Talk II : the quantum hyperbolic invariants.
Plan of the talk :
1. QH State sums over weakly branched triangulations
2. Some relations with other quantum invariants
3. Examples
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1. QH State sums over weakly branched triangulations
2. Some relations with other quantum invariants
3. Examples
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The quantum hyperbolic invariants of a one-cusped manifold M
are defined by state sums
HN (T , b) : ZN → C/µN
over weakly branched ideal triangulations (T , b) of M carrying
the spaces ZN defined previously.
The state sums HN (T , b) are non-commutative analogues of the
function H1 : Z∞ → C∗ that defined the Chern-Simons function
S(M) in Talk 1.
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Definition
A 3-dim. pseudo-manifold triangulation is pre-branched if each
2-face is co-oriented and two co-orientations point inwards and two
outwards each tetrahedron. The triangulation is weakly branched
if its tetrahedra are branched and have compatible pre-branchings.
A
B
B
A
Figure: A pre-branched tetrahedron with its square edges oriented.
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A
B
Figure: Branched tetrahedra inducing the same pre-branching. Positive
orientation on the left (“∗b = +1”), negative on the right (“∗b = −1”).
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Simple facts:
1. Any triangulation has some pre-branchings, hence also weak
branchings.
2. The pre-branchings correspond to solutions of the gluing eqs
of the form (1, 1, −1) on each tetrahedron (‘Z/2-taut angle
structures”).
2
0
3
1
Figure: Graph encoding of a branched tetrahedron with ∗b = +1.
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A graph representing a weak branching on a triangulation T of the
“figure eight sister” cusped manifold:
2
1
NB. There is no system of edge orientations on T (a “branching”)
that define a branching by restriction to each tetrahedron of T .
NB. The gluing maps between two 2-faces is given by an even
permutation. This gives a color r ∈ Z/3Z for each 2-face of T .
Take a weakly branched triangulation (T , b) of M. Assign copies
V0 , . . . , V3 of CN to the faces of the (branched) tetra as follows:
1
0
2
0
2
3
3
1
−1
+1
Then, given an automorphism R ∈ GL(CN ⊗ CN ) and a branched
tetrahedron ∆, set
(
i,j
(Rk,l
) : V3 ⊗ V1 → V2 ⊗ V0 if ∗b = +1
R(∆) =
k,l
(R̄i,j ) : V2 ⊗ V0 → V3 ⊗ V1 if ∗b = −1.
Similarly, given a 3-periodic automorphism Q ∈ GL(CN ), assign
Qr (e) to each (co-oriented) 2-face of T with color r (e) ∈ Z/3.
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Definition of the state sums
Let M be a one-cusped manifold, and (T , b) a weakly branched
triangulation of M such that its gluing variety G (T ) has a rich
component Z , as defined in Talk 1.
Recall the space ZN ⊂ C3s of Nth-roots of the shape parameters
in Z . A point w ∈ ZN has coordinates
w = (w01 , w11 , w21 , . . . , w0s , w1s , w2s )
| {z }
| {z }
w1
ws
where wj , j ∈ {1, . . . , s}, is associated to the tetrahedron ∆j of T .
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For any point w = (wj )j ∈ ZN we set:
HN (T , b)(w) :=
s
X Y
σ
j=1
RN (∆j )(wj )σ
Y
r (e)
(QN )σ
faces e
where:
I
for every j and every face e, RN (·)(wj ) ∈ GL(CN ⊗ CN ) and
r (e)
QN ∈ GL(CN ) are associated to ∆j and e as we explained.
I
σ runs over all maps {2−faces of T } → {0, 1, . . . , N − 1};
I
(. . .)σ means that we take the entry of the tensor (...) which
is determined by σ.
RN (·)(·) is a tensor valued function of wj = (w01 , w11 , w21 ) called
matrix dilogarithm; its behaviour under tetrahedral symmetries
determine the matrix QN .
