Dual equivalence between self-dual and topologically massive $B\wedge F$ models coupled to matter in $3+1$ dimensions

In this work, we revisit the duality between a self-dual non-gauge invariant theory and a topological massive theory in $3+1$ dimensions. The self-dual Lagrangian is composed by a vector field and an antisymmetric field tensor whereas the topological massive Lagrangian is build using a $B \wedge F$ term. Though the Lagrangians are quite different, they yield to equations of motion that are connected by a simple dual mapping among the fields. We discuss this duality by analyzing the degrees of freedom in both theories and comparing their propagating modes at the classical level. Moreover, we employ the master action method to obtain a fundamental Lagrangian that interpolates between these two theories and makes evident the role of the topological $B \wedge F$ term in the duality relation. By coupling these theories with matter fields, we show that the duality holds provided a Thirring-like term is included. In addition, we use the master action in order to probe the duality upon the quantized fields. We carried out a functional integration of the fields and compared the resulting effective Lagrangians.


I. INTRODUCTION
Dualities are a main theme in nowadays physics. By connecting different theories or opposite regimes of a same model, dualities are powerful tools to seek and understand new 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 a strong coupling regime of a conformal field theory at the boundary [3]. Among the duality processes, the so-called bosonisation is of special importance and widely used to investigate nonperturbative properties in quantum field theory and condensed matter systems in low dimensions [4]. In 1+1 dimension, it is possible to establish a fermion-boson correspondence based on the properties of the Fermi surfaces [5]. This duality can be further generalized for non-abelian fields [6] and even for higher dimensions [7,8]. Recently, the bosonization lead to new 2 + 1 relations called web of dualities [9,10].
Another example of duality involves topologically massive gauge theories. A well-known duality 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 dimensional Minkowski space-time. Nevertheless, only the MCS model is gauge-invariant. The equivalence between the SD and MCS models was initially proved by Deser and Jackiw [12], and over the years, several studies of this equivalence have been carried out in the literature [13][14][15][16][17][18][19][20]. Particularly, by considering couplings with fermionic fields, it was shown in [18] that the models are equivalent provided that a Thirring-like interaction is included. In addition, supersymmetric [21][22][23] and noncommutative [24] extensions to the duality involving the SD and MCS models have been studied in different contexts.
At the heart of this duality, the Chern-Simons term plays a key role. An alternative topological term in 3 + 1 dimensions can be formed from a U(1) vector gauge field A µ and a rank-2 antisymmetric tensor field B µν , also known as the Kalb-Ramond field [25,26].
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 classic equivalence between the models, i.e., at the level of the equations of motion, through the gauge embedding procedure [20]. In addition, when interactions with fermionic fields are considered, the duality mapping only is preserved if Thirring-like terms are taken into account, analogously to the SD/MCS case in 2 + 1 dimensions. Yet, the issues regarding the generalization for arbitrary non-conserved matter currents and the proof of quantum duality have 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 SD sector couple linearly with non-conserved currents, composed by arbitrary dynamic fields of matter. The master action approach has the advantage of providing a fundamental theory that interpolates between the two models and allows a more direct demonstration of duality at the quantum level. Besides, the master action method is a natural trail for the supersymmetric 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 classic duality by comparing their equations of motion. Moreover, we built a master Lagrangian density from the T M B∧F model, introducing auxiliary fields in order to obtain a first-order derivative theory. In section III, we included matter couplings in the SD sector and verify whether the equivalence is still compatible. We apply our results to the case of minimal coupling with fermionic matter and compare it with those found in the literature. In section IV, we investigate the equivalence at the quantum level within the path-integral framework. Finally in section V we provide our conclusions and perspectives concerning further investigations.

