Gauge field theories: various mathematical approaches
aa r X i v : . [ m a t h - ph ] A p r Gauge field theories: various mathematical approaches
Jordan François, Serge Lazzarini and Thierry Masson
Aix Marseille Université, Université de Toulon, CNRS,CPT, UMR 7332, Case 907, 13288 Marseille, France.To be published in the book
Mathematical Structures of the Universe (Copernicus Center Press, Kraków, Poland, 2014)
Abstract
This paper presents relevant modern mathematical formulations for (classical) gauge fieldtheories, namely, ordinary differential geometry, noncommutative geometry, and transitive Liealgebroids. They provide rigorous frameworks to describe Yang-Mills-Higgs theories or gravita-tion theories, and each of them improves the paradigm of gauge field theories. A brief comparisonbetween them is carried out, essentially due to the various notions of connection. However theyreveal a compelling common mathematical pattern on which the paper concludes. ontents Since its inception in 1918, 1927 and 1929, through the pioneering work of Weyl, London, andFock on electromagnetism, the idea of local symmetries, or gauge symmetries, has proven to be adecisive insight in the structure of fundamental interactions (for a general account see e.g. [67] andreferences therein). The elaboration of these theories provides a spectacular example of convergencebetween physics and mathematics. In the early 1950’s, while Yang and Mills proposed their idea ofnon abelian gauge fields (generalization of electromagnetism), Ehresmann developed the notion ofconnections on principal fiber bundles, which turns out to be the natural mathematical frameworkfor Yang-Mills field theories.In the 1960’s, the elaboration of the Standard Model (SM) of particle physics has shown thatthree of the four fundamental interactions (electromagnetism, weak and strong interactions) can bemodeled as abelian and Yang-Mills gauge fields, supplemented by a C -valued scalar field whichgenerates masses for the gauge bosons through the spontaneous symmetry breaking mechanism(SSBM). The discovery of the massive vector bosons Z µ , W ± µ in 1983, and of the massive Higgs bosonin 2012, confirms the relevance of the SM in its present formulation, in terms of the mathematicsof connections. In this formulation, one has the correspondence between physical objects andmathematical structures given in Table 1. However it remains a weakness in this mathematicalscheme. Indeed, the C -valued scalar field involved in the SSBM is, at the same time, a section ofa (suitable) associated vector bundle [71; 73; 74], and a boson, so that it is an “hybrid structure”belonging to the two rows of the table. Moreover, in this scheme, its scalar potential does notemerge from a natural mathematical construction.The theory for the fourth fundamental interaction, gravitation, has been elaborated by Einsteinwithin the framework of (pseudo-)riemannian geometry, and not as a gauge field theory. Later,this theory has been reformulated using connections on the frame bundle of space-time. Thesereformulations have a richer structure (originating in the notion of soldering form) than bare Yang-Mills theories based on Ehresmann connections. 2 ind of Particle Statistics Mathematical Structure interaction particle ↔ boson ↔ connection on principal fiber bundlematter particle ↔ fermion ↔ section of associated vector bundleTable 1: Correspondence between physical objects and mathematical structures in the usual formu-lation of the SM (see 2.1).In this paper, we review and compare three mathematical frameworks suited to formulate gaugefield theories, namely, ordinary differential geometry, noncommutative geometry, and the frameworkof transitive Lie algebroids.Noncommutative geometry has been the first attempt to develop gauge field theories beyondthe usual geometry of fiber bundles and connections. One of its first successes has been to proposegauge field theories in which scalar fields are part of the generalized notion of connection, and inwhich a naturally constructed Lagrangian produces a quadratic potential for these (new) fields,providing a SSBM in these models [17; 35; 36]. Despite the success of the (re)construction of anoncommutative version of the SM [11], noncommutative geometry has never been widely adoptedas a new framework to model physics beyond the SM. The most important reason for the rejectionof noncommutative geometry is certainly due to the mathematical skill required to master this newconceptual framework.On the other side, the newly proposed framework of gauge field theories on transitive Lie alge-broids [39; 54], while giving rise to Yang-Mills-Higgs theories following the same successful recipesdiscovered in noncommutative geometry, is close enough to ordinary differential geometry and tousual algebraic structures, to permit to a wider audience to master this scheme. Moreover, contraryto noncommutative geometry in which the gauge group is the group of automorphisms of an asso-ciative algebra (this disqualifies U (1)-gauge field theories in noncommutative geometry), any gaugegroup of a principal fiber bundle can be used, since it is possible to consider the transitive AtiyahLie algebroid associated to this principal fiber bundle.An important point of this review paper is to compare, in both schemes, how scalar fieldssupplements naturally Yang-Mills fields in the corresponding notions of “generalized” connections.Then these fields belong to the first row of (a generalized version of) Table 1, as bosons and as part of(generalized) connections. These schemes also provide a SSBM without requiring some extra (Godgiven) inputs in the model: the associated scalar potential is given by the Lagrangian describingthe dynamics of the fields of these generalized connections.Another main point on which this paper focuses is the modeling of gravitation theories in termsof gauge field theories. A new recent way of thinking about symmetry reduction [40] permits toclearly understand how the geometrical objects of the Einstein’s theory of gravity are reconstructedafter the decoupling of a gauge symmetry modeled in terms of Cartan connections.Let us emphasize an essential characterization of gauge field theories, as they are described andconsidered in this review paper. Any theory written as the integral of a Lagrangian globally definedon a (space-time) manifold M is necessarily invariant under diffeomorphisms. This is due to thefact that this Lagrangian must be invariant under any change of local coordinates on M (at theprice to introduce a non Minkowskian metric if necessary), and a diffeomorphism on M is locallyequivalent to a change of coordinates. This “basic” symmetry, in the sense that it is always requiredand also in the mathematical sense that it is governed by the “base” manifold M , is consistentwith the terminology “natural geometry” put forward in [49]. In addition to this basic symmetry,the Lagrangian can be symmetric under more general transformations. Among them are the gaugesymmetries, which, in the usual point of view, require an extra (non basic) structure. This extrastructure is a principal fiber bundle P in ordinary geometry: it is defined on top of M , but it can3ot be reconstructed from M only. This is the essence of gauge symmetries, which are constrainson modeling of physical systems supplementing the basic space-time constrains.Gauge field theories are based on physical ideas which require essential mathematical structuresin order to be elaborated. These basic ingredients can be listed as follows:1. A space of local symmetries, (local in the sense that they depend on points in space-time): forinstance this space is usually given by a so-called gauge group (finite gauge transformations)or a Lie algebra (infinitesimal gauge transformations).2. An implementation of the symmetry on matter fields: it takes the form of a representationtheory associated to the natural mathematical structures of the theory.3. A notion of derivation: that is the differential structure on which equations of motion arewritten.4. A replacement of ordinary derivations: this is the covariant derivative , which encodes thephysical idea of “minimal coupling” between matter fields and gauge fields.5. A way to write a gauge invariant Lagrangian density (up to a divergence term): this is the action functional , from which the equations of motion are deduced.Let us emphasize that the three mathematical frameworks under consideration in this paperfulfill all these main features. In order to get right away a direct comparison between the three, letus gather the listed items just down below.In ordinary differential geometry which will be recalled in Section 2, the fundamental mathe-matical structure is that of a G -principal fiber bundle P over a smooth m -dimensional manifold M which is usually expressed as the sequence G / / P π / / M . Then the ingredients are: The gauge group: this is the group of vertical automorphisms of P , denoted by G ( P ). The representation theory: any (linear) representation ℓ of G on a vector space E defines anassociated vector bundle E = P × ℓ E , and there is a natural action of G ( P ) on the space of sectionsof E . The differential structure: it consists into the (ordinary) de Rham differential calculus.
The covariant derivative: any (Ehresmann) connection 1-form ω on P induces a covariant deriva-tive ∇ on sections of any associated vector bundles. The action functional: in order to define an action functional, one needs an integration on thebase manifold M , a Killing form on the Lie algebra g of G , and the Hodge star operator associatedto a metric on M . Then the action functional is written using the curvature of ω .An important aspect of this framework is that the connection, which contains the Yang-Mills gaugefields, is defined on the main structure (the principal fiber bundle P ), and these fields couple tomatter fields only when a representation is given. In the same way, gauge transformations aredefined on P , and they act on any object naturally introduced in the theory.We will see in 2.3 that Cartan connections can also be used, in replacement of Ehresmannconnections, in particular to model gravitation as a gauge field theory.In noncommutative geometry dealt with in Section 3, the basic ingredient is an associativealgebra A . Think of it as a replacement for the (commutative) algebra C ∞ ( M ). Then one has: The representation theory: it consists to a right module M over A . It is often required to bea projective finitely generated right module such that the theory is not empty. The gauge group: this is the group Aut( M ) of automorphisms of the right module. Contrary toordinary differential geometry, it does depend on the representation space. The differential structure: any differential calculus defined on top of A can be used. There isno canonical construction here, and one has to make an explicit choice at this point. At least two4mportant directions can be followed: consider the derivation-based differential calculus canoni-cally associated to the algebra A (see 3.3), or introduce supplementary structure to constitute aspectral triple ( A , H , D ) (see 3.2). The covariant derivative: it is a noncommutative connection, which is defined on M with thehelp of the chosen differential calculus. In many situations, as in ordinary differential geometry,this covariant derivative can be equivalently described by a 1-form in the chosen space of forms. The action functional: it heavily depends on the choice of the differential calculus. For instance,using a derivation-based differential calculus, one can use some noncommutative counterparts ofintegration and Hodge star operator to construct a gauge invariant action based on the curvatureof the connection. When a spectral triple is given, it is convenient to consider the spectral actionassociated to the Dirac operator D , which requires the Dixmier trace as a substitute for theintegration.Here, the connection (the generalized Yang-Mills fields) and the gauge transformations are defined acting on matter fields, not at the level of the primary object A . This implies that the constructionof a gauge field theory must take into account, at the very beginning , the matter content. This wayof thinking departs from the one in ordinary differential geometry, where gauge theories withoutmatter fields can be considered. A way out is to particularize the right module as the algebra itself.In the framework of transitive Lie algebroids to which Section 4 is devoted, the basic structureis a short exact sequence of Lie algebras and C ∞ ( M )-modules, / / L ι / / A ρ / / Γ( T M ) / / , where ρ satisfies some axioms (see 4.1). Think of it as an infinitesimal version of a principal fiber bundle G / / P π / / M . Then one has: The differential structure: it consists on a space of “forms” defined as multilinear antisymmetricmaps from A to L , equipped with a differential which takes into account the Lie structure on A . The representation theory: to any vector bundle E over M , one can associate its transitive Liealgebroid of derivations, denoted by D ( E ) (first order differential operators on E whose symbol isthe identity). Then a representation of A is a morphism of Lie algebroids A → D ( E ). The gauge group: given a representation as above, the gauge group is the group Aut( E ) of verticalautomorphisms of E . This depends on the vector bundle E . But infinitesimal gauge transforma-tions can be defined as elements of L (see 4.3), independently on any representation of A . The covariant derivatives: there is a good notion of “generalized connections”, which are definedas 1-forms b ω : A → L . Then a representation on E induces an element in D ( E ) associated to ω .This is the covariant derivative. The action functional: one can write a gauge invariant action functional using natural objectson A , which consists into a metric, its Hodge star operator, a notion of integration along L , andan integration on M .As in noncommutative geometry, finite gauge transformations are only defined once a representationis given. But, as in ordinary differential geometry, a connection is intrinsically associated to the mainstructure (the short exact sequence), as well as are infinitesimal gauge transformations. Moreover,finite gauge transformations can be defined on any Atiyah Lie algebroid (see 4.3), which are thenatural transitive Lie algebroids to consider to get the closer generalizations of Yang-Mills fieldtheories in this framework.Althrough the three frameworks look quite different, it will be shown that they present simi-larities from which a general scheme emerges. The latter can be summarized under the form ofthe sequence (5.1), which ought to provide a general setting for treating gauge field theories at theclassical level. 5 Ordinary differential geometry
Many textbooks explain in details the theory of fiber bundles and connections (see for instance[4; 42; 48; 66]). We will suppose that the reader is quite familiar with these notions. Here, we willconcentrate on ordinary (Ehresmann) connections on principal fiber bundles, a notion that will begeneralized in the next two sections, and on the geometry of Cartan connections, which permits toconsider Einstein theory of gravitation as a gauge theory.
