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We consider (effective) Quantum General Relativity coupled to the Standard Model (QGR-SM) and study the corresponding BRST double complex. This double complex is generated by infinitesimal diffeomorphisms and infinitesimal gauge transformations. To this end, we define the respective BRST differentials $P$ and $Q$ and their corresponding anti-BRST differentials $\overline{P}$ and $\overline{Q}$. Then we show that all differentials mutually anticommute, which allows us to introduce the total BRST differential $D := P + Q$ and the total anti-BRST differential $\overline{D} := \overline{P} + \overline{Q}$. Furthermore, we characterize all non-constant Lagrange densities that are essentially closed with respect to the diffeomorphism differentials $P$ and $\overline{P}$ as scalar tensor densities of weight $w = 1$. As a consequence thereof, we obtain that graviton-ghosts decouple from matter if the Yang--Mills theory gauge fixing and ghost Lagrange densities are tensor densities of weight $w = 1$. In particular, we show that every such Lagrange density can be modified uniquely to satisfy said condition. Finally, we introduce a total gauge fixing fermion $\Upsilon$ and show that we can generate the complete gauge fixing and ghost Lagrange densities of QGR-SM via $D \Upsilon$., Comment: 18 pages, article

The standard model (SM) is viewed as a variation on the Yang-Mills theory with gauge symmetry $u(1) \times su(2) \times su(3)$, in which the flavour symmetry is framed and to which 3 generations of quarks and leptons are appended as inputs from experiment. The framed standard model (FSM) is then a further variation on the SM in which the colour symmetry is also framed, where the 3 generations now follow as consequences together with their characteristic mass and mixing patterns. In addition, the FSM yields as a bonus a solution to the strong CP problem leading to a unified treatment seemingly of all known CP physics for both quarks and leptons. It predicts, however, also a "hidden" sector populated by particles yet unknown, hence full both of promises (e.g.\ dark matter?) and of threats, which is just beginning to be explored (i) with a modified Weinberg mixing and (ii) with the $g-2$ and some other anomalies., Comment: Invited contribution to the "Festschrift for the Yang Centenary". edited FC Chen et al

The extraordinary success of the Standard Model asks for a more rigorous control beyond the perturbative approach, which is affected by mathematical problems (interaction picture and canonical quantization, non-convergence of the perturbative series, trivialy results for the $\ph^4$ model and related models etc.). We shall briefly discuss sone non-perturbative results concerning the crucial role and realization of symmetry breakings in the Standard Model., Comment: 8

The N=4 superconformal Yang-Mills theory on flat four-dimensional Minkowski space is a de-confined gauge theory in the sense that the string tension for fundamental representation coloured quarks vanishes. In fact, static fundamental representation quarks which lie in certain half-BPS super-multiplets do not interact at all. An interesting question asks whether such quarks would carry a well-defined global colour charge which, when the gauge is fixed, should have the status of an internal symmetry. We shall present a simple paradigmatic model which suggests that the answer to this question lies in the way in which infrared divergences are dealt with., Comment: 28 pages, 2 figures, typos fixed and some clarification added

Massless Dirac equation for spinor multiplets is minimally coupled with a unitary representation of an arbitrary compact semisimple gauge group. The spectrum of the second quantized interaction Hamiltonian has a positive mass gap running along the classical energy scale., Comment: 13 pages. Reedited version

In this article we describe the relation between the Chern-Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional., Comment: 37 pages; 2 references added

We compute the semiclassical formulas for the partition functions obtained using two different Lagrangians: the Chern-Simons functional and the symplectic action functional., Comment: 14 pages; additional references added

We present a Hamiltonian formulation of the Schwinger model on the circle in Coulomb gauge as a semi-bounded self-adjoint operator which is invariant under the modular group ${\cal M}=\Z$ of large gauge transformations. There is a nontrivial action of $\cm$ on fermionic Fock space $\mcH_0$ and its vacuum which plays a role analogous to that of the spectral flow in the formalism involving the infinite Dirac sea. The formulation allows (i) a description of the anomaly and its relation to this group action, and (ii) an explicit identification of the interacting vacuum which arises after the destabilization of the non-interacting vacuum in $\mcH_0$.

