Dual equivalence between self-dual and topologically massive B∧F models coupled to matter in 3+1 dimensions
R. V. Maluf, F. A. G. Silveira, J. E. G. Silva, C. A. S. Almeida
aa r X i v : . [ h e p - t h ] J un Dual equivalence between self-dual and topologically massive B ∧ F models coupled to matter in dimensions R. V. Maluf, ∗ F. A. G. Silveira, † J. E. G. Silva, ‡ and C. A. S. Almeida § Universidade Federal do Cear´a (UFC), Departamento de F´ısica,Campus do Pici, Fortaleza, CE, C.P. 6030, 60455-760 - Brazil. Universidade Federal do Cariri (UFCA),Av. Tenente Raimundo Rocha, Cidade Universit´aria,Juazeiro do Norte, Cear´a, CEP 63048-080, Brazil. (Dated: June 5, 2020)
Abstract
In this work, we revisit the duality between a self-dual non-gauge invariant theory and a topo-logical massive theory in 3 + 1 dimensions. The self-dual Lagrangian is composed by a vectorfield and an antisymmetric field tensor whereas the topological massive Lagrangian is build usinga B ∧ F term. Though the Lagrangians are quite different, they yield to equations of motion thatare connected by a simple dual mapping among the fields. We discuss this duality by analyzingthe degrees of freedom in both theories and comparing their propagating modes at the classicallevel. Moreover, we employ the master action method to obtain a fundamental Lagrangian thatinterpolates between these two theories and makes evident the role of the topological B ∧ F termin the duality relation. By coupling these theories with matter fields, we show that the dualityholds provided a Thirring-like term is included. In addition, we use the master action in order toprobe the duality upon the quantized fields. We carried out a functional integration of the fieldsand compared the resulting effective Lagrangians. ∗ Electronic address: r.v.maluf@fisica.ufc.br † Electronic address: adevaldo.goncalves@fisica.ufc.br ‡ Electronic address: [email protected] § Electronic address: carlos@fisica.ufc.br . INTRODUCTION Dualities are a main theme in nowadays physics. By connecting different theories oropposite regimes of a same model, dualities are powerful tools to seek and understandnew effects. Notably, string theories are connected by T and S dualities [1, 2] and the AdS/CF T correspondence links low-energy gravitational theory in
AdS spacetime with astrong coupling regime of a conformal field theory at the boundary [3]. Among the dualityprocesses, the so-called bosonisation is of special importance and widely used to investigatenonperturbative properties in quantum field theory and condensed matter systems in lowdimensions [4]. In 1 + 1 dimension, it is possible to establish a fermion-boson correspondencebased on the properties of the Fermi surfaces [5]. This duality can be further generalizedfor non-abelian fields [6] and even for higher dimensions [7, 8]. Recently, the bosonizationlead to new 2 + 1 relations called web of dualities [9, 10].Another example of duality involves topologically massive gauge theories. A well-knownduality occurs between the self-dual (SD) [11] and the Maxwell-Chern-Simons (MCS) [12]models. These two theories describe a single massive particle of spin-1 in 2 + 1 dimensionalMinkowski space-time. Nevertheless, only the MCS model is gauge-invariant. The equiva-lence between the SD and MCS models was initially proved by Deser and Jackiw [12], andover the years, several studies of this equivalence have been carried out in the literature [13–20]. Particularly, by considering couplings with fermionic fields, it was shown in [18] thatthe models are equivalent provided that a Thirring-like interaction is included. In addition,supersymmetric [21–23] and noncommutative [24] extensions to the duality involving the SDand MCS models have been studied in different contexts.At the heart of this duality, the Chern-Simons term plays a key role. An alternativetopological term in 3 + 1 dimensions can be formed from a U (1) vector gauge field A µ anda rank-2 antisymmetric tensor field B µν , also known as the Kalb-Ramond field [25, 26].Such a massive topological term is commonly called the B ∧ F term [27–30]. Therefore, anatural generalization of the MCS model in four dimensions consists of the Maxwell andKalb-Ramon fields coupled by a B ∧ F term [31]. This topologically massive gauge-invariant B ∧ F theory ( T M B ∧ F ) is unitary and renormalizable when minimally coupled to fermions,and represents a massive particle of spin-1 [27]. Models involving the Kalb-Ramond fieldhave been extensively studied in the literature, specially in connection with string theories232], quantum field theory [33, 34], supersymmetry [35], Lorentz symmetry violation [36–39],black hole solutions [40], cosmology [41], and brane words scenarios [42, 43].A self-dual version of the T M B ∧ F model was studied in Ref. [44]. It involves the B ∧ F term in a non-gauge invariant, first-order model ( SD B ∧ F ). Such work showed the classicequivalence between the models, i.e., at the level of the equations of motion, through thegauge embedding procedure [20]. In addition, when interactions with fermionic fields areconsidered, the duality mapping only is preserved if Thirring-like terms are taken into ac-count, analogously to the SD/MCS case in 2 + 1 dimensions. Yet, the issues regarding thegeneralization for arbitrary non-conserved matter currents and the proof of quantum dualityhave not yet been fully elucidated.The main goal of this work is to provide an alternative method, via master action [45],to prove the duality between the SD B ∧ F and T M B ∧ F theories, when the fields of the SDsector couple linearly with non-conserved currents, composed by arbitrary dynamic fieldsof matter. The master action approach has the advantage of providing a fundamentaltheory that interpolates between the two models and allows a more direct demonstration ofduality at the quantum level. Besides, the master action method is a natural trail for thesupersymmetric generalization of the duality studied here [23].The present work is organized as follows. In section II, we present the SD B ∧ F and T M B ∧ F theories in the free case, review their main physical characteristics, and check the classicduality by comparing their equations of motion. Moreover, we built a master Lagrangiandensity from the T M B ∧ F model, introducing auxiliary fields in order to obtain a first-orderderivative theory. In section III, we included matter couplings in the SD sector and verifywhether the equivalence is still compatible. We apply our results to the case of minimalcoupling with fermionic matter and compare it with those found in the literature. In sectionIV, we investigate the equivalence at the quantum level within the path-integral frame-work. Finally in section V we provide our conclusions and perspectives concerning furtherinvestigations. II. THE DUALITY AT THE CLASSICAL LEVEL.
