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Preprint submitted on 29 Aug 2006
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The invariant of Turaev-Viro from Group category
Jerome Petit
To cite this version:
Jerome Petit. The invariant of Turaev-Viro from Group category. 2006. �hal-00089825v2�
ccsd-00089825, version 2 - 29 Aug 2006
JÉRÔMEPETIT
Abstrat. AGroupategory isa spherialategory whose simpleobjets
areinvertible. TheinvariantofTuraev-Viro withthispartiular ategory is
infattheinvariantofDijkgraaf-Wittenwhosethegroupandthe3-oyleis
givenbythesimpleobjetsandtheassoiativityonstraintoftheategory.
Contents
Introdution 2
1. TheinvariantofDijkgraaf-Witten 2
2. Groupategory 3
2.1. Denition 4
2.2. SomeresultsonGroupategory 5
2.3. 6j-symbol 6
2.4. 6j-symbolfrom Groupategory 7
3. TheinvariantofTuraev-Viro 9
4. Theequality 10
5. Topologialinterpretationofadmissible oloring 12
5.1. ThefundamentalgroupoïdofT 12
5.2. Thegaugeation 13
6. ConstrutionofTQFT 14
6.1. TriangulateTQFT 14
6.2. ConstrutionofTuraev-Viro 14
7. Examples 16
7.1. α= 1 16
7.2. G=Zn 17
7.3. G=S3 20
7.4. ExamplesofTQFTs 20
AppendixA. Someomputation ofH3(g, k⋆) 22
A.1. Example 22
Aknowledgements 23
Aknowledgements 23
Referenes 23
2000MathematisSubjetClassiation. 57N10,18D10,20J06.
Keywordsandphrases. Quantuminvariant,invariantofTuraev-Viro,monoidalategory.
Introdution
In 1992 M. Wakui [15℄ reformulated the invariant of Dijkgraaf-Witten [4℄ and
he proved the topologial invariane in a rigorous way. The invariane is based
upon the triangulationand the Pahner moves. One given a nite groupand a
3-oyle the Dijkgraaf-Witten invariant is dened ombinatorially. Moreover in
thispaperhebuiltatopologialquantumeldtheory(TQFT)fromthisinvariant.
ThesameyearV.TuraevandO.Viro[12℄ builtaninvariantof3manifoldthanks
to 6-j symbol to prove the topologial invariane they showed a relative version
of atheoremof Alexander[1℄on equivaleneof triangulation. This invariantwas
reformulatedinaategoriallanguages[11℄andtheTQFTwasbuilt. Inthesame
spiritof[11℄J.W.BarretandB.W.Westburry[2℄havebuilta3-manifoldinvariant
usingspherialategories. Inthisonstrutionthetopologialinvarianeputsbak
downthetriangulation andthePahner moves. IndependentlyI.Gelfand andD.
Kazhdan [6℄ have built a 3-manifold using spherial ategories and in 1993 D.N.
YetterhasstudiedanuntwistedversionoftheinvariantofDijkgraaf-Wittenin[16℄
the Turaev-Viro style in . In fat these onstrutions are reformulations of the
Turaev-Viroinvariant. Intherest ofthepaperwewill allsuh kindofinvariant
theinvariantofTuraev-Viroanditwillbedenoted : T VC where C istheategory
usedtobuild theinvariant.
Themaingoalofthispaperistogivearelationbetweenthistwoapproahesbased
on triangulation. That's why we utilize a "speial" spherial ategory. Roughly
speaking,itis aspherialategorysuhthat everysimpleobjetis invertibleand
hasadimension equalto one. Thedimension isgivenbythe spherialstruture.
In[10℄,F.Quinnalledthisategory: "Groupategory". In[7℄invertibleobjets
arealledsimplesurrentsandthetensorategorywhoseeverysimpleobjetsare
invertibleisdenoted Pointedategory. Theauthorshavedenoted Piardategory
ofCthefulltensorategoryofCwhoseobjetsarediretsumofinvertibleobjets
ofC. Thusifthere isanite numberof simpleobjetand ifeveryobjetis nite
diretsumofsimpleobjetthenapointedategoryisequaltoitsPiardategory.
In this paper we will use the terminology of F. Quinn [10℄. L. Crane and D.N.
