Gauge symmetry of the 3BF theory for a generic Lie 3-group
aa r X i v : . [ h e p - t h ] J a n Gauge symmetry of the BF theory for a generic Lie -group Tijana Radenkovi´c ∗ and Marko Vojinovi´c † Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia
The higher category theory can be employed to generalize the BF action to the so-called 3 BF action, by passing from the notion of a gauge group to the notion of a gauge 3-group. In this workwe determine the full gauge symmetry of the 3 BF action. To that end, the complete Hamiltoniananalysis of the 3 BF action for a general Lie 3-group is performed, by using the Dirac procedure.This analysis is the first step towards a canonical quantization of a 3 BF theory. This is an importantstepping-stone for the quantization of the complete Standard Model of elementary particles coupledto Einstein-Cartan gravity, formulated as a 3 BF action with suitable simplicity constraints. Weshow that the resulting gauge symmetry group consists of the already familiar G -, H -, and L -gauge transformations, as well as additional M - and N -gauge transformations, which have not beendiscussed in the existing literature. Keywords: quantum gravity, higher gauge theory, higher category theory, 3-group, BF action, 3 BF action,gauge symmetry I. INTRODUCTION
One of the most important open problems in contemporary theoretical physics is the quantization of gravity. Withinthe framework of Loop Quantum Gravity (LQG), one of the most prominent candidates for the quantum theory ofgravity, the study of nonperturbative quantization has evolved in two directions: the canonical and the covariantapproach. See [1–4] for an overview and a comprehensive introduction to the theory.The covariant quantization approach focuses on defining the gravitational path-integral of the theory: Z gr = Z D g e iS gr [ g ] . (1)In order to give the rigorous definition of the path integral, the classical action of the theory S gr is written as a sum ofthe topological BF action, i.e., the action with no propagating degrees of freedom, and the part featuring the simplicityconstraints, i.e., sum of products of Langrange multipliers and the corresponding simplicity constraints imposed onthe variables of the topological part of the action. Next, one defines the path-integral of the topological theory givenby the BF action, using the Topological Quantum Field Theory (TQFT) formalism. Once a path-integral is definedfor the topological sector, it is deformed into a non-topological theory, by imposing the simplicity constraints. Thisquantization technique is known as the spinfoam quantization method.The spinfoam quantization procedure has been successfully employed in various theories, including the three-dimensional topological Ponzano-Regge model of quantum gravity [5], the four-dimensional topological Ooguri model[6], the Barrett-Crane model of gravity in four dimensions [7, 8], and others. The most successful among these isthe renowned EPRL/FK model [9, 10], which had been specifically formulated to correspond to the quantum theoryof gravity obtained by the canonical loop quantization , where a state of the gravitational field is described by theso-called spin network .However, note that all mentioned models, formulated as constrained BF actions, are theories of pure gravity,without matter fields. Recently, as an endeavor to formulate a theory that unifies all the known interactions, oneinteresting new avenue of research has been opened, based on a categorical generalization of the BF action in thecontext of Higher Gauge Theory (HGT) formalism [12]. One novel candidate discussed in the literature [13], uses the3-group structure to formulate the 3 BF action as a categorical generalization of the BF theory. Then, modifying thepure 3 BF action by adding the appropriate simplicity constraints, one obtains the constrained BF action, describingthe theory of all the fields present in the Standard model coupled to Einstein-Cartan gravity in a standard way.In addition to the covariant approach, one can also study the constrained 3 BF action, using the canonical quantiza-tion . This approach focuses on defining the quantum theory via a triple ( H , A , W ), i.e., the Hilbert space of states H ,the algebra of observables A , and the dynamics W given by the transition amplitudes. Specifically, in canonical LQG, ∗ Electronic address: [email protected] † Electronic address: [email protected] the algebra of fields that are promoted to the quantum operators is chosen to be the algebra based on the holonomiesof the gravitational connection. However, in the case of the 3 BF theory, the notion of connection is generalized to thenotion of 3-connection, which makes its canonical quantization approach an interesting avenue of research. The firststep toward the canonical quantization of the theory is the Hamiltonian analysis, resulting in the algebra of first-classand second-class constraints. The first-class constraints become conditions on the physical states determining theHilbert space, while the Hamiltonian constraint determines the dynamics.The results presented in this paper are the natural continuation of the results presented in [13]. The main result isthe calculation of the full symmetry group of the pure 3 BF action. To that end, the complete Hamiltonian analysisof the 3 BF action for a general Lie 3-group is performed by using the Dirac procedure (see [14] for an overview anda comprehensive introduction to the Hamiltonian analysis). It is a generalization of the Hamiltonian analysis of a2 BF action done in [15–18], and of the Hamiltonian analysis for the special case of a 2-crossed module correspondingto the theory of scalar electrodynamics, carried out in [19]. The analysis of the Hamiltonian structure of the theorygives us the algebra of first-class constraints and second-class constraints present in the theory. As usual, the first-class constraints generate gauge transformations, which do not change the physical state of the system. Using theCastellani’s procedure, one can find the generator of the gauge transformations in the theory on a spatial hypersurface.Then, the results obtained by this method are generalized to the whole spacetime. The complete gauge symmetry,consisting of five types of finite gauge transformations, and the proofs that they are indeed the gauge symmetries of3 BF action, are presented. With these results in hand, the structure of the full gauge symmetry group is analyzed,and its corresponding Lie algebra is determined.The obtained results give rise to a connection between the gauge symmetry group of the 3 BF action, and itsunderlining 3-group structure, establishing a duality between the two. This analysis is an important step towards thestudy of the gauge symmetry group of theory of gravity with matter, formulated as the constrained 3 BF action [13],as well as the canonical quantization of the theory. Furthermore, it is important for the overall understanding of thephysical meaning of the 3-group structure and its interpretation as the underlining symmetry of the pure 3 BF action,which represents a basis for the constrained 3 BF action describing the physical theory.The layout of the paper is as follows. In Section II, we give a brief overview of BF and 2 BF theories, andintroduce the 3 BF action. Section III contains the Hamiltonian analysis for the 3 BF theory. In subsection III A,the canonical structure of the theory is obtained, while in subsection III B the resulting first-class and second-classconstraints present in the theory, as well as the algebra of constraints, are presented. In the subsection III C we analyzethe Bianchi identities that the first-class constraints satisfy, which enforce restrictions in the sense of Hamiltoniananalysis, and reduce the number of independent first-class constraints present in the theory. We then proceed withthe counting of the physical degrees of freedom. Finally, this section concludes with the subsection III D where weconstruct the generator of the gauge symmetries for the topological theory, based on the calculations done in sectionIII B.Section IV contains the main results of our paper and is devoted to the analysis of the symmetries of the 3 BF action. Having results of the subsection III D in hand, we find the form variations of all variables and their canonicalmomenta, and use that result to determine all gauge transformations of the theory. This is done in four steps. Thesubsection IV A deals with the gauge group G , and the already familiar G -gauge transformations. In subsectionIV B we discuss the gauge group ˜ H L which consists of the familiar H -gauge and L -gauge transformations, while thesubsection IV C examines the novel M -gauge and N -gauge transformations which also arise in the theory. The resultsof the subsections IV A, IV B, and IV C are summarized in subsection IV D, where the complete structure of thesymmetry group is presented, including its Lie algebra. Our concluding remarks are given in section V, containing asummary and a discussion of the obtained results, as well as possible future lines of investigation. The Appendicescontain various technical details concerning 3-groups, additional relations of the constraint algebra, the computationof the generator of gauge symmetries, form-variations of all fields and momenta, and some other technical details.Our notation and conventions are as follows. Spacetime indices, denoted by the mid-alphabet Greek letters µ, ν, . . . , are raised and lowered by the spacetime metric g µν . The spatial part of these is denoted with lo-wercase mid-alphabet Latin indices i, j, . . . , and the time component is denoted with 0. The indices that arecounting the generators of groups G , H , and L are denoted with initial Greek letters α, β, . . . , lowercase ini-tial Latin letters a, b, c, . . . , and uppercase Latin indices A, B, C, . . . , respectively. The antisymmetrization overtwo indices is denoted as A [ a | a ...a n − | a n ] = (cid:0) A a a ...a n − a n − A a n a ...a n − a (cid:1) , while the total antisymmetriza-tion is denoted as A [ a ...a n ] = n ! P σ ∈ S n ( − sgn( σ ) A a σ (1) ...a σ ( n ) . Likewise, the symmetrization over two indicesis denoted as A ( a | a ...a n − | a n ) = (cid:0) A a a ...a n − a n + A a n a ...a n − a (cid:1) , while the total symmetrization is denoted as A ( a ...a n ) = n ! P σ ∈ S n A a σ (1) ...a σ ( n ) . We work in the natural system of units, defined by c = ~ = 1 and G = l p , where l p is the Planck length. All additional notation and conventions used throughout the paper are explicitly defined inthe text where they appear. II. THE BF THEORY
Given a Lie group G and its corresponding Lie algebra g , one can introduce the so-called BF action as S BF = Z M h B ∧ F i g , (2)where F ≡ d α + α ∧ α is the curvature 2-form for the algebra-valued connection 1-form α ∈ A ( M , g ) on some4-dimensional spacetime manifold M , and B ∈ A ( M , g ) is a Lagrange multiplier 2-form. The h , i g denotes the G -invariant bilinear symmetric nondegenerate form on g . For more details see [20–22].Varying the action (2) with respect to the Lagrange multiplier B and the connection α , one obtains the equationsof motion, F = 0 , ∇ B ≡ d B + α ∧ B = 0 . (3)These equations of motion imply that α is a flat connection, while the Lagrange multiplier B is a constant field.Therefore, the theory given by the BF action has no local propagating degrees of freedom, i.e., the theory is topological.Within the framework of Higher Gauge Theory, one can define the categorical generalization of the BF action tothe so-called 2 BF action, by passing from the notion of a gauge group to the notion of a gauge 2-group, see [23–25].In the category theory, a 2-group is defined as a 2-category consisting of only one object, where all the morphisms and2-morphisms are invertible. It has been shown that every strict 2-group is equivalent to a crossed module ( H ∂ → G , ✄ ),where G and H are groups, δ is a homomorphism from H to G , while ✄ : G × H → H is an action of G on H . Givena crossed-module ( H ∂ → G , ✄ ), one can introduce a generalization of the BF action, the 2 BF action [23, 24]: S BF = Z M h B ∧ Fi g + h C ∧ Gi h , (4)where the 2-form B ∈ A ( M , g ) and the 1-form C ∈ A ( M , h ) are Lagrange multipliers, and h is a Lie algebraof the Lie group H . The variables F ∈ A ( M , g ) and G ∈ A ( M , h ) define the fake -curvature ( F , G ) for the2-connection ( α , β ), given by g -valued 1-form α ∈ A ( M , g ) and an h -valued 2-form β ∈ A ( M , h ): F = d α + α ∧ α − ∂β , G = d β + α ∧ ✄ β . (5)Also, h , i g and h , i h denote the G -invariant bilinear symmetric nondegenerate forms for the algebras g and h ,respectively. See [23, 24] for review and references. Varying the 2 BF action (4) with respect to variables B and C one obtains the equations of motion F = 0 , G = 0 , (6)while varying with respect to α and β one obtainsd B α − f αβγ B γ ∧ α β − ✄ αab C b ∧ β a = 0 , (7)d C a − ∂ aα B α + ✄ αab C b ∧ α α = 0 . (8)Similarly to the case of the BF action, the 2 BF action defines a topological theory, i.e., a theory with no propagatingdegrees of freedom, see [15, 18].Continuing the categorical generalization one step further, one can generalize the 2 BF action to the 3 BF action, bypassing from the notion of a 2-group to the notion of a 3-group. Similarly to the definition of a group and a 2-groupwithin the category theory formalism, a 3-group is defined as a 3-category with only one object, where all morphisms,2-morphisms, and 3-morphisms are invertible. Moreover, analogously as a 2-group is equivalent to a crossed-module,it has been proved that a 3-group is equivalent to a 2-crossed module [26].A Lie 2-crossed module, denoted as ( L δ → H ∂ → G , ✄ , { , } pf ) (see Appendix A for the precise definition), is analgebraic structure specified by three Lie groups G , H , and L , together with the homomorphisms δ : L → H and ∂ : H → G , an action ✄ of the group G on all three groups, and a G -equivariant map, called the Peiffer lifting: { , } pf : H × H → L .