Let G / / P π / / M (2.1)be a G -principal fiber bundle for a Lie group G and a m -dimensional smooth manifold M . Denoteby e R the right action of G on P : e R g ( p ) = p · g for any g ∈ G and p ∈ P . Let g be the Lie algebra of G . A connection on P can be characterized following two points of view.The geometer says that a connection is a G -equivariant horizontal distribution H P in the tangentbundle T P : for any p ∈ P , one supposes given a linear subspace H p P ⊂ T p P such that H p · g P = T p e R g ( H p P ). The curvature of the connection measures the failure for the distribution H P to beintegrable.However, a dual equivalent algebraic setting is better suited to field theory. The distribution H P is thus defined as the kernel of a 1-form ω on P with values in g . The algebraist then says thata connection on P is an element ω ∈ Ω ( P ) ⊗ g such that ω ( ξ P ) = ξ, ∀ ξ ∈ g , e R ∗ g ω = Ad g − ω, ∀ g ∈ G . (2.2)where ξ P is the fundamental vector field on P associated to the action e R e tξ . The curvature of ω is defined as the 2-form Ω ∈ Ω ( P ) ⊗ g given by the Cartan structure equation Ω = d ω + [ ω, ω ](where the graded bracket uses the Lie bracket in g ). The space of connections is an affine space.The gauge group G ( P ) is the group of vertical automorphisms of P , which are diffeomorphismsΞ : P → P which respect fibers and such that Ξ( p · g ) = Ξ( p ) · g for any p ∈ P and g ∈ G . Thisgroup acts by pull-back on forms, and it induces an action on the space of connections, i.e. Ξ ∗ ω satisfies also (2.2). The gauge group can also be described as sections of the associated fiber bundle P × α G for the action α g ( h ) = ghg − of G on itself, and also as covariant maps Υ : P → G satisfyingΥ( p · g ) = g − Υ( p ) g . Then a direct computation shows that (with d the de Rham differential on P )Ξ ∗ ω = Υ − ω Υ + Υ − dΥ , Ξ ∗ Ω = Υ − ΩΥ . (2.3)The Lie algebra of infinitesimal gauge transformations is the space of sections of the associatedvector bundle in Lie algebras Ad P = P × Ad g for the Ad representation of G on g .The theory of fiber bundles tells us that P can be locally trivialized by using a couple ( U , φ ),where U ⊂ M is an open subset and φ : U × G → π − ( U ) is a isomorphism such that φ ( x, gh ) = φ ( x, g ) · h for any x ∈ U and g, h ∈ G . Then s : U → π − ( U ) defined by s ( x ) = φ ( x, e ) is a localtrivializing section, and one defines the local trivializations of ω and Ω on U as A = s ∗ ω ∈ Ω ( U ) ⊗ g , F = s ∗ Ω ∈ Ω ( U ) ⊗ g . (2.4)As section of P × α G , an element of the gauge group, can be trivialized into a map γ : U → G , andits action A A γ and F F γ is given by the local versions of (2.3): A γ = γ − Aγ + γ − d γ, F γ = γ − F γ. (2.5)6et { ( U i , φ i ) } i ∈ I be a family of trivializations of P such that S i ∈ I U i = M . On any U i ∩ U j = ∅ ,there is then a map g ij = U i ∩ U j → G such that φ i ( x, g ij ( x )) = φ j ( x, e ). Let us define the family A i = s ∗ i ω and F i = s ∗ i Ω for any i ∈ I . These forms satisfy the gluing relations A j = g − ij A i g ij + g − ij d g ij , F j = g − ij F i g ij . (2.6)These local expressions are those used in field theory, namely A is the gauge potential and F thefield strength.Note the similarity between active gauge transformations (2.5) and gluing relations (2.6) (whichare called “passive gauge transformations”). A Lagrangian written in terms of F and A which isinvariant under active gauge transformations is automatically compatible with the gluing relations,so that it is well defined everywhere on the base manifold.The 2-form Ω is horizontal, in the sense that it vanishes on vertical vector fields, and its is( e R , Ad)-equivariant, e R ∗ g Ω = Ad g − Ω for any g ∈ G . We say that Ω is tensorial of type ( e R , Ad).Such a form defines a form F ∈ Ω ( M , Ad P ). The existence of F can also be deduced from thehomogeneous gluing relations (2.6) for the F i ’s. The 1-form ω is not tensorial (the gluing relationsof the A i ’s are inhomogeneous) so that it does does define a global form on M with values in anassociated vector bundle.What we end up with is an equivalent description of these algebraic structures at three levels: Globally on P : ω ∈ Ω ( P ) ⊗ g is the connection 1-form on P , which satisfies (2.2) (equivarianceand a vertical normalization), and Ω ∈ Ω ( P ) ⊗ g its curvature, which is tensorial of type ( e R , Ad).This is in general the preferred description for mathematicians.
Locally on M : on any local trivialization ( U i , φ i ) of P , with associated local section s i ( x ) = φ i ( x, e ), the pull-back by s i defines the local descriptions A i and F i as in (2.4). These localdescriptions are related from one trivialization to another by the gluing relations (2.6). This isthe preferred description for physicists, who define field theories in term of maps on space-time(the manifold M ) to write down local Lagrangian. Globally on M : The curvature is also a 2-form F globally defined on M , with values in an asso-ciated vector bundle. This description is not complete: the connection does not define a global1-form on M . Nevertheless, notice that the difference of two connections belongs to Ω ( M , Ad P ).While incomplete, this description is the one that will be generalized in 3.1 and 4.3. As we willsee then, if one accepts to depart from ordinary differential geometry, a convenient space can bedefined to consider a “1-form” to represent the connection ω in this description.The connection ω defined on P induces a “connection” on any associated vector bundle E = P × ℓ E . From a geometric point of view, such a connection is a notion of parallel transport inthe fibers along paths on the base manifold. Looking at the infinitesimal version of this paralleltransport, one can define a derivation on the space Γ( E ) of smooth sections of E : this is the covariantderivative. In physics, matter fields are represented as elements in Γ( E ). Notice that the gauge groupacts naturally on sections of any associated vector bundle, so that the symmetry is automaticallyimplemented on any space of matter fields. Contrary to what will be described in 3.1, the gaugegroup is independent of the space Γ( E ).Recall that in a vector bundle E , there is no canonical way to define a derivation of ψ ∈ Γ( E )along a vector field X ∈ Γ( T M ). The connection is precisely the structure needed to define this“derivation along X ”. The covariant derivative defined by ω associates to any X ∈ Γ( T M ) a linearmap ∇ X : Γ( E ) → Γ( E ) such that ∇ X ( f ψ ) = ( X · f ) ψ + f ∇ X ψ, ∇ X + Y ψ = ∇ X ψ + ∇ Y ψ, ∇ fX ψ = f ∇ X ψ. (2.7)These relations are sufficient to define a covariant derivative on the space of smooth sections of anyvector bundle E . The quantity [ ∇ X , ∇ Y ] − ∇ [ X,Y ] is a C ∞ ( M )-linear map Γ( E ) → Γ( E ) which isthe multiplication by F ( X, Y ) (modulo a representation).7onsider a gauge transformation Ξ. Then the theory tells us that it induces an invertible map σ : Γ( E ) → Γ( E ) such that σ ( f ψ ) = f σ ( ψ ) for any f ∈ C ∞ ( M ). The covariant derivative ∇ σ associated to Ξ ∗ ω is then given by ∇ σX ψ = σ − ◦ ∇ X ◦ σ ( ψ ).Using a trivialization of E (induced by a trivialization ( U , φ ) of P ), the section ψ is a map ϕ : U → E , and the covariant derivative takes the form D X ϕ = X · ϕ + η ( A ( X )) ϕ. where η is the representation of g on E induced by ℓ . The action of a gauge group element γ : U → G on ϕ is given by ϕ γ = ℓ ( γ ) − ϕ . In order to simplify notations, let us omit the representations ℓ and η in the following.Using a local coordinate system ( x µ ) on U , this defines the differential operator D µ = ∂ µ + A µ where D µ = D ∂ µ and A µ = A ( ∂ µ ) ∈ g . This is the ordinary covariant derivative used inphysics, which gives rise, in Lagrangians, to the minimal coupling “ A µ ϕ ”. The field ϕ supportsthe representation ϕ γ − ϕ , and from a physical point of view, this is its characterization as agauge field. Then the operator ∂ µ does not respect this representation, because γ , being “local”(it depends on x ∈ U ), one has ∂ µ ϕ γ = ( ∂ µ γ − ) ϕ + γ − ∂ µ ϕ = γ − ( ∂ µ + ( γ∂ µ γ − )) ϕ , where inthe last expression we make apparent a well defined object γ∂ µ γ − with values in g , so that thislast expression has a general meaning. On the contrary, the differential operator D µ respects therepresentation, since D µ ϕ γ = ( D µ ϕ ) γ = γ − D µ ϕ . This is the heart of the usual formulation ofgauge field theories: promote a global symmetry ( γ constant) to a local symmetry ( γ a function)by replacing ∂ µ everywhere in the Lagrangian by a differential operator D µ which is compatiblewith the action of the gauge group. This requires to add new fields A µ in the game with gaugetransformations (2.5). But one needs also to introduce a gauge invariant term which describes thedynamics of the fields A µ . The simplest solution is the so-called Yang-Mills action S Gauge, YM [ A ] = Z tr( F ∧ ⋆F ) = Z tr( F µν F µν ) dvol (2.8)where dvol is a metric volume element on M , F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] (2.9)is the local expression of the curvature Ω, and tr is a Killing metric on g .Notice that (2.8) is not the only admissible action functional for the fields A . The Chern-Simonsaction can be defined for space-times of dimension 3 as S Gauge, CS [ A ] = Z tr (cid:16) A ∧ d A + A ∧ A ∧ A (cid:17) when the structure group G is non abelian. This action is not gauge invariant, and only e iκ S Gauge, CS [ A ] can be made gauge invariant by a suitable choice of κ .Let us consider a more formal point of view about connections, which will be the key forgeneralizations of this notion in 3.1 and 4.3. The Serre-Swan theorem [69; 72] tells us that a vectorbundle E on a smooth manifold M is completely characterized by its space of smooth sections M = Γ( E ), which is a projective finitely generated module over the (commutative) algebra C ∞ ( M ).Then, the assignment X
7→ ∇ X defines a map ∇ : M → Ω ( M ) ⊗ A M , such that ∇ ( f ψ ) = d f ⊗ ψ + f ∇ ψ. (2.10)This map can be naturally extended into a map ∇ : Ω • ( M ) ⊗ A M → Ω • +1 ( M ) ⊗ A M ,
