It is well known that three-dimensional gravity is a good test theory for trying to understand the difficulties that emerge in the quantization of four-dimensional gravity. Since the work by Achucarro, Towsend, Witten and other authors [1, 2], there has been a huge effort for understanding the classical and quantum connection between gravity and gauge theories, such as three-dimensional Chern–Simons theory [3]; then it is expected that the insight gained in the three-dimensional case could be useful for constructing better tools and apply them in the quantization of the four-dimensional theory. In this respect, it is common to obtain in three dimensions a relation between the Palatini and the Chern–Simons theories, which are equivalent at the Lagrangian level up to a total derivative [2–4]. Such a relation is not the only one, since the so-called exotic action with a Barbero–Immirzi like parameter [we call it from now on the Bonzom–Livine action (BL)] turns out to be classically equivalent to Palatini’s theory as well [5]. In fact, the BL model describes a set of actions sharing the equations of motion with Palatini’s theory; however, the symplectic structure is different. The symplectic structure in the BL model depends on a Barbero–Immirizi like parameter, from which one expects that the quantum theories will be different [4]. In this respect, something similar happens in the four-dimensional case with the Holst action [6], which provides a set of actions classically equivalent to Einstein’s theory; it depends on the so-called Barbero–Immirizi (γ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\gamma )$$\end{document} parameter whose contribution can be appreciated at the classical level in the symplectic structure of the theory and the coupling of fermionic matter with gravity. In fact, it determines the coupling constant of a four-fermion interaction [7]. From the quantum point of view, the parameter gives a contribution in the quantum spectra of the area and volume operators in the context of loop quantum gravity [8–10]. Furthermore, the term added by Holst to Palatini’s action facilitates the canonical description of General Relativity, and depending of the values of that parameter we can reproduce the different scenarios found in canonical gravity. For instance, it is possible to obtain the ADM, Ashtekar and Barbero formulations in a straightforward way [6]. Nevertheless, in spite of the Holst action providing a general action for gravity, the γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} parameter is still controversial [8–10]. In this sense, the BL action comes to be the three-dimensional equivalent model of Holst’s action. In fact, the equivalence is not given only with the presence of a γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} parameter, but at the classical level if we perform a partial gauge fixing in the canonical description of BL, it is possible obtain a full Ashtekar connection dynamics [4]. In this respect, the analysis of the symmetries of the BL action has been performed in [4, 5]; in this work the canonical analysis was performed by using the Dirac method. However, the analysis was developed on a smaller phase space and the complete structure of the constraints on the full phase space was not reported.

It is important to remark that if some of the Dirac steps is omitted, then it is possible to obtain incomplete results [11, 12]. In this manner, an analysis developed on the full phase space and following all the Dirac steps is mandatory. Nevertheless, in some cases, to develop the Dirac method for gauge theories is a large and tedious task, hence, it is necessary to use alternative formulations that could give us a complete canonical description of the theory, in this sense, there is a different approach for studying gauge theories, the so-called Faddeev–Jackiw [FJ] formalism [13]. The FJ method is a symplectic approach, namely, all relevant information of the theory can be obtained through an invertible symplectic tensor, which is constructed by means of the symplectic variables identified as the degrees of freedom. Because of the theory is singular there will be constraints, and the FJ framework has the advantage that all constraints of the theory are to be treated on the same footing, namely, it is not necessary to perform the classification of the constraints in primary, secondary, first class or second class as in Dirac’s method [14, 15]. When a symplectic tensor is obtained, then its components are identified with the FJ generalized brackets; at the end Dirac’s brackets and FJ brackets coincide.

In this paper a complete Dirac and FJ analysis of the BL model is performed. In fact, in order to compare the two approaches, it is necessary to work out a complete Dirac analysis. Hence, we need to know the complete structure of the constraints over the full phase space for constructing the Dirac brackets and to compare these brackets with the generalized FJ ones [16]. Furthermore, we shall prove that the FJ approach is more economic than Dirac’s one.

It is important to comment that our results have not been reported in the literature, and as a special case those results reported in [4, 5] are reproduced. In addition, we would also remark that for BL theory we shall construct the Dirac brackets by eliminating the second class constraints and there will remain first class ones. Furthermore, at the end of the paper, we have added an appendix where the analysis of an Abelian BL theory is performed; in that appendix, we construct the Dirac brackets by fixing the gauge and also we reproduce all those results by means of FJ formalism.

The paper is organized as follows; in Sect. 2 a detailed canonical analysis of BL is performed. We report the complete structure of the constraints defined on the full phase space, then we eliminate the second class constraints for constructing the Dirac brackets. In Sect. 3, we study the relation between BL and Chern–Simons theory. We reproduce the results of the previous section by performing a pure Dirac analysis of a generalized Chern–Simons theory. In Sect. 4, a detailed FJ analysis of BL action is developed. In order to reproduce all the Dirac results, we work with the configuration space fields as symplectic variables, and identifying all the constraints of the theory we show that the FJ generalized and Dirac brackets coincide. In Sect. 5 we add some remarks and conclusions.

