A quantum ergodic theorem for mapping class groups action on character variety

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by  Julien Korinman

A quantum ergodic theorem for mapping class groups action
on character variety
Julien Korinman
UniversidadeFederal de São Carlos
Rodovia Washington Luís, Km 235, s/n
São Carlos - SP, 13565-905
email: [email protected]
6
1
0 Abstract
2
Westateatheoremrelatingtheergodicityoftheactionofagivensubgroupofthemapping
n
class group of a surface on the character variety, to the asymptotic of its invariant subspaces
a
throughtheReshetikhin-Turaevrepresentations. Asapplication wegiveanasymptoticresult
J
on thespin decomposition arising in [BHMV95]
2
2 Keywords: Reshetikhin-Turaev representations, mapping class group, character variety,
quantumchaos.

]
h
p 1 Introduction
-
h
t
a Wittengavein[Wit89]convincingargumentsfortheexistenceofTopologicalFieldTheories,as
m
defined in [Ati88, Wit88], giving a three dimensional interpretation of the Jones polynomial when
[ thegaugegroupisSU(2). Soonafter,ReshetikhinandTuraevgavearigorousconstructionofthese

TQFTs [RT91] using a modular categorybasedon the representationsof certainquantum groups.
1
v We canextractfrom eachof these TQFT a projective unitary representationof the mapping class
6 group Mod(Σ ) of a genus g closed oriented surface Σ .
g g
0
1 Even though our results work in a more general context, we will focus on the representations
6 associated to the gauge group SU(2), that we will denote by:
0
. ρ :Mod(Σ ) PU(V (Σ )) , for p 3
1 p g → p g ≥
0
where V (Σ ) is a finite dimensional complex vector space and p 3 an integer, called level,
6 p g
≥
1 indexing the representations.
:
v These representations are closely related to the action of Mod(Σ ) on the character variety:
g
i
X
r g :=Hom(π1(Σg),SU(2)) SU(2)
a X
.
equipped with its natural symplectic form ω.
The spaces End(V (Σ )) can be thought as quantizations of the commutative algebra of
p g OXg
regularfunctions on . We now describe the relationbetween these algebrasthatwe willneed in
g
X
this paper.
Denote by the set of isotopy classes of multicurves in Σ , that is of one dimensional
g g
M
submanifolds of Σ (including the empty one), with no contractible component.
g
Given γ , we can define a regular function f by the formula:
∈Mg γ ∈OXg
f ([ρ])= Tr(ρ(γ)) , for [ρ]
γ g
− ∈X
1
It is shown in [Bul97, CM09] that the f form a vectorial basis of . However, the algebra
γ OXg
is equipped with a natural tracial state τ given by:
OXg
1
τ(f):= fdV
Vol( )
Xg ZXg
wheredV isthemeasureassociatedtothesymplecticformωandVol( )isthesymplecticvolume.
g
X
Ontheotherhand,wecanalsoassociatetoγ a(curve)operatorAdd (γ) End(V (Σ )),
g p p g
∈M ∈
which we be defined in the next section, such that the set of curve operatorsis a generating set of
End(V (Σ )). This algebra has a natural tracial state B defined by:
p g p
1
B (M):= Tr(M)
p
dim(V (Σ ))
p g
In [MN08], the authors proved the following relation between these tracial algebras which is
the key property of this paper:
Marché - Narrimanejad [MN08] For all γ , we have:
g
∈M
B (Add (γ)) τ(f ) (1)
p p γ
−p−→−∞→
We now state the main theorem of this paper:
Theorem 1.1. .
Let G be a subgroup of Mod(Σ ) acting ergodically on .
g g
X
For each p 3, let
≥
V (Σ )=W ... W
p g 1,p⊕ ⊕ Np,p
be a decomposition of V (Σ ) into G-invariant subspaces.
p g
Denote by P the orthogonal projection on W .
i,p i,p
Then there exist sets J 1,...,N such that:
p p
⊂{ }
1. We have:
1
dim(W ) 1
i,p
dimVp(Σg) −p−→−∞→
iX∈Jp
2. For any sequence j =(j ) with j J , we have:
p p≥3 p p
∈
B P Add (γ) τ(f )
p jp,p p −p−→−∞→ γ
(cid:0) (cid:1)
In section 2 we will state a more general version of this theorem that we wish to apply to the
geometric quantization of general character variety Hom(π1(Σg),H)/H, where H is a compact,
simply laced Lie group, or the group U(1).