8y the derivation rule ∇ ( η ⊗ ψ ) = d η ⊗ ψ + ( − r η ∧ ∇ ψ, for any η ∈ Ω r ( M ).The curvature of ∇ is then defined as ∇ : M → Ω ( M ) ⊗ A M , and it can be shown that ∇ ( f ψ ) = f ∇ ψ , and also that ∇ ψ is the multiplication by F (modulo the representation η of g on E mentioned before). The gauge transformation σ : M → M can be extended as an invertiblemap σ : Ω • ( M ) ⊗ A M → Ω • ( M ) ⊗ A M , and one has ∇ σ = σ − ◦ ∇ ◦ σ. (2.11)From the three levels of description of connections given above, it is clear that the covariantderivative ∇ is related to the last one (“Globally on M ”) because it acts on section of E (which areglobally defined on M ). The differential operator D corresponds to the second one, and there is athird description (not given here) which makes use of (equivariant) maps defined on P . As a vector bundle over M , the tangent bundle T M is an essential structure for studying M and itsdifferential geometry. The global topology of this bundle is not arbitrary as could be the topologyof the vector bundles considered above, and, as an associated vector bundle, it is related to thetopology of the principal fiber bundle L M of frames on M .Then, the theory of connections defined in this situation is quite different from the one definedabove on arbitrary principal fiber bundle. A linear connection is a connection defined on L M . Itis usual to look at this connection as a covariant derivative ∇ X : Γ( T M ) → Γ( T M ) which satisfies(2.7). In addition to the curvature defined as [ ∇ X , ∇ Y ] −∇ [ X,Y ] , it is possible to introduce a 2-form inΩ ( M , T M ) which is specific to linear connections, the torsion T ( X, Y ) = ∇ X Y − ∇ Y X − ∇ [ X,Y ] ∈ Γ( T M ). Another singular and important object is the soldering form θ ∈ Ω ( M , T M ) defined by θ ( X ) = X for any X ∈ Γ( T M ). The torsion is related to θ by T = ∇ θ .Following what have been explained in 2.1, let us consider L M as a principal fiber bundle withstructure group GL ( n, R ), and let us introduce a connection ω on it. Then this connection inducescovariant derivatives on any associated vector bundles, for instance ∇ on T M , but also on thebundle of forms, and generally on any bundle of tensors on M . Doing that, the gauge group isdefined as G ( L M ), so that, locally, a gauge transformation is a map γ : U → GL n ( R ).It is also natural to look locally at the covariant derivative ∇ using a coordinate system ( x µ ) onan open subset U ⊂ M . Then the local derivations ∂ µ induce natural basis on each tangent spaceover U , and, using (2.7), ∇ is completely determined by the quantities Γ ρµν , the Christoffel symbols,defined by ∇ ∂ µ ∂ ν = Γ ρµν ∂ ρ . The Γ ρµν ’s define the local trivialization of ω if one uses the ∂ µ as a local trivialization of L M .Straightforward computations then give the curvature and the torsion as R ρσ µν = ∂ µ Γ ρνσ − ∂ ν Γ ρµσ + Γ ρµη Γ ηνσ − Γ ρνη Γ ηµσ , T ρµν = Γ ρµν − Γ ρνµ . The expression of the curvature is (2.9) in this specific situation. The Ricci tensor is then definedas a contraction of the curvature: R σν = R ρσ ρν .The Christoffel symbols determine completely the linear connection ω , so defining ∇ is sufficientto introduce a linear connection. In the following we will identify a linear connection with itscovariant derivative on Γ( T M ).Let us now introduce a metric g on M . A linear connection ∇ is said to be metric if it satisfies X · g ( Y, Z ) = g ( ∇ X Y, Z ) + g ( Y, ∇ X Z ) for any X, Y, Z ∈ Γ( T M ). It is well-known that there is a9nique torsionless metric linear connection ∇ LC . It is the Levi-Civita connection, whose Christoffelsymbols are Γ ρµν = g ρσ ( ∂ ν g σµ + ∂ µ g σν − ∂ σ g µν ) . (2.12)The metric g can be used to contract indices of the Ricci tensor R σν to produce the scalar curvature R = g σν R σν .The Einstein’s theory of gravitation has been formulated historically in terms of these structures,using the action S [ g ] = − πG Z R q | g | d x. (2.13)on a 4-dimensional space-time manifold M .Written in this form, i.e. starting from the metric g as the primary field, this theory is not agauge field theory. Since the Lagrangian L = R is a scalar in terms of natural structures on M (see[49] for this notion), the theory is only invariant under the action of the group of diffeomorphismsof M : there is no gauge group in the theory, in the sense of 2.1. This is also confirmed by the factthat the primary object in the theory is the metric field g , and not a connection 1-form ω , whichderives from g by (2.12). The covariant derivative is a byproduct, whose purpose is for instance towrite equations of motion of point-like objects, in the form of geodesic equations.Nevertheless, if one wants to stress the importance of the linear connection ∇ LC , one is temptedto look at this theory as a gauge theory for the gauge group G ( L M ), which is related to thediffeomorphisms group of M by the short exact sequence: / / G ( L M ) / / Aut( L M ) / / Diff( M ) / / , (2.14)where Aut( L M ) is the group of all automorphisms of the principal fiber bundle L M . But thisgroup Aut( L M ) is superfluous, since all the symmetries of S [ g ] are already in the group Diff( M ). Provided one departs from the theory of connections described in 2.1, it is possible to write Einstein’stheory of gravitation as a gauge field theory.The original idea of Klein, formulated in his
Erlangen program of 1872, is to characterize ageometry as the study of the invariants of an homogeneous and isotropic space. In modern language,a Klein geometry is a couple of Lie groups (
G, H ) such that H is a closed subgroup of G , and G/H is the homogeneous space.The purpose of Cartan geometry is to consider a global manifold which can be locally modeledon a Klein geometry (
G, H ). In the following, we use the bundle definition of a Cartan geometry,described as follows [70].Let (
G, H ) be as before, and denote by ( g , h ) the associated Lie algebras. A Cartan geometryconsists in the following data:1. a principal fiber bundle P with structure group H on a smooth manifold M ;2. a g -valued 1-form ̟ on P such that:a. e R ∗ h ̟ = Ad h − ̟ for any h ∈ H ,b. ̟ ( ξ P ) = ξ for any ξ ∈ h ,c. at each point p ∈ P , the linear map ̟ p : T p P → g is an isomorphism.Notice that the dimension of G is exactly the dimension of P , i.e. the dimension of M plus thedimension of H . Condition 2.c is a strong requirement: the principal fiber bundle P is “soldered”to the base manifold, which constrains its global topology.The curvature Ω ∈ Ω ( P ) ⊗ g is defined as Ω = d ̟ + [ ̟, ̟ ], it vanishes on vertical vectorfields. Denote by ρ : g → g / h the quotient map, then the torsion is ρ (Ω). For any ξ ∈ g and10 ∈ P , ̟ − p ( ξ ) ∈ T p P . But if ξ ∈ h , then ̟ − p ( ξ ) ∈ V p P (vertical vectors in T p P ), so thatΩ p ( ̟ − p ( ξ ) , X p ) = 0 for any X p ∈ T p P .A simple example consists to consider P = G , M = G/H and ̟ = θ G , the Maurer-Cartan 1-form on G . Then the curvature is zero. This is the Klein model on which general Cartan geometriesare based.Let Ξ ∈ G ( P ) be a vertical automorphism of P . Then it acts by pull-back on ̟ and ( P , Ξ ∗ ̟ )defines another Cartan geometry. The gauge group of a Cartan geometry is then the (ordinary)gauge group of P .A reductive Cartan geometry corresponds to a situation when one has a H -module decomposition g = h ⊕ p , where p is a Ad H -module (reductive decomposition of g ). Then the Cartan connection ̟ splits as ̟ = ω ⊕ β , where ω takes its values in h and β in p . From the hypothesis on ̟ , onecan check that ω is an Ehresmann connection on P . The 1-form β is called a soldering form on P :for any p ∈ P , it realizes an isomorphism β p : H p P → p , where H p P = Ker ω p is the horizontalsubspace of T p P associated to ω . In particular, it vanishes on V p P . The curvature 2-form splits aswell into Ω = Ω + τ (Ω), where Ω is the curvature of ω in the sense of 2.1, and τ (Ω) is the torsion.Let ( U , φ ) be a local trivialization of P , and let s : U → P |U be its associated local section.Denote by Γ = s ∗ ω and Λ = s ∗ β the local trivializations of ω and β . Then Λ x : T x U → p isan isomorphism. Let η be a Ad H -invariant bilinear form on p . Then, for any X , X ∈ T x U , g x ( X , X ) = η (Λ x ( X ) , Λ x ( X )) defines a metric on M . Using local coordinates ( x µ ) on U , anda basis { e a } of p , one has Λ = Λ aµ d x µ ⊗ e a , and g µν = η ab Λ aµ Λ bν with obvious notations. Thisrelation is well known in the tetrad formulation of General Relativity.Consider now a reductive Cartan geometry, on an orientable manifold M , based on the groups G = SO (1 , m − ⋉ R m and H = SO (1 , m − p = R m . Let ̟ = ω ⊕ β be a Cartanconnection on P , and denote as before Γ and Λ the local trivializations of ω (the spin connection)and β . Let us introduce a basis { e a } ≤ a ≤ m of R m , so that any element ξ ∈ h is a matrix ( ξ ab ) ≤ a,b ≤ m .The local 1-form Γ can be written as Γ = (Γ abµ d x µ ) ≤ a,b ≤ m and the local 1-form Λ is vector-valued(Λ aµ d x µ ) ≤ a ≤ m . Denote by R = ( R abµν d x µ ∧ d x ν ) ≤ a,b ≤ m the local expression of the curvature of ω . Finally, define g as the metric on M induced by β and the Ad H -invariant Minkowski metric η on R m , and denote by ⋆ its Hodge star operator.The action functional of General Relativity can be written as S Gauge + S Matter where S Gauge [ ω, β ] = − πG Z R ab ∧ ⋆ (Λ b ∧ Λ a ) (2.15)is the (tetradic) Palatini action functional and S Matter is the action functional of the matter, whichdepends also on β and ω by minimal coupling. The equations of motion are obtained by varying β and ω independently: the first one gives the usual Einstein’s equations, and varying the spinconnection relates the torsion τ (Ω) to the spin of matter (see [42] for details). When the spin ofmatter is zero, the torsion vanishes, and one gets the usual Einstein’s theory of gravitation. Themodel described above is the Einstein-Cartan version of General Relativity, which takes into accountspin of matter.In a reductive Cartan geometry, the principal fiber bundle P is necessarily a reduction of the GL + n ( R )-principal fiber bundle L M to the subgroup H [70, Lemma A.2.1]. In the present case,the soldering form realizes an isomorphism between P and the SO (1 , m − M for the metric g (induced by β ). The usual point of view is toconsider that the theory defined on P is induced, through a metric g , by a symmetry reduction GL + n ( R ) → SO (1 , m − ω is obtained from a g -compatible linear connection ∇ on L M [73].But, following the general scheme presented in [40], one can consider the linear connection ∇ on L M as the result of the decoupling of the gauge symmetry on P to “nothing”. Indeed, from the local11rivializations Γ and Λ, one can construct the gauge invariant composite fields Λ − ΓΛ + Λ − dΛ,which turn out to behave geometrically as the Christoffel symbols of a linear connection ∇ on L M .This amounts to decoupling completely the SO (1 , m −
