We quantize the three-dimensional $BF$-model using axial gauge conditions.
Exploiting the rich symmetry-structure of the model we show that the
Green-functions correspond to tree graphs and can be obtained as the unique
solution of the Ward-Identities. Furthermore, we will show that the theory can
be uniquely determined by symmetry considerations without the need of an action
principle.Comment: one reference added, transmission errors correcte

Brandhuber, A.

Emery, S.

Landsteiner, K.

Schweda, M.

English

arXiv

We quantize the three-dimensional BF-model using axial gauge conditions. Exploiting the rich symmetry-structure of the model we show that the Green-functions correspond to tree graphs and can be obtained as the unique solution of the Ward-Identities. Furthermore, we will show that the theory can be uniquely determined by symmetry consideration without the need of an action principle

Brandhuber, A

Emery, S

Landsteiner, K

Schweda, Manfred

CERN Document Server

PostScript〉  processed by the SLAC/DESY Libraries on 28 Feb 1995.HEP-TH-9502147CERN-TH/95-29TUW 95-04arch-ive/9502147The three-dimensional BF Model with Cosmological Term in theAxial GaugeBy A. Brandhuber#, S. Emery1, K. Landsteiner#and M. Schweda#CERN, CH-1211 Geneve 23 (Switzerland)Institut fur Theoretische Physik, Technische Universitat WienWiedner Hauptstrae 8-10, A-1040 Wien (Austria)Abstract.We quantize the three-dimensional BF -model using axial gauge conditions. Exploiting the rich symmetry-structure of the model we show that the Green-functions correspond to tree graphs and can beobtained as the unique solution of the Ward-Identities. Furthermore, we willshow that the theory can be uniquely determined by symmetry considerationwithout the need of an action principle.CERN-TH/95-29TUW 95-04February 19951On leave from the Departement de Physique1 IntroductionTopological eld models [1] of the Schwarz-type [2] have been the subject of continous investigationsover the recent years. These theories are characterized by an invariant action which does not dependon the metric structure of the manifold. Therefore, they are devoid of any local observables.Nevertheless, the metric appears inthe gauge-xing term which is itself a BRST-variation. The variation of the gauge-xing termwith respect to the metric is a BRST-exact quantity implying the existence of a linear vector-likesupersymmetry [4] in an elegant manner. Together withthe BRST-symmetry, the symmetry forms an algebra of the formfs; g = @(1.1)stating that translations are no physical operations and thus reecting the topological nature ofthe theory. Thus one might say that this relation lies at the heart of their topological properties.The most prominent example of these theories is of course the three-dimensional Chern-Simonstheory which has led to the powerful connection between link-invariants and the vacuum expectationvalue of Wilson lines [3]. The Chern-Simons theory has also been studied extensively from a purelyeld theoretical point of view. It turns out to be a completely ultraviolet nite theory and thatthis niteness is a direct consequence of the topological supersymmetry (1.1). Originally, thissupersymmetry has been found using the Landau-gauge [4]. In a serie of papers it has beengeneralized to other gauge-conditions as well [5, 6, 7]. Of particular interest was the case