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ON THE B\/ QUANTIZATION OF GAUGE GRAVITATION THEORY
D. BASHKIROV* and G. SARDANASHVILYT Department of Theoretiu,l Phgsics Moscow State Un'iuersity, 117234, Moscow, Russia *[email protected] Isard@ grau -phy s.msu. su
Received 30 December 2004 Accepted 10 February 2005
Quantization of gravitation theory as gauge theory of general covariant transformations in the framework of Batalin-Vilkoviski (BV) formalism is considered. Its gauge-fixed Lagrangian is constructed.
Keywords: Gauge gravitation theory; Batalin-Vilkoviski quantization; BRST symmetry.
L. Introduction
Classical gauge gravitation theory has been developed in different variants since the 1960s (see [1-3] for a survey). We describe gravitation theory as a standard gauge theory on fiber bundles where parameters of gauge transformations are vector fields on a base manifold X. The well-known BV quantization technique [4,5] can be applied to this theory. Since this quantization technique fails to provide a functional measure, v/e restrict ourselves to constructing a gauge-fixed Lagrangian. Recall that an r-order Lagrangian of a Lagrangian system on a smooth fiber bundle Y ---+X is defined as a density
-+ : L: Lu:J'Y i\r.", u d,rrn... Ad,rn, n-dimx,, (1)
---+ on the r-order jet manifoLd J'Y of sections of Y X [6,7]. Given bundle coordi- nates (r^,,Ai) on y, the jet manifold J'Y is endowed with the adapted coordinates (*\,yo,gi), whereA - ()r...)r), k:!,...,r, is asymmetricmulti-index. We use the notation .\ * A : (lln ...)1) and
d,x--o.r*\,Y\*nof, drr.:d,\.o"'od.l,. Q) 0 203 2O4 D. Baslilsilu U G. Satdanashuily A Lagrangian system on a fiber bundle is said to be a gauge system if its Lagrangi an L admits a gauge syrrmetry depending on parameter functions and their derivatives. In order to describe a gauge system, let us consider Lagrangian formalism on the bundle Product E -Y;W, (3) where W --+X is a vector bundle whose sections are gauge parameter functions l8]. Let W -+ X be coordinated by ("), €'). Then a gauge symmetry is represented by a differential operator u- t ,?n(*^,,t'){nan (4) 0SlAl 2. Gauge Gravitation Theories Classical theory of gravity in the absence of matter fields can be formulated as a gauge theory on natural bundles over an oriented four-dimensional manifold X. On the BV Quanti,zation of Gauge Graai,tation Theory 2O5 The group of automorphisms of a natural bundle contains the subgroup of general covariant transformations defined as the canonical lift of diffeomorphisms of X. They are gauge transformations of metric-affine gravitation theory on natural bun- dles whose variables are linear connections and pseudo-Riemannian metrics on X. Linear connections on X (henceforth world connections) are connections on the principal bundle LX of linear frames in the tangent bundle TX of X with the structure group GL4: GL+(4,R). These connections are representedby sections of the quotient bundle C x : Jr LX lGL4, Q) -+ where J|LX is the first-order jet manifold of sections of LX X 16,7]. This fiber bundle is providedwith bundle coordinates (r),kx"o) such that, for any sec- tion K of.Cx --aX, its coordinates k^'o o K: K^'o are coefficients of the linear connection K-d"r^g (D^+ Ksp,r"0p) onTX with respect to the holonomic bundle coordinates (r),i)). Note that the first-order jet manifold JLC7 of the fiber bundle (7) possessesthe canonical decom- position taking the coordinate form 'r(Rx*o1 kxpo.p: B * sxr"p), Rxpo - kxpo - kpxop * kpo p - k^ouk*" p, (8) B B "kx' - Sxp" B - k^po p * kpxo B kro rkx" p * kxo rkpt B. (e) If /(isasection of Cx -- X, then Ro K isthecurvatureof theworldconnectionK. In order to describe gravity, we assume that the linear frame bundle ^LX admits a Lorentz structure, i.e., a principal subbundle LhX C LX whose structure group is the proper Lorentz group L : SO0(1,3). By virtue of the well-known theorem [10], there is one-to-one correspondence between these subbundles and the global sections h of the quotient bundle Er : LXIL. (10) Namely, LhX : h*LX is the bull-back of LX -- Er by h. Wb agree to call Xr the tetrad bundle. This is the twofold of the metric bundle E : LX/SO(1,3), (11) whose sections are pseudo-Riemannian (henceforth world) metrics on X. Being an open subbundle of the tensor bundle 7fX, the bundle E (11) is provided with bundle coordinates (r) ,oP'). 