `P is the Planck length....
Spin Foam Models Of Riemannian Quantum Gravity (2002)
John C. Baez, J. Daniel Christensen, Thomas R. Halford, David C. Tsang
Using numerical calculations, we compare three versions of the Barrett{ Crane model of 4-dimensional Riemannian quantum gravity. In the version with face and edge amplitudes as described by De...
Higher Yang-Mills Theory (2002)
Electromagnetism can be generalized to Yang{Mills theory by replacing the group U(1) by a nonabelian Lie group. This raises the question of whether one can similarly generalize 2-form...
Positivity of Spin Foam Amplitudes (2001)
Baez, John C., Christensen, J. Daniel
The amplitude for a spin foam in the Barrett-Crane model of Riemannian quantum gravity is given as a product over its vertices, edges and faces, with one factor of the Riemannian 10j symbols...
The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics....
The Meaning of Einstein's Equation (2001)
This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain...
Integrability for Relativistic Spin Networks (2001)
Baez, John C., Barrett, John W.
The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph...
Integrability for Relativistic Spin Networks (2001)
The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph...
Integrability for Relativistic Spin Networks (2001)
Barrett, John W., Baez, John C.
The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph...
From Finite Sets to Feynman Diagrams (2001)
John C. Baez, James Dolan, Björn Engquist, Wilfried Schmid
‘Categorification ’ is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin...
John C. Baez, Craig Callender, Nick Huggett, Cambridge U. Press
This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a...
Integrability for Relativistic Spin Networks (2001)
Barrett, John W., Baez, John C.
The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph...
From Finite Sets to Feynman Diagrams (2000)
`Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with...
From Finite Sets to Feynman Diagrams (2000)
John C. Baez, James Dolan, Bjorn Engquist, Wilfried Schmid
`Categorication' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorication can simplify and unify mathematics. We begin with...
Spin Foam Perturbation Theory (1999)
We study perturbation theory for spin foam models on triangulated manifolds. Starting with any model of this sort, we consider an arbitrary perturbation of the vertex amplitudes, and write the...
An Introduction to Spin Foam Models of Quantum Gravity and BF Theory (1999)
In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of `spin foam' is intended to serve as a similar...
The Quantum Tetrahedron in 3 and 4 Dimensions (1999)
Baez, John C., Barrett, John W.
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries...
Higher-Dimensional Algebra and Planck-Scale Physics (1999)
This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a...
The Quantum Tetrahedron in 3 and 4 Dimensions (1999)
Baez, John C., Barrett, John W.
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries...
The Quantum Tetrahedron in 3 and 4 Dimensions (1999)
Baez, John C., Barrett, John W.
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries...
Quantum Geometry of Isolated Horizons and Black Hole Entropy (1999)
Abhay Ashtekar, John C. Baez, Kirill Krasnov
Using the classical Hamiltonian framework of [1] as the point of departure, we carry out a non-perturbative quantization of the sector of general relativity, coupled to matter, admitting non-rotating...
The Quantum Tetrahedron in 3 and 4 Dimensions (1999)
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to...
Higher-Dimensional Algebra and Planck-Scale Physics (1999)
John C. Baez, Craig Callender, Nick Huggett, Cambridge U. Press
This is a nontechnical introduction to recent work on quantum gravity using ideas from higher-dimensional algebra. We argue that reconciling general relativity with the Standard Model requires a...
The Quantum Tetrahedron in 3 and 4 Dimensions (1999)
Baez, John C., Barrett, John W.
Recent work on state sum models of quantum gravity in 3 and 4 dimensions has led to interest in the `quantum tetrahedron'. Starting with a classical phase space whose points correspond to geometries...
Renormalization Made Easy (1999)
John C. Baez, Er E. Hoffnung, Christopher D. Walker
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed...
Higher-Dimensional Algebra IV: 2-Tangles (1998)
Baez, John C., Langford, Laurel
Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R^4 can be described as certain...
Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural...
Categorification is the process of finding category-theoretic analogs of set-theoretic concepts by replacing sets with categories, functions with functors, and equations between functions by natural...
John C. Baez and James Dolan (1998)
We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set...
While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the...
Quantization of diffeomorphism-invariant theories with fermions (1998)
John C. Baez, Kirill V. Krasnov
1
While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the...
Diffeomorphism-Invariant Spin Network States (1997)
We extend the theory of diffeomorphism-invariant spin network states from the real-analytic category to the smooth category. Suppose that G is a compact connected semisimple Lie group and P -> M is a...
An Introduction to n-Categories (1997)
An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing...
Baez, John C., Langford, Laurel
Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R^4 may be described as certain 2-morphisms in the 2-category of...
Quantization of Diffeomorphism-Invariant Theories with Fermions (1997)
Baez, John C., Krasnov, Kirill V.
We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P -> Sigma be a principal G-bundle over space and let F be...
Degenerate Solutions of General Relativity from Topological Field Theory (1997)
Working in the Palatini formalism, we describe a procedure for constructing degenerate solutions of general relativity on 4-manifold M from certain solutions of 2-dimensional BF theory on any framed...
Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes (1997)
We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set of...
Higher-Dimensional Algebra III: n-Categories and the Algebra of Opetopes (1997)
We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or `S-operads', and given such an operad O, we denote its set...
Degenerate Solutions of General Relativity from Topological Field Theory (1997)
Working in the Palatini formalism, we describe a procedure for constructing degenerate solutions of general relativity on 4-manifold M from certain solutions of 2-dimensional BF theory on any framed...
Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R 4 may be described as certain 2-morphisms in the 2-category of...
Higher-Dimensional Algebra II: 2-Hilbert Spaces (1996)
A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a...
Spin Networks in Gauge Theory (1996)
Given a real-analytic manifold M , a compact connected Lie group G and a principal G-bundle P ! M , there is a canonical `generalized measure' on the space A=G of smooth connections on P modulo...
4-Dimensional BF Theory as a Topological Quantum Field Theory (1996)
Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT...
Email: Baez@math.ucr.edu (1996)
A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects...
Higher-Dimensional Algebra I: Braided Monoidal 2-Categories (1996)
We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their relevance to 4d TQFTs and 2-tangles. Then we give concise definitions of semistrict...
Higher-Dimensional Algebra II: 2-Hilbert Spaces (1996)
A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2-Hilbert space to be an abelian category enriched over Hilb with a...
Higher-Dimensional Algebra II: 2-Hilbert Spaces (1996)
A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we de ne a 2-Hilbert space to be an abelian category enriched over Hilb with a...
Higher-Dimensional Algebra I: Braided Monoidal 2-Categories (1995)
We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded...
Functional Integration on Spaces of Connections (1995)
Let $G$ be a compact connected Lie group and $P \to M$ a smooth principal $G$-bundle. Let a `cylinder function' on the space $\A$ of smooth connections on $P$ be a continuous function of the...
4-Dimensional BF Theory as a Topological Quantum Field Theory (1995)
Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that 4-dimensional BF theory with cosmological term gives rise to a TQFT...
Spin networks in nonperturbative quantum gravity (1995)
A spin network is a generalization of a knot or link: a graph embedded in space, with edges labelled by representations of a Lie group, and vertices labelled by intertwining operators. Such objects...
Higher-dimensional Algebra and Topological Quantum Field Theory (1995)
The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of...
Quantum Gravity Hamiltonian for Manifolds with Boundary (1995)
John C. Baez, Javier P. Muniain, Dardo D. Piriz
In canonical quantum gravity, when space is a compact manifold with boundary there is a Hamiltonian given by an integral over the boundary. Here we compute the action of this `boundary...
Higher-Dimensional Algebra and Topological Quantum Field Theory (1995)
The study of topological quantum field theories increasingly relies upon concepts from higher-dimensional algebra such as n-categories and n-vector spaces. We review progress towards a definition of...
Spin Network States in Gauge Theory (1994)
Given a real-analytic manifold M, a compact connected Lie group G and a principal G-bundle P -> M, there is a canonical `generalized measure' on the space A/G of smooth connections on P modulo gauge...
An Algebraic Approach to Discrete Mechanics (1994)
John C. Baez, James W. Gilliam
Using basic ideas from algebraic geometry, we extend the methods of Lagrangian and symplectic mechanics to treat a large class of discrete mechanical systems, that is, systems such as cellular...
Knots and Quantum Gravity: Progress and Prospects (1994)
Recent work on the loop representation of quantum gravity has revealed previously unsuspected connections between knot theory and quantum gravity, or more generally, 3-dimensional topology and...
Generalized Measures in Gauge Theory (1993)
Let P -> M be a principal G-bundle. Using techniques from the loop representation of gauge theory, we construct well-defined substitutes for ``Lebesgue measure'' on the space A of connections on P...
Strings, Loops, Knots and Gauge Fields (1993)
The loop representation of quantum gravity has many formal resemblances to a background-free string theory. In fact, its origins lie in attempts to treat the string theory of hadrons as an...
The notion of a measure on the space of connections modulo gauge transformations that is invariant under diffeomorphisms of the base manifold is important in a variety of contexts in mathematical...
Link Invariants, Holonomy Algebras and Functional Integration (1993)
Given a principal G-bundle over a smooth manifold M, with G a compact Lie group, and given a finite-dimensional unitary representation of G, one may define an algebra of functions on the space of...
Strings, Loops, Knots and Gauge Fields (1993)
The loop representation of quantum gravity has many formal resemblances to a background-free string theory. In fact, its origins lie in attempts to treat the string theory of hadrons as an...
Link Invariants of Finite Type and Perturbation Theory (1992)
The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link...
Quantum Gravity and the Algebra of Tangles (1992)
In Rovelli and Smolin's loop representation of nonperturbative quantum gravity in 4 dimensions, there is a space of solutions to the Hamiltonian constraint having as a basis isotopy classes of links...
Link invariants of finite type and perturbation theory, Wellesley college preprint (1992)
The Vassiliev-Gusarov link invariants of finite type are known to be closely related to perturbation theory for Chern-Simons theory. In order to clarify the perturbative nature of such link...
Scattering and complete integrability in conformally invariant nonlinear theories (1990)
We study conformally invariant nonlinear wave equations in four dimensions corresponding to multicomponent massless scalar fields with a quartic interaction. We prove that the scattering operator S...
Conserved Quantities For The Yang-Mills Equations (1990)
There are infinitely many gauge-invariant conserved quantities for sufficiently regular solutions of the Yang-Mills equations on 4-dimensional Minkowski space, for example those that extend to C 2...
Higher-Dimensional Algebra IV: 2-Tangles
Just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions, compact surfaces smoothly embedded in R 4 can be described as certain...
Functional Integration on Spaces of Connections
Suppose that G is a compact connected Lie group and P !M is a smooth principal G-bundle. We define a `cylinder function' on the space A of smooth connections on P to be a continuous complex...