of theaxial gauge where it turnedout that the topological supersymmetry is not only responsible for the niteness ofthe theory, but also allows to compute the Green functions without the use ofan action principle [8]. Let us also mention that it has been shown that the topological super-symmetry exists also in string theories [9] and in two-dimensional chiral W3-gravity [10] and thatit turned out to be an extremely useful tool for solving the descent equations associated with theintegrated BRST-cohomology [11].Another class of Schwarz-type Topological theories [2] arethe BF models. Despite of their simple form they reveal a surprisingly richsymmetry structure. Indeed, they allow for reducible invariances[12]. In the particular case of three dimensions, another interesting featureis that the B-eld is a 1-form and therefore allows the addition of a cubic term in B into theaction. This term is usually referred to as a cosmological constant term since then the model isrelated to three-dimensional Einstein-Hilbert gravity with such a term. A detailed investigation ofthe symmetry stucture and niteness properties of this model in the Landau-gauge has been givenin [13].Theorique, Universite de Geneve. Supported by the "Fonds Turrettini" and the "Fonds F. Wurth"1In the case of the Chern-Simons model in the axial gauge, we found thatthe supersymmetry has two main consequences. Indeed, it turned out to bestrong enough for xing all the Green functions of the theory (in this sense, onecan say that it can be substituted to the action principle) and it also imposesthe principal value prescription for the propagators. Therefore, it would be desirable to knowwether this remains valid for the BF system in the axial gauge. This is precisely thequestion we will address in this paper and we will show that the answer ispositive.The work is organized as follows. In section 2 we introduce the action and x the notation andconventions. Section 3 presents the symmetries and all the functional identities. We investigatetheirconsequences for the equations of motion in section 4.1 and for the calculation of the propagatorsin section 4.2. At the end, we propose some conclusion.2 The 3D BF model with cosmological term in the axial gaugeThe complete action of the 3D BF model containing a cosmological term with an axial gauge xingis given byS = Sinv+ Sgf; (2.1)withSinv=  12TrZd3x(FB+23BBB);Sgf= TrZd3xbnA+ dmB+ cn(Dc+ [B; ]) +m(D+ [B; c]):(2.2)and D: : := @: : :+g[A; : : :] denoting the gauge covariant derivative. Fis the eld strengthof the gauge eld A. Further b, d are the Langrange multipliers imposing the gauge-conditionsnA= 0 and mB= 0 where nand mare it a priori two independent gauge xing directions.c, c and,  are theanti-ghost and ghost elds corresponding to the two gauge symmetries of Sinv1A=  D ; 1B=   [B; ] ;2A=   [B; ] ; 2B=  D :(2.3)We choose the gauge group to be simple, all elds belong to the adjoint representation and arewritten as Lie algebra matrices '(x) = 'a(x)ta, with2[ta; tb] = fcabtc; Tr(tatb) = ab: (2.4)Finally  is some numerical constant. We summarize the canonical dimensions and the ghostnumbers of the various elds in Table 1.A B b d c c Dimension 1 1 2 2 0 2 0 2Ghost number 0 0 0 0 1  1 1  1Table 1: Dimensions and ghost numbers.3 Symmetries of the action and Ward identitiesThe action (2.1) is invariant under the