In order for this structure to be a 3-group, the structure constants of algebras g , h , and l , together with the maps ∂ and δ , the action ✄ , and the Peiffer lifting, must satisfy certain axioms, see [13]. Here g , h , and l denote the Liealgebras corresponding to the Lie groups G , H , and L .Given a 2-crossed module, one can define a 3-connection, an ordered triple ( α, β, γ ), where α , β , and γ are algebra-valued differential forms, α ∈ A ( M , g ), β ∈ A ( M , h ), and γ ∈ A ( M , l ). The corresponding fake 3-curvature( F , G , H ) is defined as: F = d α + α ∧ α − ∂β , G = d β + α ∧ ✄ β − δγ , H = d γ + α ∧ ✄ γ + { β ∧ β } pf . (9)Fixing the bases in algebras g , h , and l as τ α ∈ g , t a ∈ h , and T A ∈ l , one defines the structure constants[ τ α , τ β ] = f αβγ τ γ , [ t a , t b ] = f abc t c , [ T A , T B ] = f ABC T C , (10)maps ∂ : H → G and δ : L → H as ∂ ( t a ) = ∂ aα τ α , δ ( T A ) = δ Aa t a , (11)and an action of g on the generators of g , h , and l as τ α ✄ τ β = f αβγ τ γ , τ α ✄ t a = ✄ αab t b , τ α ✄ T A = ✄ αAB T B , (12)respectively. To define the Peiffer lifting in a basis, one specifies the coefficients X abA : { t a , t b } pf = X abA T A . (13)Writing the curvature in the bases of the corresponding algebras and differential forms F = 12 F αµν τ α d x µ ∧ d x ν , G = 13! G aµνρ t a d x µ ∧ d x ν ∧ d x ρ , H = 14! H Aµνρσ T A d x µ ∧ d x ν ∧ d x ρ ∧ d x σ , one obtains the corresponding components: F αµν = ∂ µ α αν − ∂ ν α αµ + f βγα α βµ α γν − β aµν ∂ aα , G aµνρ = ∂ µ β aνρ + ∂ ν β aρµ + ∂ ρ β aµν + α αµ β bνρ ✄ αba + α αν β bρµ ✄ αba + α αρ β bµν ✄ αba − γ Aµνρ δ Aa , H Aµνρσ = ∂ µ γ Aνρσ − ∂ ν γ Aρσµ + ∂ ρ γ Aσµν − ∂ σ γ Aµνρ +2 β aµν β bρσ X ( ab ) A − β aµρ β bνσ X ( ab ) A + 2 β aµσ β bνρ X ( ab ) A + α αµ γ Bνρσ ✄ αBA − α αν γ Bρσµ ✄ αBA + α αρ γ Bσµν ✄ αBA − α ασ γ Bµνρ ✄ αBA . (14)Then, similarly to the construction of BF and 2 BF actions, one can define the gauge invariant topological 3 BF action,with the underlying structure of a 3-group. For the manifold M and the 2-crossed module ( L δ → H ∂ → G , ✄ , { , } pf ),that gives rise to 3-curvature (9), one defines the 3 BF action as S BF = Z M h B ∧ Fi g + h C ∧ Gi h + h D ∧ Hi l , (15)where B ∈ A ( M , g ), C ∈ A ( M , h ), and D ∈ A ( M , l ) are Lagrange multipliers. The forms h , i g , h , i h , and h , i l are G -invariant bilinear symmetric nondegenerate forms on g , h , and l , respectively. Fixing the basis in algebras g , h , and l , as defined in (10), the forms h , i g , h , i h , and h , i l map pairs of basis vectors of algebras g , h , and l , tothe metrics on their vector spaces, g αβ , g ab , and g AB : h τ α , τ β i g = g αβ , h t a , t b i h = g ab , h T A , T B i l = g AB . (16)As the symmetric maps are nondegenerate, the inverse metrics g αβ , g ab , and g AB are well defined, and are used toraise and lower indices of the corresponding algebras.Varying the action (15) with respect to Lagrange multipliers B α , C a , and D A one obtains the equations of motion F α = 0 , G a = 0 , H A = 0 , (17)while varying with respect to the 3-connection variables α α , β a , and γ A one gets:d B α − f αβγ B γ ∧ α β − ✄ αab C b ∧ β a + ✄ αBA D A ∧ γ B = 0 , (18)d C a − ∂ aα B α + ✄ αab C b ∧ α α + 2 X ( ab ) A D A ∧ β b = 0 , (19)d D A − ✄ αAB D B ∧ α α + δ Aa C a = 0 . (20)For further details see [26, 27] for the definition of the 3-group, and [13] for the definition of the pure 3 BF action.Choosing the convenient underlying 2-crossed module structure and imposing the appropriate simplicity constraintsonto the degrees of freedom present in the 3 BF action, one can obtain the non-trivial classical dynamics of thegravitational and matter fields. A reader interested in the construction of the constrained 2 BF actions describingthe Yang-Mills field and Einstein-Cartan gravity, and 3 BF actions describing the Klein-Gordon, Dirac, Weyl andMajorana fields coupled to gravity in the standard way, is referred to [13, 25]. One can also introduce higherdimensional, nBF actions, see for example [28]. III. HAMILTONIAN ANALYSIS OF THE BF THEORY
In this section, the canonical structure of the theory is presented, with the resulting first-class and second-classconstraints present in the theory. The algebra of Poisson brackets between all, the first-class and the second-classconstraints, is obtained. We will use this result to calculate the total number of degrees of freedom in the theory,and in order to do that, we will have to analyse the Bianchi identities that the first-class constraints satisfy, whichenforce restrictions in the sense of Hamiltonian analysis. They reduce the number of independent first-class constraintspresent in the theory, thus increasing the number of degrees of freedom. We will obtain that the pure 3 BF theoryis topological, i.e., there are no local propagating degrees of freedom. Finally, we will finish this section with theconstruction of the generator of gauge symmetries of the 3 BF action, which is used to calculate the form-variationsof all the variables and their canonical momenta. This result will be crucial for finding the gauge symmetries of 3 BF action, which will be a topic of section IV. A. Canonical structure and Hamiltonian
Assuming that the spacetime manifold M is globally hyperbolic, the Lagrangian on a spatial foliation Σ ofspacetime M corresponding to the 3 BF action (15) is given as: L BF = Z Σ d ~x ǫ µνρσ (cid:0) B αµν F βρσ g αβ + 13! C aµ G bνρσ g ab + 14! D A H Bµνρσ g AB (cid:1) . (21)For the Lagrangian (21), the canonical momenta corresponding to all variables B αµν , α αµ , C aµ , β aµν , D A , and γ Aµνρ are: π ( B ) αµν = δLδ∂ B aµν = 0 ,π ( α ) αµ = δLδ∂ α αµ = 12 ǫ µνρ B ανρ ,π ( C ) aµ = δLδ∂ C aµ = 0 ,π ( β ) aµν = δLδ∂ β aµν = − ǫ µνρ C aρ ,π ( D ) A = δLδ∂ D A = 0 ,π ( γ ) Aµνρ = δLδ∂ γ Aµνρ = ǫ µνρ D A . (22)These momenta give rise to the six primary constraints of the theory, as none of them can be inverted for the timederivatives of the variables, P ( B ) αµν ≡ π ( B ) αµν ≈ ,P ( α ) αµ ≡ π ( α ) αµ − ǫ µνρ B ανρ ≈ ,P ( C ) aµ ≡ π ( C ) aµ ≈ ,P ( β ) aµν ≡ π ( β ) aµν + ǫ µνρ C aρ ≈ ,P ( D ) A ≡ π ( D ) A ≈ ,P ( γ ) Aµνρ ≡ π ( γ ) Aµνρ − ǫ µνρ D A ≈ . (23)Employing the following fundamental Poisson brackets, { B αµν ( ~x ) , π ( B ) βρσ ( ~y ) } = 2 δ αβ δ ρ [ µ | δ σ | ν ] δ (3) ( ~x − ~y ) , { α αµ ( ~x ) , π ( α ) β ν ( ~y ) } = δ αβ δ νµ δ (3) ( ~x − ~y ) , { C aµ ( ~x ) , π ( C ) bν ( ~y ) } = δ ab δ νµ δ (3) ( ~x − ~y ) , { β aµν ( ~x ) , π ( β ) bρσ ( ~y ) } = 2 δ ab δ ρ [ µ | δ σ | ν ] δ (3) ( ~x − ~y ) , { D A ( ~x ) , π ( D ) B ( ~y ) } = δ AB δ (3) ( ~x − ~y ) , { γ Aµνρ ( ~x ) , π ( γ ) Bστξ ( ~y ) } = 3! δ AB δ σ [ µ δ τν δ ξρ ] δ (3) ( ~x − ~y ) , (24)one obtains the algebra of primary constraints : { P ( B ) αjk ( ~x ) , P ( α ) βi ( ~y ) } = ǫ ijk g αβ ( ~x ) δ (3) ( ~x − ~y ) , { P ( C ) ak ( ~x ) , P ( β ) bij ( ~y ) } = − ǫ ijk g ab ( ~x ) δ (3) ( ~x − ~y ) , { P ( D ) A ( ~x ) , P ( γ ) Bijk ( ~y ) } = ǫ ijk g AB ( ~x ) δ (3) ( ~x − ~y ) . (25)Note that all other Poisson brackets vanish. The canonical, on-shell Hamiltonian is given by the following expression: H c = Z Σ d ~x (cid:20) π ( B ) αµν ∂ B αµν + π ( α ) αµ ∂ α αµ + π ( C ) aµ ∂ C aµ + 12 π ( β ) aµν ∂ β aµν + π ( D ) A ∂ D A + 13! π ( γ ) Aµνρ ∂ γ Aµνρ (cid:21) − L . (26)Employing the definition of the curvature components (14), the Hamiltonian (26) can be written as the sum of termsthat are equal to the product of the primary constraints and time derivatives of the variables, and the remainder.As the primary constraints are zero on-shell, the terms multiplying the time derivatives vanish, and the canonicalHamiltonian becomes: H c = − Z Σ d ~x ǫ ijk (cid:20) B α i F αjk + 16 C a G aijk + β a i (cid:18) ∇ j C ak − ∂ aα B α jk + β bjk D A X ( ab ) A (cid:19) + 12 α α (cid:18) ∇ i B α jk − C ai ✄ αb a β bjk + 13 D A ✄ αB A γ Bijk (cid:19) + 12 γ A ij (cid:18) ∇ k D A + C ak δ Aa (cid:19)(cid:21) . (27)Adding to the canonical Hamiltonian the product of the Lagrange multipliers λ and the primary constraints, for everyprimary constraint, one gets the total, off-shell Hamiltonian : H T = H c + Z Σ d ~x (cid:20) λ ( B ) αµν P ( B ) αµν + λ ( α ) αµ P ( α ) αµ + λ ( C ) aµ P ( C ) aµ + 12 λ ( β ) aµν P ( β ) aµν + λ ( D ) A P ( D ) A + 13! λ ( γ ) Aµνρ P ( γ ) Aµνρ (cid:21) . (28) B. Consistency conditions and algebra of constraints
In order for primary constraints to be preserved during the evolution of the system, they must satisfy the consistencyconditions, ˙ P ≡ { P , H T } ≈ , (29)for every primary constraint P . Imposing this condition on primary constraints P ( B ) α i , P ( α ) α , P ( C ) a , P ( β ) a i ,and P ( γ ) A ij , one obtains the secondary constraints S , S ( F ) αi ≡ ǫ ijk F αjk ≈ , S ( ∇ B ) α ≡ ǫ ijk (cid:0) ∇ [ i B α jk ] − C a [ i ✄ αb a β bjk ] + 13 D A ✄ αB A γ Bijk (cid:1) ≈ , S ( G ) a ≡ ǫ ijk G aijk ≈ , S ( ∇ C ) ai ≡ ǫ ijk (cid:0) ∇ [ j | C a | k ] − ∂ aα B α jk + β bjk D A X ( ab ) A (cid:1) ≈ , S ( ∇ D ) Aij ≡ ǫ ijk (cid:0) ∇ k D A + C ak δ Aa (cid:1) ≈ , (30)while in the case of the constraints P ( α ) αk , P ( B ) αjk , P ( β ) ajk , P ( C ) ak , P ( γ ) Aijk , and P ( D ) A the correspondingconsistency conditions determine the following Lagrange multipliers: λ ( B ) αij ≈ ∇ i B α j −∇ j B α i + C a β bij ✄ αba + C bi ✄ αba β a j − C bj ✄ αba β a i + g βγα α β B γij + D B γ A ij ✄ αBA ,λ ( α ) αi ≈ ∇ i α α + ∂ aα β a i ,λ ( C ) ai ≈ ∇ i C a + C bi ✄ αab α α − β b i D A X ( ba ) A + B α i ∂ aα ,λ ( β ) aij ≈ ∇ i β a j −∇ j β a i − β bij ✄ αba α α + γ A ij δ Aa ,λ ( D ) A ≈ α α D B ✄ αAB − C a δ Aa ,λ ( γ ) Aijk ≈ − β a i β bjk X ( ab ) A +2 β a j β bik X ( ab ) A − β a k β bij X ( ab ) A − α α ✄ αBA γ Bijk + ∇ i γ A jk −∇ j γ A ik + ∇ k γ A ij . (31)Note that the rest of the Lagrange multipliers λ ( B ) α i , λ ( α ) α , λ ( C ) a , λ ( β ) a i , λ ( γ ) A ij , (32)remain undetermined.Further, as the secondary constraints must also be preserved during the evolution of the system, the consistencyconditions of secondary constraints must be enforced. However, no tertiary constraints arise from these conditions(see equation (B1) in Appendix B), leading the iterative procedure to an end. Finally, the total Hamiltonian can bewritten in the following form: H T = Z Σ d ~x (cid:20) λ ( B ) α i Φ( B ) αi + λ ( α ) α Φ( α ) α + λ ( C ) a Φ( C ) a + λ ( β ) a i Φ( β ) ai + 12 λ ( γ ) A ij Φ( γ ) Aij − B α i Φ( F ) ai − α α Φ( ∇ B ) α − C a Φ( G ) a − β a i Φ( ∇ C ) ai − γ A ij Φ( ∇ D ) Aij (cid:21) , (33)where Φ( B ) αi = P ( B ) α i , Φ( α ) α = P ( α ) α , Φ( C ) a = P ( C ) a , Φ( β ) ai = P ( β ) a i , Φ( γ ) Aij = P ( γ ) A ij , Φ( F ) αi = S ( F ) αi − ∇ j P ( B ) αij − P ( C ) ai ∂ aα , Φ( G ) a = S ( G ) a + ∇ i P ( C ) ai − β bij ✄ α ba P ( B ) αij + P ( D ) A δ Aa , Φ( ∇ C ) ai = S ( ∇ C ) ai − ∇ j P ( β ) aij + C bj ✄ αba P ( B ) αij − ∂ aα P ( α ) αi + 2 D A X ( ab ) A P ( C ) bi + β bjk X ( ab ) A P ( γ ) Aijk , Φ( ∇ B ) α = S ( ∇ B ) α + ∇ i P ( α ) αi − f αγβ B βij P ( B ) γij − C bi ✄ αa b P ( C ) ai − β bij ✄ αa b P ( β ) aij − P ( D ) A D B ✄ αA B + 13! P ( γ ) Aijk γ Bijk ✄ αB A , Φ( ∇ D ) Aij = S ( ∇ D ) Aij + ∇ k P ( γ ) Aijk − P ( β ) aij δ Aa − P ( B ) αij ✄ α BA D B , (34)are the first-class constraints. The second-class constraints in the theory are: χ ( B ) αjk = P ( B ) αjk , χ ( C ) ai = P ( C ) ai , χ ( D ) A = P ( D ) A ,χ ( α ) αi = P ( α ) αi , χ ( β ) aij = P ( β ) aij , χ ( γ ) Aijk = P ( γ ) Aijk . (35)The PB algebra of the first-class constraints is given by { Φ( F ) αi ( ~x ) , Φ( ∇ B ) β ( ~y ) } = f βγα Φ( F ) γ i ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ B ) α ( ~x ) , Φ( ∇ B ) β ( ~y ) } = f αβγ Φ( ∇ B ) γ ( ~x ) δ (3) ( ~x − ~y ) , { Φ( G ) a ( ~x ) , Φ( ∇ C ) bi ( ~y ) } = − ✄ αba Φ( F ) αi ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ C ) ai ( ~x ) , Φ( ∇ C ) bj ( ~y ) } = − X ( ab ) A Φ( ∇ D ) Aij ( ~x ) δ (3) ( ~x − ~y ) , { Φ( G ) a ( ~x ) , Φ( ∇ B ) α ( ~y ) } = ✄ αba Φ( G ) b ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ C ) ai ( ~x ) , Φ( ∇ B ) α ( ~y ) } = ✄ αba Φ( ∇ C ) bi ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ B ) α ( ~x ) , Φ( ∇ D ) Aij ( ~y ) } = ✄ αAB Φ( ∇ D ) Bij ( ~x ) δ (3) ( ~x − ~y ) . (36)The algebra between the first and the second class constraints is given in the Appendix B, equation (B2).With the algebra of the constraints in hand, one can proceed to calculate the generator of gauge symmetries of theaction. The generator will be used to calculate the form-variations of all the variables and their canonical momenta,which will help us find the finite gauge symmetries of the action. Additionally, we can determine the number ofindependent parameters of gauge transformations, since usually all the first class constraints generate unphysicaltransformations of dynamical variables, i.e., that to each parameter of the gauge symmetry there corresponds onefirst-class constraint. However, before we embark on the construction of the symmetry generator, we will devote someattention to the number of local propagating degrees of freedom in the theory, in order to determine if the 3 BF actionis topological or not. C. Number of degrees of freedom