In a 2 + 1 flat spacetime Townsend, Pilch, and Nieuwenhuizen proposed a first-orderderivative theory self-dual to the topological Chern-Simons theory [11]. In four dimensions,3his kind of duality can be built through a topological B ∧ F term. In fact, consider a gaugenon-invariant SD B ∧ F model composed by a vector field A µ and an antisymmetric 2-tensorfield B µν governed by the Lagrangian density [27, 44] L SD = m A µ A µ − B µν B µν + χθ ǫ µναβ B µν F αβ , (1)where m is a parameter with dimension of mass, θ is a dimensionless coupling constantand χ = ± − ) to the theory.The field strengths associated with the vector and tensor fields are defined respectively by F µν = ∂ µ A ν − ∂ ν A µ and H µνα = ∂ µ B να + ∂ ν B αµ + ∂ α B µν . The equations of motion for the A µ and B µν fields are, respectively, m A β − χθ ǫ µναβ ∂ α B µν = 0 , (2) B µν − χθǫ µναβ ∂ α A β = 0 , (3)and satisfy the constraint relations ∂ µ A µ = 0 , (4) ∂ µ B µν = 0 . (5)Eqs. (2) and (3) form a set of coupled first-order differential equations that can be rewritten,with the help of relations (4) and (5), in the form of a wave equation given by (cid:20) (cid:3) + m θ (cid:21) ϕ = 0 , (6)where ϕ denotes A µ or B µν fields. This implies that the first-order Lagrangian density L SD describes the dynamics of a massive vector field. In fact, the field B µν is auxiliary and canbe removed from the action leading to [36] L SD = m A µ A µ − θ F µν F µν , (7)which is the Lagrangian density for a massive vector field with three propagating degrees offreedom.In the context of the present work, we are interested in investigate the equivalence betweenthe self-dual model (1) and a second-order gauge-invariant theory. For this purposes, let usconsider a topologically massive B ∧ F model defined as [28, 44] L T M = θ m H µνα H µνα − θ F µν F µν − χθ ǫ µναβ B µν F αβ . (8)4ote that the first two terms of L T M are invariant under the gauge transformations A µ → A µ + ∂ µ λ and B µν → B µν + ∂ µ β ν − ∂ ν β µ , whereas the variation of the last term yields toa total divergence. The gauge parameter β µ still has a subsidiary gauge transformation β µ → β µ + ∂ µ α that leaves B µν unchanged. The equations of motions derived from thisLagrangian density are θ m ∂ µ H µνλ + χθ ǫ νλαβ F αβ = 0 , (9) θ ∂ µ F µλ + χθ ǫ µναλ H αµν = 0 . (10)In general, the two fields A µ and B µν have four and six independent degrees of freedom,respectively. However, due to the gauge symmetry in the theory described by L T M , some ofthem can be eliminated. In order to identify which ones propagate as massive physical modesor which are spurious (gauge dependent) modes, it is instructive to perform a decompositionin time-space on the equations of motions (9) and (10). For this purpose, let us split B µν into the independent components B i and B ij and to introduce spatial vectors ~ X and ~ Y defined by X i ≡ − B i , Y i ≡ ǫ ijk B jk , (11)where ǫ ijk = ǫ ijk . With these definitions, we obtain a set of coupled second order differentialequations in the form ∇ A + ∂ ∂ i A i + χθ ∂ i Y i = 0 , (12) (cid:3) A i − ∂ i (cid:0) ∂ A + ∂ j A j (cid:1) + χθ (cid:0) ǫ ijk ∂ k X j + ∂ Y i (cid:1) = 0 , (13) −∇ X i − ∂ i ∂ j X j + ǫ ijk (cid:18) ∂ ∂ j Y k − χm θ ∂ j A k (cid:19) = 0 , (14) ∂ Y k + ∂ i ∂ k Y i + ǫ ijk ∂ ∂ j X i + χm θ (cid:0) ∂ k A − ∂ A k (cid:1) = 0 . (15)After some manipulation of these equations, we can formally solve the temporal compo-nent A and the 3-vector ~ X in terms of the other components according to A = − ∇ (cid:16) ∂ ∂ i A i ( L ) + χθ ∂ i Y i ( L ) (cid:17) , (16) X ( T ) i = 1 ∇ ǫ ijk (cid:18) ∂ ∂ j Y k ( T ) − χm θ ∂ j A k ( T ) (cid:19) , (17)where v i ( T ) ≡ θ ij v j and v i ( L ) ≡ ω ij v j are the transversal ( T ) and longitudinal ( L ) componentsof a 3-vector ~v , respectively, with the projectors θ ij and ω ij defined by θ ij ≡ δ ij − ω ij , ω ij ≡ − ∂ j ∂ i ∇ . (18)5imilar procedures can be applied to the components of the ~A and ~ Y , such that (cid:20) (cid:3) + m θ (cid:21) A i ( T ) = 0 , (19) (cid:20) (cid:3) + m θ (cid:21) Y