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The main things to know about RN are:
I
I
RN is derived from the 6j-symbols of the cyclic reps of Uq sl2
(first computed by Kashaev);
b N −→ Aut(CN ⊗ CN ),
RN eventually reduces to a rat. map C
b N is the Riemann surface of the Nth root map
where C
C \ {0, 1} −→ C2 ,
z 7−→ (e
log(z)
N
,e
j
j
k
Figure: R(e
log(z)+πip
N
,e
l
− log(1−z)+πiq
N
i
).
k
i
*b = 1
− log(1−z)
N
*b = −1
l
), where z := w0 and 1/(1 − z) = w1 .
I
Set l0,N :=
log(z)+πip
, l1,N
N
RN (e l0,N , e l1,N )i,j
k,i+j ∝
ϕ(e
l0,N
,e
l1,N
:=
− log(1−z)+πiq
.
N
We have
√
√
√
2 −1π 2 −1π 2 −1π
i,
j,
k
)
N
N
N
where
N
ϕ(e l0,N , e l1,N )(u, v , t) = e 2iπ ((t−u)l1,N +vt+t
2 /2)
× Sπ/N (−i(l0,N + u − t) + (1 − N)π/N)
and...
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...Sπ/N (. . .) is a value of the meromorphic function
Z
1
e zx dx
.
Sγ (z) = exp
4 R+i0 x sinh(πx) sinh(γx)
where 0 < arg γ < π/2 and |Re(z)| < π + |Re(γ)| (Barnes’ double
sine 1900, Faddeev ’95).
Write Sγ = exp(Iγ (z)). One has a uniform asymptotic expansion
on compact subsets of |Re(z)| < π:
∞
1
1 X 2n−1 22n B2n (1/2) 2n
iz
Iγ (z) =
Li2 (−e )+
γ
∂z Li2 (−e iz )
2iγ
2i
(2n)!
n=1
as γ → 0.
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1
NB. By the Jacobi triple product formula, Sγ is a multiplier 2 :
Θ(y , τ ) = ξγ Sγ (z)Sγ (−z)Θ(y /τ, −1/τ )
where Θ(y , τ ) :=
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P∞
n=0 q
n2 x n ;
, ξγ ∈ C.
I
RN satisfies fundamental “5-term” identities called pentagon
relations, and RN (∆) has some tetrahedral symmetries.
Here is a graph representing a pentagon relation between matrix
dilogarithms (NB: it is a non Abelian 3-cocycle relation):
x3
y
=
x1
x2
x
Figure: x1 = y /x, x2 = y (1 − x)/x(1 − y ), x3 = (1 − x)/(1 − y ), where
0 < y > x < 1. Each of the arguments x, y , x1 , x2 and x3 stands for the
shape parameter z0j of the corresponding tetrahedron. The products of
N-th roots wrj are required to be equal along the edges on both sides.
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Any such identity corresponds to a move (T , b, w) → (T 0 , b 0 , w0 )
which preserves the branchings locally, between weakly branched
triangulations having their tetrahedra equipped with the edge
variables wrj . The above identity corresponds to:
(1-x)/(1-y)
y
x
y(1-x)/x(1-y)
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y/x
Recall that:
I
any point w in ZN defines a holonomy ρ(w) ∈ X and a weight
hZ ,N (w) ∈ H 1 (∂ M̄; C∗ );
I
the set of such weights is a Zariski open subset of the
covering AN of the A-polynomial curve A = h(Z ).
Theorem (B-Benedetti 2012) The state sum HN (T , b)(w) is an
invariant of (M, ρ(w), hZ ,N (w)) up to multiplication by N-th roots
of 1.
By varying w in ZN , these invariants define rational functions
HN (M) : AN → C/µN .
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Strategy of the proof:
I
The state sums are invariant under any change of the weak
branching (⇒ one needs a “calculus” for such objects).
I
One can define moves between triangulations of the form
(T , b, w), which preserve holonomy and weight, and are
compatible with the pre-branching.
