Lie Algebroids and the Geometry of Off-shell BRST
aa r X i v : . [ h e p - t h ] J a n Lie Algebroids and the Geometry of Off-shell BRST
Luca Ciambelli a and Robert G. Leigh b a Physique Math´ematique des Interactions Fondamentales & InternationalSolvay Institutes, Universit´e Libre de Bruxelles, Campus Plaine - CP 231, 1050Bruxelles, Belgium b Illinois Center for Advanced Studies of the Universe & Department of Physics,University of Illinois, 1110 West Green St., Urbana IL 61801, U.S.A.
Abstract
It is well-known that principal bundles and associated bundles underlie the geometric structure ofclassical gauge field theories. In this paper, we explore the reformulation of gauge theories in terms ofLie algebroids and their associated bundles. This turns out to be a simple but elegant change, mathe-matically involving a quotient that removes spurious structure. The payoff is that the entire geometricstructure involves only vector bundles over space-time, and we emphasize that familiar concepts such asBRST are built into the geometry, rather than appearing as adjunct structure. Thus the formulationof gauge theories in terms of Lie algebroids provides a fully off-shell account of the BRST complex. Weexpect that this formulation will have appealing impacts on the geometric understanding of quantizationand anomalies, as well as entanglement in gauge theories. The formalism covers all gauge theories, andwe discuss Yang-Mills theories with matter as well as gravitational theories explicitly. ontents ˆ d for sections of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 The map φ E for general vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 The action of ˆ d on E -valued extended forms . . . . . . . . . . . . . . . . . . . . . . . 142.3 The BRST interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 TP / G . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 The physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ˆ d and extended forms 35 Introduction
The geometrization of gauge theories in terms of principal fibre bundles, originally initiated in [1–4], iswell-established and gives a precise mathematical structure that has many applications in both physics andmathematics. In this interpretation, a principal connection on a principal bundle corresponds to a gaugefield and charged matter fields to sections of associated bundles. One of the central ideas in gauge theoriesis BRST symmetry [5, 6], which underlies quantization and both the physical spectrum of the theories aswell as the structure of quantum anomalies [7–11]. There is a deep mathematical structure underlyingeach of these involving the cohomology of the nilpotent operator generating the BRST symmetry, deeplyrelated to Lie algebra cohomology [18]; a necessarily non-exhaustive list of references concerning BRSTcohomology includes [19–30].In physics discussions, the BRST structure is described by employing Grassmann-valued “ghost” fields.However, it has long been appreciated that there is an alternative, more geometric, viewpoint in whichthe ghost fields are nothing but partners of the gauge fields that arise naturally in the principal bundleframework. This idea was originally formulated by Thierry-Mieg and collaborators, [31–36]. Subsequently,this idea was extended and improved upon [37–46]. The main point here is that (the reader should thinkhere of Yang-Mills theories, but we will present the general case including gravity in the body of the paper)a connection on a principal fibre bundle P → M can be thought of as a section of T ∗ P valued in the Liealgebra, whereas in physics language, a gauge field is thought of as a section of T ∗ M (a one-form) valuedin the Lie algebra. Thus the gauge field is only the “horizontal” part of the connection, the “vertical” partproposed to correspond to the ghost field. This works out because the Grassmann algebra is isomorphicto an exterior differential algebra, and so the ghosts can be alternatively represented as differential formson the principal bundle.Although we regard this as a beautiful suggestion, nevertheless the Grassmann language is almostalways used in place of this geometric construction. Presumably, this is because the mathematics involvedis somewhat formidable, and the Grassmann formulation simpler to understand. One of the things that wewould like to emphasize in this paper (and indeed repair) is that the principal bundle construction obscuressome physics because of a technical point: many of the interesting features of gauge theories involve inthis formalism the tangent and cotangent bundles TP and T ∗ P , which are bundles over P itself ratherthan space-time M . This presents complications for understanding local physics on M , and in addition insome sense there is superfluous structure present. This comes about because the gauge group acts bothon the left and on the right. The left action corresponds to the familiar gauge transformations, while theright action is redundant. While this redundancy is well-understood, it complicates the presentation of thetheory. The central subject that we want to emphasize and explore in this paper involves the replacementof the principal bundle (and its tangent and cotangent bundles) by the corresponding Atiyah Lie algebroid ,first developed in [47, 48]. The Atiyah Lie algebroid is precisely the quotient of TP by the right action,and so understanding it has the benefit of the redundancies being removed. Lie algebroids in general havemany rather technical applications in mathematical physics [49–55], but are perhaps unfamiliar to mostreaders. In the interest of making the paper self-contained, we will carefully review the subject of Liealgebroids. This is by no means the first time that Lie algebroids have been advocated for formulatinggauge theories [56–61] (or for deforming them [62–64]), but we do not believe that the significance of BRSTin this framework has been appreciated. It is part of our agenda in this paper to make the material, whilemathematically rigorous, more accessible to the physics reader, while correcting and extending some of theprevious work. The primary mathematical foundations of the subject are contained in the two books [65,66]by Kirill Mackenzie. We have several motivations for advocating Lie algebroids in this context. First, they resolve the The literature on this subject is vast. A list of reviews, focussed on different aspects of BRST, may be found in [12–17]. We were saddened to hear of the recent passing of Professor Mackenzie. The present work would not have been possiblebut for his important contributions to this mathematical field. P . Thus the Lie algebroid efficientlyencodes both the diffeomorphisms of the base space-time M as well as gauge transformations, and leads toa fully covariant formalism. The significance of the fact that the Atiyah Lie algebroid is a vector bundleover M (as is any associated bundle) rather than P should not be underestimated. It makes possible theunderstanding of all physically relevant concepts in terms of maps between vector bundles, which then havea direct local interpretation on the base manifold M . This includes both connections on the Lie algebroidsas well as matter fields, represented as sections of associated bundles. As we will establish, connections ona Lie algebroid acquire a simple and effective geometrical meaning as Ehresmann connections. Here, noextra requirement of right invariance is imposed, for the latter concept is already taken care of by definitionof Atiyah Lie algebroids.We will show how BRST transformations are encoded in the structure of the Lie algebroid: theyare part of the exterior derivative, which we call ˆ d , on the Lie algebroid, going hand in hand with theusual exterior derivative on M . This structure aligns well with the above-stated usual BRST formalism,in which one considers extended forms, formal sums of differential forms on M and Grassmann-valuedghosts. Indeed, it is well-known that the corresponding extended exterior derivative combining ordinaryderivatives and BRST transformations is nilpotent and thus has a cohomology of physical significance. Inthe formalism that we will review and develop in this paper, the corresponding concept, ˆ d , is shown toinclude the standard BRST transformations and as such the latter are geometrically well-defined. We willshow that the Fadeev-Popov gauge ghost fields [67] arise as part of the connections on the Lie algebroidand all of the familiar properties of the aforementioned extended formalism, such as the “Russian formula”,are automatic, rather than being assumptions. It should be emphasized that this BRST structure, sinceit is encoded in the geometrical structure, is “off-shell”, rather than being associated with any particular(classical) action or gauge-fixing mechanism. Given a particular geometric structure, we can instead askwhat (classical) actions are consistent with it, being well-defined top forms on M that can be integrated.Such possibilities are in fact encoded in the cohomological structure of ˆ d . Therefore, the key result ofour work is the construction of a framework where classical gauge theories and their BRST structureare completely geometrized, without having extra unphysical degrees of freedom to remove with externalrestrictions on the geometry.The paper is organized as follows. In Section 2, we review the construction of transitive Lie algebroids.This section includes discussions of the maps involved in defining the Lie algebroid and their relation to theconnection. We emphasize the (global) split of the Lie algebroid into horizontal and vertical sub-bundles.As an example, we review the construction of derivations on vector bundles as having the structure of aLie algebroid. This construction serves to define a representation of a given Lie algebroid on associatedvector bundles, leading to the definition of the exterior derivative ˆ d on the Lie algebroid and associatedbundles. We emphasize in this discussion the horizontal-vertical split of ˆ d and we have chosen notationsuch that this split corresponds to the usual physics notation of a covariant derivative of a section of anassociated bundle along a direction in M (the horizontal part) and the BRST transformation (the verticalpart). We complete this section with a discussion of this novel BRST interpretation.In Section 3, we provide an account of the standard mathematical formulation of gauge theories,employing principal connections on principal gauge bundles. We emphasize that this formulation suffersfrom the presence of the redundant right action of the structure group, and that taking the quotientof TP by this right action leads to an Atiyah Lie algebroid over M . As an example of a transitive Liealgebroid, all of the results of Section 2 carry over immediately. We finish this section with a discussionof the significance of the construction to physics, briefly reviewing the aforementioned Thierry-Mieg ideaand starting a discussion on physical data that will culminate at the end of Section 4, once local fields arecarefully introduced. 4ections 2 and 3 were presented in a coordinate- and basis-independent language by making use ofarbitrary auxiliary sections of the various vector bundles. The benefit of this is that the formalism isautomatically well-defined, and fully diffeomorphism- and gauge-invariant. In Section 4, we describe thelocal structure of Atiyah Lie algebroids, to make contact with the usual physics “index” notation. We fullydescribe how the connection on an Atiyah Lie algebroid encodes locally a gauge field as well as a gaugeghost field and show how, by carefully constructing the local trivialization, the usual gauge transformationsand diffeomorphisms are recovered. We end Section 4 with a discussion of the physics of classical gaugetheories and how the Lie algebroid structures neatly encode them. We should acknowledge at this pointthat essentially all of the geometric structure described here rests on the standard notion of connectionson vector bundles (and we have made a significant effort to clarify that fact). However, with the extrageometric structure associated with the Lie algebroid, the physical interpretation is apparently novel, andnaturally includes off-shell BRST.In Section 5, we show how the Atiyah Lie algebroid structure can be used to describe standard grav-itational theories in the first order formalism. The basic geometric object is an Atiyah Lie algebroidconstructed from the frame bundle (or more precisely, a related G -structure). Along with the connection,we describe how the usual notion of a solder form can be carried over naturally to a section of a bundleassociated to this Atiyah Lie algebroid. From this point of view, the gravitational theory is essentiallyequivalent to any other Yang-Mills theory, albeit with a non-compact structure group. Finally in Section5.2, we discuss the absence of diffeomorphism ghosts in the Lie algebroid formalism and how this fits withthe more traditional BRST constructions.In Section 6, we conclude with remarks and discussion of planned followup works. Appendix A isdevoted to an account of the conventions used throughout the paper, while Appendix B discusses thehorizontal-vertical split of ˆ d acting on general forms valued in associated vector bundles. In this section, we introduce the mathematical structure behind the notion of transitive Lie algebroidsand begin the study of its consequences in physics. We start by defining the abstract notion of transitiveLie algebroids and how a horizontal distribution is constructed via an Ehresmann connection. Then, weintroduce a specific such algebroid: the bundle of derivations of an associated bundle E . This in turn allowsus to discuss the cochain complex and the action of its exterior derivative on forms valued in E . Singling outthe horizontal and complementary vertical pieces of the exterior derivative is crucial for physical application,as we carefully establish at the end of this section, where we show that the BRST transformation of physicalfields is built into this mathematical construction. The conventions adopted in this section are summarizedin Appendix A, to which we direct the reader unfamiliar with the general differential geometry abstractnotation that we employ. The theory of transitive Lie algebroids is extensively studied in Refs. [65, 66],which are useful references for this section, together with [56, 57]. A transitive Lie algebroid is a vector bundle A → M over a manifold M along with a (surjective) anchormap ρ : A → TM . For now we will not specify A , but we will consider specific cases relevant to physicslater in the paper. We will refer to sections of A as X , Y , ... ; these are not themselves vector fields, but theanchor map serves to project them onto ordinary vector fields, as ρ ( X ) ∈ Γ( TM ) . Note that we will usethe sections X , ... and vector fields X , ... as auxiliary objects; they will not be interpreted as physical fields.In TM , we have the usual Lie bracket L X Y = [ X , Y ] , and we suppose that we also have a Lie bracketon A , denoted (cid:2) X , Y (cid:3) A . The Lie bracket on A satisfies the Leibniz rule via the anchor map. That is, for Original accounts of G -structures are [68–70], see also the standard book reference [71]. , g ∈ C ∞ ( M ) , (cid:2) f X , g Y (cid:3) A = fg (cid:2) X , Y (cid:3) A + f ρ ( X )( g ) Y − g ρ ( Y )( f ) X , (1)where, since ρ ( X ) ∈ Γ( TM ) , ρ ( X )( g ) ∈ C ∞ ( M ) is the ordinary derivative of g along ρ ( X ) . It then followsfrom the Jacobi identity that the anchor map gives a representation of TM in A ; equivalently ρ is amorphism, satisfying R ρ ( X , Y ) = (cid:2) ρ ( X ), ρ ( Y ) (cid:3) − ρ ( (cid:2) X , Y (cid:3) A ) = 0. (2)Therefore the Lie bracket on A is not linear (with respect to multiplication of X or Y by functions), butonly in the sense that the usual Lie bracket on TM is non-linear, involving differentiation of the functions f , g .The image of the anchor map ρ is TM , and in general, ρ will have a kernel, ker ρ = n X ∈ Γ( A ) (cid:12)(cid:12)(cid:12) ρ ( X ) = 0 o , which is itself a vector bundle over M . Often ker ρ is referred to as the vertical sub-bundle V ⊂ A . Formally,there is an inclusion map of a vector bundle L over M , ι : L → A , whose image is ker ρ , and thus ρ ◦ ι = 0 (i.e., ρ ( ι ( µ )) = 0 , ∀ µ ∈ Γ( L ) ). Thus we have all the elements of a short exact sequence L A TM ι ρ . (3)Given that this is exact, the tangent bundle is a quotient, TM = A / V . We will denote sections of L as µ , ν and the Lie bracket on L as [ · , · ] L . We require the inclusion ι to be a morphism, R ι ( µ , ν ) = (cid:2) ι ( µ ), ι ( ν ) (cid:3) A − ι ( (cid:2) µ , ν (cid:3) L ) = 0, ∀ µ , ν ∈ Γ( L ). (4)Note that (1) then implies that the bracket on L is linear, (cid:2) f µ , g ν (cid:3) L = fg (cid:2) µ , ν (cid:3) L (5)and so the Lie bracket on L defines a Lie algebra (above each point in M ). So we see that A can be regardedas a vector bundle whose fibres at each point contain ordinary vectors tangent to M , as well as vectors ina Lie algebra L . There is an interesting integrability problem in which a Lie algebroid is “exponentiated”to a Lie groupoid, but as far as we understand, this remains an active area of mathematical research [53].In the present paper, we will not make use of Lie groupoids, but we expect that they will be of interest inphysics for certain problems at least. We have seen above that given a Lie algebroid A , the tangent bundle may be thought of as the quotient TM = A / V . Vector fields in TM are thus in one-to-one correspondence with equivalence classes of sectionsof A , with the equivalence being X ∼ X + Y V , with Y V ∈ Γ( V ) . There is no unique choice of a representativeof each class; making such a choice is equivalent to introducing a connection on a Lie algebroid, in thesense of Ehresmann. This defines a split of the short exact sequence, specifying how to lift vector fields in TM to sections of A . As such, it defines the horizontal sub-bundle H ⊂ A such that A = H ⊕ V globally, Here, and throughout the paper, when we refer to a map being a morphism, we mean that it preserves the Lie bracket,as in eq. (2). As we will see, this is a crucial concept in this subject. The quantity R ρ will be referred to as the curvature ofthe map ρ , and we will employ similar notation for other bundle maps that will arise in the following. V is the aforementioned vertical sub-bundle. We will call σ : TM → H the map realizing this lift.As such, we have ρ ◦ σ : X ρ ◦ σ ( X ) = ρ ( σ ( X )) = X , ∀ X ∈ Γ( TM ). (6)That is, ρ ◦ σ = Id TM , (7)whereas σ ◦ ρ : A → A acts as a projector on A , σ ◦ ρ ( X ) ≡ X H ∈ Γ( H ), H ⊂ A . Generally, the connection σ will have curvature, which we can express as = R σ ( X , Y ) = [ σ ( X ), σ ( Y )] A − σ ([ X , Y ]) ∈ Γ( A ). (8)So for Lie algebroids, one way to express the concept of curvature of a connection is simply the failure of σ to be a morphism. We will soon introduce other notions of curvature for a Lie algebroid, but as we willshow, they will all turn out to be equivalent.Notice that R σ is vertical, that is R σ ( X , Y ) ∈ Γ( V ), ∀ X , Y ∈ Γ( TM ) . Indeed, if we map it to TM using ρ , we find ρ ( R σ ( X , Y )) = ρ ([ σ ( X ), σ ( Y )] A ) − ρ ◦ σ ([ X , Y ]) = [ ρ ◦ σ ( X ), ρ ◦ σ ( Y )] − [ X , Y ] = 0, (9)where we used (2) and (7), and so R σ ( X , Y ) is in the kernel of ρ .Another important ingredient in the theory of Lie algebroids is the connection reform ω : A → L .Equivalently we can regard ω ∈ Γ( A ∗ × L ) , where A ∗ is the dual bundle to A , whose sections are “extendedforms”, dual to sections of A . Regarded as a map from A to L , we require the connection reform to have ker( ω ) = im ( σ ) = H ⊂ A . Thus we can now draw L A TM ι ρω σ . (10)The composition ω ◦ ι is (minus) the identity on L , whereas ι ◦ ω is the projector on A whose image is V . Thus, we now have two projectors on A , one for H and one for V . For any section X of A , we candecompose it into horizontal and vertical parts, X = σ ◦ ρ ( X ) − ι ◦ ω ( X ) ≡ X H + X V , (11)with X H ≡ σ ◦ ρ ( X ), X V ≡ − ι ◦ ω ( X ). (12) Here, we are are implying that σ ◦ ρ acts as an isomorphism on H . Consequently, since ( σ ◦ ρ ) = σ ◦ ρ , we have σ ◦ ρ (cid:12)(cid:12) H = Id H . The term reform is sometimes used in the mathematics literature [65, 66], and we use it here to help distinguish it fromthe usual notion of differential forms, sections of T ∗ M . We will later see in detail how the connection reform is related to theusual notion of connection forms in gauge theory. We refer to sections of ∧ n A ∗ as extended forms because they consist of both horizontal and vertical forms, the horizontalforms being in one-to-one correspondence with differential forms on M . This terminology is also used in the BRST formalismwhere the extension is to formally add differential forms and Grassmann quantities. Here, the vertical parts of our extendedforms will play the role of the Grassmann quantities. The minus sign appearing here is chosen so as to ensure consistency with standard notation, such as eq. (40) below. ω to (11) gives ω ( X ) on the left and ω ◦ σ ◦ ρ ( X ) − ω ◦ ι ◦ ω ( X ) on the right, which coincides with ω ( X ) since ω ◦ σ = 0 and ω ◦ ι = − Id L . (13)These formulae are a basis-independent way to project onto the horizontal and vertical, and we will makeextensive use of them throughout the paper. The split performed here is also discussed in [56, 57], with asimilar set of conventions for the various bundle maps. It is important to note the following properties ofthe Lie bracket on A with respect to this decomposition [ X H , Y V ] A ∈ Γ( V ), [ X V , Y V ] A ∈ Γ( V ), (14)which follow directly from (2) and ρ ◦ ι = 0 . The Lie bracket of two horizontal sections, [ X H , Y H ] A hasno vertical part if and only if H is an integrable distribution in A ; we will see later that integrability isequivalent to the vanishing of the curvature of the connection.The connection reform ω has a curvature which we will be able to define once we have introduced the(nilpotent) exterior derivative ˆ d on A . As we will explicitly see below, this exterior derivative is not the usualde Rham exterior derivative d associated with M , but is its analogue for A , mapping ˆ d : ∧ r A ∗ → ∧ r +1 A ∗ . To work towards an understanding of ˆ d , we introduce a first example of a Lie algebroid, the derivationsof a vector bundle. Given any vector bundle E → M , we can introduce a sequence of related bundlesof maps denoted Diff n ( E ) as follows. By Diff ( E ) is just meant the endomorphisms of the bundle, Diff ( E ) = End ( E ) , which have the property of linearity; that is, for f ∈ C ∞ ( M ) , ψ ∈ Γ( E ) and ϕ ∈ End ( E ) , ϕ ( f ψ ) = f ϕ ( ψ ). (16)The bundle End ( E ) has a Lie bracket given by the commutator of endomorphisms, which is skew andlinear, [ ϕ , f ϕ ′ ] = f [ ϕ , ϕ ′ ] . In what follows, End ( E ) will play the role of L .A first order differential operator Γ( Diff ( E )) ∋ D : E → E is such that D ( f ψ ) − fD ( ψ ) = ϕ f ( ψ ), ϕ f ∈ End ( E ). (17)We can interpret this as Leibniz if the endomorphism ϕ f involves a derivative of the function f associatedto D . To do so, one introduces an (anchor) map ρ E to TM , defining a sub-bundle Der ( E ) of Diff ( E ) .That is, we require ρ E ( D ) , for D ∈ Γ( Der ( E )) , to be an ordinary derivative on functions, ϕ f = ρ E ( D )( f ) ∈ C ∞ ( M ) ⊂ End ( E ) . The bundle Der ( E ) together with the anchor map ρ E has the structure of a Liealgebroid. So we define a derivation D ∈ Γ( Der ( E )) such that D ( f ψ ) = f D ( ψ ) + ρ E ( D )( f ) ψ . (18)The Lie bracket on Der ( E ) is just given by composition (cid:2) D , D ′ (cid:3) ( ψ ) = D ( D ′ ( ψ )) − D ′ ( D ( ψ )). (19) Given a basis of sections e a for E and e a ⊗ f b for End ( E ) , we have ϕ ( ψ ) = ϕ ab e a ⊗ f b ( ψ c e c ) = ( ϕ ab ψ b ) e a , ϕ ∈ End ( E ), ψ ∈ Γ( E ) (15)so the endomorphisms act as linear transformations of the components of the section of E . Here, we have taken f b as the dualbasis of sections of E ∗ so that f b ( e c ) = δ bc . For more details and conventions concerning bases, see Sec. 4 and App. A. Notethat for brevity we do not distinguish between the bundle End ( E ) and its space of sections Γ( End ( E )) . We also note thatconventionally, we do not underline sections of the bundles End ( E ) , Diff ( E ) and Der ( E ) . (cid:2) D , D ′ (cid:3) ( f ψ ) = D ( D ′ ( f ψ )) − D ′ ( D ( f ψ ))= D ( f D ′ ( ψ )) + D ( ρ E ( D ′ )( f ) ψ ) − D ′ ( f D ( ψ )) − D ′ ( ρ E ( D )( f ) ψ )= f (cid:2) D , D ′ (cid:3) ( ψ ) + (cid:2) ρ E ( D ), ρ E ( D ′ ) (cid:3) ( f ) ψ = f (cid:2) D , D ′ (cid:3) ( ψ ) + ρ E ( (cid:2) D , D ′ (cid:3) )( f ) ψ , (20)where in the last line we used the fact that the anchor map is a morphism. Finally, notice that sections of ker ρ E are endomorphisms (in that case, (18) becomes (16)), and so we see that indeed Der ( E ) has the Liealgebroid structure End ( E ) Der ( E ) TM ι E ρ E . (21)As remarked upon above, we will make use of this construction extensively, once we have an idea of whichvector bundles E may be relevant to a given situation. That Der ( E ) is a Lie algebroid has been explainedin [65, 66]; see also e.g., [56]. In fact, the bundle Der ( E ) gives a representation of a Lie algebroid A ifwe supply maps φ E : A → Der ( E ) and v E : L → End ( E ) . In such a situation there will be a derivationassociated to each section X of A , as graphically shown hereafter L A TM End ( E ) Der ( E ) ι v E ρφ E ι E ρ E . (22)We will later discuss our prime examples, Atiyah Lie algebroids, which are related to principal bundles,and in that case, the vector bundles E will be associated bundles carrying a representation of the structuregroup. The map v E gives, for each section of L , an endomorphism, expressing the infinitesimal action ofthe group on sections of E via the usual matrix representations. But before specializing, let us exploresome important properties of such representations.The central property of the maps v E , φ E is that they are morphisms. That is, [ φ E ( X ), φ E ( Y )] = φ E ([ X , Y ] A ), [ v E ( µ ), v E ( ν )] = v E ([ µ , ν ] L ). (23)Using the notation for bases introduced in footnote 9, for an element µ = µ A t A , with { t A } a basis ofsections of Γ( L ) (we again refer to App. A for further details), we write v E ( µ ) = µ A v E ( t A ) = µ A ( t A ) ab e a ⊗ f b ≡ µ ab e a ⊗ f b . (24)The fact that v E is a morphism simply translates into the fact that the ( t A ) ab give a matrix representationof the Lie algebra (see footnote 12 below).Similarly, since φ E ( X ) ∈ Γ( Der ( E )) for any X ∈ Γ( A ) , it acts as a derivation on sections of E . So wehave φ E ( X )( f ψ ) = f φ E ( X )( ψ ) + ρ ( X )( f ) ψ , ∀ ψ ∈ Γ( E ), f ∈ C ∞ ( M ), X ∈ Γ( A ), (25)where ρ E ◦ φ E = ρ . Note that although we have described some of its features, we have not actually fixeda definition of the map φ E ; we will supply such a definition later in the paper. In the rest of this section,we will show, as we claimed above, that φ E must be a morphism of Lie brackets.9iven that φ E ( X ) is a derivation, it is natural to introduce a corresponding exterior derivative on E ,defined as φ E ( X )( ψ ) ≡ (ˆ d ψ )( X ), ψ ∈ Γ( E ), (26)where we interpret ˆ d ψ : A → E , or equivalently, as a section ˆ d ψ ∈ Γ( A ∗ × E ) where A ∗ is the bundle dualto A . We will now show that the fact that φ E is a morphism translates into the nilpotency of ˆ d . To extendthe action of ˆ d to E -valued (extended) forms, we note that the latter are elements ψ n ∈ Γ( ∧ n A ∗ × E ) .Therefore, in analogy with the de Rham complex, we refer to Γ( ∧ n A ∗ × E ) as Ω n ( A , E ) and introduce thecochain complex of (extended) forms in A valued in E as Ω • ( A , E ) = M Ω n ( A , E ), (27)where the sum goes from to the rank of A , and Ω ( A , E ) = Γ( E ) . The exterior derivative ˆ d is then themap ˆ d : Ω n ( A , E ) → Ω n +1 ( A , E ) , and its action is explicitly given by the Koszul formula (ˆ d ψ n )( X , ..., X n +1 ) ≡ n +1 X r =1 ( − r +1 φ E ( X r )( ψ n ( X , ..., c X r , ..., X n +1 )) (28) + X r < s ( − r + s ψ n ([ X r , X s ] A , X , ..., c X r , ..., c X s , .., X n +1 ), where overhats refer to omission. The right-hand side is computable from (26) because ψ n ( X , ..., X n ) ∈ Γ( E ) . It reduces to eq. (26) for n = 0 , and the second line of (28) is such as to make the formula linear inall the sections X , ..., X n +1 (that is, only ψ n is differentiated). We can immediately establish that ˆ d = 0 as follows: we simply write (28) for ψ n = ˆ d ψ n − , and iterate. Let us do this explicitly for n = 1 : (ˆ d ˆ d ψ )( X , X ) = φ E ( X )(ˆ d ψ ( X )) − φ E ( X )(ˆ d ψ ( X )) − ˆ d ψ ([ X , X ] A )= [ φ E ( X ), φ E ( X )] ( ψ ) − φ E ([ X , X ] A )( ψ ). (29) Thus we see that the nilpotency of ˆ d is equivalent to φ E being a morphism. Similarly, we can extend thiscomputation to rank- n E -valued forms, although the computations become tedious. Here we give just onemore example, (ˆ d ˆ d ψ )( X , X , X ) = φ E ( X )(ˆ d ψ ( X , X )) − φ E ( X )(ˆ d ψ ( X , X )) + φ E ( X )(ˆ d ψ ( X , X )) − ˆ d ψ ([ X , X ] A , X ) + ˆ d ψ ([ X , X ] A , X ) − ˆ d ψ ([ X , X ] A , X )= ψ ([[ X , X ] A , X ] A ) + ψ ([[ X , X ] A , X ] A ) + ψ ([[ X , X ] A , X ] A ) = 0, which follows again from the facts that φ E is a morphism and that the Lie bracket on A satisfies the Jacobiidentity. This can be carried out to higher order, and so we conclude that ˆ d = 0 on the full cochaincomplex Ω • ( A , E ) , if φ E is a morphism. ˆ d for sections of L Later, we will define φ E for arbitrary associated vector bundles. But given a connection on A , we havean L -valued form, the connection reform ω . The above technology can be applied by regarding L as anexample of a vector bundle E . Applying ˆ d to ω , we get an L -valued two-form (i.e., an element of Ω ( A , L ) );contracting with sections X , Y we have a section of L which can be mapped to A using ι . To do all of this10xplicitly, we need to understand φ L . In this case it is simply given by the Lie bracket on A [65, 66]; given µ ∈ Γ( L ) and X ∈ Γ( A ) we have ι ( φ L ( X )( µ )) = [ X , ι ( µ )] A . (30)That this defines a morphism is equivalent to the Jacobi identity for the Lie bracket on A , that is ι ([ φ L ( X ), φ L ( Y )]( µ )) = ι ( φ L ( X )( φ L ( Y )( µ ))) − ι ( φ L ( Y )( φ L ( X )( µ ))) (31) = [ X , [ Y , ι ( µ )] A ] A − [ Y , [ X , ι ( µ )] A ] A (32) = [[ X , Y ] A , ι ( µ )] A (33) = ι ( φ L ([ X , Y ] A )( µ )). (34)Applying ˆ d to ω gives ι ((ˆ d ω )( X , Y )) = ι (cid:16) φ L ( X )( ω ( Y )) − φ L ( Y )( ω ( X )) − ω ([ X , Y ] A ) (cid:17) (35) = [ X , ι ◦ ω ( Y )] A − [ Y , ι ◦ ω ( X )] A − ι ◦ ω ([ X , Y ] A ), (36)where we used (28) for n = 1 and (30) repeatedly. Making use of eqs. (12-14) along with R ι = 0 , we thencompute ι ((ˆ d ω )( X , Y ) + [ ω ( X ), ω ( Y )] L ) = − ι ◦ ω ([ X H , Y H ] A ). (37)Note that the right-hand side is the vertical part of the bracket [ X H , Y H ] A , which encodes the non-integrability of H ⊂ A . It is well-known (see e.g. [65, 66]) that this non-integrability is proportional tothe curvature of the Ehresmann connection σ . Indeed, we have immediately using (11) and R ρ = 0 − ι ◦ ω ([ X H , Y H ] A ) = [ σ ◦ ρ ( X ), σ ◦ ρ ( Y )] A − σ ◦ ρ ([ X H , Y H ] A ) = R σ ( ρ ( X ), ρ ( Y )). (38)An explicit realization of the Ehresmann connection (in the context of Atiyah Lie algebroids discussed inthe upcoming sections) corresponds to the physics notion of a gauge field, and the usual notion of thecurvature of the gauge field coincides with the curvature of the Ehresmann connection.Furthermore, we note that we can rewrite (37) as Ω( X , Y ) = − ω ([ X H , Y H ] A ), (39)if we define Ω = ˆ d ω + 12 [ ω , ω ], Ω ∈ Γ( ∧ A ∗ × L ) (40)(given [ ω , ω ]( X , Y ) = [ ω ( X ), ω ( Y )] L ). We will refer to Ω as the curvature reform (that is, the curvature2-form of the connection reform). In the context of Atiyah Lie algebroids discussed in the next section, wewill see that the connection reform ω contains not only the physics notion of gauge field, but also encodesthe notion of a BRST ghost field.It is convenient to now introduce R ω ( X , Y ) ≡ [ ω ( X ), ω ( Y )] L + ω ([ X , Y ] A ), (41) Eq. (30) is essentially the ad action; in [65, 66], the ι map on both sides of the equation was implicit. Here we haveincluded it for consistency. We use the notation R ω here because it vanishes if − ω is a morphism.