Yetterhavestudied groupoyleto desribemonoidalategorywithdualsin [3℄.
Hereisanoutlineofthepaper. InSetion1wereallthedenitionoftheDijkgraaf-
Invariant[15℄. In Setion2we givethedenitionand wereall somefats onthe
Group ategory. In Setion 3 we givethe denition of the Turaev-Viroinvariant
of 3-manifold. InSetion 4,we omputethe Turaev-Viroinvariantin the aseof
Group-ategory with other onditionsand we show the main theorem (4.2) . In
setion5wegiveatopologialinterpretationoftheadmissibleolorings. InSetion
6wegivetheonstrutiontheTQFTwhiharisesfrom thisinvariant. Weendthe
paperbydisussingafewexamples.
1. The invariantof Dijkgraaf-Witten
Throughoutthis paperk will beaommutativeeld suh thatcar(k) = 0and k=k.
Weuse thedesriptionof[15℄. LetGbeanite group,this groupwillbealways
amultipliativegroup. MoreoverkisarepresentationofGwiththetrivialation.
Thenwean deneZ3(G, k⋆)thesetof 3-oyleofGwithoeientsin k⋆ and
wexα∈Z3(G, k⋆). LetT bean-simplexwithn≥1,aolorofT isthefollowing
data:
(1.1) γ: {orientededgesofT} →G,
whihsatisestheonditions:
(i) foranyorientededgee: γ(e) =γ(e)−1, whereeis theorientededgewith
theoppositeorientation.
(ii) For anyoriented2-simplex(012)ofT wehave: γ(01)γ(12)γ(20) = 1.
WedenoteCol(T)thesetofallolorsofT,ifT isatriangulationofan-manifold
M,withn≥2, wedenoteCol(M, T)theset ofallolorsofM givenbyT. When
thereisnoambiguityonthehoieofatriangulation,wedenoteCol(M)thesetof
olorsofM.IfM isamanifoldwithboundary: ∂M,then∂M is endowedwith a
triangulationwhihomesfromthetriangulationofM. Ifτ isaolorof∂M then
thesetofallolorsofM whihextendτ,isdenotedCol(M, τ). Wegiveanorderto
theset ofvertiesofatriangulationofM, theneah3-simplexhasanorientation givenby theasending order. Thenfor γ∈Col(M)and forthe3-simplex (0123)
weput:
α(∆, γ) =α(γ(01), γ(12), γ(20)),
withα∈Z3(G, k⋆).
Theorem 1.1 (Wakui (92)[15℄). Let G be anite group, we x a3-oyle α∈ Z3(G, k⋆). Let M be a ompat oriented triangulated 3-manifold, T is a trian-
gulation of M. We denote the number of verties of T by n0 and T3 the set of
3-simplex in T. All the 3-simplex are oriented by a numbering of the verties.
Given τ∈Col(∂M),wedene the Dijkgraaf-Witten invariantby :
ZM(τ) =|G|−n0 X
γ∈Col(M,τ)
Y
∆∈T3
α(∆, γ)ǫ∆,
where
ǫ∆=
1 if∆andM havethe sameorientation,
−1 otherwise.
Then ZM(τ)doesnotdependonthe hoieoftriangulationofM andthehoieof
orderof vertiesinM wheneverwex atriangulationof∂M andτ.
Thankstotheindependeneofthehoieofnumbering,weanonsideranum-
beringofthetriangulationsuhthatthe3-simplexhavethesameorientationofM.
Thentheinvariantis:
ZM(τ) =|G|−n0 X
γ∈Col(M,τ)
Y
∆∈T3
α(∆, γ),
whereall3-simplex∆havethesameorientationofM.
Remark 1.2. If we onsider M without boundary, then ZM(∅) is a 3-manifold
invariant andwedenote it: ZM.
2. Group ategory
2.1. Denition. LetCbeamonoidalategory,byasalarobjet[14℄ofCweshall
meananobjetofC suhthat: End(X) =k. IfC isabelianandkisalgebraially losed then an objet is salar i it is an simple objet. We denote the set of
isomorphismlassesofsalarobjetsofCbyΛC.
