The Cohomology of Algebras over Moduli Spaces

arXiv:hep-th/9410108v27Nov1994THECOHOMOLOGYOFALGEBRASOVERMODULISPACESTAKASHIKIMURAANDALEXANDERA.VORONOVOctober28,1994Thepurposeofthispaperistointroducethecohomologyofvariousalgebrasoveranoperadofmodulispacesincludingthecohomologyofconformalfieldtheories(CFT’s)andvertexoperatoralgebras(VOA’s).ThiscohomologytheoryproducesanumberofinvariantsofCFT’sandVOA’s,oneofwhichisthespaceoftheirinfinitesimaldeformations.ThepaperisinspiredbytheideasofDrinfeld[5],Kontsevich[16]andGinzburgandKapranov[10]onKoszuldualityforoperads.Anoperadisagadgetwhichpa-rameterizesalgebraicoperationsonavectorspace.Theywereoriginallyinvented[21]inordertostudythehomotopytypeofiteratedloopspaces.Recently,operadshaveturnedouttobeaneffectivetoolindescribingvariousalgebraicstructuresthatariseinmathematicalphysicsintermsofthegeometryofmodulispacesparticularlyinconformalfieldtheory(see[8],[12],[14],[19],[26],[27])andtopologicalgravity(see[15],[18]).Infact,a(treelevelc=0)conformalfieldtheoryisnothingmorethanarepresentationoftheoperad,P,consistingofmodulispacesofconfigurationsofholo-morphicallyembeddedunitdisksintheRiemannsphere.Alternately,aconformalfieldtheoryissaidtobeaP-algebraoranalgebraoverP.Inthefirstpartofthispaper,weusetheideaofGinzburg-Kapranov[10]ofho-mologyofanalgebraoveraquadraticoperadtodefinethecohomologyofanalgebraoveraquadraticoperadwithvaluesinanarbitrarymodule.WeprovethatPisaquadraticoperadandconstructitsKoszuldualoperad.Wethenconstructthecohomologytheoryassociatedtoaconformalfieldtheory.Wedemonstratethatthesecondcohomologygroup,asitshould,parameterizesdeformationsofconformalfieldtheories.OurapproachtodeformationsismorallythesameastheonedevelopedbyDijkgraafandE.andH.Verlinde[3,4]andRanganathan,SonodaandZwiebach[22]using(1,1)-fields,exceptthatwefixtheactionoftheVirasoroalgebra.Itwouldbeveryinterestingtofindanexplicitconnectionbetweenthetwoapproaches.Byanalogywiththecaseofassociativealgebras,oneexpectsthethirdcohomologygrouptocontainobstructionstoextendinganinfinitesimaldeformationtoaformalneigh-borhood.AnotherplausibleinterpretationofhigherdimensionalcohomologyisthatResearchofthefirstauthorwassupportedinpartbyanNSFPostdoctoralResearchFellowshipResearchofthesecondauthorwassupportedinpartbyanNSFgrant12TAKASHIKIMURAANDALEXANDERA.VORONOVtheyshouldparameterizeinfinitesimaldeformationsinsidealargercategoryof“A∞-conformalfieldtheories”whereoneallowsmultilinearvertexoperatorswhicharenotcompositionsofbilinearones.Thesecondpartofthispaperisstructuredintheoppositeway.Weconstructthecohomologytheoryassociatedtoalgebrasoverthelittleintervalsoperad,B–thespaceofconfigurationsofintervalsembeddedintheintervalviatranslationsanddilations–bystudyingdeformationsofB-algebrasinanalogywiththemannerinwhichHochschildcohomologyarisesfromdeformationsofassociativealgebras.Wedosobyrealizingassociativealgebrasasonedimensionaltopologicalfieldtheories.ThelittleintervalsoperadisanimportantsuboperadoftherealanalyticanalogofPwhichcanberegardedasasuboperadofP.Therefore,aconformalfieldtheoryisaB-algebra.Thisleadsustoacomplexwhich,inthecasewheretheconformalfieldtheoryisaso-calledvertexoperatoralgebra,canbewrittenexplicitly.ThiscomplexcloselyresemblesHochschildcohomologyandlookslikewhatonewouldobtainfromformallydeformingtheoperatorproductinavertexoperatoralgebra.Throughoutthepaper,werestrictourattentiontotreeleveltheories,thosewhichcorrespondtoRiemannsurfacesofgenuszero.Thereshouldbeacyclicversionofthecohomologytheoryconsideredhere,asithappensforthehomologyofalgebrasovercyclicoperads,see[9].Acomputationofthecohomologyofaconformalfieldtheorywouldprovidevalu-ableinformationaboutthestructureofthemodulispaceofconformalfieldtheorieswhichplaysanimportantroleinrelatedphenomenasuchasmirrorsymmetry,cf.KontsevichandManin[18].Weexpecttechniquesfromthetheoryofvertexoperatoralgebrastobeusefulinperformingthiscomputation.1.Algebrasandoperads.AssumeforsimplicitythatallvectorspacesareoverC.PerhapsthesimplestexampleofanoperadistheendomorphismoperadofavectorspaceV,denotedbyEndV={EndV(n)}n≥1,whichisacollectionofspacesEndV(n)=Hom(V⊗n,V),eachwiththenaturalactionofthepermutationgroup,Sn,whichactsbypermutingthefactorsofthetensorproduct,andwhichhavenaturalcompositionsbetweenthem,namely,thecomposition,f◦if′,oftwoelementsfinO(n)andf′inO(n′)isobtainedby(f◦if′)(v1⊗···⊗vn+n′−1)=f(v1,⊗···⊗vi−1⊗f′(vi⊗···vi+n′−1)⊗···⊗vn+n′−1)foralli=1,...,n,andthepermutationgroupsactequivariantlywithrespecttothecompositions.Finally,thereisanelementIinEndV(1),theidentitymapI:V→V,whichisaunitwithrespecttothecompositionmaps.Thisstructurecanbeformalisedinthefollowingway.AnoperadO={O(n)}n≥1withunitisacollectionofobjects(topologicalspaces,vectorspaces,etc.–elementsofanysymmetricmonoidalcategory)suchthateachTHECOHOMOLOGYOFALGEBRASOVERMODULISPACES3O(n)hasanactionofSn,thepermutationgrouponnelements,andacollectionofoperationsforn≥1and1≤i≤n,O(n)×O(n′)→O(n+n′−1)givenby(f,f′)7→f◦if′suchthat(1)iff∈O(n),f′∈O(n′),andf′′∈O(n′′)where1≤ij≤nthen(f◦if′)◦j+n′−1f′′=(f◦jf′′)◦if′(2)iff∈O(n),f′∈O(n′),andf′′∈O(n′′)wheren,n′≥1andi=1,...,nandj=1,...,n′then(f◦if′)◦i+j−1f′′=f