1) internal gauge degrees of freedom, and theresult is a purely geometric theory. Accordingly, the gauge invariant Einstein-Cartan action (2.15)reduces to the geometrically well-defined Einstein-Hilbert action (2.13) constructed on tensorialquantities only. This procedure can be interpreted with the help of the short exact sequence ofgroups / / SO (1 , m − / / SO (1 , m − ⋉ R m / / R m / / , (2.16)where the decoupling of the SO (1 , m −
1) part of the symmetry reduces the total symmetry basedon SO (1 , m − ⋉ R m to the diffeomorphisms only (encoded in the R m part).Thus, the original geometric formulation of Einstein’s theory of gravitation can be lifted to agauge field theory in the framework of reductive Cartan geometries. But this construction doesnot make apparent new fundamental symmetries: following the procedure introduced in [40], a fulldecoupling of the gauge group can be realized as a mere change of variables in the space of fields,and it gives rise to the original formulation of the theory. Noncommutative geometry is not a physical theory, contrary to string theories or quantum loopgravity. It a mathematical research activity which has emerged in the 80’s [12; 14; 18; 27; 29; 43; 53]at the intersection of differential geometry, normed algebras and representation theory. In particular,as a generalization of ordinary differential geometry, noncommutative geometry has shed new lightson gauge field theories. A notion of connections can be defined in terms of modules and differentialcalculi, which is the natural language of noncommutative geometry.The main difficulty to get a clear view of these achievements comes from the fact that manyapproaches have been proposed to study the differential structure of noncommutative spaces. Twoof them will be of interest here. The theory of spectral triples, developed by Connes, emphasizesthe metric structure [13; 18; 43], which is encoded into a Dirac operator. On the other hand, manynoncommutative spaces can be studied through a more canonical differential structure [6; 19–21; 27–29; 58; 59; 61; 62; 64], based on the space of derivations of associative algebras.In spite of that, many of the noncommutative gauge field theories that have been developed andstudied so far use essentially the same ideas and the same building blocks. Independently of theirexact constitutive elements, many of these gauge theories share some common or similar features,among them the origin of the gauge group and the possibility to naturally produce Yang-Mills-HiggsLagrangians.The following review on gauge field theories in noncommutative geometry can be completedby [64].
Noncommutative geometry relies on fundamental theorems which identify the good algebra of func-tions on specific spaces that encode all the structure of the space. For instance, the Gelfand-Naïmarktheorem tells us that a unital commutative C ∗ -algebra is always the commutative algebra of continu-ous functions on a compact topological space, equipped with the sup norm. Studying a commutative C ∗ -algebra is studying the underlying topological space. In the same way, one can study a mea-surable space using the commutative von Neumann algebra of bounded measurable functions. Fordifferentiable manifolds, no such theorem has been established. Nevertheless, the reconstructiontheorem by Connes [16] is an attempt to characterize spin manifolds using commutative spectraltriples. 12hese fundamental theorems do not only identify the good (category of) algebras, they alsoproduce a collection of tools to study these spaces using only these algebras. And these toolsare defined on, or can be generalized to, noncommutative algebras in the same category. Amongthe fundamental tools, two of them must be mentioned: K -theory [5; 44; 68; 75], and it dual K -homology which is at the heart of the mathematical motivation for spectral triples, and cyclichomology [25; 55]. These tools permits to revoke the assumption about the commutativity of thealgebra describing the space under study, and to consider “noncommutative” versions of these spacesas noncommutative algebras in the same category.The theory of vector bundles plays an essential role on gauge field theories, as mentioned inSection 2. The good noncommutative notion of vector bundle is played by projective finitely gener-ated modules over the algebra. This characterization relies on theorems by Serre and Swan [69; 72]which identify in this algebraic way sections of vector bundles.Connections use also some notion of differentiability, for instance the de Rham differential orthe covariant derivative. In noncommutative geometry, it is accustomed to replace the de Rhamspace of forms by a differential calculus associated to the associative algebra we want to study. Inthe following, we will assume that the reader is familiar with certain basic algebraic notions, suchas associative algebras, modules, graduations, involutions (see [45] for instance).A differential calculus on an associative algebra A is a graded differential algebra (Ω • , d) suchthat Ω = A . The space Ω p is called the space of noncommutative p -forms (or p -forms in short),and it is automatically a A -bimodule. By definition, d : Ω • → Ω • +1 is a linear map which satisfiesd( ω p η q ) = (d ω p ) η q + ( − p ω p (d η q ) for any ω p ∈ Ω p and η q ∈ Ω q . We will suppose that A has aunit , and then this property implies d = 0. When A is equipped with an involution a a › , wecan suppose that the graded algebra Ω • has also an involution, denoted by ω p ω › p , which satisfies( ω p η q ) › = ( − pq η › q ω › p for any ω p ∈ Ω p and η q ∈ Ω q , and we suppose that the differential operatord is real for this involution: (d ω p ) › = d( ω › p ).There are many ways to define differential calculi, depending on the algebra under investigation.Two of them will be of great interest in the following. The first one is the de Rham differentialcalculus (Ω • ( M ) , d) on the algebra A = C ∞ ( M ), where M is a smooth manifold. This one neednot be described further. It is the “commutative model” of noncommutative geometry.The second one can be attached to any unital associative algebra: it is the universal differentialcalculus, denoted by (Ω • U ( A ) , d U ) (see for instance [28] for a concrete construction). It is definedas the free unital graded differential algebra generated by A in degree 0. The unit in Ω • U ( A ) is alsoa unit for Ω U ( A ) = A , so that it coincides with the unit of A . This differential calculus has anuniversal property (so its name) formulated as follows: for any unital differential calculus (Ω • , d)on A , there exists a unique morphism of unital differential calculi φ : Ω • U ( A ) → Ω • (of degree 0)such that φ ( a ) = a for any a ∈ A = Ω U ( A ) = Ω . This universal property permits to characterizeall the differential calculi on A generated by A in degree 0 as quotients of the universal one. Evenif A is commutative, Ω • U ( A ) need not be graded commutative.An explicit construction of (Ω • U ( A ) , d U ) describes Ω nU ( A ) as finite sum of elements a d U b · · · d U b n for a, b , . . . , b n ∈ A , where the notation d U b can be considered as formal, except that it takes intoaccount the important relation d U = 0. Then if A is involutive, the involution on Ω • U ( A ) is definedas ( a d U b · · · d U b n ) › = ( − n ( n − (d U b › n ) · · · (d U b › ) a › . This differential calculus is strongly related to Hochschild and cyclic homology [25; 55; 63].Noncommutative connections are defined using the characterization (2.10) of ordinary connec-tions, and the fact that, due to the Serre-Swan theorem, the good notion of “noncommutative vectorbundle” is the notion of (projective finitely generated) module.13et M be a right A -module, and let (Ω • , d) be a differential calculus on A . Then a noncom-mutative connection on M is a linear map b ∇ : M → M ⊗ A Ω , such that b ∇ ( ma ) = ( b ∇ m ) a + m ⊗ d a, (3.1)for any m ∈ M and a ∈ A . This map can be extended as b ∇ : M ⊗ A Ω p → M ⊗ A Ω p +1 , for any p ≥