In this section, we will study the Hamiltonian dynamics of the action proposed by BL [5]. We will perform our analysis by using a pure Dirac method, namely, we will find all constraints defined on the full phase space. As was commented on above, there is an analysis of the BL action developed on a smaller phase space reported in [4, 5]. However, in this work the structure of the constraints is not complete, thus, in order to compare the FJ method with the Dirac one it is mandatory to perform the Dirac analysis on the full phase space by following all Dirac steps.

It is well known that three-dimensional gravity with a cosmological constant can be written as a Chern–Simons theory [1–5]. In fact, if the principal gauge bundle G over M is given by G=SU(2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=\mathrm{SU}(2)$$\end{document} for 3d Euclidean gravity, then we can enlarge the group G to G~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{G}$$\end{document}, where G~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{G}$$\end{document} could be SO(4), ISO(3) or SO(3,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{SO}(3,1)$$\end{document}, depending on the sign of the cosmological constant Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}, it being positive, zero or negative respectively. Hence, the algebra of the generators of G~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{G}$$\end{document} will satisfy the following commutation relations [4, 5]:1[Ji,Jj]=ϵijkJk[Ji,Kj]=ϵijkKkKi,Kj=sϵijkJk,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&[ J_i, J_j ]= \epsilon _{ij}{^{k}}J_k \quad [ J_i, K_j ]= \epsilon _{ij}{^{k}}K_k \nonumber \\&\left[ K_i, K_j \right] =s \epsilon _{ij}{^{k}}J_k, \end{aligned}$$\end{document}where s=-1,0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=-1, 0, 1$$\end{document}, corresponding to the sign of the cosmological constant i,j,k=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i, j, k=1, 2, 3$$\end{document} and Ji\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_i$$\end{document}, Ki\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_i$$\end{document} are rotations and boosts respectively. In order to construct a Chern–Simons theory being equivalent to standard Einstein’s action of gravity, we choose the following non-degenerate invariant bilinear form:2⟨Ji,Kj⟩=δij,⟨Ji,Jj⟩=⟨Ki,Kj⟩=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \langle J_i, K_j \rangle =\delta _{ij}, \quad \langle J_i, J_j \rangle = \langle K_i, K_j \rangle =0, \end{aligned}$$\end{document}in this manner, the 3d Palatini’s action with cosmological constant can be written as3SPalatini′=1|Λ|∫Mϵμνρ⟨Aμ,∂νAρ⟩+13⟨Aμ,[Aν,Aρ]⟩.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S'_{\mathrm{Palatini}}= \frac{1}{\sqrt{|\Lambda |}} \int _M \epsilon ^{\mu \nu \rho } \left( \langle A_\mu , \partial _\nu A_\rho \rangle + \frac{1}{3} \langle A_\mu , [A_ \nu , A_\rho ] \rangle \right) .\nonumber \\ \end{aligned}$$\end{document}On the other hand, if Λ≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda \ne 0$$\end{document}, then there is another invariant non-degenerate bilinear form, given by4(Ji,Jj)=δij(Ki,Kj)=sδij(Ji,Kj)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (J_i, J_j)=\delta _{ij} \quad (K_i, K_j)=s\delta _{ij} \quad (J_i, K_j)=0, \end{aligned}$$\end{document}in this case, we can obtain from the following Chern–Simons action:5S~Exotic=1|Λ|∫MϵμνρAμ,∂νAρ+13Aμ,[Aν,Aρ],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{S}_{\mathrm{Exotic}}= & {} \frac{1}{\sqrt{|\Lambda |}} \int _M \epsilon ^{\mu \nu \rho } \left( \left( A_\mu , \partial _\nu A_\rho \right) \right. \nonumber \\&\left. + \, \frac{1}{3} \left( A_\mu , [A_ \nu , A_\rho ] \right) \right) , \end{aligned}$$\end{document}the so-called exotic action for gravity,6S~[A,e]=1∣Λ∣∫MAi∧dAi+13ϵijkAi∧Aj∧Ak+s∣Λ∣∫Mei∧dAei,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \widetilde{S}[A,e]= & {} \frac{1}{\sqrt{\mid \Lambda \mid }}\left[ \int _M A^{i} \wedge d A_{i} + \frac{1}{3}\epsilon _{ijk}A^{i}\wedge A^{j} \wedge A^{k}\right] \nonumber \\&+ \, s\sqrt{\mid \Lambda \mid }\int _M e^{i}\wedge d_{A}e_{i}, \end{aligned}$$\end{document}where the 1-form Ai=Aμidxμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^{i}=A{_{\mu }}^{i}dx^{\mu }$$\end{document}, (dAv)i=dvi+[A,v]i=dvi+ϵijkAj∧vk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(d_{A}v)^{i}= dv^{i}+ [A,v]^{i}=dv^{i}+\epsilon {^{i}}_{jk}A^{j}\wedge v^{k} $$\end{document} and Fi=dAi+12ϵijkAj∧Ak\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ F^{i}=dA^{i} +\frac{1}{2} \epsilon ^{i}{_{jk}}A{^{j}} \wedge A{^{k}}$$\end{document} is the strength 2-form. In this manner, the BL model consists of considering the combination of Palatini’s and exotic action through the γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} parameter, this being a kind of Barbero–Immirzi parameter,7Sγ[A,e]=SPalatini′[A,e]+1γS~Exotic[A,e].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S{_{\gamma }}[A,e]=S_{\mathrm{Palatini}}'[A,e]+\frac{1}{\gamma }\widetilde{S}_{\mathrm{Exotic}}[A,e]. \end{aligned}$$\end{document}In fact, from the action (7) we obtain a family of theories classically equivalent to 3d gravity in the sense that Palatini’s theory with cosmological constant and BL actions share the equations of motion, which can be seen from the variation of the action (7),8δSγ[A,e]δeμi:ϵμνρFiνρ[A]+s∣Λ∣2ϵijkeνjeρk+s∣Λ∣γϵμνρDνeiρ=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\delta S{_{\gamma }}[A,e]}{ \delta e_{\mu i}}: \qquad \epsilon ^{\mu \nu \rho }\left[ F{^{i}}_{\nu \rho }[A] +s\frac{\mid \Lambda \mid }{2} \epsilon {^{i}}_{jk}e{_{\nu }}^{j} e{_{\rho }}^{k}\right] \nonumber \\&\quad +s\frac{\sqrt{\mid \Lambda \mid }}{\gamma }\epsilon ^{\mu \nu \rho } D_\nu e{^{i}}_{\rho }=0, \end{aligned}$$\end{document}9δSγ[A,e]δAμi:ϵμνρDνeiρ+1γ∣Λ∣ϵμνρ×Fiνρ[A]+s∣Λ∣2ϵijkeνjeρk=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\frac{\delta S{_{\gamma }}[A,e]}{ \delta A_{\mu i}}: \qquad \epsilon ^{\mu \nu \rho }D_\nu e{^{i}}_{\rho }+\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\epsilon ^{\mu \nu \rho }\nonumber \\&\times \, \quad \left[ F{^{i}}_{\nu \rho }[A]+s\frac{\mid \Lambda \mid }{2} \epsilon {^{i}}_{jk}e{_{\nu }}^{j} e{_{\rho }}^{k}\right] =0, \end{aligned}$$\end{document}and Eqs. (8) and (9) are equivalent to Einstein’s equations. Hence, in order to develop the Hamiltonian analysis, we perform the 2+1 decomposition of the action (7) obtaining10Sγ[e,A]=∫d3x[2ϵ0abδije0i+1γ∣Λ∣A0i×Fjab+s∣Λ∣2ϵjkleakebl+2ϵ0abδijDaebi×A0j+sΛγe0j+2ϵ0abδijebi∂0Aaj+12γ∣Λ∣Abi∂0Aaj+s∣Λ∣2γebi∂0eaj],\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S{_{\gamma }}[e,A]= & {} \int \mathrm{d}^{3}x \bigg [ 2\epsilon ^{0ab}\delta _{ij}\left( e{_{0}}^{i}+\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}A{_{0}}^{i}\right) \nonumber \\&\times \left( F^{j}{_{ab}}+s\frac{\mid \Lambda \mid }{2}\epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b}\right) +2\epsilon ^{0ab}\delta _{ij}D_{a}e{_{b}}^{i}\nonumber \\&\times \left( A^{j}_{0}+s\frac{\sqrt{\Lambda }}{\gamma }e^{j}_{0}\right) + 2\epsilon ^{0ab}\delta _{ij}\left( e^{i}_{b}\partial _{0}A^{j}_{a}\right. \nonumber \\&\left. + \, \frac{1}{2\gamma \sqrt{\mid \Lambda \mid }}A^{i}_{b}\partial _{0}A^{j}_{a}+s\frac{\sqrt{\mid \Lambda \mid }}{2\gamma }e^{i}_{b}\partial _{0}e^{j}_{a}\right) \bigg ],\nonumber \\ \end{aligned}$$\end{document}where a,b,c=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a, b, c= 1, 2$$\end{document}. The definition of the momenta (παi,Παi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\pi ^{\alpha }{_{i}},\Pi ^{\alpha }{_{i}})$$\end{document} canonically conjugate to (eiα,Aαi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$( e{^i}_{\alpha }, A_{\alpha } {^{i}})$$\end{document} is given by11Παi=δLδA˙αi,παi=δLδe˙iα.