In this subsection, we will show that the structure of the constraints implies that there are no local degrees offreedom in a 3 BF theory. To that end, let us first specify all the Bianchi identities (BI) present in the theory.The 2-form curvatures corresponding to 1-forms α and C , given by F α = d α α + f βγα α β ∧ α γ , T a = d C a + ✄ αba α α ∧ C b , (37)satisfy the BI: ǫ λµνρ ∇ µ F ανρ = 0 , (38) ǫ λµνρ (cid:0) ∇ µ T aνρ − ✄ αba F αµν C bρ (cid:1) = 0 . (39)Similarly, the 3-form curvatures corresponding to 2-forms B and β , given by S α = d B α + f βγα α β ∧ B γ , G a = d β a + ✄ αba α α ∧ β β , (40)satisfy the BI: ǫ λµνρ (cid:18) ∇ λ S αµνρ − f βγα F βλµ B γνρ (cid:19) = 0 , (41) ǫ λµνρ (cid:18) ∇ λ G aµνρ − ✄ αba F αλµ β bνρ (cid:19) = 0 . (42)Finally, defining the 1-form curvature for D , Q A = d D A + ✄ αBA α α ∧ D B , (43)one can write the corresponding BI for Q A : ǫ λµνρ (cid:18) ∇ ν Q Aρ − ✄ αB A F ανρ D B (cid:19) = 0 . (44)These Bianchi identities play an important role in determining the number of degrees of freedom present in thetheory.As the general theory states, if there are N fields in the theory, F independent first-class constraints per spacepoint, and S independent second-class constraints per space point, the number of independent field components, i.e.,the number of the physical degrees of freedom present in the theory, is given by: n = N − F − S . (45)Let p denote the dimensionality of the group G , q the dimensionality of the group H , and r the dimensionality ofthe group L . Determining the number of initial fields present in the 3 BF theory, by counting the field componentslisted in Table I, one obtains the number N = 10( p + q ) + 5 r . Similarly, one determines the number of independent α αµ β aµν γ Aµνρ B αµν C aµ D A p q r p q r TABLE I: Initial fields present in the 3 BF theory. components of the second-class constraints by counting the components listed in Table II and obtains the number S = 6( p + q ) + 2 r . However, when counting the number of the first-class constraints F one notes they are not allmutually independent. Namely, one can prove the following identities, as a consequence of the BI.Taking the derivative of Φ( F ) αi one obtains ∇ i Φ( F ) αi + ∂ aα Φ( G ) a = 12 ǫ ijk ∇ i F αjk − f βγα F βij P ( B ) ij . (46)0 χ ( B ) αjk χ ( C ) ai χ ( D ) A χ ( α ) αi χ ( β ) aij χ ( γ ) Aijk p q r p q r TABLE II: Second-class constraints in the 3 BF theory. This relation gives ∇ i Φ( F ) αi + ∂ aα Φ( G ) a = 0 , (47)as the first term on the right-hand side of (46) is zero off-shell because ǫ ijk ∇ i F ajk = 0 are the λ = 0 components ofBI (38), and the second term on the right-hand side is also zero off-shell, since it is a product of two constraints:12 f βγα F βij P ( B ) ij = 12 f βγα ǫ ijk S ( F ) β k P ( B ) ij = 0 . (48)The relation (47) means that p components of the first-class constraints Φ( F ) αi and Φ( G ) a are not independent ofthe others. Furthermore, taking the derivative of Φ( ∇ C ) ai one obtains ∇ i Φ( ∇ C ) ai + C bi ✄ αab Φ( F ) αi + ∂ aα Φ( ∇ B ) α − β bij X ( ab ) A Φ( ∇ D ) Aij − D A X ( ab ) A Φ( G ) b = 12 ǫ ijk (cid:0) ∇ i T ajk − ✄ αba F αjk C bi (cid:1) − ǫ ijk ✄ αa b P ( B ) αij S ( ∇ C ) bk + ǫ ijk X ( ab ) A P ( C ) bi S ( ∇ D ) Ajk + 13 ǫ ijk X ( ab ) A P ( γ ) Aijk S ( G ) b + 12 ǫ ijk ✄ αa b P ( β ) bij S ( F ) αk . (49)Noting that the right-hand side of (49) is zero off-shell as the λ = 0 components of the BI (39), and the remainingterms on the right-hand side are zero off-shell as products of two constraints, one obtains the following relation: ∇ i Φ( ∇ C ) ai + C bi ✄ αab Φ( F ) αi + ∂ aα Φ( ∇ B ) α − β bij X ( ab ) A Φ( ∇ D ) Aij − D A X ( ab ) A Φ( β ) b = 0 . (50)This relation means that q components of the constraints Φ( ∇ C ) ai , Φ( F ) αi , Φ( ∇ B ) α , Φ( ∇ D ) Aij , and Φ( β ) b , arenot independent of the others, further lowering the number of the independent first-class constraints. Finally, thefollowing relation is satisfied ∇ j Φ( ∇ D ) Aij − ✄ αBA D B Φ( F ) αi − δ Aa Φ( ∇ C ) ai = ǫ ijk ( ∇ j Q Ak + 12 ✄ αA B F αjk D B ) + 12 ǫ jkl ✄ αB A P ( γ ) Bijk S ( F ) αl − ǫ jkl ✄ αB A P ( B ) αij S ( ∇ D ) Bkl . (51)As the first term on the right-hand side is precisely the λ = 0 component of the BI (44), while the second and thirdterm are equal to zero as products of two constraints, this gives: ∇ j Φ( ∇ D ) Aij − ✄ αBA D B Φ( F ) αi − δ Aa Φ( ∇ C ) ai = 0 . (52)This relation suggests that 3 r components of the primary constraints Φ( ∇ D ) Aij , Φ( F ) αi , and Φ( C ) ai are not indepen-dent of the others. However, this is slightly misleading, since the covariant derivative of the BI (44) is automaticallysatisfied as a consequence of the BI (38), ǫ λµνρ D B ✄ αB A ∇ µ F ανρ = 0 , (53)which means that there are in fact only 2 r components of the constraint (52). A formal proof of this statement wouldinvolve evaluating the Wronskian of all first-class constraints, and is out of the scope of this paper.The number of independent components of first-class constraints is determined by counting the components listedin Table III, and then subtracting the number of independent relations (47), (50), and (52).Bearing the previous analysis in mind, one obtains the number of independent first-class constraints: F = 8( p + q ) + 6 r − p − q − r = 7( p + q ) + 4 r . Finally, using the definition (45), the number of degrees of freedom in the 3 BF theory is: n = 10( p + q ) + 5 r − p + q ) − r − p + q ) + 2 r . (54)Therefore, there are no local propagating degrees of freedom in a 3 BF theory.1 Φ( B ) αi Φ( C ) a Φ( α ) α Φ( β ) ai Φ( γ ) Aij Φ( F ) αi Φ( G ) a Φ( ∇ C ) ai Φ( ∇ B ) α Φ( ∇ D ) Aij p q p q r p − p q q − q p r − r TABLE III: First-class constraints in the 3 BF theory. D. Symmetry generator
The unphysical transformations of dynamical variables are often referred to as gauge transformations. The gaugetransformations are local , meaning that the parameters of the transformations are arbitrary functions of space andtime. We shall now construct the generator of all gauge symmetries of the theory governed by the total Hamiltonian(33), using the Castellani’s algorithm (see Chapter 5 in [14] for a comprehensive overview of the procedure). Thedetails of the construction are given in Appendix C, and the following result is obtained G = Z Σ d ~x (cid:18) ( ∇ ǫ g α ) ( ˜ G ) α + ǫ g α ( ˜ G ) α + ( ∇ ǫ h ai ) ( ˜ H ) ai + ǫ h ai ( ˜ H ) ai + 12 ( ∇ ǫ l Aij ) ( ˜ L ) aij + 12 ǫ l Aij ( ˜ L ) aij + ( ∇ ǫ m αi ) ( ˜ M ) αi + ǫ m αi ( ˜ M ) αi + ( ∇ ǫ n a ) ( ˜ N ) a + ǫ n a ( ˜ N ) a (cid:19) , (55)where( ˜ G ) α = − Φ( α ) α , ( ˜ G ) α = − (cid:0) f αγβ B β i Φ( B ) γ i + C a ✄ αb a Φ( C ) b + β a i ✄ αb a Φ( β ) b i − γ A ij ✄ αA B Φ( γ ) Bij − Φ( ∇ B ) α (cid:1) , ( ˜ H ) ai = − Φ( β ) ai , ( ˜ H ) ai = C b ✄ αa b Φ( B ) αi − β b j X ( ab ) A Φ( γ ) Aij + Φ( ∇ C ) ai , ( ˜ L ) aij = Φ( γ ) Aij , ( ˜ L ) aij = − Φ( ∇ D ) Aij , ( ˜ M ) αi = − Φ( B ) αi , ( ˜ M ) αi = Φ( F ) αi , ( ˜ N ) a = − Φ( C ) a , ( ˜ N ) a = β b i ✄ αa b Φ( B ) αi + Φ( G ) a , (56)and ǫ g α , ǫ h ai , ǫ l Aij , ǫ m αi , and ǫ n a are the independent parameters of the gauge transformations.The obtained gauge generator (55) is then employed to calculate the form variations of variables and their corre-sponding canonical momenta, denoted as A ( t, ~x ), using the following equation, δ A ( t, ~x ) = { A ( t, ~x ) , G } . (57)The form variations of all fields and canonical momenta are given in Appendix E, equations (E2), while the algebraof the generators (56) is obtained in the Appendix B, equations (B4)-(B7). However, one must bear in mind that thegauge generator (55) is the generator of the symmetry transformations on a slice of spacetime, i.e., on a hypersurfaceΣ . Having in hand all these results, specifically the form variations of all variables and their canonical momenta(E2), we can determine the full gauge symmetry of the theory, which will be done in the next section. IV. SYMMETRIES OF THE BF ACTION
In order to systematically describe all gauge transformations of the 3 BF action, we will discuss in turn each set ofgauge parameters ǫ g α , ǫ h ai , ǫ l Aij , ǫ m αi , and ǫ n a , appearing in (55). The subsection IV A deals with the gauge group G ,2and the already familiar G -gauge transformations. In subsection IV B we discuss the gauge group ˜ H L which consistsof the already familiar H -gauge and L -gauge transformations, while the subsection IV C examines the M -gauge and N -gauge transformations which are also present in the theory. Finally, the results of the subsections IV A, IV B, andIV C will be summarized in the subsection IV D, where we will present the complete structure of the gauge symmetrygroup. A. Gauge group G First, consider the infinitesimal transformation with the parameter ǫ g α , given by the form variations δ α αµ = − ∂ µ ǫ g α − f βγα α β µ ǫ g γ , δ B αµν = f βγα ǫ g β B γµν ,δ β aµν = ✄ αba ǫ g α β bµν , δ C aµ = ✄ αb a ǫ g α C bµ ,δ γ Aµνρ = ✄ αB A ǫ g α γ Bµνρ , δ D A = ✄ αB A ǫ g α D B , (58)which is analogous to writing the transformation as: α → α ′ = α − ∇ ǫ g , B → B ′ = B − [ B, ǫ g ] ,β → β ′ = β + ǫ g ✄ β , C → C ′ = C + ǫ g ✄ C ,γ → γ ′ = γ + ǫ g ✄ γ , D → D ′ = D + ǫ g ✄ D . (59)Based on these infinitesimal transformations, one can extrapolate the finite symmetry transformations, defined in theTheorem 1.
Theorem 1 ( G -gauge transformations) In the BF theory for the -crossed module ( L δ → H ∂ → G, ✄ , { , } pf ) ,the following transformation is a gauge symmetry, α → α ′ = Ad g α + g d g − , B → B ′ = gBg − ,β → β ′ = g ✄ β , C → C ′ = g ✄ C ,γ → γ ′ = g ✄ γ , D → D ′ = g ✄ D , (60) where g = exp( ǫ g · ˆ G ) = exp( ǫ g α ˆ G α ) ∈ G , and ǫ g : M → g is the parameter of the transformation. Proof.
Note that if one considers an element of the group, g ∈ G , the transformations of the Theorem 1 give rise tothe following 3-curvature transformation F → F ′ = g F g − , G → G ′ = g ✄ G , H → H ′ = g ✄ H , (61)and the invariance of the 3 BF action under this transformation follows from the G -invariance of the symmetricbilinear forms on g , h , and l .Let us consider two subsequent infinitesimal G -gauge transformations, determined by the small parameters ǫ g α and ǫ g β . To calculate the commutator between the generators of the G -gauge transformations, we will make use ofthe Baker-Campbell-Hausdorff formula in the case when the parameters of the transformations are smalle ǫ g α ˆ G α e ǫ g β ˆ G β = e ǫ g α ˆ G α + ǫ g β ˆ G β + ǫ g α ǫ g β [ ˆ G α , ˆ G β ]+ O ( ǫ g ) , (62)from which it follows: e ǫ g α ˆ G α e ǫ g β ˆ G β − e ǫ g β ˆ G β e ǫ g α ˆ G α = ǫ g α ǫ g β [ ˆ G α , ˆ G β ] + O ( ǫ g ) . (63)Using the equation (63), we obtain that the generators of the G -gauge transformations defined in the Theorem 1satisfy the following commutation relations: [ ˆ G α , ˆ G β ] = f αβγ ˆ G γ , (64)where f αβγ are the structure constants of the algebra g . By noting that there exists an isomorphism between generatorsˆ G α ∼ = τ α , one establishes that the group of the G -gauge transformations from the Theorem 1 is the same as the group G of the 2-crossed module ( L δ → H ∂ → G, ✄ , { , } pf ). This is an important result, which will not be true for theremaining symmetry transformations, as we shall see below.3 B. The gauge group ˜ H L Let us now consider the form variations of the variables corresponding to the parameter ǫ h ai . For example, one cansee from the equations (E2) that the form-variation of the variables α α and α αi are: δ α αo = 0 , δ α αi = − ∂ aα ǫ h ai . (65)Taking into account that the action of the generator (55) gives the symmetry transformations on one hypersurfaceΣ with the time component of the parameter equal to zero, ǫ h a = 0, one can extrapolate that for parameter of thespacetime gauge transformations ǫ h aµ , the form-variation of the variable α αµ is given as: δ α αµ = − ∂ aα ǫ h aµ , (66)and similarly for the rest of the variables. Thus, the infinitesimal symmetry transformations in the whole spacetimecorresponding to the parameter ǫ h aµ are given by the form variations: δ α αµ = − ∂ aα ǫ h aµ , δ B αµν = 2 C a [ µ | ǫ h b | ν ] ✄ βb a g αβ ,δ β aµν = − ∇ [ µ | ǫ h a | ν ] , δ C aµ = 2 D A X ( ab ) A ǫ h bµ ,δ γ Aµνρ = 3! β a [ µν ǫ h bρ ] X ( ab ) A , δ D = 0 . (67)For these infinitesimal transformations one obtains the finite symmetry transformations given in Theorem 2. Theorem 2 ( H -gauge transformations) In the BF theory for the -crossed module ( L δ → H ∂ → G, ✄ , { , } pf ) ,the following transformation is a symmetry: α → α ′ = α − ∂ǫ h , B → B ′ = B − C ′ ∧ T ǫ h − ǫ h ∧ D ǫ h ∧ D D ,β → β ′ = β − ∇ ′ ǫ h − ǫ h ∧ ǫ h , C → C ′ = C − D ∧ X ǫ h − D ∧ X ǫ h ,γ → γ ′ = γ + { β ′ , ǫ h } pf + { ǫ h , β } pf , D → D ′ = D , (68) where ǫ h ∈ A ( M , h ) is an arbitrary h -valued -form, and ∇ ′ denotes the covariant derivative with respect to theconnection α ′ . The maps T , D , X , and X are defined in Appendix D. Proof.