The point-splitting computation of the gauge invariant Hamiltonian for the Schwinger model on the circle in a positive energy representation is presented., Comment: 10 pages

We make an extensive use of BLM approach to study details of predicting higher order corrections based on the approach. This way we are able to test two procedures to improve the prediction process. Beside the main line of the two procedures it is found out that overall normalization could change BLM patterns effectively. Finally we try to find out whether a BLM pattern is sufficient for a prediction or not, and how one should use such a pattern., Comment: 8 pages

A heatbath algorithm is proposed for pure SU(N) lattice gauge theory based on the Manton action of the plaquette element for general gauge group N. Comparison is made to the Metropolis thermalization algorithm using both the Wilson and Manton actions. The heatbath algorithm is found to outperform the Metropolis algorithm in both execution speed and decorrelation rate. Results, mostly in D=3, for N=2 through 5 at several values for the inverse coupling are presented., Comment: 9 pages, 10 figures, 1 table, major revision, final version, to appear in PRD

The classical matter fields are sections of a vector bundle E with base manifold M. The space L^2(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of C_b^\infty(M) on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum context, we show that this module action defines an automorphism group on the algebra A, of the canonical anticommutation relations on L^2(E), with which we can perform the analogous localization. That is, the net structure of the CAR, A, w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction maps. As a corollary, we prove a well-known "folk theorem," that the algebra A contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E., Comment: 15 pages

We study a discrete model of the SU(2) Yang-Mills equations on a combinatorial analog of $\Bbb{R}^4$. Self-dual and anti-self-dual solutions of discrete Yang-Mills equations are constructed. To obtain these solutions we use both techniques of a double complex and the quaternionic approach. Interesting analogies between instanton, anti-instanton solutions of discrete and continual self-dual, anti-self-dual equations are also discussed., Comment: 17 pages

We derive the universal collapse law of degree 1 equivariant wave maps (solutions of the sigma-model) from the 2+1 Minkowski space-time,to the 2-sphere. To this end we introduce a nonlinear transformation from original variables to blowup ones. Our formal derivations are confirmed by numerical simulations., Comment: 1 figure

This article studies the nonabelian localization results of Beasley and Witten, and considers the analogue of these results when the gauge group is U(1). It compares these results with results of Manoliu on abelian Chern-Simons theory, showing that the dependence on the coupling constant is the same., Comment: 14 pages; for conference in honour of Hans Duistermaat's 65th birthday

We show that Chern-Simons gauge theory with appropriate cutoffs is equivalent, term by term in perturbation theory, to a Fermionic theory with a nonlocal interaction term. When an additional cutoff is placed on the Fermi fields, this Fermionic theory gives rise to a convergent perturbation expansion. This leads us to conjecture that Chern-Simons gauge theory also gives rise to convergent perturbation expansions, which would give a mathematically well-defined construction of the theory., Comment: Minor errors corrected

The paper discusses the mathematical consequences of the application of derived variables in gauge fields. Physics is aware of several phenomena, which depend first of all on velocities (like e.g., the force caused by charges moving in a magnetic field, or the Lorentz transformation). Applying the property of the second Noether theorem, that allowed generalised variables, this paper extends the article by Al-Kuwari and Taha (1991) with a new conclusion. They concluded that there are no extra conserved currents associated with local gauge invariance. We show, that in a more general case, there are further conserved Noether currents. In its method the paper reconstructs the clue introduced by Utiyama (1956, 1959) and followed by Al-Kuwari and Taha (1991) in the presence of a gauge field that depends on the co-ordinates of the velocity space. In this course we apply certain (but not full) analogies with Mills (1989). We show, that handling the space-time coordinates as implicit variables in the gauge field, reproduces the same results that have been derived in the configuration space (i.e., we do not lose information), while the proposed new treatment gives additional information extending those. The result is an extra conserved Noether current., Comment: 14 pages

We use Hamilton-Jacobi theory to construct a gauge-invariant zero-energy candidate ground state for canonically quantized Yang-Mills theory with a "nonlinear normal" factor ordering, generalizing an analogous ordering introduced by Moncrief and Ryan for problems with finitely many degrees of freedom. Invariance under spatial rotations and translations is immediate; boost invariance remains under investigation. The motivation is to find a model for constructing a candidate ground state in general relativity, canonically quantized a la the Ashtekar variables. We seek to avoid replicating some of the more troublesome features of the Kodama state, inherited from the Chern-Simons state., Comment: 20 pages, 0 figures. Submitted to Communications in Mathematical Physics