BRST transformation s :sA=  Dc  [B; ]; sB=  D  [B; c];sc = c2+ 2; s = f; cg;sc = b; s = d;sb = 0; sd = 0:(3.1)Since we are dealing with a topological eld theory of Schwarz type, the only metric dependencearises from the gauge xing part of the action. Therefore, the energy momentum tensor is BRSTexact:T= s(3.2)with= Tr(cnA  cnA  cnA+ nB nB nB): (3.3)Using the equations of motion, one gets for the divergenceof (3.3) the following expression@= Tr(@cSb ASc  nc"SB+ @Sb BS  m"SA+ nc"md  n"mb) + tot: der:(3.4)Integrating (3.4) allows to derive the usual form for the topological supersymmetry only for thecase where1n= mwhich wewill assume for the rest of the paper. Thus we have the following form for1Actually onecould also insist in keeping dierent gauge vectors since the breaking term is BRST exact. This breaking could becontrolled by coupling it to anadditional source and adding it to the action [7]3the vector supersymmetry transformations :A=  "n; B=  "nc;c = A;  = B;c = 0;  = 0;b =  @c; d =  @:(3.5)The transformations s and the supersymmetry transformations form an algebra which closeson-shell:s2= f; g = 0; fs; g = @+ Eq: of motion: (3.6)In addition there exist two discrete symmetries2which leave the action (2.1) invariant:c ! c ;  ! (3.7)andc ! ; c !  (3.8)At the level of the generating functional of the connected Greenfunctions ZC, all these symmetries leads to a set of WI. The one which correspond tothe vector supersymmetry takes the formVZC= TrZd3xJb@ZCJc+ "nJBZCJc+ JcZCJA++Jd@ZCJ+ "nJAZCJ+ JZCJB!= 0:(3.9)In this formalism, the axial gauge is imposed by the two gaugeconditions :Jb+ nZCJA= 0;Jd+ nZCJB= 0:(3.10)As in any linear gauge there exist antighost equations which in the case of the axial gauge are local[5]. In our case we havetwo of them and they take the following form:Jc  n@ZCJc+Jb;ZCJc+ Jd;ZCJ= 0;J  n@ZCJ+Jb;ZCJ+Jd;ZCJc= 0:(3.11)2In fact, (3.7) and (3.8) imply the existence of additional anti-BRST-likesymmetries and anti-vector-like supersymmetries as in [4, 5].4Finally, it is well known that in the axial gauge, due to thedecoupling of the ghosts, the Slavnov identity which express the invariance of the theory underthe BRST-transformation (3.1) takes the form of alocal WI. Therefore, one get the two following local gauge WI's@JA JA;ZCJA JB;ZCJB Jb;ZCJb Jd;ZCJd  Jc;ZCJc Jc;ZCJc J;ZCJ (J;ZCJ)+ (n  @)ZCJb= 0(3.12)and@JB  JA;ZCJB JB;ZCJA Jb;ZCJd  Jd;ZCJb  Jc;ZCJ (Jc;ZCJ) J;ZCJc  J;ZCJc+ (n  @)ZCJd= 0:(3.13)4 Consequences of the Symmetries4.1 Equations of motionLet us now investigate in some detail the meaning of therelations of the last section, starting with the projection of the WI for the supersymmetry(3.9),along the gauge xed direction. Without loss of generality we can choose the gauge vector ntobe (0; 0; 1). Wewill denote this gauge xed direction by u and the transverse coordinates by xtr= (xi); i = 1; 2.Then,VuZC= TrZd3x Jb@uZCJc+ JcZCJuA+ Jd@uZCJ+ JZCJuB!= 0: (4.1)Taking into accout the gauge conditions (3.10), the latter canbe written asTrZd3x JbX + JdY= 0:where X and Y are the most general forms compatiblewith (4.1)X = @uZCJc+ 1Jd;ZCJc+ 2"Jd;ZCJ#+ 3Jb;ZCJc+ 4"Jb;ZCJ#  Jc= 0Y = @uZCJ+ 1Jd;ZCJc+ 2"Jd;ZCJ#+ 3Jb;ZCJc+ 4"Jb;ZCJ#  J= 05At this level, one can use the consistency condition between the twoequations we just found and the ghost equation (4.2) in orderto x the arbitrary parameters. Then, the result isJc  n@ZCJc+Jb;ZCJc+"Jd;ZCJ#= 0;J  n@ZCJ+"Jb;ZCJ#+ Jd;ZCJc= 0:(4.2)which are nothing else than the ghost equations. Thus, theequations of motion for the ghost sector are a consequence of thegauge-xed component of the supersymmetry WI, the gauge condition and the Slavnov identity.For the gauge sector, let us consider the transverse componentof (3.9)ViZC= TrZd3xJb@iZCJc+ "jiJjBZCJc+ JcZCJiA++Jd@iZCJ+ "jiJjAZCJ+ JZCJiB!