11n terms of which we may write (39) as Ω( X , Y ) = − R ω ( X H , Y H ). (42)We will see below that R ω also contains further information.Consequently, there are multiple ways to express the curvature, R σ ( ρ ( X ), ρ ( Y )) = ι (Ω( X , Y )) = − ι ( R ω ( X H , Y H )). (43)Given either expression, we see immediately that only the horizontal parts of the sections X , Y contribute;that is, Ω is a horizontal form in that Ω( X V , Y ) = 0 because ρ ◦ ι = 0 . For gauge theories, this resultwill translate to the Russian formula which in the usual discussions of BRST geometry is introducedas a requirement (see for instance [31–35, 72–74, 36]); here we see that it follows immediately from theconditions R ι = 0 = R ρ , which are built-in properties of any Lie algebroid. Curvature of a Lie algebroidcan be understood as the failure of σ and − ω to be morphisms.To understand how we might choose φ E for general bundles, let us explore further φ L . First we notethat (30) implies φ L ( X )( µ ) = − ω ([ X , ι ( µ )] A ), (44)since ω ◦ ι = − Id L . It is useful to split X appearing in (44) into horizontal and vertical parts, φ L ( X )( µ ) = − ω ([ X H , ι ( µ )] A ) − ω ([ X V , ι ( µ )] A ) (45) = − ω ([ X H , ι ( µ )] A ) − [ ω ( X V ), µ ] L . (46)We can rewrite the second term in (46) in terms of the map v L : L → End ( L ) as [ ω ( X V ), µ ] L = v L ◦ ω ( X V )( µ ). (50)Here, we regard v L ◦ ω ( X ) ∈ End ( L ) . From eq. (41), the first term in (46) is equal to − R ω ( X H , ι ( µ )) , andwe introduce the notation ω ([ X H , ι ( µ )] A ) = R ω ( X H , ι ( µ )) ≡ −∇ L X H µ . (51)Note that we have written the section X H ∈ Γ( A ) as a subscript to ∇ L to denote a directional derivative;since this section is horizontal, we could equally well have written ρ ( X H ) ∈ Γ( TM ) to emphasize that thederivative is along a tangent to M in the spirit of eq. (1). We use the ∇ notation here because it dependson the connection and, as we will see later, in gauge theories it will become the gauge covariant derivativealong ρ ( X H ) . It is also natural to write ∇ L µ ∈ Γ( A ∗ × L ) , with ∇ L µ ( X ) ≡ ∇ L X H µ . For a bundle E , we have the morphism v E : L → End ( E ) , as in eq. (22). For { t A } a basis of sections of L , see App. A, v E ( t A ) is the corresponding matrix representation, and we have [ v E ( t A ), v E ( t B )] = (( t A ) c d ( t B ) d f − ( t B ) cd ( t A ) d f ) e c ⊗ f f (47) v E ([ t A , t B ] L ) = f AB C v E ( t C ) = f AB C ( t C ) c f e c ⊗ f f . (48)So if v E is a morphism, then we have the matrix equation [ t A , t B ] ab = f AB C ( t C ) ab . (49)In the special case where E is the bundle L , we have the result v L ( t A ) = f AC B t B ⊗ t C , the adjoint representation. It is easy to see that R ω ( X V , Y V ) = 0 which we can interpret as the statement that the Lie algebra has no curvature.We also saw that R ω ( X H , Y H ) is related to curvature, and so R ω ( X H , Y V ) is the only remaining component, and makes itsappearance in (51).
12o summarize, we have that the derivation on L , defined by the Lie bracket on A as in eq. (30), reads φ L ( X )( µ ) = ∇ L X H µ − v L ◦ ω ( X V )( µ ) (52)and so we have split φ L into a horizontal and vertical part, the horizontal part having an interpretation asa covariant derivative, the vertical being an endomorphism. Given the definition of ˆ d in (26), applied to L , ˆ d µ ( X ) = φ L ( X )( µ ), (53)we can interpret the above result as ˆ d µ ( X H ) = ∇ L X H µ , ˆ d µ ( X V ) ≡ s µ ( X ) = − v L ◦ ω ( X V )( µ ). (54)Here we have introduced the notation s µ to denote the vertical part of ˆ d µ . Later we will generalize theseexpressions from µ ∈ Γ( L ) to general E -valued n -forms.Before moving on, let us note that ∇ L has a curvature which we define in the usual way, as a map R L : H × H × L → L , R L ( X H , Y H )( µ ) = ∇ L X H ( ∇ L Y H µ ) − ∇ L Y H ( ∇ L X H µ ) − ∇ L [ X H , Y H ] H µ , (55)which is linear in X H , Y H with respect to multiplication by functions on M . The notation [ X H , Y H ] H refersto the horizontal part of the Lie bracket. We compute, making repeated use of (51), R L ( X H , Y H )( µ ) = −∇ L X H ( ω ([ Y H , ι ( µ )] A )) + ∇ L Y H ( ω ([ X H , ι ( µ )] A )) + ω ([[ X H , Y H ] H , ι ( µ )] A )= ω ([ X H , ι ◦ ω ([ Y H , ι ( µ )] A )] A ) − ω ([ Y H , ι ◦ ω ([ X H , ι ( µ )] A )] A )+ ω ([[ X H , Y H ] A , ι ( µ )] A ) − ω ([[ X H , Y H ] V , ι ( µ )] A )= − ω (cid:16) [ X H , [ Y H , ι ( µ )] A ] A + [ Y H , [ ι ( µ ), X H ] A ] A + [ ι ( µ ), [ X H , Y H ] A ] A (cid:17) − R ω ([ X H , Y H ] V , ι ( µ )) − [ ω ([ X H , Y H ] V ), µ ] L )= − ( v L ◦ ω ([ X H , Y H ] V ))( µ )= v L (Ω( X H , Y H ))( µ ), (56)where we used Jacobi, the fact that R ω ( X V , Y V ) = 0 , eq. (50) and introduced the notation [ X H , Y H ] V to refer to the vertical part of the Lie bracket. Referring back to eq. (43), we see that R L involvesjust the previous notions of curvature; the presence of the map v L simply expresses the curvature as anendomorphism on L . φ E for general vector bundles Returning to the general case of ψ ∈ Γ( E ) , the Lie bracket can no longer be used to define φ E , but wepropose that φ E must be written in the same form as (52), φ E ( X )( ψ ) = (ˆ d ψ )( X ) = ∇ E X H ψ − v E ◦ ω ( X V )( ψ ) = ∇ E X H ψ + s ψ ( X V ). (57)In the context of gauge theories, we will interpret ∇ E as the induced connection on the associated bundle E and ∇ E X H ψ then as the covariant derivative of ψ along ρ ( X H ) . We must check that the given φ E is infact a morphism, which as shown will be equivalent to the nilpotency of ˆ d ; using (57), we find [ φ E ( X ), φ E ( Y )]( ψ ) − φ E ([ X , Y ] A )( ψ ) = φ E ( X )( φ E ( Y )( ψ )) − φ E ( Y )( φ E ( X )( ψ )) − φ E ([ X , Y ] A )( ψ )= v E ( R ω ( X , Y ))( ψ ) + R E ( X H , Y H )( ψ )+ ∇ E Y H ( v E ◦ ω ( X )( ψ )) − v E ◦ ω ( X )( ∇ E Y H ψ ) −∇ E X H ( v E ◦ ω ( Y )( ψ )) + v E ◦ ω ( Y )( ∇ E X H ψ ), (58)13here we define the curvature of ∇ E analogously to (55), R E ( X H , Y H )( ψ ) ≡ ∇ E X H ( ∇ E Y H ψ ) − ∇ E Y H ( ∇ E X H ψ ) − ∇ E [ X H , Y H ] H ψ . (59)It is simplest to understand (58) by considering separately horizontal and vertical sections. First note thatif X , Y are both vertical this vanishes identically. If we take X to be horizontal and Y = ι ( µ ) , then, using(51), we obtain [ φ E ( X H ), φ E ( ι ( µ ))]( ψ ) − φ E ([ X H , ι ( µ )] A )( ψ ) = − v E ( ∇ L X H µ )( ψ ) + ∇ E X H ( v E ( µ )( ψ )) − v E ( µ )( ∇ E X H ψ ). (60)The vanishing of this equation is essentially a compatibility condition between ∇ L and ∇ E . The remainingpart of (58) has X and Y both horizontal, and reads [ φ E ( X H ), φ E ( Y H )]( ψ ) − φ E ([ X H , Y H ] A )( ψ ) = v E ( R ω ( X H , Y H ))( ψ ) + R E ( X H , Y H )( ψ ) (61) = v E ◦ ω ([ X H , Y H ] A )( ψ ) + R E ( X H , Y H )( ψ ). (62)Thus the statement that φ E is a morphism can be understood as the condition that “curvature is curvature”.The curvature (59) acts on sections of E as R E ( X H , Y H )( ψ ) = − v E ◦ ω ([ X H , Y H ] A )( ψ ) (63)which is of precisely the same form as (56). Recalling the result (42), we can alternatively write this as R E ( X H , Y H )( ψ ) = v E (Ω( X , Y ))( ψ ). (64)This is consistent with ∇ E being an induced connection, and furthermore shows that each notion ofcurvature that we have encountered is ultimately the same, geometrically associated to a non-trivial lift of TM into A . ˆ d on E -valued extended forms We have given above, eq. (57), the essential formula for derivations on an associated vector bundle E . TheKoszul formula (28) instructs how to extend the action of ˆ d on sections ψ n of ∧ n A ∗ × E , and furthermore,this may be split into vertical and horizontal pieces which we write as ˆ d ψ n = ( ∇ ∧ n A ∗ × E ψ n ) + s ψ n . (65)The two pieces of this equation may be written more explicitly as ( ∇ ∧ n A ∗ × E ψ n )( X , ..., X n +1 ) = n +1 X j =1 ( − j +1 ( ∇ ∧ n A ∗ × E X jH ψ n )( X , ..., c X j , ..., X n +1 ), (66) ( s ψ n )( X , ..., X n +1 ) = − n +1 X j =1 ( − j +1 v E ◦ ω ( X jV )( ψ n )( X , ..., c X j , ..., X n +1 ) (67) + X j < k ( − j + k ψ n ([ X jV , X kV ], X , ..., c X j , ..., c X k , .., X n +1 ). The induced connection ∇ ∧ n A ∗ × E can be systematically constructed via Leibniz and linearity. This isexplained for the case n = 1 in Appendix B, where we show how the compact result (65) is obtained fromthe Koszul formula. This completes the horizontal-vertical split for arbitrary E -valued extended forms,and in particular defines the vertical part s generally, which is a key result for physical applications.14 .3 The BRST interpretation So far, we have discussed the mathematics of transitive Lie algebroids. In the following section, we willdescribe a special case, the Atiyah Lie algebroids, which are associated with principal bundles and will beof direct relevance to gauge theories. Let us pause here though to explain how the Lie algebroids can beused in physical theories emphasizing some of the important structure that we have already established.In physics, fields will correspond either to gauge fields (which we will extract below) or to sections ofassociated bundles. We have understood how ˆ d implements (through the horizontal parts) a connection ∇ on these bundles. As such the horizontal part tells us how to move sections of associated bundles along M .But we have also seen that ˆ d contains more, through the vertical part, which we referred to as s above.That is, ˆ d combines displacements on the manifold M with those associated with the bundle L . In fact,the vertical part of ˆ d , which we called s above, should be interpreted as the BRST transformation.For sections of E , we wrote previously in (57), s ψ ( X ) = ˆ d ψ ( X V ) = − v E ◦ ω ( X V )( ψ ) ≡ − c E ( ψ )( X V ), (68)and the nilpotency of ˆ d implies the nilpotency of s . Let us recall that ω : A → L , so ω ( X V ) ∈ Γ( L ) . Further,since v E : L → End ( E ) , we can regard c E = v E ◦ ω ∈ Γ( A ∗ × End ( E )) , and so v E ◦ ω ( X V ) ∈ End ( E ) acts asmatrix multiplication on the components of ψ in a basis for E . Equivalently, we can say that v E ◦ ω canbe regarded as an End ( E ) -valued (that is, matrix-valued) vertical 1-form.The form of (68) is precisely that of a BRST transformation of the field ψ , with s playing the role of theBRST generator, and c E = v E ◦ ω playing the role of the ghost field. In the usual physics (BRST) discussions,as for instance reviewed in [12–17], ghost fields are said to be Grassmann-valued. The important aspectof this representation is that they are anti-commuting. Here, the role of the Grassmann algebra is playedby the vertical exterior algebra, which shares the anti-commuting property through the wedge product offorms. What we have here is an invariant way of describing this geometric structure. In the followingsection, we will discuss the specific case of Atiyah Lie algebroids which correspond more directly to gaugetheories (as there is a group action involved), and this same BRST structure will carry over unchanged. InSection 4, we will introduce local trivializations, which will allow us to make even more direct contact withphysics notions. We will see then that the connection reform ω can be thought of as having two aspects:first, its components in a given basis will be interpreted as the components of the ghost field, and second,it will carry the information of the connection on the Lie algebroid through the basis form. Thus the Liealgebroid formalism for gauge theories will have the remarkable property that the Grassmann nature ofBRST is geometrized. We now come to the central example of Lie algebroids for physics: the
Atiyah Lie algebroid . The traditionalaccount of the geometric formulation of gauge theories [1–4] involves equivariant connections on principalbundles. It has been previously suggested [57, 58, 61] that that formalism may be replaced by consideringconnections on Atiyah Lie algebroids. As we will review, the passage from (the tangent bundle of) aprincipal bundle to the corresponding Atiyah Lie algebroid involves taking the quotient by the right actionof the group. It is this right action that is a redundancy in the principal bundle formulation and in therequired equivariance of the principal connections.In this section, we will provide a review of the geometry of principal bundles and their principalconnections, and explain the effect of the quotient. Our presentation, although perhaps technical andcertainly not novel (e.g., see again [65, 66]), is not meant to be entirely rigorous, but instead to give anintroduction to how an Atiyah Lie algebroid can be constructed, and why it is the right bundle to fullygeometrize gauge theories. In particular, we will emphasize that through taking the quotient of the right15ction, we obtain a description of the physics of gauge theories involving only bundles over space-time itselfand the gauge connections become simply Ehresmann connections without an equivariance requirement.In the previous section, we have considered transitive Lie algebroids in general, and have noted thatthe exterior derivative on the Lie algebroid may be interpreted to contain, in its vertical part, the BRSToperator. Consequently, as we pass to Atiyah Lie algebroids with their proposed physical interpretation,the usual physics notion of BRST also acquires a geometric interpretation, and this interpretation is presentregardless of details of dynamics (that is, it is off-shell). Although, as we mentioned, Lie algebroids havebeen advocated before as alternative formulations of gauge theory, we do not believe that the relationshipwith BRST has been previously made. A relation of BRST to principal bundles was made long ago [31–36],and so one might regard the presence of BRST in the Lie algebroid formalism as a natural consequence.This similarity however is only skin deep, and in this section and particularly in Section 4, we will explainin detail how the BRST ghosts are encoded in the geometry. A reader familiar with the mathematics ofprincipal connections and principal bundles may wish to skip to the end of the following subsection.