206 D. Bashki,wu€i G. Sardanashuilg The configuration space of gauge gravitation theory in the absence of matter fields is the bundle product E x Cx. This is a natural bundle admitting the canon- ical lift ?.trKl: uPL,"+ (o'9 Lruo+ oo'Lrul) #, + (Lruok*' p - ilBu' k1"o,- 0*u' lerop * LrBuo) (12) # of vector fields u: u\0x on X [6]. This lift is a Lie algebra homomorphism, and the vector fields (12) are infinitesimal generators of local one-dimensional groups of general covariant transformations, whose gauge parameters are vector fields on X. Dealing with non-vertical automorphisms of non-natural bundles, one also comes to gravitation theory. Fbr instance, let us consider the gauge theory on a principal bundle P ---+X with a structure Lie group G. In a general setting, the gauge. natural prolongations of. P and the associated natural-gauge bundles are called into play [11, 12]. Principal connections on P are represented by sections of the quotient C - JLP/G -- x, (13) called the bundle of principal connections [6]. This is an affine bundle coordinated by ("), o!) such that, given a section A of.C ---+X, its components Ai - a\o A are coefficients of the familia^r local connection form (i.e., gauge potentials). Infinitesi- mal generators of local one.parameter groups of automorphisms of a principal bun- dle P are G-invariant projectable vector fields on P --+X. They are represented by sections of the vector bundle TcP - TPIG -+ X. This fiber bundle is endowed with - the coordinates (r^,r\ k\,{') with respect to the fiber bases{0r,er} for T6P, where {e,} is the basis for the right Lie algebra g of G such that leo,es]:$ner. The bracket of sections of. TcP --+X reads ?.Lp: u^0x * ?tr'err'up :ul0.r * t)'er, - - - lup,rpl (up\ru\ up\pu^)a^ + (u^}su' u^}su' * fiouoro)",. (14) Any sectiarL up of the vector bundle TcP + X yields the vector field ltrc : u^ ox + ($oae^uq* }xu' - aiopq)ol (15) on the bundle of principal connections C (13). For instance, this vector field is a gauge symmetry of the global Chern-Simons Lagrangian in gauge theory on a principal bundle with a structure semi-simple Lie group G over a three.dimensional base X [13]. In order to obtain a gauge symmetry of the Yang-Mills Lagrangian, one has to extend the vector field (15) to the vector field 'tccE: - u^Lx+ ($nae^uq* Lxu' af,Lsup)ai + 7o'9L,ua i oo'0,u\# (16) on the bundle product C x E of C and the metric bundle X (11). On the BV Quantizati,on of Gauge Graaitation Theory 207 The physical underlying reasons for the existence of a Lorentz structure and, consequently, a world metric are Dirac fermion fields. They are described by sections of Dirac spinor bundles, which fail to be natural. In order to develop gauge theory of gravity in the presence of Dirac fermion fields, one has to define general covariant transformations of these fields. Recall that Dirac spinors are described in terms of Clifford algebras as follows [14,15]. Let M be the Minkowski space equipped with the Minkowski metric 11of - - -), signature (+ and let {t"} be a fixed basis for M. The complex Clifford algebra C1,3 is defined as the complexified quotient of the tensor algebra 8M of M by the two-sided ideal generated by elements tgtt +{ &t-2q(t,t') e &M, t,t' € M. The Dirac spinor space % is introduced as a minimal left ideal of C1,3 on which this algebra acts by left multiplications. There is the representation .y : M 8V -- V, .y(t") :.yo,, (17) of elements of the Minkowski space M C Cr,s by the Dirac 7-matrices on V". Different ideals % lead to equivalent representations (17). The complex Clifford algebra C1,3 is isomorphic to the real Clifford algebr& IR2,3,whose generating space is IR5 endowed with the metric of signature (* *). Its subalgebra generated by elements of M c lR5 is the real Clifford algebralRr,g.The Clifford group G1,3 consists of the invertible elements l" of IR1,3such that the inner automorphisms defined by these elements preserve the Minkowski space M C lRr,g. Since this action of G1,3on M is not effective,one considersits spin subgroup L" : SL(2,C), acting on the spinor space V" by means of the generators 1' l Loo : (18) Zl'Ya'''YuJ. This action preserves the representation (17) and the spinor metric a(u,u')- !r{r*^r'r'+ u'+-you),, u,u' € v". (1e) The spin group L" is the universal twofold of the proper Lorentz group L. Given the universal twofold GLq of the group GL+, we have the commutative diagram GL+ - GL+ ll tl (20) tl L" ------+ L Note that the group dLa admits spinor representations, but they are infinite. dimensional 12,16]. Let us consider the G.L-principal bundLe LX ---+X which is the twofold of the linear frame bundle LX lL7-L9]. We agree to call it the spin 208 D. Bashkirou €i G. Sardanashaily frame bundle. One says that i* admits a Dirac spin structure if there exists its L"-principal subbundle Ph. Then we have the commutative diagram LX -:--+ LX (21) tttl Ph ---- LhX where LhX is some Lorentz structure 120,21]. As a consequence,,LXfL" : lr (10), and there is one-to-one correspondence Ph : h*i* between the Dirac spin structures Ph CTX and the global sections h of the tetrad bundle Er - X. We agree to call these sections the tetrad fields because of the following. With respect to the holonomic frames {A*} in the tangent bundle TX,, every : element {H"} of the linear frame bundle LX takes the form Ho H!0*. The matrix elements Hf constitute the bundle coordinates on LX --. X such that the transition functions and the canonical right action of.GL+ on LX take the form A'rtlt H'f' : nl,, GL+) $ Ht ,-. H{ 9bo. Tofrn Let collections of local sections V - {rj} and V : {zu: C o zi} yield associated atlases of the principal bundles T* -. Er and LX --+E7, respectively. Then the Er --+ X can be provided with bundle coordinates (r) ,,ot : tetrad