A Q-manifold is a graded manifold endowed with a vector field of degree one squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of ``gauge fields'' (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. Any principal bundle yielding canonically a Q-bundle, this construction generalizes Chern-Weil classes. Novel examples include cohomology classes that are locally the de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras., Comment: 23 pages, LaTeX, uses diagrams.sty

In this paper we consider an Abelian Gauge Theory in R^4 equipped with the Minkowski metric. This theory leads to a system of equations, the Klein-Gordon-Maxwell equations, which provide models for the interaction between the electromagnetic field and matter. A three dimensional vortex is a finite energy solution of these equations in which the magnetic field looks like the field created by a finite solenoid. Under suitable assumptions, we prove the existence of vortex-solutions., Comment: 31 pagine

Abelian gauge theories consist of a class of field equations which provide a model for the interaction between matter and electromagnetic fields. In this paper we analyze the existence of solitary waves for these theories. We assume that the lower order term W is positive and we prove the existence of solitary waves if the coupling between matter and electromagnetic field is small. We point out that the positiveness assumption on W implies that the energy is positive: this fact makes these theories more suitable to model physical phenomena., Comment: 27 pages

We give a new interpretation of the Faddeev-Mickelsson anomaly in certain Yang-Mills theories in terms of S^1-central extensions of Lie groupoids., Comment: v2. 27 pages, content reorganized

We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces., Comment: 12 pp.; the Introduction has been rewritten and mention of cohomology dropped in Proposition 3.2; material slightly reorganized

We prove that monotonicity of density and energy inequality imply the rectifiability of the singular sets for Yang-Mills flow., Comment: 11 pages

This work is a continuation of the former paper in which principal bundles are given by compact spin toric manifolds and compact connected semisimple Lie groups. In this paper, ambient manifolds are assumed to be compact toric manifolds and Lie groups are compact connected. The main result is that locally smooth manifolds are the instanton moduli spaces of the noncommutative vector bundles over noncommutative 4-manifolds associated with highest weights of the compact Lie groups. Moreover their dimensions are described explicitly in terms of given principal bundles and highest weights. As a corollary, it is shown that a smooth 4-manifold is the instanton moduli space of the canonocal noncommutative vector bundle over the noncommutative comples projective 2-space associated with the highest weight 1 of the 1-torus., Comment: Latex2e 19 pages

In this paper we study the holomorphic Euler characteristics of determinant line bundles on moduli spaces of rank 2 semistable sheaves on an algebraic surface X, which can be viewed as $K$-theoretic versions of the Donaldson invariants. In particular, if X is a smooth projective toric surface, we determine these invariants and their wallcrossing in terms of the K-theoretic version of the Nekrasov partition function (called 5-dimensional supersymmetric Yang-Mills theory compactified on a circle in the physics literature). Using the results of math.AG/0606180 we give an explicit generating function for the wallcrossing of these invariants in terms of elliptic functions and modular forms., Comment: 72 pages, 2 figures

An intrinsically defined gauge-invariant discrete model of the Yang-Mills equations on a combinatorial analog of $\Bbb{R}^4$ is constructed. We develop several algebraic structures on the matrix-valued cochains (discrete forms) that are analogs of objects in differential geometry. We define a combinatorial Hodge star operator based on the use of a double complex construction. Difference self-dual and anti-self-dual equations will be given. In the last section we discus the question of generalizing our constructions to the case of a 4-dimensional combinatorial sphere, Comment: LaTeX, 17 pages

The violation of the Noether relation between symmetries and charges is reduced to the time dependence of the charge associated to a conserved current. For the U(1) gauge symmetry a non-perturbative control of the charge commutators is obtained by an analysis of the Coulomb charged fields. From this, in the unbroken case we obtain a correct expression for the electric charge on the Coulomb states, its superselection and the presence of massless vector bosons; in the broken case, we obtain a general non-perturbative version of the Higgs phenomenon, i.e. the absence of massless Goldstone bosons and of massless vector bosons. The conservation of the (gauge dependent) current associated to the U(1) axial symmetry in QCD is shown to be compatible with the time dependence of the corresponding charge commutators and a non-vanishing eta' mass, as a consequence of the non locality of the (conserved) current., Comment: Invited contribution to ``The Quantum Universe'', dedicated to G. Ghirardi for his 70th birthday