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Pi ^{\alpha }{_{i}}= \frac{\delta {\mathcal {L}} }{ \delta \dot{A}_{\alpha } {^{i}} }, \quad \pi ^{\alpha }{_{i}}= \frac{\delta {\mathcal {L}} }{ \delta \dot{e}{^i}_\alpha } . \end{aligned}$$\end{document}The matrix elements of the Hessian12∂2L∂(∂μeαi)∂(∂μeβi),∂2L∂(∂μeαi)∂(∂μAβi),∂2L∂(∂μAαi)∂(∂μAβi),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned}&\frac{\partial ^2{\mathcal {L}} }{\partial (\partial _\mu e^{i}_\alpha ) \partial (\partial _\mu e^{i}_\beta )} , \quad \frac{\partial ^2{\mathcal {L}} }{\partial ( \partial _\mu e^{i}_\alpha ) \partial (\partial _\mu A_{\beta } {^{i}} )}, \\&\quad \frac{\partial ^2{\mathcal {L}} }{\partial (\partial _\mu A_{\alpha } {^{i}} ) \partial (\partial _\mu A_{\beta } {^{i}} ) }, \end{aligned} \end{aligned}$$\end{document}are identically zero, thus we expect 18 primary constraints. From the definition of the momenta (11) we identify the following 18 primary constraints:13ϕi0:=πi0≈0,ϕia:=πia-s∣Λ∣γϵ0abδijebj≈0,Φi0:=Πi0≈0,Φia:=Πia-2ϵ0abδijebj+12γ∣Λ∣Abj≈0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \phi {_{i}}^{0}:= & {} \pi {_{i}}^{0} \approx 0 ,\nonumber \\ \phi {_{i}}^{a}:= & {} \pi {_{i}}^{a}- s\frac{\sqrt{\mid \Lambda \mid }}{\gamma }\epsilon ^{0ab}\delta _{ij} e{_{b}}^{j} \approx 0 ,\nonumber \\ \Phi {_{i}}^{0}:= & {} \Pi {_{i}}^{0} \approx 0 ,\nonumber \\ \Phi {_{i}}^{a}:= & {} \Pi {_{i}}^{a}- 2\epsilon ^{0ab}\delta _{ij}\left( e{_{b}}^{j}+\frac{1}{2\gamma \sqrt{\mid \Lambda \mid }}A{_{b}}^{j}\right) \approx 0.\nonumber \\ \end{aligned}$$\end{document}The canonical Hamiltonian takes the form14Hc=∫dx2-2ϵ0abδijDaeibA0i+sΛγe0i-2ϵ0abδije0i+1γ∣Λ∣A0i×Fjab+s∣Λ∣2ϵjkleakelb,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H_\mathrm{c}= & {} \int \mathrm{d}x^2 \left[ -2\epsilon ^{0ab}\delta _{ij} D_{a}e{^{i}}_{b}\left( A^{i}_{0}+s\frac{\sqrt{\Lambda }}{\gamma }e^{i}_{0}\right) \right. \nonumber \\&\left. - \, 2\epsilon ^{0ab}\delta _{ij}\left( e{_{0}}^{i}+\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}A{_{0}}^{i}\right) \right. \nonumber \\&\times \left. \left( F^{j}{_{ab}}+s\frac{\mid \Lambda \mid }{2}\epsilon ^{j}{_{kl}}e^{k}_{a}e{^{l}}_{b}\right) \right] , \end{aligned}$$\end{document}and the primary Hamiltonian is given by15HP=Hc+∫dx2λiαϕiα+ξiαΦiα,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} H_\mathrm{P}= H_\mathrm{c} + \int \mathrm{d}x^2 \left[ \lambda ^{i}{_{\alpha }} \phi {_{i}}^{\alpha } + \xi ^{i}{_{\alpha }} \Phi {_{i}}^{\alpha } \right] , \end{aligned}$$\end{document}where λiα,ξiα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \lambda ^{i}{_{\alpha }}, \xi ^{i}{_{\alpha }}$$\end{document} are Lagrange multipliers enforcing the constraints (ϕiα,Φiα)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\phi {_{i}}^{\alpha }, \Phi {_{i}}^{\alpha })$$\end{document}. The fundamental Poisson brackets of the theory are given16eαi(x),πβj(y)=δβαδijδ2(x-y),Aαi(x),Πβj(y)=δβαδijδ2(x-y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left\{ e_{\alpha } {^{i}}(x),\pi ^{\beta }{_{j}}(y) \right\} = \delta {^ \beta } _\alpha \delta {^ i} _j \delta ^2(x-y), \nonumber \\&\left\{ A_{\alpha } {^{i}}(x),\Pi ^{\beta }{_{j}}(y) \right\} =\delta {^ \beta } _\alpha \delta {^ i} _j \delta ^2(x-y), \end{aligned}$$\end{document}where we can observe that in these fundamental brackets there is not any contribution of the γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document} parameter; in the Dirac brackets, however, there will be a non-trivial contribution. In order to determine the presence of more constraints, we calculate the following 18 ×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 18 matrix whose entries are the Poisson brackets among the constraints (13):17ϕia(x),ϕjb(y)=-2s∣Λ∣γϵ0abδijδ2(x-y),ϕia(x),Φjb(y)=-2ϵ0abδijδ2(x-y),Φia(x),Φjb(y)=-21γ∣Λ∣ϵ0abδijδ2(x-y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left\{ \phi {_{i}}^{a} (x),\phi {_{j}}^{b} (y) \right\} = -2s\frac{\sqrt{\mid \Lambda \mid }}{\gamma }\epsilon ^{0ab} \delta _{ij}\delta ^2(x-y), \nonumber \\&\left\{ \phi {_{i}}^{a} (x),\Phi {_{j}}^{b} (y) \right\} = -2\epsilon ^{0ab}\delta _{ij}\delta ^2(x-y),\\&\left\{ \Phi {_{i}}^{a} (x),\Phi {_{j}}^{b} (y) \right\} = -2\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\epsilon ^{0ab}\delta _{ij}\delta ^2(x-y),\nonumber \end{aligned}$$\end{document}we appreciate that this matrix has rank = 12 and six null vectors. By using the six null vectors and consistency conditions we obtain the following six secondary constraints:18γi0=πi0≈0,γ~i0=Πi0≈0,ϕ˙i0={ϕi0(x),HP}≈0⇒ψi:=2ϵ0abs∣Λ∣γDaeib+2ϵ0abFiab+s∣Λ∣2ϵijkejaekb≈0,Φ˙i0={Φi0(x),HP}≈0⇒Ψi:=2ϵ0abDaeib+2ϵ0ab1γ∣Λ∣Fiab+s∣Λ∣2ϵijkejaekb≈0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma {_{i}}^{0}= & {} \pi {_{i}}^{0} \approx 0, \nonumber \\ \tilde{\gamma }{_{i}}^{0}= & {} \Pi {_{i}}^{0} \approx 0, \nonumber \\ \dot{\phi }{_{i}}^{0}= & {} \{\phi {_{i}}^{0} (x), {H}_{\mathrm{P}} \} \approx 0 \quad \Rightarrow \quad \psi _{i}:= 2\epsilon ^{0ab}s\frac{\sqrt{\mid \Lambda \mid }}{\gamma } D_{a}e_{ib}\nonumber \\&+ \, 2\epsilon ^{0ab}\left( F_{iab}+\frac{s\mid \Lambda \mid }{2}\epsilon {_{ijk}} e{^{j}}_{a}e{^{k}}_{b}\right) \approx 0, \nonumber \\ \dot{\Phi }{_{i}}^{0}= & {} \{\Phi {_{i}}^{0} (x), {H}_{\mathrm{P}} \} \approx 0 \quad \Rightarrow \quad \Psi _{i}:=2\epsilon ^{0ab} D_{a}e_{ib}\nonumber \\&+ \, 2\epsilon ^{0ab}\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\left( F_{iab} +\frac{s\mid \Lambda \mid }{2}\epsilon {_{ijk}}e^{j}{_{a}}e^{k}{_{b}}\right) \approx 0,\nonumber \\ \end{aligned}$$\end{document}and the rank allows us to fix the following values for the Lagrangian multipliers:19ϕ˙ia={ϕia,HP}≈0⇒2ϵ0abs-γ2γ∣Λ∣(-λbi+Dbe0i+ϵlimebmA0l)≈0,Φ˙ia={Φia,HP}≈0⇒2ϵ0abs-γ2γ2(-ξbi+DbA0i+s∣Λ∣ϵlimebme0l)≈0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\dot{\phi }{_{i}}^{a}=\{\phi {_{i}}^{a},H_{\mathrm{P}}\}\approx 0 \nonumber \\&\quad \Rightarrow 2\epsilon ^{0ab}\frac{s-\gamma ^{2}}{\gamma }\sqrt{\mid \Lambda \mid } (-\lambda _{bi}+D_{b}e_{0i}+\epsilon _{lim}e^{m}_{b}A^{l}_{0})\approx 0, \nonumber \\&\dot{\Phi }{_{i}}^{a}=\{\Phi {_{i}}^{a},H_{\mathrm{P}}\}\approx 0 \nonumber \\&\quad \Rightarrow 2\epsilon ^{0ab}\frac{s-\gamma ^{2}}{\gamma ^{2}}(-\xi _{bi}+D_{b}A_{0i} +s\mid \Lambda \mid \epsilon _{lim}e^{m}_{b}e^{l}_{0})\approx 0.