Gauging without Initial Symmetry
The gauge principle is at the heart of a good part of fundamental physics: Starting with a group of so-called rigid symmetries of a functional defined over space-time , the original functional is extended appropriately by additional -valued 1-form gauge fields so as to lift the symmetry to . Physically relevant quantities are then to be obtained as the quotient of the solutions to the Euler-Lagrange equations by these gauge symmetries.
In this article we show that one can construct a gauge theory for a standard sigma model in arbitrary space-time dimensions where the target metric is not invariant with respect to any rigid symmetry group, but satisfies a much weaker condition: It is sufficient to find a collection of vector fields on the target satisfying the extended Killing equation for some connection acting on the index . For regular foliations this is equivalent to requiring the conormal bundle to the leaves with its induced metric to be invariant under leaf-preserving diffeomorphisms of , which in turn generalizes Riemannian submersions to which the notion reduces for smooth leaf spaces .
The resulting gauge theory has the usual quotient effect with respect to the original ungauged theory: in this way, much more general orbits can be factored out than usually considered. In some cases these are orbits that do not correspond to an initial symmetry, but still can be generated by a finite-dimensional Lie group . Then the presented gauging procedure leads to an ordinary gauge theory with Lie algebra valued 1-form gauge fields, but showing an unconventional transformation law. In general, however, one finds that the notion of an ordinary structural Lie group is too restrictive and should be replaced by the much more general notion of a structural Lie groupoid.
The Standard Model of elementary particle physics, but also General Relativity and String Theory, are gauge theories. In the former case, for example, gauging of an SU(3) rigid symmetry rotation between the three quarks leads to the introduction of the eight gluons that mediate the interaction between those elementary particles. Mathematically the resulting theory is described by connections in a principle bundle (in the above example with the structure group SU(3), the connection 1-forms representing the SU(3) gluons) with the matter fields being sections in appropriate associated vector bundles.
The procedure can be generalized to matter fields being sections in arbitrary fiber bundles, the fibers being equipped with geometric structures invariant w.r.t. some group G, the structural or “rigid” symmetry group. In the case of a trivial bundle, sections correspond to maps from the base manifold to the fiber and one obtains a sigma model, such as e.g. the “standard” one:
This is a functional on smooth maps . The -dimensional spacetime and the -dimensional target manifold carry a (possibly Lorentzian signature) metric and , respectively, entering by means of .
Symmetries of the geometrical data on the source manifold or on the target manifold lift to symmetries of functionals using only such data. So, in the case of (1) an invariance of and leads to an invariance of . By the Noether procedure this gives rise to conserved quantities. For example, if is a flat, these conserved quantities yield the energy momentum tensor .
Let us now suppose that the metric has a nontrivial isometry group , which infinitesimally implies , valid for the vector fields on corresponding to arbitrary elements . In this case, there is a canonical procedure to lift the induced -symmetry of to a gauge symmetry on an extended functional called minimal coupling: After introducing -valued 1-forms , denoting any basis of , ordinary derivatives on the scalar fields are replaced by covariant ones,
where . The new functional
is now invariant with respect to the combined infinitesimal gauge symmetries generated by
for arbitrary . Here are the structure constants of the Lie algebra in the chosen basis.
In the space of (pseudo) Riemannian metrics, those permitting a non-trivial invariance or isometry group are the big exception. A generic metric does not permit any non-vanishing vector field satisfying .
It is conventional belief that an isometry is necessary to gauge the functional (1). It is our intention to show that this is far from true: First, there may be group actions on the target that are not isometries but still can be gauged using Lie algebra valued 1-forms. Second, and maybe more important, one does not need to restrict to the action of finite-dimensional Lie groups. It is sufficient to have a Lie groupoid over . In fact, the use of Lie groupoids (and their associated Lie algebroids) in the context of gauge theories is even suggested by the present analysis as the much more generic one.
Ii The case of 1-dimensional leaves
For a conceptual orientation, we first consider the highly simplified situation of a (regular) foliation of into one-dimensional, hyper-surface-orthogonal leaves for a positive-definite metric . In this case we can choose an adapted local coordinate system such that generates these leaves and yields orthogonal hyper-planes. This implies that for all or, if we denote those indices by Greek letters from the beginning of the alphabet, that (while certainly ). not generating an isometry is tantamount to , for at least some components.
According to standard folklore, it should not be possible to extend by gauge fields such that becomes a direction of gauge symmetries, i.e. such that will leave the extended action invariant for an arbitrary choice of the parameter function (together with an appropriate transformation of the gauge field).
Since the leaves are 1-dimensional, we will introduce also only one gauge field and consider the action functional
If we postulate the conventional , we achieve that is strictly gauge invariant. It is then easy to see that with this transformation of the gauge field, we necessarily need , which would imply that generates isometries of . However, one notices that changes of under the flow of can be compensated by means of a modified transformation of the gauge field . It is sufficient to require merely for gauge invariance of (6) if is amended by
This makes sense also geometrically: To factor out the one-dimensional leaves equipped with the metric , it is not necessary that this metric, disappearing in the quotient, is invariant along the leaves, but only that the transversal metric is: .
A conceptual understanding and straightforward generalization of this idea is obvious: Consider the Cartesian product of two Riemannian manifolds and equipped with , the sum of the pullback of the 2-tensors on each of the two factors by the respective projection map. To construct a sigma model with target in terms of a “quotient construction” for a gauge theory with target , it should not play any role that the total metric is not invariant along the leaves . Decisive is that has this “transversal-to- part” (the pullback of ) invariant under diffeomorphisms along the leaves (the invariance following precisely from the fact that it is a pullback). In fact, in (6) is not a function of only , but it can depend on all the coordinates of : Correspondingly, the fiber-metric on can variy along the fibers. In general we will not have to even require a fibration, but permit singular foliations.
Let us confirm our expectation that in the case of equipped with the adapted coordinates such that the metric tensor on satisfies , the gauge invariant content of (6) is indeed described by a sigma model of the type (1) with target : Variation of (6) w.r.t. the gauge field leads to , i.e. to , determining completely in terms of . Since moreover is purely gauge by construction, neither nor contain any physical information. Varying the action (6) w.r.t. the remaining fields , terms from the first line evidently vanish on-shell, while the second line gives the Euler Lagrange equations of the expected “reduced functional”.
In general the physical degrees of freedom cannot be separated that easily from the unphysical ones. This is the main point of the use of gauge theories. In some sense they provide a smooth definition of an otherwise very singular quotient space. So while the theory has to be constructed so as to give the expected results in the simplest situations of clearly separable physical and gauge degrees of freedom, the real interest lies in those situations where this separation is either hidden or not even possible (on a global level and in a smooth manner).
Iii Arbitrary foliations
Consider the neighborhood of a point in in which the leaves of the foliation are generated (over smooth functions) by a set of vector fields , . Clearly they must be involutive, i.e. there will exist functions such that
We want to promote arbitrary deformations along the leaves to a gauge symmetry, Equation (4) with arbitrary functions on , at least locally. The functional (6) corresponds to a special case of (3), only (7) deviates from the conventional transformation behavior. It is easy to verify that, as a consequence of involutivity, also in the more general situation where are structure functions over , the expressions (2) transform covariantly, , if (5) holds true.111We remark that is not yet uniquely defined by means of (8) since the set of s does not form a basis in general. At the moment we only assume a smooth, consistent choice being made. Thus, in generalization of (7), we make the ansatz
This vanishes for all if and only if there exist some coefficients , corresponding to an -matrix the coefficients of which are locally 1-forms on , such that the following holds true:
where denotes the symmetric tensor product (for 1-forms . It is comforting to verify that this condition is independent of the chosen generators along the leaves. For example, using a change of generating vector fields,
where are the components of an matrix that are locally functions on , this only changes the matrix : Indeed, (12) yields
In fact, in the case of a regular foliation on a Riemannian manifold there is a more geometrical formulation of the condition (12), which is presented in the first part of the following Theorem, the second part summarizing the main findings of this letter up to this point:
Given a (regular) foliation of a Riemannian manifold , then its conormal bundle with its canonical metric is invariant with respect to leaf-preserving diffeomorphisms on , iff for any locally defined generating set of vector fields of there exists a local matrix such that (12) holds true or, equivalently, iff
Here is the -orthogonal complement to the tangent distribution , , and the annihilator subbundle of inside .
Let be a pseudo-Riemannian manifold and a possibly singular foliation of . Suppose that for any local choice of and satisfying Eq. (8) there exists an such that (12) holds true. Then the action functional (3) is gauge invariant with respect to the infinitesimal gauge transformations generated by and .
The conormal bundle is the annihilator-subbundle inheriting a fiber-metric as a subspace of . We did not restrict to merely foliation-preserving diffeomorphisms, but those which preserve each leaf of the foliation separately.
Proof: Since , any symmetric tensor field can be uniquely decomposed into three parts, belonging to the space of sections of , , and . Let and be the first and the second components of in this decomposition, respectively; thus . In general, has all three components. (15) expresses that its second component vanishes, , which clearly follows from (12); but also vice versa: being regular, one can always choose a smooth collection of such -forms under the condition (15).
On the other hand, equips with a metric, as well as its dual . Thus, what remains to be shown is that for any (local) -tangent vector field , iff
But since for any and
. Using involutivity of , one shows similarly .
Iv Lie groupoids versus Lie groups
Most of the investigations of Sophus Lie were concerned with transformations satisfying infinitesimally conditions of the form (8). It was a highly non-trivial step to arrive from this to the abstract notion of a group and a Lie algebra. In the latter case it amounts to first restricting to cases where the structure functions in (8) can be chosen as constant, then postulating elements having their proper own life: they are to generate an algebra structure on the vector space spanned by the s via the product relation .
