# Gauge theory deformations and novel Yang-Mills Chern-Simons field theories with torsion

###### Abstract

A basic problem of classical field theory, which has attracted growing attention over the past decade, is to find and classify all nonlinear deformations of linear abelian gauge theories. The physical interest in studying deformations is to address uniqueness of known nonlinear interactions of gauge fields and to look systematically for theoretical possibilities for new interactions. Mathematically, the study of deformations aims to understand the rigidity of the nonlinear structure of gauge field theories and to uncover new types of nonlinear geometrical structures.

The first part of this paper summarizes and significantly elaborates a field-theoretic deformation method developed in earlier work. Some key contributions presented here are, firstly, that the determining equations for deformation terms are shown to have an elegant formulation using Lie derivatives in the jet space associated with the gauge field variables. Secondly, the obstructions (integrability conditions) that must be satisfied by lowest-order deformations terms for existence of a deformation to higher orders are explicitly identified. Most importantly, a universal geometrical structure common to a large class of nonlinear gauge theory examples is uncovered. This structure is derived geometrically from the deformed gauge symmetry and is characterized by a covariant derivative operator plus a nonlinear field strength, related through the curvature of the covariant derivative. The scope of these results encompasses Yang-Mills theory, Freedman-Townsend theory, Einstein gravity theory, in addition to their many interesting types of novel generalizations that have been found in the past several years.

The second part of the paper presents a new geometrical type of Yang-Mills generalization in three dimensions motivated from considering torsion in the context of nonlinear sigma models with Lie group targets (chiral theories). The generalization is derived by a deformation analysis of linear abelian Yang-Mills Chern-Simons gauge theory. Torsion is introduced geometrically through a duality with chiral models obtained from the chiral field form of self-dual 2+2 dimensional Yang-Mills theory under reduction to 2+1 dimensions. Field-theoretic and geometric features of the resulting nonlinear gauge theories with torsion are discussed.

## I Introduction and summary

Classical gauge field theories are widely studied in many areas of mathematics, and their importance in physics almost needs no comment. They provide the starting point for formulations of physically important theories of quantum fields; in particular, for the case of electromagnetic and gravitational fields, they describe the dynamics of macroscopic quantum states in the classical regime where quantum effects are not significant.

Looked at from the view point of geometry, classical gauge theories have a rich nonlinear structure — field variables typically are sections of a vector bundle or group bundle over spacetime, equations of motion for the fields usually involve a connection, covariant derivative, and curvature tensor on this bundle, while gauge symmetries of the fields are described by an infinite-dimensional Lie group action or local bundle automorphisms. One way to characterize all this structure is by the idea of deformations of linear abelian gauge theories.

A prototype example is non-abelian Yang-Mills theory. The Yang-Mills field is mathematically defined as a connection 1-form in a semisimple Lie group bundle on spacetime, in which the Yang-Mills gauge symmetry is a local automorphism given by the group action of the bundle. If a linearization around a flat connection is considered, the theory reduces to a set of linear electromagnetic field equations and corresponding spin-one gauge symmetries. Non-abelian Yang-Mills theory describes a nonlinear deformation of the structure of this linear abelian (electromagnetic field) theory YMth ; Des ; Wal . Another prototypical example is Einstein gravity theory. The gravitational field is a metric tensor on spacetime to which is naturally associated a metric-connection in the tangent bundle, and the gauge symmetries arise from diffeomorphisms acting on the spacetime manifold. Linearization around a flat metric reduces the theory to the linear gravity wave equation and its accompanying abelian spin-two gauge symmetry. Einstein gravity theory represents a highly nonlinear deformation of the structure of this linear abelian (graviton field) theory OgiPol ; Des ; FanFro ; Wal . In both these examples, the gauge covariant field strength in the theory is simply the curvature defined geometrically from the field.

An interesting problem of classical field theory is to find and classify all nonlinear deformations of a given linear abelian gauge theory AMSpaper ; Hen1 ; Sta . Deformations, in general, refer here to adding linear and higher power local terms to the abelian gauge symmetry, while also adding quadratic and higher power local terms to the linear field equation, such that there exists a gauge invariant local action functional for the deformed theory. Such a deformation is essentially nonlinear if the deformed Lagrangian is not equivalent, to within a local divergence, to the Lagrangian of the linear theory under any invertible local nonlinear field redefinitions. Deformations in which the abelian gauge symmetry remains undeformed (for example, by adding gauge invariant Pauli-type interaction terms to the Lagrangian) are regarded as trivial since the abelian gauge invariance (and, hence, geometrical structure) is unchanged. Importantly, the condition of gauge invariance can be used to formulate determining equations to solve for all allowed deformation terms order-by-order in powers of the fields, without any need for special assumptions or ansatzes on possible forms of the deformed field equations, gauge symmetries, and gauge group structure. There are two main formulations of the deformation determining equations — a direct, field-theoretic approach AMSpaper where gauge invariance is expressed elegantly by Lie derivative equations for gauge symmetries, using a jet space formalism in terms of the gauge field variables; and a powerful BRST approach Hen1 based on the field-antifield formalism BarHen ; Sta ; StaFulLad1 in which the condition of a gauge invariant Lagrangian is encoded by the cohomology of a BRST operator.

