Cohomological Analysis of Gauged{Fixed Gauge Theories

The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Cohomological Analysis of GaugedFixed Gauge Theories Glenn Barnich Marc Henneaux Tobias Hurth Kostas Skenderis Vienna Preprint ESI Octob er Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat ESI ULBTH FTUV CERNTH MPIPhT SPIN hepthyymmnnn Cohomological analysis of gaugedxed gauge theories ab bc de Glenn Barnich Marc Henneaux Tobias Hurth f and Kostas Skenderis a Departament de Fsica Teorica Universitat de Valencia E Burjassot Valencia Spain b Physique Theorique et Mathematique Universite Libre de Bruxelles Campus Plaine CP B Brussels Belgium c Centro de Estudios Cientcos de Santiago Casilla Santiago Chile d CERN Theory Division CH Geneva Switzerland e MaxPlanckInstitute for Physics WernerHeisenb ergInstitute Fohringer Ring D Munich Germany f Spinoza Insitute University of Utrecht Leuvenlaan CE Utrecht The Netherlands Abstract The relation b etween the gaugeinvariant lo cal BRST cohomology involving the antields and the gaugexed BRST cohomology is claried It is shown in particular that the co cycle conditions b ecome equivalent once it is imp osed on the gaugexed side that the BRST co cycles should yield deformations that preserve the nilp otency of the gaugexed BRST dierential This shows that the restrictions imp osed on lo cal counterterms by the Quantum No ether condition in the EpsteinGlaser construction of gauge theories are equivalent to the restrictions imp osed by BRST invariance on lo cal counterterms in the standard Lagrangian approach gbarnichulbacbe henneauxulbacbe tobiashurthcernch KSkenderisphysuunl Intro duction In the BRST approach to p erturbative gauge theories the p ossible counterterms are restricted by WardSlavnovTaylor identities which have a cohomological interpretation If one follows the pathintegral approach and takes into account the renor malization of the BRST symmetry a la ZinnJustin by intro ducing sources coupled to the BRST variation of the elds and the ghosts it may b e shown that the counterterms must full the BRST invariance condition sA where s is the BRST dierential acting in the space of elds ghosts and asso ciated sources antields The counterterms are lo cal so A in is given by the integral of a lo cal nform a in terms of which the BRST invariance condition b ecomes sa db for some n form b Following an initial investigation by Joglekar and Lee the general solution of for YangMills gauge mo dels has b een determined in where it was shown that up to trivial terms of the form sc de the counterterm a in is equal to a strictly gaugeinvariant op erator plus ChernSimons terms in o dd spacetime dimensions in the absence of U factors for which there are further solutions also dealt with in This guarantees renormalizability of the theory in the mo dern sense in any numb er of spacetime dimensions and in the standard p owercounting sense in dimensions If one follows instead the op erator formalism and the Quantum No ether metho d based on the gaugexed BRST formulation one nds that the counterterms are con strained by the condition g a db g where is the gaugexed BRST dierential acting on the elds and where b oth a and b involve only the elds no antield The symb ol means equal when the gaugexed equations of motion hold The question then arises as to whether and are equivalent It may b e shown that the antield and gaugexed lo cal cohomologies are equivalent so that 0 g 0 any solution a of sa denes a solution a of a and vice versa This is not true however for the cohomologies mo dulo d In particular there are solutions of that have no analogue in the antield cohomology and which therefore do not corresp ond to an integrated gaugeinvariant op erator An example is given by the CurciFerrari mass term a a A A C C a a which is a solution of in the gauge where the equation of motion for the auxiliary b b c eld is b A C f C but which do es not dene an integrated gaugeinvariant a b a ac op erator The prop erties of have b een studied in Thus and are in general not equivalent If however the co cycle condition is supplemented by the requirement of nilp o tency of the deformed BRST dierential which is required if we want the theory to b e unitary the CurciFerrari mass term is excluded It is the purp ose of this letter to show that quite generally the gaugedxed co cycle condition supplemented by the requirement that the deformation generated by the p ermissible counterterms should pre serve onshell nilp otency of the BRST symmetry is equivalent to the antield co cycle condition which