i ( L ) = 0 . (20)The form of these solutions reveals that the only physical components are A i ( T ) and Y i ( L ) ,while the other are auxiliary or gauge modes. Furthermore, as the longitudinal part of ~ Y iscurl-free, it propagates as a massive scalar field, i.e., ~ Y = ∇ φ , whose mass depends on thecoupling constant θ . Thus, the results above show that the T M B ∧ F theory defined in (8),like the SD B ∧ F model, contains three massive propagating modes.To make explicit the hidden duality between the models described above, it is conve-nient to introduce the dual fields associated with the field strength tensors H µνα and F µν ,respectively by ˜ H µ ≡ − χθ m ǫ µναβ H ναβ , (21)˜ F µν ≡ χθ ǫ µναβ F αβ . (22)In terms of ˜ H µ and ˜ F µν , the equations of motion (9) and (10) become m ˜ H β − θ χ ǫ µναβ ∂ µ ˜ F να = 0 , (23)˜ F µν − θχ ǫ µναβ ∂ α ˜ H β = 0 . (24)A direct comparison between the pairs of equations (2,3) and (23,24) shows that the dualfields ˜ H β and ˜ F µν satisfy exactly the same equations obtained for SD B ∧ F model whenwe identify A µ → ˜ H µ and B µν → ˜ F µν . Therefore, the basic fields of the SD B ∧ F modelcorrespond to the dual fields of the T M B ∧ F model. This proves the classical equivalence viaequations of motion in the free field case.However, despite having established the dual connection, the mapping A µ → ˜ H µ and B µν → ˜ F µν leads to L T M ( ˜ H, ˜ F ) = − m H µ ˜ H µ + 14 ˜ F µν ˜ F µν − B µν ˜ F µν , (25)wherein the identities F µν F µν = − /θ ˜ F µν ˜ F µν and H µνα H µνα = − m /θ ˜ H µ ˜ H µ were used.Note that (25) does not recover (1) and the equivalence between the two models is notevident. The common origin of these Lagrangian densities can be better addressed bymeans of the master Lagrangian method, which we will formulate in the sequel.6 . Classic Duality via Master Lagrangian The study of dual equivalence among four-dimensional models containing a topological B ∧ F term was carried out for the first time in Ref. [44], whereby the authors used thedynamical gauge embedding formalism to show the classic duality between (1) and (8). Here,we employ the master Lagrangian method [12, 18] that extends and interpolates those twostudied models. Moreover, this method allows us to study the duality at the quantum levelmore directly.Let us start from Lagrangian density L T M in the form (25) written explicitly in terms ofthe fundamental fields A µ and B µν . Following [12], we will introduce auxiliary fields Π µ andΛ µν in order to obtain a first-order derivative theory such that L M = a Π µ ǫ µρσδ ∂ ρ B σδ + b Π µ Π µ + c Λ µν ǫ µνρσ ∂ ρ A σ + d Λ µν Λ µν − χθ ε µναβ B µν ∂ α A β , (26)where a , b , c and d are constant coefficients to be determined. Note that the presence ofmass terms for Π µ and Λ µν ensures the auxiliary character of these fields.The functional variation of L M with respect to the auxiliary fields Π µ and Λ µν allows usto write Π µ = − a b ǫ µναβ ∂ ν B αβ , (27)Λ µν = − c d ǫ µναβ ∂ α A β . (28)Substituting (27) and (28) in (26) and imposing L M = L T M , we obtain the relations a b = θ m , (29) c d = − θ . (30)The same procedure can be performed for the fields A µ and B µν , and we can immediatelysolve their equations of motion, obtaining the following solutions: A µ = 2 aχθ Π µ + ∂ µ φ, (31) B µν = 2 cχθ Λ µν + ∂ µ Σ ν − ∂ ν Σ µ , (32)being φ and Σ µ arbitrary fields. Now, replacing (31) and (32) in (26) and imposing L M =7 SD , we obtain b = m , (33) d = − , (34)such that we can immediately fix a = c = χθ/ L M = χθ µ ǫ µρσδ ∂ ρ B σδ + m µ Π µ + χθ µν ǫ µνρσ ∂ ρ A σ −
14 Λ µν Λ µν − χθ ε µναβ B µν ∂ α A β . (35)Accordingly, the Lagrangian density (35) describes both (1) and (8). This mechanismtransforms models without gauge invariance into models with this symmetry by adding termswhich does not appear on-shell. Note that the gauge invariance of L M under δA µ = ∂ µ λ and δB µν = ∂ µ β ν − ∂ ν β µ with δ Π µ = δ Λ µν = 0 is now evident, while it was a hidden symmetry inthe self-dual formulation. With the master method, we were able to establish the relation ofequivalence when the coupling to other dynamical fields is considered and we have a simpleformalism which account for the investigation of the theory at the quantum level. III. DUALITY MAPPING WITH A LINEAR MATTER COUPLING