The physical interest in studying deformations is to address uniqueness of known nonlinear interactions of gauge fields and to look systematically for theoretical possibilities for new interactions. Mathematically, the study of deformations aims to understand the rigidity of the nonlinear structure of gauge field theories and to uncover new types of nonlinear geometrical structures. Sparking this subject is the rich interplay between, on the one hand, differential geometry and deformations of graviton theory, connections on vector bundles and deformations of electromagnetic theory, and on the other hand, their obvious significance for many developments in mathematics and physics.

Indeed, the last decade has seen a large body of work on the study of nonlinear gauge theory deformations, which has yielded many interesting new kinds of generalizations of Yang-Mills theory and Einstein gravity theory both as classical theories of spin-one and spin-two fields and as geometrical theories of Lie-algebra valued 1-forms and algebra valued vielbeins or metrics. Rigidity results for these generalizations have been obtained as well HenKna ; BarBraHen ; Hen2 ; Annalspaper ; CQGpaper ; JMPpaper

The first gravity generalizations to be found were motivated by aspects of classical supergravity theory classicalSG and describe nonlinear multi-graviton theories in four dimensions involving the introduction of various algebras in which the fields take values CutWal ; Annalspaper . Supersymmetric extensions of these generalizations have also been obtained, along with more novel extensions that involve non-commuting classical graviton fields (and non-anticommuting gravitino fields) CQGpaper . Recently, an exotic type of multi-graviton theory with a parity-violating interaction that exists for commuting graviton fields only in three and five dimensions has been investigated BouGua ; PhysRevpaper . This exotic gravity generalization is most remarkable in that its gauge invariance is completely different than the familiar diffeomorphism invariance of Einstein gravity, in contrast to the previous generalizations which all feature diffeomorphism invariance extended to an algebra-valued setting.

Yang-Mills generalizations were first obtained for 1-form potentials in three dimensions abelianth ; nonabelianth and come from a Yang-Mills interaction combined with a Freedman-Townsend type interaction available for 1-forms only when the number of dimensions equals three. Recall, Freedman-Townsend theory FTth exists for -form potentials in dimensions, with the dual nonlinear field strength of the potential serving as a connection 1-form whose curvature vanishes due to the field equations; as is well known, because the connection is flat, Freedman-Townsend theory geometrically describes a dual form of principal chiral models. Extensions of this type of generalization of Yang-Mills theory to interacting 1-form and 2-form potentials in four dimensions Dra , including a Chern-Simons type mass term, were later obtained JMPpaper . (It should also be noted that gauge theories containing 2-form potentials are of active interest in supergravity contexts Bra1 .) Subsequently a different type of generalization in four dimensions was derived in recent work LMPpaper involving a Chapline-Manton type interaction between 1-form and 2-form potentials combined with a Yang-Mills interaction and an extended Freedman-Townsend interaction. The gauge invariance in all these generalizations unifies the form of the familiar Yang-Mills and Freedman-Townsend gauge symmetries, leading to a nonlinear structure that mixes geometrical features of pure Yang-Mills and pure Freedman-Townsend theories in an interesting way. In particular, the combined Yang-Mills Chapline-Manton generalization exhibits a dual formulation that describes a principal chiral field with an exotic dilaton coupling to the Yang-Mills fields through a generalized Chern-class term LMPpaper .

In another direction, a novel Yang-Mills generalization involving a gravity-like interaction of 1-form potentials has been recently derived Bra2 , in which the interaction is constructed using conserved currents associated with Killing vectors of spacetime. Its gauge invariance unites the familiar Yang-Mills type with the vielbein type found in Einstein gravity, giving a gauge theory of spacetime symmetries (Killing vectors) in a direct sense. However, the geometrical structure and gravity-like aspects of the theory remain to be explored more fully.

