Lie Algebroid Yang Mills with Matter Fields
aa r X i v : . [ h e p - t h ] A ug Lie Algebroid Yang Mills with Matter Fields
C. Mayer and T. StroblOctober 29, 2018
Abstract
Lie algebroid Yang-Mills theories are a generalization of Yang-Millsgauge theories, replacing the structural Lie algebra by a Lie algebroid E . In this note we relax the conditions on the fiber metric of E forgauge invariance of the action functional. Coupling to scalar fieldsrequires possibly nonlinear representations of Lie algebroids. In allcases, gauge invariance is seen to lead to a condition of covariantconstancy on the respective fiber metric in question with respect toan appropriate Lie algebroid connection.The presentation is kept in part explicit so as to be accessible alsoto a less mathematically oriented audience. Contents Introduction
In Ref. [1] pure Yang Mills (YM) gauge theories have been generalized to asetting where the structural Lie algebra is replaced by a Lie algebroid. Thisis a vector bundle E → M with, among others, a Lie algebra structure onits sections, thus reducing to a Lie algebra for M being a point, in whichcase also the Lie Algebroid Yang Mills (LAYM) gauge theory reproducesjust an ordinary YM theory in d spacetime dimensions. Simultaneously, itconstitutes a nonlinear type of gauge theory, which in contrast to topologicalprototypes like the Poisson Sigma Model [13, 14] has propagating degrees offreedom. At least on the classical level, moreover, these propagating degreesseem to be those of ordinary YM theories, albeit of potentially different typeand with potentially different structure groups, glued together over somefinite dimensional moduli space [1].In this paper we first reconsider these LAYM theories, using a second typeof gauge symmetries (one that is induced by an auxiliary connection chosenon E ). For actions of type F + F , where F (1) and F (2) denote the 1-formand 2-form field strengths of the gauge field, respectively, and the squareis understood as denoting an appropriate norm square, we find that gaugeinvariance restricts E to be an action Lie Algebroid, which from the physicalpoint of view corresponds to ordinary YM theory coupled to “Higgs fields”possibly taking values in some curved target manifold. We then show thatthe action [1] of the form B F (1) + F , B denoting Lagrange multiplier fields,is gauge invariant if the respective fiber metric on E is covariantly constantw.r.t. a Lie algebroid or E -connection (induced by the auxiliary ordinary oneon E ) that (at least when flat) can be thought of generalizing the adjointrepresentation of a Lie algebra.Then we turn to the main subject of the paper, the coupling of scalarmatter fields to the LAYM theory. We first assume that these scalar fieldstake values in some vector bundle V → M . Starting with some elementaryansatz for gauge transformations of the scalar fields, we are lead rather di-rectly to flat E -connections on V . This is the mathematical generalizationof a Lie algebra representation on a vector space, to which it reduces for M being a point. Gauge invariance of the kinetic term requires that the fibermetric on V , needed for its construction, is covariantly constant w.r.t. this E -connection.This is then, in a second step, generalized to scalar fields with target spacean arbitrary bundle p : ˜ M → M over the base of the Lie algebroid. In this2ontext it is helpful to observe that all the needed data can be reassambledinto a Lie algebroid structure on e E = p ∗ E . It is the generalization of anaction Lie algebroid to the context of Lie algebroids. Gauge invariance ofthe more general kinetic term in this sigma model context now requires afiber metric on V f M (the bundle of vertical vectors on f M ) that is covariantlyconstant w.r.t. a canonically induced flat e E -connection on V f M .This perspective suggests a reinterpretation of the scalar fields. Namely,one may have started right away by considering the Lie algebroid e E → f M .The previously constructed coupled LAYM-matter Lagrangian is then seenas a functional of a (pure, if one likes) LAYM gauge theory for e E of theform that only part of the 1-form field strengths enter the functional withLagrange multipliers while the “remaining” ones are squared by means ofan appropriate symmetric covariant two-tensor on f M (a partially degenerate“metric” tensor on f M ).Finally, we exploit generalized Bianchi identities to further relax the con-dition on the fiber metric E g on E in a pure LAYM-theory of the typeconsidered in [1]. In fact, for gauge invariance it turns out to be sufficientthat the restriction of E g to the kernel of the anchor map of E is invariantw.r.t. a canonical Bott-type E -connection. Here we start recalling the basic elements of a Lie algebroid Yang-Mills (YM)theory in a rather explicit, elementary fashion. The d -dimensional spacetimemanifold we denote as (Σ , h ), where h is a fixed (possibly pseudo-) Rie-mannian metric. The structural Lie algebra entering the construction of anordinary YM algebra is generalized to a Lie algebroid ( E → M, [ · , · ] , ρ ), thebasic definition of which (together with other background material) can befound in Appendix A. Using some local coordinates x i on M and a local frame e a of E , all the structural quantities of E can be described by functions ρ ia ( x )and C cab ( x ), satisfying the differential equations ρ ja ρ ib,j − ρ jb ρ ia,j = C cab ρ ic , C ead C dbc + ρ ia C ebc,i + cycl ( abc ) = 0 . (1)Clearly, if M is a point, thus C cac not depending on x i , and ρ ia ≡