= 0:(4.3)together with the antighost equations (3.11) written asfunctional operators acting on ZC@uJc Jb;Jc  Jd;JZC= Jc;@uJ Jb;J Jd;JcZC= J:(4.4)Their consistency gives rise to the following identities@uJiA Jd;JiA  Jb;JiBZC= "jiJjB  @iJd;@uJiB Jd;JiB Jb;JiAZC= "jiJjA  @iJb:(4.5)which correspond to the equations of motion for the gauge elds. Thisconcludes the analysis of the consequences of the supersymmetry for the equationsof motion.64.2 Calculation of the PropagatorsGauge conditionsWe begin by looking at the gauge conditions (3.10) which imply the vanishing of the connectedGreen functions involving the components A3or B3hAa3(x)Qi'i(xi)i = 0hBa3(x)Qi'i(xi)i = 08'i(4.6)with two exceptionshAa3(x)bb(y)i =  ab(3)(x  y);hBa3(x)db(y)i =  ab(3)(x  y):(4.7)Antighost equationsThe antighost equations (4.2) give the following dierential equations for the connected Greenfunctions involving one pair of ghost elds:@x3hca(x)cb(y)i = ab(3)(x  y); (4.8)@x3ha(x)b(y)i = ab(3)(x  y) (4.9)and@x3hdc1(z1)::dcr(zr)bb1(y1)::bbr(yr)cb(y)ca(x)i =sXj=1fabjchdc1(z1)::dcr(zr)bb1(y1)::cbbj(yj)::bbr(yr)cb(y)cc(yj)i (x  yj) +rXi=1facichdc1(z1)::cdci(zi)::dcr(zr)bb1(y1)::bbr(yr)cb(y)c(zi)i (x  zi)(4.10)and@x3hdc1(z1)::dcr(zr)bb1(y1)::bbr(yr)cb(y)a(x)i =sXj=1fabjchdc1(z1)::dcr(zr)bb1(y1)::cbbj(yj)::bbr(yr)cb(y)c(yj)i (x  yj) +rXi=1facichdc1(z1)::cdci(zi)::dcr(zr)bb1(y1)::bbr(yr)cb(y)cc(zi)i (x  zi):(4.11)The solutions of the equations (4.8) and(4.9) arehca(x)cb(y)i = ab[(x3  y3) + 1](2)(xtr  ytr); (4.12)ha(x)b(y)i = ab[(x3  y3) + 2](2)(xtr  ytr) (4.13)The form of the terms proportional to 1, 2is dictated by transverse two dimensional Poincare7invariance and scale invariance. Indeed, the latter forbids solutions of the type 1=(xtr ytr)2becausethis term is not a well dened distribution. To give it a meaning would need the introduction ofUV subtraction point, i.e., of a dimensionful parameter which would break scale invariance. Theintegration constant can be xed with the help of the discrete symmetry of the action (3.7) to be1= 2=  12.Integration of the equations (4.10) and (4.11) yields the following recursion relations for the Greenfunctions with one pair of ghosts:hdc1(z1)::dcr(zr)bb1(y1)::bbs(ys)cb(y)ca(x)i ==sXj=1fabjc[(x3  y3j) + (r; s)](2)(xtr  ytrj)hdc1(z1)::dcr(zr)bb1(y1)::cbbj(yj)::bbs(ys)cb(y)cc(yj)i ++rXi=1facic[(x3  z3i) + (r; s)](2)(xtr  ztri)hdc1(z1)::cdci(zi)::dcr(zr)bb1(y1)::bbs(ys)cb(y)c(zi)i(4.14)andhdc1(z1)::dcr(zr)bb1(y1)::bbs(ys)cb(y)a(x)i ==sXj=1fabjc[(x3  y3j) + (r; s)](2)(xtr  ytrj)hdc1(z1)::dcr(zr)bb1(y1)::cbbj(yj)::bbs(ys)cb(y)c(yj)i ++rXi=1facic[(x3  z3i) + (r; s)](2)(xtr  ztri)hdc1(z1)::cdci(zi)::dcr(zr)bb1(y1)::bbs(ys)cb(y)cc(zi)i :(4.15)Using the discrete symmetry one could produce two additonal recursion relations which are notwritten explicitly since they are not needed for our calculations. The integration constants are allxed to (r; s) =  12by the discrete symmetry (3.7) and