II. THE DUALITY AT THE CLASSICAL LEVEL.
In a 2 + 1 flat spacetime Townsend, Pilch, and Nieuwenhuizen proposed a first-order derivative theory self-dual to the topological Chern-Simons theory [11]. In four dimensions, this kind of duality can be built through a topological B ∧ F term. In fact, consider a gauge non-invariant SD B∧F model composed by a vector field A µ and an antisymmetric 2-tensor field B µν governed by the Lagrangian density [27,44] where m is a parameter with dimension of mass, θ is a dimensionless coupling constant and χ = ±1 defines either the self-duality (+) or the anti self-duality (−) to the theory.
The field strengths associated with the vector and tensor fields are defined respectively by The equations of motion for the A µ and B µν fields are, respectively, and satisfy the constraint relations 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 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 can be removed from the action leading to [36] which is the Lagrangian density for a massive vector field with three propagating degrees of freedom.
In the context of the present work, we are interested in investigate the equivalence between the self-dual model (1) and a second-order gauge-invariant theory. For this purposes, let us consider a topologically massive B ∧ F model defined as [28,44] Note 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 to a 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 this Lagrangian density are 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 of them can be eliminated. In order to identify which ones propagate as massive physical modes or which are spurious (gauge dependent) modes, it is instructive to perform a decomposition in time-space on the equations of motions (9) and (10). For this purpose, let us split B µν into the independent components B 0i and B ij and to introduce spatial vectors X and Y defined by where ǫ 0ijk = ǫ ijk . With these definitions, we obtain a set of coupled second order differential equations in the form After some manipulation of these equations, we can formally solve the temporal component A 0 and the 3-vector X in terms of the other components according to where v i (T ) ≡ θ i j v j and v i (L) ≡ ω i j v j are the transversal (T ) and longitudinal (L) components of a 3-vector v, respectively, with the projectors θ i j and ω i j defined by Similar procedures can be applied to the components of the A and Y, such that 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 is curl-free, it propagates as a massive scalar field, i.e., Y = ∇φ, whose mass depends on the coupling 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 convenient to introduce the dual fields associated with the field strength tensors H µνα and F µν , In terms ofH µ andF µν , the equations of motion (9) and (10) become A direct comparison between the pairs of equations (2,3) and (23,24) shows that the dual fieldsH β andF µν satisfy exactly the same equations obtained for SD B∧F model when we identify A µ →H µ and B µν →F µν . Therefore, the basic fields of the SD B∧F model correspond to the dual fields of the T M B∧F model. This proves the classical equivalence via equations of motion in the free field case.
Note that (25) does not recover (1) and the equivalence between the two models is not evident. The common origin of these Lagrangian densities can be better addressed by means of the master Lagrangian method, which we will formulate in the sequel.

A. 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 the dynamical 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 two studied models. Moreover, this method allows us to study the duality at the quantum level more directly.
Let us start from Lagrangian density L T M in the form (25) written explicitly in terms of the fundamental fields A µ and B µν . Following [12], we will introduce auxiliary fields Π µ and Λ µν in order to obtain a first-order derivative theory such that where a, b, c and d are constant coefficients to be determined. Note that the presence of mass 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 us to write Substituting (27) and (28) in (26) and imposing L M = L T M , we obtain the relations The same procedure can be performed for the fields A µ and B µν , and we can immediately solve their equations of motion, obtaining the following solutions: being φ and Σ µ arbitrary fields. Now, replacing (31) and (32) in (26) and imposing L M = such that we can immediately fix a = c = χθ/2 so that our master Lagrangian takes the final form Accordingly, the Lagrangian density (35) describes both (1) and (8). This mechanism transforms models without gauge invariance into models with this symmetry by adding terms which 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 δΠ µ = δΛ µν = 0 is now evident, while it was a hidden symmetry in the self-dual formulation. With the master method, we were able to establish the relation of equivalence when the coupling to other dynamical fields is considered and we have a simple formalism 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 free theories. However, it is fundamental to ensure that this dual equivalence is also valid in the presence 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 are composed only of matter fields, represented generically by ψ. The cases involving nonlinear couplings or when the currents depend explicitly on the gauge or self-dual fields are beyond our present scope.
Let us consider the master Lagrangian (35) added by dynamical matter fields ψ linearly coupled to the self-dual sector: where L(ψ) represents a generic Lagrangian density responsible for the dynamics of the matter fields, with the corresponding currents being denoted by J µ and J µν . Note that due 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 not assume any specific form to the matter sector for now.
First, we will remove the dependency on the gauge fields in Eq. (36). Varying the actioń M with respect to the fields A µ and B µν , we obtain their corresponding equations of motion whose solutions are given by and substituting these solutions into Eq. (36) we find L (1) Then, L (1) SD is equivalent to the self-dual theory (1) linearly coupled to the matter, as expected.
Next, we will eliminate the fields Π µ and Λ µν from the master Lagrangian L Replacing Eqs. (40) and (41) into the master Lagrangian then implies L (1) From the above result, it is clear that the Lagrangian density L T M has appeared before in [44]. However, the approach used in [44] was based on the gauge embedding method, different from the one developed here. Also, one may verify that the equations of motion 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 identification It is worth noting that the duality symmetry between SD B∧F /T M B∧F theories exchanges linear couplings Π µ J µ and Λ µν J µν , involving currents not necessarily conserved in the selfdual sector into derivative dual couplings A µ ǫ µναβ ∂ ν J αβ and B µν ǫ µναβ ∂ α J β in the gauge sector, whose associated currents are automatically conserved. Moreover, self-interaction matter terms are naturally generated, which will play a decisive role in ensuring the duality in 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 Lagrangian densities (39) and (42) are equivalent since the SD B∧F and T M B∧F fields obey the same equations of motion in the presence of external sources. However, for this equivalence between the models to be complete, it is also necessary to verify what happens in the matter sector, when these sources are dynamics.
To this end, we now consider the equation of motion for the matter field ψ. First, let us focus our attention on the SD B∧F model described by (39), so δ δψˆd 4 xL (1) where δL(ψ) δψ is the Lagrangian derivative. On the other hand, the equations of motion for the fields Π µ and Λ µν are: and obey the constraints Inserting (47) into (46), we can eliminate Λ µν in favor of Π µ and obtain a second-order differential equation as where we used the constraint m 2 ∂ µ Π µ = −∂ µ J µ . Defining the wave-operator asR −1 = + m 2 θ 2 , we can write A similar procedure for the field Λ µν results in Replacing the solutions (51) and (52) back in the matter equation (45), we come to the This is a non-local differential equation, expressed only in terms of the matter fields. (1) 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 , These equations can be decoupled, and after some algebraic manipulations we get the following resultsH Substituting these solutions in Eq. (54) we obtain Using the definitionR −1 = + m 2 θ 2 , we can write = R −1 − m 2 θ 2 which implies By comparing Eqs. (53) and (60), we conclude that the matter sectors of the two models give rise to the same equations of motion. Thus, we have shown that the Lagrangians L In order to liken our results with the literature, let us consider, as a particular case, a fermionic matter field minimally coupled to the self-dual field Π µ . Assuming the following identifications: where M is the Dirac field mass, and the fermionic currents are with e being a dimensionless coupling constant. The equation of motion for ψ (60) takes the simple form 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 equations of motion, we now check whether this duality is preserved at the quantum level. For this purpose, we adopt the path-integral framework and define the master generating functional as where N is a overall normalization constant. Our aim is to evaluate the effective Lagrangian resulting from the integration over the fields. Firstly, let us integrate out the contribution of the SD B∧F fields.
To integrate over the fields configurations A µ and B µν , let us first note that the master Lagrangian L M can be rewritten, up to surface terms, as 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, such that the A µ and B µν fields decouple. Then, performing the function integration yields to the following generating functional with L (2) corresponding to the same Lagrangian density (39) previously obtained. It is worth highlighting the physical implications contained in (68). We clearly see that the master Lagrangian L M obtained in (35) is equivalent to self-dual Lagrangian L SD added by a purely topological B ∧ F term, which makes evident the role of the master Lagrangian on the duality symmetry.
The implications of the above results at the quantum level can be explored by considering the 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 where the wave operator,Ô, form a 2 × 2 matrix, X, and J represent vector-tensor duplet of type To accomplish the functional integration, we use the Gaussian path integral formula over a bosonic field X, In our case, the determinant Det −iÔ is field-independent and can be absorbed by the normalization constant. The calculation of propagatorsÔ −1 is rather lengthy, and the details are in Appendix A. Here we just write the results and where Θ µν , ω µν , S µνα , P µν,αβ and P (2) µν,αβ are projection operators whose definitions and closed algebras 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 2 + m 2 = 0, and confirm that the particle spectrum of both theories are equivalent, so that we may consider the 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 functional integration in (66) and (70). After completing all tensorial contractions, we obtain the same effective Lagrangian for the matter field It is easy to verify that the equation of motion for the matter field obtained from L It is worth mentioning that the dynamics of the matter fields is preserved in the functional integration in (66) only if the Thirring-like interactions are added to the diagonal elements ofÔ −1 T M matrix. Besides, the gauge-dependent parts involving the gauge fixing parameters are canceled, as it should be.