0, onereobtains the structure constants of a Lie algebra g . This is formulated more intrinsically in the last section of the present article. A a = A aµ ( u µ )d u µ , where u µ are coor-dinates on Σ, together with 0-form fields X i ( u µ ). The latter ones describe amap X i from Σ to M , X and A together a vector bundle map a : T Σ → E ( cf. [2] or [1] for further details). Associated to these “gauge fields” are the“field strengths” F i = d X i − ρ ia A a F a = d A a + c abc A b ∧ A c + Γ iab F i ∧ A b , (2)where Γ iab are the coefficients of a fixed background connection ∇ i in E ; theyare necessary if one wants to define the 2-form field strengths F a covariantlywith respect to E -frame changes. This becomes most transparent whenrewriting the second equation according to F a = ( D Γ A ) a − T abc A b ∧ A c , (3)where ( D Γ A ) a ≡ d A a + Γ iab d X i ∧ A b , (4) T cab ≡ − C cab + ρ ia Γ icb − ρ ib Γ ica . (5)Here D Γ A is the exterior covariant derivative on A ∈ Ω(Σ , X ∗ E ) and T isthe E -torsion of the E -connection ∇ ρ ( · ) , both being induced by the chosenconnection ∇ on E ( cf. Appendix A for further details); the 2-form fieldstrength is then an element in Ω (Σ , X ∗ E ). In the specific case described atthe end of the previous paragraph, E = g , one is back to the usual YM setting(with a Lie algebra valued 2-form curvature and no 1-form field strength) .This also applies to the gauge transformations, which we will now address.Infinitesimally the gauge transformations are taken to be of the form δ ǫ X i = ρ ia ǫ a (6) δ ǫ A a = d ǫ a + C abc A b ǫ c + Γ aib ǫ b F i , (7) Both field strengths together can be given a meaning also without introducing a con-nection ( cf. , e.g. [15]); it is only the separation of the 2-form part which requires theconnection.—The fixed connection on E is not to be confused with the gauge fields, which,in the case of an ordinary YM theory are connections in a principal bundle; the formerones correspond to structures needed to be fixed for defining a functional, while the latterones are dynamical, i.e. they are the argument of that functional. We discuss trivial bundels over Σ here only, cf. [15] for how to generalize to nontrivialones. F a . There is also an alternative, geometrically motivated, off-shell closed version of gauge symmetries, not using an auxiliary connectionand also generalizing the usual YM ones ( cf. [2, 1]); as mentioned already inthe Introduction, in this note we want instead to focus on this connection-induced type of gauge symmetries. In any case, in the variation of A a a termproportional to F i is needed for E -covariance again. Note, however, that theterms in (7) do not combine completely into covariant objects following thepattern of (3): δ ǫ A a = D Γ ǫ a − T abc A b ǫ c − ρ ic ǫ c Γ iab A b . (8)The reason is that infinitesimal gauge transformations are a derivative-typeobject and the extra term is needed for compatibility with (6), cf. [2] as wellas the likewise discussion following Eq. (21) below.On the A -fields the variations (7) close only modulo a term proportionalto F i , (cid:16)(cid:2) δ ǫ , δ ǫ (cid:3) − δ ǫ (cid:17) A a = ǫ b ǫ c F i S iabc , (9) ǫ a ≡ C abc ǫ b ǫ c (10) S iabc ≡ ∇ i T abc + ρ jc R ij ab − ρ jb R ij ac (11)where ∇ i denotes the covariant derivative with respect to the fixed back-ground connection on E and R ij ba its curvature. As a consequence, the gaugesymmetries provide a representation of the Lie algebroid on the fields X i , A a only if either F i = 0 or S iabc = 0. For later use we provide the gauge variationof the field strengths: δ ǫ F i = ǫ a (cid:0) ∇ j ρ ia (cid:1) F j , (12) δ ǫ F a = − ǫ c (cid:0) c abc + Γ iac ρ ib (cid:1) F b , + ǫ b R ij ab F i ∧ F j + ǫ c S iabc F i ∧ A b . (13)where ∇ j ρ ia ≡ ρ ia,j − Γ bja ρ ib denotes the covariant derivative w.r.t. the index a only. These equations hold in a frame where the parameters ǫ a depend on coordinates ofΣ only, but not also on the fields X or even X and A . We intend to provide a morecoordinate independent interpretation elsewhere. cf. [3, 1, 4]): S LABF = Z Σ B i ∧ F i + B a ∧ F a , (14)where B i and B a are ( d −
1) and ( d − F i = 0 and F a = 0 require( X, A ) to correspond to a Lie algebroid morphism from T Σ to E , while theabove gauge transformations reduce to Lie algebroid homotopies in that case( cf. [2] for further details).So as to construct a gauge invariant Lie Algebroid YM action, one wouldnaturally be lead to square both field strengths, Z Σ − F a ∧ ⋆F b g ab − F i ∧ ⋆F j g ij , (15)using a metric g ∼ g ij on M , a fibre metric E g ∼ g ab on E , and the metric h on Σ for the Hodge dual of differential forms. The condition of gaugeinvariance of the action should then imply some meaningful conditions onthe additional structures g ij , g ab (generalizing ad-invariance of the metric onthe Lie algera in the ordinary YM case) and their existence then possiblya restriction on the possible Lie algebroids E (quadratic Lie algebras in theYM situation).In the context of the above functional, however, the restrictions turnout to be enormous, bringing one back implicitely to the realm of ordinaryYM gauge theories: The variation of the field stregth F a in the first termproduces terms proportional to F i ∧ F j ∧ ⋆F a and F i ∧ A a ∧ ⋆F b , both of whichcannot be compensated for by variations of other parts of the actions andthus have to vanish individually. The vanishing of the first term implies that ∇ is a flat connection on E , R ij ba = 0, the second constraint, S iabc = 0, thenreduces to ∂ i C abc = 0 ( cf. Eqs. (11) and (5)). This in turn implies that we canidentify E with M × g , g being the Lie algebra with the respective structureconstants C abc and ρ : E → T M can be identified with a representation of iton M ( cf. Eq. (1)). From a physical perspective, then, the theory reduces tostandard YM theory (first term in (15)) with structural Lie algebra g , coupled Such an E is called an action Lie algebroid.