controls the counterterms in the ZinnJustin approach This letter is organized as follows In the next section we recall some salient prop erties of the gaugexed action The equivalence of the two co cycle conditions is shown in section while a discussion of trivial solutions is presented in section In section we review the analysis of counterterms in the Quantum No ether metho d and show how nilp otency of the deformed BRST dierential arises in that context Finally in an app endix we present an analysis of the relation b etween the antield and the weak gaugedxed cohomology using metho ds of homological algebra Gaugexed action The starting p oint is the solution S of the master equation L L R R S S S S S S A A A A We use DeWitts condensed notations The solution S is a lo cal functional as are all func A tionals without free indices o ccurring b elow The elds include the original elds the ghosts as well as the auxiliary elds and the antighosts of the nonminimal sector We assume that the canonical transformation necessary for gaugexing has already b een p erformed so that the gaugexed action is simply obtained by setting the antields equal g g to zero S S The gaugexed BRST dierential is dened by R S g A j A R L where right and left derivatives are dened by F F z z z F z We use the conventions of but the derivations are taken to act from the left so g sF S F etc The transformation generated by leaves the gaugexed action g invariant b ecause of the master equation As a result the functional derivatives of S transform into themselves R R g L R g S S S g A B A B 1 We keep here the auxiliary beld but similar considerations can b e made if one eliminates the auxiliary elds since the gaugexed BRST cohomological groups are invariant under such an elimination here and b elow it is understo o d that is set equal to zero after the second derivatives have b een computed The gaugexed BRST dierential is weakly nilp otent R g R L S S g A B B A Both and are direct consequences of the denition and the master equa tion The BRST dierential in the space of the elds and the antields is dened by sF S F g A g A for any F It is related to as s antielddep endent terms or which g A A is the same s j It is strictly nilp otent s It will b e useful in the A sequel to give a sp ecial name to the terms linear in in the expansion of s A g A g A s O with g A A S j linear in L R S B B A The action of s on the antields can also b e expanded in p owers of the antields One g g g O where is the Koszul dierential asso ciated with the has s A A A gaugexed stationary surface R g S g j S A A A and where R L S g S j A A B linear in A B g g g g g g g g g A g A One easily veries the relations g g g from the denitions of the derivations and These relations are actually the rst ones to arise in the expansion of s in p owers of the antields The canonical transformation appropriate to gaugexing do es not mo dify the coho mology of s either in the space of lo cal functions or in the space of lo cal functionals b ecause it is just a change of variables So in the case of YangMills theory the coho mology group H s F of the BRST dierential in the space F of lo cal functionals is still given by the analysis of In H s F the sup erscript is the total ghost numb er g g g Note however that the expansion s is not the standard expansion arising prior to gaugexing since the degree involved here is the total antield numb er that gives equal weight to each antield irresp ective of its antighost numb er This is why it is the Koszul resolution asso ciated with the gaugexed stationary surface that arises in the present analysis and not the KoszulTate resolution asso ciated with the gaugeinvariant equations of motion Since the equations of motion following from the gaugexed action have no gauge invariance by assumption one may invoke the general results of to assert that g H F k k where k is the total antield numb er used in the ab ove expansions In words any lo cal R g g functional A F A a that solves A a db and is at least quadratic g g in the antields has the form A C a c dm g We have of course H L for k in the space L of lo cal functions but k we shall need the version valid for lo cal functionals b elow This is a direct consequence of Theorem or of which states that there can b e no nontrivial higher order conservation laws for an action having no gauge symmetries This theorem is also known as the Vinogradov twoline theorem Because higherorder conservation g g laws and elements of H F also denoted H jd are in bijection the prop

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