V. CONCLUSION
In this work, we revisited the duality between the self-dual and topologically massive models involving the B ∧ F term in 3 + 1 spacetime dimensions. The study of this duality when couplings with fermionic matter are included was first carried out in [44], through the gauge embedding formalism. Here, we considered another approach, namely the master action method, whereby we obtained a fundamental Lagrangian density that interpolates between the two models and provides direct proof of dual equivalence at both the classical and quantum level. The master action enabled us to relate the equations of motion of these models via a dual map among fields and currents of both theories, which ensures that they are equivalent at the classical level. In addition, we demonstrated the duality at quantum through the path-integral framework. We defined a master generating functional wherein the integration over the different fields provided effective Lagrangians that are the same as those obtained classically. Moreover, after a last functional integration over the bosonic fields, we obtained an effective non-local Lagrangian for the matter fields, which proves the equivalence between the matter sectors of the analyzed models.
We assumed that the external currents are linearly coupled with the self-dual fields and are constituted exclusively of the matter fields. We show that these interactions induce "magnetic" couplings involving the gauge fields, in addition to current-current Thirring-like interactions. These types of couplings are, in general, non-renormalizable by direct power counting [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 1 /N perturbative expansion when the matter field is an N-component fermionic field, such that the theory becomes renormalizable. An explicit verification of this issue, as well as a possible extension of our results to the supersymmetric case [23,35], are themes for forthcoming works.
where the first two terms represent a gauge-invariant Maxwell-Kalb-Ramond theory, while the last is a topological B ∧ F term. The calculation of the Feynman propagator for the theory (A1) can be performed as follows.
First, let us rewrite the integrand in Eq. (A1) on the matrix form with the wave operator,Ô T M , being a 2×2 matrix, and X represents a column vector-tensor as Adding convenient gauge-fixing terms in (A2), namely, − 1 2λ (∂ µ A µ ) 2 and 1 2ξ (∂ µ B µν ) 2 , we can explicitly write the operatorÔ T M +gf , in the form, where we have introduced the set of spin-projection operators as
From these preliminary definitions, we obtain a system of four equations, whose solutions can be written as we get