A main purpose of this paper will be to present a new geometrical type of Yang-Mills generalization in three dimensions motivated from considering torsion in the context of nonlinear sigma models torsion . Recall a nonlinear sigma field (also known as a wave map) is a function on spacetime taking values in a Riemannian target space e.g. an -sphere or a compact simple Lie group (note the latter defines a chiral field model). The nonlinear structure of a sigma model is given jointly in terms of the spacetime metric and the symmetric (Riemannian) metric-connection on the target. More specifically, the field equation is a semilinear geometrical wave equation (sometimes called a wave map equation) consisting of the spacetime wave operator on the sigma field plus a quadratic interaction given by a product of sigma field gradients contracted with the target metric-connection and the spacetime metric tensor. This interaction readily generalizes to include torsion in the antisymmetric part of the connection on the target, if at the same time a skew tensor is added to the spacetime metric.

Torsion first arose in this context for 1+1 dimensional sigma models, motivated by topological Wess-Zumino-Witten terms which describe the winding number of instanton solutions in three dimensional sigma models. This is analogous to the relation between three dimensional Chern-Simons terms and four dimensional instantons in self-dual Yang-Mills theory. Indeed, torsion sigma models in 2+1 dimensions with Lie group targets were later found to arise from the chiral field form of self-dual Yang-Mills theory under dimensional reductions Yan ; Pol ; War . As important motivation, such reductions yield integrable chiral field equations that possess many of the same integrability features in 2+1 dimensions as are known for chiral models in 1+1 dimensions integrable . These features geometrically stem from tuning the torsion in the connection on the Lie group target space so as to flatten its generalized Riemannian curvature torsion . Torsion more generally is of interest in the analytical study of the Cauchy problem for sigma models in 2+1 dimensions AncIse , since this is the critical dimension in which blow-up may occur for solutions with smooth initial data of large energy ShaStr .

The torsion gauge theories described in this paper will be geometrical generalizations of the dual form of general 2+1 dimensional torsion chiral models. In particular, as a main new result, torsion will be shown to be consistent with a Yang-Mills interaction of 1-form potentials in 2+1 spacetime dimensions provided a Chern-Simons mass term is included in the full theory. Because the introduction of a skew tensor needed to support the torsion interaction breaks Lorentz covariance at each point in spacetime, the field equations in these new torsion gauge theories have a semi-relativistic form resembling that of 2+1 dimensional torsion chiral models War .

To begin, a strengthened geometrical version of the field-theoretic approach to gauge theory deformations developed in the course of earlier work AMSpaper ; Annalspaper ; JMPpaper will be summarized in Sec. II. The approach given here simplifies the steps for finding deformation terms up to second order and clearly identifies the obstructions (integrability conditions) that must be satisfied on 1st order deformations for existence of a deformation to higher orders. In particular, it is shown how the Noether current and commutator associated with the 1st order deformed gauge symmetries explicitly determine the quadratic deformation terms in both the field equation and the gauge symmetry. This enables finding all obstructions, which is a key contribution. As further main results of this comprehensive approach, firstly, a simple general quasilinear form for the deformation terms is shown to be required by preserving the number of initial-data and gauge degrees of freedom order by order. Secondly, a remarkable universal geometrical structure derived in terms of a covariant derivative operator associated with the gauge symmetry, along with a nonlinear field strength connected to the curvature of this covariant derivative, is shown to be common to a large class of deformations. The scope of these results is quite general and encompasses the gauge theory examples discussed earlier.

In Sec. III, a generalization of nonabelian Yang-Mills Chern-Simons theory incorporating a Freedman-Townsend interaction without torsion is constructed by a deformation analysis of linear abelian Yang-Mills theory with a Chern-Simons term for 1-form potentials in three spacetime dimensions. Geometrical aspects of this generalization are highlighted and shown to display a duality with a principal chiral field theory that is coupled in a novel manner to nonabelian Chern-Simons theory with a Proca mass term. The torsion generalization of three-dimensional nonabelian Yang-Mills Chern-Simons gauge theory is then derived in Sec. IV. The derivation applies a deformation analysis to linear abelian Yang-Mills Chern-Simons theory with the inclusion of torsion based on the dual form of torsion chiral models for abelian Lie group targets. Field-theoretic and geometric features of the resulting nonlinear gauge theories with torsion are discussed. In Sec. V an extension of torsion to the gravity-like generalization of Yang-Mills theory is obtained by a similar derivation.

## Ii Deformation method

For the purpose of a general deformation theory, all gauge theories of primary interest in mathematics and physics can be formulated in common by gauge field variables taken to be vector-valued -form potentials in dimensions, with , for some appropriate internal vector space . Furthermore, these field variables can be chosen such that the field equations have a zero potential as a solution, and also such that the gauge symmetries when linearized around this solution have the form of a set of gauge transformations involving an arbitrary vector-valued -form as the parameter.