\nonumber \\ \end{aligned}$$\end{document}Consistency requires conservation in time of the secondary constraints, however, for this theory there are no tertiary constraints. At this point, we need to identify from the primary and secondary constraints which ones correspond to first and second class. To this aim, we need to calculate the rank and the null vectors of the 24×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times $$\end{document} 24 matrix whose entries will be the Poisson brackets between primary and secondary constraints[ this is20{ϕia(x),ϕjb(y)}=-2s∣Λ∣γϵ0abδijδ2(x-y),{ϕia(x),Φj(y)}=-2ϵ0abδijδ2(x-y),{ϕia(x),ψj(y)}=-2ϵ0abs∣Λ∣γδij∂b-ϵijk×s∣Λ∣γAkb+s∣Λ∣ekbδ2(x-y),{ϕia(x),Ψj(y)}=-2ϵ0abδij∂b-ϵijkAkb+s∣Λ∣γekbδ2(x-y),{Φia(x),Φjb(y)}=-21γ∣Λ∣ϵ0abδijδ2(x-y),{Φia(x),ψj(y)}=-2ϵ0abδij∂b-ϵijk×Akb+s∣Λ∣γekbδ2(x-y),{Φia(x),Ψj(y)}=-2ϵ0ab1γ∣Λ∣δij∂b-ϵijk×1γ∣Λ∣Akb+ekbδ2(x-y).{ψi(x),ψj(y)}=0,{Ψi(x),Ψj(y)}=0,{ψi(x),Ψj(y)}=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\{\phi {_{i}}^{a} (x),\phi {_{j}}^{b} (y) \} = -2s\frac{\sqrt{\mid \Lambda \mid }}{\gamma }\epsilon ^{0ab}\delta _{ij}\delta ^{2}(x-y), \nonumber \\&\{\phi {_{i}}^{a} (x),\Phi _{j} (y)\} = -2\epsilon ^{0ab}\delta _{ij}\delta ^{2}(x-y),\nonumber \\&\{\phi {_{i}}^{a} (x),\psi {_{j}}(y) \} =-2\epsilon ^{0ab}\left[ \frac{s\sqrt{\mid \Lambda \mid }}{\gamma }\delta _{ij}\partial _{b}-\epsilon _{ijk}\right. \nonumber \\&\quad \times \left. \left( \frac{s\sqrt{\mid \Lambda \mid }}{\gamma }A{^{k}}_{b}+s\mid \Lambda \mid e{^{k}}_{b}\right) \right] \delta ^2(x-y) , \nonumber \\&\{\phi {_{i}}^{a} (x), \Psi _{j}(y) \} = -2\epsilon ^{0ab}\left[ \delta _{ij}\partial _{b}-\epsilon _{ijk} \left( A{^{k}}_{b}\right. \right. \nonumber \\&\quad \left. \left. + \frac{s\sqrt{\mid \Lambda \mid }}{\gamma }e{^{k}}_{b}\right) \right] \delta ^2(x-y) , \nonumber \\&\{\Phi {_{i}}^{a} (x),\Phi {_{j}}^{b}(y) \} = -2\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\epsilon ^{0ab} \delta _{ij}\delta ^2(x-y),\\&\{\Phi {_{i}}^{a} (x),\psi _{j}(y) \} =-2\epsilon ^{0ab}\left[ \delta _{ij}\partial _{b} -\epsilon _{ijk}\right. \nonumber \\&\quad \times \left. \left( A{^{k}}_{b}+ \frac{s\sqrt{\mid \Lambda \mid }}{\gamma }e{^{k}}_{b}\right) \right] \delta ^2(x-y),\nonumber \\&\{\Phi {_{i}}^{a} (x),\Psi _{j}(y) \} =-2\epsilon ^{0ab}\left[ \frac{1}{\gamma \sqrt{\mid \Lambda \mid }} \delta _{ij}\partial _{b}-\epsilon _{ijk}\right. \nonumber \\&\quad \times \left. \left( \frac{1}{\gamma \sqrt{\mid \Lambda \mid }}A{^{k}}_{b} +e{^{k}}_{b}\right) \right] \delta ^{2}(x-y).\nonumber \\&\{\psi _{i} (x),\psi _{j}(y) \} = 0,\nonumber \\&\{\Psi _{i} (x),\Psi _{j}(y) \} = 0,\nonumber \\&\{\psi _{i} (x),\Psi _{j}(y) \} =0.\nonumber \end{aligned}$$\end{document}This matrix has a rank = 12 and 12 null vectors, thus the theory presents a set of 12 first class constraints and 12 second class constraints. By using the contraction of the null vectors with the constraints (13) and (18), we identify the following 12 first class constraints:21γ0i=πi0≈0,Γ0i=Πi0≈0,ωi=Daχia-s∣Λ∣ϵijkekaΞja+2ϵ0abs∣Λ∣γDaeib+2ϵ0abFiab+s∣Λ∣2ϵijkejaekb≈0,Γi=DaΞia-ϵijkekaχja+2ϵ0abDaeib+2ϵ0ab1γ∣Λ∣Fiab+s∣Λ∣2ϵijkejaekb≈0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \gamma ^{0}{_{i}}= & {} \pi {_{i}}^{0} \approx 0, \nonumber \\ \Gamma ^{0}{_{i}}= & {} \Pi {_{i}}^{0} \approx 0, \nonumber \\ \omega _{i}= & {} D_a\chi {_i}^a-s\mid \Lambda \mid \epsilon {_{i}}{^{j}}_{k}e{^{k}}_{a}\Xi {_{j}}^{a}+2\epsilon ^{0ab}s\frac{\sqrt{\mid \Lambda \mid }}{\gamma } D_{a}e_{ib}\nonumber \\&+\, 2\epsilon ^{0ab}\left( F_{iab}+\frac{s\mid \Lambda \mid }{2} \epsilon {_{ijk}}e{^{j}}_{a}e{^{k}}_{b}\right) \approx 0, \nonumber \\ \Gamma _{i}= & {} D_a\Xi {_i}^a-\epsilon {_{i}}{^{j}}_{k} e{^{k}}_{a}\chi {_{j}}^{a}+2\epsilon ^{0ab}D_{a}e_{ib}\nonumber \\&+ \, 2\epsilon ^{0ab}\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\left( F_{iab}+\frac{s\mid \Lambda \mid }{2} \epsilon {_{ijk}}e{^{j}}_{a}e{^{k}}_{b}\right) \approx 0,\nonumber \\ \end{aligned}$$\end{document}and the following 12 second class constraints:22χia=πia-s∣Λ∣γϵ0abδijebj≈0,Ξia=Πia-2ϵ0abδijebj+12γ∣Λ∣Abj≈0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{aligned} \chi {_{i}}^{a}&= \pi {_{i}}^{a}- s\frac{\sqrt{\mid \Lambda \mid }}{\gamma }\epsilon ^{0ab}\delta _{ij} e{_{b}}^{j} \approx 0, \\ \Xi {_{i}}^{a}&= \Pi {_{i}}^{a}- 2\epsilon ^{0ab}\delta _{ij}\left( e{_{b}}^{j}+\frac{1}{2\gamma \sqrt{\mid \Lambda \mid }}A{_{b}}^{j}\right) \approx 0. \end{aligned} \end{aligned}$$\end{document}It is important to remark that these constraints have not been reported in the literature, and their complete structure defined on the full phase space will be relevant in order to know the fundamental gauge transformations and for constructing the Dirac brackets. On the other hand, the constraints will play a key role to make the progress in the quantization. We have commented above that the structure of the constraints (21) is obtained by means of the null vectors, for instance, a set of null vectors of the matrix (20) are given byVi1=0,-δji∂aδ2(x-y)-ϵjkiAakδ2(x-y),0,-s∣Λ∣ϵijkeakδ2(x-y),δjiδ2(x-y),0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} V^{1}_i= & {} \left( 0,-\delta ^{i}_{j}\partial _{a}\delta ^{2}(x-y) -\epsilon ^{i}_{jk}A^{k}_{a}\delta ^{2}(x-y),0,\right. \nonumber \\&\left. -s\mid \Lambda \mid \epsilon _{i}{^{j}}_{k}e^{k}_{a}\delta ^{2}(x-y), \delta ^{i}_{j}\delta ^{2}(x-y),0\right) , \end{aligned}$$\end{document}hence, by performing the contraction of these null vectors with the primary and secondary constraints, the first class constraint ωi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _i$$\end{document} given in (21) is obtained. Now, we will calculate the algebra of the constraints23{χia(x),χjb(y)}=-2s∣Λ∣γϵ0abδijδ2(x-y),{χia(x),Ξjb(y)}=-2ϵ0abδijδ2(x-y),{Ξia(x),Ξjb(y)}=-21γ∣Λ∣ϵ0abδijδ2(x-y),{χia(x),ωj(y)}=s∣Λ∣ϵijkΦakδ2(x-y)≈0,{Ξia(x),ωj(y)}=ϵijkϕakδ2(x-y)≈0,{χia(x),Γj(y)}=ϵijϕkaδ2(x-y)≈0,{Ξia(x),Γj(y)}=ϵijkΦkaδ2(x-y)≈0,{ωi(x),ωj(y)}=s∣Λ∣ϵijkΓkδ2(x-y)≈0,{Γi(x),Γj(y)}=ϵijkΓkδ2(x-y)≈0,{ωi(x),Γj(y)}=ϵijkωkδ2(x-y)≈0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \{\chi {_{i}}^{a} (x),\chi {_{j}}^{b} (y) \}= & {} -2s\frac{\sqrt{\mid \Lambda \mid }}{\gamma }\epsilon ^{0ab} \delta _{ij}\delta ^{2}(x-y), \nonumber \\ \{\chi {_{i}}^{a} (x),\Xi {_{j}}^{b} (y) \}= & {} -2\epsilon ^{0ab}\delta _{ij}\delta ^{2}(x-y), \nonumber \\ \{ \Xi {_{i}}^{a} (x), \Xi {_{j}}^{b}(y) \}= & {} -2\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\epsilon ^{0ab}\delta _{ij} \delta ^2(x-y),\nonumber \\ \{\chi {_{i}}^{a} (x),\omega _{j} (y) \}= & {} s\mid \Lambda \mid \epsilon {_{ij}}^{k}\Phi {^{a}}_{k} \delta ^{2}(x-y)\approx 0,\nonumber \\ \{\Xi {_{i}}^{a} (x),\omega _{j} (y) \}= & {} \epsilon {_{ij}}^{k}\phi {^{a}}_{k}\delta ^{2}(x-y)\approx 0, \\ \{\chi {_{i}}^{a} (x),\Gamma {_{j}}(y) \}= & {} \epsilon {_{ij}}\phi ^{a}_{k}\delta ^2(x-y)\approx 0 , \nonumber \\ \{ \Xi {_{i}}^{a} (x), \Gamma _{j}(y) \}= & {} \epsilon _{ijk}\Phi ^{ka}\delta ^{2}(x-y)\approx 0, \nonumber \\ \{\omega _{i} (x),\omega _{j}(y) \}= & {} s\mid \Lambda \mid \epsilon _{ijk}\Gamma ^{k}\delta ^{2}(x-y)\approx 0, \nonumber \\ \{\Gamma {_{i}} (x),\Gamma _{j}(y) \}= & {} \epsilon _{ijk}\Gamma ^{k} \delta ^2(x-y)\approx 0 , \nonumber \\ \{ \omega _{i} (x), \Gamma _{j}(y) \}= & {} \epsilon _{ijk}\omega ^{k}\delta ^2(x-y)\approx 0.\nonumber \end{aligned}$$\end{document}where we can observe that the algebra is closed. Furthermore, with all the information obtained until now, we can construct the Dirac brackets. In fact, there are two options for constructing them. The first one is by eliminating the second class constraints and keeping on the first class, and the second option is by fixing the gauge and converting the first class constraints into second class ones. In this section we will eliminate the second class constraints, the first class ones remaining; in the FJ approach we will discuss both. Hence, in order to construct the Dirac brackets, we shall use the matrix whose elements are only the Poisson brackets among second class constraints, namely Cαβ(u,v)={ζα(u),ζβ(v)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C{_{\alpha \beta }}(u,v)=\{\zeta ^{\alpha }(u),\zeta ^{\beta }(v)\}$$\end{document}, given by24[C(αβ)(x,x′)]=-2s∣Λ∣γ111γ∣Λ∣δijϵ0abδ2(x-x′),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{}[C_{(\alpha \beta )}(x,x')]= -2\left( \begin{array}{cccc} \frac{s\sqrt{\mid \Lambda \mid }}{\gamma }&{}1\\ 1&{}\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\\ \end{array} \right) \delta _{ij}\epsilon ^{0ab}\delta ^2(x-x'),\nonumber \\ \end{aligned}$$\end{document}its inverse is given by25[C-1(αβ)(x,x′)]=γ22(s-γ2)1γ∣Λ∣-1-1s∣Λ∣γ×δijϵ0baδ2(x-x′).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}{}[C{^{-1}}_{(\alpha \beta )}(x,x')]{^{}}_{}= & {} \frac{\gamma ^{2}}{2(s-\gamma ^{2})}\left( \begin{array}{cc} \frac{1}{\gamma \sqrt{\mid \Lambda \mid }}&{}-1\\ -1&{}\frac{s\sqrt{\mid \Lambda \mid }}{\gamma }\\ \end{array} \right) \nonumber \\&\times \delta ^{ij}\epsilon {_{0ba}}\delta ^2(x-x'). \end{aligned}$$\end{document}The Dirac brackets for two functionals A, B are expressed by26{A(x),B(y)}D={A(x),B(y)}P-∫dudv{A(x),ζα(u)}C-1αβ(u,v){ζβ(v),B(y)},\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\{A(x),B(y)\}_{D}=\{A(x),B(y)\}_{P}\nonumber \\&\quad -\int \mathrm{d}u\mathrm{d}v\{A(x),\zeta ^{\alpha }(u)\}C{^{-1}}_{\alpha \beta }(u,v) \{\zeta ^{\beta }(v),B(y)\},\nonumber \\ \end{aligned}$$\end{document}where {A(x),B(y)}P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{A(x),B(y)\}_{P}$$\end{document} is the usual Poisson bracket between the functionals A, B and ζα(u)=(χia,Ξia)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta ^{\alpha }(u)=(\chi {_{i}}^{a},\Xi {_{i}}^{a})$$\end{document} is the set of second class constraints. Hence, by using (25) and (26) we obtain the following Dirac brackets of the theory:27{eia(x),πbj(y)}D=s2(s-γ2)δbaδijδ2(x-y),{eia(x),ejb(y)}D=12∣Λ∣γs-γ2δijϵ0abδ2(x-y),{πai(x),πbj(y)}D=s22γ∣Λ∣s-γ2δijϵ0abδ2(x-y),{Aia(x),Πbj(y)}D=s-2γ22(s-γ2)δbaδijδ2(x-y),{Aia(x),Ajb(y)}D=s∣Λ∣2γs-γ2δijϵ0abδ2(x-y),{Πai(x),Πbj(y)}D=s2γ∣Λ∣1s-γ2δijϵ0abδ2(x-y),{Aia(x),ejb(y)}D=12γ2γ2-sδijϵ0abδ2(x-y),{eia(x),Πbj(y)}D=12∣Λ∣γs-γ2δbaδijδ2(x-y),{Aia(x),πbj(y)}D=-s∣Λ∣2γs-γ2δbaδijδ2(x-y),{πai(x),Πbj(y)}D=12ss-γ2δijϵ0abδ2(x-y).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \{e{^{i}}_{a}(x),\pi {^{b}}_{j}(y)\}_{D}= & {} \frac{s}{2(s-\gamma ^{2})} \delta {^{b}}_{a}\delta {^{i}}_{j} \delta ^{2}(x-y),\nonumber \\ \{e{^{i}}_{a}(x),e{^{j}}_{b}(y)\}_{D}= & {} \frac{1}{2\sqrt{\mid \Lambda \mid }} \frac{\gamma }{s-\gamma ^{2}}\delta {^{ij}}\epsilon _{0ab}\delta ^{2}(x-y), \nonumber \\ \{\pi {^{a}}_{i}(x),\pi {^{b}}_{j}(y)\}_{D}= & {} \frac{s^{2}}{2\gamma }\frac{\sqrt{\mid \Lambda \mid }}{s-\gamma ^{2}} \delta _{ij}\epsilon {^{0ab}}\delta ^{2}(x-y),\nonumber \\ \{A{^{i}}_{a}(x),\Pi {^{b}}_{j}(y)\}_{D}= & {} \frac{s-2\gamma ^{2}}{2(s-\gamma ^{2})}\delta {^{b}}_{a}\delta {^{i}}_{j}\delta ^{2}(x-y), \nonumber \\ \{A{^{i}}_{a}(x),A{^{j}}_{b}(y)\}_{D}= & {} \frac{s\sqrt{\mid \Lambda \mid }}{2}\frac{\gamma }{s-\gamma ^{2}}\delta ^{ij}\epsilon {_{0ab}}\delta ^{2}(x-y), \nonumber \\ \{\Pi {^{a}}_{i}(x),\Pi {^{b}}_{j}(y)\}_{D}= & {} \frac{s}{2\gamma \sqrt{\mid \Lambda \mid }}\frac{1}{s-\gamma ^{2}}\delta _{ij}\epsilon {^{0ab}} \delta ^{2}(x-y),\nonumber \\ \{A{^{i}}_{a}(x),e{^{j}}_{b}(y)\}_{D}= & {} \frac{1}{2}\frac{\gamma ^{2}}{\gamma ^{2}-s}\delta ^{ij}\epsilon _{0ab}\delta ^{2}(x-y),\nonumber \\ \{e{^{i}}_{a}(x),\Pi {^{b}}_{j}(y)\}_{D}= & {} \frac{1}{2\sqrt{\mid \Lambda \mid }} \frac{\gamma }{s-\gamma ^{2}}\delta {^{b}}_{a}\delta {^{i}}_{j}\delta ^{2}(x-y), \nonumber \\ \{A{^{i}}_{a}(x),\pi {^{b}}_{j}(y)\}_{D}= & {} -\frac{s\sqrt{\mid \Lambda \mid }}{2}\frac{\gamma }{s-\gamma ^{2}}\delta {^{b}}_{a}\delta {^{i}}_{j}\delta ^{2}(x-y), \nonumber \\ \{\pi {^{a}}_{i}(x),\Pi {^{b}}_{j}(y)\}_{D}= & {} \frac{1}{2}\frac{s}{s-\gamma ^{2}}\delta _{ij}\epsilon {^{0ab}}\delta ^{2}(x-y). \end{aligned}$$\end{document}We can observe that the Dirac brackets depend on the constants (s, γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}), and we can reproduce several scenarios depending of specific values. Hence, for s=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s=1$$\end{document} and the limit γ→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow \infty $$\end{document}, we recover the Dirac canonical structure of Palatini’s action [11]. It is important to remark that in the BL model the fields e, A and their canonical momenta have become non-commutative, while in Palatini’s action they are commutative. This is a classical difference between the BL model and Palatini’s theory. Moreover, at the quantum level this difference is fundamental for constructing the Ashtekar representation of the BL model [4].