The discussion on the duality developed in the previous section deals only with freetheories. However, it is fundamental to ensure that this dual equivalence is also valid in thepresence of external sources coupled to the fields in L M . Here and throughout the paper,we will assume only linear couplings with external fields, whose associated currents arecomposed only of matter fields, represented generically by ψ . The cases involving nonlinearcouplings or when the currents depend explicitly on the gauge or self-dual fields are beyondour present scope.Let us consider the master Lagrangian (35) added by dynamical matter fields ψ linearlycoupled to the self-dual sector: L (1) M = χθ µ ǫ µρσδ ∂ ρ B σδ + m µ Π µ + χθ µν ǫ µνρσ ∂ ρ A σ −
14 Λ µν Λ µν − χθ ε µναβ B µν ∂ α A β + Π µ J µ + Λ µν J µν + L ( ψ ) , (36)where L ( ψ ) represents a generic Lagrangian density responsible for the dynamics of thematter fields, with the corresponding currents being denoted by J µ and J µν . Note that8ue to the lack of gauge symmetry in the self-dual sector, the matter currents J µ and J µν are generally not conserved. Also, to make our analysis as general as possible, we will notassume any specific form to the matter sector for now.First, we will remove the dependency on the gauge fields in Eq. (36). Varying the action ´ d x L (1) M with respect to the fields A µ and B µν , we obtain their corresponding equations ofmotion whose solutions are given by A µ = Π µ + ∂ µ φ, (37) B µν = Λ µν + ∂ µ Σ ν − ∂ ν Σ µ , (38)and substituting these solutions into Eq. (36) we find L (1) M = L (1) SD , with L (1) SD = m µ Π µ −
14 Λ µν Λ µν + χθ µ ǫ µναβ ∂ ν Λ αβ + Π µ J µ + Λ µν J µν + L ( ψ ) . (39)Then, L (1) SD is equivalent to the self-dual theory (1) linearly coupled to the matter, as ex-pected.Next, we will eliminate the fields Π µ and Λ µν from the master Lagrangian L (1) M . Theequations of motion for these fields areΠ µ = − χθ m ǫ µναβ ∂ ν B αβ − m J µ , (40)Λ µν = χθǫ µναβ ∂ α A β + 2 J µν . (41)Replacing Eqs. (40) and (41) into the master Lagrangian then implies L (1) M = L (1) T M , with L (1) T M = θ m H µνα H µνα − θ F µν F µν − χθ ǫ µναβ B µν ∂ α A β − χθ m B µν ǫ µναβ ∂ α J β − m J µ J µ + χθA µ ǫ µναβ ∂ ν J αβ + J µν J µν + L ( ψ ) . (42)From the above result, it is clear that the Lagrangian density L (1) T M represents the
T M B ∧ F theory (8) interacting with the matter through “magnetic” currents plus Thirring-like termsinvolving only the matter fields. A similar Lagrangian density to the L (1) T M has appearedbefore in [44]. However, the approach used in [44] was based on the gauge embeddingmethod, different from the one developed here. Also, one may verify that the equations of9otion for the fields Π µ and Λ µν in the SD B ∧ F model (39) and for the gauge fields A µ and B µν in the T M B ∧ F model (42) can be cast in the same form by means of the identificationΠ µ → ˜ H µ − m J µ , (43)Λ µν → ˜ F µν + 2 J µν . (44)It is worth noting that the duality symmetry between SD B ∧ F /T M B ∧ F theories exchangeslinear couplings Π µ J µ and Λ µν J µν , involving currents not necessarily conserved in the self-dual sector into derivative dual couplings A µ ǫ µναβ ∂ ν J αβ and B µν ǫ µναβ ∂ α J β in the gaugesector, whose associated currents are automatically conserved. Moreover, self-interactionmatter terms are naturally generated, which will play a decisive role in ensuring the dualityin the matter sector, as we shall see in what follows. A. The matter sector