For a smooth projective toric surface we determine the Donaldson invariants and their wallcrossing in terms of the Nekrasov partition function. Using the solution of the Nekrasov conjecture math.AG/0306198, hep-th/0306238, math.AG/0409441 and its refinement math.AG/0311058, we apply this result to give a generating function for the wallcrossing of Donaldson invariants of good walls of simply connected projective surfaces with $b_+=1$ in terms of modular forms. This formula was proved earlier in alg-geom/9506018 more generally for simply connected 4-manifolds with $b_+=1$, assuming the Kotschick-Morgan conjecture and it was also derived by physical arguments in hep-th/9709193., Comment: 45pages, typos corrected, update the reference to a new version of Mochizuki's paper math.AG/0210211

These notes aim at a pedagogical introduction to recent work on deformation of spaces and deformation of vector bundles over them, which are relevant both in mathematics and in physics, notably monopole and instanton bundles. We first decribe toric noncommutative manifolds (also known as isospectral deformations) and give a detailed introduction to gauge theories on a toric four-sphere. This includes a Yang-Mills action functional with associated equations of motion and self-duality equations. We construct a particular class of instanton solutions on a SU(2) bundle with a suitable use of twisted conformal symmetries. In the second part, we describe a different deformation of an instanton bundle over the classical four-sphere by constructing a quantum group SU_q(2) bundle on a sphere which is different from the toric one., Comment: 34 pages; AMS-Latex. v2: Several minor changes. Based on lectures delivered at the 2005 Summer school on ``Geometric and Topological Methods for Quantum Field Theory'', July 11-29 2005, Villa de Leyva, Colombia

We give a brief review of a research made in the field of differential geometry in Estonia in the period from the beginning of the 19th century to the present time. The biographic data of mathematicians who made a valuable contribution to the development of differential geometry in Estonia in mentioned period are presented. The material presented in the introduction covers the period from the beginning of the 19th century to the middle of the 20th century and it can be considered as a brief historical sketch of the development of differential geometry in Estonia in this period. The next sections give an idea of the modern trends of development of differential geometry in Estonia., Comment: 22 pages, based on the talk presented at the Finnish-Estonian Colloquium held in Tallinn (Estonia), May 27-29, 2002

This article provides an explicit construction for a family of singular instantons on S^4 S^2 with arbitrary real holonomy parameter \alpha. This family includes the original \alpha = 1/4, c_2 = 3/2 solution discovered by P. Forgacs, Z. Horvath, and L. Palla, and our approach is modeled on that of their 1981 paper. Our primary tool is the ansatz due to Corrigan, Fairlie, Wilczek, and 't Hooft that constructs a self-dual Yang-Mills connection using a positive real-valued harmonic super-potential. Here we reformulate this harmonic function ansatz in terms of quaternionic notation, and we show that it arises naturally from the Levi-Civita connection of a conformally Euclidean metric. To simplify the construction, we introduce an SO(3)-action on S^4, and we show by dimensional reduction that the symmetric self-duality equation on S^4 is equivalent to the vortex equations over hyperbolic space H^2. We thus obtain a similar harmonic function ansatz for hyperbolic vortices, which we also derive using conformal transformations of H^2. Using this ansatz, we construct the vortex equivalents of the symmetric 't Hooft instantons, and we prove using the equivariant ADHM construction that they provide a complete description of all hyperbolic vortices. We also analyze when two vortices constructed by this ansatz are gauge equivalent, obtaining the surprising result that two such vortices are completely determined by the gauge transformation between them., Comment: 26 pages

We present Noether's second theorem for graded Lagrangian systems of even and odd variables on an arbitrary body manifold X in a general case of BRST symmetries depending on derivatives of dynamic variables and ghosts of any finite order. As a preliminary step, Noether's second theorem for Lagrangian systems on fiber bundles over X possessing gauge symmetries depending on derivatives of dynamic variables and parameters of arbitrary order is proved., Comment: 31 pages, to be published in J. Math. Phys

