Observables and Symmetries of n-Plectic Manifolds

54 Pages·2016·0.656 MB·English

by  Leonid Ryvkin (auth.)

BestMasters
Springer awards „BestMasters“ to the best master’s theses which have been com-
pleted at renowned universities in Germany, Austria, and Switzerland.
Th e studies received highest marks and were recommended for publication by su-
pervisors. Th ey address current issues from various fi elds of research in natural
sciences, psychology, technology, and economics.
Th e series addresses practitioners as well as scientists and, in particular, off ers guid-
ance for early stage researchers.
Leonid Ryvkin
Observables and
Symmetries of n-Plectic
Manifolds
Leonid Ryvkin
Bochum, Germany
BestMasters
ISBN 978-3-658-12389-5 ISBN 978-3-658-12390-1 (eBook)
DOI 10.1007/978-3-658-12390-1
Library of Congress Control Number: 2015960848
Springer Spektrum
© Springer Fachmedien Wiesbaden 2016
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or
part of the material is concerned, speci(cid:191) cally the rights of translation, reprinting, reuse of illus-
trations, recitation, broadcasting, reproduction on micro(cid:191) lms or in any other physical way, and
transmission or information storage and retrieval, electronic adaptation, computer software, or by
similar or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a speci(cid:191) c statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained
herein or for any errors or omissions that may have been made.
Printed on acid-free paper
Springer Spektrum is a brand of Springer Fachmedien Wiesbaden
Springer Fachmedien Wiesbaden is part of Springer Science+Business Media
(www.springer.com)
Acknowledgements
IwouldliketoexpressmydeepestappreciationtomyadvisorProfessorDr.TilmannWurzbacher,
whohasbeenanidealadvisorforvariousreasons. Notonlyhasheproposedthehighlyinter-
estingtopicofthethesisandgivenmeoptimalpreparationtoattackingitthroughthecourseshe
offeredduringthelastfewyears,buthealsoinvestedatremendousamountoftimeinguiding
andsupportingmeinthecourseofthecompositionofthisthesis. Withouthispatience,mathe-
matical(anddidactic)skillIwouldhaveneverbeenabletomasterthisthesis. Iwouldalsolike
tothankhimforhisenthusiasmandsenseofhumourwhichprovided“alightinthedark”during
themoretechnicalpartsofwork.
IwouldliketoexpressmygratefulnesstotheGerhardC.StarckStiftungforthefinancialand
moralsupportduringmystudies. Thanksforgivingmethepossibilitytofullyconcentrateon
thesubject,thatIamsopassionateabout,andfortheexceptionalpeoplethatIhadtheprivilege
tomeetthroughtheStiftung.
IwouldalsoliketothankmyfamilyandLeonie,whohavebeenagreatemotionalandpractical
supportduringthecourseofthethesis,whohadtoendurealotofvacuum,whereIshouldhave
beeninthelastfewmonths.Theircareandsympathywereofgreatvalueforme.
LastbutnotleastIwouldliketothankmyfriends,whohaveturnedthepast4yearsofstudies
intoanunforgettabletime. Itisthankstothem,thatstudyingwasfunfromthefirstdaytothe
last. Special thanks goes to Stefan Heuver, who has shown great understanding and kindness
duringthelastfewmonths.Thanksforhisgreatadvices,asforinstance“Ifyouworry,juststop
worrying”. Thanksforshowingmethatmyproblemswerenotallthatbad, byexplaininghis
problemsandlastbutnotleast,thanksforforcingmetohavealittlebreakonceinawhile.
LeonidRyvkin
V
Contents
1 Introduction 1
2 L∞-algebras 3
2.1 GeneralizingdifferentialgradedLiealgebras. . . . . . . . . . . . . . . . . . . . 3
2.2 L∞-algebrasascoalgebraswithdifferentials . . . . . . . . . . . . . . . . . . . . 6
2.3 MorphismsofL∞-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 RepresentationsofL∞-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Liem-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Observablesofpre-n-plecticmanifolds 29
3.1 Generalizingsymplecticmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Symmetriesofpre-n-plecticmanifolds 34
4.1 WeaklyHamiltoniangroupactions . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 StronglyHamiltoniangroupactions . . . . . . . . . . . . . . . . . . . . . . . . 36
A Universalcoalgebraproperties 47
References 49
VII
1 Introduction
Theaimofthisthesisistheanalysisandunderstandingofn-plectic(alsocalledmultisymplectic)
manifolds,theirobservablesandtheirsymmetries.