Classically, the duality mapping established in Eqs. (43-44) ensures that the Lagrangiandensities (39) and (42) are equivalent since the SD B ∧ F and T M B ∧ F fields obey the sameequations of motion in the presence of external sources. However, for this equivalencebetween the models to be complete, it is also necessary to verify what happens in thematter sector, when these sources are dynamics.To this end, we now consider the equation of motion for the matter field ψ . First, let usfocus our attention on the SD B ∧ F model described by (39), so δδψ ˆ d x L (1) SD = 0 ⇒ δ L ( ψ ) δψ = − Π µ δJ µ δψ − Λ µν δ J µν δψ , (45)where δ L ( ψ ) δψ is the Lagrangian derivative.On the other hand, the equations of motion for the fields Π µ and Λ µν are: m Π µ + χθ ǫ µναβ ∂ ν Λ αβ = − J µ , (46)12 Λ µν − χθ ǫ µναβ ∂ α Π β = J µν , (47)and obey the constraints m ∂ µ Π µ = − ∂ µ J µ , (48) ∂ µ Λ µν = 2 ∂ µ J µν . (49)10nserting (47) into (46), we can eliminate Λ µν in favor of Π µ and obtain a second-orderdifferential equation as (cid:0) θ (cid:3) + m (cid:1) Π µ = − J µ − θ m ∂ µ ∂ ν J ν − χθǫ µναβ ∂ ν J αβ , (50)where we used the constraint m ∂ µ Π µ = − ∂ µ J µ . Defining the wave-operator as ˆ R − = (cid:3) + m θ , we can write Π µ = − ˆ Rθ (cid:18) J µ + θ m ∂ µ ∂ ν J ν + χθǫ µναβ ∂ ν J αβ (cid:19) . (51)A similar procedure for the field Λ µν results inΛ µν = − ˆ Rθ (cid:2) − m J µν + 2 θ ∂ α ( ∂ µ J να − ∂ ν J µα ) + χθǫ µναβ ∂ α J β (cid:3) . (52)Replacing the solutions (51) and (52) back in the matter equation (45), we come to theresult δ L ( ψ ) δψ = ˆ Rθ (cid:20) J µ + θ m ∂ µ ( ∂ ν J ν ) + χθǫ µναβ ∂ ν J αβ (cid:21) δJ µ δψ + ˆ Rθ (cid:2) − m J µν + 2 θ ∂ α ( ∂ µ J να − ∂ ν J µα ) + χθǫ µναβ ∂ α J β (cid:3) δ J µν δψ . (53)This is a non-local differential equation, expressed only in terms of the matter fields.Now, if we start from L (1) T M , the equation of motion for the matter field takes the form δδψ ˆ d x L (1) T M = 0 ⇒ δ L ( ψ ) δψ = (cid:18) m J µ − ˆ H µ (cid:19) δJ µ δψ + (cid:16) − J µν − ˆ F µν (cid:17) δ J µν δψ , (54)where we have used the definitions (21-22) for the dual fields.To eliminate the dual fields in (54), we write the equations of motion for A µ and B µν ,obtained from L (1) T M , as m ˜ H µ + θ χ ǫ µναβ ∂ ν ˜ F αβ = − χθǫ µναβ ∂ ν J αβ , (55) − ˜ F µν + θχ ǫ µναβ ∂ α ˜ H β = χθm ǫ µναβ ∂ α J β . (56)These equations can be decoupled, and after some algebraic manipulations we get the fol-lowing results ˜ H µ = ˆ Rθ (cid:20) θ m ( (cid:3) J µ − ∂ µ ∂ ν J ν ) − χθǫ µναβ ∂ ν J αβ (cid:21) , (57)˜ F µν = − R (cid:3) J µν − ˆ Rθ ² (cid:2) θ ∂ α ( ∂ µ J να − ∂ ν J µα ) + χθǫ µναβ ∂ α J β (cid:3) . (58)11ubstituting these solutions in Eq. (54) we obtain δ L ( ψ ) δψ = " m (cid:16) − ˆ R (cid:3) (cid:17) J µ + ˆ Rθ (cid:18) θ m ∂ µ ∂ ν J ν + χθǫ µναβ ∂ ν J αβ (cid:19) δJ µ δψ + " (cid:16) ˆ R (cid:3) − (cid:17) J µν + ˆ Rθ (cid:0) θ ∂ α ( ∂ µ J να − ∂ ν J µα ) + χθǫ µναβ ∂ α J β (cid:1) δ J µν δψ . (59)Using the definition ˆ R − = (cid:3) + m θ , we can write (cid:3) = R − − m θ which implies δ L ( ψ ) δψ = ˆ Rθ (cid:20) J µ + θ m ∂ µ ( ∂ ν J ν ) + χθǫ µναβ ∂ ν J αβ (cid:21) δJ µ δψ + ˆ Rθ (cid:2) − m J µν + 2 θ ∂ α ( ∂ µ J να − ∂ ν J µα ) + χθǫ µναβ ∂ α J β (cid:3) δ J µν δψ . (60)By comparing Eqs. (53) and (60), we conclude that the matter sectors of the two modelsgive rise to the same equations of motion. Thus, we have shown that the Lagrangians L (1) SD and L (1) T M are equivalent and have established the classical duality between the SD B ∧ F and T M B ∧ F theories when couplings with dynamical matter fields are considered.In order to liken our results with the literature, let us consider, as a particular case, afermionic matter field minimally coupled to the self-dual field Π µ . Assuming the followingidentifications: L ( ψ ) → L Dirac = ¯ ψ ( iγ µ ∂ µ − M ) ψ, (61)where M is the Dirac field mass, and the fermionic currents are J µ → − eJ µ = − e ¯ ψγ µ ψ, (62) J µν → , (63)with e being a dimensionless coupling constant. The equation of motion for ψ (60) takesthe simple form ( iγ µ ∂ µ − M ) ψ = e θ ˆ RJ µ γ µ ψ, (64)which agrees with the result obtained in [44]. IV. THE DUALITY AT THE QUANTUM LEVEL