bundle 4 " 0. Components of a tetrad field h with respect to these coordinates are tetrad functions ht : of o h. The familiar relation gu' - h?hirfb between tetrad functions and h metric functions of the world metric A , X rDr - E holds. There is the well-known topological obstruction to the existence of a Dirac spin structure. Let us restrict our consideration to non-compact manifolds X. Then a - Dirac spinor structure exists if and only if X is parallelizable 122](e.g., X IR x Z where Z is a compact orientable three-dimensional manifold 123]).In this case, the spin frame bundle is unique and all Dirac spin structures are equivalent 122,241, i.e., all L"-principal subbundles Ph of LX are pairwise isomorphic by means of automorphisms otL* and isomorphic to the trivial L"-principal bundle P0 ---+X. Since LX --+ Er is an L"-principal bundle, one can consider the associated spinor bundle 5 : 1TrtxV")lL" -* Xr (22) over E7 [6,21]. We agree to call it the universal spinor bundle because, given a tetrad field h, the pull-back Sh - h* S --+X of ,S onto X is a spinor bundle on X which is associated to the L"-principal bundle Ph. Sections of ^9h describe Dirac fermion fields in the presence of a tetrad gravitational field h. Given an atlas 0 otT-X and the corresponding bundle coordinates (*^,ot) on 87, the universal On the BV Quanti'zationof GaugeGmaitation Theory 2Og spinor bundle S (22) is endowed with bundle coordinates (rI, ot,AA), where yA are coordinates on the spinor space I/". The universal spinor bundle ,S -+ X7 is a subbundle of the bundle of Clifiord algebras which is generated by the bundle of Minkowski spaces E*r: (LX x M)/L + Xr (23) associatedwith the L-principal bundle LX + X7. Then due to the representation 1 GT),,there exists the representation --.(tX -, - (24) 12:T*X@S x (M eV))lL" 1Tx x'y(M&V))lL" 5, E7 defined by the coordinate expression 12(dr^) - ol7". Given a world connection K on X, the universal spinor bundle ,S -+ E7 is provided with the connection At- d,r^I (u^ -I(ruuof,- no"our)of,Ksp,L"bABaBaA\ \ 4' / / + d,of* (ui +f,{nouoi- qo"4)LouAeuB a") (25) Given a tetrad field h, its restriction to ,Sh is the familiar spin connection tnt Kn : d,r^8[r^'* uuni- rfohur)(axn|- h'uK^u,)Loro u^),, Q6) "yt definedby a world connectionK 125,261.The connection(25) yieldsthe first order differential operator D: JLs->T*x$s, Er ou D : d,r\ - - ,f" 4) @Ku- o1Kxp,) LouA nuBfuo, (zr) " lrf lt "i on the fiber bundle S -+ X. Its restriction to ../1Sh c JLS recovers the familiar covariant differential on the spinor bundle ,Sh * X relative to the spin connec- tion (26). Combining (24) and (27) gives the first-order differential operator +T*X ---,S, D:.yEo D: JLS @S Xr l^.o- 1, - - ABoD-,\,-o,Bra'r A rf"4)(oKo o|Kxp,)r,ro.v"], (28) Le.\ a(nkboi" on the fiber bundle ,S -+ X. Its restriction to JlSh c JL S is the familiar Dirac operator on the spinor bundle Sh in the presence of a background tetrad field h and a world connection K. 2LO D. Bashhilru U G. Sardanashail,gl Thus, we come to the model of the metric-affine gravity and Dirac fermion fields. Its configuration space is the bundle product Y:Cx XS, (2e) Er coordinat"d by (rp,o!,,krop,yA). Let JLrY denote the first-order jet manifold of the fiber bundle Y + Xr. This fiber bundle can be endowed with the spin connection Av : d,r^I - o'ri - ,t*"o'r) ox,re (r^ f,t xp,Loa" ,a" ao) + d,ofu + - ,r*"our)LouA (30) (4 )@rori naBa") With this connection, we obtain the first-order difierential operator 'n{rou Dv:dr^8 - ri - qu" (o^u- otkxp,) LouAauB]ro, (31) lr* 4) and the total Dirac operator yB oDv - oljoB tlr* - *u - qk"obr) (o^k - o1kxp,)L"uo |t "i tyt), e2) on the fiber bundle Y + X. Given a world connection K: X --+C6, the restrictions of the spin connection ,4y (30), the operators Dy (31) and Dy Q2) to K"Y are exactly the spin connection (25), the operators (27) and (28), respectively. The total Lagrangian of the metric-affine gravity and fermion fields is the sum of a metric- affi.ne Lagrangian Iy4 depending on the variabres op" - ototT"b, kro p and the Dirac Lagrangian constructed from the Dirac operator (32) and the spinor metric (1e)[6,21]. Turn now to gauge transformations in this gauge theory. The spin frame bundle T* - X just like the linear frame one LX --+ X is natural, and it admits the canonical lift of any vector field on X. Hence, the universal spinor bundle ^9+ X is also natural, and the canonical lift us onto ^9of vector fields on X exists. The goal is to find its coordinate expression. Difficulties arise because the tetrad coordinates o! on E7 depend on the choice of an atlas itr of the bundle LX. Therefore, non- canonical components appear in the coordinate expression of us 12I,26]. Since the fiber bundles LX - X, GLa - GL+IL and, accordingly, the fiber bundles irt -* X, da -, L" are trivial, the fiber bundles EX -- !7 and ^9 --+ E7 are also trivial. Let i[ be an atlas of ffi -- Er with transition functions constant on fibers of Er. This atlas provides a trivialization Tft = P0 x E7 and the corresponding trivialization S=SoxXr (33) On the