Now, we can calculate the gauge transformations on the full phase space. In fact, the correct gauge symmetry is obtained according to Dirac’s conjecture by constructing a gauge generator using the first class constraints, and the structure of the constraints defined on the full phase space will give us the fundamental gauge transformations. To this aim, we will apply the Castellani algorithm for constructing the gauge generator; we define the generator of gauge transformations as28G=∫∑D0εi0γ0i+D0τ0iΓ0i+εiωi+τiΓi.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} G=\int _{\sum }\left[ D{_{0}}\varepsilon {^{i}}_{0}\gamma {^{0}}_{i} +D{_{0}}\tau {_{0}}^{i}\Gamma {^{0}}_{i}+\varepsilon ^{i}\omega {_{i}} +\tau ^{i}\Gamma {_{i}}\right] . \end{aligned}$$\end{document}Therefore, we find that the gauge transformations on the phase space are29δ0ei0=D0εi0,δ0eia=-Daεi+ϵijkeakτj,δ0A0i=D0τ0i,δ0Aai=-Daτi+s∣Λ∣ϵijkekaεj,δ0π0i=0,δ0πai=-Ωϵ0abDbεi+ϵijkπakτj+s∣Λ∣ϵijkΞak,δ0Π0i=-ϵijkπ0kεj-Π0kτj,δ0Πai=-2ϵ0abDbεi+ϵijkχakεj-1γ∣Λ∣ϵ0abDbτi+ϵijkΞakτj+2ϵ0abϵijkekbτj+Ωϵ0abϵijkebkεj.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \delta {_{0}} e{^{i}}_{0}= & {} D_{0}\varepsilon {^{i}}_{0},\nonumber \\ \delta {_{0}}e{^{i}}_{a}= & {} -D{_{a}}\varepsilon ^{i}+\epsilon {^{i}}_{jk} e{_{a}}^{k}\tau ^{j},\nonumber \\ \delta {_{0}} A_{0}{^{i}}= & {} D_{0}\tau {_{0}}{^{i}},\nonumber \\ \delta {_{0}}A_{a}{^{i}}= & {} -D{_{a}}\tau ^{i}+s\mid \Lambda \mid \epsilon {^{i}}_{jk} e{^{k}}_{a}\varepsilon ^{j},\nonumber \\ \delta {_{0}} \pi {^{0}}_{i}= & {} 0 ,\nonumber \\ \delta {_{0}} \pi {^{a}}_{i}= & {} -\Omega \epsilon ^{0ab}D_{b}\varepsilon _{i} + \epsilon {_{ij}}^{k}\pi {^{a}}_{k}\tau ^{j}+s\mid \Lambda \mid \epsilon {_{ij}}^{k} \Xi {^{a}}_{k} ,\nonumber \\ \delta {_{0}}\Pi {^{0}}_{i}= & {} -\epsilon {_{ij}}^{k}\left( \pi {^{0}}_{k}\varepsilon ^{j} -\Pi {^{0}}_{k}\tau ^{j}\right) ,\nonumber \\ \delta {_{0}}\Pi {^{a}}_{i}= & {} -2\epsilon ^{0ab}D_{b}\varepsilon _{i} +\epsilon {_{ij}}^{k}\chi {^{a}}_{k}\varepsilon ^{j}\nonumber \\&-\frac{1}{\gamma \sqrt{\mid \Lambda \mid }}\epsilon {^{0ab}}D_{b}\tau _{i} +\epsilon {_{ij}}^{k}\Xi {^{a}}_{k}\tau ^{j}\nonumber \\&+ \, 2\epsilon ^{0ab}\epsilon _{ijk}e{^{k}}_{b}\tau ^{j} +\Omega \epsilon ^{0ab}\epsilon _{ijk}e^{k}_{b}\varepsilon ^{j}. \end{aligned}$$\end{document}We realize that the fundamental gauge transformations of the BL action are given by (29) and they are Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}-deformed ISO(3) transformations. It is important to remark that the internal group of the theory is SU(2) (or SO(3)), however, the fundamental gauge symmetry corresponds to Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}-deformed ISO(3) transformations, all these results were not reported in [4, 5]. On the other hand, any theory with background independence is diffeomorphisms covariant, and this symmetry must be obtained from the fundamental gauge transformations. Hence, the diffeomorphisms can be found by redefining the gauge parameters as ε0i=-εi=ξρeiρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon {_{0}}^{i}=-\varepsilon ^{i}=\xi ^{\rho }e{^{i}}_{\rho }$$\end{document}, τ0i=-τi=ξρAρi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau {_{0}}^{i}=-\tau ^{i}=\xi ^{\rho }A_{\rho }{^{i}}$$\end{document}, and the gauge transformation (29) takes the following form:30e′iα→eiα+Lξeiα+ξρDαeiρ-Dρeiα,A′αi→Aαi+LξAαi+ξρ∂αAρi-∂ρAiα+ϵijkAjαAkρ+s∣Λ∣ϵijkejαekρ.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} e'{^{i}}{_{\alpha }}\rightarrow & {} e{^{i}}{_{\alpha }}+\mathfrak {L}{_{\xi }} e{^{i}}{_{\alpha }}+\xi ^{\rho }\left[ D{_{\alpha }}e{^{i}}{_{\rho }}-D{_{\rho }} e{^{i}}{_{\alpha }}\right] , \nonumber \\ A'{_{\alpha }}^{i}\rightarrow & {} A{_{\alpha }}^{i} +\mathfrak {L}{_{\xi }}A{_{\alpha }}^{i}+\xi ^{\rho }\left[ \partial _{\alpha } A{^{i}_{\rho }}-\partial _{\rho }A{^{i}}_{\alpha }\right. \nonumber \\&+\left. \epsilon _{ijk}A{^{j}}_{\alpha } A{^{k}}_{\rho }+s\mid \Lambda \mid \epsilon {^{i}}_{jk}e{^{j}}_{\alpha } e{^{k}}_{\rho }\right] . \end{aligned}$$\end{document}Therefore, the diffeomorphisms are obtained (on shell) from the fundamental gauge transformations as an internal symmetry of the theory. With the correct identification of the constraints, we can carry out the counting of degrees of freedom in the following form: there are 36 canonical variables (eiα,Aαi,παi,Παi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(e{^i}_{\alpha }, A_{\alpha } {^{i}}, \pi ^{\alpha }{_{i}},\Pi ^{\alpha }{_{i}})$$\end{document}, 12 first class constraints (γi0,Γi0,ωi,Γi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\gamma {_{i}}^{0}, \Gamma {_{i}}^{0}, \omega _{i}, \Gamma _{i})$$\end{document} and 12 second class constraints (χia,Ξia)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\chi {_{i}}^{a}, \Xi {_{i}}^{a})$$\end{document} and one concludes that the Sγ[A,e]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\gamma }[A,e]$$\end{document} action for gravity in three dimensions is devoid of degrees of freedom; therefore, the theory is topological.

As a conclusion of this part, we have obtained the extended action, the extended Hamiltonian, the complete structure of the constraints on the full phase space and the algebra among them. The price to pay for using the complete phase space is that the theory presents a set of first and second class constraints; by using the second class constraints we have constructed the Dirac brackets and they will be useful in the quantization of the theory [4].

We have seen in previous sections that either Palatini’s theory or an exotic action for gravity can be expressed as a Chern–Simons theory, however, it will be interesting to express the BL action as a Chern–Simons theory as well; is this possible? The answer is yes. In fact, the action analyzed in the previous section can be written in an elegant form in terms of a Chern–Simons theory. By introducing the variables ω±i=Ai±σ|Λ|ei\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ^{\pm i}= A^i \pm \sigma \sqrt{|\Lambda |} e^i$$\end{document} [5], where σ2=s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^2=s$$\end{document} with σ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =1$$\end{document} for Λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda >0$$\end{document} and σ=i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma = i$$\end{document} for Λ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda <0$$\end{document}, we see that the action (7) can be written as31Sγ=|Λ|2γ-1+σsSCS(ω+)+|Λ|2γ-1-σsSCS(ω-),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S{_{\gamma }}= & {} \frac{\sqrt{|\Lambda |}}{2} \left( \gamma ^{-1} + \frac{\sigma }{s}\right) S_{\mathrm{CS}} (\omega ^{+}) \nonumber \\&+\frac{\sqrt{|\Lambda |}}{2} \left( \gamma ^{-1} - \frac{\sigma }{s} \right) S_{\mathrm{CS}} (\omega ^{-}), \end{aligned}$$\end{document}whereSCS(ω±)=∫Mω±i∧dωi±+13ϵijkω±i∧ω±j∧ω±k.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} S_{\mathrm{CS}}(\omega ^{\pm })= \int _M \omega ^{\pm i}\wedge d \omega ^{\pm }_{i} + \frac{1}{3} \epsilon _{ijk} \omega ^{\pm i} \wedge \omega ^{\pm j} \wedge \omega ^{\pm k}. \end{aligned}$$\end{document}The equations of motion obtained from (31) imply that the connections ω±i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ^{\pm i }$$\end{document} are flat, and it is easy to prove that these flatness conditions are equivalent to the equations of motion given in (8) and (9). On the other hand, if we develop a pure Dirac analysis of the action (31) we will reproduce the results given in the previous section, in particular we will reproduce the Dirac brackets. In fact, in summary, by performing a pure Dirac method we obtain the following results.