For example, the gauge field variable in -dimensional Yang-Mills theory is a connection 1-form represented by a potential that takes values in the Lie algebra of the Yang-Mills gauge group e.g. . As discussed in the introduction, a linearization about a flat connection yields the abelian Yang-Mills gauge symmetry, given by

For generalizations such as multi-graviton theories, the vector space is just enlarged Annalspaper ; CQGpaper by a tensor product with some kind of internal algebra ; and for the gravity-like generalization of Yang-Mills theory, is any Lie subalgebra of the isometry group admitted by the spacetime metric Bra2 . In addition, supersymmetric gauge theories such as supergravity are accommodated by enlarging as a direct sum to include an appropriate spinor space CQGpaper .

In general it will be convenient to write the gauge field variables and their derivatives in local coordinates on spacetime and assume the underlying spacetime manifold is equipped with a fixed metric and volume form for which the coordinate derivative is metric-compatible, , . As in all the preceding examples, for typical gauge theories the geometrical content is independent of this background spacetime structure.

For the sequel, a multi-index notation will be used to denote collections of spacetime indices as ; index symmetrization and antisymmetrization will be denoted by and as usual.

### ii.1 Linear abelian gauge theory

So consider a linear abelian gauge theory on a -dimensional spacetime manifold using vector-valued -form potentials as field variables whose differential gauge invariance is given by

(1) |

The theory is specified by giving a quadratic Lagrangian top-form that is locally constructed from the fields and their derivatives , along with a fixed spacetime metric and volume form on , an inner product fixed on , as well as any extra structure which may be available on or (for example, a flat vielbein). Gauge invariance is expressed by the curl condition

(2) |

for some locally constructed -form . If is of compact support on then the action functional is gauge invariant since vanishes by Stokes’ theorem. Equivalently, under the abelian gauge symmetry, the scalar Lagrangian varies into a divergence where .

The stationary points of the action functional yield the linear field equation on . Gauge invariance of the theory implies this field equation depends on only through the abelian field strength -forms

(3) |

and their derivatives.

In typical theories, the linear field equation is at most second order in derivatives and thus has the form

(4) |

where the coefficients depend only on the structure available on and . Correspondingly, the Lagrangian is at most first order in derivatives, to within a curl.

### ii.2 Preliminaries

A precise mathematical setting for writing down deformation terms and analyzing the deformation determining equations is provided by the field configuration space , defined as the set of all smooth sections of the vector bundle of -valued -forms on . Associated with is the jet space formally defined using local coordinates

(5) |

where represents a point in ; represents the value of the -form field at ;

(6) |

where for stands for . This setting makes clear what will be meant by locally constructed deformation terms, and it allows the use of tools of variational calculus Olv ; And ; AMSpaper that will be relevant for formulating gauge symmetries, field equations, and the condition of gauge invariance in a general deformation theory.

Geometrically, a field variation is a vector field on . Corresponding vector fields on the jet space that involve no motion on the spacetime coordinates and that preserve the derivative relations among the field coordinates represent field variations that are locally constructed from the fields and their derivatives as well as from any structure available on and . Given such a field variation, there is a natural Lie derivative operator defined as follows. On scalar functions on , it acts as a total variation

(7) |

For covector functions on , the Lie derivative action is given by

(8) |

This action extends via the Leibniz rule to vector functions and, more generally, any tensor functions on . The Lie derivative operator on scalar and covector functions will be central here to the formulation of deformation determining equations.

There are two useful identities that hold for the Lie derivative operator. Firstly, Lie derivatives with respect to any locally constructed field variations and satisfy the familiar commutation relation

(9) |

where defines the commutator of the field variations. Secondly, the Jacobi relation is satisfied, , since for any three locally constructed field variations,

(10) |

holds identically. These properties are direct consequences of the representation of field variations as vector fields on jet space.

Another important variational operator will be the Euler-Lagrange operator acting on scalar functions on by

(11) |

This operator takes functions into covector functions

The Euler-Lagrange and Lie derivative operators are related through integration by parts. In particular, for scalar functions on , repeated integration by parts on a Lie derivative with respect to any locally constructed vector field on yields

(12) |

where

(13) |

### ii.3 Determining equations for nonlinear deformations

A deformation of a linear abelian gauge theory for fields consists of adding linear and higher power terms to the abelian gauge symmetry (1),

(14) |

while simultaneously adding quadratic and higher power terms to the linear field equation (4),

(15) |

such that there exists a top-form Lagrangian

(16) |

required to be gauge invariant to within a curl. Here the deformation terms in the field equation and gauge symmetry are to be locally constructed from powers of the fields and their derivatives up to some finite differential order, with coefficients allowed to depend on the spacetime coordinates , the spacetime metric and volume form , the internal vector-space inner product , and any other available structure on or . As well, the deformation terms in the gauge symmetry are to be linear in the gauge parameter and its derivatives to a finite differential order. Thus, the expressions for and will be, respectively, vector and covector functions on . Natural restrictions on the order of derivatives appearing in these expressions will be addressed shortly (see proposition 2.4).