BV Quantizationof GaugeGro,uitation Theory 211, of the universal spinor bundle. With respect to such an atlas, one can define the lift ?.cS : u^ 0X * 1ruo of, ,fi + ) kt*uoi" - rto"our) (a,uu oi - u" of,n) Loud a&+ LouAeaB 0a), (84) 4\', ? "oft Lood.: rlu.6*,* rto"6t, onto ,S of vector fields u : u\0x on X. This lift however is not a vector field because it depends on the jet coordinates of,*. It is a generalized vector field (see the next section). Its projection onto ,Shin accordance with the splitting (33) is the well-known Lie derivative 1.th: u^0x + n7,- ,f" nor)(a,uu ni - u' a,hff)Looo ao (35) ){nru ua" of spinor fields in the presence of a tetrad field h 127,,28].Note that the lift (34) fails to be a Lie algebra homomorphism. At the same time, one can think of the fiber bundle ^9 -+ X (33) as being associated with the L"-principal bundle P0 - X. Infinitesimal generators of local onepaxameter groups of automorphisms of P0 are represented by sections of the vector bundle TPo /L" -+ X which take the coordinate form LLp: u^Ox * u*e^, where {"*} is the basis for the Lie algebra of L". Any vector field up yields the vector field ltrs: u^ 0x + (}ru"o"" - u* I^o * u* I^A naB 0e (36) "rfr)03, on the associated bundle S. These vector fields can also be regarded as infinitesimal gauge transformations of gauge theory of gravity in the presence of Dirac fermion fields. The total configuration space of this theory is the bundle product Y:SxCx (37) Er coordinated by (rx,ot,k*o p,grA). The generalized vector fields (34) and the vector fields (36) are extended to the fiber bundle (37) as 4 ttrSK:u^}X*\ru"o""Ofi t, + - - u"oh)(- Louo""fra&*LouAsutao) )(nkb"i, n*"our)(a"uuoi - - + l}ru"kr'p ilBu'k1ro, Lru'kro p * Lrpuof (38) #, 'ttrsK : u^ 0x + (\ru" o"" - u* Irno a& + u* rrnAsyB 04 "oft) - - + lLruotcr'B ilBu'le1rq, Lru'kro B * LpBu')*T; (3e) 2L2 D. Bashhiwa U G. Sardonoshui,IE 3. Gauge S;rmmetries In a general setting, gauge systems on fiber bundle Y -+ X are described as follows [8]. With the inversesystem of jet manifolds X J-y & JLy - ...J,-ry LL J,y <- ... , (40) one has the direct system n* jL nl-'* (2* X ' O*Y OiY ------+' " Oi-rY ' O;Y ---+ " ' (41) of graded differential algebras (henceforth GDAs) 0;Y of exterior forms on jet manifolds J'Y with respect to the pull-back monomorphisms rl-r*. Its direct limit OUY is a GDA consisting of all exterior forms on finite order jet manifolds modulo the pull-back identification. The projective limit (J*Y,,rf;: J*Y - J'Y) of the inverse system (a0) is a Fr6chet manifold. A bundle atlas {(Uv;r^,yd)} of.Y + X yields the coordinate atlas (uv);n^,aL)}, < (42) {((rr3")-t a'\+tr:ffiarr|, o lAl, of J*Y, where d* arc the total derivatives(2). Then OUY can be written in a coordinate form where the horizontal one.forms {dr^} and the contact one-forms {0L : dyL - a\*nd.r^I are local generatingelements of the OlY-algebra O\Y. There is the canonicaldecomposition O:"Y - aOf{Y of 0\Y into Of Y-modules O*{Y of k-contact and m-hofizontal forms together with the correspondingpro- jectors hp: OiY - O$iY. Accordingly,the exterior differentialon 0\Y is split into the sum d - d,n * dv of the nilpotent total and vertical differentials dn(il - d,x\Adx(d),, dv(il - eLn*6 S e o\Y. Any finite order Lagrangian (1) is an element of 0!Y, while - 6L t,ili Aa: f (-rlt^tan@{t)Onnu e oY,y (43) 0 rt v"YAA r*x, (44) Y where V"Y denotes the vertical cotangent bundle of Y + X. Given a Lagrangian .L and its Euler-Lagrange operator 5L (43), we further abbreviate A = 0 with an equality which holds on-shell. This means that .4 is an element of a module over the ideal ^I1,of the ring(C!Y which is locally generated by the variational derivatives t,i, and their total derivations dt&. We say that Ir is a differential ideal because, if a local function / belongs to 17, then every total derivative dll does as well. On the BV Quantizationof GaugeGruuitation Theory 2L3 A Lagrangian system on a fiber bundle Y -+ X is said to be a gauge theory if its Lagrangian L admits a family of variational symmetries parameterized by elements of a vector bundle W + X and its jet manifolds as follows. Let aOo*Y be the O\Y-module of derivations of the lR-ring 0%Y. Any t9 e oQ$Y yields the graded derivation (the interior product) S)Q of the GDA OUY given by the relations ,e)d,f-8(f), s)(OAo)-(slilAo* (-gt0tfA(reJo),f e0\v, Q,o€oLY, and its derivation (the Lie derivative) Lod:s)d,o+d(s)il, Lo(d^O') -Lo(Q) Ad'+d^Lo(d'),, Q,d'eo\Y. (45) Relative to an atlas (42), a derivation tg e OOL reads tg- d}ar*giil+ t s'naf, (46) lAl>0 where the tuple of derivations {01,4} ir defined as the dual of that of the exterior forms {d*^,dAL} with respect to the interior product I [29]. A derivation t9 is called contact if the Lie derivative L,e (45) preserves the contact ideal of the GDA OiY generated by contact forms. A derivation u (46) is contact if and only if Slr=dn(8n-aLr9*)*aL+nflr, 0 < lAl. (47) Any contact derivation admits the horizontal splitting /\ g-8n*gv:rg^d.:, --8i -yi9u. (48) +lui&+ I apiaf l, ui \ o L,sL- ul6L + d,H(hs(flEr))+ Ldy(8n )w), (50) where E; is a Lepagean equivalent of L 1291.A contact derivation t9 ( S) is called va,riational if the Lie derivative (50) is ds-exact, i.e., LreL: dno, o e 0!'