Denition 2.1. A nitely semisimple monoidal ategory is a monoidal ategory
(C,⊗, I, a, l, r)suhthat :
(a) C isanabelian k-ategory and⊗isabifuntork-linear,
(b) every objetofC isanite diretsum ofsalarobjets ofC,
() ♯ΛC <∞andI isasalarobjet,
(d) C issovereign1
IfCisanitelysemisimplemonoidalategory,theneveryobjetX ofC admits
arightduality: (X, X∨, eX, hX)andaleftduality: (X∨, X, ǫX, ηX),weantake
thesameobjetbeauseCis sovereign. Bydenition ofduality: eX : X⊗X∨→I
ǫX : X∨⊗X →I ηX : I→X⊗X∨ hX : I→X∨⊗X.
andwehavethefollowingequalities:
(eX⊗idX)(idX⊗hX) =idX
(idX⊗ǫX)(ηX⊗idX) =idX
(idX∨⊗eX)(hX⊗idX∨) =idX∨ (ǫX⊗idX∨)(idX∨⊗ηX) =idX∨.
Theleftquantum traeofanendomorphismf ∈EndC(X)isdened by: trl(f) =eX(f⊗idX∨)ηX,
therightquantumtraeofanendomorphismf ∈EndC(X)isdened by: trr(f) =ǫX(idX∨⊗f)hX.
foranyendomorphismsf, ginC wehave:
trl(f ⊗g) =trl(f)trl(g), trr(f ⊗g) =trr(f)trr(g),
trr(f) =trl(f∨),
themultipliationisgivenbythemultipliationofk=End(I).
Denition 2.2. A spherial ategory is a nitely semisimple monoidal ategory
suhthat,for allendomorphism f inC wehave : trl(f) =trr(f).
1Cadmitsarightandaleftdualitywhihareisomorphiasmonoidalfuntor.
Inaspherialategorywedenotethelefttraebytrandsowehavetr=trl=trr.
Thequantum dimensionof anobjetX inaspherialategoryC isdened by: dim(X) =tr(idX),
sowehavedim(X) =dim(X∨).
Denition2.3.
(i) AnobjetX ofamonoidal ategoryC isalledinvertibleithereexistsan
objetY suhthatX⊗Y ∼=I,whereI isthe tensorunit ofC.
(ii) Amonoidal ategory isalledpointedievery salar objetisinvertible.
(iii) TheGroup ategoryPi(C) ofthemonoidal ategoryC isthefullmonoidal
subategoryof C whose objetsarediretsumsof invertibleobjets of C.
(iv) AGroupategoryisapointedspherial ategory.
(v) Aθ-ategoryisabraided,pointednitely semisimplemonoidal ategory.
2.1.1. Example ofGroupategory. Gisanitegroup,wedenotek[G]theategory
whose objets are G-graded nite dimensional k-vetor spaes2 and whose mor-
phismsarek-linearmorphismthat preservesthegrading. IfV andW areobjets
ofk[G]themonoidalstrutureofk[G]isgivenby: (V ⊗W)g = X
h, k hk=g
Vh⊗Wk.
theassoiativityistheidentityandtheisomorphismlassesofsalarobjetsarein
bijetionwithG: g↔δg whereδg isdenedin thefollowingway:
(δg)h=
k ifg=h, 0 otherwise.
and every salar objet is invertible, thus k[G] is a Group ategory. k[G] is a θ-ategoryiGisanabeliangroup.
2.2. Some results on Group ategory. Whenever C is a Group ategory, it
followsfrom thedenition ofa Groupategoryand thequantum dimension that
for allX ∈ ΛC : dim(X)2 = 1. The Grothendiek ring ofC is isomorphito the
groupalgebraofthenite groupΛC,itisdenoted K0(C)∼=Z[ΛC].
Proposition2.4. If C isaGroup ategory then:
(i) allinvertible objetsare inΛC,
(ii) (ΛC,⊗, I)isanitegroup.
Proof(i): IfX isinvertibleinC thenthereexists anobjetY inC suhthat : X⊗Y ∼=I,thuswehave:
X
Z∈ΛC
µZ(X)Z⊗Y =I,
whereµZ(X) =dimk(HomC(X, Z))andsowehave: X
Z′,Z∈ΛC
µZ(X)µZ′(Z⊗Y)Z′=I,
2 G A A
sineIisasalarobjet X
Z∈ΛC
µZ(X)µZ′(Z⊗Y) =
1 ifZ′=I, 0 otherwise.