6o a Higgs-type sigma model with the Higgs fields taking values in M (thesecond term in (15) reduces to the usual kinetic term of such a theory).In fact, part of these conditions, namely S iabc = 0, can already be deducedfrom (9), taking into account that obviously F i = 0 are not field equationsfor the action functional (15). This consideration, however, provides also ahint for a way to avoid the above no-go-type result: One may want to ensurethat F i = 0 are part of the field equations of the strived for generalizationof the YM-action (note that for an ordinary YM theory, M is a point and F i vanishes identically). In this way one is lead to [1] S LAY M = Z Σ B i ∧ F i − F a ∧ ⋆F b g ab , (16) B i being ( d −
1) forms on Σ like in (14) above.Now again we ask for the conditions on the structural ingredients, i.e. E , ∇ i , and g ab , such that the above functional is gauge invariant w.r.t. thesymmetries generated by Eqs. (6, 7) (for some transformation induced onthe B i -fields). The action functional (16) is gauge invariant w.r.t. thosegauge transformations, if the fiber metric E g ∼ g ab is covariantly constantw.r.t. a certain Lie algebroid (“ E -”) connection E ˜ ∇ , E ˜ ∇ E g = 0 . (17)This E -connection is one induced by the ordinary connection ∇ on E anddefined via E ˜ ∇ ψ ˜ ψ = ∇ ρ ( e ψ ) ψ + [ ψ, ˜ ψ ] . (18)In local components the coefficients of this E -connection read as e Γ abc = ρ ic Γ aib + c abc = ρ ib Γ aic − T abc . From the first equality one obtains (17) at once,observing that the first line of (13) contains precisely e Γ abc (while the twoterms in the second line, which resulted in the unwanted severe restrictionon E in the case of (15), now can be absorbed by the variation of B i since theyare both proportional to F i ); the second equality shows that E ˜ ∇ differs fromthe more obvious E -connection ∇ ρ ( · ) by subtraction of its own E -torsion.Note that for M being a point, the first term in (18) is absent since ρ vanishes and one reobtains the usual condition of an ad-invariant metric This geometric interpretation was observed already shortly after completion of [1] andreported e.g. in [5]. The concept of a Lie algebroid connection and corresponding generalizations of cur-vature and torsion is recalled in Appendix A.
7n the Lie algebra. The existence of an ordinary connection ∇ and a fibermetric E g such that (17) is fulfilled, poses a restriction on E . In the case ofintegrable Lie algebroids and for E g having definite signature, this restrictionis conjectured by Fernandes to precisely give Lie algebroids E coming from proper Lie groupoids (a notion coinciding with compactness in the Lie groupcase) [6]. It is amusing that this particular E -connection pops out naturallyfrom invariance of the functional (16) and the simple ansatz (6,7) for thegauge symmetries.In fact, it turns out that a condition like (17) (or the likewise one foundin [1]) is sufficient but not also necessary for gauge invariance of the actionfunctional S LAY M . We will discuss this issue in detail in section 5 below.Before closing this section, we make a remark on some geometric inter-pretation of the tensor (11); in fact it is related to the E -curvature of E ˜ ∇ a by contraction with the anchor map ρ ( cf. Appendix A): E ˜ R abdc = ρ ic S idab . (19)Hence, if the gauge transformatons close off-shell, i.e. if S iabc = 0, then E ˜ ∇ a is flat. The converse statement is not true. We will encounter flat E -connections in the subsequent section when considering the issue of couplingmatter fields to the above action functional S LAY M . Flat E -connections onvector bundles over M are the natural generalization of a (linear) represen-tation of a Lie algebra to the context of Lie algebroids ( cf. , e.g. , [9]). A flat E -connection E ˜ ∇ on E can then be considered as a possible generalizationof the adjoint representation of a Lie algebra. In this section we address the issue of coupling scalar fields to the YM-typetheory of the previous section. Since we address trivial bundles over Σ onlywithin this note, in the ordinary YM situation this would correspond to somefunctions on Σ taking values in a vector space which carries a representationof the structural Lie algebra. Representations of Lie algebroids are knownin the mathematical literature as flat E -connections. Here we will, however, We will in the following section, however, not assume familiarity with such a math-ematical concept. Instead, we will start in a pedastrian style for the construction ofa coupling to matter fields and be lead automatically to the mathematical concepts bymeans of gauge invariance. φ σ of functions on Σ. We expect/wantformulas to be covariant w.r.t. φ σ → φ σ ≡ M στ φ τ (20)for arbitrary matrices M στ . In the usual YM setting this corresponds to achange of basis in the representation space of the Lie algebra. In the presentmore general setting the gauge fields contain not only 1-forms A on Σ, butalso 0-forms X i and it is thus natural to permit M στ to depend on x . Moreabstractly, this implies that the Higgs-type scalar fields φ σ correspond tosections of X ∗ V , where V is a vector bundle over M , the same base as theLie algebroid E (and X the previous map from Σ to M ).Now we make the following ansatz for infinitesimal gauge transformations: δ ǫ φ σ = − ǫ a Γ aστ φ τ , (21)where Γ aστ are some at this point not further specified fixed parameters de-pending on X . Covariance restricts them further, however: We want that for f φ σ we have a likewise formula. On the other hand, using Eq. (6) and the factthat δ ǫ ( M στ φ τ ) = δ ǫ ( M στ ) φ τ + M στ δ ǫ φ τ , we can determine the transformationproperty of the above coefficients,Γ aστ = M σσ ′ Γ aσ ′ τ ′ M − τ ′ τ − ρ ia M σσ ′ ,i M − σ ′ τ . (22)This implies that these coefficients have the geometrical interpreation of an E -connection E ∇ on the vector bundle V . Finally we demand that the gauge transformations close on the newlyintroduced fields, [ δ ǫ , δ ǫ ] φ σ = δ ǫ φ σ , (23) We usually drop the extra upper E in the E -connection coefficients, since their indicesalready make clear of what nature they are. Solely