-t. A glance at the expression (50) shows that: (i) ,t (48) is variational only if it is projected onto X; (ii) ?t is variational if and only if its vertical part tly is well; (iii) it is variational if and only if. u)dL is ds-exact. Therefore, we can restrict our consideration to vertical contact derivations d - 8v. A generalized vector field u 2I4 D. Basl/cirva U G. Sodanashailg (49) is called a variational symmetry of a Lagrangian Z if it generates a variational contact derination. In order to define a gauge symmetry [8], Iet us consider the bundle product E (3) coordinated by (r\,,Ai,,{"). Given a Lagrangian L onY,let us consider its pull-back, s&y again .L, onto E. Let t9r be a contact derivation of the lR-ring OLE, whose restriction 8-ynloy":lan"'af (51) 0 Then one says that u ( ) is a gauge symmetry of a Lagrangi an L. In accordance with Noether's second theorem [8], if a Lagrangian .L (1) admits a gauge symmetry u (4), its Euler-Lagrange operator (43) obeys the Noether identity Ao6.L-0, | _ r r \ . .,a,_.:^ I (53) I f L';^d1,t6le'"-lI (-r;lrt4n@?nln)l€'r-0, Lo< lll A_ (,a,w- t €'ai^("),ft)gmr:{'I (-r;lnt4n@?nilfr, 0 is the Noether operator on the fiber bundle (44), coordinated by ("), Ui,!t), with values in the fiber bundle E.I f\r-*. (55) Y For instance, if a gauge sYmmetrY u * (r!,€' + u?p{t" * uf"p(i)ai (56) is of second jet order in parameters, the corresponding Noether operator (54) reads A',: ui - dpu?p* d,pu?"p, L?* : -ri' a 2d,ul:"p', L?"' - 'i"'' (57) On the BV Quantizationof GaugeGrauitation Theory 215 4. Gauge Symmetries in Gauge Gravitation Theories Turn now to gauge symmetries of gauge gravitation theories mentioned in Sec. 2. Let us start with the metric-affine gravitation theory on the fiber bundle ExCx with the infinitesimal gauge transformations (12) parameterized by vector fields on X. In this case, the configuration space (3) of the gauge system is the bundle product E:ExCx xTX, (58) xx coordinated by (n^,ou',k^'o,z)), and the gauge symmetry ( ) is the generalized vector field p - (r| ou| + ,f oo' - ,'oog) #, * (rikr" p - r"pkrro,- r'*krop * rtB - rukvrt t) (5e) h Let us consider the gauge theory of principal connections in the presence of a metric gravitational field on the fiber bundle C x E with the infinitesimal gauge transformations (16). The configuration space of the corresponding gauge systern is the bundle product E-CxExTcP,, (60) xx coordinated by (rx ,,a\, ooq , r\ ,,{'), and its gauge symmetry is given by the gener- alized vector field u - (cf,oae^€q+{i-rroL-rroix)a} +(rjo"e *rf oo'-r^oi\#, (61) The configuration space of the gauge theory of gravity in the presence of Dirac fermion fields is the fiber bundle (37). In the case of infinitesimal gauge transfor- mations (38), the corresponding gauge system is defined on the bundle product E: S (62) iC* ;r*, coordinated by (r^,of ,k^'o,AA,,r^).Its gauge symmetry reads - oft+ rfrrf)?fof - r"ofo)aft " |ifdqo"ot - ,f"4)?roi - ruof,o)LouA - *lf,u-'ofi naB ""a!]a^ * (rfkr" p ' rbkr"o,- ,ilk,o B + ,tp - ,v kvtt t)# (63) If infinitesimal gauge transformations are vector fields (39), the configuration space of this gauge system is the bundle product E - S I Cx / Tt (64) Er lr "Po, 2l6 D. Bo.;slr'/rircu& G. Sardanashtrily coordinatud by (r^,of,,le^'o,aA,T\,{-), and the gauge symmetry is p - (rf;o!. - ," o3. - €* I*d .ofr)A&* (€* I,nAByB - ," af;)On * (rf kr' B - rbkr"o,- ,'rkrop + ,tB - ,u kvrt ,)# (65) 5. BRST Symmetries In order to introduce BRST symmetries, let us consider Lagrangian systems of even and odd variables. We describe odd variables and their jets on a smooth manifold X as generating elements of the structure ring of a graded manifold whose body is X [8,29,30]. This definition reproduces the heuristic notion of jets of ghosts in the fietd-antifield BRST theory [31,32]. Recall that any graded manifold (A,, X) with a body X is isomorphic to the one whose structure sheaf 2lq is formed by germs of sections of the exterior product AQ":Rog.o[O.(E..., (66) xxx where Q* is the dual of some real vector bundle Q -+ X of fiber dimension rn. In field models, a vector bundle Q is usually given from the beginning. Therefore, we consider graded manifolds (X,%d where the above.mentioned isomorphism holds, and call (X,?Id the simple graded manifold constructed from Q. The structure fing Aq of sections of 2lq consists of sections of the exterior bundle (66) called graded functions. Let {c"} be the fiber basis for Q* - X, together with transition functions Co : pt"b.It is called the local basis for the graded manifold (X;21d. With respect to this basis, graded functions read f : "' con, " ?-o'"'t f,f",...'*co' where for...ooare local smooth real functions on X. graded (X,,2Ie), of Z2-graded Given a manifold let nAq be the " q-module derivations of the Z2-graded ring of Aq, i.