0, using the derivation rule b ∇ ( m ⊗ ω p ) = ( b ∇ m ) ⊗ ω p + m ⊗ d ω p for any ω p ∈ Ω p .The curvature of b ∇ is then defined as b R = b ∇ = b ∇ ◦ b ∇ : M → M ⊗ A Ω , and it satisfies b R ( ma ) = ( b Rm ) a for any m ∈ M and a ∈ A . The space A ( M ) of noncommutative connectionson M is an affine space modeled on the vector space Hom A ( M , M ⊗ A Ω ) of right A -modulesmorphisms.Suppose now that A has an involution. A Hermitian structure on M is a R -bilinear map h− , −i : M ⊗ M → A such that h ma, nb i = a › h m, n i b and h m, n i › = h n, m i for any a, b ∈ A and m, n ∈ M . There is a natural extension h− , −i to ( M ⊗ A Ω p ) ⊗ ( M ⊗ A Ω q ) → Ω p + q defined by h m ⊗ ω p , n ⊗ η q i = ω › p h m, n i η q . A noncommutative connection b ∇ is said to be compatible with h− , −i , or Hermitian, if, for any m, n ∈ M , h b ∇ m, n i + h m, b ∇ n i = d h m, n i . In this context, the gauge group G is then defined as the group of automorphisms of M as aright A -module: Φ ∈ G satisfies Φ( ma ) = Φ( m ) a for any m ∈ M and a ∈ A . It depends on thechoice of the right module M . We can extend a gauge transformation Φ to a right Ω • -moduleautomorphism on M ⊗ A Ω • by Φ( m ⊗ ω ) = Φ( m ) ⊗ ω . Generalizing (2.11), we can show that themap b ∇ Φ = Φ − ◦ b ∇ ◦ Φis a noncommutative connection on M . This defines the action of gauge transformation on A ( M ).A gauge transformation Φ is said to be compatible with the Hermitian structure h− , −i if h Φ( m ) , Φ( n ) i = h m, n i for any m, n ∈ M . Denote by U ( G ) the subgroup of G of gauge transforma-tions which preserve h− , −i . This subgroup defines an action on the subspace of noncommutativeconnections compatible with h− , −i .A natural question is to ask if the space A ( M ) is not empty. There is a natural condition on M (suggested by the Serre-Swan theorem) which solves this problem. If M is a projective finitelygenerated right module, then A ( M ) is not empty. Indeed, the condition means that there is aninteger N > p ∈ M N ( A ) such that M ≃ p A N . Then p extends to a map (Ω • ) N → (Ω • ) N which acts on the left by matrix multiplication and one has M ⊗ A Ω • = p (Ω • ) N . Let b ∇ bea noncommutative connection on the right module A N . Then it is easy to show that m p ◦ b ∇ m is a noncommutative connection on M , where m ∈ M ⊂ A N . Notice then that b ∇ m = d m is anoncommutative connection on A N , so that A ( M ) is not empty. The associated noncommutativeconnection is given by b ∇ m = p ◦ d m on M , and its curvature is the left multiplication on M ⊂ A N by the matrix of 2-forms p d p d p .By construction, this definition of noncommutative connections is a direct generalization ofcovariant derivatives on associated vector bundles. There is a way to introduce algebraic structures(noncommutative forms) to replace this noncommutative covariant derivative. In order to simplifythe presentation, we will consider the particular case M = A . See [64] for the more generalsituation.With M = A , one has M ⊗ A Ω • = Ω • , and since A is unital, one has b ∇ ( a ) = b ∇ ( a ) = b ∇ ( ) a + ⊗ d a = b ∇ ( ) a + d a . This implies that b ∇ ( ) = ω ∈ Ω characterizes completely b ∇ . We14 mod 8 0 1 2 3 4 5 6 7 ǫ − − − − ǫ ′ − − ǫ ′′ − − ǫ, ǫ ′ , and ǫ ′′ according to the dimension n of the spectral triple.call ω the connection 1-form of b ∇ , and the curvature of b ∇ is the left multiplication by the 2-formΩ = d ω + ωω ∈ Ω . An element Φ of the gauge group is completely determined by Φ( ) = g ∈ A (invertible element). It acts on M by left multiplication: Φ( a ) = ga . A simple computation showsthat the connection 1-form associated to b ∇ Φ is ω g = g − ωg + g − d g and its curvature 2-form is g − (d ω + ωω ) g = g − Ω g . These relations can be compared to (2.3) or (2.5). When A is involutive, h a, b i = a › b defines a natural Hermitean structure on M , and one has U ( G ) = U ( A ), the group ofunitary elements in A . In order to simplify the presentation, we will restrict ourselves to compact spectral triples, i.e. thealgebras will be unital.A spectral triple ( A , H , D ) is composed of a unital C ∗ -algebra A , a faithful involutive represen-tation π : A → B ( H ) on a Hilbert space H , and an unbounded self-adjoint operator D on H , calleda Dirac operator, such that:– the set A = { a ∈ A / [ D , π ( a )] is bounded } is norm dense in A ;– (1 + D ) − has compact resolvent.The main points of this definition is that the representation makes H into a left A -module,and the Dirac operator D defines a differential structure (more on this later). The sub algebra A identifies with the “smooth functions” on the noncommutative space and the differential of a ∈ A is more or less d a = [ D , a ]. Be aware of the fact that this can only be an heuristic formula since thecommutator with D cannot be used to define a true differential.The spectral property of D is used to define the dimension n of the spectral triple through thedecreasing rate of the eigenvalues of |D| − . The Dirac operator gives also a geometric structure tothe spectral triple, in the sense that it gives a way to measure “lengths” between states. See [13]for further details.A spectral triple is said to be even when its dimension n is even and when there exists asupplementary operator γ : H → H such that γ › = γ , D γ + γ D = 0, γπ ( a ) − π ( a ) γ = 0, and γ = 1,for any a ∈ A . This operator is called chirality.A spectral triple is said to be real when there exists an anti-unitary operator J : H → H suchthat [
J π ( a ) J − , π ( b )] = 0, J = ǫ , J D = ǫ ′ D J and J γ = ǫ ′′ γJ for any a, b ∈ A . The coefficients ǫ, ǫ ′ , and ǫ ′′ take their values according to the dimension n of the spectral triple as given in Table 2.By definition, J π ( a ) › J − commutes with π ( A ) in B ( H ) (bounded operators on H ), so theinvolutive representation a J π ( a ) › J − of A on H induces a structure of A -bimodule on H . Wedenote it by ( a, b ) π ( a ) J π ( b ) › J − Ψ ≃ π ( a )Ψ π ◦ ( b ) for any Ψ ∈ H (the presence of J in thisformula implies the use of π ( b ) › instead of π ( b )). Then the operator D is required to be a first orderdifferential operator for this bimodule structure [30]: (cid:2) [ D , π ( a )] , J π ( b ) J − (cid:3) = 0 for any a, b ∈ A .We have presented here a restricted list of axioms for a spectral triple, but it is sufficient tounderstand the principles of the gauge theories constructed in this approach.15et us give a first example, which is the commutative model. Let M be a smooth compactRiemannian spin manifold of dimension m , and let A = C ( M ) be the commutative algebra ofcontinuous functions on M . With the sup norm, this is a (commutative) C ∗ -algebra. Let /S be aspin bundle given by the spin structure on M , and let H = L ( /S ) be the associated Hilbert space.The Dirac operator D = /∂ = iγ µ ∂ µ is the (usual) Dirac operator on /S associated to the Levi-Civitaconnection (spin connection in this context), where the γ µ ’s are the Dirac gamma matrices satisfying { γ µ , γ ν } = 2 g µν . The dimension of the spectral triple ( A , H , D ) is m , and the sub algebra A is C ∞ ( M ). When m is even, the chirality is given by γ M = − γ γ · · · γ m . The charge conjugationdefines a real structure J M on this spectral triple.We will say that two spectral triples ( A , H , D ) and ( A ′ , H ′ , D ′ ) are unitary equivalent if thereexists a unitary operator U : H → H ′ and an algebra isomorphism φ : A → A ′ such that π ′ ◦ φ = U πU − , D ′ = U D U − , J ′ = U J U − , and γ ′ = U γU − (the last two relations are required onlywhen the operators J , J ′ , γ and γ ′ exist).Then we define a symmetry of a spectral triple as a unitary equivalence between two spectraltriples such that H ′ = H , A ′ = A , and π ′ = π . In that case, U : H → H and φ ∈ Aut( A ). Asymmetry acts only on the operators D , J and γ . Let us consider the symmetries for which theautomorphisms φ are A -inner: there is a unitary u ∈ U ( A ) (unitary elements in A ) such that φ u ( a ) = uau › for any a ∈ A . Such a unitary defines all the symmetry, with U = π ( u ) J π ( u ) J − : H → H .Considering the bimodule structure on H , U is the conjugation with π ( u ): π ( u ) J π ( u ) J − Ψ ≃ π ( u )Ψ π ◦ ( u ) › . A direct computation shows that inner symmetries leave invariant J and γ , while theoperator D is modified as D u = D + π ( u )[ D , π ( u ) › ] + ǫ ′ J ( π ( u )[ D , π ( u ) › ]) J − . (3.2)We define a gauge transformation as a unitary u ∈ U ( A ) which acts on the spectral triple asdefined above. This looks different from the definition proposed in 3.1, but we will show how thetwo points of view can be reconciled.As in ordinary differential geometry, the ordinary derivative, here played by D = iγ µ ∂ µ , isnot invariant by gauge transformations, and we need an extra field to compensate for the inhomo-geneous terms in (3.2). Differential forms P i a i d U b i · · · d U b ni in the universal differential calculus(Ω • U ( A ) , d U ) defined in 3.1 can be represented on H as π D X i a i d U b i · · · d U b ni ! = X i π ( a i )[ D , π ( b i )] · · · [ D , π ( b ni )] . This suggest to interpret [ D , π ( b )] as a differential, but this is impossible: the map π D is not arepresentation of the graded algebra Ω • U ( A ), and d U is not represented by the commutator [ D , − ]as a differential. For instance [ D , [ D , π ( b )]] is not necessarily 0 as required if it were a differen-tial. Using J , there is also a representation π ◦D of Ω • U ( A ) on the right module structure of H :Ψ π ◦D (cid:0)P i a i d U b i · · · d U b ni (cid:1) = J π D (cid:0)P i a i d U b i · · · d U b ni (cid:1) › J − Ψ.Let b ∇ : A → Ω U ( A ), with ω = b ∇ ∈ Ω U ( A ), be a noncommutative connection on the right A -module A for the universal differential calculus. Using the bimodule structure on H , we have thenatural isomorphism of bimodules H ≃ A ⊗ A H ⊗ A A , where Ψ ∈ H is identified with ⊗ Ψ ⊗ .We define the modified Dirac operator D ω on H by D ω (Ψ) = π D ( ω )Ψ ⊗ + ⊗ D Ψ ⊗ + ǫ ′ ⊗ Ψ π ◦D ( ω ) › , for any Ψ ∈ H . This operator can also be written D ω = D + π D ( ω ) + ǫ ′ J π D ( ω ) J − .Let u ∈ U ( A ). As a gauge transformation (defined as an inner symmetry) of the spectral triple,it acts on D ω as( D ω ) u = D + π ( u )[ D , π ( u ) › ] + ǫ ′ J π ( u )[ D , π ( u ) › ] J − | {z } D u + π ( u ) π D ( ω ) π ( u ) › + ǫ ′ J π ( u ) π D ( ω ) π ( u ) › J − . u is also a gauge transformation as an automorphism of the right module A , a ua . Thegauge transformation of the connection 1-form ω is ω u = uωu › + u d U u › , and the associated modifiedDirac operator D ω u on H is then given by D ω u = D + π D ( uωu › + u d U u › ) + ǫ ′ J π D ( uωu › + u d U u › ) J − . Developing this relation shows that it is ( D ω ) u . This means that the two implementations of gaugetransformations coincide.It can be shown that ( A , H , D ω ) is a spectral triple. The replacement of D by D ω is called aninner fluctuation in the space of Dirac operators associated to the couple ( A , H ). In this approach,gauge fields are inner fluctuations in the space of Dirac operators. Notice that the original Diracoperator D is (in general) an unbounded operator on H , while inner fluctuations π D ( ω ) are boundedoperators by hypothesis. This implies in particular that the K -homology class defined by D and D ω are the same, since inner fluctuations then reduce to compact perturbations of the Fredholmoperator associated to D .Let consider the case of a spectral triple associated to a spin geometry, where locally D = iγ µ ∂ µ .Then an inner fluctuation corresponds to the twist of the Dirac operator by a connection defined ona vector bundle E . This procedure consists to replace /S by /S ⊗ E and to define D A = iγ µ ( ∂ µ + A µ )on this space using a connection A on E . This is the minimal replacement ∂ µ D µ explained in2.1.The spectral properties of the Dirac operator D ω is used to define a gauge invariant actionfunctional S [ D ω ] using the spectral action principle [8]: S [ D ω ] = tr χ ( D ω / Λ) , where tr is the trace on operators on H , χ is a positive and even smooth function R → R , and Λis a real (energy) cutoff which helps to make this trace well-behaved. For asymptotically large Λ,this action can be evaluated using heat kernel expansion. The action functional S [ D ω ] produces thedynamical part for the gauge fields of the theory, and one has to add the minimal coupling withfermions in the form h Ψ , D ω Ψ i for Ψ ∈ H to get a complete functional action.This procedure has been applied to propose a noncommutative Standard Model of particlephysics, which gives a clear geometric origin for the scalar fields used in the SSBM [9–11; 15]. Thismodel relies on a so-called “almost commutative geometry”, which consists to use an algebra of thetype A = C ∞ ( M ) ⊗ A F for a spin manifold M and finite dimensional algebra A F , for instance asum of matrix algebras. The total spectral triple ( A , H , D ) is the product of a commutative spectraltriple ( C ( M ) , L ( /S ) , /∂ ) with a “finite spectral triple” ( A F , H F , D F ): A = C ( M ) ⊗ A F , H = L ( /S ) ⊗ H F , D = /∂ ⊗ γ M ⊗ D F , γ = γ M ⊗ γ F , J = J M ⊗ J F . For the Standard Model, one takes A F = C ⊕ H ⊕ M ( C ) and H F = M ( C ) ⊕ M ( C ) ≃ C ,which contains exactly all the fields of a family of fermions in a see-saw model. At the end, thefull Hilbert space is taken to be H F to account for the 3 families of particles. Using this geometry,inner fluctuations can be described, and the most general Dirac operator is D ω = /∂ + iγ µ A µ + γ D F + γ Φ , where the A µ ’s contain all the U (1) × SU (2) × SU (3) gauge fields, and Φ is a doublet of scalarfields, which enters into the SSBM. Note that this (re)formulation of the Standard Model is moreconstrained than the original one (see for instance [46]).This model describes in the same Lagrangian the Standard Model of particle physics, and theEinstein’s theory of gravitation. Indeed, the group of symmetries on which this Lagrangian isinvariant is Aut( A ), which fits in the short exact sequence of groups / / Inn( A ) / / Aut( A ) / / Out( A ) / / (3.3)17here Inn( A ) are inner automorphisms, the gauge transformations, and Out( A ) are outer automor-phisms, the diffeomorphisms of M . Derivation-based noncommutative geometry was defined in [26], and it has been studied for variousalgebras, for instance in [6; 31–36; 58; 59]. See also [29; 61; 62] for reviews. The idea is to introducea natural differential calculus which is based on the derivations of the associative algebra.Let A be an associative algebra with unit , and let Z ( A ) = { a ∈ A / ab = ba, ∀ b ∈ A } itscenter. The space of derivations of A isDer( A ) = { X : A → A / X linear , X · ( ab ) = ( X · a ) b + a ( X · b ) , ∀ a, b ∈ A } . This vector space is a Lie algebra for the bracket [ X , Y ] a = XY a − YX a for all X , Y ∈ Der( A ), anda Z ( A )-module for the product ( f X ) · a = f ( X · a ) for all f ∈ Z ( A ) and X ∈ Der( A ). The subspaceInt( A ) = { ad a : b [ a, b ] / a ∈ A } ⊂ Der( A )is called the vector space of inner derivations: it is a Lie ideal and a Z ( A )-submodule. The quotientOut( A ) = Der( A ) / Int( A ) gives rise to the short exact sequence of Lie algebras and Z ( A )-modules / / Int( A ) / / Der( A ) / / Out( A ) / / . (3.4)Out( A ) is called the space of outer derivations of A . This short exact sequence is the infinitesimalversion of (3.3). If A is commutative, there are no inner derivations, and the space of outerderivations is the space of all derivations.In case A has an involution, a derivation X ∈ Der( A ) is said to be real when ( X a ) › = X a › forany a ∈ A , and we denote by Der R ( A ) the space of real derivations.Let Ω n Der ( A ) be the vector space of Z ( A )-multilinear antisymmetric maps from Der( A ) n to A ,with Ω ( A ) = A . Then the total spaceΩ • Der ( A ) = M n ≥ Ω n Der ( A )gets a structure of N -graded differential algebra for the product( ωη )( X , . . . , X p + q ) = 1 p ! q ! X σ ∈ S p + q ( − sign( σ ) ω ( X σ (1) , . . . , X σ ( p ) ) η ( X σ ( p +1) , . . . , X σ ( p + q ) )for any X i ∈ Der( A ). A differential b d is defined by the so-called Koszul formula b d ω ( X , . . . , X n +1 ) = n +1 X i =1 ( − i +1 X i · ω ( X , . . . i ∨ . . . . , X n +1 )+ X ≤ i 7→ ∇ X is asplitting of (3.8) as C ∞ ( M )-modules, but not as Lie algebras, since the obstruction R ( X, Y ) =21 ∇ X , ∇ Y ] − ∇ [ X,Y ] is precisely the curvature of ∇ . For any X ∈ Der( A ), let X = ρ ( X ). Then ρ ( X − ∇ X ) = 0, so that there is a α ( X ) ∈ A such that X = ∇ X − ad α ( X ) . The map X α ( X )belongs to Ω ( A ) and satisfies the normalization α (ad γ ) = − γ for any γ ∈ A .This map realizes an isomorphism between the space of SU ( n )-connections ∇ E on E and thespace of traceless anti-Hermitian noncommutative 1-forms α on A such that α (ad γ ) = − γ . Thenoncommutative 1-form α is defined globally on M , and it completes (in a new space of forms) thelast description proposed in 2.1 on connections and curvatures in ordinary differential geometry.It can be shown that α is defined in terms of the local trivializations A i of the connection 1-form associated to ∇ E (see [59; 62]). In the same way, the noncommutative 2-form Ω( X , Y ) = b d α ( X , Y ) + [ α ( X ) , α ( Y )] depends only on the projections X and Y of X and Y : as a section of V T ∗ M ⊗ End( E ), it identifies with F , i.e. the curvature R E of ∇ E .Notice that the gauge group G ( P ) of P is precisely SU ( A ) ⊂ A , the unitary elements in A withdeterminant 1. The action of u ∈ G ( P ) = SU ( A ) on ∇ E induces the action α α u = u − αu + u − b d u on α .Let us now consider noncommutative connections on the A -module M = A equipped withthe Hermitian structure h a, b i = a › b . From the general theory we know that any noncommutative1-form b ω defines a noncommutative connection by b ∇ X a = X · a + b ω ( X ) a for any a ∈ M = A and X ∈ Der( A ). In particular, the noncommutative 1-form α associated to ∇ E defines a noncommutativeconnection b ∇ α which can be written as b ∇ α X a = ∇ X a + aα ( X ). Then b ∇ α is compatible with theHermitian structure, its curvature is b R α ( X , Y ) = R E ( X, Y ) = b d α ( X , Y ) + [ α ( X ) , α ( Y )], and a SU ( A )-noncommutative gauge transformation on b ∇ α is exactly a (ordinary) gauge transformationon ∇ E (here we use the fact that the two gauge groups are the same). The main result of thisconstruction is that the space of noncommutative connections on the right module A compatiblewith the Hermitian structure ( a, b ) a ∗ b contains the space of ordinary SU ( n )-connections on E ,and this inclusion is compatible with the corresponding definitions of curvature and SU ( A ) = G ( P )gauge transformations.A noncommutative connection b ω describes an ordinary connection if and only if it is normalizedon inner derivations: b ω (ad γ ) = − γ . This implies that noncommutative connections have moredegrees of freedom that ordinary connections. In gauge field models, these degrees of freedomdescribe scalar fields which induce (as in the case A = C ∞ ( M ) ⊗ M n ( C )) a SSBM. We refer to [62]for more details. Lie algebroids have been defined and studied in relation with classical mechanics and its variousmodern mathematical formulations, like Poisson geometry and symplectic manifolds (see [1; 47; 50;77], [51; 57] and references in [23; 56]). This approach considers a Lie algebroid as a generalization ofthe tangent bundle, on which a Lie bracket is defined. Our approach departs from this geometricalpoint of view. We would like to consider a Lie algebroid (more precisely a transitive Lie algebroid)as an algebraic replacement for a principal vector bundle, from which it is possible to constructgauge field theories. This program has been proposed in [54] where the useful notion of connectionhas been studied, and it has been pursued in [39], where the necessary tools to build gauge fieldstheories have been defined. The usual definition of Lie algebroids consists in the following geometrical description. A Liealgebroid ( A , ρ ) is a vector bundle A over a smooth m -dimensional manifold M equipped with twostructures: 22. a structure of Lie algebra on the space of smooth sections Γ( A ),2. a vector bundle morphism ρ : A → T M , called the anchor, such that ρ ([ X , Y ]) = [ ρ ( X ) , ρ ( Y )] , [ X , f Y ] = f [ X , Y ] + ( ρ ( X ) · f ) Y , for any X , Y ∈ Γ( A ) and f ∈ C ∞ ( M ).The following (equivalent) algebraic definition of Lie algebroids will be used in the following. Thegeometric structure is ignored in favor of the algebraic structure, as in noncommutative geometry,from which we will borrow some ideas and constructions in the following.A Lie algebroid A is a finite projective module over C ∞ ( M ) equipped with a Lie bracket [ − , − ]and a C ∞ ( M )-linear Lie morphism, the anchor ρ : A → Γ( T M ), such that [ X , f Y ] = f [ X , Y ] +( ρ ( X ) · f ) Y for any X , Y ∈ A and f ∈ C ∞ ( M ).A Lie algebroid A ρ −→ Γ( T M ) is transitive if ρ is surjective. The kernel L = Ker ρ of a transitiveLie algebroid is itself a Lie algebroid with null anchor. Moreover, there exists a locally trivial bundlein Lie algebras L such that L = Γ( L ). A transitive Lie algebroid defines a short exact sequence ofLie algebras and C ∞ ( M )-modules / / L ι / / A ρ / / Γ( T M ) / / . (4.1)From a gauge field theory point of view, this short exact sequence must be looked at as an infinites-imal version of the sequence (2.1) defining a principal fiber bundle.The kernel L will be referred to as the “inner” part of A . This terminology is inspired by thephysical applications we have in mind, where Γ( T M ) will refer to (infinitesimal) symmetries onspace-time (“outer” symmetries) and L to (infinitesimal) inner symmetries i.e. infinitesimal gaugesymmetries. Compare this with the physical interpretation of the short exact sequences (3.3) and(3.8).A morphism between two Lie algebroids ( A , ρ A ) and ( B , ρ B ) is a morphism of Lie algebras and C ∞ ( M )-modules ϕ : A → B compatible with the anchors: ρ B ◦ ϕ = ρ A .The following example of transitive Lie algebroid is fundamental to define the correct notionof representation. Let E be a vector bundle over M , and let A ( E ) be the associative algebra ofendomorphisms of E as in the end of 3.3. Denote by D ( E ) the space of first-order differentialoperators on E with scalar symbols. Then the restricted symbol map σ : D ( E ) → Γ( T M ) producesthe short exact sequence / / A ( E ) ι / / D ( E ) σ / / Γ( T M ) / / . D ( E ) is the transitive Lie algebroid of derivations of E [50; 52]. A representation of a transitive Liealgebroid A ρ −→ Γ( T M ) on a vector bundle E → M is a morphism of Lie algebroids φ : A → D ( E )[56]. This can be summarized in the commutative diagram of exact rows / / L ι / / φ L (cid:15) (cid:15) A ρ / / φ (cid:15) (cid:15) Γ( T M ) / / / / A ( E ) ι / / D ( E ) σ / / Γ( T M ) / / (4.2)where φ L : L → A ( E ) is a C ∞ ( M )-linear morphism of Lie algebras.The second example permits to embed the ordinary theory of connections on principal fiberbundle into this framework. Let P be a G -principal fiber bundle over M with projection π . We usethe notations of 2.1. The two spacesΓ G ( T P ) = { X ∈ Γ( T P ) / e R g ∗ X = X for all g ∈ G } , Γ G ( P , g ) = { v : P → g / v ( p · g ) = Ad g − v ( p ) for all g ∈ G } , C ∞ ( M )-modules. Γ G ( T P ) is the space of vector fields on P which areprojectable as vector fields on the base manifold, and Γ G ( P , g ) is the space of ( e R , Ad)-equivariantmaps v : P → g , which is also the space of sections of the associated vector bundle Ad P .Denote by ξ P the fundamental (vertical) vector field on P associated to ξ ∈ g . The map ι : Γ G ( P , g ) → Γ G ( T P ) defined by ι ( v )( p ) = − v ( p ) P| p = (cid:18) ddt p · e − tv ( p ) (cid:19) | t =0 is an injective C ∞ ( M )-linear morphism of Lie algebras. The short exact sequence of Lie algebrasand C ∞ ( M )-modules / / Γ G ( P , g ) ι / / Γ G ( T P ) π ∗ / / Γ( T M ) / / (4.3)defines Γ G ( T P ) as a transitive Lie algebroid over M . This is the Atiyah Lie algebroid associatedto P [3].Consider the case where P = M× G is trivial. The associated transitive Lie algebroid is denotedby TLA ( M , g ) = Γ G ( T P ), and called the Trivial Lie Algebroid on M for g . It is the space of sectionsof the vector bundle A = T M ⊕ ( M × g ), equipped with the anchor and the bracket ρ ( X ⊕ γ ) = X, [ X ⊕ γ, Y ⊕ η ] = [ X, Y ] ⊕ ( X · η − Y · γ + [ γ, η ]) , for any X, Y ∈ Γ( T M ) and γ, η ∈ Γ( M × g ) ≃ C ∞ ( M ) ⊗ g . The kernel is the space of sections ofthe trivial vector bundle L = M × g . The short exact sequence (4.1) is split as Lie algebras and C ∞ ( M )-modules. The importance of this notion relies on the fact that any transitive Lie algebroidcan be described locally as a trivial Lie algebroid TLA ( U , g ) over an open subset U ⊂ M . Given a representation φ : A → D ( E ), one can define an associated differential calculus in thefollowing way [56, Definition 7.1.1]. For any p ∈ N , let Ω p ( A , E ) be the linear space of C ∞ ( M )-multilinear antisymmetric maps A p → Γ( E ). For p = 0 one has Ω ( A , E ) = Γ( E ). The graded spaceΩ • ( A , E ) = L p ≥ Ω p ( A , E ) is equipped with the natural differential b d φ : Ω p ( A , E ) → Ω p +1 ( A , E )defined on ω ∈ Ω p ( A , E ) by the Koszul formula( b d φ ω )( X , . . . , X p +1 ) = p +1 X i =1 ( − i +1 φ ( X i ) · ω ( X , . . . i ∨ . . . . , X p +1 )+ X ≤ i 1, there is a map i X : Ω p ( A , E ) → Ω p − ( A , E ) such that therelations i f X = f i X , i X i Y + i Y i X = 0 , [ L X , i Y ] = i [ X , Y ] , [ L X , L Y ] = L [ X , Y ] , (4.4)hold for any X , Y ∈ B , f ∈ C ∞ ( M ), where L X = b d φ i X + i X b d φ . We will denote by ( B , i, L ) such aCartan operation on (Ω • ( A , E ) , b d φ ). A Cartan operation of a Lie algebra can also be defined in thesame way.Given a Cartan operation, one can define horizontal, invariant and basic elements in Ω • ( A , E ):Ω • ( A , E ) Hor is the graded subspace of horizontal elements (kernel of all the i X , for X ∈ B ), Ω • ( A , E ) Inv is the graded subspace of invariant elements (kernel of all the L X , for X ∈ B ), and Ω • ( A , E ) Basic =Ω • ( A , E ) Hor ∩ Ω • ( A , E ) Inv is the graded subspace of basic elements.The kernel L of A defines a natural Cartan operation when the map i is the restriction to ι ( L )of the ordinary inner operation on forms.Let A = TLA ( M , g ) be a trivial Lie algebroid. Then the graded commutative differential algebra(Ω • ( A ) , b d A ) is the total complex of the bigraded commutative algebra Ω • ( M ) ⊗ V • g ∗ equipped withthe two differential operatorsd : Ω • ( M ) ⊗ V • g ∗ → Ω • +1 ( M ) ⊗ V • g ∗ , s : Ω • ( M ) ⊗ V • g ∗ → Ω • ( M ) ⊗ V • +1 g ∗ , where d is the de Rham differential on Ω • ( M ), and s is the Chevalley-Eilenberg differential on V • g ∗ ,so that b d A = d + s. In the same way, the graded differential Lie algebra (Ω • ( A , L ) , b d) is the totalcomplex of the bigraded Lie algebra Ω • ( M ) ⊗ V • g ∗ ⊗ g equipped with the differential d and theChevalley-Eilenberg differential s ′ on V • g ∗ ⊗ g for the adjoint representation of g on itself, so that b d = d + s ′ . We will use the compact notation (Ω • TLA ( M , g ) , b d TLA ) for this graded differential Liealgebra.Let A be the Atiyah Lie algebroid of a G -principal fiber bundle P over M , and denote by(Ω • Lie ( P , g ) , b d) its associated differential calculus of forms with values in its kernel. Let g equ = { ξ P ⊕ ξ / ξ ∈ g } ⊂ Γ( T P ⊕ ( P × g )): it is a Lie sub algebra of the trivial Lie algebroid TLA ( P , g ),and, as such, it induces a natural Cartan operation on the differential complex (Ω • TLA ( P , g ) , b d TLA ).Let us denote by (Ω • TLA ( P , g ) g equ , b d TLA ) the differential graded subcomplex of basic elements.It has been proved in [54] that when G is connected and simply connected, (Ω • Lie ( P , g ) , b d) and(Ω • TLA ( P , g ) g equ , b d TLA ) are isomorphic as differential graded complexes. This describes the differen-tial calculus of an Atiyah Lie algebroid as the subspace of basic forms in Ω • ( P ) ⊗ V • g ∗ ⊗ g .This description must be compared with the description of sections of an associated fiber bundleas equivariant maps on the principal fiber bundle with valued in the space which is the fiber modelof the associated fiber bundle. Let us consider the theory of connections in this framework. There is first an ordinary notion ofconnection [56], defined on a transitive Lie algebroid A ρ −→ Γ( T M ), as a splitting ∇ : Γ( T M ) → A of the short exact sequence (4.1) as C ∞ ( M )-modules. Its curvature is defined to be the obstructionto be a morphism of Lie algebras: R ( X, Y ) = [ ∇ X , ∇ Y ] − ∇ [ X,Y ] .To such a connection there is an associated 1-form defined as follows: for any X ∈ A , let X = ρ ( X ), then X − ∇ X ∈ Ker ρ , so that there is an element α ( X ) ∈ L such that X = ∇ X − ι ◦ α ( X ) . (4.5)25he map α : A → L is a morphism of C ∞ ( M )-modules, so that α ∈ Ω ( A , L ). It is normalized on ι ◦ L by the relation α ◦ ι ( ℓ ) = − ℓ for any ℓ ∈ L . Conversely, any 1-form α ∈ Ω ( A , L ) normalized as beforedefines a connection on A . The 2-form b R = b d α + [ α, α ] is horizontal for the Cartan operation of L on (Ω • ( A , L ) , b d), and with obvious notations, one has ι ◦ b R ( X , Y ) = ι (cid:16) ( b d α )( X , Y ) + [ α ( X ) , α ( Y )] (cid:17) = R ( X, Y ). b R ∈ Ω ( A , L ) is called the curvature 2-form of ∇ . It satisfies the Bianchi identity b d b R +[ α, b R ] = 0.For the transitive Lie algebroid D ( E ) of derivations of a vector bundle, a connection ∇ E associatesto any X ∈ Γ( T M ) a map ∇ E X : Γ( E ) → Γ( E ), and all the relations of (2.7) are satisfied. This isthen an ordinary covariant derivative on E .More generally, let φ : A → D ( E ) be a representation of A and α the connection 1-form of aconnection ∇ on A . Then for any ψ ∈ Γ( E ), A ∋ X φ ( X ) · ψ + φ L ( α ( X )) ψ vanishes for X = ι ( ℓ ) forany ℓ ∈ L , so that ∇ E X ψ = φ ( X ) · ψ + φ L ( α ( X )) ψ is well-defined with X = ρ ( X ), and it is a covariantderivative on E , in the sense of (2.7).A connection on the Atiyah Lie algebroid of a principal fiber bundle P associates to X ∈ Γ( T M )a right invariant vector field ∇ X ∈ Γ G ( T P ). This corresponds to the usual horizontal lift X X h defined by a connection ω on P . Suppose now that G is connected and simply connected. Theconnection 1-form ω on P is an element of Ω ( P ) ⊗ g , and so of Ω ( P ) ⊗ V g ∗ ⊗ g , which satisfies(2.2). Let θ ∈ V g ∗ ⊗ g be the Maurer-Cartan 1-form on G , which can be considered as an elementin C ∞ ( P ) ⊗ V g ∗ ⊗ g . The difference ω − θ then belongs to Ω TLA ( P , g ), and using the properties of ω and θ , it is easy to show that it is g equ -basic. As a basic element in Ω TLA ( P , g ), it identifies with α ∈ Ω ( P , g ) in the correspondence described before (see [54] for details).As a global object on M , the generalized 1-form α associated to the connection 1-form ω on P completes the last description proposed in 2.1 on connections and curvatures in ordinary differentialgeometry. Using local descriptions of the Atiyah transitive Lie algebroid Γ G ( T P ) (in terms of localtrivializations of P ), the local descriptions of α are given by A i − θ , where the A i ’s are the localtrivialization 1-forms of ω .The three above examples show that this ordinary notion of connections on transitive Lie alge-broid is close to the geometric notion of connections described in 2.1.This notion of connections on Lie algebroids admits generalizations under different names ([37]and references therein): A -connections or A -derivatives. Here we introduce a definition proposedin [54], which is more restrictive than the other ones, but which fits perfectly with the ambition topromote