The condition of gauge invariance is stated by

(17) |

holding for some -form function on , where the Lagrangian (16) is related to the field equation (15) through the Euler-Lagrange operator,

(18) |

Necessary and sufficient conditions for to arise as an Euler-Lagrange equation are that the Frechet derivative of must be a self-adjoint operator Olv ; AncBlu . These conditions can be shown to determine

(19) |

to within a curl. In terms of the Euler-Lagrange operator, the condition of gauge invariance is expressed through the integration by parts identity (12) by the equation

(20) |

on . This constitutes the determining equation for all allowed deformations of the linear abelian gauge theory (1) and (4).

The deformation determining equation (20) can be reformulated more usefully and geometrically as Lie derivative equations, as shown by results in AMSpaper .

Theorem 2.1: Gauge invariance holds iff the Lie derivative of the field equation with respect to the gauge symmetry vanishes

(21) |

This equation asserts that the gauge symmetry for each parameter is a vector field tangent to the surface in corresponding to the field equation (and all its derivatives), . Due to invariance of the action functional, the commutators of these vector fields for all parameters have the same tangency property.

Theorem 2.2: Gauge invariance holds only if the Lie derivative of the field equation with respect to the gauge symmetry commutators vanishes

(22) |

It is important to remark that here no conditions are assumed or required on the possibilities allowed for the form of the commutators of the deformed gauge symmetries. Nevertheless, when the gauge symmetries are restricted to the solution space of the deformed field equations, closure of the commutators will be seen to arise order by order, stemming from the fact that the abelian gauge symmetries generate all of the differential gauge invariance present in the solution space of the linear field equations (i.e. modulo any auxiliary gauge freedom). Any deformation therefore will automatically determine an associated infinitesimal gauge group structure. As will be stated in precise form later (see also Ref. StaFulLad2 ), the commutators characterizing this group structure may involve local structure functions that depend on the fields (and their derivatives) and may fail to close other than when the fields satisfy .

A related remark is that gauge invariance as expressed by the Lie derivative equation (21) implies the deformed gauge symmetry formally generates an infinitesimal symmetry group of the deformed field equation, since

(23) |

Geometrically, this equation is precisely the condition for the associated vector field in to lie tangent to the solution space surface . (This general notion of symmetries defined as locally constructed tangent vector fields to comprises classical point symmetries as well as generalized or higher order symmetries of a field equation ; see Refs. Olv ; AncBlu ; book .) However, it is worthwhile to emphasize here that the symmetry determining equation (23) is strictly weaker than the Lie derivative equation (21), as not every symmetry of an Euler-Lagrange field equation will necessarily generate an invariance of its action functional; for example, symmetries that scale the coordinates of typically leave invariant the Lagrangian only in certain spacetime dimensions.

There is a related formulation of gauge invariance from this point of view, which is also useful. Introduce the gauge parameter space defined as the vector bundle of all smooth -valued -forms on . The gauge symmetry (14) then can be viewed as a linear differential operator

(24) |

from to that is locally constructed from , derivatives of , and any structure available on and . Note via integration by parts, this operator has a formal adjoint that acts as a linear differential operator from into (i.e. the dual vector bundles of and ). Now, an application of the Euler-Lagrange operator with respect to in the statement of gauge invariance (17) gives a Noether divergence identity

(25) |

Conversely, contraction of onto this identity followed by use of the Euler-Lagrange relation (18) along with repeated integration by parts then gives back equation (17).

Proposition 2.3: Gauge invariance holds iff the field equation satisfies the Noether identity (25) derived from the gauge symmetry.

Now to proceed, an expansion of the Lie derivative equations (21) and (22) in powers of the field coordinates in gives a hierarchy of determining equations whose solutions yield all allowed deformation terms for the field equation and gauge symmetry. Before looking at this hierarchy, it is important to consider the order of derivatives on and allowed in the deformation terms.