Note that the 3-curvature transforms as
F → F ′ = F , G → G ′ = G − F ∧ ✄ ǫ h , H → H ′ = H + {G ′ , ǫ h } pf − { ǫ h , G} pf . (69)Taking into account the transformations of the 3-curvature (69) and the transformations of the Lagrange multipliers,the action S BF transforms as: S ′ BF = S BF + Z M (cid:16) − h C ′ ∧ T ǫ h , Fi g − h ǫ h ∧ D ǫ h ∧ D D, Fi g − h C ′ , F ∧ ✄ ǫ h i h − h D ∧ X ǫ h , Gi h − h D ∧ X ǫ h , Gi h + h D, {G , ǫ h } pf i l − h D, {F ∧ ✄ ǫ h , ǫ h } pf i l − h D, { ǫ h , G} pf i l (cid:17) . (70)Using the definitions of the maps T , D , X , and X , given in Appendix D, one sees that the terms in the parenthesescancel, specifically the first term with the third, second with seventh, fourth with eighth, and fifth with the sixthterm.The H -gauge transformations do not form a group. Namely, one can check that the two consecutive H -gaugetransformations do not give a transformation of the same kind, i.e., the closure axiom of the group is not satisfied.This is analogous to the well-known structure of Lorentz group, where boost transformations are not closed, and thusdo not form a group. Indeed, one must consider both rotations and boosts to obtain the set of transformations thatforms the Lorentz group. In the case of the H -gauge transformations, we will show that together with the H -gaugetransformations one needs to consider the transformations corresponding to the parameter ǫ l Aij . From the equations(E2) one reads the form-variations on a space hypersurface Σ corresponding to this parameter. Similarly as it is donein the case of the H -gauge trasformations, one extrapolates that the form-variations for all the variables corresponding4to the parameter ǫ l Aµν are given as: δ α αµ = 0 , δ B αµν = − D A ✄ βB A ǫ l Bµν g αβ ,δ β aµν = δ Aa ǫ l Aµν , δ C aµ = 0 ,δ γ Aµνρ = ∇ µ ǫ l Aνρ − ∇ ν ǫ l Aµρ + ∇ ρ ǫ l Aµν , δ D A = 0 . (71)These infinitesimal transformations correspond to the finite symmetry transformations defined in Theorem 3. Theorem 3 ( L -gauge transformations) In the BF theory for the -crossed module ( L δ → H ∂ → G, ✄ , { , } pf ) ,the following transformation is a symmetry α → α ′ = α , B → B ′ = B + D ∧ S ǫ l ,β → β ′ = β + δǫ l , C → C ′ = C ,γ → γ ′ = γ + ∇ ǫ l , D → D ′ = D , (72) where ǫ l ∈ A ( M , l ) is an arbitrary l -valued -form, and the map S is defined in Appendix D. Proof.
Note that the 3-curvature transforms as
F → F ′ = F , G → G ′ = G , H → H ′ = H + F ∧ ✄ ǫ l . (73)Taking into account the transformations (73) and the transformations of the Lagrange multipliers, the action trans-forms as: S ′ BF = S BF + Z M (cid:16) h D ∧ S ǫ l , Fi g + h D, F ∧ ✄ ǫ l i l (cid:17) . (74)According to the definition of the map S , the terms in the parentheses cancel.Let us denote the generators of the H -gauge transformations given by the Theorem 2 and the L -gauge transfor-mations given by the Theorem 3 as ˆ H aµ and ˆ L Aµν , respectively. As we have commented above, one can now checkthat the transformations defined in the Theorem 2, i.e., the H -gauge transformations, do not form a group. If oneperforms two consecutive H -gauge transformations, defined with parameters ǫ h and ǫ h , one obtainse ǫ h · ˆ H e ǫ h · ˆ H − e ǫ h · ˆ H e ǫ h · ˆ H = 2 ( { ǫ h ∧ ǫ h } pf − { ǫ h ∧ ǫ h } pf ) · ˆ L , (75)where ǫ h · ˆ H = ǫ h aµ ˆ H aµ and ǫ l · ˆ L = ǫ l Aµν ˆ L Aµν . Using the equation analogous to BCH formula (63), one obtainsthat the commutator of the generators of two H -gauge transformations is the generator of an L -gauge transformation(see Appendix F for the details of the calculation):[ ˆ H aµ , ˆ H bν ] = 2 X ( ab ) A ˆ L Aµν . (76)Next, note that the transformations defined in Theorem 3 are the linear transformations, and the two subsequent L -gauge transformations give one L -gauge transformation with the parameter ǫ l + ǫ l . Formally, one can write theprevious statement as e ǫ l · ˆ L e ǫ l · ˆ L = e ( ǫ l + ǫ l ) · ˆ L , (77)which leads to the conclusion that the generators of the L -gauge transformations are mutually commuting:[ ˆ L Aµν , ˆ L Bρσ ] = 0 . (78)Thus, the L -gauge transformations form an Abelian group, which will be denoted as ˜ L . According to the indexstructure of the parameters and generators, we can conclude that the group ˜ L is isomorphic to R r , where r is thedimension of the group L : ˜ L ∼ = R r . (79)5Our analogy with the case of the Lorentz group can once again prove useful, since the closing of the L -gauge trans-formations resembles the fact that the composition of two rotations is a rotation. The Abelian group ˜ L should notbe confused with the non-Abelian group L of the 2-crossed module ( L δ → H ∂ → G, ✄ , { , } pf ).Let us now examine the relationship between H -gauge transformations and L -gauge transformations. The followingresult, e ǫ h · ˆ H e ǫ l · ˆ L = e ǫ l · ˆ L e ǫ h · ˆ H , (80)leads to the conclusion that the commutator of generators of the H -gauge transformations and generators of the L -gauge transformations vanishes: [ ˆ H aµ , ˆ L Aνρ ] = 0 . (81)From the closure of the algebra (76), (78), and (81), one can conclude that the H -gauge transformations togetherwith the L -gauge transformations form a group, which will be denoted as ˜ H L . Lastly, the action of the group G onthe H -gauge and L -gauge transformations is examined by calculating the expressions:[ ǫ g · ˆ G, ǫ h · ˆ H ] = ( ǫ g ✄ ǫ h ) · ˆ H , [ ǫ g · ˆ G, ǫ l · ˆ L ] = ( ǫ g ✄ ǫ l ) · ˆ L , (82)which lead to the following commutators: [ ˆ G α , ˆ H aµ ] = ✄ αab ˆ H bµ , [ ˆ G α , ˆ L Aµν ] = ✄ αAB ˆ L Bµν . (83)Theorems 1, 2, and 3 represent the G -, H -, and L -gauge transformations, which are already familiar from theprevious literature (see for example [27]). C. The gauge groups M and N Next, consider the infinitesimal transformation with the parameter ǫ m αi , given by the form variations in AppendixE. In a similar manner as done in the previous subsection, one establishes that the form variations obtained as a resultof the Hamiltonian analysis are transformations on one hypersurface Σ , from which one can guess the symmetryin the whole spacetime. Keeping in mind that the variations on the hypersurface have the time component of theparameter set to ǫ m α = 0, one extrapolates the form-variations of the whole spacetime for the parameter ǫ m αµ to be: δ α αµ = 0 , δ B αµν = − ∇ [ µ | ǫ m α | ν ] ,δ β aµν = 0 , δ C aµ = − ∂ aα ǫ m αµ ,δ γ Aµνρ = 0 , δ D A = 0 . (84)Based on this result, one obtains the finite symmetry transformations in the whole spacetime, as defined in Theorem4, which we will refer to as the M -gauge transformations. Theorem 4 ( M -gauge transformations) In the BF theory for the -crossed module ( L δ → H ∂ → G, ✄ , { , } pf ) ,the following transformation is a symmetry α → α ′ = α , B → B ′ = B − ∇ ǫ m ,β → β ′ = β , C a → C ′ a = C a − ∂ aα ǫ m α ,γ → γ ′ = γ , D → D ′ = D , (85) where ǫ m ∈ A ( M , g ) is an arbitrary g -valued -form. Proof.
Consider the transformation of the 3 BF action under the transformations of the variables defined in theTheorem 4. One obtains: S ′ BF = S BF + Z M d x ǫ µνρσ (cid:18) −
12 ( ∇ µ ǫ m αν ) F αρσ − ∂ aα ǫ m αµ G aνρσ (cid:19) . (86)6Using the definition of 3-curvature, given by the expressions (14), one obtains: S ′ BF = S BF + Z M d x ǫ µνρσ (cid:16) −
12 ( ∇ µ ǫ m αν ) ( F αρσ − ∂ aα β aρσ ) − ∂ aα ǫ m αµ (cid:0) ∇ ν β aρσ − δ Aa γ Aνρσ (cid:1) (cid:17) . (87)Taking into account that the second and the third term cancel, while the last term is zero because of the identity(A1), the expression reduces to: S ′ BF = S BF − Z M d x ǫ µνρσ ǫ m αµ ∇ ν F αρσ . (88)Finally, the term ǫ µνρσ ∇ ν F αρσ = 0 is the BI (38). One concludes that the action S BF is invariant under thetransformation defined in Theorem 4.Note that the transformations defined in Theorem 4 are linear transformations, and the two subsequent M -gaugetransformations give one M -gauge transformation with the parameter ǫ m + ǫ m . Denoting the generators of the M -gauge transformations as ˆ M αµ , one can now write the previous statement formally as:e ǫ m · ˆ M e ǫ m · ˆ M = e ( ǫ m + ǫ m ) · ˆ M , (89)where ǫ m · ˆ M = ǫ m αµ ˆ M αµ , leading to the conclusion that:[ ˆ M αµ , ˆ M βν ] = 0 . (90)Thus, the M -gauge transformations form an Abelian group, which will be denoted as ˜ M . According to the indexstructure of its parameters and generators, we see that this group is isomorphic to R p , where p is the dimension ofthe group G : ˜ M ∼ = R p . (91)Next, one can examine the relationship of M -gauge transformations with the G , H , and L -gauge transformationsdefined in the previous subsections. Specifically, considering the G -gauge symmetry generators, one finds[ ǫ g · ˆ G, ǫ m · ˆ M ] = ( ǫ g ✄ ǫ m ) · ˆ M , (92)obtaining the result: [ ˆ G α , ˆ M βµ ] = f αβγ ˆ M γµ . (93)Considering the H - and L -gauge transformations, one obtainse ǫ h · ˆ H e ǫ m · ˆ M = e ǫ m · ˆ M e ǫ h · ˆ H , e ǫ l · ˆ L e ǫ m · ˆ M = e ǫ m · ˆ M e ǫ l · ˆ L , (94)leading to the conclusion that the generators of the M -gauge transformations commute with both the generators of H -gauge transformations and the generators of the L -gauge transformations:[ ˆ H a , ˆ M αµ ] = 0 , [ ˆ L Aµν , ˆ M αρ ] = 0 . (95)Finally, examining the infinitesimal transformation corresponding to the parameter ǫ n a , given by the form-variationsas calculated in (E2), δ α αµ = 0 , δ B αµν = β bµν ✄ α ′ a b ǫ n a g αα ′ ,δ β aµν = 0 , δ C aµ = −∇ µ ǫ n a ,δ γ Aµνρ = 0 , δ D A = δ Aa ǫ n a . (96)one obtains the Theorem 5, the symmetry transformations which will be referred to as N -gauge transformations. Notethat the N -gauge transformations are simultaneously the transformations in the whole spacetime, since the parameterdoes not carry spacetime indices.7 Theorem 5 ( N -gauge transformations) In the BF theory for the -crossed module ( L δ → H ∂ → G, ✄ , { , } pf ) ,the following transformation is a symmetry α → α ′ = α , B → B ′ = B − β ∧ T ǫ n ,β → β ′ = β , C → C ′ = C − ∇ ǫ n ,γ → γ ′ = γ , D A → D ′ A = D A + δ Aa ǫ n a , (97) where ǫ n : M → h is an arbitrary h -valued -form. Proof.