I
The state sums are invariant under such moves.
I
Any two triangulations of M of the form (T , b, w) and having
the same holonomy and weight are related by such moves.
A similar proof applies to show the invariance of H1 mentionned at
the end of Talk 1.
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Example of invariance under moves preserving the pre-branching:
(0321)
(01)
(12)
(12)
(12)
(01)
(12)
(01)
(0123)
(01) (12)
(01)
(12)
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(01)
1. QH State sums over weakly branched triangulations
2. Some relations with other quantum invariants
3. Examples
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Relation with quantum Teichmüller theory
The following papers imply that the QHI of mapping cylinders of
punctured surfaces contain the (finite dim.) q-Teichmüller theory:
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I
F. Bonahon, X. Liu, Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms, Geom.
Topol. 11 (2007) 889–937
I
H. Bai, Quantum Teichmüller spaces and Kashaev’s
6j-symbols, Alg. Geom. Topol. 7 (2007) 1541–1560
I
H. Bai, F. Bonahon, X. Liu, Local representations of the
quantum Teichmüller space, ArXiv:math.GT/?? (2007)
I
J. Toulisse, Irreducible decomposition for local representations
of the quantum Teichmüller spaces, ArXiv:math.GT/14044938
Relations with invariants of closed manifolds
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The QHI are defined also for links in arbitrary compact oriented
3-mfds with PSL(2, C)-characters. In the case of S 3 we have:
Theorem (B-Benedetti 2011). For every link L in S 3 and every
odd integer N ≥ 3 we have
HN (S 3 , L, ρtriv , 0) ≡N JN (L)(e 2iπ/N )
where JN (L) is the normalized N-th colored Jones polynomial of L
such that JN (unknot) = 1, ρtriv is the trivial holonomy, and 0 is
the null weight.
Here ≡N denotes equality up to multiplication by N-th roots of 1.
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Similarly to the behaviour of Vol + iCS we have:
Theorem (B-Benedetti 2007) For any sequence of closed hyp.
Dehn fillings Mn of M with surgery cores Ln and holonomies ρn
converging to ρhyp in the character variety, we have
lim HN (Mn , Ln , ρn , 0) ≡N HN (M)(ρhyp , 0)
n→∞
where 0 denotes the 0 weights.
1. QH State sums over weakly branched triangulations
2. Some relations with other quantum invariants
3. Examples
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Summarizing, in order to compute HN (M) one needs:
I
an ideal triangulation T of M having a solution zhyp ∈ G (T )
of the gluing eqs with d.f. holonomy and coordinates with
non-negative imaginary part (for example a subdivision of the
EP cellulation); this provides rich components Z of G (T ).
I
Explicit formulas for the solutions w ∈ Z of the gluing eqs,
and for their N-th roots w ∈ ZN .
This is essentially provided by Snappea, snap, and Zickert’s algo.
Moreover one needs:
I
a weak-branching b of T (immediate);
I
the state sum formula of HN (M) on (T , b) (immediate);
I
a very good computer for large values of N !
Numerical computations show that:
I
The invariants depend on the characters and the weights.
I
This dependence persists as N → +∞.
I
For some weights the invariants grow exponentially with N,
and yield instances of the volume conjecture, and for some
other weights they do not.
I
There are symmetries of the invariants at the hyperbolic
holonomy, which disappear when moving away from it.
I
For knot complements, the absolute values of the invariants
do not depend on the weights of the meridian as N → ∞.
In practice, any example offers a whole world to explore !
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Example 1:
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M = complement of the figure eight knot. We take the canonical
triangulation. It has a branching. We fix one, and find the state
sum expression (up to a scalar which is negligeable as N → +∞):
HN (M)(u, v) ∝
N−1
X
ζβ
2 −α2
α,β=0
ω(u0 , u−1
1 |N − β)
ω(v0 /ζ, v1−1 |N − α)
Here u := (u0 , u1 , u2 ) and v := (v0 , v1 , v2 ) are the triangulation’s
2πi
N-th roots of shape parameters, and we set ζ := e N and
ω(x, y |n) :=
n
Y
j=1
y (1 − xζ j )−1 .