We have the following first class constraints:32Gi±=s±σγsγ∣Λ∣ϵ0ab∂aωbi(±)+12ϵijkωb(±)jωc(±)k+Da(±)χi(±)a≈0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathfrak {G}^{\pm }_{i}= & {} \frac{s\pm \sigma \gamma }{s\gamma \sqrt{\mid \Lambda \mid }}\epsilon ^{0ab}\left( \partial _{a}\omega ^{(\pm )}_{bi}+\frac{1}{2}\epsilon _{ijk}\omega ^{(\pm )j}_{b}\omega ^{(\pm )k}_{c}\right) \nonumber \\&+D^{(\pm )}_a\chi ^{(\pm )a}_{i}\approx 0, \end{aligned}$$\end{document}and the following second class constraints:33χi(±)a=πi(±)a-12s±σγsγ∣Λ∣ϵ0abω(±)bi≈0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \chi ^{(\pm )a}_{i}=\pi ^{(\pm )a}_{i}-\frac{1}{2}\frac{s \pm \sigma \gamma }{s\gamma \sqrt{\mid \Lambda \mid }}\epsilon ^{0ab}\omega _{(\pm )bi}\approx 0, \end{aligned}$$\end{document}here, πi(±)a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi ^{(\pm )a}_{i}$$\end{document} are the conjugate canonical momenta of the connection ωa±i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ^{\pm i}_a$$\end{document} and Da±λi=∂aλi+ϵijkωa±jλk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_a^{\pm } \lambda ^i = \partial _a \lambda ^i+ \epsilon ^{i}{_{jk}} \omega ^{\pm j}_a\lambda ^k $$\end{document}. Furthermore, by using the second class constraints (33) the following Dirac brackets are obtained:34ωa(±)i(x),ωb(±)j(y)D=sγ∣Λ∣s±σγδijϵ0abδ2(x-y),ω(±)ia(x),πj(±)b(y)D=12δjiδabδ2(x-y),πi(±)a(x),πj(±)b(y)D=14s±σγsγ∣Λ∣δijϵ0abδ2(x-y);\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\left\{ \omega ^{(\pm )i}_{a}(x),\omega ^{(\pm )j}_{b}(y)\right\} _{D}= \frac{s\gamma \sqrt{\mid \Lambda \mid }}{s \pm \sigma \gamma }\delta ^{ij}\epsilon _{0ab}\delta ^{2}(x-y), \nonumber \\&\left\{ \omega {^{(\pm )i}}_{a}(x),\pi ^{(\pm )b}_{j}(y)\right\} _{D}=\frac{1}{2} \delta ^{i}_{j}\delta ^{b}_{a}\delta ^{2}(x-y),\nonumber \\&\left\{ \pi ^{(\pm )a}_{i}(x),\pi ^{(\pm )b}_{j}(y)\right\} _{D}=\frac{1}{4}\frac{s \pm \sigma \gamma }{s\gamma \sqrt{\mid \Lambda \mid }}\delta _{ij}\epsilon {^{0ab}} \delta ^{2}(x-y);\nonumber \\ \end{aligned}$$\end{document}we can observe that the Dirac brackets between the dynamical variables given in (34) depend on the constants (s,γ,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(s,\gamma ,\sigma )$$\end{document}. This fact will be important because, by using the definition of ωa±i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega ^{\pm i}_a$$\end{document} given in terms of Aai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A^i_a$$\end{document} and eai\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^i_a$$\end{document} in (34), the Dirac brackets given in (27) are reproduced. It is important to comment that our results are given in a general form and contain the cases for Λ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda >0$$\end{document} and Λ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda <0$$\end{document}, thus in particular we can reproduce the results given in [17] where the case Λ<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda <0$$\end{document} was studied. On the other hand, in the limit γ→∝\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \rightarrow \propto $$\end{document} the action (31) is reduced to SCS(ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_{\mathrm{CS}}(\omega )$$\end{document} and the Dirac brackets are reduced to those reported in [11], where Palatini’s theory was analyzed. Finally, we can observe that in this section we have proved the equivalence between the BL model and the Chern–Simons theory, thus the standard quantization procedure can be performed. In the following section we will study the action (7) by using the FJ approach and we will obtain all the Dirac results in an alternative way.

By rewriting the action (10) in the following form:35L=2ϵ0abδijebiA˙aj+βϵ0abδijAbiA˙aj+Ωϵ0abδijebie˙aj-V(0),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {L}= & {} 2\epsilon ^{0ab}\delta _{ij}e^{i}_{b}\dot{A}^{j}_{a} +\beta \epsilon ^{0ab}\delta _{ij}A^{i}_{b}\dot{A}^{j}_{a}\nonumber \\&+ \, \Omega \epsilon ^{0ab}\delta _{ij}e^{i}_{b}\dot{e}^{j}_{a}-V^{(0)}, \end{aligned}$$\end{document}where V(0)=-2ϵ0abδij(e0i+βA0i)(Fabj+s∣Λ∣2ϵjkleakebl)+(Ωe0i+A0i)Daebj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V^{(0)}=-2\epsilon ^{0ab}\delta _{ij}\left[ (e^{i}_{0}+\beta A^{i}_{0})(F^{j}_{ab}+s\frac{\mid \Lambda \mid }{2}\epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b})\right. \left. + \, (\Omega e^{i}_{0}+A^{i}_{0})D_{a}e^{j}_{b}\right] $$\end{document} is called the symplectic potential and we have introduced the following constants Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}:36Ω=s∣Λ∣γ,β=1γ∣Λ∣.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega =\frac{s\sqrt{\mid \Lambda \mid }}{\gamma }, \quad \beta =\frac{1}{\gamma \sqrt{\mid \Lambda \mid }} . \end{aligned}$$\end{document}In the FJ framework, the Euler–Lagrange equations of motion are given by [13]37fab(0)ξ˙b=∂V(0)(ξ)∂ξa,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^{(0)}_{ab}\dot{\xi }^{b}=\frac{\partial V^{(0)}(\xi )}{\partial \xi ^{a}}, \end{aligned}$$\end{document}where the symplectic matrix fab(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{(0)}_{ab}$$\end{document} takes the form38fab(0)(x,y)=δab(y)δξa(x)-δaa(x)δξb(y),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^{(0)}_{ab}(x,y)=\frac{\delta \mathrm {a}_{b}(y)}{\delta \xi ^{a}(x)}-\frac{\delta \mathrm {a}_{a}(x)}{\delta \xi ^{b}(y)}, \end{aligned}$$\end{document}with ξ(0)a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi {^{(0)a}}$$\end{document} and a(0)a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {a}{^{(0)}}{_{a}}$$\end{document} are representing a set of symplectic variables. It is important to comment that in the FJ framework we are free to choose the symplectic variables; we can choose the field configuration variables or the phase space variables. In previous sections, we have constructed the Dirac brackets by eliminating the second class constraints, hence, in order to obtain these results by means of the FJ method we will use the configuration space as symplectic variables [16]. To this aim, we choose from the symplectic Lagrangian (35) the symplectic variables ξ(0)a(x)={eai,e0i,Aai,A0i}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \xi {^{(0)a}}(x)=\{e^{i}_{a},e^{i}_{0},A^{i}_{a},A^{i}_{0}\}$$\end{document}, and the components of the symplectic 1-forms are a(0)a(x)={Ωϵ0abebi,0,2ϵ0abebi+βϵ0abAbi,0}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {a}{^{(0)}}{_{a}}(x)=\{\Omega \epsilon ^{0ab}e_{bi},0,2\epsilon ^{0ab}e_{bi}+\beta \epsilon ^{0ab} A_{bi},0\}$$\end{document}. Hence, by using our set of symplectic variables, the symplectic matrix (38) takes the form39fab(0)(x,y)=-2Ωϵ0abδij0-2ϵ0abδij00000-2ϵ0abδij0-2βϵ0abδij00000δ2(x-y).