with the respective E -curvatures wekeep it for clarity also in the components. Note that in the present section ordinaryconnections as well as E -connections always refer to the vector bundle V → M , in contrastto the previous section where they both referred to E → M itself—for notational simplicitywe use the same symbols. The representation space V can be chosen as E itself, certainly;the notations are chosen such that they coincide in that particular case. ǫ is given by formula (10). Note that in this case it is not naturalto permit a contribution proportional to F i as in (9), although using themetric h on Σ one could produce also 0-form contributions from F i . Thecondition (23) is equivalent to E ∇ having vanishing E -curvature. Thus withthese physical considerations we indeed find that to couple matter fieldsto a Lie algebroid Yang-Mills theory S LAY M for some given structural Liealgebroid E → M we need a vector bundle V → M carrying a flat E -connection E ∇ , a “Lie algebroid representation” on V in the mathematicalsense. Note that vanishing E -curvature by definition means [ E ∇ ψ , E ∇ e ψ ] = E ∇ [ ψ, e ψ ] , with the Lie algebroid bracket on the r.h.s.; thus this indeed impliesthat the differential operators E ∇ ψ are a representation of the Lie algebradefined by the Lie algebroid bracket.The generalization of a covariant derivative on Higgs fields in ordinaryYM-theory takes the form Dφ σ = d φ σ + Γ aστ A a φ τ + Γ iστ φ τ F i , (24)where Γ iστ and Γ aστ are coefficients of an ordinary connection ∇ and theabove E -connection E ∇ , respectively, both defined on V (it is certainly theirpullback by X that enters in such an expression, φ being a section in X ∗ V —following physics conventions such identifications are understood). The firsttwo terms are familiar ones if, following Eq. (21), one identifies Γ aστ with thecoefficients of a representation; the contribution proportional to F i is againneeded for covariance (under changes of E - and V -frames). Indeed, the termsin (24) may be recombined into( Dφ ) σ = ( D Γ φ ) σ − A a T aστ φ τ (25)where ( D Γ φ ) σ = d φ σ +d X i Γ iστ φ τ is the canonical exterior covariant derivativein X ∗ V induced by the connection ∇ on V . T , on the other hand, is (thepullback by X of) a section in E ∗ ⊗ End ( V ), defined, for any ψ ∈ Γ( E ), bymeans of the difference of two E -connections (on V ), namely T ψ = ∇ ρ ( ψ ) − E ∇ ψ . (26)In the particular case of V = E and E ∇ = E ˜ ∇ it coincides with the E -torsiontensor of ∇ ρ ( · ) , cf. the text following Eq. (18). Thus Dφ is indeed a sectionof T ∗ Σ ⊗ X ∗ V , as it should be. For a more general ansatz of a covariant derivative in the Lie algebroid setting cf. [7],with however the same result. D “commutes” withgauge transformations, but only modulo a term proportional to the fieldstrength F i , ( δ ǫ Dφ ) σ − ( Dδ ǫ φ ) σ = ǫ a F i e S aiστ φ τ + ǫ a A b E R baστ φ τ . (27)Indeed, the second term vanishes identically since E ∇ is a flat E -connection.Here e S aiστ ≡ (cid:0) ∇ i T a − ρ ja R ij (cid:1) στ . We remark in parenthesis that for the adjoint E -connection E ˜ ∇ on E the tensor S parametrizing the non-closure of gaugetransformations on A I , cf. Eq. (9), and e S do, for R ij ac = 0, not coincide, S aicb = e S aicb + ρ ja R ij cb .The action of LAYM theory coupled to matter fields φ σ is then the sumof the LAYM action and a kinetic term for the mattter fields, S LAY M + matter = S LAY M − Z Σ 12 ( Dφ ) σ ∧ ⋆ ( Dφ ) τ g στ ( X ) , (28)where g στ ∼ V g is (the pullback by X of) a non-degenerate metric on V . Theaction is invariant under the gauge symmetries, if g στ is compatible with the E -connecton E ∇ on V , i.e. if E ∇ ( V g ) = 0 . (29)The terms proportional to ˜ S , coming from the variation of the kinetic term byuse of eq. (27), are proportional to F i and thus can be absorbed by redefining δ ǫ B i correspondingly.It is easy to add e.g. a mass term for φ to this, using V g : R Σ φ σ φ τ g στ vol Σ isalready by itself invariant under gauge transformations (here vol Σ denotes thevolume form on Σ induced by h ). This can be generalized in a straightforwardmanner to higher powers in φ , including thus self-interactions of the scalarfields, by means of completely symmetric tensors I σ ...σ n ∼ I ∈ Γ( S n V ) whichare E -covariantly constant, E ∇ ( I ) = 0: X n Z Σ φ σ . . . φ σ n I σ ...σ n ( X ) vol Σ . (30)Another way of obtaining a coupling of a LAYM theory (16) to scalarfields is to perform a Kaluza-Klein dimensional reduction from Σ d to Σ d − S , which we may take to be along the direction µ = 0. Thenthe vector field A a on Σ d decomposes into a vector field ˆ A a on Σ d − and into ascalar field, φ a coming from the 0-component of A a . As shown in Appendix B,the dimensional reduction of both, the gauge symmetries and the action,shows that φ a transforms according to E ˜ ∇ a , Eq. (18); moreover the (0 , m )component of F a coincides with the covariant derivative D for the particular E -covariant derivative E ∇ I = E ˜ ∇ I . One might ask how the condition ofa flat E -connection found in this Section is compatible with dimensionalreduction where E ˜ ∇ is arbitrary. However, the dimensional reduction of thezero-component of F i restricts φ a to ker ρ on-shell—as a relict from the B i F i -term in S LAY M — where the curvature of E ˜ ∇ vanishes, cf. (19). Dimensionalreduction therefore leads to a rather restricted setting. The E -connection ispermitted to be nonflat, but at the price of restricting the scalar fields totaking values in ker ρ only. One of the possible perspectives on a LAYM theory is that it generalizesordinary YM gauge theories to the realm of sigma models, cf. , e.g. , [10].In the usual YM setting, scalar fields, like the Higgs field, take values invector bundels associated to the principal bundle in which the gauge fieldsare connections. From the present perspective, such a restriction to linearity,as present in the formulas (20), (21), (24) for example, seems unnecessary andnon-natural. In the ordinary Lie algebra situation, E = g , this correspondedto linear representations of the Lie algebra on a vector space