First, a natural requirement is that the number of dynamical degrees of freedom of the fields in the linear abelian theory should be preserved order by order in a nonlinear deformation. Otherwise, severe consistency conditions could arise on solutions of the field equation, and the deformed theory would not be physically or mathematically satisfactory. (These degrees of freedom can be identified as the number of initial-data functions modulo the gauge symmetry freedom, provided that the linear field equation is a well-posed system of PDEs after suitable gauge constraints have been imposed on the fields.) With this condition, for the typical situation in which the linear field equation is a system of second order PDEs, the most general possible form then allowed for deformed field equations will be a quasilinear second order PDE system, namely, highest derivatives are of second order while the coefficients of these terms depend on no higher than first order derivatives,

(26) |

Here, for later notational ease, the lower derivative terms have been written in a factored form where contains no derivatives of and contains no higher than first derivatives of . Note that the form of the linear field equation gives

(27) |

Next, as a consequence of the quasilinear second-order derivative form sought for the deformed field equations, a bound arises on the order of derivatives allowed in the deformed gauge symmetry through the Noether divergence identity (25), as follows. The deformed gauge symmetry has the form of a linear differential operator on , with coefficients of and of

(28) |

When this identity is expanded in powers of the field coordinates in , taking into account the form of the abelian gauge symmetry, it gives divergence equations of the form

(29) |

for each . In general, as increases, these equations imply an unbounded escalation in the order of derivatives on in the deformation terms compared with , unless vanishes for . Moreover, for a balance to hold in the order of derivatives appearing on both sides of the divergence equations under the condition that is quasilinear second order in derivatives of , for should contain no second or higher order derivatives of while should contain second derivatives of at most linearly.

Therefore, given a general quasilinear second-order derivative form for the deformed field equation, the most general possible form allowed for the deformed gauge symmetry will be a first-order linear differential operator on

(30) |

whose coefficients and respectively have no dependence on higher than first and second order derivatives of , while is at most linear in second derivatives of . Note, from the form of the abelian gauge symmetry, the coefficients at lowest order are given by

(31) |

However, the divergence equations (29) show that, with , any dependence on first derivatives of in or second derivatives of in will lead to an escalating total number of derivatives in as increases Wal ; Annalspaper , in which case the deformed field equation will be non-polynomial in first (lowest order) derivatives of . In particular, count one derivative as a first derivative appearing linearly, two derivatives as a first derivative appearing quadratically or a second derivative appearing linearly, and so on. Now, if is to be only polynomial in first derivatives (and at most linear in second derivatives), then clearly the number of derivatives on both sides of

(32) |

will balance only if contains no derivatives of and contains first order derivatives of at most linearly and no second order derivatives of , for every order . Furthermore, at each successive order , this balance in the number of derivatives will imply should only contain the same number of derivatives as contained in .

Proposition 2.4: With a polynomial restriction on lowest-order derivatives of in the quasilinear form (26) sought for the deformed field equation, the most general compatible form for the deformed gauge symmetry will be linear in first derivatives of both and . The deformed field equation, in turn, must be only of a semilinear second-order derivative form, with no derivatives of allowed to appear in the coefficient of the second derivative (highest order) term and at most quadratic dependence on derivatives of allowed for the first derivative (lowest order) term.

Next, the Noether divergence identity (32) leads to an interesting relation connecting the gauge symmetry and the field equation. Associated with the deformed gauge symmetry is a rigid symmetry defined by the field variation

(33) |

Due to gauge invariance, this field variation leaves the deformed Lagrangian invariant to within a curl,

(34) |

and hence it generates a conserved current of the deformed field equation by Noether’s theorem Olv ; AncBlu . The current arises from the identity

(35) | |||||

where is the integration by parts term (13) relating the Lie derivative and Euler-Lagrange operators. This yields

(36) |

from which the Noether current is seen to be conserved on the solution space, . Now, a comparison of the conservation equation (36) with the divergence identity (32) reveals the relation

Proposition 2.5: The deformed field equation can be written in terms of a rigid Noether current

(37) |

which is associated with the deformed gauge symmetry

(38) |

where denotes the formal left-inverse of defined in powers of the field coordinates through equation (31).

Finally, deformations that are related by a locally constructed invertible change of field variables

(39) |

or of gauge parameters

(40) |

will be regarded as equivalent. In order to stay within the general form given for deformation terms (26) and (30), these transformation terms and for must contain no derivatives of , and must also depend linearly on (and contain no derivatives of ).

### ii.4 Solving the determining equations

Steps will now be outlined for how to solve for the deformation terms in an efficient, explicit manner from the hierarchy of determining equations given by theorems 2.1 and 2.2. In this hierarchy

(42) | |||||

(44) | |||||

observe that the unknowns and for are partly decoupled since they first enter the equations at th and th orders, respectively. Also, notice the bottom equation (42) holds as a gauge-invariance identity on and from the given linear abelian theory.