€., u(f f') -u(f)f' + (-r;t"Jlflfuff'),, u e tAs,, f ,f' e Aq, where [.] denotes the Grassmann parity. Its elements are called Z2-graded (or, sim- ply graded) vector fields on (X, 2Id.Due to the canonical splitting VQ : Q x Q, the vertical tangent bundle VQ * Q of Q --+ X can be provided with the fiber bases {d"} which is the dual of {c"}. Then a graded vector field takes the local On the BV Quantization of Gauge Gra,uitation Theory 2L7 form u - u^A^ * uo1o, where u\,,uo are local graded functions. It acts on Aq by the rule u(fo...uc". . . - u^Ax(fo...b)c". . - * ud fo...u1a)("". . . (67) "b) "b "\. This rule implies the corresponding transformation law ?..0,\: r.L\, yto : Oiui +u^Tx@?)C. Then one can show that graded vector fields on a simple graded manifold can be represented by sections of the vector bundle Ve - X, locally isomorphic to nQ* 8x (Q exTX). Accordingly, graded exterior forms on the graded manifold (X,,2Ld are intro- duced as sections of the exterior bundle nV[, where Vb - X is the AQ*-dual of Vq. Relative to the dual local bases {d"^} for T"X and {dcb} for Q*, graded one-forms read 6 : Ssdr^+ Ood"", O!,: p-r\du, 6'x: dx + p-t|1^(p?QuC. The duality morphism is given by the interior product u)Q - u^dx* (-f;td")u"6o- Graded exterior forms constitute the bigraded differential algebra (henceforth BGDA) C[ with respect to the bigraded exterior product A and the exterior differ- ential d. Since the jet bundle J'Q - X of a vector bundle Q ---+X is a vector bundle, let us consider the simple graded manifold (X,2Lyg) constructed from J'Q -+ X. ( Its local basis is {r}, cfi},0 < lAl r, together with the transition functions - d,x(p|C), d^:o.r * (68) "'f+r. I "i*no*, lAlcr where 0f are the duals of. cfr. Let Cj"q be the BGDA of graded exterior forms on the gradd manifold (X,2Iyg). A linear bundle morphism r[-{ J'Q - J'-rQ yieldsthe correspondingmonomorphism of BGDAs Cj,-rq -- C\,e.Hence, there is the direct systemof BGDAs cAg Ci,e U Ci,e------+.. . . (6e) Its direct limit CLQ consists of graded exterior forms on graded manifolds (X,211"g), e N, modulo the pull-back identification, and it inherits the BGDA " operations intertwined by the monomorphisms rf-r*. It is a C-(X)-algebra locally - generatedby theelements (L,cfr,dr^,07 d.fr- c\*ndr^), 0 < lAl. In order to regard even and odd dynamic variables on the same footing, let Y ---+X be hereafter an affine bundle (or a subbundle of an affine bundle which need not be affine), and let PLY C OUY be the C*(X)-subalgebra of exterior 2L8 D. Basll/circa U G. Sodanashaily forms whose coefficients a,re polynomial in the fiber coordinates gl, on jet bundles J'Y -- X. Let us consider the product sL : crQ^P;Y (70) of graded algebras CtQ and PiY over their common graded subalgebr a 0* X of exterior forms on X [29]. It consists of the elements Drlr&6t,, Dor&*t, $ecLQ, oePLY,, modulo the commutation relations the0_(-1)ldltot464;, ,heCLe, O€piy, tn1\ \'r'' 0bno)86:th8(oAO), oeO*X' They are endowedwith the total form degree lrll + l@land the total Grassmann parity [T/].Their multiplication khs il ^ Qh'I O) :- (-1;14'tt6t1E^rh')s (d^ O') (72) obeys the relation p A g' - (-t;klls'l+lPllv'lp' A g, p,Q' € S;, and makes SL (70) into a bigraded C""(X)-algebra. For instance, elements of the ring S$ a,re polynomials of cff and yi with coefficients in C*(X). The algebra Si is provided with the exterior differential d(rhs d),: (d,c$)8d + (-t;lut ?he (d,p4), rhe CL, d ePJ", (73) where ds and dp arc exterior differentials on the differential algebrasCiQ and PiY,, respectively.It obeys the relations d(p ng'):dp Ag'+eDlvlp Ad,P', v,e'€ s;, and makesSL into a BGDA. We agreeto call elementsof Si the graded exterior forms on X. Herea,fter,let the collective symbols sf stand both for even and odd generatingelements cf , aL of the C€(X)-ring .S$. Then the BGDA S; is locally generatedby (1, sf,dr^,|f;.: dtf. - rf*rr&^), l^l > 0. Sincethe generatingele. ments sf and 0f are derived from sA, the set {sA} is called the local basis for the BGDA SL. Similarly to O\Y, the BGDA Si is decomposedinto S$-modules S$' of k-contact and r-horizontal graded forms together with the correspondingprojec- tions hn and h'. Accordingly, the exterior difierential d (73) on Si is split into the sum d - dn * dv of the total and vertical difierentials d,n(il- dn\ ndx(d), dv(il - ef n4,0,, d € s:. One can think of the elements L - Lu € sg', 6(L)- I (-tlt^t7Atd^@IL) € sg" lAl>0 as being a graded Lagrangian and its Euler-Lagrange operator, respectively. On the BV Quantizationof GaugeGmuitation Theory 279 A graded derivation d e O5$ of the lR-ring S$ is said to be contact if the Lie derivative L15rpreserves the ideal of contact graded forms of the BGDA S;. With respect to the local basis ("),"f, dr^,,?f) for the BGDA 5f, any contact graded derivation takes the form $:8n-l$v-dld.r+ (o"a^*,F_dns"4,), (74) where 8^ ,,8A are local graded functions [29]. The interior product t9l/ and the Lie derivative LoQ, d e 51, are defined by the formulas 8)Q-,9^d^ +(-r;td"l8A6e. d € SL, s)(OAa)- (0)O)Ao * (-t;ldl+tolt"ldA(,9_1"), g,o e Si, Lod : s)d,O+ d(rj)il, Lo(dAo) : Lo(il A o *(-r;lollol4 A L.e(a). The