Under the transformations defined in Theorem 5, the action is transformed as follows: S ′ BF = S BF + Z M d x ǫ µνρσ (cid:18) β bµν ✄ αa b ǫ n a F αρσ −
13! ( ∇ µ ǫ n a ) G aνρσ + 14! δ Aa ǫ n a H A µνρσ (cid:19) . (98)Using the expressions for the 3-curvature defined in (9), one obtains S ′ BF = S BF + Z M d x ǫ µνρσ (cid:16) β bµν ✄ αa b ǫ n a ( F αρσ − ∂ cα β cρσ ) −
13! ( ∇ µ ǫ n a ) (cid:0) ∇ ν β aρσ − δ Aa γ A νρσ (cid:1) + 14! δ Aa ǫ a (cid:0) ∇ µ γ A νρσ + 6 X ( bc ) A β bµν β cρσ (cid:1) (cid:17) . (99)Here, after one partial integration the last term in the first row of the equation (99) cancels with the first term in thesecond row, while taking into account the identity12 ǫ µνρσ ( ∇ ν ∇ µ ǫ n a ) β aρσ = − ǫ µνρσ β bρσ ✄ αa b ǫ n a F αµν , (100)the first term and the third term also cancel, leading to the following expression: S ′ BF = S BF + Z M d x ǫ µνρσ (cid:16) ǫ n a ✄ α ( b | a ∂ | c ) α β bµν β cρσ + 14 ǫ n a δ Aa X ( bc ) A β bµν β cρσ (cid:17) . (101)Here, the remaining two terms vanish because of the symmetrized form of the identity (A6): ✄ α ( b | a ∂ | c ) α + δ Aa X ( bc ) A = f ( bc ) a = 0 , as a consequence of the antisymmetric property of the structure constants. One concludes that the S BF action isinvariant under the transformations defined in Theorem 5.The N -gauge transformations defined in Theorem 5 define the group which will be denoted as ˜ N . Note that thesetransformations are also linear, and the composition of two N -gauge transformations gives one N -gauge transformationwith the parameter ǫ n + ǫ n . The generators of the group ˜ N will be denoted with ˆ N a , and one can write these resultsas: e ǫ n · ˆ N e ǫ n · ˆ N = e ( ǫ n + ǫ n ) · ˆ N , (102)where ǫ n · ˆ N = ǫ n a ˆ N a , leading to the conclusion that:[ ˆ N a , ˆ N b ] = 0 . (103)It follows that the group ˜ N is Abelian, and the index structure of parameters and generators indicates that it isisomorphic to R q , where q is the dimension of the group H . Therefore,˜ N ∼ = R q . (104)Next, one can examine the relationship of the N -gauge transformations with the G , H , L , and M -gauge transfor-mations. First, considering the G -gauge transformations one obtains:[ ǫ g · ˆ G, ǫ n · ˆ N ] = ( ǫ g ✄ ǫ n ) · ˆ N , (105)from which it follows: [ ˆ G α , ˆ N a ] = ✄ αab ˆ N b . (106)8Let us now examine the relationship between N -gauge transformations and H -gauge transformations, calculatingthe following expression: e ǫ h · ˆ H e ǫ n · ˆ N − e ǫ n · ˆ N e ǫ h · ˆ H = − ( ǫ n ∧ T ǫ h ) · ˆ M , (107)where the proof is given in Appendix F. One obtains that the commutator between the generators of H -gaugetransformation and N -gauge transformation is the generator of M -gauge transformation:[ ˆ H aµ , ˆ N b ] = ✄ αab ˆ M αµ . (108)Analogously, one can check that the following is satisfiede ǫ l · ˆ L e ǫ n · ˆ N = e ǫ n · ˆ N e ǫ l · ˆ L , e ǫ m · ˆ M e ǫ n · ˆ N = e ǫ n · ˆ N e ǫ m · ˆ M , (109)leading to the conclusion that the generators of L -gauge, M -gauge, and N -gauge transformations mutually commute,i.e., [ ˆ M αµ , ˆ N a ] = 0 , [ ˆ L Aµν , ˆ N a ] = 0 . (110)This concludes the calculation of the algebra of generators. D. Structure of the symmetry group
Summarizing the results of the previous subsections, one can write the algebra of the generators of the full gaugesymmetry group as follows. • The algebra g of the group G of the 2-crossed module ( L δ → H ∂ → G , ✄ , { , } pf ):[ ˆ G α , ˆ G β ] = f αβγ ˆ G γ . (111) • The algebra of the group ˜ H L consisting of the generators of H - and L -gauge transformations:[ ˆ H aµ , ˆ H bν ] = 2 X ( ab ) A ˆ L Aµν , [ ˆ L Aµν , ˆ L Bρσ ] = 0 , [ ˆ H aµ , ˆ L Aνρ ] = 0 . (112) • The algebra of the generators of M -gauge transformations:[ ˆ M αµ , ˆ M βν ] = 0 . (113) • The algebra of the generators of N -gauge transformations:[ ˆ N a , ˆ N b ] = 0 . (114) • The commutators between the generators of the groups ˜ M and ˜ N :[ ˆ M αµ , ˆ N a ] = 0 . (115) • The action of the generators of the group ˜ H L on the generators of M - and N -gauge transformations:[ ˆ H aµ , ˆ N b ] = ✄ αab ˆ M αµ , [ ˆ H aµ , ˆ M αν ] = 0 , [ ˆ L Aνρ , ˆ M αµ ] = 0 , [ ˆ L Aµν , ˆ N a ] = 0 . (116)9 • The action of the generators of the group G on the generators of H -, L -, M -, and N -gauge transformations:[ ˆ G α , ˆ H aµ ] = ✄ αab ˆ H bµ , [ ˆ G α , ˆ L Aµν ] = ✄ αAB ˆ L Bµν , [ ˆ G α , ˆ M βµ ] = f αβγ ˆ M γµ , [ ˆ G α , ˆ N a ] = ✄ αab ˆ N b . (117)Based on the equations (111)-(117), one can investigate the symmetry group structure. On the Hesse-like diagramshown in Figure 1, we have included only the relevant subgroups of the whole symmetry group G BF , where theinvariant subgroups are boxed.Let us remember that the subgroup is an invariant subgroup , or equivalently a normal subgroup , if is invariantunder conjugation by members of the group of which it is a subgroup. Formally, one says the group H is an invariantsubgroup of the group G if H is a subgroup of G , i.e., H ≤ G , and for all h ∈ H and g ∈ G , the conjugation of theelement of H with the element of G is an element of H , i.e., ∃ h ′ ∈ H such that ghg − = h ′ . On the level of algebra,the corresponding object is an ideal . Formally written, an algebra A is a subalgebra of an algebra L with respect tothe multiplication in L , i.e., [ A, A ] ⊂ A . Then, a subalgebra A of L is an ideal in L if its elements, multiplied withany element of the algebra, give again an element of the subalgebra, i.e., [ A, L ] ⊂ A . ˜ G ⋉ ( ˜ H L ⋉ ( ˜ N × ˜ M ))˜ H L ⋉ ( ˜ N × ˜ M )˜ H L ˜ N × ˜ M ˜ L ˜ M ˜ N ˜ G { } FIG. 1: Relevant subgroups of the symmetry group G BF . The invariant subgroups are boxed. With the above definitions in mind, note first that the groups ˜ L , ˜ M , and ˜ N , are subgroups of the full symmetrygroup G BF . The groups ˜ L and ˜ M are invariant subgroups, as the only nontrivial commutators between the generatorsˆ L Aµν , and ˆ M αµ , are with the generators of the group ˜ G , and are equal to some linear combinations of the generatorsof ˜ L , and ˜ M , respectively. The group ˜ N is not an invariant subgroup, as the commutator between the generatorsˆ N a and ˆ H aµ are linear combinations of the generators ˆ M αµ . However, the generators of the groups ˜ N and ˜ M aremutually commuting, and the group ˜ N is an invariant subgroup of the product of the groups ˜ M and ˜ N , which makesthis product a direct product. The obtained group ˜ N × ˜ M is an invariant subgroup of the whole symmetry group.On the other hand, we saw that the H -gauge transformations together with the L -gauge transformations form thegroup ˜ H L . This group is not an invariant subgroup of the whole symmetry group G BF , because of the commutator ofthe generators ˆ H aµ and ˆ N b . Similarly as before, one can join these two subgroups, of which one is invariant and oneis not, using a semidirect product to obtain a subgroup ˜ H L ⋉ ( ˜ N × ˜ M ), that will as a result be an invariant subgroupof the complete symmetry group G BF . Here, the product is semidirect because the group ˜ H L is not an invariantsubgroup of the group ˜ H L ⋉ ( ˜ N × ˜ M ), due to the commutator between the generators ˆ H aµ and ˆ N b .0Finally, following the same line of reasoning, one adds the G -gauge transformations and obtains the complete gaugesymmetry group G BF as: G BF = ˜ G ⋉ ( ˜ H L ⋉ ( ˜ N × ˜ M )) . (118)This concludes the analysis of the group of gauge symmetries for the 3 BF action. V. CONCLUSIONSA. Summary of the results
Let us summarize the results of the paper. In Section II, we have introduced a generalization of the BF theoryin the framework of higher category theory, the 3 BF theory. Section III contains the Hamiltonian analysis for the3 BF theory. In subsection III A, the basic canonical structure and the total Hamiltonian are obtained, while insubsection III B the complete Hamiltonian analysis of the 3 BF theory is performed, resulting in the first-class andsecond-class constraints of the theory, as well as their Poisson brackets. In the subsection III C we have discussed theBianchi identities and also the generalized Bianchi identities, since they enforce restrictions and reduce the numberof independent first-class constraints present in the theory, and having those identities in mind, the counting of thedynamical degrees of freedom has been performed. As expected, it was established that the considered 3 BF action isa topological theory. Finally, this section concludes with the subsection III D where we have constructed the generatorof the gauge symmetries for the topological theory, based on the calculations done in section III B, and we have foundthe form-variations for all the variables and their canonical momenta, listed in the Appendix E, equations (E2).In section IV, the main results of our paper are presented. With the material of the subsection III B in hand,after obtaining the form variations of all variables and their canonical momenta, we proceeded to find all the gaugesymmetries of the theory. The subsection IV A examined the gauge group G , and the already familiar G -gaugetransformations. In subsection IV B we discussed the gauge group ˜ H L which gives the already familiar H -gauge and L -gauge transformations, while in the subsection IV C we analyzed the M -gauge and N -gauge transformations whichrepresent a novel result. The results of the subsections IV A, IV B, and IV C are summarized in subsection IV D, wherethe complete structure of the symmetry group had been presented. The known G -, H -, and L -gauge transformationshave been rigorously defined in Theorems 1, 2, and 3, while the two novel M - and N -gauge transformations, havebeen defined in Theorems 4 and 5. The Lie algebra of the full gauge symmetry group G BF has also been obtained. B. Discussion
From the fact that the 3 BF action is formulated in a manifestly covariant way, using differential forms, it shouldbe obvious that the diffeomorphisms are a symmetry of the theory. However, by looking at the structure of the gaugegroup G BF , one does not immediately see whether Diff ( M , R ) is its subgroup. In fact, this issue is subtle, and itdeserves some discussion.It is easy to see that every action, which depends on at least two fields φ ( x ) and φ ( x ), is invariant underthe following transformation, determined by the Henneaux-Teitelboim (HT) parameter ǫ HT (see [29] for details andnaming), δ φ = ǫ HT ( x ) δSδφ , δ φ = − ǫ HT ( x ) δSδφ , (119)which can be easily verified by calculating the variation of the action: δ HT S [ φ , φ ] = δSδφ δ φ + δSδφ δ φ = 0 . (120)As this invariance is present even in theories with no gauge symmetry, it is not associated with constraints, and thusnot present in the generator of gauge symmetries (55), see [29] for details.Now, let us consider the diffeomorphism transformation x µ → x ′ µ = x µ + ξ µ ( x ) , (121)where the parameter ξ µ ( x ) is an arbitrary function, which we will consider to be infinitesimal. Also, let us denoteall parameters of the gauge group collectively as ǫ i ( x ). If diffeomorphisms are a symmetry of the action, then for1every field φ ( x ) in the theory, and every parameter of the diffeomorphisms ξ µ ( x ), there should exist a choice of theparameters ǫ i ( x ) and ǫ HT ( x ), such that: ( δ + δ + δ ) φ = 0 . (122)In other words, if the diffeomorphisms are a symmetry of the theory, their form variations should be expressible asgauge form variations combined with HT form variations: δ φ = − δ φ − δ φ . (123)In our case, the 3 BF action depends on the fields α αµ , β aµν , γ Aµνρ , B αµν , C aµ , and D A . The HT parameters ǫ HT αβµνρ , ǫ HT abµνρ , and ǫ HT ABµνρ are defined via the following form variations, analogous to (119): δ α αµ = 12 ǫ HT αβµνρ δSδB βνρ , δ B αµν = − ǫ HT αβρµν δSδα β ρ ,δ β aµν = ǫ HT abµνρ δSδC bρ , δ C aµ = − ǫ HT abνρµ δSδβ bνρ ,δ γ Aµνρ = ǫ HT ABµνρ δSδD B , δ D A = − ǫ HT ABµνρ δSδγ
Bµνρ , (124)while the gauge parameters ǫ g α , ǫ h aµ , ǫ l Aµν , ǫ m αµ , and ǫ n a are defined in Theorems 1–5. Given these, there indeedexists a choice of these parameters, such that (122) is satisfied for all fields. Specifically, if one chooses the gaugeparameters as ǫ g α = − ξ λ α αλ , ǫ h aµ = ξ λ β aµλ , ǫ l Aµν = ξ λ γ Aµνλ , ǫ m αµ = ξ λ B αµλ , ǫ n a = − ξ λ C aλ , (125)and the HT parameters as ǫ HT αβµνρ = ξ λ g αβ ǫ µνρλ , ǫ HT abµνρ = ξ λ g ab ǫ λµνρ , ǫ HT ABµνρ = ξ λ g AB ǫ µνρλ , (126)one can obtain, using (123), precisely the standard form variations corresponding to diffeomorphisms: δ α αµ = − ∂ µ ξ λ α αλ − ξ λ ∂ λ α αµ ,δ β aµν = − ∂ µ ξ λ β aλν − ∂ ν ξ λ β aµλ − ξ λ ∂ λ β aµν ,δ γ Aµνρ = − ∂ µ ξ λ γ Aλνρ − ∂ ν ξ λ γ Aµλρ − ∂ ρ ξ λ γ Aµνλ − ξ λ ∂ λ γ Aµνρ ,δ B αµν = − ∂ µ ξ λ B αλν − ∂ ν ξ λ B αµλ − ξ λ ∂ λ B αµν ,δ C aµ = − ∂ µ ξ λ C aλ − ξ λ ∂ λ C aµ ,δ D A = − ξ λ ∂ λ D A . (127)This establishes that diffeomorphisms are indeed contained in the full gauge symmetry group G BF , up to the HTtransformations, which are always a symmetry of the theory. C. Future lines of investigation
Based on the results obtained in this work, one can imagine various additional topics for further research.First, since we have obtained that the pure 3 BF theory is a topological theory, it does not describe a realistic physicaltheory which ought to contain local propagating degrees of freedom. To build a realistic physical theory, one introducesthe degrees of freedom by imposing the simplicity constraints on the topological action. In our previous work [13], wehave formulated the classical actions that manifestly distinguish the topological sector from the simplicity constraints,for all the fields present in the Standard Model coupled to Einstein-Cartan gravity. Specifically, we have defined theconstrained 2 BF actions describing the Yang-Mills field and Einstein-Cartan gravity, and also the constrained 3 BF actions describing the Klein-Gordon, Dirac, Weyl and Majorana fields coupled to gravity in the standard way. Thenatural continuation of this line of research would be the Hamiltonian analysis of all such constrained 3 BF modelsof gravity coupled to various matter fields, and the study of their canonical quantization.On the other hand, as an alternative to the canonical quantization, one may choose the spinfoam quantizationapproach, and define the path integral of the theory as the state sum for the Regge-discretized 3 BF action. Thetopological nature of the 3 BF action, together with the structure of the gauge 3-group, should ensure that such a sumis a topological invariant, i.e., that it is triangulation independent. Unfortunately, in order to rigorously define this2state sum, one needs the higher category generalizations of the Peter-Weyl and Plancherel theorems, from ordinarygroups to the cases of 2-groups and 3-groups. These theorems ought to determine the domains of various labels livingon simplices of the triangulation, as a consequence of the representation theory of 3-groups. Until these mathematicalresults are obtained, one can try to guess the appropriate structure of the irreducible representations of a 3-groupand construct the topological invariant Z for the 3 BF topological action, in analogy with what was done in the caseof 2 BF theory, see [23, 25]. Once the topological state sum is obtained, one can proceed to impose the simplicityconstraints, and thus construct the state sum corresponding to the tentative quantum theory of gravity with matter.The classical action for gravity and matter is formulated in [13] in a way that explicitly distinguishes between thetopological sector and the simplicity constraints sector of the action, making the procedure of imposing the constraintsstraightforward.Next, it would be useful to investigate in more depth the mathematical structure and properties of the simplicityconstraints, in particular their role as the gauge fixing conditions for the symmetry group G BF . The simplicityconstraints should explicitly break the symmetry group G BF to the subgroup corresponding to the constrained 3 BF theory, which may then be further spontaneously broken by the Higgs mechanism.One of the results obtained in this work is a duality between the gauge symmetry group of the 3 BF action, G BF ,and the underlining 3-group, i.e., the 2-crossed module ( L δ → H ∂ → G , ✄ , { , } pf ). This duality should be betterunderstood. On one hand, the group G BF can provide further insight into the construction of the TQFT state sum,i.e., a topological invariant corresponding to the underlining 3-group structure. On the other hand, this duality isinteresting from the perspective of pure mathematics, since it can provide deeper insight in the structure of 3-groups.Finally, in [28] it was pointed out that it may be useful to make one more step in the categorical generalization,and consider a 4 BF theory as a description of a quantum gravity model with matter fields. One could then calculatethe gauge group of the 4 BF action, and compare the results with the results obtained for the 3 BF theory.The list is not conclusive, and there may be many other interesting topics to study. Acknowledgments
This research was supported by the Ministry of Education, Science and Technological Development of the Republicof Serbia (MPNTR), and by the Science Fund of the Republic of Serbia, Program DIASPORA, No. 6427195, SQ2020.The contents of this publication are the sole responsibility of the authors and can in no way be taken to reflect theviews of the Science Fund of the Republic of Serbia.