Take (u, v) with d.f. holonomy and longitude weight as in the left
πi
column, and deform it by u0 (A, B) := e 3 + A + iB as in the top
row. You get the values of |H15 (M)(u, v)| listed below:
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(0, 0)
(1, 0)
(1, 0.5)
(1, 1)
(A, B)(2,3)
−8πi
2.5587...
2.6504...
2.0018...
1.6118....
5.0307....
−6πi
58.5466...
58.3761...
47.0533...
39.9892...
95.0326...
−4πi
2.1356...
2.0279...
1.9058...
1.8138...
2.7491...
−2πi
77.4851...
77.5401...
77.5885...
77.6997...
76.4850...
0
0.1118...
0.1620...
0.1672...
0.1746...
0.0650...
2πi
77.4851...
77.5401...
95.8738...
112.7032...
47.4247...
4πi
2.1356...
2.0279...
2.7647...
3.5183...
1.0549...
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Take (u, v) with d.f. holonomy and longitude weight κ(l) ∈ 2πiZ.
The ratio
GN (κ(l)) := π log
|HN+2 (M)(u, v)|
|HN (M)(u, v)|
gives the real part of the exponential growth rate of (HN (M))N for
large values of N. At N = 151 we get:
G151 (−6πi) = 2.03069... , G151 (−4πi) = −0.49036...
G151 (−2πi) = 2.02968... , G151 (0) = 0.48922...
Further numerical computations show that GN (2πi) converges to
Vol(M) ≈ 2.02988321... as N >> 1.
The values of GN at points of 4πiZ behave differently.
Example 2:
Msister = the figure eight knot “sister” manifold. again we consider
the weakly branched triangulation dually represented by the graph:
2
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1
On this weakly branched triangulation we have the state sums:
HN (Msister )(u, v) ∝
N−1
X
ζ (i+k)(j+I )−ik+
j 2 +I 2
2
−1
ω(u0 , u−1
1 |i −k)ω(v0 , v1 |I −j).
i,j,k,I =0
Here are some values of |H15 (Msister )(u, v)| (conventions as
before):
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(0, 0)
(1, 0)
(1, 1)
(2, 7)
2
4.7755...
4.5346...
2.6295...
1.9024...
1
173.2621...
173.3850...
126.3516...
101.7810...
0
0.2500...
0.3624...
0.1498...
0.3701...
−1
173.2621...
173.3850...
236.1129...
302.5352...
−2
4.7755...
4.5346...
9.3949...
14.6569...
Some immediate perspectives:
Recall that the volume conjecture for the QHI deals with
H∞ ((xN )) := limsupN→ ∞ log
|HN (xN )|
N
∈ R ∪ {+∞}.
In order to get a robust formulation of the conjecture one should:
I
Study numerically the invariants of many one-cusped mfds;
I
When does (HN (xN ))N have an exponential growth rate ?
When this growth rate is given by complex Chern-Simons
invariants ? Or what measures the discrepancy ?
I
What can we say about the variation of the growth rate as a
function of the sequence (xN )N , about its regularity ? Etc.
NB. H∞ ((xN )) should depend only on the limit point of (xN )N in
the Hausdoff limit of the sequence of curves (XN )N (which is =
(amoeba of the A-polynomial) ×T 2 ).
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More ambitious perspectives:
I
Lift HN : XN → C/µN to a C-valued function;
I
HN is a rational function: expression in terms of augmented
characters of meridian/longitude ? Poles, periods ?
I
Find a skein theoretic construction of HN
I
Find a pure geometric quantization construction of HN from
the Chern-Simons function and line bundle.
Work in progress with Benedetti and Frohman.
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