Given a d -dimensional manifold M , with an atlas { U i } for which each U i is homeomorphic to an open set V i in R d , φ i : U i → V i , the transition functions of the manifold are defined on U i ∩ U j , φ ij = φ i ◦ φ − j , (69)with the respective maps restricted to U i ∩ U j . This allows to describe a manifold locally, and we want todo the same for principal bundles.A fibre bundle π : E → M can be described as a short exact sequence F E M π (70)where F is the fibre, which might be a vector space or a group. Locally, within an atlas on M , a fibrebundle π : E → M can be described as ( x , f ) where x ∈ U i and f ∈ F . An example of a vector bundlethat we get for free for any M is the tangent bundle TM where the fibre at a point x is the tangent plane T x M , and thus a point in TM can be described locally as ( x , v x ) where v x ∈ T x M . We also get for free thecotangent bundle T ∗ M , which is also a vector bundle, the dual of TM ; in this case the points in the fibresare 1-forms, which are maps from T x M → R .What we mean by “locally” above is as follows. A local trivialization of the fibre bundle is an atlas for M together with a homeomorphism ϕ i : π − ( U i ) → U i × F for all x ∈ U i and all i , such that π ◦ ϕ − i ( x , f ) = x for all f ∈ F . So locally, E looks like U × F : E M F U × F U πϕ . (71)The homeomorphisms define transition functions on overlaps U i ∩ U j for the bundle via ϕ ij = ϕ i ◦ ϕ − j : ( U i ∩ U j ) × F → ( U i ∩ U j ) × F . (72) We alternately think of x as a point in U i , or as its coordinates φ i ( x ) , which is a point in V i . Also, we consider the inclusionof F in E trivial so we consider them indistinguishably.
16t least in the case where F is a vector space of dimension k , we have ϕ i ◦ ϕ − j ( x , v x ) = ( x , g ij ( x ) v x ), ∀ x ∈ U i ∩ U j , v x ∈ F x , (73)where g ij : U i ∩ U j → GL ( k , R ) , often restricted to lie in a subgroup. This is a ˘Cech cocycle, in that werequire g ii ( x ) = Id , g ij ( x ) g jk ( x ) g ki ( x ) = Id , for x ∈ U i ∩ U j ∩ U k (74)A (local) section of E is a continuous map s : U i → E such that π ◦ s ( x ) = x , ∀ x ∈ U i .A principal bundle π : P → M is a fibre bundle whose fibres are a Lie group G and is endowed with acontinuous right action R : P × G → P , such that R h : P → P for h ∈ G . R acts freely: the stabilizer ofevery point is trivial. In the following we will write R h ( p ) = p · h . For the principal bundle, we can writea short exact sequence G P M π (75)and essentially all we said about vector bundles carries over here. A local trivialization (or “choice ofgauge”) of P is an atlas for M together with a map T i : π − ( U i ) → U i × G given by T i ( p ) = ( π ( p ), t i ( p )) where t i : π − ( U i ) → G is equivariant under right action in P , t i ( p · h ) = t i ( p ) h , which implies T i ( p · h ) = ( π ( p · h ), t i ( p · h )) = ( π ( p ), t i ( p ) h ), (76)where we used that π is invariant under the right action of the group, π ( p · h ) = π ( p ) . So under these defini-tions, the right action of the group acts along the fibres and acts on the group itself as right multiplicationvia t i . On overlaps, we take the transition functions, as we wrote above, to be given by left multiplication, ( π ( p ), t i ( p )) = T i ◦ T − j ( π ( p ), t j ( p )) = ( π ( p ), ψ ij ( p ) t j ( p )), ψ ij ( p ) = t i ( p ) t j ( p ) − . (77)Thus the gauge transformations are understood to be associated with left multiplication, and the ψ ij areinvariant under the right action of G ψ ij ( p · h ) = t i ( p · h ) t j ( p · h ) − = t i ( p ) hh − t j ( p ) − = ψ ij ( p ). (78)Hence it is conventional to write ψ ij ( p ) as ψ ij ( x ) . This is already a good indication that, for physicalapplications, the right action of the group is a redundancy, i.e., transition functions are invariant.Since we are interested in connections on principal bundles, the main object of interest is not P itself butits tangent bundle ˆ π : TP → P . The bundle projection π : P → M pushes forward to a map π ∗ : TP → TM ,and π ∗ will have a kernel called the vertical subspace V p P ⊂ T p P . A section X ∈ Γ( TP ) is then vertical if X p ∈ V p P for all points p . As we recalled above, the right action of the group acts vertically on P → M .Given X p ∈ T p P , we can generate a section by pushing forward the right action. This is a special class ofsections, which are automatically right-invariant. That is R ∗ h : T p P → T p · h P , (79)and the map R ∗ h is a morphism of the Lie bracket on TP , R ∗ h ([ X , Y ] p ) = [ R ∗ h ( X p ), R ∗ h ( Y p )] p · h , ∀ X , Y ∈ Γ( TP ), ∀ p ∈ P . (80) A local trivialization T i is equivalent to specifying a local section σ i : U i → π − ( U i ) such that π ◦ σ = Id U i . We have simply σ i ( x ) = T − i ( x , e ) and/or T i ( σ i ( x ) · h ) = ( x , h ) , where x ∈ U i , h ∈ G and e is the identity in G . X ∈ Γ( TP ) is right-invariant if for all p ∈ P it satisfies R ∗ h ( X p ) = X p · h . (81)The vertical vector space, at each point p , is isomorphic to the Lie algebra g of the group G . That is,there is a map j p : g → T p P given by j p ( γ ) = ddt ( p · e t γ ) (cid:12)(cid:12) t =0 , γ ∈ g , (82)which satisfies π ∗ j p ( γ ) = 0 for all p ∈ P and thus j ( γ ) ∈ Γ( VP ) for all γ ∈ g . The right action on V gives R ∗ h j p ( γ ) = j p · h ( Ad h − γ ), with Ad h ( γ ) = ddt ( he t γ h − ) (cid:12)(cid:12) t =0 . (83)Thus the vertical sub-bundle VP ⊂ TP is an invariant distribution under the group right action.A principal connection on P can be defined as a G -equivariant 1-form on P valued in the Lie algebra g , i.e., ω ∈ Γ( T ∗ P × g ) . For a right-invariant section X ∈ Γ( TP ) , a principal connection ω satisfies ( R ∗ h ω p · h )( X p ) = ω p · h ( R ∗ h X p ) = Ad h − ( ω p ( X p )), (84)or more briefly Ad h ( R ∗ h ω ) = ω . Here we used the pull back R ∗ h : T ∗ p · h P → T ∗ p P .Another notion of connection on P is introduced `a la Ehresmann specifying a distribution HP on TP ,complementary to VP . This connection is associated to a principal connection ω only when the distribution HP is right invariant. If this is the case, then ω ( X ) = 0 if X ∈ Γ( HP ) and ω ( j ( γ )) = − γ , (85)where the minus sign conforms with our conventions. Using (84) for j ( γ ) we simply have ( R ∗ h ω p · h )( j p ( γ )) = − Ad h − ( γ ). (86)At this point, this construction starts to strongly resemble the more general one discussed in theprevious section for Lie algebroids. Nonetheless, the results expressed here apply only to right invariantsections in Γ( TP ) . Moreover, it is important to observe that TP is a bundle over P , whereas the generalLie algebroids discussed previously are bundles over M . Note that the set of these sections is closed underLie bracket, since [ X p · h , Y p · h ] = [ R ∗ h ( X p ), R ∗ h ( Y p )] = R ∗ h ([ X , Y ] p ) = [ X , Y ] p · h . (87)So the set of right-invariant sections on P , denoted Γ G ( TP ) , is a closed subset of Γ( TP ) . While TP can beidentified with Γ( TP ) , one may ask what bundle we can identify with Γ G ( TP ) . The answer is the bundle TP / G , that is, the quotient of TP by the right action of the group. This means that the set Γ( TP / G ) is inone to one correspondence with Γ G ( TP ) . Furthermore, as we will show shortly, TP / G is a vector bundleover M , and the principal connection on P descends to a connection on TP / G , which is the so-called AtiyahLie algebroid. In a local trivialization, if we write π ∗ ( X p ) = X π ( p ) ∈ T π ( p ) M , we have γ p = X p − π − ∗ ( X π ( p ) ) ∈ ker( π ∗ ) . We then regardlocally X p ≃ ( X π ( p ) , γ p ) . Right invariance therefore means that R ∗ h ( X π ( p ) , γ p ) ≡ ( X π ( p · h ) , R ∗ h ( γ p )) = ( X π ( p ) , γ p · h ). .2 Constructing the Atiyah Lie algebroid TP / G Given a principal bundle π : P → M with a structure group G , there is a corresponding Atiyah Liealgebroid TP / G that we may systematically construct. To define TP / G , recall that a point in TP islocally described by ( p , v p ) with v p ∈ T p P , and the group acts (freely and transitively) on the right as ( p , v p ) ( p · h , R ∗ h ( v p )) . Thus TP / G is defined by identifying ( p , v p ) ∼ ( p · h , R ∗ h ( v p )), ∀ h ∈ G . (88)The resulting equivalence classes can be thought of in two ways, the first leading to an interpretation interms of a vector bundle over M , and the second leading to right-invariant sections over P . Referring tothe local description in footnote 16, we know that we can regard locally v p ≃ ( X π ( p ) , γ p ) . For each pointin TP / G , we can simply select a convenient representative of the equivalence class in eq. (88) for which p ≃ ( π ( p ), e ) and thus v p ≃ ( X π ( p ) , γ ( π ( p ), e ) ) . Then denoting π ( p ) = x , the point in TP / G has a localdescription ( p , v p ) ≃ (( x , e ), ( X x , γ ( x , e ) )) , or more simply ( x , X x , γ ( x , e ) ) with γ ( x , e ) recognized as an elementof the Lie algebra g associated to G . This implies that TP / G is a vector bundle over M rather than P , withthe fibre over x described locally by ( X x , γ ( x , e ) ) , consisting of a tangent vector and a Lie algebra element.Thus, TP / G is a vector bundle of rank d + dim G over M .Alternatively, starting from a point ( p , v p ) ∈ TP , we can regard (88) as simply defining a right-invariantsection over P : since R h acts transitively, the equivalence simply defines a section for which v p · h is givenby R ∗ h v p for all h (and thus all points in P ). This interpretation of the quotient makes direct contact with Γ G ( TP ) , the subset Γ G ( TP ) ⊂ Γ( TP ) of right invariant sections over P , as previously claimed.The interpretation of TP / G as a vector bundle over M can be taken further noticing that π ∗ : TP / G → TM , in the local description, has a kernel consisting of vectors of the form (0, γ ( x , e ) ) . So we see that at eachpoint x , ker π ∗ ≃ g , as expected. The union over all points x ∈ M gives a bundle of Lie algebras which canbe identified with the adjoint bundle L G ≡ P × Ad G g , which is a vector bundle over M , associated to P ,whose fibre is the Lie algebra of G . The map j introduced above, extended to all points p ∈ P , is thenthe inclusion ι : L G → TP / G , where the adjoint representation comes about thanks to (83). We eventuallyhave all of the structure necessary to interpret TP / G as a Lie algebroid with π ∗ being the anchor map and L G being the kernel of π ∗ via ι , L G TP / G TM ι π ∗ . (89)This is the Atiyah Lie algebroid corresponding to the principal bundle P .An Ehresmann connection on this algebroid defines a complementary sub-bundle H to ker π ∗ in TP / G ,which is now automatically right invariant. A principal connection on P passes through the quotient toan Ehresmann connection on TP / G . Thus, as far as physics is concerned, a connection on an Atiyah Liealgebroid serves as an appropriate geometric interpretation of a gauge field, the structure of which we willexplore in detail in what follows. The idea of formulating gauge theories geometrically on principal bundles was described in [1–4]. Later on,with the advent of BRST [5, 6], the question of how to encompass this new formalism on principal bundlesarose. In the ′ s, Thierry-Mieg and collaborators [31–36] observed that it is possible to identify the gauge Locally, we can understand L G as being the space of equivalence classes (( x , g ), µ x ) ∼ (( x , gh ), ad h µ x ) , a representative ofwhich can be taken to be (( x , e ), ad g − µ x ) , which we associate to ( x , ν x ) with ν x ∈ g . P . We briefly review theirresults here. The idea is to choose local coordinates along the fibres of P and write extended quantitiesin P . Defining a section in P from the base M selects a gauge choice. The procedure was then to “glue”together space-time forms with Lie algebra valued vertical ones in extended objects on P . Consider theextended gauge one form: its pullback to M is the Yang-Mills one form while its vertical part was recognizedto be the Faddeev-Popov ghost [67]. The interpretation was pushed even further identifying the BRSToperator s as the vertical part of the extended exterior derivative in local coordinates. There, conditionslike the horizontality of the curvature -form (i.e., the Russian formula) were imposed in order to findthe known BRST transformations. Furthermore, the discussion was fully local, since their descriptionrequired coordinates to be introduced along the fibres. One of the main results was the understanding ofthe Grassmann algebra geometrically as the algebra of vertical forms.In this work, we have