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^{(0)}_{ab}(x,y)= \left( \begin{array}{cccc} -2\Omega \epsilon ^{0ab}\delta _{ij}&{}0&{}-2\epsilon ^{0ab}\delta _{ij}&{}0\\ 0&{}0&{}0&{}0\\ -2\epsilon ^{0ab}\delta _{ij}&{}0&{}-2\beta \epsilon ^{0ab}\delta _{ij}&{}0\\ 0&{}0&{}0&{}0\\ \end{array} \right) \delta ^2(x-y).\nonumber \\ \end{aligned}$$\end{document}The symplectic matrix fab(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{(0)}_{ab}$$\end{document} is of dimension [18×18]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[18\times 18]$$\end{document} and it is a singular matrix. In fact, in the FJ method this means that there are constraints present. In order to obtain these constraints, we calculate the zero modes of the symplectic matrix; the modes are given by (va(0))1T=(0,ve0i,0,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v_{a}^{(0)})_{1}^{T}=(0,v^{e^{i}_{0}},0,0)$$\end{document} and (va(0))2T=(0,0,0,vA0i)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v_{a}^{(0)})_{2}^{T}=(0,0,0,v^{A^{i}_{0}})$$\end{document}, where ve0i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^{e^{i}_{0}}$$\end{document} and vA0i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v^{A^{i}_{0}}$$\end{document} are arbitrary functions. In this manner, by using the zero modes and the symplectic potential V(0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V^{(0)}$$\end{document} we obtain40Ωi(0)=∫d2x(v(0))aT(x)δδξ(0)a(x)∫d2yV(0)(ξ)=-∫d2xve0i(x)2ϵ0abδij×Fjab+s∣Λ∣2ϵjkleakebl+ΩDaejb→-2ϵ0abδijFjab+s∣Λ∣2ϵjkleakebl+ΩDaejb=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega ^{(0)}_{i}= & {} \int \mathrm{d}^{2}x(v^{(0)})^{T}_{a}(x)\frac{\delta }{\delta \xi ^{(0)a}(x)}\int \mathrm{d}^{2}y V^{(0)}(\xi ) \nonumber \\= & {} -\int \mathrm{d}^{2}x v^{e^{i}_{0}}(x)2\epsilon ^{0ab}\delta _{ij}\nonumber \\&\times \left[ \left( F{^{j}}_{ab} +s\frac{\mid \Lambda \mid }{2}\epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b}\right) +\Omega D_{a}e{^{j}}_{b}\right] \nonumber \\\rightarrow & {} -2\epsilon ^{0ab}\delta _{ij}\left[ \left( F{^{j}}_{ab}+s\frac{\mid \Lambda \mid }{2} \epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b}\right) +\Omega D_{a}e{^{j}}_{b}\right] =0,\nonumber \\ \end{aligned}$$\end{document}41βi(0)=∫d2x(v(0))aT(x)δδξ(0)a(x)∫d2yV(0)(ξ)=-∫d2xvA0i(x)2ϵ0abδij×βFjab+s∣Λ∣2ϵjkleakebl+Daejb→-2ϵ0abδijβFjab+s∣Λ∣2ϵjkleakebl+Daejb=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta ^{(0)}_{i}= & {} \int \mathrm{d}^{2}x(v^{(0)})^{T}_{a}(x)\frac{\delta }{\delta \xi ^{(0)a}(x)}\int \mathrm{d}^{2}y V^{(0)}(\xi ) \nonumber \\= & {} -\int \mathrm{d}^{2}x v^{A^{i}_{0}}(x)2\epsilon ^{0ab}\delta _{ij}\nonumber \\&\times \left[ \beta \left( F{^{j}}_{ab} +s\frac{\mid \Lambda \mid }{2}\epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b}\right) + D_{a}e{^{j}}_{b}\right] \nonumber \\\rightarrow & {} -2\epsilon ^{0ab}\delta _{ij}\left[ \beta \left( F{^{j}}_{ab} +s\frac{\mid \Lambda \mid }{2}\epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b}\right) + D_{a}e{^{j}}_{b}\right] =0.\nonumber \\ \end{aligned}$$\end{document}thus we identify the following constraints:42Ωi(0)=2ϵ0abδijFjab+s∣Λ∣2ϵjkleakebl+ΩDaejb=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Omega ^{(0)}_{i}=2\epsilon ^{0ab}\delta _{ij}\left[ \left( F{^{j}}_{ab} +s\frac{\mid \Lambda \mid }{2}\epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b}\right) +\Omega D_{a}e{^{j}}_{b}\right] =0,\nonumber \\ \end{aligned}$$\end{document}43βi(0)=2ϵ0abδijβFjab+s∣Λ∣2ϵjkleakebl+Daejb=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \beta ^{(0)}_{i}= 2\epsilon ^{0ab}\delta _{ij}\left[ \beta \left( F{^{j}}_{ab} +s\frac{\mid \Lambda \mid }{2}\epsilon {^{j}}_{kl}e^{k}_{a}e^{l}_{b}\right) + D_{a}e{^{j}}_{b}\right] =0,\nonumber \\ \end{aligned}$$\end{document}these constraints are the secondary constraints found by means of Dirac’s method in the previous sections. In order to observe if there are more constraints, we calculate [18–21]44fcb(1)ξ˙b=Zc(ξ),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f^{(1)}_{cb}\dot{\xi }^{b}=Z_{c}(\xi ), \end{aligned}$$\end{document}where45Zc(ξ)=∂V(0)(ξ)∂ξa00\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&Z_{c}(\xi )= \left( \begin{array}{cccc} \frac{\partial V^{(0)}(\xi )}{\partial \xi ^{a}}\\ 0\\ 0\\ \end{array} \right) \end{aligned}$$\end{document}and46fcb(1)=fab(0)∂Ω(0)∂ξb∂β(0)∂ξb=-2Ωϵ0abδij0-2ϵ0abδij00000-2ϵ0abδij0-2βϵ0abδij000002ϵ0ab(Ωδij∂a-ϵijk(ΩAak+s∣Λ∣eak))02ϵ0ab(δij∂a-ϵijk(Aak+Ωeak))02ϵ0ab(δij∂a-ϵijk(Aak+Ωeak))02ϵ0ab(βδij∂a-ϵijk(βAak+eak))0δ2(x-y).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&f^{(1)}_{cb}= \left( \begin{array}{cccc} f^{(0)}_{ab}\\ \frac{\partial \Omega ^{(0)}}{\partial \xi ^{b}}\\ \frac{\partial \beta ^{(0)}}{\partial \xi ^{b}}\\ \end{array} \right) =\left( \begin{array}{cccc} -2\Omega \epsilon ^{0ab}\delta _{ij}&{}0&{}-2\epsilon ^{0ab}\delta _{ij}&{}0\\ 0&{}0&{}0&{}0\\ -2\epsilon ^{0ab}\delta _{ij}&{}0&{}-2\beta \epsilon ^{0ab}\delta _{ij}&{}0\\ 0&{}0&{}0&{}0\\ 2\epsilon ^{0ab}(\Omega \delta _{ij}\partial _{a}-\epsilon _{ijk}(\Omega A^{k}_{a}+s\mid \Lambda \mid e^{k}_{a}))&{}0&{} 2\epsilon ^{0ab}(\delta _{ij}\partial _{a}-\epsilon _{ijk}( A^{k}_{a}+ \Omega e^{k}_{a}))&{}0\\ 2\epsilon ^{0ab}(\delta _{ij}\partial _{a}-\epsilon _{ijk}(A^{k}_{a}+\Omega e^{k}_{a}))&{}0&{} 2\epsilon ^{0ab}(\beta \delta _{ij}\partial _{a}-\epsilon _{ijk}(\beta A^{k}_{a}+e^{k}_{a}))&{}0\\ \end{array} \right) \delta ^2(x-y).\qquad \qquad \end{aligned}$$\end{document}We can observe that the matrix (46) is not a square matrix as expected, however, it has linearly independent modes given by (v(1))1T=(δji∂avλ+ϵikjAakvλ,δjive0i,-s∣Λ∣ϵijkeak,0,δjivλ,0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v^{(1)})_{1}^{T}=(\delta ^{i}_{j}\partial _{a}v^{\lambda } +\epsilon {^{i}}_{kj}A^{k}_{a}v^{\lambda },\delta ^{i}_{j}v^{e^{i}_{0}}, -s\mid \Lambda \mid \epsilon {^{i}}_{jk}e^{k}_{a},0,\delta ^{i}_{j}v^{\lambda },0)$$\end{document} and (v(1))2T=(-ϵijkeak,0,δji∂avβ+ϵikjAakvβ,δjivA0i,0,δjivβ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v^{(1)})_{2}^{T}=(-\epsilon {^{i}}_{jk}e^{k}_{a},0,\delta ^{i}_{j} \partial _{a}v^{\beta }+\epsilon {^{i}}_{kj}A^{k}_{a}v^{\beta }, \delta ^{i}_{j}v^{A^{i}_{0}},0,\delta ^{i}_{j}v^{\beta })$$\end{document}; these modes are used in order to obtain further constraints. In fact, by calculating the following contraction [18–21]:47(v(1))cTZc=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (v^{(1)})_{c}^{T}Z_{c}=0, \end{aligned}$$\end{document}where c=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=1,2$$\end{document}, we find that (47) is an identity, thus in the FJ formalism there are not more constraints for the theory under study.