V (used in theconstruction of the associated bundle). However, we may be interested also innonlinear, sigma-model like couplings of the scalar fields to the LAYM-part.Towards this goal it is useful to note that the data used in the previoussection, a Lie algebroid E → M together with a flat E -connection E ∇ on p : V → M can be put together into a bigger Lie algebroid e E : As a vec-tor bundle this Lie algebroid is just e E ≡ p ∗ E → V , i.e. E considered asliving over the bundle V as base manifold. The Lie bracket between sec-tions coming from sections of E is the old one, [ p ∗ ψ , p ∗ ψ ] := p ∗ [ ψ , ψ ].It remains to define what happens when p ∗ ψ is multiplied by a functionover V that is fiber-linear (the rest follows by the Leibniz rule), i.e. by sec-tions α ∈ Γ( V ∗ ). It is here where the E -connection enters: [ p ∗ ψ , αp ∗ ψ ] := αp ∗ [ ψ , ψ ] + ( E ∇ ψ α ) p ∗ ψ . The flatness condition of E ∇ comes in when12hecking the Jacobi condition of that bracket.Now it is straightforward to generalize to the nonlinear setting. Let us justreplace the vector bundle p : V → M by a general fiber bundle p : f M → M .Again we can consider the vector bundle e E := p ∗ E → f M . Instead of a repre-sentation on V we want to consider the structure of a Lie algebroid definedon e E , satisfying an appropriate compatibility condition with E certainly:There is always a natural projection π : p ∗ E → E induced by p : f M → M . e E ≡ p ∗ E −−−→ f M π y y p E −−−→ M (31)One can check that in the linear situation above, π is a Lie algebroid mor-phism ( cf. e.g. [2] for a convenient way of checking this). This is what wenow want to require also in the present more general situation: by definition,an E -action on f M is a Lie algebra structure on p ∗ E such that the projection π is a morphism of Lie algebroids.It is important in this context that e E really is the bundle p ∗ E and not justisomorphic to it and that π is the corresponding canonical projection. Onecan check, furthermore, that for M being a point the Lie algebroid e E → f M reduces to the action Lie algebroid e E = g × f M of a Lie algebra action g on a manifold f M . So, e E = E × M f M is the “action Lie algebroid” of a Liealgebroid ( E → M )-action on f M → M .Part of the Lie algebroid morphism property of π : e E → E is the commu-tativity of the following diagram p ∗ E e ρ −−−→ T f M π y y p ∗ E −−−→ ρ T M . (32)This permits us to identify the anchor map e ρ : e E → T f M with an E -connectionon the fiber bundle p : f M → M , which, by definition as given in [11], is pre-cisely a map e ρ such that the above diagram is commutative. A map e ρ permitsto lift a vector ψ x ∈ E x at the point x ∈ M to the corresponding vector in T u f M at the point u ∈ f M with p ( u ) = x . Commutivity of the diagram meansthat this “horizontal lift” should be such that the projection down to M by13 ∗ of the lifted vector agrees with the vector ρ ( ψ x ). For E = T M , ρ = id, thestandard Lie algebroid, this reproduces the standard condition of an ordinaryconnection in p : f M → M that the composition of the projection with thelift is the identity on T x M .Since e E is a Lie algebroid, its anchor is a morphism of Lie brackets, e ρ (cid:0) [ e ψ , e ψ ] (cid:1) − (cid:2)e ρ ( e ψ ) , e ρ ( e ψ ) (cid:3) = 0 (33)which, for e ρ being viewed as an E -connection on f M , is tantamount to itsflatness. In fact, a flat E -connection e ρ : p ∗ E → T f M on f M can be seen tobe equivalent to our definition of an E -action on f M . In this formulation weeasily reproduce the results of the previous section, where the connection wasfurther restricted to respect the linear structure on the bundle f M = V .We now put this into explicit formulas, generalizing the respective onesof the previous section. In bundle coordinates ( X i , φ σ ) on f M , the anchor of e E applied to the ( f M -fiberwise constant) basis e e a := p ∗ e a induced by a localbasis of sections on E , takes the form e ρ ( e e a ) = ρ ia ( x ) ∂∂x i + e ρ aσ ( x, φ ) ∂∂φ σ , (34)where instead of e ρ aσ we could have written also Γ aσ , stressing the interpre-tation of these components as an E -connection on f M . Equation (21) for thegauge transformations now turns into δ ǫ φ σ = − ǫ a e ρ aσ ( X, φ ) , (35)while for the exterior covariant derivative of φ ∈ C ∞ (Σ , X ∗ f M ) we get Dφ σ = d φ σ + e ρ aσ ( X, φ ) A a + Γ iσ ( X, φ ) F i . (36)Here Γ iσ denote the components of an ordinary connection on p : f M → M .Requiring linearity in φ , we recover the context of the previous section in allthese formulas.Now we are in the position of considering the coupling of a kinetic sigmamodel term to the pure gauge part of the action. This gives S LAY M + matter = S LAY M − Z Σ 12 ( Dφ ) σ ∧ ⋆ ( Dφ ) τ g στ ( X, φ ) . (37)14he allegedly small change of permitting g στ to depend also on φ in compari-son to (28) implies some conceptual complications: Before, g στ correspondedto a fiber metric on V , which we could also view as a quadratic function on V = f M . Now, V g is a fiber metric on V f M ⊂ T f M , the subbundle over f M con-sisting of vertical tangent vectors. A condition of the type (29) does not yetmake any sense thus, we first need a Lie algebroid-connection on V f M , whichcan be viewed also as the foliation Lie algebroid T F of the foliation/fibrationof f M by its fibers.However, in fact there is a canonical lift of the flat E -connection e ρ : e E → T f M to a flat e E -connection ee ρ : ee E → T ( V f M ) with ee E = e p ∗ e E and e p : V f M → f M : ee E ≡ e p ∗ e E ee ρ −−−→ T ( V f M ) e π y y e p ∗ e E ≡ p ∗ E e ρ −−−→ T f M π y y p ∗ E −−−→ ρ T M . (38)In other words, there exists a Lie algebroid structure on ee E = e p ∗ e E such that e π : ee E → e E is a Lie algebroid morphism: e p ∗ e E −−−→ V f M e π y y e p e E −−−→ f M . (39)In order to show that the Lie algebroid structure on e E induces a Lie algebroidstructure on ee E , it suffices to specify the anchor ee ρ of the latter, since thebracket on ee E is fixed already uniquely by means of the bracket on e E or E when applied to sections coming from e E and E , respectively. Denoting by ϕ σ = d φ σ fiber linear coordinates on V f M , the anchor map ee ρ of ee E applied to ee e a := e p ∗ e e a ≡ e p ∗ p ∗ e a reads as ee ρ a = ρ ia ( x ) ∂∂x i + e ρ aσ ( x, φ ) ∂∂φ σ + ∂ e ρ σa ( x, φ ) ∂φ τ ϕ τ ∂∂ϕ σ . (40)15his also corresponds to a flat e E -connection on V f M with components e Γ aστ ( x, φ )= ∂∂φ τ e ρ σa ( x, φ ).Let us now provide a coordinate independent construction of this canoni-cal lift, which also shows that its definition is independent of the chosen basisin E , and that the construction depends crucially on restriction to verticalvector fields on f M (equipped itself with a flat E -connection). We want todefine a bundle map ee ρ : ee E → T ( V f M ). Extend a point ee ψ ∈ ee E to some fiber-wisely constant section ee ψ ∈ Γ( ee E ) coming from a section ψ ∈ Γ( E ); so, in thepreviously introduced local basis of sections in ee E , ee ψ = ψ a ee e a with ψ a depend-ing on coordinates x i of M only and with ee ψ evaluated at the projection of ee ψ to M agreeing with ee ψ . This induces also a section e ψ = ψ a e e a in e E , whoseimage with respect to e ρ gives a vector field on f M . Consider the (local) flowΦ tψ of this vector field and lift it to T f M by means of the pushforward map(Φ tψ ) ∗ : T f M → T f M , a vector bundle morphism covering the flow Φ tψ on f M .This lift is thus generated by a vector field on T f M covering the vector field e ρ ( e ψ ). We can restrict the vector field viewed as a section in T ( T f M ) to thesubmanifold V f M of T f M . Two things happen in this context: Firstly, whilethe vector field on T f M is not C ∞ ( M ) linear in ψ ∈ Γ( E ) in general, therestriction has this property (which is essential for having the result beingindependent on the extension of ee ψ to an at least locally defined section ee ψ or ψ ). Secondly, the restriction is tangent to V f M ⊂ T f M (here the fact that e ρ ( e ψ ) is projectable to M , covering ρ ( ψ ), cf. diagram (32), enters crucially)and can thus be viewed as a vector field on V f M . Evaluate this vector fieldat the point in V f M living under ee ψ ∈ ee E and call this ee ρ ( ee ψ ). By a straight-forward calculation one may check that this geometric construction indeedyields (40).With these ingredients at hand, we are now in the position to formulate acondition on the fiber metric V g on V f M as entering the functional (37). Thefunctional becomes invariant w.r.t. gauge transformations if the followingcondition on g is satisfied (in addition to the conditions to be placed on E g ): e E ∇ ( g ) = 0 , (41)where e E ∇ is the flat e E -connection corresponding to (40) and described inthe sentence after that formula. 16n the present more general framework than in the previous section, for-mulating the conditions on some selfinteraction for the scalar fields, cf. Eq. (30)and the corresponding discussion, becomes simpler: We can add to (37) anyterm of the form Z Σ W ( X, φ )vol Σ , (42)provided only that W is a function on f M invariant along the orbits generatedby e ρ , i.e. if ( ρ ia ∂ i + e ρ σa ∂ σ ) W = 0.If the Lie algebroid E permits an integration to an source-simply con-nected Lie groupoid G ⇒ M ( cf. [12, 9] for the necessary and sufficientconditions), the above considerations have the following global reinterpreta-tion: First, given G we can consider its action on p : f M → M , where p isusually called the moment map in this context. An action is then given bya map ϕ : G × M f M → f M which is compatible with the structural maps on G . In particular this means that any g ∈ G with source x and target y islifted to an isomorphism of fibers, ϕ g : p − ( x ) → p − ( y ). This can again bemade into a new groupoid e G ⇒ f M whose elements e g consist of the maps ϕ g mapping one point in f M (the source of e g ) to another one (the targetof e g ). Finally, the diffeomorhpisms of f M -fibers ϕ g can be lifted to isomor-phisms of their tangent bundles. This induces canonically a Lie groupoid ee G ⇒ T F . As already anticipated by the notations, these two groupoidsare the integrations of e E and ee E , respectively, as we recommend the readerto check as an exercise. The condition (41) now just states that the maps( ϕ g ) ∗ : T ( p − ( x )) → T ( p − ( y )), corresponding to a collection of elements in ee G , are also isomorphisms (isometries) of T F equipped with the fiber metric V g . In this concluding section we want to discuss two more aspects of the topicspresented in this article up to here. First of all this concerns the pure gaugefield system (16), relaxing the conditions on the tensor E g needed for squaringthe 2-form field strength. Afterwards we come back to the coupled mattergauge field system, discussing it from a slightly more unified perspective. Wenow turn to the first issue. 17he field strengths F i , F a satsify some generalized version of Bianchiidentities [1]. In what follows in particular the first one of those will playan important role, for which reason we display it explicitely:d F i − ρ ia , j A a ∧ F j + ρ ia F a = 0 . (43)Primarily, this leads to a second independent gauge symmetry [1]. Sup-pose that we transform B i according to δ λ B i := d λ i + ρ ja , i A a ∧ λ j . (44)Then it is easy to see that S LAY M is invariant w.r.t. such transformations upto boundary contributions (resulting from a partial integration), if λ i ρ ia = 0—implying, more geometrically, that λ , instead of taking values arbitrarily in T ∗ M , is restricted to the conormal bundle of the tangent distribution tothe orbits generated by ρ . In fact, here, and also in what is to follow, wewill consider only regions of M where the rank of ρ is constant. Furtherinvestigations of what happens more precisely at regions where the rank of ρ jumps would be interesting though.One may employ Eq. (43) in another direction also, however: The contrac-tion of ρ with the 2-form field strength can be expressed in terms proportionalto the 1-form field strengths (and its derivative). Since, on the other hand,any term proportional to F i in (16) can be dropped by an appropriate redef-inition of the field B i , one finds that there is an equivalence relation betweenfiber metrics