MoreoverHomC(X, Y) is nitefor allobjetsin C and soµZ(X)∈N֒→ k, thus
there is only one Z0 ∈ ΛC suh that µZ0(X) 6= 0 and moreover µZ0(X) = 1 so X =Z0∈ΛC. Weannotiethat ifY istheinverseofX thenX⊗Y ∼=Iand so
Y ∼=X∨.
Proof(ii) : If X issalarthen by denition ofaGroup ategoryX isinvert-
ible and so there is Y an objet of C suh that X ⊗Y ∼= I we have seen that Y =X∨∈ΛC. Inanitely salarmonoidalategorywehave: X∨⊗X =I⊕Z
whereZ isanobjetofC,thuswehave: X∨∼=X∨⊗I
∼=X∨⊗(X⊗X∨)
∼= (X∨⊗X)⊗X∨
∼=X∨⊕Z⊗X∨
IfX is salarthen X∨ issalarthusZ⊗X∨= 0and X∨6= 0. Soitfollowsthat Z = 0 and X∨⊗X ∼=I, then X∨ isthe left andrightinverse of X. IfX and Y
aresalarobjetsthenX⊗Y isanobjetofC and:
EndC(X⊗Y)∼=HomC(X, X⊗Y ⊗Y∨)∼=EndC(X)∼=k,
thenX⊗Y isasalarobjetthus(ΛC,⊗, I)isanitegroup.
Theorem 2.5 ([5℄, setion 7.5). Suppose G isa nite group, then : Group ate-
gories withunderlying group GorrespondtoH3(G, k⋆).
InfatH3(G, k⋆)lassiesalltheassoiativityonstraint(uptomonoidalequiv- alenes). ThegroupGgivesthesetof isomorphilassesof salarobjetsandan
elementα∈H3(G, k⋆)givestheassoiativityonstraintoftheGroupategory. If wetake α, α′ ∈ Z3(G, k⋆) suh that [α] = [α′] ∈ H3(G, k⋆) then we obtain two
GroupategoriesdenotedbyC(G, α)andC(G, α′)suhthat: C(G, α)∼=⊗C(G, α′)
(monoidalequivalene).
2.3. 6j-symbol. We x D a nitely monoidal ategory then for all objet X in D we have: X =X1⊕...⊕Xn with Xi∈ΛD then for all 1 ≥ j ≥ n there are
morphismsij ∈HomD(Xj, X)and pj ∈HomD(X, Xj)suhthat pjij =idXj and P
jijpj=idX.
Lemma2.6. Wex a, b, c, d, e, f ∈ΛD thenthe following appliation
Ψ: Hom(a, e⊗d)⊗kHom(e, b⊗c→Hom(a,(b⊗c)⊗d) v⊗w7→(w⊗idd)v
induesanisomorphism between HomD(a,(b⊗c)⊗d)and L
e∈ΛHomD(a, e⊗d)⊗kHomD(e, b⊗c). In the sameveinwehave: HomD(a, b⊗(c⊗d))∼=L
f∈ΛHomD(a, b⊗f)⊗kHomD(f, c⊗d)
Proof: Bydenition ofDwehave: b⊗c=⊕e∈λDµe(b⊗c)ewith
µe(b⊗c) =dimk(Hom(e, b⊗c)). Thenforallf ∈HomD(a,(b⊗c)⊗d)wehave: f =idb⊗c⊗iddf
=X
e∈Λ
(iepe⊗idd)f
=X
e∈Λ
(ie⊗idd)(pe⊗idd)f,
andsoΨissurjetive. Moreoverthevetorspaesarenitedimensionalandthey havethesamedimensionthuswegettheisomorphism. Theseondisomorphismis
obtainedin thesameway.
