QUANTUM HYPERBOLIC INVARIANTS
Stéphane Baseilhac
Université de Montpellier
Nara, October 2014
1/26
The work presented here comes from collaborations with
Benedetti, in particular:
S. B., R. Benedetti, Analytic families of quantum hyperbolic
invariants, to appear in Algebraic & Geometric Topology.
2/26
We are going to discuss some of the mathematics behind the
volume conjecture (Kashaev ’97, Murakami-Murakami ’99):
Volume Conjecture: If K is a hyperbolic knot in S 3 , then
log |JN (K )(e
N→∞
N
lim
2πi
N
)|
=
1
Vol(S 3 \ K )
2π
Here JN (K ) ∈ Z[q ± ] is the normalized N-th colored Jones
polynomial of the knot K , such that JN (unknot) = 1.
3/26
Here are some motivations:
4/26
I
Unfold the asymptotical geometric content of quantum
invariants; the vol. conjectures are predictions for 3-dim.
quantum gravity.
I
Clarify the “quantization processes” lying in between.
I
Find new interplays between algebra and geometry: new
QFTs, new mapping class group representations, etc.
5/26
Rather than the Jones polynomials, we will consider the quantum
hyperbolic invariants HN (M) (QHI), and the following version of
the volume conjecture for the QHI:
For any one-cusped hyperbolic manifold M there exist sequences
of points xN ∈ XN such that
HN (M)(xN ) ∼N→+∞ exp(
N
(Vol(M) + iπ 2 CS(M))).
2π
Here, for each odd N ≥ 3,
HN (M) : XN → C/µN
is a rational function with complex values modulo the N-th roots
of 1, and XN a determined finite cover of the “geometric” compt
of the variety of PSL(2, C)-characters of M.
Since the quantum invariants are defined combinatorially, we will
build in a similar way spaces and maps fitting in a diagram like:
XON
X∞
HN (M), rational
/ C/µN
S(M), analytic
/C
where again we have covering maps
X∞
/Z2
/ XN
/(Z/NZ)2
/X
over the geometric component X of the variety of PSL(2, C)
characters of M, and S(M) is the Chern-Simons function.
6/26
Next talk : The quantum hyperbolic invariants; it requires a
combinatorial definition of the Chern-Simons function S(M).
This talk: The combinatorial definition of S(M), and the
modifications required by the quantum setup.
1. Gluing varieties
2. The coverings
3. The Chern-Simons function
7/26
1. Gluing varieties
2. The coverings
3. The Chern-Simons function
8/26
Denote by X the geometric component of augmented PSL(2, C)
valued characters of M, and X (∂ M̄) the character variety of ∂ M̄.
The restriction map res : X → X (∂ M̄) is regular.
Theorem (Dunfield) The map res : X → X (∂ M̄) is birational
onto its image.
Fixing a cusp basis (l, m), denote the induced map and image by
h : X → C∗ × C∗
A := h(X )
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Let T be an ideal triangulation of M without null-homotopic
edges. Then:
I
The gluing variety G (T ) is non empty (Segerman-Tilmann);
I
G (T ) is a complex curve (Neumann-Zagier);
I
There exists a point zhyp ∈ G (T ) with d.f. holonomy ρhyp .
A question: Is Dunfield’s theorem true by replacing the geometric
compt X of the character variety with some component of G (T ) ?
Some issues:
I
zhyp may not be a regular point of G (T ); hence it may be
contained in several components.
I
Dunfield’s proof uses the volume rigidity for closed hyperbolic
Dehn fillings of M, and the variation formula of the volume
function of characters.
10/26
Let T be an ideal triangulation of M without null-homotopic
edges. Then:
I
The gluing variety G (T ) is non empty (Segerman-Tilmann);
I
G (T ) is a complex curve (Neumann-Zagier);
I
There exists a point zhyp ∈ G (T ) with d.f. holonomy ρhyp .
A question: Is Dunfield’s theorem true by replacing the geometric
compt X of the character variety with some component of G (T ) ?
Some issues:
I
zhyp may not be a regular point of G (T ); hence it may be
contained in several components.
I
Dunfield’s proof uses the volume rigidity for closed hyperbolic
Dehn fillings of M, and the variation formula of the volume
function of characters.