Once we proved the duality between SD B ∧ F and T M B ∧ F models at the level of equationsof motion, we now check whether this duality is preserved at the quantum level. For this12urpose, we adopt the path-integral framework and define the master generating functionalas Z ( ψ ) = N ˆ D A µ D B µν D Π µ D Λ µν exp (cid:26) i ˆ d x [ L M + J µ Π µ + J µν Λ µν + L ( ψ )] (cid:27) , (65)where N is a overall normalization constant. Our aim is to evaluate the effective Lagrangianresulting from the integration over the fields. Firstly, let us integrate out the contributionof the SD B ∧ F fields.After the shifts, Π µ → Π µ + ˜ H µ − m J µ and Λ µν → Λ µν + ˜ F µν + 2 J µν , we perform thefunctional integration in Eq. (65) over the fields Π µ and Λ µν , thereby producing Z ( ψ ) = N ˆ D A µ D B µν exp (cid:20) i ˆ d x L (1) eff ( A, B, ψ ) (cid:21) , (66)where L (1) eff ( A, B, ψ ) = θ m H µνα H µνα − θ F µν F µν − χθ ǫ µναβ B µν ∂ α A β − χθ m B µν ǫ µναβ ∂ α J β − m J µ J µ + χθA µ ǫ µναβ ∂ ν J αβ + J µν J µν + L ( ψ ) , (67)is the same Lagrangian density found in Eq. (42).To integrate over the fields configurations A µ and B µν , let us first note that the masterLagrangian L M can be rewritten, up to surface terms, as L M = χθ ǫ µναβ (Λ µν − B µν ) ∂ α ( A β − Π β ) + L SD . (68)In this way, we can make a shift in the gauge fields through B µν → B µν + Λ µν and A β → A β + Π β , which allows us to rewrite the generating function (65) as, Z ( ψ ) = N ˆ D A µ D B µν D Π µ D Λ µν exp (cid:26) i ˆ d x (cid:20) − χθ ǫ µναβ B µν ∂ α A β + L SD + J µ Π µ + J µν Λ µν + L ( ψ ) (cid:21)(cid:27) , (69)such that the A µ and B µν fields decouple. Then, performing the function integration yieldsto the following generating functional Z ( ψ ) = N ˆ D Π µ D Λ µν exp (cid:20) i ˆ d x L (2) eff (Π , Λ , ψ ) (cid:21) , (70)with L (2) eff (Π , Λ , ψ ) = m µ Π µ −
14 Λ µν Λ µν + χθ µ ǫ µναβ ∂ ν Λ αβ + Π µ J µ + Λ µν J µν + L ( ψ ) , (71)13orresponding to the same Lagrangian density (39) previously obtained. It is worth high-lighting the physical implications contained in (68). We clearly see that the master La-grangian L M obtained in (35) is equivalent to self-dual Lagrangian L SD added by a purelytopological B ∧ F term, which makes evident the role of the master Lagrangian on theduality symmetry.The implications of the above results at the quantum level can be explored by consideringthe functional derivatives of (66) and (70) with respect to the sources. Setting J µ = J µν = 0,we can establish the following identities to the correlation functions h Π µ ( x ) · · · Π µ N ( x N ) i SD = D ˜ H µ [ B ( x )] · · · ˜ H µ N [ B ( x N )] E T M + contact terms , (72) h Λ µ ν ( x ) · · · Λ µ N ν N ( x N ) i SD = D ˜ F µ ν [ A ( x )] · · · ˜ F µ N ν N [ A ( x N )] E T M + contact terms . (73)These relations show that the classical dual map (43-44) is satisfied by all quantum correla-tion functions of those fields, up to contact terms.Finally, we now complete the proof of quantum duality between the SD B ∧ F /T M B ∧ F models by performing the path integration over A µ and B µν gauge fields in Eq. (66), andover Π µ and Λ µν self-dual fields in Eq. (70). For this goal, it is convenient to organize theeffective Lagrangians (67) and (71) in a matrix-form according to the L = 12 X T ˆ O X + X T J , (74)where the wave operator, ˆ O , form a 2 × X , and J represent vector-tensor dupletof type X = A µ B µν . (75)To accomplish the functional integration, we use the Gaussian path integral formula overa bosonic field X , ˆ D X exp (cid:20) i ˆ d x (cid:18) X T ˆ O X + X T J (cid:19)(cid:21) = h Det (cid:16) − i ˆ O (cid:17)i − × exp (cid:20) − i ˆ d x J T ˆ O − J (cid:21) . (76)In our case, the determinant Det (cid:16) − i ˆ O (cid:17) is field-independent and can be absorbed bythe normalization constant. The calculation of propagators ˆ O − is rather lengthy, and thedetails are in Appendix A. Here we just write the results (cid:16) ˆ O − SD (cid:17) µ,αβ ; ν,λσ = θ (cid:3) + m Θ µν + m ω µν θ (cid:3) + m S µλσ − θ (cid:3) + m S αβν − m θ (cid:3) + m (cid:0) P (1) (cid:1) αβ,λσ − (cid:0) P (2) (cid:1) αβ,λσ , (77)14nd (cid:16) ˆ O − T M (cid:17) µ,αβ ; ν,λσ = θ (cid:3) + m Θ µν + λ (cid:3) ω µν − m θ (cid:3) ( θ (cid:3) + m ) S µλσ m θ (cid:3) ( θ (cid:3) + m ) S αβν − m θ (cid:3) + m (cid:0) P (1) (cid:1) αβ,λσ − ξ (cid:3) (cid:0) P (2) (cid:1) αβ,λσ , (78)where Θ µν , ω µν , S µνα , P (1) µν,αβ and P (2) µν,αβ are projection operators whose definitions and closedalgebras are shown in Appendix A. Also, λ and ξ are convenient gauge fixing parameters.Note that the physical poles of the two propagators are equal, i.e., θ (cid:3) + m = 0, andconfirm that the particle spectrum of both theories are equivalent, so that we may considerthe self-dual theory equivalent to T M B ∧ F theory with the fixed gauge.The above propagators, together with formula (76), enable us to perform the functionalintegration in (66) and (70). After completing all tensorial contractions, we obtain the sameeffective Lagrangian for the matter field L (3) eff ( ψ ) = L ( ψ )+ 12 (cid:16) J µ J αβ (cid:17) − θ (cid:3) + m η µν − θ m θ (cid:3) + m ∂ µ ∂ ν − χθθ (cid:3) + m ǫ µλσδ ∂ δχθθ (cid:3) + m ǫ αβνδ ∂ δ θ (cid:3) + m (cid:0) θ (cid:3) P (2) + m I (cid:1) αβ,λσ J ν J λσ . (79)It is easy to verify that the equation of motion for the matter field obtained from L (3) eff (79) is precisely that found in the previous section (see Eqs. (53) or (60)). Thus, weprove the quantum equivalence between the matter sector of the SD B ∧ F /T M B ∧ F models.It is worth mentioning that the dynamics of the matter fields is preserved in the functionalintegration in (66) only if the Thirring-like interactions are added to the diagonal elementsof ˆ O − T M matrix. Besides, the gauge-dependent parts involving the gauge fixing parametersare canceled, as it should be.
V. CONCLUSION
In this work, we revisited the duality between the self-dual and topologically massivemodels involving the B ∧ F term in 3 + 1 spacetime dimensions. The study of this dualitywhen couplings with fermionic matter are included was first carried out in [44], throughthe gauge embedding formalism. Here, we considered another approach, namely the masteraction method, whereby we obtained a fundamental Lagrangian density that interpolates15etween the two models and provides direct proof of dual equivalence at both the classicaland quantum level. The master action enabled us to relate the equations of motion of thesemodels via a dual map among fields and currents of both theories, which ensures that theyare equivalent at the classical level. In addition, we demonstrated the duality at quantumthrough the path-integral framework. We defined a master generating functional whereinthe integration over the different fields provided effective Lagrangians that are the sameas those obtained classically. Moreover, after a last functional integration over the bosonicfields, we obtained an effective non-local Lagrangian for the matter fields, which proves theequivalence between the matter sectors of the analyzed models.We assumed that the external currents are linearly coupled with the self-dual fields andare constituted exclusively of the matter fields. We show that these interactions induce“magnetic” couplings involving the gauge fields, in addition to current-current Thirring-likeinteractions. These types of couplings are, in general, non-renormalizable by direct powercounting [18, 22]. However, as in 2 + 1 dimensional case involving the Maxwell-Chern-Simons model, we may expect which this weakness can be overcome by a / N perturbativeexpansion when the matter field is an N -component fermionic field, such that the theorybecomes renormalizable. An explicit verification of this issue, as well as a possible extensionof our results to the supersymmetric case [23, 35], are themes for forthcoming works. Acknowledgments
The authors thank the Funda¸c˜ao Cearense de Apoio ao Desenvolvimento Cient´ıfico eTecnol´ogico (FUNCAP), the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior(CAPES), and the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq),Grants no 312356/2017-0 (JEGS), no 305678/2015-9 (RVM) and no 308638/2015-8 (CASA)for financial support.