Bose symmetry of the Lagrange multipliersb and d.Now we will discuss the recursion relations for some special values of r and s.The case r = 0For this discussion we will use a more symbolic notion i.e. we will drop indices and variables becausewe only want to nd the vanishing Green functions whereas the non vanishing Green functions canalways be obtained from the explicit recursion relations (4.14) and (4.15). From (4.14) we geth(b)scci =Ph(b)s 1cci which ends up after s steps with the hcci propagator dened in (4.8). Onthe contrary h(b)sci = 0 since the recursion relation stops with the vanishing Green function hci.Using again the discrete symmetry we obtain h(b)sci = 0 and h(b)si =Ph(b)s 1i.8The case s = 0In this case we have have to use (4.14) and (4.15) iteratively e.g. h(d)rcci =Ph(d)r 1ci =h(d)r 2cci = : : : . The nal result depends on wether the recursion relation ends up with hcci whichgives a non vanishing result or with hci which gives zero. Here we only want to compile the zeroresults:h(d)rcci = 0 and h(d)ri = 0 if r is an odd integer.h(d)rci = 0 and h(d)rci = 0 if r is an even integer.The Green functions with additional ghosts, gauge elds or Lagrange multipliers vanish ingeneral as a consequence of the antighost equations (4.2):hXcci = hXci = hXci = hXci = 0 ; unless X = (b)m(d)n: (4.16)Transverse supersymmetryFrom equation (4.3) we get further relations along the same lines. For reasons of simplication weuse the sloppy notation from the discussion above whenever possible.The results for the two-point functions are:hAai(x)Bbj(y)i = "ijab[(x3  y3) 12](2)(xtr  ytr) ; (4.17)hba(x)Abi(y)i =  ab[(x3  y3) 12]@xi(2)(xtr  ytr) ; (4.18)hda(x)Bbi(y)i =  ab[(x3  y3) 12]@xi(2)(xtr  ytr) (4.19)and hAAi = hBBi = hbbi = hddi = hdAi = hbBi = hci = hci = 0. Furthermore we observe thatall two-point functions with one ghost and one bosonic eld vanish.For the higher Green functions we obtain the recursion relationshbd1(w1)::bdr(wr)dc1(z1)::dcs(zs)Ab1l1(y1)::Abtlt(yt)Ba1n1(x1)::Baunu(xu)(Abi(y)Bbi(y))i ==rXk=1@wikhbd1(w1)::cbdk(wk)::bdr(wr)(d)s(A)t(B)u(cb(y)b(y))cdk(wk)i ++sXk=1@zikh(b)rdc1(z1)::cdck(zk)::dcs(zs)(A)t(B)u(cb(y)b(y))ck(zk)i ++tXk=1"ilkh(b)r(d)sAb1l1(y1)::dAbklk(yk)::Abtlt(yt)(B)u(cb(y)b(y))bk(yk)i ++uXk=1"inkh(b)r(d)s(A)tBa1n1(x1)::dBaknk(xk)::Baunu(xu)(cb(y)b(y))cbk(yk)i:(4.20)In the following we want to specify these recursion relations for special values of r, s, t and uto demonstrate that all Green functions can be obtained from our recursion relations. All Greenfunctions with one pair of ghosts have been obtained in the previous subsection. Now we want tocalculate the remaining Green functions with only bosonic elds and at least one A or B eld.9The case t = u = 0We obtain from (4.20):h(b)r(d)sAi =X@h(b)r 1(d)scci+X@h(b)r(d)s 1ciso the calculations breaks down to summing over already known Green functions. The same holdsfor h(b)r(d)sBi .The case r = s = 0We obtainh(A)t(B)uAi = "h(A)t(B)u 1cci+ "h(A)t 1(B)uciUsing (4.16) we nd h(A)a(B)bi = 0 unless a = 1 and b = 1 which yields the two-point functionhABi .The case s = u = 0In this case the relation (4.20) takes the form:h(b)r(A)t AB!i =X@h(b)r 1(A)t c!ci+X"h(b)r(A)t 1 c!iFor t = 0 we get the results:h(b)rAi =P@h(b)r 1cci and h(b)rBi = 0For t = 1:h(b)rABi =P"h(b)ri and h(b)rAAi = 0.The case r = u = 0In this case the relation (4.20) takes the