We consider a Lagrangian system on a fiber bundle and its gauge transformations depending on derivatives of dynamic variables and gauge parameters of arbitrary order. We say that gauge transformations form an algebra if they generate a nilpotent BRST operator., Comment: 12 pages

We consider bosons on Euclidean R^4 that are minimally coupled to an external Yang-Mills field. We compute the logarithmically divergent part of the cut-off regularized quantum effective action of this system. We confirm the known result that this term is proportional to the Yang-Mills action. We use pseudodifferential operator methods throughout to prepare the ground for a generalization of our calculation to the noncommutative four-dimensional Moyal plane (also known as noncommutative flat space). We also include a detailed comparison of our cut-off regularization to heat kernel techniques. In the case of the noncommutative space, we complement the usual technique of asymptotic expansion in the momentum variable with operator theoretic arguments in order to keep separated quantum from noncommutativity effects. We show that the result from the commutative space R^4 still holds if one replaces all pointwise products by the noncommutative Moyal product., Comment: 37 pages, v2 contains an improved treatment of the theta function in Appendix A.3

Let G be one of the local gauge groups C(X,U(n)), C^\infty(X,U(n)), C(X,SU(n)) or C^\infty(X,SU(n)) where X is a compact Riemannian manifold. We observe that G has a nontrivial group topology, coarser than its natural topology, w.r.t. which it is amenable, viz the relative weak topology of C(X,M(n)). This topology seems more useful than other known amenable topologies for G. We construct a simple fermionic model containing an action of G, continuous w.r.t. this amenable topology., Comment: 8 pages, Latex

Let P be a principal bundle with semisimple compact simply connected structure group G over a compact simply connected four-manifold M. In this note we give explicit formulas for the rational homotopy groups and cohomology algebra of the gauge group and of the space of (irreducible) connections modulo gauge transformations for any such bundle., Comment: 12 pages, to appear in Compositio Mathematica

By using supermanifold techniques we prove a generalization of the localization formula in equivariant cohomology which is suitable for studying supersymmetric Yang-Mills theories in terms of ADHM data. With these techniques one can compute the reduced partition functions of topological super Yang-Mills theory with 4, 8 or 16 supercharges. More generally, the superlocalization formula can be applied to any topological field theory in any number of dimensions., Comment: 22 pages, Latex2e

In this expository review we discuss various aspects of gauge theory. While the focus is on mathematics, wherever possible we make contact with theoretical high energy physics. Particular emphasis is placed on instantons and monopoles, which admit physical interpretation, and yield interesting and nontrivial mathematics. We give a clear and essentially self-contained exposition of the mathematical structure of the Seiberg-Witten monopole equations. Other topics include Donaldson's theorem on moduli spaces of monopoles, compactification of spaces of ASD connections, The Abelian monopole equations, and Abelian Higgs vortices., Comment: 16 pages, no figures

It is well-known that the perturbative prepotentials of four-dimensional N=2 supersymmetric Yang-Mills theories satisfy the generalized WDVV equations, regardless of the gauge group. In this paper we study perturbative prepotentials of the five-dimensional theories for some classical gauge groups and determine whether or not they satisfy the WDVV system., Comment: 15 pages, no figures, web address of paper added in bibliography. Misprints corrected

The so-called covariant Poincare lemma on the induced cohomology of the spacetime exterior derivative in the cohomology of the gauge part of the BRST differential is extended to cover the case of arbitrary, non reductive Lie algebras. As a consequence, the general solution of the Wess-Zumino consistency condition with a non trivial descent can, for arbitrary (super) Lie algebras, be computed in the small algebra of the 1 form potentials, the ghosts and their exterior derivatives. For particular Lie algebras that are the semidirect sum of a semisimple Lie subalgebra with an ideal, a theorem by Hochschild and Serre is used to characterize more precisely the cohomology of the gauge part of the BRST differential in the small algebra. In the case of an abelian ideal, this leads to a complete solution of the Wess-Zumino consistency condition in this space. As an application, the consistent deformations of 2+1 dimensional Chern-Simons theory based on iso(2,1) are rediscussed., Comment: 39 pages Latex file, 1 eps figure, typos and proof of lemma 5 corrected