shown how their local construction can be implemented intrinsically on AtiyahLie algebroids. The consequences are far-reaching: the so far mostly algebraic BRST discussion is nowgeometrized, requirements like the Russian formula are just by-products of the Lie algebroid formalism,everything is defined at a global – coordinate independent – level, and the question of which physicalobjects should be extended vertically develops into the simpler and unambiguous task of recognizing wherethese objects are geometrically defined. Lastly, the Faddeev-Popov ghost and the BRST operator for gaugetheories become Atiyah’s counterparts of the general ghost and BRST transformation introduced in Sec.2.3. To make further contact with gauge theories and their local degrees of freedom, we will explore in thenext section a local description involving an atlas for M in which specific choices of local frames are made,along with transformations between such local data. This will allow us, in Section 4.5, to return to a moredetailed discussion of the physical interpretation. In this section, our intent is to bring the abstract (but intrinsic) notation of the previous sections a littlemore down to earth. In particular, we will consider the local trivializations of the Lie algebroid and thevector bundles E , L and TM . For a gauge theory application, one must make a choice of structure group G and a selection of associated bundles. The physical fields then correspond to the connection as well asthe sections of the associated bundles (i.e., the charged matter fields). Each of the associated bundles E has a left G action in a representation R , whose dimension is the rank of E . This left action is the gaugetransformation, which we will be able to define once the bundles are trivialized. A local trivialization willinvolve a choice of atlas for M , and a local choice of basis for the fibres of each bundle. We will see thatthe extraction of the physical gauge field as well as the gauge ghosts is a little intricate: the gauge fieldin particular appears in several guises. We will generally use the notation A = TP / G for the Atiyah Liealgebroid and we will write L = L G , leaving the appellation G implicit. We begin the discussion with somegeneral comments about bases of sections for the various vector bundles, which will be used in the localdescription.Let us begin by recalling the properties of the various maps involved, ρ : A → TM , ι : L → A , σ : TM → A and ω : A → L , with the properties: ρ ◦ σ = Id TM , ω ◦ ι = − Id L , (90) ρ ◦ ι = 0, ω ◦ σ = 0. (91)We will choose specific sets of indices to label bases of sections of each vector bundle: we write { E M } ( M = 1, ... , dim A ) as a basis for A , { ∂ µ } ( µ = 1, ... , dim M ) as a (coordinate) basis for TM and { t A } ( A = 1, ... , dim G ) as a basis for L (consult Appendix A for further details on conventions). We then have20xplicitly ρ ( E M ) = ρ µ M ∂ µ , σ ( ∂ µ ) = σ M µ E M (92) ι ( t A ) = ι N A E N , ω ( E M ) = ω AM t A . (93)Relationships (90-91) read then ρ ν M σ M µ = δ ν µ , ω AM ι M B = − δ AB , (94) ρ µ M ι M A = 0, ω AM σ M µ = 0. (95)Correspondingly, contractions σ M µ ρ µ N and ι M A ω AN are horizontal and vertical projectors, respectively,for sections of A . That is, for a section X of A , we have X = X M E M = X H + X V = X M σ N µ ρ µ M E N − X M ι N A ω AM E N . (96)It is clear that we can change basis on A at will, E M = J N M E ′ N . (97)Correspondingly, the components of X transform as X ′ M = J M N X N , (98)such that X is invariant, that is a tensor. Given a basis E M for A , we have a dual basis for A ∗ , which wewrite as E M satisfying E M ( E N ) = δ M N . In the next subsection, we will make use of such transformations J M N to specify further adapted bases that block-diagonalize the projectors.
The vertical distribution V ⊂ A is integrable, because it is an ideal with respect to Lie brackets in A . Ithas rank dim L and enjoys two properties: it is the image of the inclusion ι , and defines the kernel of theanchor ρ . The horizontal distribution H ⊂ A , complementary to V , is defined as the image of the map σ .We can consequently introduce in A a separation of indices M = ( α , A ) with A = (1, ... , dim G ) such thatthe basis E A spans V and α = (1, ... , dim M ) such that E α spans H . We will refer to this as a split frame .This will in general involve the use of (97), and will lead in particular to a block-diagonalization of theprojectors referred to in the previous subsection. The maps ι and σ can be written as ι ( t A ) = ι AA E A + ι α A E α , (99) σ ( ∂ µ ) = σ αµ E α + σ A µ E A , (100)and requiring that their images are V and H respectively corresponds to setting ι α A = 0, σ A µ = 0. (101)Given that V is the kernel of ρ , and H the kernel of ω , we have also ρ ( E A ) = 0, ⇒ ρ µ A = 0, ω ( E α ) = 0, ⇒ ω A α = 0. (102)The index structure responsible for (101) and (102) can be alternatively seen as a specific solution of theequations ρ ◦ ι = 0 and ω ◦ σ = 0 . So in a split frame (at each point x ∈ M ), the maps satisfy (101-102),21ut the remaining components, namely ι AA , ω AA , ρ µα and σ αµ are unrestricted, apart from the relations ρ ν α σ αµ = δ ν µ , ω AA ι AB = − δ AB . (104)We can define the projected sections as above X H = σ ◦ ρ ( X ) ∈ Γ( H ), X V = − ι ◦ ω ( X ) ∈ Γ( V ), (105)and in the split basis we have X H = X α H E α = σ αµ ρ µ M X M E α = σ αµ ρ µβ X β H E α , (106) X V = X AV E A = − ι AA ω AM X M E A = − ι AA ω AB X BV E A . (107)Although such a split basis may always be chosen, the price to pay is to reduce the admissible basistransformations (97) in A to the ones preserving this structure, E A = J B A E ′ B ⇒ J α A = 0, (108) E α = J βα E ′ β ⇒ J A α = 0. (109)Therefore the residual transformation matrices J N M are block diagonal. Note that the dual basis for A ∗ isalso split, and we write the basis forms as E α and E A , with E α ( E β ) = δ αβ , E α ( E A ) = 0, E A ( E β ) = 0, E A ( E B ) = δ AB . (110)Here we have spoken in linear algebra terms, and so one should think in terms of this happeningseparately at each point x ∈ M . But given that specifying a connection on the Lie algebroid implies thatglobally A = H ⊕ V , a split basis may be chosen at every point. In Section 4.3, we will make explicit thetrivialization of each of the vector bundles and show how the J M N matrices may be used in constructingtransition functions.
In addition, we have an algebraic structure on A . Given a basis of sections E M , we define the rotationcoefficients C MN P as (cid:2) E M , E N (cid:3) A ≡ C MN P E P . (111)In a split basis, this can be decomposed into the collection (cid:2) E α , E β (cid:3) A = C αβ γ E γ + C αβ A E A , (112) (cid:2) E α , E A (cid:3) A = C α AB E B , (113) (cid:2) E A , E B (cid:3) A = C AB C E C , (114)where we made use of eqs. (14) to eliminate terms. Each of the rotation coefficients has a direct gaugetheory interpretation, as we now discuss. Generally, the bracket satisfies the Leibniz property, eq. (1). Eqs. (104) are just the split frame expressions of eqs. (7) and (13). Note further that ι ◦ ω and σ ◦ ρ being projectors on A to V and H respectively, we have ι ◦ ω (cid:12)(cid:12) V = − Id V and σ ◦ ρ (cid:12)(cid:12) H = Id H . Thus, in a split frame we also have σ αµ ρ µβ = δ αβ , ι AA ω AB = − δ AB . (103) ρ ( E A ) = 0 , we have that (114) is linear. Indeed, given two vertical sections X V , Y V ∈ Γ( A ) wehave [ X AV E A , Y BV E B ] A = X AV Y BV [ E A , E B ] A = X AV Y BV C AB C E C . (115)The rotation coefficients C AB C are related to the Lie algebra structure constants of L via the ι map, C AB C ι AA ι B B = f AB C ι C C , (cid:2) t A , t B ] L = f AB C t C , (116)which follows from ι being a morphism of the Lie brackets. This realizes the fact that at each point x ∈ M ,there is a copy of the Lie algebra included into A .This linearity does not extend to the rest of the rotation coefficients. For example, for a horizontalsection X H ∈ Γ( A ) and a vertical one ι ( µ ) ∈ Γ( A ) with µ ∈ Γ( L ) , applying (1) to (113) we obtain (cid:2) X H , ι ( µ ) (cid:3) A = X α H (cid:16) ι AC µ C C α AB + ρ ( E α )( ι B C µ C ) (cid:17) E B . (117)Now, given eq. (51), we have [ X H , ι ( µ )] A = ι ( ∇ L X H µ ). (118)Generally, given a connection ∇ E (a directional derivative along a horizontal vector) on a vector bundle E and a basis e a for E , we would define the connection coefficients in the given basis as ∇ E X H e a = A ba ( X H ) e b . (119)What we will show in the next sub-section is that these connections coefficients are determined, for eachassociated bundle, by the same local data supplied by a trivialization of the Lie algebroid. This is a familiarresult in gauge theory, but here it is a little more complicated, because we have a bundle of Lie algebras(rather than the same Lie algebra at each point in M ).For the adjoint bundle specifically, we have ∇ L X H t A = A B A ( X H ) t B , (120)so that ι ( ∇ L X H µ ) = ι ( ∇ L X H ( µ A t A )) = X α H (cid:16) ρ ( E α )( µ B ) + µ A A α B A (cid:17) ι B B E B , (121)where we have written A α B A ≡ A B A ( E α ) . Comparing to (117), we see that C α AB ι AC + ρ ( E α )( ι B C ) = ι B B A α B C , (122)and thus ι , in its role as a map between L and V , induces a non-linear relationship between the rotationcoefficient and the connection coefficients.Now using (39), we see also that the vertical part of (112) gives C αβ A = Ω A ( E α , E β ) ι AA ≡ ι AA Ω A αβ (123)which is linear and determined just by the curvature. As already noticed, this is the curvature of thedistribution H that makes it non-integrable. Finally, supplying two horizontal sections σ ( X ), σ ( Y ) ∈ Γ( A ) with X , Y ∈ Γ( TM ) , the horizontal part of (112) evaluates to (cid:2) σ ( X ), σ ( Y ) (cid:3) α A = σ β µ X µ σ γν Y ν C βγ α + X µ ∂ µ ( σ αν Y ν ) − Y ν ∂ ν ( σ αµ X µ )= X µ Y ν (cid:16) ∂ µ σ αν − ∂ ν σ αµ + C βγ α σ β µ σ γ ν (cid:17) + σ αµ (cid:2) X , Y (cid:3) µ (124)23n the other hand, we know that R σ is vertical (recall eq. (9)), so C βγα σ βµ σ γν = − ∂ µ σ αν + ∂ ν σ αµ . (125)Similar to ι , the σ αµ are involved non-linearly in the relationship between the rotation coefficients in H and those of TM (which in a coordinate basis are zero). The connection implicit in the map σ comesnot from the σ αµ , but from the nature of the lift. We will explore the latter in the following subsection,which requires the precise notion of trivialization, and show how it is related to the connection coefficientsintroduced above. In this section we give some details on the trivialization of transitive Lie algebroids. As A and its associatedbundles are vector bundles over M , we need only to specify the nature of the transition functions employedon overlaps of coordinate patches in an atlas for M . While what discussed so far is valid for any transitiveLie algebroid, we will focus here on the specific case of Atiyah Lie algebroids.The trivialization of A involves specifying transition functions on overlaps of coordinate patches on M .Since globally A = H ⊕ V , H and V are themselves vector bundles over M and thus it is sensible to taketransition functions that act on each bundle separately. On each coordinate patch U i ⊂ M , we are to makea choice of basis on the fibre. If we choose a split basis for A , then the transition functions will be of theform given in eqs. (108-109), i.e., E U i A = J ij B A E U j B , E U i α = J ij βα E U j β . (126)The transition functions on TM are just the Jacobians associated with diffeomorphisms. That is ∂ U i µ = J ij νµ ∂ U j ν , J ij νµ ≡ ∂ x ν j ∂ x µ i , (127)where x µ i are coordinates on the patch U i , etc. The transition functions on any associated bundle are givenby the corresponding group representation. For a vector bundle E associated to a representation R of thegroup G , choosing a basis for the fibre in each patch, we have e U i a = R ( g ij ) ba e U j b , g ij ∈ G . (128)Correspondingly, the components of a section satisfy ψ bj = R ( g ij ) ba ψ ai . (129)The latter comes simply from the tensorial property, ψ U i = ψ U j ; eq. (129) is the familiar gauge transfor-mation of a charged matter field. This applies equally well to the adjoint bundle, for which we write t U i A = t ( g ij ) B A t U j B . (130)For any associated bundle E , we have discussed the construction of ˆ d and the separation into horizontal andvertical parts, each of which is separately tensorial. For ψ ∈ Γ( E ) , we had ˆ d ψ ( X ) = ∇ E X H ψ − v E ◦ ω ( X V )( ψ ) and given a trivialization we must have ( ∇ E X H ψ ) U i = ( ∇ E X H ψ ) U j . (131) The convention