Lie derivative L6L of a Lagrangian .t along a contact graded derivation I (74) fulfils the first variational formula L,sL - 8v)5L + dH(ho(r9lEz,))* dv(fs)u)L, (75) where Ez, is a Lepagean equivalent of a graded Lagrangian L 1291. A contact graded derivation t9 is said to be variational if the Lie derivative (75) is ds-exact A glance at the expression (75) shows that: (i) a contact graded derivation t9 is variational only if it is projected onto X, and (ii) d is va,riational if and only if its vertical part tly is well. Therefore, we restrict our consideration to vertical contact graded derivations - I | an"Aa!. (76) 0 Such a derivation is completely defined by its first summand 'u: uA(r^, sf;)06, o < lAl < /c, (77) which is also a graded derivation of 5$. It is called the generalized graded vector field. A glance at the first variational formula (75) shows that I (76) is variational if and only if u)6L is ds-exact. A vertical contact graded derivation I (76) is said to be nilpotent if L"(Luf) : ("""a8@il4+(-1)r"'lr""luf ufala|)d :0 (78) lrl>o,lAl>o for any horizontal graded form d € Sg. or, equivalently, (rl o d)(/) : 0 for any graded function / e S3". One can show that t9 is nilpotent only if it is odd and if and only if the equality 8(uA1: - o (7e) D, "$a}(ro) lrl>0 holds for all uA lZ0]. 22O D. Basl*irca E G. Sad,anasfufily Return now to the original gauge system on a fiber bundle Y with a Lagrangian L (L) and a gauge symmetry u ( ).Let us consider the BGDA sl[w;Y] : cLW^PLY whose local basis is {c', yi}.Let L e {7ffY be a polynomial in aL,0 < lrl. Then it is a graded Lagrangian L €Py'Y c sg"lw;yl in silw;y). Its gauge slrnme6ly u (4) gives rise to the generalized vector field uB _ u on .8, and the latter defines the generalized graded vector field u (77) by the formula (5). It is easily justified that the contact graded derivation ?9(76) generated by u (5) is variational for L. k is odd, but need not be nilpotent. However, one can try to find a nilpotent contact graded derivation (76) generated by some generalized graded vector field (6) which coincides with t9 on PLY.Then u (6) is called a BRST symmetry. In this case, the nilpotency conditions (79) read /\ - I a"( t r?E"?.)t af;pt,^yci+ t d,1,(u,)ui,n0, (80) r \E / A A + d,r(u')ff),':o ' (81) T ( Dan@i'="?)o! for all indices j and q. They are equations for graded functions u" € sglw;Yl. Since these functions are polynomials -- y,r uiol+D"?i,e"+ I +... (82) I f r,fz "Ti\;l;,q,,fi in cfl, Eqs. (80) and (81) take the form /\ f a"t t u!;E"L,lt a?("'",n)ci, + I an(u121)ul,^ -s, (83) f an@ft,q2)r/"'A - o, (84) I t d,n(ul*cL)af"o6-r1+I an("T*)tu\t -0. (85) A 3 m*n-]:k One can think of the equalities (83) and (85) as being the generalized commuta- tion relations and generalized Jacobi identities of original gauge transformations, respectively [9]. 6. BRST Symmetries in Gauge Gravitation Theories Following the procedure in Sec. 5, let us obtain BRST symmetries of gauge gtavi- tation theories listed in Sec. 2. Given a gauge system on the fiber bundle (58), let us consider the BGDA sJ" E;cKl - c; (86) frx' lrxl Ap*lzic*], On the BV Quantizationof GaugeGrnuitation Theory 221 whose basis consists of even elements oo?, k,oB and odd elements c), which are ghosts substituting for paxameters in the gauge symmetry (59). We obtain the ") nilpotent BRST symmetry A ' 0 o,-..):,) o,ogJ r^.e ffi+rr*gM*u^6 .A - (o,Pc]+ oo,4 - "^o7\ ,-9, I * (clkr"p - - * cl,p-.^lexuo t)# + (82) te. ""Bkr"o,"Lk,'B "1"rfi. The BGDA of the gauge system on the fiber bundle (00) is SJ"["cP; C x E] whose odd basis elementsc" and d substitute for parameters {' and zI in the gaugesymmetry (61). The correspondingBRST symmetry reads a (^'9c| u - (&oox"o* c\- aic\- cunr.*1r.tt \ t, + oo,4 - 0a\ \" "^o1;\#, .(- *qn*"n-*";)#.4*#. (88) T\rrn now to the gauge theory of gravity in the presence of fermion field. Since gauge transformations (35) fail to form a closed algebra, one cannot associate to them a nilpotent BRST symmetry. Therefore, let us consider the gauge gravitation system on the fiber bundle (64). Its BGDA is s: [r",po;,Si,"*),, whose odd basis elements c* and d play the role of ghosts of the gauge symmetry (65). The associated BRST symmetry is - - p - (clo!, c* I*d."il-". v-"r ! * (c* I,nA naB- al)6o "'oi. )ot "" - - - * (clk*" B cbkuo, c|k,op + c\.p d "^kxuo # . (-iq,*"'-*"7)#*"i"# (8e) 7. The BV Quantization As was mentioned above, we restrict our consideration to the BV quantization of the metric-affine gravitation theory. Its original LagrangianIya on the fiber bundle E x Cx need not be specified, but it possessesthe gauge symmetry u (59). It is important that this gauge symmetry is complete and irreducible. 222 D. Boslr/cirouE G. Sadanashuily Let us note that any Lagrangian .L has gauge symmetries. In particular, there always exist trivial gauge symmetries qi,j/t,ES-g. ,7-j,i,It,E - _rpi,j,E,lt u - M;'n:- F wLv?,t Lr -r t't(W?