The geometry of minimal surfaces generated by charge 2 Bogomolny monopoles on 3-dimensional Euclidean space is described in terms of the moduli parameter k. We find that the distribution of Gaussian curvature on the surface reflects the monopole structure. This is elucidated by the behaviour of the Gauss maps of the minimal surfaces., Comment: 23 pages, 2 figures

It is shown that the definition of physical integration measures via ``exponential of minus the action times kinematical integration measure'' typically contradicts properties of physical models. In particular, theories with uncountably many non-vanishing Wilson-loop expectation values cannot be gained this way. The results are rigorous within the Ashtekar approach to gauge field theories., Comment: 4 pages, LaTeX

It is proven that the physical measure for the two-dimensional Yang-Mills theory is purely singular with respect to the kinematical Ashtekar-Lewandowski measure. For this, an explicit decomposition of the gauge orbit space into supports of these two measures is given. Finally, the results are extended to more general (e.g. confining) theories. Such a singularity implies, in particular, that the standard method of determining the physical measure via ``exponential of minus the action times kinematical measure'' is not applicable., Comment: 42 pages, LaTeX

It is indicated that the definition of physical measures via ``exponential of minus the action times kinematical measure'' contradicts properties of certain physical models. In particular, theories describing confinement typically cannot be gained this way. The results are rigorous within the Ashtekar approach to gauge field theories., Comment: 5 pages, LaTeX

We prove the conjecture that a monopole in three-dimensional anti-de Sitter space can be completely determined by its ``holographic'' image on the conformal boundary two-sphere., Comment: 18 pages. Complete revision of the paper, focusing on clearer exposition. Although sometimes stated differently, the results remain the same as in the original version

We consider the quantum effective action of Dirac fermions on four dimensional flat Euclidean space coupled to external vector- and axial Yang-Mills fields, i.e., the logarithm of the (regularized) determinant of a Dirac operator on flat R^4 twisted by generalized Yang-Mills fields. According to physics folklore, the logarithmic divergent part of this effective action in the pure vector case is proportional to the Yang-Mills action. We present an explicit computation proving this fact, generalized to the chiral case. We use an efficient computation method for quantum effective actions which is based on calculation rules for pseudo-differential operators and which yields an expansion of the logarithm of Dirac operators in local and quasi-gauge invariant polynomials of decreasing scaling dimension., Comment: LaTex, 26 pages

In this paper, using the Atiyah-Ward equivalence and a theorem of Hitchin, one makes to correspond to certain bundles on the projective space, which are extensions of instanton bundles (in particular, these new bundles may have the first Chern class not congruent to 0 modulo the rank), bundles on the grassmanian Gr(2,4) or on the sphere S4, equiped with certain connections constructed from data associated to the bundle. Monads for these extensions are described., Comment: latex file, 12 pages, one uses xypic

The construction of appropriate jet space coordinates for calculating local BRST cohomology groups is discussed. The relation to tensor calculus is briefly reviewed too., Comment: 11 pages, correction in eq. (29), to appear in Lett.Math.Phys

We construct and study the Yang-Mills measure in two dimensions. According to the informal description given by the physicists, it is a probability measure on the space of connections modulo gauge transformations on a principal bundle with compact structure group. We are interested in the case where the base space of this bundle is a compact orientable surface. The construction of the measure in a discrete setting, where the base space of the fiber bundle is replaced by a graph traced on a surface, is quite well understood thanks to the work of E. Witten. In contrast, the continuum limit of this construction, which should allow to put a genuine manifold as base space, still remains problematic. This work presents a complete and unified approach of the discrete theory and of its continuum limit. We give a geometrically consistent definition of the Yang-Mills measure, under the form of a random holonomy along a wide, intrinsic and natural class of loops. This definition allows us to study combinatorial properties of the measure, like its Markovian behaviour under the surgery of surfaces, as well as properties specific to the continuous setting, for example, some of its microscopic properties. In particular, we clarify the links between the Yang-Mills measure and the white noise and show that there is a major difference between the Abelian and semi-simple theories. We prove that it is possible to construct a white noise using the measure as a starting point and vice versa in the Abelian case but we show a result of asymptotic independence in the semi-simple case which suggests that it is impossible to extract a white noise from the measure., Comment: LaTeX, 113 pp, 12 figures