a transitive Lie algebroid to an infinitesimal version of a principal fiber bundle. Such aprincipal fiber bundle supports the primary notion of connection and defines completely the gaugegroup, and all these notions are transferred to associated vector bundle (“representations”). Wewill do the same for transitive Lie algebroids.A generalized connection 1-form on the transitive Lie algebroid A is then a 1-form b ω ∈ Ω ( A , L ),and its curvature is the 2-form b R = b d b ω + [ b ω, b ω ] ∈ Ω ( A , L ).Since A is a kind of infinitesimal version of a principal fiber bundle, there is no notion of gaugetransformations as “finite” transformations, but we can identify a Lie algebra of infinitesimal gaugetransformations to be L . There is at least two motivations for that. First, let φ : A → D ( E ) be arepresentation of A . Then, for any ξ ∈ L , φ L ( ξ ) defines an infinitesimal gauge transformation on E . Notice that the gauge group of E is well defined as Aut( E ), the (vertical) automorphisms of E .Secondly, for an Atiyah transitive Lie algebroid associated to a principal fiber bundle P , the kernel L = Γ G ( P , g ) identifies as the Lie algebra of infinitesimal gauge transformations on P . These twoexamples motivate also the following definition.The action of an infinitesimal gauge transformation ξ ∈ L on a generalized connection 1-form b ω is defined to be the 1-form b ω ξ = b ω + ( b d ξ + [ b ω, ξ ]) + O ( ξ ).An ordinary connection ∇ on A defines a 1-form α normalized on ι ( L ). This 1-form then definesa generalized connection on A . This implies that the space of ordinary connections on A is contained26n the space of generalized connection 1-forms, and this inclusion is compatible with the notions ofcurvature and (infinitesimal) gauge transformations.Let φ : A → D ( E ) be a representation of A , and let b ω ∈ Ω ( A , L ) be a generalized connection on A . Then its covariant derivative b ∇ on E is defined for any X ∈ A as b ∇ E X = φ ( X ) + ι ◦ φ L ◦ b ω ( X ) . This is a differential operator on Γ( E ), and one has [ b ∇ E X , b ∇ E Y ] − b ∇ E [ X , Y ] = φ L ◦ b R ( X , Y ) for any X , Y ∈ A .To summarize, we get the following diagram with the structures defined above: / / L ι / / φ L (cid:15) (cid:15) A b ω w w ρ / / φ b ∇ E (cid:15) (cid:15) Γ( T M ) / / / / A ( E ) ι / / D ( E ) σ / / Γ( T M ) / / b ∇ E is often called a generalized representation, in the sense that it is not compatible with the Liebrackets.Generalized connections of Atiyah Lie algebroids can be described as g equ -basic 1-forms inΩ TLA ( P , g ): b ω = ω + φ ∈ Ω TLA ( P , g ) = (Ω ( P ) ⊗ g ) ⊕ ( C ∞ ( P ) ⊗ V g ∗ ⊗ g ) . (4.6)where ω and φ are g equ -invariant, but ω is not necessarily a connection 1-form on P and φ is notnecessarily related to the Maurer-Cartan form on G . Atiyah Lie algebroids admit a notion of (finite)gauge transformations as elements in the ordinary gauge group G ( P ) of P . This shows that thetheory of generalized connections on Atiyah Lie algebroids is a close extension of the theory ofordinary connections on P .A general theory of metrics, Hodge star operators, and integrations on transitive Lie algebroidshas been developed in [39], which permits to write explicit gauge invariant actions for generalizedconnections and its coupling, via a covariant derivative, to matter fields in a representation E of A .A metric on A is a symmetric, C ∞ ( M )-linear map b g : A ⊗ C ∞ ( M ) A → C ∞ ( M ). Under certainnon degeneracy conditions, such a metric decomposes in a unique way into three pieces:1. a metric g on M ,2. a metric h on the vector bundle L such that L = Γ( L ),3. an ordinary connection ˚ ∇ on A , with associated generalized 1-form ˚ ω .A Hodge star operator ⋆ can then be defined, as well as an integration along the inner part L , which,combined with the integration on M against the measure dvol g , produces a global integration R A .The gauge invariant action is then defined as S Gauge [ b ω ] = Z A h ( b R, ⋆ b R ) , (4.7)where b R is the curvature of b ω . In order to understand the content of this action functional, one hasto introduce the following elements:– τ = b ω ◦ ι + Id L is an element of End( L ), which contains the degrees of freedom of b ω along L ,that is the algebraic part of b ω ;– R τ : L × L → L is the obstruction for τ ∈ End( L ) to be an endomorphism of Lie algebras: R τ ( γ, η ) = [ τ ( γ ) , τ ( η )] − τ ([ γ, η ]) for any γ, η ∈ L ;27 ω = b ω + τ (˚ ω ) is the generalized 1-form of an ordinary connection ∇ on A , which contains thedegree of freedom of b ω along M , that is the geometric part of b ω ;– ( D X τ )( γ ) = [ ∇ X , τ ( γ )] − τ ([˚ ∇ X , γ ]) is a covariant derivative of τ along the ordinary connec-tions ∇ , for any X ∈ Γ( T M ) and γ ∈ L ;– b F = R − τ ◦ ˚ R ∈ Ω ( M , L ), in which ˚ R, R ∈ Ω ( M , L ) are the curvature 2-forms of theordinary connections ˚ ω and ω .Then the curvature of b ω decomposes into three parts: b R = ρ ∗ b F − ( ρ ∗ D τ ) ◦ ˚ ω + ˚ ω ∗ R τ , and S Gauge [ b ω ]is a sum of the squares of these three pieces.When b ω is an ordinary connection (normalized on L ), τ = 0, so that R τ = 0, ω = b ω , D X τ = 0, b F = R . On an Atiyah Lie algebroid, the action functional then reduces exactly to the Yang-Millsaction (2.8).The gauge theories obtained in this way are of Yang-Mills-Higgs type: the fields in the ordinaryconnection ω are Yang-Mills-like fields, and the τ ’s fields behave as scalar fields which exhibit aSSBM. Indeed, the potential for these fields is the square of ˚ ω ∗ R τ , and it vanishes when τ is a Liealgebra morphism. This can occur for instance when τ = Id L , and this non zero configuration, oncereported into the square of the covariant derivative ( ρ ∗ D τ ) ◦ ˚ ω , induces mass terms for the (geometric)fields contained in ω . There is a similar decomposition of the action functional associated to theminimal coupling with matter fields, and the algebraic part τ of b ω induces also mass terms for thesematter fields. It is worthwhile to notice similarities between some of the constructions presented at the end ofsection 3.3 on the endomorphism algebra of a SU ( n )-vector bundle, and some of the construc-tions presented in section 4.3 on transitive Lie algebroids. In particular, they share the followingstructures:– both constructions make apparent a short exact sequence of Lie algebras and C ∞ ( M )-modules,(3.8) and (4.1);– the notion of ordinary connections corresponds in both situations to a splitting of these shortexact sequences;– the connection 1-form associated to such an ordinary connection uses in both situations thedefining relation (4.5);– gauge field theories written in both situations are of the Yang-Mills-Higgs type, and theyremain close to ordinary gauge field theories in their formulation.These similarities are not pure coincidence. They reflect a result proved in [54], where we use the factthat (3.8) defines Der( A ) as a transitive Lie algebroid. The following three spaces are isomorphic:1. The space of generalized connection 1-forms on the transitive Lie algebroid Der( A ).2. The space of generalized connection 1-forms on the Atiyah Lie algebroid Γ G ( T P ), where P isthe principal fiber bundle underlying the geometry of the endomorphism algebra A .3. The space of traceless noncommutative connections on the right A -module M = A The isomorphisms are compatible with curvatures and (finite) gauge transformations. Moreover,these spaces contain the ordinary connections on P , and the inclusion is compatible with curvaturesand gauge transformations.This result shows that the two generalizations of the ordinary notion of connections proposedin 3.3 and 4.3 are more or less the same, and they extend in the same “direction” the usual notionof connection introduced in the geometrical framework described in 2.1. In both constructions, the28eneralized connections split into two parts, see (3.6) and (4.6): a Yang-Mills type vector field a or ω , and some scalar fields b or φ . The corresponding Lagrangians (3.7) and (4.7) provide forfree a (purely algebraic) quadratic potential for the scalar fields which allows a SSBM with massgeneration. This cures the mathematical weakness of the SM stressed in the introduction.As explained before, noncommutative geometry restricts the possible gauge group to the auto-morphism group of the algebra A , but using Atiyah Lie algebroids, this restriction is no more true,since any principal fiber bundle can be considered.More generally, the approaches described in subsections 2.1, 3.2, 3.3 and 4.3 share a commonstructure which appears under the form of “sequences” such as (2.1), (2.14), (2.16), (3.3), (3.4),(4.1) and (4.3). All of them reproduce the same following pattern:Algebraicstructure Globalstructure Geometricstructureinclusion projection (5.1)This pattern embodies the characterization of gauge field theories exposed in the introduction. The“geometric structure” in this diagram represents the basic symmetries induced by the base (space-time) manifold M (diffeomorphisms, change of coordinate systems), while the “algebraic structure”is a supplementary ingredient on top of M (a group, a Lie algebra. . . ) from which emerges thecharacterization of gauge fields in the theory (mainly through representation theory). The “globalstructure” in the middle encodes all the symmetries of the theory, under a structure which can not besplited in general (group of all the automorphisms of a principal fiber bundle, automorphisms of anassociative algebra, transitive Lie algebroid A . . . ). The local dependance of gauge transformationsin a gauge field theory is then the result of the (geometric) implementation of an algebraic structureon top of a base manifold.Einstein’s theory of gravitation can be written in terms of purely geometric structures, on theright of the diagram. Using a suitable formalism, for instance reductive Cartan geometries, thisconstruction can be lifted to a “global structure”, in a theory which contains some extra degrees offreedom in new fields, submitted to a (new) gauge symmetry with the same amount of degrees offreedom. 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