^{^, L-r A t Furthermore, given a gauge symmetry u (4),let h be a linear differential operator on some vector bundle E' --. Y, coordinated by ("), Ui,,€'"), with values in the vector bundle ^8. Then the composition u o h : u'!'L€KOl is a variational symmetry of the pull-back onto Et of.a Lagrangian L on Y, i.e., a gauge symmetry of.L.In view of this ambiguity we agree to say that a gauge symmetry u (4) of a Lagrangian .t is complete if a different gauge symmetry u' of. -L factors through u as uto:uoh+7,, T=0. In the case of a generic metric-affine Lagrangian .Lya on the jet manifold .fl(I x Cx), the gauge symmetry u (59) is complete. Although there are Lagrangians, e.g., the Hilbert-Einstein scalar curvature and the Yang-Mills type Lagrangian which admit more wide gauge symmetries. Note that a metric-affine Lagrangian tru.q. usually depends on kro B through the curvature R1r"p (8). The Euler-Lagrange operator dtrvra : tapd,oa? A, + tt" oPdkro p A, (e0) of a metric-affine Lqgrangian takes its values in the fiber bundle 4 (u."trv"cK) A A".x, (e1) ExC x coordinated by (r^,6u',k^'o,opr,E ro). Due to the canonical splittings v*E: x x x* - E x \/ r"x, v"Cx- cx cxi (8 ,t x x Y yc;r: qr"r),(e2) the fiber bundle (91) is isomorphic to 4 ("; ci,) r*x "") ;(". i a A A complete gauge symmetry u * 0 (4) is called reducible if there exists a vector bundle Wo -+ X and a E-valued differential operator u0 rt 0 on the bundle product Eo : Y x Wo which is linear on Ws and obeys the equality u o u0 n: 0. If. u'o is another differential operator possessing these properties, then u'0 factors through uo. The gauge symmetry (59) is irreducible becauseits term of maximal jet order is ,frpl?p.Indeed, let us assumethat there exist a vector bundle Wo ---+X coordinated On the BV Quantizationof GaugeGmuitation Theory 223 by ("^, toA) and the above-mentioned differential operator u0 on the bundle product Eo: t,trCx trWo with values in the vector bundle E (5S) such that u o u0 = 0. Given the mucimal jet order term ur?=r/# * o (e3) lEl:k of the operator u0, the equality u o uo e; 0 results in the condition uslad,pw!ffi*o. lEl:k It follows that all coefficients uol'E and, consequently, the term (93) vanish on-shell. By virtue of Noether's second theorem, the gauge symmetry (59) defines the Noether operator A (54) on the fiber bundle (91) with values in the fiber bundle 4 (> Cx x".X) (e4) \xx/" A Ar*x, ExCx coordinated by>y (n^,oF'(r^, ,k\'o, fr). The formula(57) gives : o A-r)T) t lLl9'nand.,p I L^t d^dnb'o7), lAl;0,1,2 - : r\ l- @78+ 2oio6i)a,p 2o"Pd,,osB - - + ( kxpoB krr' plfl+ kBr"x * kpxoB)Er.P + ( - k*" 857* kpoxlfi * kx" B6i)d,E*,F+ drpTu1F)u. (9b) Following the standard BV quantization procedure [4,5], let us construct the extended Lagrangian which obeys certain conditions and depends on ghosts and antifields. In order to introduce antifields of even fields oo9,, kroB and odd ghosts rA, let us extend the BGDA (36) to the BGDA 4 s; (r. r*x;E xA ,. *f (e6) *Lfrx x\ x x\ci,)S A "x A , x ;cxx ;r- x where the splittings (92) have been used. The local basis for the BGDA (96) is {o",, kro p, c^, drr,Eu oF,,ex},, (97) , where the antifields opr, E\ro are odd, while the antifields cr are even. It is con- venient to use right graded derivations F with respect to antifields. They act on graded functions and forms / on the right by the rule 5 tOl- dOLF+ 4@LE), 5 tO^d') - (-r;td't5 fOl^0, + 6n*a(d,). 224 D. BoshkircaE G- SorfunoshuilY in vari- An original metric-affine Lagrangian,y4 is assumed to be polynomial in the BGDA (96)' ables otrr, k^, o. Therefore, it is obviously a graded Lagrangian Given the BRST symmetry u (87), Iet us consider the graded density L - fr! : Lwt.t * uo7 ooBa * uro pk* o9' * ulclt..''' (e8) one can show that it obeys the following conditions L ^.a-- aL .,^-0L'u : (ee) ,rovc --5 up*0: 0A&, 66A) 6' u,)u--l#9 *' **Y*l' * d,no: d,Ho'|, (1oo) LuL H+)kpo? 6k*op )ex 6c" " laafl\o! 1 (101) Ll-o:E:e:o: trua. the master A glance at these conditions shows that L (98) is a proper solution of Lagrangian. equation (100) [4,5]. Thus, it is the desired extended for One can write different variants of a gauged-fixed Lagrangian (see, e.g., [33] with the the problem of the general cova,riance in quantum gravity). By analogy quantity Lorenz gauge in Yang-Mills gauge theory, let us choose the gaugefixing sE : o\'sxro p : o\tr(k^ro p + kpxoo), (102) density' where Sxr"op is given by the expression (9). In order to defrne a gauge'fixing to the local we add even elements 7ft,, oddelements Cf and,their even antifields eg basis for the BGDA (96). Then this gaugefixing density reads a: c\(i"u- 'E)" (103). The desired gauged-fixed Lagrangian -L6p comes from the graded Lagrangian L + BfrOEa: LMA* uo7ooBuI uroBkp oga * u^esw+ Bftefu by means of replacement of antifields with the variational derivatives do 5o -2dx(c|o^*)' -CY(ko.'*+kBo'r")', E''q P - 6o9:ffi: 5k*" p - do g (lan_ c):;A:r' _o. eg\ra-W-: : "")sf). \2"' One obtains tg) Lcp: Lva.- uogCI (lroB', + kpo' p) + 2uo,P d,r(C\o'u) + BE (i"f- - Lwrt- (o^9ci + o"^f^ - (k.B" , + lepo" p) "^I|P)CN - r) + 2(clkr' - c'Bkpo,- P + d dr(C|o" B "'rkro "7,8 "\kxuo + Bfr(i"g - o\'(k^,go* r,^u")). 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