Appendix A: -crossed module Definition (Differential 2-crossed module)
A differential -crossed module is given by an exact sequence of Liealgebras: l δ → h ∂ → g , together with left action ✄ of g on g , h , and l , by derivations, and on itself via adjoint representation, and a g -equivariant bilinear map called the Peiffer lifting : { , } pf : h × h → l . Fixing the basis in the algebras as T A ∈ l , t a ∈ h and τ α ∈ g : [ T A , T B ] = f ABC T C , [ t a , t b ] = f abc t c , [ τ α , τ β ] = f αβγ τ γ , one defines the maps ∂ and δ as: ∂ ( t a ) = ∂ aα τ α , δ ( T A ) = δ Aa t a , and the action of g on the generators of l , h , and g is, respectively: τ α ✄ T A = ✄ αAB T B , τ α ✄ t a = ✄ αab t b , τ α ✄ τ β = ✄ αβγ τ γ . The coefficients X abA are introduced as: { t a , t b } pf = X abA T A . The maps ∂ and δ satisfy the following identity: ∂ aα δ Aa = 0 . (A1) Note that when η is a g -valued differential form and ω is l − , h − , or g − valued differential form, the previous actionis defined as: η ∧ ✄ ω = η α ∧ ω A ✄ αA B T B , η ∧ ✄ ω = η α ∧ ω a ✄ αa b t b , η ∧ ✄ ω = η α ∧ ω β f αβγ τ γ , where the forms are multiplied via the wedge product ∧ , while the generators of G act on the generators of the threegroups via the action ✄ .The following identities are satisfied:1. In the differential crossed module ( L δ → H , ✄ ′ ) the action ✄ ′ of h on l is defined for each h ∈ h and l ∈ l as: h ✄ ′ l = −{ δ ( l ) , h } pf , or written in the basis where t a ✄ ′ T A = ✄ ′ aAB T B the previous identity becomes: ✄ ′ aAB = − δ Ab X baB ; (A2)
2. The action of g on itself is via adjoint representation: ✄ αβγ = f αβγ ; (A3)
3. The action of g on h and l is equivariant, i.e., the following identities are satisfied: ∂ aβ f αβγ = ✄ αab ∂ bγ , δ Aa ✄ αa b = ✄ αAB δ Bb ; (A4)
4. The Peiffer lifting is g -equivariant, i.e., for each h , h ∈ h and g ∈ g : g ✄ { h , h } pf = { g ✄ h , h } pf + { h , g ✄ h } pf , or written in the basis: X abB ✄ αBA = ✄ αac X cbA + ✄ αbc X acA ; (A5) δ ( { h , h } pf ) = h h , h i p , ∀ h , h ∈ h .The map ( h , h ) ∈ h × h → h h , h i p ∈ h is bilinear g -equivariant map called the Peiffer paring , i.e., all h , h ∈ h and g ∈ g satisfy the following identity: g ✄ h h , h i p = h g ✄ h , h i p + h h , g ✄ h i p . Fixing the basis the identity becomes: X abA δ Ac = f abc − ∂ aα ✄ αbc ; (A6) [ l , l ] = { δ ( l ) , δ ( l ) } pf , ∀ l , l ∈ l , i.e., f ABC = δ Aa δ Bb X abC ; (A7) { [ h , h ] , h } pf = ∂ ( h ) ✄ { h , h } pf + { h , [ h , h ] } pf − ∂ ( h ) ✄ { h , h } pf −{ h , [ h , h ] } pf , ∀ h , h , h ∈ h , i.e., { [ h , h ] , h } pf = { ∂ ( h ) ✄ h , h } pf − { ∂ ( h ) ✄ h , h } pf − { h , δ { h , h } pf } pf + { h , δ { h , h } pf } pf ,f abd X dcB = ∂ aα X bcA ✄ αAB + X adB f bcd − ∂ bα ✄ αAB X acA − X bdB f acd ; (A8)4 { h , [ h , h ] } pf = { δ { h , h } pf , h } pf − { δ { h , h } pf , h } pf , ∀ h , h , h ∈ h , i.e., X adA f bcd = X abB δ Bd X dcA − X acB δ Bd X dbA ; (A9) { δ ( l ) , h } pf + { h, δ ( l ) } pf = − ∂ ( h ) ✄ l , ∀ l ∈ l , ∀ h ∈ h , i.e., δ Aa X abB + δ Aa X baB = − ∂ bα ✄ αAB . (A10)A reader interested in more details about 3-groups is referred to [27].The structure constants satisfy the Jacobi identities f αγδ f βǫγ = 2 f α [ β | γ f γ | ǫ ] δ , f adc f bed = 2 f a [ b | d f d | e ] c , f ADC f BED = 2 f A [ B | D f D | E ] C . (A11)Also, the following relations are useful: f βγα ✄ αb a = 2 ✄ [ β | c a ✄ | γ ] bc , f βγα ✄ αB A = 2 ✄ [ β | C A ✄ | γ ] BC . (A12) Appendix B: Additional relations of the constraint algebra
In this Appendix the useful technical results used in the subsection III B are given. First, as the secondary con-straints, given by the equations (30), must be preserved during the evolution of the system, the consistency conditionsof secondary constraints must be enforced. However, no tertiary constraints arise from these conditions, as one obtainsthe following PB: {S ( F ) αi , H T } = f βγα S ( F ) β i α γ , {S ( ∇ B ) α , H T } = f βγα B γ k S ( F ) β k + f βαγ α β S ( ∇ B ) γ + C a ✄ αb a S ( G ) b − ✄ αa b β a k S ( ∇ C ) bk + 12 ✄ α BA γ A jk S ( ∇ D ) Bjk , {S ( G ) a , H T } = ✄ αba β b k S ( F ) αk − α α ✄ αb a S ( G ) b , {S ( ∇ C ) ai , H T } = C b ✄ α ba S ( F ) αi + ✄ αab α α S ( ∇ C ) bi + 2 X ( ab ) A β b j S ( ∇ D ) Aij , {S ( ∇ D ) Aij , H T } = α α ✄ αA B S ( ∇ D ) B ij . (B1)The PB between the first-class constraints, given by the equations (34), and the second-class constraints, given by5the equations (35), are given by: { Φ( F ) αi ( ~x ) , χ ( α ) β j ( ~y ) } = − f βγα χ ( B ) γ ij ( ~x ) δ (3) ( ~x − ~y ) , { Φ( G ) a ( ~x ) , χ ( α ) αi ( ~y ) } = ✄ αba χ ( C ) bi ( ~x ) δ (3) ( ~x − ~y ) , { Φ( G ) a ( ~x ) , χ ( β ) bij ( ~y ) } = − ✄ αba χ ( B ) αij ( ~x ) δ (3) ( x − y ) , { Φ( ∇ C ) ai ( ~x ) , χ ( α ) αj ( ~y ) } = − ✄ αba χ ( β ) bij ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ C ) ai ( ~x ) , χ ( β ) bjk ( ~y ) } = 2 X ( ac ) A g bc χ ( γ ) Aijk ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ C ) ai ( ~x ) , χ ( C ) bj ( ~y ) } = ✄ αba χ ( B ) αij ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ C ) ai ( ~x ) , χ ( D ) A ( ~y ) } = 2 X ( ab ) A χ ( C ) bi ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ B ) α ( ~x ) , χ ( α ) β i ( ~y ) } = f βγα χ ( α ) γi ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ B ) α ( ~x ) , χ ( β ) aij ( ~y ) } = g αβ ✄ βab χ ( β ) bij ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ B ) α ( ~x ) , χ ( γ ) Aijk ( ~y ) } = g αβ ✄ βAB χ ( γ ) Bijk ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ B ) α ( ~x ) , χ ( B ) βij ( ~y ) } = f βγα χ ( B ) γij ( ~x ) δ (3) ( ~x − ~y ) . { Φ( ∇ B ) α ( ~x ) , χ ( C ) ai ( ~y ) } = − ✄ αba χ ( C ) bi ( ~x ) δ (3) ( ~x − ~y ) . { Φ( ∇ B ) α ( ~x ) , χ ( D ) A ( ~y ) } = g αβ ✄ βAB χ ( D ) B ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ D ) Aij ( ~x ) , χ ( α ) αk ( ~y ) } = ✄ αBA χ ( γ ) Bijk ( ~x ) δ (3) ( ~x − ~y ) , { Φ( ∇ D ) Aij ( ~x ) , χ ( D ) B ( ~y ) } = − ✄ αB A χ ( B ) αij ( ~x ) δ (3) ( ~x − ~y ) . (B2)Finally, it is useful to calculate PB between the first-class constraints, given by the equations (34), and the totalHamiltonian, given by the equation (33): { Φ( F ) αi , H T } = f βγα Φ( F ) β i α γ , { Φ( ∇ B ) α , H T } = f βγα B γ k Φ( F ) βk + f βαγ α β Φ( ∇ B ) γ + C a ✄ αb a Φ( G ) b − ✄ αa b β a k Φ( ∇ C ) bk + 12 ✄ α BA γ A jk Φ( ∇ D ) Bjk , { Φ( G ) a , H T } = ✄ αba β b k Φ( F ) αk − α α ✄ αb a Φ( G ) b , { Φ( ∇ C ) ai , H T } = C b ✄ α ba Φ( F ) αi + ✄ αab α α Φ( ∇ C ) bi + 2 X ( ab ) A β b j Φ( ∇ D ) Aij , { Φ( ∇ D ) Aij , H T } = α α ✄ αA B Φ( ∇ D ) Bij . (B3)The calculated PB brackets given by the equation (B3) will be useful for calculation of the generator of gaugesymmetries (55). With these results one can proceed to the construction of the gauge symmetry generator on onehypersurface Σ given in the equation (55), and ultimately obtain the finite gauge symmetry of the whole spacetime.The PB algebra of gauge symmetry generators ( ˜ M ) αi , ( ˜ M ) αi , ( ˜ G ) α , ( ˜ G ) α , ( ˜ H ) ai , ( ˜ H ) ai , ( ˜ N ) a , ( ˜ N ) a , ( ˜ L ) Aij ,and ( ˜ L ) Aij , as defined in (56), is: { ( ˜ G ) α ( ~x ) , ( ˜ G ) β ( ~y ) } = f αβγ ( ˜ G ) γ δ (3) ( ~x − ~y ) , (B4) { ( ˜ H ) ai ( ~x ) , ( ˜ H ) bj ( ~y ) } = 2 X ( ab ) A ( ˜ L ) Aij δ (3) ( ~x − ~y ) , { ( ˜ H ) ai ( ~x ) , ( ˜ H ) bj ( ~y ) } = 2 X ( ab ) A ( ˜ L ) Aij δ (3) ( ~x − ~y ) , (B5)6 ( ˜ G × ˜ G ) ⋉ (cid:16) ˜ H L ⋉ (( ˜ N × ˜ N ) × ( ˜ M × ˜ M )) (cid:17) ˜ H L ⋉ (( ˜ N × ˜ N ) × ( ˜ M × ˜ M ))˜ H L Σ ( ˜ N × ˜ N ) × ( ˜ M × ˜ M )˜ L × ˜ L ˜ M × ˜ M ˜ N × ˜ N ˜ G × ˜ G { } FIG. 2: The symmetry group G Σ of the Poisson bracket algebra in the phase space. The invariant subgroups are boxed. { ( ˜ H ) ai ( ~x ) , ( ˜ N ) b ( ~y ) } = ✄ αab ( ˜ M ) αi δ (3) ( ~x − ~y ) , { ( ˜ H ) ai ( ~x ) , ( ˜ N ) b ( ~y ) } = ✄ αab ( ˜ M ) αi δ (3) ( ~x − ~y ) , { ( ˜ H ) a ( ~x ) , ( ˜ N ) bi ( ~y ) } = ✄ αab ( ˜ M ) αi δ (3) ( ~x − ~y ) , (B6) { ( ˜ G ) α ( ~x ) , ( ˜ M ) βi ( ~y ) } = f αβγ ( ˜ M ) γi δ (3) ( ~x − ~y ) , { ( ˜ G ) α ( ~x ) , ( ˜ M ) βi ( ~y ) } = f αβγ ( ˜ M ) γi δ (3) ( ~x − ~y ) , { ( ˜ G ) α ( ~x ) , ( ˜ H ) ai ( ~y ) } = ✄ αab ( ˜ H ) bi ( ~x ) δ (3) ( ~x − ~y ) , { ( ˜ G ) α ( ~x ) , ( ˜ H ) ai ( ~y ) } = ✄ αab ( ˜ H ) bi ( ~x ) δ (3) ( ~x − ~y ) , { ( ˜ G ) α ( ~x ) , ( ˜ N ) a ( ~y ) } = ✄ αab ( ˜ N ) b ( ~x ) δ (3) ( ~x − ~y ) , { ( ˜ G ) α ( ~x ) , ( ˜ N ) a ( ~y ) } = ✄ αab ( ˜ N ) b ( ~x ) δ (3) ( ~x − ~y ) , { ( ˜ G ) α ( ~x ) , ( ˜ L ) Aij ( ~y ) } = ✄ αAB ( ˜ L ) Bij ( ~x ) δ (3) ( ~x − ~y ) . (B7)The gauge symmetry group has the following structure. First, the groups ˜ M × ˜ M , ˜ N × ˜ N and ˜ L × ˜ L with thecorresponding algebras a , a and a , respectively, where: a = span { ( ˜ M ) αi } ⊕ span { ( ˜ M ) αi } , a = span { ( ˜ N ) a } ⊕ span { ( ˜ N ) a } , a = span { ( ˜ L ) Aij } ⊕ span { ( ˜ L ) Aij } , (B8)are the subgroups of the full symmetry group ˜ G Σ . Besides, the subgroups ˜ L × ˜ L and ˜ M × ˜ M are the invariantsubgroups. The group ˜ N × ˜ N is not an invariant subgroup of the whole symmetry group, as the Poisson brackets7 { ( ˜ H ) ai ( ~x ) , ( ˜ N ) b ( ~y ) } and { ( ˜ H ) ai ( ~x ) , ( ˜ N ) b ( ~y ) } are equal to some linear combinations of the generators of ˜ M × ˜ M .Nevertheless, one can form a direct product ( ˜ N × ˜ N ) × ( ˜ M × ˜ M ), as the generators of these groups are mutuallycommuting, giving a group which is an invariant subgroup of the complete symmetry group.Next, consider a subgroup ˜ H L Σ determined by the algebra spanned by the generators ( ˜ L ) Aij , ( ˜ L ) Aij , ( ˜ H ) ai , and( ˜ H ) ai . This group is not invariant subgroup of the whole symmetry group, because of the PB { ( ˜ H ) ai ( ~x ) , ( ˜ N ) b ( ~y ) } and { ( ˜ H ) ai ( ~x ) , ( ˜ N ) b ( ~y ) } , due to the same argument as before. Now, one can join these two subgroups, of whichone is invariant and one is not, using a semidirect product into an invariant subgroup H L ⋉ (( N × N ) × ( M × M )),determined by the algebra a : a = span { ( ˜ M ) αi , ( ˜ M ) αi , ( ˜ H ) ai , ( ˜ H ) ai , ( ˜ N ) a , ( ˜ N ) a , ( ˜ L ) Aij , ( ˜ L ) Aij } . . Finally, following the same line of reasoning, one adds the group ˜ G × ˜ G and obtains the full gauge symmetrygroup ˜ G Σ to be equal to: ˜ G Σ = ( ˜ G × ˜ G ) ⋉ (cid:16) ˜ H L ⋉ (( ˜ N × ˜ N ) × ( ˜ M × ˜ M )) (cid:17) . The complete symmetry group structure is shown in the Figure B. Here, the invariant subgroups of the wholesymmetry group are boxed.