Thedevelopmentofageometrictheoryofn-plecticmanifoldsnaturallystartswiththeclassical
caseofsymplecticmanifolds. InthatsettingoneconsidersamanifoldMequippedwithanon-
degenerateclosedtwo-formω. Thistwo-formrelatesC∞(M),thespaceofsmoothfunctionson
Mwiththevectorfieldson M. ThisrelationinducesaLiealgebrastructureonC∞(M),which,
endowed with this structure, is called the Lie algebra of observables of M. The infinitesimal
versionζ ofasymplecticgroupactionon(M,ω)canoftenbelinearlyliftedtothisLiealgebra.
SuchgroupactionsarecalledweaklyHamiltonianandarethebasisofwhatmightbecalled“mo-
mentumgeometry”,whichisalsoofgreatinterestforphysics. Theexistenceofanequivariant
momentmapisobstructedbyacertaincohomologyclass.
Inthegeneraln-plecticcaseweconsideramanifoldMwithaclosed,non-degeneraten+1-form.
Unlikethesymplectic(1-plectic)case,thenon-degeneracyofωonlyguaranteesinjectivenessof
themapv (cid:3)→ ι ωbutnosurjectiveness. Alsothespaceofobservablescarriesamoregeneral
v
algebraicstructurewhichturnsitintoa“Lien-algebra”,whichisaLiealgebraonlywhenn=1.
Lien-algebrasareaspecialcaseofL∞-algebras,whosestudyconstitutesthefirstchapterofthis
thesis.Lien-algebrasfromn-plecticmanifoldsarethenintroducedinChaptertwo.Thegeneral-
izationofthenotionofa(co-)momentmapisthemainthemeofthethirdchapter,culminatingin
anexplicitcalculationoftheobstructionclassestostronglyHamiltoniann-plecticgroupactions.
Thethesisisstructuredinthefollowingway:
• Sections1.1and1.2developtwopossibleperspectivesonL∞-algebras.Theearlieronede-
scribesanL∞-algebraasaZ-gradedvectorspace,withafamilyofbrackets{lk|1≤k}satis-
fyinganinfiniteseriesofmulti-bracketidentites.ThelatterstandpointviewsanL∞-algebra
asagradedvectorspaceVendowedwithadegreeonecoderivationD:S•(sV)→S•(sV),
whichsquarestozero,whereS•(sV)denotesthecofreegradedcommutativecoalgebraof
thevectorspacesV,wheresVisVwithdegreesshiftedby1.Themainresultofthesesec-
tions,theequivalenceofthesedescriptions(Theorem2.14)isbasedontheproofsketched
in[7],butweworkoutthefulldetailsoftheproofhere.
• Section 1.3 then discusses the right notion of morphisms of L∞-algebras. Though L∞-
morphisms more tangible from the coalgebra standpoint, the multi-bracket formulation
is more practical for calculations. Hence, the main result of this section is the multi-
bracketformulationofL∞-morphisms(Lemma2.18).Itisbasedonthecalculationsmade
inAppendixAof[5]andresultsinaformulaalreadystatedinin[1],butwithoutproof.
We giva a detailed proof and we are also more explicit about the precise range of the
summationsinvolved.
1
L. Ryvkin, Observables and Symmetries of n-Plectic Manifolds, BestMasters,
DOI 10.1007/978-3-658-12390-1_1, © Springer Fachmedien Wiesbaden 2016
• Section1.4givesanintroductiontotherepresentationtheoryofL∞-algebrasaccordingto
[6]. Itpresentstwoperspectivesonrepresentations: viaL∞-morphismsintotheendomor-
phismspaceofadifferentialgradedvectorspaceandviaL∞-modules.Theequivalenceof
bothapproachesisonlyquotedfrom[6]inthisthesis.
• Section1.5thenreducesthe“heavymachinery”developedinthepreceedingsectionsto
thecaseofgoundedLiem-algebras,whichareofspecialinterestastheyincludetheL∞-
algebrasformedbytheobservablesofamultisymplecticmanifold. Thissectionmostly
reproducesresultsfrom[5],wheretheproperty“tobegrounded”isreferredtoas“having
Property(P)”.
• Section2.1introducesthenotionofn-plectic(ormultisymplectic)manifoldsandderives
their basic properties in accordance with [11]. The discussion of the pre-n-plectic case
mostlyrelieson[5].TheLien-algebraofobservablesisdescribedforbothcases.