When the gauge group H is U(1), the associated representations are the so-called Weil repre-
sentations and the phase space is the symplectic torus Tg. When the genus is one and G = Z
whose generator acts on the torus by an Anosov elements, then we recover the Schnirelman-type
theorem of Bouzouina and De Bièvre of [BDB96]. Our theorem is thus a generalization of this
quantum ergodic theorem when both G and H are non-abelian.
To apply this theorem, we need groups acting ergodically on for the symplectic measure.
g
X
In[Gol97], Goldman showedthat the actionofMod(Σ ) is ergodic. In [FM13], Funar andMarché
g
showed that the action of the first Johnson subgroup of Mod(Σ ) is also ergodic. In [BHMV95],
g
nontrivialinvariantsubspacesforbothMod(Σ )andtheTorelligroupwherefoundwhen4divides
g
p.
2
Denote by the group:
H
H (Σ ,Z/2Z) , if p 4 (mod 8)
H:= H11(Σgg,Z/2Z)⋊Z/2Z , if p≡0 (mod 8)
(cid:26) ≡
wherethe semi-directproductisthe onlynontrivialone. In[BHMV95],theauthorsdefinedanon
trivial decomposition:
V (Σ )= V (Σ ,χ)
p g p g
χ∈HomM(H,Z/2Z)
where each V (Σ ,χ) is invariant under the Torelli group and V (Σ ,χ=0) is invariant under
p g p g
Mod(Σ ). We can deduce from Theorem 1.1 the following:
g
Corollary 1.2. Let (p ) be an increasing sequence of non-negative integers, all of which being
n n
congruent to either 4 or 0 modulo 8, and let χ Hom( ,Z/2Z). Denote by P the orthogonal
χ
∈ H
projector on V (Σ ,χ). Then we have:
p g
B (P Add (γ)) τ(f ) , for all γ
pn χ pn −n−→−−∞→ γ ∈Mg
The paper is organized as follows. In section 2 we briefly define the Reshetikhin-Turaev rep-
resentations using skein theory following [Lic91, BHMV95]. Nothing new is claimed here. Section
3 is devoted to the statement and the proof of the main theorem. In the last section we prove
Corollary 1.2.
Acknowledgements: The author is thankful to L.Charles, L.Funar, J.Marché and F.Paulin
for useful discussions. He also thanks C.Oliveira, A.E.Presotto, F.Ruffino, D.Vendruscolo and the
mathematic departement of UFSCar for their kind hospitality during the redaction of this paper.
He acknowledgesupport from the grantANR 2011 BS 0102001ModGroup, the GDR Tresses, the
GDR Platon and the GEAR Network.
2 Skein construction of the Reshetikhin-Turaev representa-
tions
Following [BHMV95], we will briefly define the representations ρ and fix some notations.
p
2.1 The spaces V (Σ )
p g
Given an even integer p 6, we note A:= exp(iπ/p) C, a primitive 2p th root of unity.
≥ − ∈ −
UsingtheKauffmanskeinrelationofFigure1,weassociatetoanyframedlinkL S3 aninvariant
L C. ⊂
h ip ∈
A A
-1
A A
( 2 -2)
Figure 1: Skein relations defining the framed link invariants.