form:h(d)s(A)t AB!i =X@h(d)s 1(A)t c!i+X"h(d)s(A)t 1 c!iFor t = 0 we get the results:h(d)sAi = 0 for s odd and h(d)sBi = 0 for s even.For t = 1:h(d)sAAi = 0 for s even and h(d)sABi = 0 for s odd.The case r = t = 0In this case the relation (4.20) takes the form:h(d)s(B)u AB!i =X@h(d)s 1(B)u c!i+X"h(d)s(B)u 1 c!ci10For t = 0 we get the results:h(d)sAi = 0 for s odd and h(d)sBi = 0 for s even.For t = 1:h(d)sBAi = 0 for s odd and h(d)sBBi = 0 for s even.Gauge invarianceThe two gauge symmetries (3.12) and (3.13) do not give a lot of new information besides consistencychecks and the fact that all correlators consisting only of the Lagrange multipliers vanish:hba1(x1)::bam(xm)db1(y1)::dbn(yn)i = 0 8 m; nThis is the unique solution obeying dimensional and scaling arguments.5 Concluding RemarksThe main result of our study is that the Green functions of the model are the unique solutionsof the Ward-identities dening the theory. Furthermore it turned out that the topological vectorsupersymmetry imposed a rather unexpected restriction on thea priori independent gauge vectors. It is also worth noticing that the Green functions correspondto tree graphs only. Note also that in principle there are loop graphs with external b and d eldsonly, however, as in Chern-Simons theory [14], the gauge-eld and the ghost eld contributions tothese graphs cancel exactly due to the topological supersymmetry. Having investigated here thethree-dimensional BF model it is now natural to apply the axial gauge also to higher dimensionalBF models. It would be highly interesting to know wether the methods developpedhere and in [8] are also applicable to these cases.6 AcknowledgementsThe authors would like to thank O. Piguet for helpful discussions and comments.11References[1] D. Birmingham, M. Blau, M. Rakowski and G. Thompson,Phys. Reports 209 (1991)129.[2] A.S. Schwarz, Baku International Topological Conference, Abstracts, vol. 2, p.345 (1987);[3] E. Witten, Comm. Math. Phys. 121 (1989)351.[4] F. Delduc, F. Gieres and S. P. Sorella Phys. Lett. 225 (1989)367.[5] A. Brandhuber, M. Langer, O. Piguet, M. Schweda and S. P. Sorella,Phys.Lett. B300 (1993)92.[6] T. Gajdosik and W. Kummer, Phys. Rev. D48 (1993)2897.[7] K. Landsteiner, M. Langer, M. Schweda and S. Sorella, Phys. Lett. B337 (1994)294.[8] A. Brandhuber, M. Langer, M. Schweda, S. Sorella, S. Emery and O. Piguet,Helv. Phys. Acta 66 (1993)551.;[9] M. Schweda, S.P. Sorella and M. Werneck de Oliveira, Phys. Lett. B315 (1993)93.A. Boresch, M. Schweda and S. Sorella, Phys. Lett. B328 (1994)36.[10] M. Carvalho, L. C. Queiroz Vilar and S.P. Sorella Algebraic characterization of anomalies inchiral W(3) gravity preprint CBPF-NF-042-94, hep-th:9408127[11] S. P. Sorella Comm. Math. Phys. 157 (1993)231.S. P. Sorella and M. Werneck de Oliveira, Int. J. Mod. Phys. A9 (1994)2979.O. Moritsch, S. P. Sorella and M. Schweda, Class. Quantum Grav. 11 (1994)1225.[12] C. Lucchesi, O. Piguet, S. P. Sorella, Nucl. Phys. B395 (1993)325.[13] N. Maggiore and S.P. Sorella, Nucl. Phys. B377 (1992)236.[14] C. P. Martin, Phys. Lett. B263 (1991)69.12

The three-dimensional BF model with cosmological term in the axial gauge

The three-dimensional BF Model with Cosmological Term in the Axial Gauge

The three-dimensional BF Model with Cosmological Term in the Axial Gauge

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