adopted here is that the bundles, their bases and sections carry the patch subscript U i while the componentsof the various maps and sections have patch index i only. (cid:16) ρ ( X H )( ψ ai ) + A ai b ( X H ) ψ bi (cid:17) e U i a = (cid:16) ρ ( X H )( ψ aj ) + A aj b ( X H ) ψ bj (cid:17) e U j a . (132)Given (126) and (128-129), it is straightforward to show that A bi c ( E U i α ) = J ij β α (cid:16) ( R ( g ij ) − ) bd ρ ( E U j β )( R ( g ij ) d c ) + ( R ( g ij ) − ) bd A dj b ( E U j β ) R ( g ij ) bc (cid:17) , (133)which is the expected transformation for a gauge connection. We will discuss the tensorial nature of thevertical part below. This result is true for any bundle associated to the Atiyah Lie algebroid A = TP / G ,including the adjoint bundle L itself.Although the above discussion constitutes a trivialization of the Lie algebroid, we have not made explicitthe Ehresmann connection which σ gives rise to. To do so, we note that locally (i.e., within any coordinatepatch), the bundle A can be thought of in terms of the bundle L ⊕ TM . That is, on the patch U i , weintroduce a morphism τ i : A U i → L U i ⊕ TU i , with the property τ i ( X H ) = X α i , H τ i µα ( ∂ U i µ + b i A µ t U i A ), τ i ( X V ) = X Ai , V τ i AA t U i A . (134)The map τ allows us to add sections of TM and L together, and gives an explicit expression for the splitbasis vectors. The b i A µ are the components of the Ehresmann connection – they parameterize the lift from TM to H given by the map σ . Indeed, as we will now show, the b i A µ are equivalent to the connectioncoefficients discussed above. In fact, these properties follow because τ is a morphism. Expressions in thefollowing discussion being local, they should be written with patch subscripts U i , i . However, since herewe work exclusively in the patch U i , we will for brevity drop these subscripts.To begin, we first note that (cid:2) τ ( X V ), τ ( Y V ) (cid:3) L ⊕ TM = τ ( (cid:2) X V , Y V (cid:3) A ), (135)implies that τ AA τ B B f AB C = C AB C τ C C , (136)which, making use of (116), can be written as ( τ ◦ ι ) AD ( τ ◦ ι ) B E f AB C = ( τ ◦ ι ) C F f DE F , (137)where ( τ ◦ ι ) AB = τ AA ι AB . Clearly, τ ◦ ι is a local endomorphism of L and simply reflects the possibilityof there being a change of basis for L associated with the map τ . Note that although τ AA has the sameindex structure as ω AA , they are unrelated. Without loss of generality, we can take τ ◦ ι = Id L (which since τ ◦ ι : L → L ⊕ TM means that we are taking this to be the trivial identification), at which point (137)gives no further information, and the second of eq. (134) implies that τ : ι ( µ ) ( µ , 0) .Next we note that, again writing Y V = ι ( µ ) , and using (118), (cid:2) τ ( X H ), τ ( Y V ) (cid:3) L ⊕ TM = τ µα X α H (cid:16) ∂ µ µ C + f AB C b A µ µ B (cid:17) t C , (138) τ ( (cid:2) X H , Y V (cid:3) A ) = τ ◦ ι ( ∇ L X H µ ) = (cid:16) ρ ( X H )( µ C ) + X α H A α C B µ B (cid:17) t C , (139)25o for τ to be a morphism, we are to set τ µα = ρ µα , and then we obtain ρ µα f AB C b A µ = A α C B . (140)We see that the connection coefficients are equivalent to the Ehresmann connection components.Finally, we note (cid:2) τ ( X H ), τ ( Y H ) (cid:3) L ⊕ TM = (cid:2) X , Y (cid:3) µ D µ + X µ Y ν F A µν t A (141) τ ( (cid:2) X H , Y H (cid:3) A ) = τ ( σ ( (cid:2) X , Y (cid:3) )) + τ ( R σ ( X , Y )), (142)where for brevity we have introduced D µ ≡ ∂ µ + b A µ t A and have taken X H = σ ( X ) , Y H = σ ( Y ) . In eq.(141), we have denoted the components of the curvature of b A µ by F A µν = ∂ µ b A ν − ∂ ν b A µ + f BC A b B µ b C ν , (143)and the fact that τ is a morphism simply relates that to the other notions of curvature of the Lie algebroid,in agreement with (140).So far, we have demonstrated that the connection on the Lie algebroid appears within the geometricstructure in several equivalent ways, consistent with our previous demonstration that the curvature alsoappears in several equivalent ways as well. The local trivialization has just uncovered one more way to seethis, based on the explicit extraction of the Ehresmann connection. There is one remaining aspect thatwe did not directly address in the above discussion, which is the vertical part of E -valued extended forms.Consider the one form ˆ d ψ ∈ Γ( A ∗ × E ) . Its vertical part is tensorial, and given by (see (68)) s ψ ( X ) = ˆ d ψ ( X V ) = − v E ◦ ω ( X V )( ψ ) = − c E ( ψ )( X V ). (144)In this expression, ω is an L -valued 1-form. In a local trivialization, restoring momentarily the patchsubscripts, we can then write ω U i = ω Ai A E AU i ⊗ t U i A . (145)To understand this, we need to understand the local forms E AU i , the dual basis in A ∗ | U i . Given the map τ i introduced above, we can deduce their structure from eq. (110). The result is τ ∗ i ( E α U i ) = ( τ − i ) αµ dx µ i , τ ∗ i ( E AU i ) = ( τ − i ) AA ( t AU i − b i A µ dx µ i ), (146)where τ ∗ : A ∗ → L ∗ ⊕ T ∗ M is the map dual to τ (preserving the pairing (110)) and t AU i is a basis formfor L ∗ U i . So we see that locally, the form ω U i contains two things: the coefficient ω Ai A , and the connectioncoefficients b i A µ . The latter is responsible for the fact that Ω coincides with the curvature R σ . Again,in what follows we will drop the patch subscripts for brevity. We interpret the coefficients ω AA as thecomponents of the ghost field, which is in line with (144), encoding the BRST transformation of the field ψ . Locally, this appears as s ψ ( E A ) = − ω AA v E ( t A )( ψ ) = − ( t A ) ab ω AA ψ b e a , (147) Consequently, the map τ is determined by the local values of the ι and ρ maps. However, we should emphasize that the ρ and τ maps are different, as they have different image spaces, but the coefficients τ µα and ρ µα are equal locally. The differenceis precisely the Ehresmann connection coefficients b A µ . Had we not assumed τ ◦ ι = Id L , then A and b would have been equal up to a shift proportional to ∂ µ ( τ ◦ ι ) AB . Notethat locally we can model L as M × g and consequently in a particular coordinate patch we can set f AB C to be constant in aninfinitesimal neighbourhood of a point. A similar statement can be made of ( t A ) ab for any associated bundle. s ψ = s ( ψ a e a ) = − c ab ψ b e a = − c E ( ψ ), (148)with the vertical form gauge ghost field c ab given by c ab ≡ c Aab E A = ( t A ) ab ω AA E A . (149)Eq. (148) is the usual form of a BRST transformation in terms of the ghost field, with its usual Grassmannnature replaced by it being a vertical form in A ∗ . This aligns with the idea of Thierry-Mieg and collaborators[31–36] discussed in Section 3.3 but arises as a matter of course in the framework of Lie algebroids. Eq.(149) can be equivalently recast as an expression for an extended one form valued in End ( E ) : v E ◦ ω = ( t A ) ab ω AA E A ⊗ e a ⊗ f b = c Aab E A ⊗ e a ⊗ f b = c ab e a ⊗ f b = c E . (150)To recap, we have identified locally the gauge connection and have shown that it is equivalently describedas an Ehresmann connection. We then proved that the gauge ghost field is encoded in the components ofthe connection reform. We will expand more on the physics of this in subsection 4.5. In the previous section we have related the local component field ω Ai A with the gauge ghost, and noted thatthe BRST transformation of any section of a bundle E properly involves it. In this section, we will showthat the BRST transformations of the ghost and gauge fields are implicitly built into the formalism as well.The BRST transformation of the ghost field itself comes directly from the result that Ω is horizontal. Thatis, from (40), the vertical part of ˆ d ω is simply − [ ω , ω ] . Applying the morphism v E to (40) and using thelinearity of s we obtain that the BRST transformation of c E is given by sc E = s ( v E ◦ ω ) = v E ◦ ( s ω ) = v E ( − (cid:2) ω , ω ]) = − (cid:2) v E ◦ ω , v E ◦ ω ] = − (cid:2) c E , c E ]. (151)For its End ( E ) components this equation reads sc ab = − (cid:2) c , c ] ab , (152)which is the well-known BRST transformation of the gauge ghost in gauge theories [5, 6].The BRST transformation of the gauge field follows from the nilpotency of ˆ d and thus is also automatic.From (65) for n = 1 , consider the case ˆ d ψ ( X V , Y H ) = ( ∇ A ∗ × E ψ )( X V , Y H ) + s ψ ( X V , Y H ). (153)If we take ψ = ˆ d ψ then this is just automatically zero by nilpotency, that is, ( s ˆ d ψ )( X V , Y H ) = − ( ∇ A ∗ × E ˆ d ψ )( X V , Y H ). (154)We now compute the two sides separately. For the left-hand side, we have ( s ˆ d ψ )( X V , Y H ) = − s (ˆ d ψ ( Y H ))( X V ) = − s ( ∇ E Y H ψ )( X V ). (155)For the right-hand side we use (B.13) and compute ( ∇ A ∗ × E ˆ d ψ )( X V , Y H ) = − ( ∇ A ∗ × E Y H ˆ d ψ )( X V ) = −∇ E Y H (ˆ d ψ ( X V )) + (ˆ d ψ )( (cid:2) Y H , X V (cid:3) A ). (156) In standard physics notation, one dispenses with the basis vectors and writes s ψ a = − c ab ψ b . We should note though thatthis is an interpretation: our s actually acts linearly, so s ψ = s ( ψ a e a ) = ψ a s ( e a ) = − ψ a c ba e b . ˆ d ψ contributes ( ∇ A ∗ × E ˆ d ψ )( X V , Y H ) = ∇ E Y H ( c E ( ψ )( X V )) − c E ( ψ )( (cid:2) Y H , X V (cid:3) A ) = ( ∇ A ∗ × E Y H c E ( ψ ))( X V ). (157)Notice that, using the nilpotency of ˆ d for two horizontal sections of A , we can replace X V with the full X in this expression. Since we can do the same also in (155), eq. (154) becomes s ( ∇ E Y H ψ )( X ) = ( ∇ A ∗ × E Y H c E ( ψ ))( X ). (158)Specializing to ψ → e a and Y H → E α , and writing the corresponding form equation in A ∗ , we obtain s ∇ EE α e a = ∇ A ∗ × EE α c E ( e a ). (159)Given eq. (119), we have s ( A α ba e b ) = ∇ A ∗ × EE α c ba e b , (160)where we used (148) and the notation A α ba = A ba ( E α ) . This is the familiar result that the BRST transfor-mation of a gauge field is given by the covariant derivative of the gauge ghost [5, 6]. We remark again thatthis equation and the one establishing the BRST transformation of the ghost are an automatic consequenceof our geometric framework, where we have not imposed any external condition, in contrast with the origi-nal accounts on geometrization of BRST [31–36], where these equations follow from the externally imposedrequirement of the extended curvature being horizontal. This is clearly an advantage of our geometryshowing how it already contains all the useful structure for physical applications. In this section, we have translated much of the important structure of field theories based on Atiyah Liealgebroids into a local index notation. As mentioned earlier in Sec. 3.3, one can then express any gauge-matter theory in its terms. We have seen that BRST is simply built into the formalism geometrically,and all of the familiar properties are obtained directly. We should note that we have made no mentionso far of a classical action, or a gauge fixing prescription, which are often given as important ingredientsfor BRST constructions [5–11]. Here, we have a very different attitude: the BRST structure, that is thegauge covariance, is simply built into the geometry of the theory, and any sensible choice of action must beinvariantly defined. In local terms, this translates into the action being gauge invariant. We should alsonote that the construction is inherently diffeomorphism invariant as well, which again is possible becausewe have not specified auxiliary structure such as a metric. Of course, traditionally in Yang-Mills theories,in order to write an action one must supply additional geometric details, usually a choice of metric. Thediffeomorphism invariance of the resulting action is only broken in the sense that we usually think ofthe metric as fixed. In this sense then, diffeomorphism invariance is “spontaneously broken”, and it isimportant that we have an off-shell notion of the full geometric structure.Before moving on to gravitational theories, we would like to offer here a summary of our results andoffer some guidance to their use. Thanks to the index notation just introduced we can now make fullcontact with typical discussions on gauge theories in physics. The novelty is that, on top of seeing howusual gauge theory quantities are related to various geometric objects on Atiyah Lie algebroids, we cando the same also for BRST data.The basic ingredient of a gauge theory is its gauge Lie group G . A gauge group is accounted for ona principal fibre bundle P , as described in Section 3.1. The second key ingredient is a gauge connection. As stated earlier, the formulation of gauge theories on Atiyah Lie algebroids is also addressed in [56–61].