Choose g 1 and denote by C the set of isotopy classes of framed links (including the empty
g
≥
link)inanorientedgenusg handlebodyH . We fix agenusg Heegaardsplitting ofthe sphere,i.e.
g
3
an element S Mod(Σ ) and two handlebodies so that :
g
∈
H1 H2 (cid:27)S3
g g
S:∂H1→∂H2
[g g
Take L ,L C and embed L in H1 and L in H2 . The above gluing defines a link
1 2 ∈ g 1 g 2 g
L L S3. We call Hopf pairing the bilinear form:
1 S 2 ⊂
S (, )H :C[C ] C[C ] C
· · g,p g × g →
defined by
(L ,L )H := L L
1 2 p,g 1 2
* +
[S p
Eventually we define the spaces V (Σ ) as the quotients:
p g
Vp(Σg):=C[Cg] ker (, )H
· · g,p
(cid:30)
(cid:16) (cid:17)
The vector spaces V (Σ ) are finite dimensional ([BHMV95]).
p g
2.2 The invariant form ,
h· ·ip
We now define a definite positive form , : V (Σ )⊗2 C which will be invariant for the
h· ·ip p g →
action of Mod(Σ ). Using the identification H (cid:27)D2 [0,1], where D2 denotes the g-holed disc,
g g g × g
we define a ’stack product’ H H (cid:27)H by using the homeomorphism[0,1] [0,1](cid:27)[0,1] given
g g g
× ×
by (x,y) 1(x+y).
→ 2
NowwhenidentifyingD2 0withD2 1inH usingtheidentitymap,wegetgconnectedsums
g× g× g
(D2 S1)#...#(D2 S1). Composing with the preceding stack product, we get a continuous
× ×
map:
T :H H (D2 S1)#...#(D2 S1)
g g
× → × ×
DenotebyK thesetofisotopyclassesofframedlinksin(D2 S1)#...#(D2 S1). Itresults
g
from classical surgery properties of reduced skein modules, that ×the space C[Kg]/× , obtained by
quotienting the C vector space generated by elements of K by the skein relations∼of Figure 1, is
g
one dimensional. We thus have an isomorphism C[Kg]/ (cid:27)C which sends the empty link to 1.
∼
Now the map:
T˜ :C C K
g g g
× →
sending (L ,L ) to T(L ,L ), induces by linearity a form:
1 2 1 2
, :V (Σ ) V (Σ ) T˜ K (cid:27)C
h· ·ip p g × p g −→ g
In [BHMV95], the authors exhibit an orthogonal basis for this form and proved it is definite
positive.
2.3 The Reshetikhin-Turaev representations
Denote by x ,...,x ,y ,...,y ,z ,...,z the oriented curves of Σ drawn in Figure 2. In
1 g 1 g 1 g−1 g
[Hum77], Humphries showed that their associated Dehn twists generate Mod(Σ ).
g
4
Figure 2: Humphries’ generators of Mod(Σ ).
g
We fix an orientation preserving homeomorphism
α:Σ ∂H
g g
→
such that the x ’s and z ’s are contractible in H through α and the y ’s are contractible through
i i g i
α S.
◦
Ifφ Mod(Σ )isaclassassociatedtoahomeomorphismwhichextendstoH throughα,such
g g
∈
as the T ’s and the T ’s, then φ acts on C and preserves the kernel of the Hopf pairing so acts
xi zi g
on V (Σ ) by passing to the quotient. Denote by ρ (φ) GL(V (Σ )) the resulting operator.
p g p p g
∈
Nowchooseφ Mod(Σ )suchthatthe correspondinghomeomorphismsextendto H through
g g
∈
α S, such as the T ’s. This extension also defines, by quotient, an operator on V (Σ ). We
◦ yi p g
denote by ρ (φ) the dualofthis operatorforthe Hopf pairing. Figure3 illustrates these operators
p
when g =1.
Figure 3: Action of the Humphries’ generators on V (Σ ).
p 1
We will now show that the classes of the operators ρ (φ) generate a projective representation:
p
ρ :Mod(Σ ) PGL(V (Σ ))
p g p g
→
Since each of the preceding generating operators preserve < , > , this representation is
p
· ·
however unitary.