a,theassoiativityonstraintofD,induesanaturalisomorphism:
(X⊗Y)⊗Z∼=X⊗(Y ⊗Z),forallX, Y, Z ∈ob(D). Thenwehavethefollowing
ommutativesquare:
L
e∈ΛHomD(a, e⊗d)⊗kHomD(e, b⊗c) //
∼=
L
f∈ΛHomD(a, b⊗f)⊗kHomD(f, c⊗d)
∼=
HomD(a,(b⊗c)⊗d) ∼
= //HomD(a, b⊗(c⊗d))
thepreviousommutativesquareindues twolinearappliations :
a b c d e f
: HomD(a, e⊗d)⊗kHomD(e, b⊗c)→HomD(a, b⊗f)⊗kHomD(f, c⊗d) a b c
d e f
inv
: HomD(a, b⊗f)⊗kHomD(f, c⊗d)→HomD(a, e⊗d)⊗kHomD(e, b⊗c),
thesearethe6j-symbolofD.
Wedeneabilinearform inthefollowingway: forallobjetsX, Y, ωX,Y : HomD(X, Y)⊗HomD(Y, X)→k
f⊗g7→trg(f g).
By denition D doesn't admit negligible morphism so ω_,_ is a non-degenerate bilinearformanditdenesanadjointof
a b c d e f
,forall(a, b, c, de, f)∈ΛD,
thisadjointisdenotedby:
λ(a, b, c, d, e, f)∈(HomD(e⊗d, a)⊗HomD(b⊗c, e)⊗HomD(a, b⊗f)⊗HomD(f, c⊗d))⋆.
2.4. 6j-symbolfromGroupategory. IfCisaGroupategorythenforallX, Y
salarobjetsX⊗Y isasalarobjet. ThusifX, Y, Z aresalarobjetsthen:
(2.1) Hom(Z, X⊗Y)∼=
k ,ifX⊗Y ∼=Z 0 , otherwise
IntheaseofGroupategorytheisomorphisms(lemma2.6)beome:
Lemma2.7. Forallsalarobjets (a, b, c, d, e, f)we have:
HomC(a, e⊗d)⊗kHomC(e, b⊗c)∼=HomC(a,(b⊗c)⊗d)
(2.2)
HomC(a, b⊗f)⊗kHomC(f, c⊗d)∼=HomC(a, b⊗(c⊗d))
(2.3)
(i)
a b c d e f
6= 0 ie∼=b⊗c,a∼= (b⊗c)⊗dandf ∼=c⊗d
(ii)
a b c d e f
inv
6= 0i e∼=b⊗c,a∼=b⊗(c⊗d)andf ∼=c⊗d
Proof : The assertions (i), (ii) and the isomorphisms (2.2), (2.3) ome from
(2.1). ThusintheaseoftheGroupategorythe6j-symbol
a b c d e f
only
depends on b, c, d. For all salarobjets b, c, d we put α(b, c, d) =
a b c d e f
.
WeandeneαinΛC. ΛCisagroupandforg∈ΛC wedenoteXgarepresentation ofthis isomorphismlass,and soforallg, h∈ΛC wehave: Xg⊗Xh ∼=Xgh. By
(2.1)weknowthat HomC(Xgh, Xg⊗Xh) is aonedimensional vetorspae. For allg, h∈ΛC weputφ(g, h)abasisofHomC(Xgh, Xg⊗Xh).
Weputg, h, k∈ΛC andwedenoteXg, Xh, Xk theirrepresentations. Byonstru- tionα(Xg, Xh, Xk)isanisomorphismofonedimensionalvetorspaesthusinthe basisφwehave:
(2.4) α(Xg, Xh, Xk)(φ(gh, k)⊗φ(g, h)) =α(g, h, k)(φ(g, hk)⊗φ(h, k)),
with α: ΛC×ΛC ×ΛC → k⋆. With thesame notationsthe ommutativesquare
whihdenesαindues thefollowingequality:
α(g, h, k)(idXg⊗φ(h, k))φ(g, hk) =a(Xg, Xh, Xh)(φ(g, h)⊗idXk)(φ(gh, k)).