28s we reviewed, although one can define it in P , the natural arena for gauge connections is in fact theAtiyah Lie algebroid TP / G . There, the redundant right group action is modded out giving rise to abundle over M . In physical applications, a gauge connection is usually thought of simply as a one form on M transforming (non-linearly) in the adjoint representation of the Lie algebra g associated to the gaugeLie group G . This quantity in our construction is b A µ , which enters the discussion locally in eq. (134).The Atiyah Lie algebroid has by construction a Lie algebra adjoint action at each point in M , related tothe bundle L G in the short exact sequence defining TP / G . The gauge connection is truly an Ehresmannconnection needed to single out a horizontal distribution on the Lie algebroid. As shown in (140), thegauge connection is equivalent to the connection appearing in the gauge covariant derivative. In additionto the bundle L G , on which the connection coefficients A α B A are defined via (120), we also find the inducedconnection coefficients on any associated bundle. We see thus that the gauge connection manifests itselfin various places: as an Ehresmann connection σ , locally as a building block of the basis in H and as theconnection in the gauge covariant derivative on associated bundles. The curvature of a gauge connectionis its field strength, which we typically write as in (143). It is the gauge covariant quantity used to buildinvariant actions. The geometric picture developed here relates it to the curvature of the map σ . This isfurthermore encoded in the curvature two form of the connection reform ω . A therefore equivalent way toexpress the gauge curvature is as the non-integrability of the distribution H . Since there are various mapsand bundles involved in our geometry, there are various notions of curvature. Part of our agenda was toshow that they are all intertwined, as different realizations of the gauge curvature, the geometric originbeing in the curvature of the Ehresmann connection, controlling the non-integrability of H in TP / G .In the last paragraph, we have discussed a gauge theory without matter – the sections X of A and µ of L G are not fields, but just useful tools employed in the description of the Lie algebroid and its connection.Physically, charged matter fields are quantities transforming linearly in some representation R of the gaugegroup G . Geometrically, these fields are just the components ψ a of an associated bundle E via the repre-sentation R , as they are in the usual formalism of principal bundles, see e.g. [4]. There is then an inducedgauge covariant derivative ∇ E acting on the associated bundle, which we have found as the horizontal partof the exterior derivative ˆ d defining the complex Ω • ( A , E ) . The split into vertical and horizontal parts ofthe exterior derivative on extended forms in Ω • ( A , E ) was an important achievement, for the vertical part,discussed shortly, contains relevant information as well. This completes the list of ingredients necessaryto construct classical gauge theories, and all the necessary data is available geometrically in a relativelysimple way on Atiyah Lie algebroids and associated bundles.The classical geometry developed in this paper, as in [31–36], also controls the quantized theory, becausethe BRST formalism is naturally included. In the familiar formalism, as reviewed for instance in [12–17],the Fadeev-Popov ghosts appear through a gauge-fixing procedure in the quantization of gauge theories,and the BRST symmetry is the symmetry of a gauge-fixed Lagrangian. The details of the BRST procedureare usually thought of as dependent on the choice of the latter, but since the origin of the BRST operatoris geometrical, one should be able to deal with it off-shell. This is the direction we followed here, whereeverything is independent of on-shell dynamics and relies only on geometric structure. Different Lie alge-broids, with a given structure group and base M , are distinguished precisely by different connections, sointegrating over connections can be regarded as the same thing as summing over Lie algebroid geometries.One of the important properties of the BRST ghost is that it is Grassmann-valued. Here, the Grassmannalgebra is geometrically realized as an exterior algebra. Although this statement resembles that of [31–36],the details are somewhat different and in fact globally well-defined, because of our implementation of thisidea on Atiyah Lie algebroids. The Fadeev-Popov ghost has the geometric and gauge properties compatiblewith c ab introduced in (149). The Grassmann nature is accounted for by the fact that it is a vertical oneform in A ∗ . It is often said heuristically that the presence of the Fadeev-Popov ghosts is responsible forthe correct number of degrees of freedom contributing to the quantum path integral; apparently then, theconnection on an Atiyah Lie algebroid has built-in the correct degrees of freedom. All of these are packagedtogether into the connection reform; on the one hand, its components ω AA in a local basis are related to29he ghost (149), while on the other hand, it locally carries information on the gauge connection via thebasis itself in A ∗ , eq. (146). It is important that its curvature two-form is yet another manifestation of thegauge curvature, horizontal by construction.So given a gauge group, we know how to geometrically establish in TP / G a gauge field and gaugeghosts. The last ingredient we need is the BRST operator s . In the usual gauge theory quantization, thisoperator generates the BRST symmetry [5, 6]. Here, we have shown that it is encoded in the vertical partof the exterior derivative ˆ d on Lie algebroids or associated bundles, eq. (68). The globally well-defineddistinction between vertical and horizontal parts of ˆ d therefore distinguishes between BRST and covariantgauge derivative, respectively. For instance, the famous Darboux-Maurer-Cartan-Ehresmann structureequation of the principal fibre bundle [75, 76], stating that the curvature two form is horizontal, and givingrise to the Russian formula [31–36], is here just a straightforward geometrical consequence of Atiyah Liealgebroids, rather than a constraint to be imposed. The fact that s can be interpreted as the BRSToperator has been clearly established by observing that the components of the matter fields ψ a , ghosts c ab and gauge connections A α ab transform under the BRST symmetry as is well-known in physics, eqs.(148), (152) and (160). Again, the placement of s as the vertical part of an extended notion of exteriorderivative is a familiar part of the usual geometric construction of BRST [31–36], whereby one formallyadds commuting and anti-commuting quantities together. On Atiyah Lie algebroids, this construction hasa well-defined geometric origin, as we have thoroughly discussed.It is well-known that the cohomology of the extended exterior derivative controls the structure ofanomalies as well as many other features of gauge theories [8, 19–30]. Here, the same will be true for ˆ d ,and we have noted the structure of its complex throughout the paper. We will not explore this furtherin this paper, leaving it for future works. In the next section we will discuss gravitational theories fromthe point of view of Lie algebroids, and show that it is another application of the general construction wedeveloped above. Given that the Lie algebroid construction is intrinsically defined with respect to both diffeomorphismsand group invariance, it is natural to consider gauge theories that have a gravitational interpretation. Inthis section then, we discuss the features of Lie algebroids and associated bundles that are relevant togravitational theories. Necessarily, we will arrive at such theories in a “first order” form; that is, theprincipal bundle of interest is the frame bundle, and we will introduce a Lie algebroid with structure groupgiven by a subgroup of GL ( d , R ) along with a connection on the Lie algebroid. In addition, we mustintroduce an associated bundle whose sections are given by a solder form.At the level of the principal frame bundle, restricting to a subgroup G ⊂ GL ( d , R ) means that weare considering a particular G -structure, which we denote by F G . Often, we can associate a choice of G -structure to a requirement that a geometric structure remain invariant [68–71]. The usual case of interestis to take G to be the local Lorentz group, G = SO (1, d − , which is associated with the requirement ofpreserving a metric (frames remaining orthonormal). Such an interpretation is an auxiliary notion, and wewill not make direct use of it here. So for now we simply assume that we have some subgroup G ⊂ GL ( d , R ) of dimension dim G . As discussed earlier in the paper, Sec. 3, we can then construct an Atiyah Lie algebroidfrom F G by quotienting TF G by the right action of G . We will refer to this Lie algebroid as A G = TF G / G and the adjoint bundle as L G , L G A G TM ι ρ . (161)Having introduced a connection for A G , given the above results, we can anticipate that there will be anEhresmann connection with components b A µ , and a G -ghost field determined by ω AA .30e will now suppose that there is a d -dimensional representation R G which gives rise to an associatedvector bundle E = F G × R G V of rank d , where V is the vector space of the representation R G . In general,the dual bundle E ∗ is inequivalent. A solder form gives an isomorphism θ : TM → E . This defines thefamiliar notion of a moving frame. In simple terms, suppose we take a coordinate basis { ∂ µ } of sections of TM , and a basis of sections { e a } for E . Then, we may write θ ( ∂ µ ) = θ a µ e a . (162)We think of the coefficients as giving a 1-form on M transforming in R ∗ G , θ a = θ a µ dx µ . (163)That is, we can equivalently regard θ ∈ Γ( T ∗ M × E ) , and write θ = θ a e a . (164)As such, θ is a section of a bundle of a type that we have not considered before. However, given that E isan associated bundle to the Lie algebroid A G , we can make use of the anchor map ρ for A G to introducethe closely related map ˆ θ = θ ◦ ρ , ˆ θ : H → E . (165)We can regard ˆ θ ∈ Γ( A ∗ × E ) , which is a section of a bundle of the type that we have considered throughoutthe paper. We will refer to ˆ θ generally as the solder form. We note that because of its definition, itis automatically horizontal. This is consistent with both our results and the literature. As we haveestablished, given that ˆ θ is a section of an associated bundle, we know how it transforms under the actionof the BRST operator. In some accounts of BRST in the case of G = SO (1, d − (e.g. [72–74]), it is oftenassumed that the co-frame field θ a has no vertical part, which then fixes its BRST transformation, thanksto the requirement of the extended torsion being horizontal. Here once again, this is a built-in feature ofthe Lie algebroid geometric structure, similarly to what happens with the Russian formula and the BRSTtransformation of the gauge field and ghosts, eqs. (152) and (160).In gravitational theories, we would interpret ˆ θ as a “matter” field in that it is a section of an associatedbundle, rather than being related to the connection on A G . The latter, for G = SO (1, d − , contains theLorentz ghost and the usual Lorentz spin connection, as we will shortly unveil. Together, this connectionand solder form are regarded as equivalent to a more traditional form of a gravitational theory, which ofcourse usually involves a metric on TM . In a given theory, this equivalence might come about throughassumption (by introducing constraints) or classically as a result of equations of motion. In metric theories,it is well-known that there is a unique connection on TM (the Levi-Civita connection [80]), which is bothmetric-compatible and torsion-free. In the present context, the metric compatibility can be interpreted asa consequence of choosing G = SO (1, d − , while the torsion-free condition comes about if it is possiblefor the solder form to be covariantly constant.Given the solder form, we can also consider the dual bundle E ∗ . Here, we mean the vector bundleassociated with A G through the dual representation R ∗ G , and we introduce P : TM → E ∗ , as well as ˆ P = P ◦ ρ : H → E ∗ , which we refer to as the Schouten form . Depending on the choice of G , E and E ∗ may or may not be equivalent. Nevertheless, since E and E ∗ are dual bundles, we regard E ∗ : E → C ∞ ( M ) .Following the general notation introduced earlier (Sec. 4 and Appendix A), we will denote by { f a } a localbasis for E ∗ which is dual to the basis { e a } for E , f a ( e b ) = δ ba . Here we are supposing that G at least contains SO (1, d − . There is no obstruction to selecting smaller subgroups.Fashionable examples would include those with non-relativistic (Galilean) or ultra-relativistic (Carrollian) symmetry [77–79]. We use this name because the usual notion of the Schouten tensor can be incorporated into the Lie algebroid language asa section of A ∗ × E ∗ . Indeed in this context, ˆ θ and ˆ P are not related, the former giving a choice of frame and the latter, infact, related to curvature. E L G A G TM E ∗ ι ρ ˆ θ ˆ P ω θ P σ . (166)The central part of this figure is identical to any Atiyah Lie algebroid, whereas the associated bundles arespecific to pure gravitational theories. In the following section, we discuss some features of the solder formand the action of the exterior derivative on it. Given that we have described the solder form as a section of the associated bundle E , it is just an applicationof the general results of the previous sections to describe how ˆ d acts here. If we regard ˆ θ as a section of A ∗ × E , and recall that ˆ θ ( X V ) = 0 , then we immediately have (ˆ d ˆ θ )( X , Y ) = φ E ( X )(ˆ θ ( Y )) − φ E ( Y )(ˆ θ ( X )) − ˆ θ ( (cid:2) X , Y (cid:3) A ) (167) = ∇ E X H ˆ θ ( Y ) − ∇ E Y H ˆ θ ( X ) − ˆ θ ( (cid:2) X , Y (cid:3) A ) − v E ◦ ω ( X V )(ˆ θ ( Y H )) + v E ◦ ω ( Y V )(ˆ θ ( X H )), (168)where v E ◦ ω ∈ Γ( A ∗ × End ( E )) and ˆ θ ( Y H ) ∈ Γ( E ) . The last two terms can then be rewritten v E ◦ ω ( X V )(ˆ θ ( Y H )) − v E ◦ ω ( Y V )(ˆ θ ( X H )) = ( v E ◦ ω ∧ ˆ θ )( X , Y ), (169)where the wedge product is taken in A ∗ . The torsion is then defined as usual as the covariant derivative ˆ T ( X , Y ) = (ˆ d ˆ θ + v E ◦ ω ∧ ˆ θ )( X , Y ) (170) = ∇ E X H ˆ θ ( Y ) − ∇ E Y H ˆ θ ( X ) − ˆ θ ( (cid:2) X , Y (cid:3) A ) (171) = ( ∇ A ∗ ×E ˆ θ )( X , Y ), (172)the last equality being an immediate consequence of the results in Appendix B. As we mentioned earlier,since ˆ θ is a section of an associated bundle (and not of A itself), it has no notion of vertical and horizontal,and simply annihilates the vertical part of a section of A . Consequently, we immediately obtain ˆ T ( X , Y ) = ∇ E X H ˆ θ ( Y H ) − ∇ E Y H ˆ θ ( X H ) − ˆ θ ( (cid:2) X H , Y H (cid:3) A ), (173)and so the torsion is manifestly and automatically horizontal. Notice that in the last term, since ˆ θ ishorizontal, the curvature does not enter. In a local trivialization, eq. (173) evaluates to ˆ T ( X , Y ) = X α H Y β H ρ µα ρ νβ (cid:16) ∂ µ θ a ν + A µ ab θ b ν − ∂ ν θ a µ − A ν ab θ b µ (cid:17) e a . (174)In the case of G being the Lorentz group, A µ ab , with A α ab = ρ µα A µ ab , are the components of the Lorentzspin connection. In previous accounts of the topic [72–74], the frame field has been endowed with aGrassmann partner, and the torsion then required to be horizontal, to obtain the BRST transformationof the frame. Here we see that geometrically, there is no sensible notion of the former, and the latter isautomatically satisfied, further indicating how the Atiyah Lie algebroid formalism is the correct one toaddress also gravitational theories. 