At each order , an important feature of the hierarchy is that the unknowns and occur only through variations and involving the undeformed gauge symmetry . Such variational expressions can be annihilated in two different ways, leading to integrability conditions in solving for the quadratic and higher power deformation terms. One integrability condition is that an antisymmetrized variation with respect to the undeformed gauge symmetry vanishes, since by commutativity,

(45) |

and similarly

(46) |

A second integrability condition arises because the undeformed gauge symmetry itself vanishes if its parameter is taken to be rigid, i.e. constant, so that . Hence, by rigidity,

(47) |

and

(48) |

These integrability conditions will be referred to as “commutativity-type” and “rigidity-type”. As a main result, because of the decoupled structure of the hierarchy, the commutativity-type integrability condition will turn out to be trivial, as at each order it will hold due to the Jacobi identity for the deformed gauge symmetry and the closure of the associated gauge group, discussed earlier. In contrast, the rigidity-type integrability condition will be central to the deformation analysis.

Connected with the identity (42) at the bottom of the hierarchy, the property that the abelian gauge symmetry generates all of the differential gauge invariance in the linear field equation leads to a very useful result.

Lemma 2.6: For any locally constructed field variation depending linearly on at least one arbitrary gauge parameter , the following three conditions are equivalent (provided auxiliary gauge freedom is excluded): (i) It has the form of an undeformed gauge symmetry

(49) |

whose parameter depends linearly on (and its derivatives). (ii) It is identically curl-free

(50) |

(iii) It identically satisfies the undeformed field equation

(51) |

Moreover, the same equivalences hold whenever satisfies .

The proof is immediate, since is gauge invariant under . For the converse, is a consequence of exhausting all the gauge invariance admitted by (under the assumption no auxiliary gauge freedom is present), while holds because on jet space any closed -form function is exact And . These arguments are not difficult to extend to the solution space by means of the jet-space tensor techniques from Ref. Annalspaper .

This lemma will be a key tool in the analysis to follow. In examples, the condition of no auxiliary gauge freedom is met by linear abelian Yang-Mills and Freedman-Townsend theories, including abelian Chern-Simons and torsion generalizations, and their extension to other -form fields. The main examples of a linear abelian theory having auxiliary gauge freedom are graviton theories. The simplest situation occurs when this auxiliary gauge freedom does not enter into solutions of the determining equations for the deformed field equation and deformed gauge symmetry, order by order. For solutions of that type the deformation terms in the gauge symmetry will satisfy the equivalences stated in the lemma. This is what happens for the deformation of linear abelian graviton theory corresponding to the Einstein gravitational theory and its multi-graviton (algebra-valued) generalizations, in dimensions Annalspaper ; Hen2 . In contrast, the exotic parity-violating multi-graviton theories in dimensions involve a more complicated situation where the auxiliary gauge freedom from the linear theory is itself deformed nontrivially in the course of solving the determining equations PhysRevpaper .

So, as the simplest case, only those deformations obtained by postulating the lemma to hold independently of any auxiliary gauge freedom in solving the determining equations will be considered hereafter. Note this assumption is trivially met whenever the abelian gauge symmetry exhausts the gauge freedom present in the linear field equation.

To begin the analysis, the first step is to solve the th order Lie derivative commutator equation (42) for the linear deformation terms in the gauge symmetry

(52) |

Through lemma 2.6, equation (42) is equivalent to

(53) |

which expands out to give

(54) | |||||

Substitution into this equation using an explicit general linear form for and in terms of and derivatives of yields determining equations on their coefficients. These equations are readily solved by using the jet-space (algebraic) tensor techniques shown in Ref. Annalspaper .

The solution for and determines the lowest order part of the infinitesimal gauge group structure where the parameter is given in terms of and to lowest order by the closure relation

(55) |

which follows from equation (53).

Closure of the gauge group at 1st order is derived as the next step from the following analysis of the determining equations at st order in the hierarchy. Consider the Lie derivative commutator equation (44) minus the Lie derivative equation (44) applied to the gauge group commutator parameter ,

(56) |

where

(57) |

denotes the field variation representing the deviation from closure of the deformed gauge symmetry group. When is taken to satisfy the linear field equation, the terms drop out of equation (56), yielding

(58) |

Lemma 2.6 implies

(59) |

for some locally constructed parameter , which determines the deformation of the gauge group structure at 1st order. Hence a closure result is obtained.

Theorem 2.7: The deformed gauge symmetry generates a gauge group that closes up to 1st order on the solution space deformed field equation

(60) |

Closure can be shown to extend in a similar way to higher orders, leading to an infinitesimal gauge group structure

(61) |

with

(62) |

depending bilinearly on (and their derivatives).