Thusαdeterminethefollowingisomorphism:
HomC(Xghk,(Xg⊗Xh)⊗Xk)∼=HomC(Xghk, Xg⊗(Xh⊗Xk)) v7→a(Xg, Xh, Xh)v,
aistheassoiativityonstraintofC,thusasatisestheMalane'spentagon: with Xg, Xh, Xk, Xlsalarobjets
((Xg⊗Xh)⊗Xi)⊗Xj
(Xg⊗(Xh⊗Xi))⊗Xj
Xg⊗((Xh⊗Xi)⊗Xj)
(Xg⊗Xh)⊗(Xi⊗Xj) Xg⊗(Xh⊗(Xi⊗Xj))
a(g,h,i)⊗id
<<
yy yy yy yy yy yy yy
a(g,hi,j)
""
EE EE EE EE EE EE EE
id⊗a(h,i,j)
a(gh,i,j)
a(g,h,ij)//
Ifweapplythelastequalityinthebasisφwehave:
α(g, h, kl)α(gh, k, l) =α(h, k, l)α(g, hk, l)α(g, h, k).
Proposition2.8. If C isa Group ategory then the 6j-symbol isdetermined bya
3-oyleonZ3(ΛC, k⋆)andabasis of Hom(Xgh, Xg⊗Xh).
Therelationof awiththeidentityonstraint(r, l)induesthat :
l(h)α(g,1, h) =r(g),forallg, h∈ΛC thusr(g) =α(g,1,1) and(g) =α(1,1, h)−1.
We anhange α suh that αis normalized without hanged theohomologous lassofα. Intermofbasisφ, itisahangeofbasis.
3. The invariantof Turaev-Viro
Weadopt theapproahof [6℄ratherthen[12℄,but weuseaspherialategory.
Beauseifweonsiderasovereignategorythereisaproblemin theonstrution.
Theproblemoursatthelevelofindependeneofthenumberingofthe3-simplex.
LetT bean-simplexwithn≥1,thenaTuraev-ViroolorofT(Turaev-Viropoint ofview)isthefollowingdata: γ : {orientededgesofT} →ΛC whihsatisesthe
onditions:
(i) for anyorientededge e: γ(e) = γ(e)∨, where e isthe orientededge with
theoppositeorientation.
Theset of all Turaev-Viro olorof T is denoted ColT V(T). letT bea n-simplex
and wex anumberingof the verties of F, everyfaes of T hasan orientation givenbytheasendingorder: (012). Foreveryfaes(012)wedenethefollowing
vetorspae: VC((012), γ) =HomC(I, γ(01)⊗γ(12)⊗γ(20)).
Lemma3.1.
VC((012), γ)∼=VC((201), γ)∼=VC((120), γ)
(3.1)
VC((012), γ)∼=VC((021), γ)⋆
(3.2)
Proof(3.1): Itomesfrom thesovereignstrutureofC.
ForallX, Y, Z∈ob(C)wehave:
HomC(I, X⊗Y ⊗Z)↔HomC(I, Y ⊗Z⊗X)
f 7→(ǫX⊗idY⊗Z⊗X)(idX∨⊗f⊗idX)(hx)
Proof(3.2): ItomesfromthefatthattheategoryCdoesn'tadmitnegligible
morphismandsothefollowingbilinearformisnon-degenerate:
contr: VC((012), γ)⊗VC((021), γ)→k
f⊗g7→f∨g=tr(f∨g) =tr(g∨f)
ThusthevetorspaeVC((012), γ)isindependentofthestartingpointandifwe hangetheorientationofthe2-simplexweobtainthedualvetorspae. Moreover
thisdualvetorspaeanbeobtainbyahangeofolorinfat:
VC((012), γ)∼=VC((021), γ)⋆∼=VC((021), γ′),
withγ′(02) =γ(01),γ′(21) =γ(12),γ′(10) =γ(20). LetT bethetriangulationof aompatorientedsurfaeΣandT2 theset of2-simplexofT,thenwedene
VC(Σ, T) = M
γ∈Col(T)
O
f∈T2
VC(f, γ),
andthisspaeisindependentofthehoieofanumberingofT.
Let∆ bea3-simplex, anumberingofthevertiesof∆ givesanorientationof∆,
withthisorientation∆ isdenoted(0123). Wetakeγ∈Col(∆)andweput : VC((132), γ)⊗VC((023), γ)⊗VC((031), γ)⊗VC((012), γ)L((0123),γ)
−→ k
(3.3)
v0⊗v1⊗v2⊗v37→dim(γ(13))−1λ(γ(03), γ(01), γ(12), γ(23), γ(02), γ(13)).
3