32s well, the BRST transformation is simply encoded in ˆ d ˆ θ , which we deduce by considering the hori-zontal/vertical parts as follows (ˆ d ˆ θ )( X H , Y H ) = ∇ E X H ˆ θ ( Y H ) − ∇ E Y H ˆ θ ( X H ) − ˆ θ ( (cid:2) X H , Y H (cid:3) A ) = ˆ T ( X H , Y H ), (175) (ˆ d ˆ θ )( X V , Y V ) = 0, (176) (ˆ d ˆ θ )( X V , Y H ) = − v E ◦ ω ( X V )(ˆ θ ( Y H )) = − c E (ˆ θ ( Y H ))( X V ). (177)Since ˆ θ is horizontal, the latter equation corresponds to the BRST transformation s ˆ θ ; in a local basis, itevaluates to s ˆ θ ( E A , E α ) = − c E (ˆ θ ( E α ))( E A ) = − ρ µα c Aab θ b µ e a , (178)or, as a form equation, s ˆ θ = − c ab ∧ ˆ θ b e a . (179)Here, ˆ θ b = ρ µα θ b µ E α , and c ab is the usual Lorentz ghost if G is the Lorentz group. Similarly, we can applyall of the above to the Schouten form ˆ P . In that case, the analogue of the torsion is the Cotton form. In BRST discussions of gravitational theories, the diffeomorphism ghost plays a prominent role. This isintroduced to account for the invariance under diffeomorphisms of such theories. However, it should benoted that at least classically, it is possible to decouple the diffeomorphism ghost by redefinitions, as wasexplained first in [72–74].Throughout the paper, we made no reference to a diffeomorphism ghost; indeed, the ghosts appearedas vertical forms encoded in the ω map, and so are associated with the structure group of the Lie algebroid.That is, we saw only gauge ghosts arise in the formalism. The construction is intrinsically well-defined,and thus independent of a choice of local coordinates. One might imagine that if diffeomorphisms weresomehow promoted to be part of the structure group, then the diffeomorphism ghosts would appear.We do not advocate this picture here. Instead, we note that the result quoted above of Refs. [72–74]is consistent with the fact that we do not have diffeomorphism ghosts. As we now describe, there is anatural place for the diffeomorphism ghosts to appear within the Lie algebroid formalism. Recall thatthis ghost, in the traditional description, is thought of as a Grassmann vector, ζ µ . Here that descriptionwould carry over to a vector-valued vertical form, a tensor with index structure ζ µ A . Whereas, the gaugeghosts appear in ω : A → L , one might expect that diffeomorphism ghosts should analogously appear in ρ : A → TM . Indeed, as we described in Section 4, the ρ map does have generally the index structure ρ µ M , and thus in the language of a split basis, there are the components ρ µ A . However, those componentswere set globally to zero in any split basis, as a result of the E A spanning the vertical sub-bundle of A , thekernel of ρ . We thus arrive at a simple picture: given a connection on the Lie algebroid, it was possible todefine a split basis; such a choice can be thought of as the choice that absorbs the diffeomorphism ghostinto a redefinition. It is important that the curvature of the map ρ vanishes, so that the split basis maybe retained globally.Thus, we have translated all of the usual notions of gravitational theories (as presented in the first-orderformalism) into the corresponding structures associated with a Lie algebroid. This again is an off-shellconstruction, and as such the off-shell symmetry has been imposed as a choice of G -structure, independentof the properties of any particular Lagrangian. 33 Conclusions
Our aim in this work has been to set the right mathematical stage to formulate the geometric structureof gauge theories, including their BRST data. We have shown that this is the Atiyah Lie algebroid. Thisefficiently encodes the proper physical degrees of freedom, not only a gauge field but also a correspondinggauge ghost field. Furthermore, we have understood carefully the exterior derivative on Lie algebroidsand on arbitrary associated bundles, and demonstrated that it encodes two fundamental notions for gaugetheories: the gauge covariant derivative in its horizontal part and the BRST operator s in its vertical part.The understanding that BRST is imprinted in (Atiyah) Lie algebroids is novel, so we have taken time toshow that it indeed gives rise locally to the well-known BRST transformations of the gauge field, the gaugeghost and charged matter fields. Contrary to previous results in the literature, there is no need to imposeexternal constraints to achieve these transformations, but instead they are just part of the geometry.It is important to realize that much of the mathematics and physics of the usual description of gaugetheories involves not just a principal bundle but the tangent and cotangent bundles thereof. A substantialdifference from previous works on geometrization of gauge theories is that the Atiyah Lie algebroid is infact a bundle over the space-time M rather than the principal bundle P . We have already seen somepayback of this fact: physical data like the gauge and ghost fields are encoded in bundle maps over M itself, removing redundant group action. We have concluded the body of the manuscript exhibiting howgravitational theories are treated on an equal footing to Yang-Mills theories in our formalism. The onlyextra ingredient is the solder form, which we have properly interpreted as related to a bundle associatedto the Atiyah Lie algebroid.There are many possible applications and extensions of our work. First, since the construction isgeometric, it is natural to apply the formalism to physical models, involving the specification of an actionand quantization. Second, it is of interest to exploit the differential complex, for example in the contextof anomalies of various types, within the Lie algebroid context.Another extension of our work that we expect to have many physical applications would involve Liealgebroids with boundaries, interfaces and corners on the base manifold M . Indeed, this was one of ourmotivations for initiating the present work, as such constructions are clearly better formulated when allof the physics is associated with bundles over M . Possible applications include holography, asymptoticsymmetries, quantum symmetries of gravity and entanglement. Acknowledgements
We are grateful to Glenn Barnich, Weizhen Jia and Laurent Baulieu for enlight-ening discussions. LC is supported by the ERC Advanced Grant
High-Spin-Grav . The work of RGL wassupported by the U.S. Department of Energy under contract DE-SC0015655.
A Conventions
This Appendix is devoted to a summary of the conventions used throughout the manuscript.Given a bundle B over a manifold M , the space of its sections is denoted Γ( B ) . Sections of a vectorbundle are written with an underline, e.g., X , Y , .. for the bundle A over M . Examples of bundles over M in the paper are A , L , E and of course TM . We use the underline to distinguish these sections fromfunctions in C ∞ ( M ) or the components of the sections, which are a set of rank B functions. One of theadvantages of this notation is that, as long as there are no explicit indices in an equation the latter iscompletely invariant (tensorial). Locally, we can introduce a basis of sections of a bundle. We carry overthe underline convention to the basis, to again distinguish it from the components. Consider for instancea vector bundle L over M . Its sections are denoted µ . We then introduce a basis { t A } with the index A running over rank L values and write µ = µ A t A . Here, µ A are just rank L functions of M . The utility ofthis convention is clear once we consider maps between bundles over M with pre-image L . An example is34he inclusion map ι mapping L to A , which is another bundle over M . While in index-free notation ι ( µ ) isjust a section of A , we can write ι ( µ ) = ι ( µ A t A ) = µ A ι ( t A ), (A.1)where we simply used the fact that ι is linear with respect to multiplication by C ∞ ( M ) functions. Havingunderlined the basis facilitates the employment of linearity of bundle maps to reduce them to maps betweenbases. Indeed, given a basis { E M } for sections of A , with the index M spanning rank A values, we have ι ( t A ) = ι M A E M , where ι M A are the components of the map ι in the specific bases for L and A . For bundlemaps, we conventionally write the target space index first and the pre-image space index second, as we didfor ι M A . All in all we thus have ι ( µ ) = ι ( µ A t A ) = µ A ι ( t A ) = ι M A µ A E M , (A.2)where again the underline convention facilitates recognizing that the output is a section of A expressed inthe local basis { E M } .Given a vector bundle B over M , we have a dual bundle B ∗ , whose sections are maps from B to C ∞ ( M ) .To avoid confusion, we do not underline bases of B ∗ . Again, the most familiar example is T ∗ M ; here we willuse as specific example the vector bundle E . Once a basis { e a } ( a = 1, .., rank E ) is specified, the dual basisof E ∗ is then denoted { f b } , defined via f b ( e a ) = δ ba . Although there could be potential tension betweenthe basis of E ∗ , f a , and the components ψ a of a section of E , ψ = ψ a e a , we decide not to underline f a butrather to simply use different letters from the ones chosen for the components ψ a . We refer to sections of A ∗ as extended forms, and introduce the basis { E M } via E M ( E N ) = δ M N . Extended forms in A ∗ may bevalued in associated bundles. Therefore, ψ n ∈ Γ( ∧ n A ∗ × E ) is an E -valued extended n -form, mapping n sections of A to E . The underline in ψ n is due to the fact that ψ n ( X , ..., X n ) , with X , ..., X n ∈ Γ( A ) , is asection of E . Thus, we write expressions like ψ ( Y ) , which simply means that the E -valued extended oneform ψ acts on a section of A to produce a section of E . Note that since ˆ d is defined to act on sectionsof ∧ n A ∗ × E via the morphism φ E , some authors include a subscript on ˆ d to emphasize this. Here wehave chosen not to, so that ˆ d should be understood by context; see for example eq. (65). In this equationwe also introduce the notation ∇ ∧ n A ∗ × E ; here we denote by ∇ B the induced connection for any bundle B associated to A . As this might be difficult to understand at first, we expand on its meaning in App.B. Extended forms on A can be seen equivalently as quantities living in ∧ n A ∗ × E , or as maps taking n sections of A to E . It is important therefore to extract from the resulting section of E the form part actingon sections of A . As an illustrative example, consider ˆ d ψ ∈ Γ( ∧ A ∗ × E ) . Acting on two sections X and Y of A it produces a section of E , ˆ d ψ ( X , Y ). Moreover, one can extract from this the extended two form ˆ d ψ = (ˆ d ψ ) aMN E M ∧ E N ⊗ e a , (A.3)where (ˆ d ψ ) aMN are just the components of this object living in Γ( ∧ A ∗ × E ) . This process is crucial inunderstanding the covariant derivative and BRST action. To characterize them, we furthermore needto discuss how to extract the horizontal and vertical parts of the extended forms themselves. This isestablished in Sec. 2.2 and further commented upon in App. B. B ˆ d and extended forms We have computed various aspects of the ˆ d -action. Since formulae can be rather involved, in this Appendixwe give full details of how ˆ d acts on extended one forms valued in E , i.e., on sections Γ( A ∗ × E ) . We havewritten the result in a compact form in the text by using the induced connection on A ∗ × E . Recall thatgiven a connection ∇ B on a vector bundle B , there is a connection induced on B ∗ via X ( α ( v )) = ( ∇ B ∗ X α )( v ) + α ( ∇ BX v ), X ∈ Γ( TM ), α ∈ Γ( B ∗ ), v ∈ Γ( B ), (B.1)35hich follows from Leibniz. For example, given (119), we have immediately ∇ E ∗ X H f a = − A ab ( X H ) f b , (B.2)since f a ( e b ) = δ ab is constant. The connections on ∧ n A ∗ × E are deduced similarly, based on Leibniz.Here, we would like to give some details on extracting the horizontal-vertical split of ˆ d ψ as an example, ˆ d ψ = ∇ A ∗ × E ψ + s ψ , (B.3)from the Koszul formula (28). Note that this process is not as immediate as it was for sections of E since ˆ d ψ ( X , Y ) involves two sections each of which have horizontal and vertical parts. The Koszul formula gives ˆ d ψ ( X , Y ) = φ E ( X )( ψ ( Y )) − φ E ( Y )( ψ ( X )) − ψ ( (cid:2) X , Y (cid:3) A )= (ˆ d ( ψ ( Y )))( X ) − (ˆ d ( ψ ( X )))( Y ) − ψ ( (cid:2) X , Y (cid:3) A )= ∇ E X H ( ψ ( Y )) − ∇ E Y H ( ψ ( X )) − ψ ( (cid:2) X , Y (cid:3) A ) − v E ◦ ω ( X V )( ψ ( Y )) + v E ◦ ω ( Y V )( ψ ( X )), (B.4)which follows since ψ ( X ) ∈ Γ( E ) . We now split the sections X , Y into horizontal and vertical parts, ˆ d ψ ( X , Y ) = ∇ E X H ( ψ ( Y H )) − ∇ E Y H ( ψ ( X H )) − ψ ( (cid:2) X H , Y H (cid:3) A )+ ∇ E X H ( ψ ( Y V )) − ∇ E Y H ( ψ ( X V )) − ψ ( (cid:2) X H , Y V (cid:3) A ) + ψ ( (cid:2) Y H , X V (cid:3) A ) − v E ◦ ω ( X V )( ψ ( Y )) + v E ◦ ω ( Y V )( ψ ( X )) − ψ ( (cid:2) X V , Y V (cid:3) A ). (B.5)Here, we organized the right-hand side in three separately tensorial expressions. Then, defining ( ∇ A ∗ × E ψ )( X H , Y H ) = ∇ E X H ( ψ ( Y H )) − ∇ E Y H ( ψ ( X H )) − ψ ( (cid:2) X H , Y H (cid:3) A ) (B.6) ( ∇ A ∗ × E ψ )( X H , Y V ) = ∇ E X H ( ψ ( Y V )) − ψ ( (cid:2) X H , Y V (cid:3) A ) (B.7) ( ∇ A ∗ × E ψ )( X V , Y H ) = −∇ E Y H ( ψ ( X V )) + ψ ( (cid:2) Y H , X V (cid:3) A ) (B.8) ( ∇ A ∗ × E ψ )( X V , Y V ) = 0, (B.9)we have ˆ d ψ ( X , Y ) = ( ∇ A ∗ × E ψ )( X , Y ) − v E ◦ ω ( X V )( ψ ( Y )) + v E ◦ ω ( Y V )( ψ ( X )) − ψ ( (cid:2) X V , Y V (cid:3) A ). (B.10)Comparing with (B.3), we then identify s ψ ( X , Y ) ≡ − v E ◦ ω ( X V )( ψ ( Y )) + v E ◦ ω ( Y V )( ψ ( X )) − ψ ( (cid:2) X V , Y V (cid:3) A ) (B.11) = − c E ( ψ ( Y ))( X V ) + c E ( ψ ( X ))( Y V ) − ψ ( (cid:2) X V , Y V (cid:3) A ), (B.12)where we introduced c E using (68).Notice finally that (B.6-B.9) give us an explicitly formula for (66): ( ∇ A ∗ × E ψ )( X , Y ) = ( ∇ A ∗ × E X H ψ )( Y ) − ( ∇ A ∗ × E Y H ψ )( X ) (B.13) = ∇ E X H ( ψ ( Y )) − ∇ E Y H ( ψ ( X )) − ψ ( (cid:2) X H , Y H (cid:3) A ) − ψ ( (cid:2) X H , Y V (cid:3) A ) + ψ ( (cid:2) Y H , X V (cid:3) A ). This result is instrumental in the derivation of the BRST transformation of the gauge field, eq. (160).36 eferences [1] M. Ikeda and Y. Miyachi, “On an Extended Framework for the Description of ElementaryParticles,”
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