Appendix C: Construction of the symmetry generator
When one substitutes the generators (56) into the equation (55), one obtains the gauge generator of the 3 BF theoryin the following form G = − Z Σ d ~x (cid:18) ( ∇ ǫ m αi )Φ( B ) αi − ǫ m αi Φ( F ) αi + ( ∇ ǫ g α )Φ( α ) α + ǫ g α (cid:0) f αγβ B β i Φ( B ) γ i + C a ✄ αb a Φ( C ) b + β a i ✄ αb a Φ( β ) b i − γ A ij ✄ αA B Φ( γ ) Bij − Φ( ∇ B ) α (cid:1) + ( ∇ ǫ n a )Φ( C ) a − ǫ n a (cid:0) β b i ✄ αa b Φ( B ) αi + Φ( G ) a (cid:1) + ( ∇ ǫ h ai )Φ( β ) ai − ǫ h ai (cid:0) C b ✄ αa b Φ( B ) αi − β b j X ( ab ) A Φ( γ ) Aij + Φ( ∇ C ) ai (cid:1) −
12 ( ∇ ǫ l Aij )Φ( γ ) Aij + 12 ǫ l Aij Φ( ∇ D ) Aij (cid:19) , (C1)where ǫ g α , ǫ h ai , ǫ l Aij , ǫ m αi , and ǫ n a are the independent parameters of the gauge transformations.The generator of gauge transformations (C1) in 3 BF theory given by the action (15), is obtained by the Castellani’sprocedure, requiring the following requirements to be met G = C P F C ,G + { G , H T } = C P F C , { G , H T } = C P F C , (C2)where C P F C denotes some first-class constraints, and assuming that the generator has the following structure: G = Z Σ d ~x (cid:16) ˙ ǫ m αi ( G ) m αi + ǫ m αi ( G ) m αi + ˙ ǫ g α ( G ) g α + ǫ g α ( G ) g α + ˙ ǫ h ai ( G ) h ai + ǫ h ai ( G ) h ai + ˙ ǫ n a ( G ) n a + ǫ n a ( G ) n a + 12 ˙ ǫ l Aij ( G ) l Aij + 12 ǫ l Aij ( G ) l Aij (cid:17) . (C3)The first step of Castellani’s procedure, imposing the set of conditions( G ) m αi = C P F C , ( G ) g α = C P F C , ( G ) h ai = C P F C , ( G ) n a = C P F C , ( G ) l Aij = C P F C , (C4)8is satisfied with a natural choice:( G ) m αi = − Φ( B ) αi , ( G ) g α = − Φ( α ) α , ( G ) h ai = − Φ( C ) αi , ( G ) n a = − Φ( β ) a , ( G ) l Aij = Φ( γ ) Aij . (C5)It remains to determine the five generators G .The Castellani’s second condition for the generator ( G ) m αi gives:( G ) m αi − { Φ( B ) αi , H T } = ( C P F C ) αi , ( G ) m αi − Φ( F ) αi = ( C P F C ) αi , (C6)that is ( G ) m αi = ( C P F C ) αi + Φ( F ) αi . Subsequently, from the Castellani’s third condition it follows { ( G ) m αi , H T } = ( C P F C ) αi , { ( C P F C ) αi + Φ( F ) αi , H T } = ( C P F C ) αi , { ( C P F C ) αi , H T } − f βγα α β Φ( F ) γi = ( C P F C ) αi , (C7)which gives ( C P F C ) αi = f βγα α β Φ( B ) γi . It follows that the generator is: ( G ) m αi = f βγα α β Φ( B ) γi + Φ( F ) αi . (C8)The Castellani’s second condition for the generator ( G ) g α gives:( G ) g α − { Φ( α ) α , H T } = ( C P F C ) α , ( G ) g α − Φ( ∇ B ) α = ( C P F C ) α , (C9)that is ( G ) g α = ( C P F C ) α + Φ( ∇ B ) α . Subsequently, from the Castellani’s third condition it follows { ( G ) g α , H T } = ( C P F C ) α , { ( C P F C ) α + Φ( ∇ B ) α , H T } = ( C P F C ) α , { ( C P F C ) α , H T } + B β i f αγβ Φ( F ) γi − α β f αβγ Φ( ∇ B ) γ + C a ✄ αb a Φ( G ) b + β a i ✄ αb a Φ( ∇ C ) bi − γ A ij ✄ αA B Φ( ∇ D ) Bij = ( C P F C ) α , (C10)which gives( C P F C ) α = − B β i f αγβ Φ( B ) γi + α β f αβγ Φ( α ) γ − C a ✄ αb a Φ( C ) b − β a i ✄ αb a Φ( β ) bi + 12 γ A ij ✄ αA B Φ( γ ) Bij . It follows that the generator is:( G ) g α = − B β i f αγβ Φ( B ) γi + α β f αβγ Φ( α ) γ − C a ✄ αb a Φ( C ) b − β a i ✄ αb a Φ( β ) bi + 12 γ A ij ✄ αA B Φ( γ ) Bij + Φ( ∇ B ) α . (C11)The Castellani’s second condition for the generator ( G ) n a gives( G ) n a − { Φ( C ) a , H T } = ( C P F C ) a , ( G ) n a − Φ( G ) a = ( C P F C ) a , (C12)9that is ( G ) n a = ( C P F C ) a + Φ( G ) a . Subsequently, from the Castellani’s third condition it follows { ( G ) n a , H T } = ( C P F C ) a , { ( C P F C ) a + Φ( G ) a , H T } = ( C P F C ) a , { ( C P F C ) a , H T } + α α ✄ αa b Φ( G ) b − β b i ✄ αa b Φ( F ) αi = ( C P F C ) a , (C13)which gives ( C P F C ) a = − α α ✄ αa b Φ( C ) b + β b i ✄ αa b Φ( B ) αi . It follows that the generator is:( G ) n a = − α α ✄ αa b Φ( C ) b + β b i ✄ αa b Φ( B ) αi + Φ( G ) a . The Castellani’s second condition for the generator ( G ) h ai gives:( G ) h ai − { Φ( β ) ai , H T } = ( C P F C ) ai , ( G ) h ai − Φ( ∇ C ) ai = ( C P F C ) ai , (C14)that is ( G ) h ai = ( C P F C ) ai + Φ( ∇ C ) ai . Subsequently, from the Castellani’s third condition it follows { ( G ) h ai , H T } = ( C P F C ) ai , { ( C P F C ) ai + Φ( ∇ C ) ai , H T } = ( C P F C ) ai , { ( C P F C ) ai , H T } + α α ✄ αa b Φ( ∇ C ) bi − C b ✄ αa b Φ( F ) αi + 2 β b j X ( ab ) A Φ( ∇ D ) Aij = ( C P F C ) ai , (C15)which gives ( C P F C ) ai = − α α ✄ αa b Φ( β ) bi + C b ✄ αa b Φ( B ) αi − β b j X ( ab ) A Φ( γ ) Aij . It follows that the generator is:( G ) h ai = − α α ✄ αa b Φ( β ) bi + C b ✄ αa b Φ( B ) αi − β b j X ( ab ) A Φ( γ ) Aij + Φ( ∇ C ) ai . The Castellani’s second condition for the generator ( G ) l Aij gives:( G ) l Aij + { Φ( γ ) Aij , H T } = ( C P F C ) Aij , ( G ) l Aij + Φ( ∇ D ) Aij = ( C P F C ) Aij , (C16)that is ( G ) l Aij = ( C P F C ) Aij − Φ( ∇ D ) Aij . Subsequently, from the Castellani’s third condition it follows: { ( G ) l Aij , H T } = ( C P F C ) Aij , { ( C P F C ) Aij − Φ( ∇ D ) Aij , H T } = ( C P F C ) Aij , { ( C P F C ) Aij , H T } − α α ✄ αA B Φ( ∇ D ) Bij = ( C P F C ) Aij , (C17)which gives ( C P F C ) Aij = α α ✄ αA B Φ( γ ) Bij . It follows that the generator is: ( G ) l Aij = α α ✄ αA B Φ( γ ) Bij − Φ( ∇ D ) Aij . (C18)0At this point, it is useful to summarize the results, and introduce the new notation:˙ ǫ m αi ( G ) m αi + ǫ m αi ( G ) m αi = −∇ ǫ m αi Φ( B ) αi + ǫ m αi Φ( F ) αi = ∇ ǫ m αi ( ˜ M ) αi + ǫ m αi ( ˜ M ) αi . (C19)Note that the time derivative of the parameter combines with some of the other terms into a covariant derivative inthe time directions.For the second part of the total generator one obtains:˙ ǫ g α ( G ) g α + ǫ g α ( G ) g α = − ˙ ǫ g α Φ( α ) α − ǫ g α (cid:0) B β i f αγβ Φ( B ) γi − α β f αβγ Φ( α ) γ + C a ✄ αb a Φ( C ) b + β a i ✄ αb a Φ( β ) bi − γ A ij ✄ αA B Φ( γ ) Bij − Φ( ∇ B ) α (cid:1) = −∇ ǫ g α Φ( α ) α − ǫ g α (cid:0) B β i f αγβ Φ( B ) γi + C a ✄ αb a Φ( C ) b + β a i ✄ αb a Φ( β ) bi − γ A ij ✄ αA B Φ( γ ) Bij − Φ( ∇ B ) α (cid:1) = ∇ ǫ g α ( ˜ G ) α + ǫ g α ( ˜ G ) α . (C20)Furthermore, it follows:˙ ǫ h ai ( G ) h ai + ǫ h ai ( G ) h ai = −∇ ǫ h ai Φ( β ) αi + ǫ h ai (cid:0) C b ✄ αa b Φ( B ) αi − β b j X ( ab ) A Φ( γ ) Aij + Φ( ∇ C ) ai (cid:1) = ∇ ǫ h ai ( ˜ H ) ai + ǫ h ai ( ˜ H ) ai , (C21)˙ ǫ n a ( G ) n a + ǫ n a ( G ) n a = −∇ ǫ n a Φ( C ) a + ǫ n a ( β b i ✄ αa b Φ( B ) αi + Φ( G ) a )= ∇ ǫ n a ( ˜ N ) a + ǫ n a ( ˜ N ) a . (C22)Finally, one gets:12 ˙ ǫ l Aij ( G ) l Aij + 12 ǫ l Aij ( G ) l Aij = 12 ˙ ǫ l Aij Φ( γ ) Aij + 12 ǫ l Aij α α ✄ αA B Φ( γ ) Bij − ǫ l Aij Φ( ∇ D ) Aij = 12 ∇ ǫ l Aij Φ( γ ) Aij − ǫ l Aij Φ( ∇ D ) Aij = 12 ∇ ǫ l Aij ( ˜ L ) Aij + 12 ǫ l Aij ( ˜ L ) Aij . (C23) Appendix D: Definitions of maps T , S , D , X , and X Given G -invariant symmetric non-degenerate bilinear forms in g and h , one can define a bilinear antisymmetricmap T : h × h → g by the rule: hT ( h , h ) , g i g = −h h , g ✄ h i h , ∀ h , h ∈ h , ∀ g ∈ g . Written in basis: T ( t a , t b ) = T abα τ α , where the components of the map T are: T abα = − g ac ✄ βb c g αβ . See [24] for more properties and the construction of 2 BF invariant topological action using this map.The transformations of the Lagrange multipliers and the 3 BF invariant topological action is defined via maps S : l × l → g , X : l × h → h , X : l × h → h , D : h × h × l → g ., S : l × l → g is defined by the rule: hS ( l , l ) , g i g = −h l , g ✄ l i l , ∀ l , ∀ l ∈ l , ∀ g ∈ g . Written in the basis: S ( T A , T B ) = S ABα τ α , the defining relation for S becomes the relation: S ABα = − ✄ β [ B C g A ] C g αβ . Given two l -valued forms η and ω , one can define a g -valued form: ω ∧ S η = ω A ∧ η B S ABα τ α . Using this map, the transformations of the Lagrange multipliers under L -gauge are defined in [13].Further, to define the transformations of the Lagrange multipliers under H -gauge transformations the bilinear map X : l × h → h is defined: hX ( l, h ) , h i h = −h l, { h , h }i l , ∀ h , h ∈ h , ∀ l ∈ l , and bilinear map X : l × h → h by the rule: hX ( l, h ) , h i h = −h l, { h , h }i l , ∀ h , h ∈ h , ∀ l ∈ l . As far as the bilinear maps X and X one can define the coefficients in the basis as: X ( T A , t a ) = X Aab t b , X ( T A , t a ) = X Aab t b . When written in the basis the defining relations for the maps X and X become: X Abc = − X baB g AB g ac , X Abc = − X abB g AB g ac . Given l -valued differential form ω and h -valued differential form η , one defines a h -valued form as: ω ∧ X η = ω A ∧ η a X Aab t b , ω ∧ X η = ω A ∧ η a X Aab t b . Finally, a trilinear map D : h × h × l → g is needed: hD ( h , h , l ) , g i g = −h l, { g ✄ h , h }i l , ∀ h , h ∈ h , ∀ l ∈ l , ∀ g ∈ g , One can define the coefficients of the trilinear map as: D ( t a , t b , T A ) = D abAα τ α , and the defining relation for the map D expressed in terms of coefficients becomes: D abAβ = − ✄ αa c X cbB g AB g αβ . Given two h -valued forms ω and η , and l -valued form ξ , the g -valued form is given by the formula: ω ∧ D η ∧ D ξ = ω a ∧ η b ∧ ξ A D abAβ τ β . With these maps in hand, the transformations of the Lagrange multipliers under H -gauge transformations aredefined, see [13].2 Appendix E: Form-variations of all fields and momenta