Appendix A: Feynman propagator for the
T M B ∧ F theory Consider the topologically massive B ∧ F model defined as S T M = ˆ d x (cid:20) − θ F µν F µν + θ m H µνα H µνα − χθ ǫ µναβ B µν F αβ (cid:21) , (A1)16here the first two terms represent a gauge-invariant Maxwell-Kalb-Ramond theory, whilethe last is a topological B ∧ F term. The calculation of the Feynman propagator for thetheory (A1) can be performed as follows.First, let us rewrite the integrand in Eq. (A1) on the matrix form L T M = 12 X T ˆ O T M X , (A2)with the wave operator, ˆ O T M , being a 2 × X represents a column vector-tensoras X = A µ B µν . (A3)Adding convenient gauge-fixing terms in (A2), namely, − λ ( ∂ µ A µ ) and ξ ( ∂ µ B µν ) , wecan explicitly write the operator ˆ O T M + gf , in the form,ˆ O µ,αβ ; ν,λσT M + gf = θ (cid:3) Θ µν + (cid:3) λ ω µν − S µλσ S αβν − θ (cid:3) m (cid:0) P (1) (cid:1) αβ,λσ − (cid:3) ξ (cid:0) P (2) (cid:1) αβ,λσ , (A4)where we have introduced the set of spin-projection operators asΘ µν = η µν − ω µν , ω µν = ∂ µ ∂ ν (cid:3) , (A5) S µνα = χθ ǫ µναβ ∂ β , (A6) P (1) µν,αβ = 12 (Θ µα Θ νβ − Θ µβ Θ να ) , (A7) P (2) µν,αβ = 12 (Θ µα ω νβ − Θ µβ ω να + Θ νβ ω µα − Θ να ω µβ ) , (A8)with (cid:3) ≡ ∂ µ ∂ µ , and η µν is the Minkowski metric with signature (+ , − , − , − ). Note that P (1) and P (2) satisfy the tensorial completeness relation: (cid:0) P (1) + P (1) (cid:1) µν,αβ = 12 ( η µα η νβ − η µβ η να ) ≡ I µν,αβ . (A9)The products between the operators defined above satisfy a closed algebra and are sum-marized in Tables I, II.The Feynman propagator is defined as ˆ O − T M + gf . In order to invert the wave operator, wewill write it and its inverse generically by: O = A BC D , and O − = A BC D , (A10)17 αν ω νν Θ µα Θ µν ω µα ω µν (cid:0) P (1) (cid:1) ρσαβ (cid:0) P (2) (cid:1) ρσαβ P (1) µνρσ P (1) µναβ P (2) µνρσ P (2) µναβ Table I: Algebra of the spin-projection operators. S αβν Θ βσ ω βσ S µαβ − θ (cid:3) Θ µν S µασ S λαβ (cid:0) P (1) (cid:1) νλρσ (cid:0) P (2) (cid:1) νλρσ S µνλ − θ (cid:3) P (1) µναβ S µρσ which fulfills the relation OO − = I , where the general identity matrix I is defined by: I = I I , (A11)with I and I are the identities to the projectors ( θ µν , ω µν ), and ( P (1) , P (2) ), respectively.From these preliminary definitions, we obtain a system of four equations, whose solutionscan be written as we get A A + B C = I A B + B D = 0 C A + D C = 0 C B + D D = I ⇒ A = ( A − BD − C ) − B = − A − B DC = − D − C AD = ( D − CA − B ) − (A12)After some algebraic manipulations with the set of the operators presented above, the T M B ∧ F gauge propagator is properly written as (cid:16) ˆ O − T M (cid:17) µ,αβ ; ν,λσ = θ (cid:3) + m Θ µν + λ (cid:3) ω µν − m θ (cid:3) ( θ (cid:3) + m ) S µλσ m θ (cid:3) ( θ (cid:3) + m ) S αβν − m θ (cid:3) + m (cid:0) P (1) (cid:1) αβ,λσ − ξ (cid:3) (cid:0) P (2) (cid:1) αβ,λσ . (A13) [1] Andrew Strominger, Shing-Tung Yau, Eric Zaslow, Mirror Symmetry is T-Duality , Nucl. Phys.B , 243-259 (1996).[2] J. Polchinski,
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