on E yielding physically equivalent gauge theories—in fact, thetensors E g can even become partially degenerate by such redefintions. Let e a denote a local frame in E ∗ , then E g = g ab e a e b . Consider replacing g ab by¯ g ab = g ab + ρ ia β ib + ρ ib β ia for some collection β a of 1-forms on M . Since inthe action functional E g is contracted with F a s, the terms proportional to β ai can be absorbed completely: we replace ρ ( F (2) ) by the corresponding twoterms according to (43), perform a partial integration in the term with d F i ,and then absorb all prefactors of the newly introduced terms proportional to F i by redefining B i appropriately. This means that a redefinition g ab ¯ g ab can be compensated by a local diffeomorphism on the field space of the func-tional (16). In other words, on the physical level, there is an equivalence In fact, such an observation may be even used as a starting point for constructingalgebroid type gauge theories, containing also nonabelian gerbes, as demonstrated in detailin [7]. E -2-tensors E g g ab ∼ g ab + ρ ia β ib + ρ ib β ia (45)for arbitrary choices of β ∈ Ω ( M, E ∗ ). The quotient of Γ( M, S E ∗ ) ∋ E g bythese orbits is in one-to-one correspondence to fiber metrics on the subbundleker ρ ⊂ E . Denote the restriction of E g to ker ρ by ρ g ; it is one-one to someequivalence class [ E g ] of a fiber metric on E .The bundle ker ρ → M carries a canonical E -connection. Let ψ ∈ Γ( E )and ˜ ψ ∈ Γ(ker ρ ) and define ρ ∇ ψ ˜ ψ := [ ψ, ˜ ψ ] . (46)Since ˜ ψ is in the kernel of ρ , this is indeed C ∞ ( M )-linear in ψ . This connec-tion is sometimes called the E -Bott connection. Comparing with equations(17) and (18), it is now obvious that ρ ∇ ρ g = 0 (47)is sufficient for gauge invariance of (16). In contrast to (17), this condition isnot only independent of any auxiliary connection ∇ on E , it is certainly alsoa weaker condition on E g , needing E g only to be in some orbit characterizedby its restriction ρ g to ker ρ such that (47) holds true.We now turn to the second issue, the coupled matter gauge field system.The emphasis on a new Lie algebroid e E → f M governing linear or nonlinearactions of Lie algebroids E → M on bundles f M → M corresponding tomatter field target spaces also resides in a possible reinterpretation of thegauge invariant coupled matter-LAYM functional (37). Who forbids oneto consider all the coordinates on f M on the same footing to start with.We had the kind of no-go theorem around (15), where a squaring of all 1-form and 2-form field strengths was taken. Eq. (37) from this perspectiveshows that squaring some of the 1-form field strengths, keeping the othersincluded via Lagrange multipliers, does not necessarily lead to likewiselystrong restrictions on admissible Lie algebroids. At the same time, some ofthe coordinates of the target Lie algebroid are promoted into propagatingdegrees of freedom typical for scalar fields from the physical point of view. This is true, when restricting to orbits that have at least one non-degenerate repre-sentative E g . Recall also that E was assumed to be regular for the moment so that underthis assumption its kernel really defines a subbundle of E .
19e make this point more explicit by rewriting (37) in this spirit. First,we denote by x I = ( x i , φ σ ) collectively all coordinates on f M and, correspond-ingly, by e X the map from Σ to all of f M . Then, the corresponding 1-form fieldstrengths F I (1) split into F i (1) , agreeing with the respective previous expression(2) except that, for clarity, we added an index in brackets to emphasize toform character of the field strength, and, by the same formula, one now has F σ (1) = d φ σ − e ρ σa A a . Note that geometrically F i (1) corresponds to elementstangent to M and F σ (1) tangent to fibers of f M → M . F (1) should be anelement of Ω (Σ , e X ∗ T f M ) on the other hand, i.e. a vector on f M . The twocomponents cannot be combined intrinsically or coordinate independentlyinto a meaningful vector on f M without a connection on that bundle. Let(d φ σ +Γ σi d x i ) ∂∂φ σ ∈ Ω ( f M , V f M ) be such a connection 1-form on f M , its kerneldetermining what is horizontal in T f M , V f M ⊕ H f M = T f M ∋ v = v ver + v hor . (48)We now see that the vertical part of F (1) , F ver (1) = F σ (1) + Γ σi F i (1) , reproducesprecisely eq. (36). On the other hand, the horizontal part is always propor-tional to F i (1) (for any choice of Γ σi ), thus the first term in (16) constrains F hor (1) to vanish. Likewisely, we could map F (1) ∈ Ω (Σ , e X ∗ T f M ) by p ∗ ◦ e X to a tangent component on the base M of f M and interpret the first LAYM-term in this way within the present setting, the Lagrange multiplier living in T ∗ M then as before. Preferring the first option, a coordinate independent,geometrical form of the total action, using e E → f M as starting Lie algebroidand a split of T f M into the two subbundles as above in (48), one finds for thecombined matter-gauge field action (37) the following form: Z Σ h B ∧ , F hor (1) i − (cid:16) V f M g ◦ e X (cid:17) (cid:0) F ver (1) ∧ , ⋆F ver (1) (cid:1) − (cid:16) e E g ◦ e X (cid:17) (cid:0) F (2) ∧ , ⋆F (2) (cid:1) , (49)where B is a ( d − e X of) H ∗ f M .This shows that partially squaring some of the 1-form field strengths iscompatible with Lie algebroids different from mere Lie algebras. Gauge in-variance of such a functional will certainly also heavily restrain the startingLie algebroid e E → f M . What we showed constructively is that such a func-tional is compatible with a Lie algebroid structure on e E coming from a Liealgebroid ( E → M )-action on p : f M → M for some E and M such that one20as the diagram (31) with π being a Lie algebroid morphism. In the languageof [8] this corresponds to a Q-bundle π : e E [1] → E [1] (which is locally trivialonly in the sense of graded but not in the category of Q-manifolds), wherein any local chart on the total space there exists a canonical isomorphism ofits degree one veriables with the degree one variables on the base.In the extreme context of squaring all necessarily lead to the Lie algebroid of a Lie algebra action on itsbase. This corresponds to the situation of M being a point in the discussionabove. It may be interesting to see if in a generalization of this observation afunctional of the form (49) with Lie algebroid e E always leads to the scenarioas in (31) above. A Some formulas on Lie algebroids