This theorem gives rise to an obstruction for existence of quadratic terms in the deformed gauge symmetry. Take the parameter to be rigid, , so then the curl of the gauge group closure equation (60) restricted to rigid symmetries yields

(63) | |||||

This equation is a rigidity-type integrability condition on the linear deformation terms, namely, further necessary determining equations on the coefficients of and derivatives of in .

An additional rigidity-type integrability condition on the linear deformation terms arises from the 1st order Lie derivative equation (44) with taken to be constant, . Factorization of the parameter out of this equation gives

(64) |

This imposes more determining equations on the coefficients of and derivatives of in .

The preceding equations are readily solved by the same algebraic techniques as discussed for the commutator equation (54). There is a useful remark to make here. If a classification of locally constructed linear symmetries depending on a rigid parameter is already known for the linear field equation, these symmetries then comprise all possible linear deformation terms for , since existence of a rigid symmetry is necessary to satisfy the integrability conditions (63) and (64). Moreover, the Noether currents associated with these symmetries comprise the corresponding possible quadratic deformation terms for , since the conservation law for a Noether current is the same as the gauge invariance identity from proposition 2.5, and hence necessarily agrees with the Noether current to within an identically divergence-free term. Thus, any symmetry of a rigid form that may be available in the linear abelian theory can be used as a starting point for solving the deformation determining equations. This observation is often referred to as the “method of gauging a rigid symmetry” or the “Noether coupling method” and it has been used extensively in supergravity contexts Des ; SG . Of course, it is unable to provide a complete classification or a uniqueness result for deformations.

Once the linear deformation terms have been determined, the next steps are to solve for the quadratic deformation terms in the gauge symmetry and in the field equation, respectively, from the 1st order closure equation (59)

(65) |

and from the 1st order Lie derivative equation (44)

(66) |

with the gauge parameters no longer being rigid. Through the closure equation (55) at lowest order, the commutativity-type integrability condition on equation (66) reduces to

(67) |

which is satisfied due to abelian gauge invariance. Then, an analysis of the commutativity-type integrability condition on equation (65) shows it is satisfied due to the lowest order part of the Jacobi identity applied to the gauge symmetries. Hence no obstructions arise for solving the st order Lie derivative equations (66) and (65).

To continue, after the expressions determined for and from prior equations are substituted, the determining equations (66) and (65) expand out to have the form

(68) |

and

(69) |

where

(70) |

Despite their complicated appearance, it is computationally straightforward to solve these equations for and by use of the jet-space (algebraic) tensor techniques in Ref. Annalspaper .

### ii.5 Geometrical structure of lowest-order deformations

Strikingly, a simple explicit solution for the quadratic deformation terms can be constructed out of the linear deformations terms and under a mild restriction on the form sought for the deformed field equation and gauge symmetry. The result relies on the solution of the th order Lie derivative commutator equation.

Theorem 2.8: Restrict deformations so the field equation and the gauge symmetry contain no more derivatives of and , in total, than the number in the linear field equation (4) and the abelian gauge symmetry (1) (namely, in notation (26) and (30), , , and have no dependence on derivatives of , while and have at most linear dependence on derivatives of , with higher order derivatives excluded). Then, up to a nonlinear change of field variable and of gauge parameter, the linear deformation terms have the form

(71) |

for some coefficients such that is constant, skew in its lower internal indices for , and vanishes for ; the quadratic deformation terms are explicitly given by

(72) | |||||

where

(76) |

Note the coefficients and are also subject to additional algebraic equations that come from the integrability conditions (63) and (64).

Remark: As a result of proposition 2.4, the restriction assumed on derivatives is necessary if deformations to all orders are required both to preserve the number of dynamical degrees of freedom of from the linear theory and to contain a bounded total number of derivatives of .

In outline, the proof of the theorem rests on using the determining equation (54) to show, firstly, that the solution for vanishes after a suitable change of field variable (39) and gauge parameter (40) are made; secondly, the solution for establishes that the curl of this expression is gauge invariant with respect to the and that the exterior product of and is an exact variation,

(77) |

This analysis is a straightforward application of the jet-space tensor techniques from Ref. Annalspaper . Then, from expressions (77), the Lie derivative equation (69) is easily solved for , as is the closure equation (68) for in terms of the commutator of . Note the expression obtained for is essentially a construction of the Noether current of the rigid symmetry , due to the Noether gauge-invariance identity stated in proposition 2.5. Uniqueness of both and will be established later (see theorem 2.10).