The obtained gauge generator (55) is employed to calculate the form variations of variables and their correspondingcanonical momenta, denoted as A ( t, ~x ), using the following equation, δ A ( t, ~x ) = { A ( t, ~x ) , G } . (E1)3The computed form variations are given as follows: δ B α i = −∇ ǫ m αi + f βγα ǫ g β B γ i δ π ( B ) α i = f αβγ ǫ g β π ( B ) γ i , + ǫ n a ✄ αab β b i + ǫ h ai ✄ αab C b ,δ B αij = − ∇ [ i | ǫ m α | j ] + f βγα ǫ g β B γij − ǫ l Aij ✄ αA B D B δ π ( B ) αij = f αβγ ǫ g β π ( B ) γ ij , + ǫ n a ✄ αab β bij +2 ǫ h a [ j | ✄ αab C b | i ] ,δ α α = −∇ ǫ g α , δ π ( α ) α = f αβγ ǫ m βi π ( B ) γ i + f αβγ ǫ g β π ( α ) γ + ✄ αba ǫ n b π ( C ) a + ✄ αba ǫ h bi π ( β ) ai − ✄ αB A ǫ l Bij π ( γ ) A ij ,δ α αi = −∇ i ǫ g α − ∂ aα ǫ h ai , δ π ( α ) αi = f αβγ ǫ m βj π ( B ) γ ij + f αβγ ǫ g β π ( α ) γ i + ✄ αbα ǫ n b π ( C ) ai + ✄ αbα ǫ h bj π ( β ) aij − ✄ αB A ǫ l Bjk π ( γ ) Aijk − ǫ ijk ∇ j ǫ m αk , − ǫ ijk ǫ n a ✄ αba β bjk ,δ C a = −∇ ǫ n a + ǫ g α ✄ αb a C b , δ π ( C ) a = − ǫ g α ✄ αab π ( C ) b + ǫ h bi ✄ αab π ( B ) α i ,δ C ai = −∇ i ǫ n a + ǫ g α ✄ αba C bi δ π ( C ) ai = − ǫ g α ✄ αab π ( C ) bi + ǫ h bj ✄ αab π ( B ) αij , − ǫ m αi ∂ aα +2 ǫ h bi D A X ( bc ) A g ac ,δ β a i = −∇ ǫ h ai + ǫ g α ✄ αba β b i , δ π ( β ) a i = − ǫ g α ✄ αab π ( β ) b i + ǫ n b ✄ αa b π ( B ) α i − ǫ h bj X ( ab ) A π ( γ ) A ij ,δ β aij = − ∇ [ i | ǫ h a | j ] + ǫ g α ✄ αb a β bij + ǫ l Aij δ Aa , δ π ( β ) aij = − ǫ g α ✄ αab π ( β ) bij + ǫ n b ✄ αab π ( B ) αij − ǫ h bk X ( ab ) A π ( γ ) Aijk + ǫ ijk ∇ k ǫ n a + ǫ ijk ǫ h αk ∂ aα ,δ γ A ij = ǫ g α γ B ij ✄ αB A + ∇ ǫ l Aij δ π ( γ ) A ij = − ǫ g α ✄ αAB π ( γ ) B ij , − ǫ h a [ i | β b | j ] X ( ab ) A ,δ γ Aijk = ǫ g α γ Bijk ✄ αB A + ∇ i ǫ l Ajk δ π ( γ ) Aijk = − ǫ g α ✄ αAB π ( γ ) Bijk + ǫ oijk δ aA ǫ n a , −∇ j ǫ l Aik + ∇ k ǫ l Aij +3! ǫ h a [ i β bjk ] X ( ab ) A ,δ D A = ǫ n a δ aA + ǫ g α D B ✄ αB A , δ π ( D ) A = − ǫ h ai X ( ab ) A π ( C ) bi − ǫ l Bij ✄ αAB π ( B ) α ij − ǫ g α ✄ αAB π ( D ) B (E2)4 Appendix F: Symmetry algebra calculations
To obtain the structure of the symmetry group of the 3 BF action, as presented in the subsection IV D, one hasto calculate the commutators between the generators of all the symmetries, i.e., the G -, H -, L -, M -, and N -gaugesymmetries. This process is described in the subsections IV A, IV B, and IV C, while details of the calculation whichare not straightforward will be given in the following.
1. Commutator [ H, H ] Let us derive the commutator of the generators of the H -gauge transformations, i.e., the equation (76). Aftertransforming the variables under H -gauge transformations for the parameter ǫ h one obtains the following α ′ = α − ∂ǫ h ,β ′ = β − α − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h ,γ ′ = γ + { β − α − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h , ǫ h } pf + { ǫ h , β } pf ,B ′ = B − ( C − D ∧ X ǫ h − D ∧ X ǫ h ) ∧ T ǫ h − ǫ h ∧ D ǫ h ∧ D D ,C ′ = C − D ∧ X ǫ h − D ∧ X ǫ h ,D ′ = D , (F1)and transforming the variables once more for the parameter ǫ h one obtains: α ′′ = α − ∂ǫ h − ∂ǫ h ,β ′′ = β − α − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h − α − ∂ǫ h − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h ,γ ′′ = γ + { β − α − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h , ǫ h } pf + { ǫ h , β } pf + { β − α − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h − α − ∂ǫ h − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h , ǫ h } pf + { ǫ h , β − α − ∂ǫ h ∇ ǫ h − ǫ h ∧ ǫ h } pf ,B ′′ = B − ( C − D ∧ X ǫ h − D ∧ X ǫ h ) ∧ T ǫ h − ǫ h ∧ D ǫ h ∧ D D − ( C − D ∧ X ǫ h − D ∧ X ǫ h − D ∧ X ǫ h − D ∧ X ǫ h ) ∧ T ǫ h − ǫ h ∧ D ǫ h ∧ D D ,C ′′ = C − D ∧ X ǫ h − D ∧ X ǫ h − D ∧ X ǫ h − D ∧ X ǫ h ,D ′′ = D . (F2)It is easy to see that for variables α αµ , C aµ and D A the following is obtained:e ǫ h · H e ǫ h · H α αµ = e ǫ h · H e ǫ h · H α αµ , e ǫ h · H e ǫ h · H C aµ = e ǫ h · H e ǫ h · H C aµ , e ǫ h · H e ǫ h · H D A = e ǫ h · H e ǫ h · H D A . (F3)5For the remaining variables, β aµν , γ Aµνρ and B αµν , after subtracting (F 1) and the corresponding equation where ǫ h ↔ ǫ h , one obtains: (cid:0) e ǫ h · H e ǫ h · H − e ǫ h · H e ǫ h · H (cid:1) β aµν = ∂ bα ǫ h b [ µ | ǫ h c | ν ] ✄ αca − ∂ bα ǫ h b [ µ | ǫ h c | ν ] ✄ αca = 2 δ Aa X ( bc ) A ǫ h b [ µ | ǫ h c | ν ] = δ Aa ( { ǫ h ∧ ǫ h } pf − { ǫ h ∧ ǫ h } pf ) Aµν , (cid:0) e ǫ h · H e ǫ h · H − e ǫ h · H e ǫ h · H (cid:1) γ Aµνρ = 2( ∂ [ µ ǫ h aν ) ǫ h bρ ] X ( ab ) A + 2 ǫ h a [ ν ( ∂ µ ǫ h bρ ] ) X ( ab ) A +2 α α [ µ ǫ h aν ǫ h bρ ] X ( ab ) B ✄ αB A = ∇ [ µ ( { ǫ h ∧ ǫ h } pf − { ǫ h ∧ ǫ h } pf ) Aνρ ] , (cid:0) e ǫ h · H e ǫ h · H − e ǫ h · H e ǫ h · H (cid:1) B αµν = D A ǫ h a [ µ | ǫ h b | ν ] ( X Aac + X Aac ) T cbα − D A ǫ h b [ µ | ǫ h a | ν ] ( X Abc + X Abc ) T caα = − D A ǫ h a [ µ | ǫ h b | ν ] ( X ( ac ) A ✄ αb c + X ( bc ) A ✄ αa c )= − D A ǫ h a [ µ | ǫ h b | ν ] X ( ab ) B ✄ αB A = ( D ∧ S ( { ǫ h ∧ ǫ h } pf − { ǫ h ∧ ǫ h } pf ) αµν . (F4)Comparing (F3) and (F4) with (72), one concludes that the commutator of two H -gauge transformation is the L -gaugetransformation with the parameter ǫ l Aµν = 4 ǫ h a [ µ | ǫ h b | ν ] X ( ac ) A :e ǫ h · H e ǫ h · H − e ǫ h · H e ǫ h · H = 2 ( { ǫ h ∧ ǫ h } pf − { ǫ h ∧ ǫ h } pf ) · ˆ L . (F5)
2. Commutator [ H, N ] Let us calculate the commutator between the generators of H -gauge transformation and N -gauge transformation,i.e., derive the equation (108). This is done by calculating the expressions (cid:0) e ǫ h · H e ǫ n · N − e ǫ n · N e ǫ h · H (cid:1) A , (F6)for all variables A present in the theory. It is easy to see that for variables α αµ , β aµν , γ Aµνρ , and D A the followingis obtained: e ǫ h · H e ǫ n · N α αµ = e ǫ n · N e ǫ h · H α αµ , e ǫ h · H e ǫ n · N β aµν = e ǫ n · N e ǫ h · H β aµν , e ǫ h · H e ǫ n · N γ Aµνρ = e ǫ n · N e ǫ h · H γ Aµνρ , e ǫ h · H e ǫ n · N D A = e ǫ n · N e ǫ h · H D A . (F7)For the remaining variables, B αµν and C aµ , after the H -gauge transformation one obtains the following: B ′ = B − ( C − D ∧ χ ǫ h − D ∧ χ ǫ h ) ∧ τ ǫ h − ǫ h ∧ D ǫ h ∧ D D ,C ′ = C − D ∧ χ ǫ h − D ∧ χ ǫ h . (F8)Next, transforming those variables with N -gauge transformation one obtains: B ′′ = B ′ − β ′ ∧ T ǫ n = B − ( C − D ∧ χ ǫ h − D ∧ χ ǫ h ) ∧ τ ǫ h − ǫ h ∧ D ǫ h ∧ D D − ( β − { α α − ∂ aα ǫ h a } ∇ ǫ h − ǫ h ∧ ǫ h ) ∧ T ǫ n ,C ′′ = C ′ − { α α − ∂ aα ǫ h a } ∇ ǫ n = C − D ∧ χ ǫ h − D ∧ χ ǫ h − { α α − ∂ aα ǫ h a } ∇ ǫ n . (F9)6Let us now exchange the order of transformations, and first transform the variables with N -gauge transformation, B · = B − β ∧ T ǫ n ,C · = C − ∇ ǫ n , (F10)and then with H -gauge transformation: B ·· = B · − ( C · − D · ∧ χ ǫ h − D · ∧ χ ǫ h ) ∧ τ ǫ h − ǫ h ∧ D ǫ h ∧ D D · = B − β ∧ T ǫ n − ( C − ∇ ǫ n − ( D + δǫ n ) ∧ χ ǫ h − ( D + δǫ n ) ∧ χ ǫ h ) ∧ τ ǫ h − ǫ h ∧ D ǫ h ∧ D ( D + δǫ n ) ,C ·· = C · − D · ∧ χ ǫ h − D · ∧ χ ǫ h = C − ∇ ǫ n − ( D + δǫ n ) ∧ χ ǫ h − ( D + δǫ n ) ∧ χ ǫ h . (F11)After subtracting (F9) and (F11) one obtains: (cid:0) e ǫ h · H e ǫ n · N − e ǫ n · N e ǫ h · H (cid:1) B α = ∇ ǫ n a ∧ ǫ h b T abα + δ Aa ǫ n a ǫ h b ∧ ǫ h d X Abc T cdα + δ Aa ǫ n a ǫ h b ∧ ǫ h d X Abc T cdα − ǫ h a ∧ ǫ h b δ Ac ǫ n c D Aabα , −∇ ǫ h a ∧ ǫ n b T abα + ∂ aβ ǫ h a ✄ βc b ǫ h c ǫ n d T bdα − ǫ h a ∧ ǫ h b f abc ǫ n d T cdα , (cid:0) e ǫ h · H e ǫ n · N − e ǫ n · N e ǫ h · H (cid:1) C c = − ( δ Aa ǫ a n ) ∧ ǫ h b X Abc − ( δ Aa ǫ a n ) ∧ ǫ h b X Abc − ∂ aβ ǫ h a ✄ βb c ǫ n b , (F12)where after using the definitions of the maps T , D , χ , and χ one obtains the result (cid:0) e ǫ h · H e ǫ n · N − e ǫ n · N e ǫ h · H (cid:1) B α = ∇ ǫ n a ∧ ǫ h b T abα − ∇ ǫ h a ∧ ǫ n b T abα = ∇ ( ǫ n ∧ T ǫ h ) α , (cid:0) e ǫ h · H e ǫ n · N − e ǫ n · N e ǫ h · H (cid:1) C c = ∂ cα ( ǫ n ∧ T ǫ h ) α , (F13)Comparing (F7) and (F13) with (85), one obtains that: (cid:0) e ǫ h · H e ǫ n · N − e ǫ n · N e ǫ h · H (cid:1) = − ( ǫ n ∧ T ǫ h ) · M . (F14) [1] C. Rovelli,
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