Let γ . We define its curve operator Add (γ) End(V (Σ )) as follows. Given L H a
g p p g g
∈M ∈ ⊂
framed link, using α:Σ (cid:27)∂H , we can draw α(L) in ∂H and then pushing inside H obtaining
g g g g
a framed link L γ. The operator Add (γ) is then defined by the assignation [L] [L γ] (see
p
→
Figure 4).
S S
Elementary surgery arguments (see e.g. [BHMV95]) imply that the set of curve operators
generate End(V (Σ )). Note that it follows from their definition that the curve operators are
p g
self-adjoint for the invariant form.
An important property verified by the curve operators is the following Egorov identity:
ρ (φ)−1Add (γ)ρ (φ)=Add (φ(γ)), γ , φ Mod(Σ ) (2)
p p p p g g
∀ ∈M ∀ ∈
5
Figure 4: Action of a curve operator in genus one.
This identity is trivially satisfied when φ is one of the T ’s or T ’s which extends to H (see
xi zi g
Figure 5). It also holds for the T ’s by duality.
yi
Figure 5: Illustration of the Egorov identity in genus 1.
We can now prove that the ρ (T )’s, the ρ (T )’s and the ρ (T )’s generate a projective
p xi p zi p yi
representation. Letφ ,...,φ beHumphries’generatorsthatformarelationR:=φ ... φ =1.
1 N 1 N
◦ ◦
Write ρ (R):=ρ (φ ) ... ρ (φ ). To show we have a projective representation, we must show
p p 1 p N
◦ ◦
thatρ (R)isscalar. BytheEgorovidentity(2),thisoperatorcommuteswithallcurvesoperators,
p
thus commutes with the whole End(V (Σ )) and is scalar. Note that by composition, the Egorov
p g
identity indeed holds for any φ Mod(Σ ).
g
∈
]
For simplicity, we consider a central extension Mod(Σ ) of Mod(Σ ) that lifts the above pro-
g g
jective representations to linear ones (see [MR95, GM13] for explicit lifts) that we still denote ρ
p
by abuse of notation:
]
ρ :Mod(Σ ) U(V (Σ ))
p g p g
→
These are the so-called (SU(2)) Reshetikhin-Turaev representations.
3 The quantum ergodic theorem
3.1 A geometric proposition
The proof of the main theorem relies on the following:
6
Proposition 3.1. .
Let (E,d) be a metric space and ∆ E be a convex compact subset.
⊂
For all p 0, fix an integer N >0 and some points τ ,...,τ ∆ together with weights
≥ p 1,p Np,p ∈
α ,...,α [0,1] such that α =1.
1,p Np,p ∈ i i,p
Denote by Bp := iαi,pτi,p tPhe barycenter of the weighted points. Eventually choose τ ∈∆ an
extremal point of ∆.
P
Suppose that:
d
B τ
p
−p−→−∞→
Then there exist subsets J 1,...,N such that:
p p
⊂{ }
1. If we note J := α , then:
k pk i∈Jp i,p
P
J 1
p
k k−p−→−∞→
2. For any sequence j =(j ) with j J , then
p p≥1 p p
∈
d
τ τ
jp,p −p−→−∞→
Figure 6 illustrates the Proposition by showing two sets of points inside a compact convex
at two different moments. When the barycenter approaches an extremal point, then ’almost all’
points approachit as well. This is our geometric interpretationof the Schnirelman theorem. Note
that it might happen that a few points with small weight do not converge to the extremal point,
such as the point τi,p′ on the right picture. Such exceptional sequence are called Scars in the
quantum chaotic literature.
Figure 6: Illustration of Proposition 3.1.
The proof of Proposition 3.1 will be deduced from the following:
Lemma 3.2. Let (E,d) be a metric space, τ ,...,τ E be some points equipped with weights
1 N
∈
α ,...,α [0,1] such that α =1.
1 N ∈ i i
Denote by B := iαiτi tPheir barycenter and by ∆ := ch(τ1,...,τN) their convex hull. For a
subset K E, we will use the notation K := α [0,1].
⊂ P | | τi∈K i ∈
The followings hold: P
7
1. There exists a linear form L E′, in the dual E′ of E, there exist constants a,b R so that
∈ ∈
L(∆)=[a,b] and L−1(a)=τ.