Let us go back to the original setting of a (possibly singular) foliation with a local description such as in (8) and let us drop the somewhat unnatural condition, which underlie implicitly the introduction of Lie groups and algebras, that there should be a global choice of s and such that the latter functions are constants over all of . First, we do not require that the vector fields need to be defined everywhere; we content ourselves with the fact that any neighborhood of a point permits a set of such vector fields generating the given (singular) foliation, keeping, however, the number fixed. On overlaps of local charts then certainly there will be matrices such that an equation of the type (13) holds true. If the s form a basis at each point of a given neighborhood, then the matrices are even unique (and satisfy automatically a cocycle condition). In general, however, the generating vector fields may be linearly dependent, and thus the transition matrices not uniquely determined. Let us assume that also in this case there is a consistent choice of these matrices such that they are invertible on each overlap and that on triple overlaps they fit together consistently.
Such a consistent choice of matrices over an atlas of is equivalent to the construction of a rank vector bundle . In the very special case where the structure functions are also constant, one has and one can separate the manifold from the abstract vector space . This is a very special situation: even as a vector bundle, may be far from trivializable.
How to obtain the algebraic structure on that, in the above particular case, would reduce to the Lie algebra structure on ? Since in general we cannot separate from inside , let us thus look at also in this case, where in principle we know the action of the Lie algebra on , , . In physics we are used to associate a nilpotent odd BRST-transformation to this:
Here the variables are considered to be odd and, in the BRST- or BV-language are called “ghosts”. then follows from the above data, but also permits reciprocally to encode the algebraic structure on as well as its action on . It is near at hand to consider the same odd vector field (17) also in the more generic case where the s are not constant. We now require that again squares to zero. (Let us remark in parenthesis that as well as the contraction of the remaining condition with follow already from (8). The additional input at this stage is mild therefore.) It is a mathematical fact Vaintrob () (cf, e.g., Gruetzmann-Strobl () for more details) that a nilpotent vector field (17) equips the above vector bundle with what is called a Lie algebroid structure!
V Extended Killing Equation
Returning to Equation (14), we now recognize that the geometrical significance of is the one of a connection on the bundle . This permits an alternative interpretation of the main equation (12) underlying the gauging. It is well-known that Killings equation can be rewritten in the form
where the indices of the vector field are lowered by means of the metric and the semicolon indicates a covariant derivative with respect to the Levi-Civita connection of . Any generating an isometry has to satisfy this equation.
We may reformulate the condition we found from gauging in a similar way: It is the condition on a collection of vector fields to satisfy the equations (8) and (12). Locally we can always consider this set of vector fields on as a single section of , where is a trivial rank bundle. In fact, in the previous section we assumed that there exists a consistent gluing over local charts so as to define a not necessarily trivial rank bundle . The collection of vector fields now corresponds to a single (at least locally defined) section where with being a local basis in . permits to be identified with a section of . Using as a connection in and the Levi-Civita connection on , we arrive at the following compact generalization of (18):
Certainly, an ordinary Lie algebra action of symmetries is a very particular case inside this: Then is the globally flat bundle , we can choose , and the above extended Killing equation (19) reduces to the standard Killing equation (18) to be satisfied for each of the individual vector fields (or 1-forms) separately.
Vi Generalized gauge fields
Since in general we cannot separate the algebraic Lie structure from the manifold , one also needs to regard the scalar and gauge fields in a more unified manner. How to do this was proposed already in Bojowald-Kotov-Strobl (); LAYM (): For a general Lie algebroid , the fields and are in one-to-one correspondence with vector bundle morphisms and the functional (3) gauging the -orbits on is a functional of such maps . Moreover, the gauge transformations found in the second part of our Theorem are precisely one of the two options of gauge symmetries developed by independent, more mathematical considerations in Bojowald-Kotov-Strobl (); LAYM (); Mayer-Strobl (). In LAYM () possible purely kinetic terms for Lie algebroid Yang-Mills theories were proposed and in Mayer-Strobl () their subsequent coupling to matter fields. The present analysis provides a complementary perspective, having started from an ordinary and standard sigma model.
We have seen that by an unbiased attempt to gauge a sigma model, not following blindly the established path of Lie groups acting as isometries on the target , we are almost automatically led to the notion of Lie algebroids (or their integtating Lie groupoids). And even in the case , we do not need the Lie algebra to be an isometry for gauging: It is sufficient that the metric satisfies (15) or, if is singular, the more general condition (19). It will be interesting to investigate this further from both sides, mathematics and physics, many new routes and problems offering themselves at this point.
Lie algebras and groups are a highly developed and important subject of mathematics and its use within physics undoubtedly indispensable. We find that in the context of gauge theories we should consider—at least also—Lie algebroids and groupoids, all the more so due to the considerable recent mathematical progress in this domain. Nature is known to have made ample use of Lie groups. It seems unlikely that Nature has restricted itself to this relatively rigid notion, not also making use of the much more flexible Lie groupoids—and this also at the level of fundamental physics. To be unravelled.
Acknowledgements.We are grateful to A. Weinstein for valuable remarks.
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