• Sections3.1and3.2developthetheoryofHamiltoniangroupactionsonn-plecticmani-
folds,guidedbytheclassicalcasedescribede.g. in[9]. The(homotopy)co-momentmap
whichisthebasicobjectofourdiscussionswasfirstintroducedin[5]. Themainresults
ofthesesectionsareLemma4.7,whichdescribestheobstructionforann-plecticactionto
beweaklyHamiltonianandTheorem4.13,thewhichgivesacohomologicaldescription
oftheobstructionforaweaklyHamiltonianactiontobestronglyHamiltonian. Thelatter
isarefinementofTheorem9.7in[5]. Furthermoreweshowthatthefirstresp. thelast
componentofastronghomotopyco-momentmapyieldsacovariantmultimomentmapin
thesenseof[3]resp.amulti-momentmapinthesenseof[8].
2
2 L -algebras
∞
LetKdenotehereafixedgroundfieldofcharacteristic0. Allvectorspaces, linearmapsand
tensor products will be defined with respect to/taken over this field, unless we explicitly state
otherwise.AgoodoverviewofthesubjectofL∞-algebrasisprovidedinthen-lab([12]).
2.1 GeneralizingdifferentialgradedLiealgebras
In this sectionwe will define L∞-algebras as thegeneralization of Lie algebras. For doingso
letusfirstreviewthenotionofaLiealgebra: ALiealgebraisavectorspace Lwithaskew-
symmetricbilinearmap[·,·]:L×L→LsatisfyingtheJacobiidentity:
[[x ,x ],x ]−[[x ,x ],x ]+[[x ,x ],x ]=0 (1)
1 2 3 1 3 2 2 3 1
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3)
Thiscanberewritteninthefollowingway,whereP={ 123 , 123 , 123 }⊂S :
(cid:4) 123 132 231 3
sgn(σ)[[xσ(1),xσ(2)],xσ(3)]=0.
σ∈P
ThesetPconsistspreciselyofthosepermutations,whichmoveoneelementtothelastposition,
withoutdistortingtheinnerorderoftheothers.Aswewilldealwithidentitiesofmulti-brackets
soon, we will have to generalize this notion of “moving one element out of three to the end”
to “moving q elements out of p+q to the end”. The permutations doing that are exactly the
(p,q)-unshuffles.
Definition2.1. Apermutationσ ∈ Sp+q isa (p,q)-unshuffleifandonlyifσ(i) < σ(i+1)for
i(cid:2) p.Wedenotethesetofof(p,q)-unshufflesbyush(p,q)⊂Sp+q.
Theconditionintheabovedefinitionguaranteesthatthefirstpandthelastqelementsstayinthe
sameinternalorder.Thesepermutationsarecalledunshuffles,becausetheirinversescorrespond
toshufflingadeckofpcardsintoadeckofqcards.
Let usn(cid:5)ow turn tothe graded context. Firstwe introduce agrading on ourvector space: We
setL= L,whereL istthevectorsubspaceofelementsofdegreei.Wewillwrite|x|=iif
i∈Z i i
x∈L.ALiestructure[·,·]onsuchavectorspaceLshouldsatisfythefollowingthreeconditions:
i
• [Li,Lj]⊂Li+j(thebracketrespectsthegrading)
• [x1,x2] = −(−1)|x1||x2|[x2,x1]forallhomogenous x1,x2 ∈ L(thebracketisgradedskew-
symmetric)
• (−1)|x1||x3|[x1,[x2,x3]]+(−1)|x1||x2|[x2,[x3,x1]]+(−1)|x2||x3|[x3,[x1,x2]] = 0forallhomoge-
nouselementsx ,x ,x ∈L(thegradedJacobiidentityholds)
1 2 3
IfwetrytobringthegradedJacobiidentityintoformofequation(1),weget:
[[x1,x2],x3]−(−1)|x2||x3|[[x1,x3],x2]+(−1)|x1||x2|+|x1||x3|[[x2,x3],x1]=0.
So[[xσ(1),xσ(2)],xσ(3)]getsanadditionalsignforeverytranspositionoftwooddelements. This
leadsustothedefinitionoftheKoszulsign(cid:6)ofapermutationσactingonelementsx ,...,x ∈L.
1 n
3
L. Ryvkin, Observables and Symmetries of n-Plectic Manifolds, BestMasters,
DOI 10.1007/978-3-658-12390-1_2, © Springer Fachmedien Wiesbaden 2016