2. Let ǫ>0 and 0<δ <1 and choose c (a,b) such that:
∈
F := x ∆, s.t L(x) c B(τ,ǫ)
c
{ ∈ ≤ }⊂
where B(τ,ǫ) denotes the ball of center τ and radius ǫ.
Noting α:= F , we have:
c
| |
(1 α)c a
d(τ,B)> − −
L
k k
3. For all ǫ>0 and for all 0<δ <1, there exists d>0 such that:
d(τ,B) d = B(τ,ǫ) δ
≤ ⇒ | |≥
Remark. In this lemma, instead of having a general convex compact as in Proposition 3.1, we
choose the convex hull of the points τ . The reasonfor it, is that for a general extremal point of a
i
convex compact subset of a metric space, the first point of Lemma 3.2 is not always satisfied. An
extremalpointhavepoint1 satisfiedis calledanexposed point in literature. The pointτ ofFigure
6 is an exemple of a not exposed extremal point.
Proof of 3.2. It is a general fact (see e.g. [Bré11]) that in any locally compact space E, we can
find a linear form L E′ separating finite sets of points, that is such that there exists c R such
∈ ∈
that:
L(τ)<c<L(τ ) , for all i 1,...,N
i
∈{ }
We obtain the first point by noting a:=L(τ) and b:=max (L(τ )).
i i
To prove the second one, note the sub-barycenter B := α and B := α .
1 i|τi∈Fc i 2 i|τi<Fc i
By convexity of ∆ F , we have L(B )>c. Moreover,since:P P
c 2
\
B =αB +(1 α)B
1 2
−
we have:
L(B)=αL(B )+(1 α)L(B )
1 2
−
Using the fact that L is continuous, we obtain:
L(B) L(τ) αL(B )+(1 α)L(B ) a
1 2
d(τ,B) | − | = − −
≥ L L
k k k k
(1 α)L(B ) a
2
− −
≥ L
k k
(1 α)c a
> − −
L
k k
which proves the second point.
To provethe lastpoint, we remarkthat sinceB(τ,ǫ) F , we have B(τ,ǫ) F =α. Thus:
c c
⊂ | |≤| |
(1 B(τ,ǫ))c a
d(τ,B)> −| | −
L
k k
So d:= (1−δ)c−a verifies the conclusion of the theorem. (cid:3)
kLk
8
Proof of Proposition 3.1. .
Apply the third point of Lemma 3.2 to the polytope ∆ := ch τ ,...,τ ,τ with ǫ = 1
p 1,p Np,p p
and δ =1 1. We obtain d >0 such that:
− p p (cid:8) (cid:9)
1 1
d(τ,B ) d = B(τ, ) 1
p p
≤ ⇒ p ≥ − p
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since (B ) converge to τ, there exists a rank r(p) such that:
p p
r r(p) = d(τ,B ) d
p p
≥ ⇒ ≤
1 1
= B(τ, ) 1
⇒ p ≥ − p
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
We can suppose that the sequence (r(p)) is strictly increasing. If r(p) r < r(p+1), we
p
≤
define:
1
J := j, s.t. d(τ,τ )
r j,r
≤ p
(cid:26) (cid:27)
1. Since r r(p), we have J = B(τ,1) 1 1. Thus:
≥ | r| p ≥ − p
(cid:12) (cid:12)
(cid:12) (cid:12) J 1
(cid:12) (cid:12) r
| |−r−→−∞→
2. If j =(j ) with j J , we have d(τ,τ ) 1. Thus:
r r r ∈ r jr,r ≤ p
d(τ,τ ) 0
jr,r −r−→−∞→
(cid:3)
3.2 The quantum ergodic theorem
WenowrestateTheorem1.1inaslightlymoregeneralform. Let( ,µ)beacompactHaussdorf
X
space equipped with a measure on the borelian σ-algebra. Let G be a group acting on by
X
preserving µ. Let
ρ :G U(V ) , for p p
p p 0
→ ≥
be a family ofrepresentationsoncomplex finite dimensionalvectorspacesV whichpreservesome
p
non-degenerate positive definite form , .