A Lie Algebroid consists of a vector bundle E → M over a manifold M ,a Lie algebra bracket, [ · , · ], between sections ψ of E , and of a bundle map ρ : E → T M , called the anchor map. The bracket satisies a Leibnitz rule,[ ψ , f ψ ] = f [ ψ , ψ ] + ρ ψ ( f ) ψ , f ∈ C ∞ , ψ , ψ ∈ Γ( E ) . (50)In local coordinates X i on M and a local basis e a of Γ( E ), this data isencoded in structural functions C cab , ρ ia ∈ C ∞ ( M ), such that the bracket andthe anchor map take the form [ e a , e b ] = C cab e c , and ρ ( e a ) = ρ ia ∂ i . As aconsequence of the definitions above, the anchor map is a morphism wrt. thebracket, i.e. [ ρ ( e a ) , ρ ( e b )] = ρ ([ e a , e b ]) . (51)Examples of Lie Algebroids include a bundle Lie Algebras ( ρ = 0), T M ( ρ = id ), and Poisson manifolds.In order to talk about E -connections E ∇ , we need to specify the Leibnitzrule: E ∇ ψ ( f ψ ) = f E ∇ ψ ψ + ρ ψ ( f ) ψ , (52)Any connection ∇ on the vector bundle E can be lifted to an E -connectionusing the anchor map: ∇ ρ ( · ) .Now that we have the concept of an E -connection on a Lie Algebroid, wecan translate concepts involving connections on vector bundles to the realmof Lie Algebroids. For a connection ∇ on a vector bundle, the curvature is21efined as R ( ∂ i , ∂ j ) = ∇ i ∇ j − ∇ j ∇ i − ∇ [ ∂ i ,∂ j ] . (53)Analogously, we define the corresponding E -curvature as E R ( ψ , ψ ) = E ∇ ψ E ∇ ψ − E ∇ ψ E ∇ ψ − E ∇ [ ψ ,ψ ] . (54)By the morphism property of the anchor map the E -curvature of an induced E -connection satisfies E R ( ψ , ψ ) = R (cid:0) ρ ( ψ ) , ρ ( ψ ) (cid:1) . (55)Given any E -connection E ∇ , we can form a tensor involving the structurefunctions C cab . This tensor T ∈ Ω ( E ) ⊗ Γ( End ( E )) is called the the E -torsiontensor corresponding to the E − connection E ∇ and is defined as T ( ψ ) ψ = [ ψ , ψ ] + E ∇ ψ ψ − E ∇ ψ ψ , (56)which in components takes the form T ( e a ) e c = T cab e b , T cab = − C cab + Γ acb − Γ bca . (57)Finally, we derive an identity which involving the E -torsion of an in-duced E -connection. In components, the induced connection is given byΓ acb = ρ ia Γ icb . As a consequence of the Jacobi identity, the E -torsion corre-sponding to this induced connection satisfies the identity: T dab T ecd + cycl ( abc ) = ρ ic ∇ i T eab + ρ ic ρ jb R ij ea + cycl ( abc ) , (58)which can be used to show that E -curvature of the “adjoint connection” E ˜ ∇ a ( cf. Eq. (13)) reduces to E ˜ R abdc = ρ ib S idab , S idab = ∇ i T dab + ρ jb R ij da − ρ ja R ij db . (59) B Dimensional Reduction of LAYM
In ordinary YM theory scalar fields in the adjoint representation can be ob-tained my performing a Kaluza-Klein dimensional reduction along a circle S . Here, we perform this dimensional reduction for the LAYM theory (6),(7), (16). Starting with a d -dimensional world sheet Σ d we perform a di-mensional reduction to a ( d − d − by splitting22 d = Σ d − × S and shrinking the radius of the circle S to zero. Then thecomponents of A along the S -direction become scalar fields in the lower-dimensional theory.On Σ d we decompose the world-sheet indices µ = 0 ... ( d −
1) into ( µ ) =(0 , m ) where µ = 0 denotes the direction along the S and m = 1 ... ( d − A a split into( A aµ ) =( A a , A am ). After the dimensional reduction, the zero components of A a be-come scalar fields φ a on Σ d − . The zero-component of the gauge transforma-tion of A I reduces to the gauge variations of φ a : δA a → δφ a = c abc φ b ǫ c − Γ iab ǫ b ρ ic φ c = − ǫ b ˜Γ abc φ c , (60)the reduction of the zero component of the F i = 0 field equations, F i →− ρ ia φ a , constrains φ a to be in ker ρ , and the reduction of the (0 , m ) componentof the field strength F a becomes the covariant derivative for φ a , F a ,m d u m → ( Dφ ) a . (61)Hence, the gauge transformations and the covariant derivative of φ a obtainedby dimensional reduction coincide with the gauge transformations and co-variant derivative of a scalar field which takes values in E and transformsaccording to the adjoint connection E ˜ ∇ . The difference between the two con-structions is that the fields generated by dimensional reduction are always inthe adjoint representation, and that they are constrained to taking values inker ρ . References [1] T. Strobl, “Algebroid Yang-Mills theories,” Phys. Rev. Lett. (2004)211601 [arXiv:hep-th/0406215].[2] M. Bojowald, A. Kotov and T. Strobl, “Lie algebroid morphisms, Pois-son Sigma Models, and off-shell closed gauge symmetries,” J. Geom.Phys. (2005) 400 [arXiv:math/0406445].[3] T. Strobl, “Gravity from Lie algebroid morphisms,” Commun. Math.Phys. (2004) 475 [arXiv:hep-th/0310168].[4] F. Bonechi and M. Zabzine, “Lie algebroids, Lie groupoids and TFT,”J. Geom. Phys. (2007) 731 [arXiv:math/0512245].235] T. Strobl, “From Poisson sigma models to general Yang-Mills type gaugetheories”, Lectures delivered at Perugia, July 2005.[6] R. L. Fernandes, “Singular reduction and integrability”, Lectures deliv-ered at ESI, August 2007.[7] M. Gruetzmann and T. Strobl, in preparation.[8] M. Bojowald, A. Kotov and T. Strobl, “Lie algebroid morphisms, Pois-son Sigma Models, and off-shell closed gauge symmetries,” J. Geom.Phys. (2005) 400 [arXiv:math/0406445].[9] M. Crainic and R. L. Fernandes, “Lectures on Integrability of Lie Brack-ets” [arXiv:math/0611259v1].[10] T. Strobl, “Algebroids and Sigma models”, Lectures delivered at Srni,Jan. 2007.[11] R. L. Fernandes, “Lie Algebroids, Holonomy and CharacteristicClasses”, Advances in Mathematics 170, (2002) 119-179.[12] M. Crainic and R. L. Fernandes, “Integrability of Lie Brackets”, Ann.of Math. (2) (2003), 575-620.[13] P. Schaller and T. Strobl, “Poisson structure induced (topological) fieldtheories,” Mod. Phys. Lett. A (1994) 3129 [arXiv:hep-th/9405110].[14] N. Ikeda, “Two-dimensional gravity and nonlinear gauge theory,” An-nals Phys.235