h ip
For each p p , we fix a decomposition:
0
≥
V =W ... W
p 1,p⊕ ⊕ Np,p
such that each W is G-invariant.
i,p
Denote by C( ) the algebra of continuous functions on and let (f ) be a dense family
n n≥0
X X
for the norm . We suppose there exist onto morphisms:
∞
k·k
Ψ :C( )։End(V )
p p
X
which are G-equivariant, where G acts on End(V ) by conjugacy. We also suppose that the oper-
p
ators Ψ(f ) are self-adjoint, that Ψ(f2)=Ψ(f )2 and that Ψ(1)=1.
n n n
We define the following states on the algebra C( ):
X
1
τ(f):= fdµ
Vol( )
X ZX
9
and:
1
B (f):= Tr(Ψ(f))
p
dim(V )
p
Also, denoting by P EndV the orthogonalprojector on W , we define:
i,p p i,p
∈
1
τ (f):= Tr(P Ψ(f))
i,p i,p
dim(W )
i,p
TheassumptionsonΨimplythatB andtheτ arestatesonthe -algebraC( ),thatis,that
p i,p
∗ X
they are linear forms verifying τ(f) 0 if f 0 and τ(1)= 1. It follows from the Riesz theorem
≥ ≥
thattheycorrespondtoprobabilitymeasureson (equippedwiththeborelianσ-algebra)andthey
X
are obviously G-invariants. Note also that if we note α := dim(Wi,p), then we have α = 1
i,p dim(Vp) i i,p
and B = α τ . We can now state the main theorem of this paper:
p i i,p i,p P
P
Theorem 3.3. .
With the preceding notations, suppose that:
– The action of G on ( ,µ) is ergodic.
X
– For all f C( ), we have:
∈ X
B (f) τ(f)
p
−p−→−∞→
Then for all p p , there exists a subset J 1,...,N such that:
0 p p
≥ ⊂{ }
1. We have:
α 1
i,p
−p−→−∞→
iX∈Jp
2. For any sequence j =(j ) with j J , we have:
p p p p
∈
τ (f) τ(f) , for all f C( )
jp,p −p−→−∞→ ∈ X
Remark. 1. Theorem 1.1, the SU(2) version, is easily deduced from this one when choosing
= , V =V (Σ ) and f =f where (γ ) is any numbering of the countable set .
X Xg p p g n γn n n Mg
The morphism Ψ is defined by Ψ (f )=Add (γ). The secondassumptionin the theoremis
p γ p
Marché-Narrimanejadtheorem (1) from [MN08].
2. ThistheoremisageneralizationofthemaintheoremofBouzouina-deBièvre[BDB96],which
inturnisavariantoftheSchnirelmantheoremfrom[Sch74](seealso[Zel87,CdV85,HMR87]
fordifferentvariantsofthistheorem). Werecoverthetheoremfrom[BDB96]whenwechoose
for the two dimensional torus T equipped with its natural symplectic measure, for V the
p
X
Segal-Shale-Weil representations which factorize through the symplectic group, and when
G = Z where 1 acts on the torus through an Anosov element. In this case, the authors
decompose the spaces V into invariant one dimensional eigenspaces and the conclusion is
p
the same thanours. Note that their proof,andthose ofallother variations,heavily relieson
the Birkhopf theorem, which is known only for a restricted class of groups, like the groups
with polynomial growthor the hyperbolic ones, but not for the mapping class groups or the
Torelli groups. Thus their proofs can not be translated in our general context.
3. Still it might happen that a few sequences (τ ) do not converge to τ if the associate
ip,p p
dimensions are not too large, as illustrated in Figure 6. Such exceptional sequences are
called Scars in the quantum chaotic literature. Such Scars have been found in the abelian
case in [FNDB03, Kel07]. The next Corollary provide a condition for not being a Scar.
We immediately deduce from Theorem 3.3 the following:
10