Definition
An irreducible component of the gluing variety G (T ) is rich if it
contains the point zhyp and also an infinite sequence of closed
hyperbolic Dehn fillings of M with shape parameters zn converging
to zhyp in G (T ).
Proposition (Petronio-Porti) For any ideal triangulation T of M
such that zhyp has coordinates with non negative imaginary part,
the gluing variety G (T ) has rich components. In particular, the
max subdivisions of the EP cellulation of M have rich components.
Using the proposition and arguments similar to Dunfield’s one can
prove:
Corollary For any rich component Z of a gluing variety G (T ) of
M, the (regular) map hZ : Z
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holonomy
/X
h
/ A is birational.
Definition
11/26
An irreducible component of the gluing variety G (T ) is rich if it
contains the point zhyp and also an infinite sequence of closed
hyperbolic Dehn fillings of M with shape parameters zn converging
to zhyp in G (T ).
Proposition (Petronio-Porti) For any ideal triangulation T of M
such that zhyp has coordinates with non negative imaginary part,
the gluing variety G (T ) has rich components. In particular, the
max subdivisions of the EP cellulation of M have rich components.
Using the proposition and arguments similar to Dunfield’s one can
prove:
Corollary For any rich component Z of a gluing variety G (T ) of
M, the (regular) map hZ : Z
holonomy
/X
h
/ A is birational.
1. Gluing varieties
2. The coverings
3. The Chern-Simons function
12/26
Now we are going to complete a square:
??
Z
hZ ,∞
hZ
/ A∞
/Z2
/A
where as before A := h(X ) is the image of the geometric cpt under
the restriction map, hZ is the map we have just defined, and we set
A∞ := {(u, v ) ∈ C2 | (e u , e v ) ∈ A}.
We need to take logarithms of the coordinates of points z ∈ Z .
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Define the analytic set (s is the number of tetrahedra of T )
Z∞ ⊂ C3s
by requiring that a point (l01 , l11 , l21 , . . . , l0s , l1s , l2s ) ∈ Z∞ satisfies:
I
I
in tetrahedra: ∀ j ∈ {1, . . . , s}, l0j + l1j + l2j = 0, and for all
r ∈ {0, 1, 2}, exp(lrj ) = ±zrj for some point z = (zrj ) ∈ Z ;
P
about edges: ∀ E ∈ E (T ), j,r lrj (E ) = 0.
If non empty, Z∞ is a space of logarithms of ± shape parameters
in Z . Note that the edge equations are the same for all edges.
Any point l = (lrj ) ∈ Z∞ determines:
I
a holonomy ρ(l) ∈ X (note that the tetrahedral relations allow
one to determine z = (zrj ) from l);
I
a class γ(l) ∈ H 1 (∂ M̄; C), defined on a loop transverse to the
cusp triangulation by summing the coordinates lrj viewed from
the loop, with signs according to the orientation (consistency
follows from by the tetrahedral and edge relations).
I
a class γ2 (l) ∈ H 1 (∂ M̄; Z/2), defined similarly, using normal
loops in T and taking the sum mod(2) of the variables
(lrj − log(zrj ))/πi
attached to the edges we face along the loops.
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Denote by dz ∈ H 1 (∂ M̄; C/2iπZ) the log of the linear part of the
restriction of ρ(l) to π1 (∂ M̄).
For all a ∈ H1 (∂ M̄; Z) we have the compatibility relations:
γ(l)(a) = dz (a) mod(iπ)
(γ(l)(a) − dz (a))/iπ = i ∗ (γ2 (l))(a) mod(2).
where i ∗ : H 1 (M̄; Z/2Z) → H 1 (∂ M̄; Z/2Z) is induced by inclusion.
Below we will assume that γ2 (l) = 0 (for simplicity).
Recall that A∞ := {(u, v ) ∈ C2 | (e u , e v ) ∈ A}. Then we can put
hZ ,∞ : Z∞ −→ A∞
l
7−→ (γ(l)(l), γ(l)(m))
Similarly define the algebraic set
ZN ⊂ C3s
by requiring that (w01 , w11 , w21 , . . . , w0s , w1s , w2s ) ∈ ZN satisfies:
I
I
N−1
in tetrahedra: ∀ j ∈ {1, . . . , s}, w01 w11 w21 = −ζ 2 , and for all
r ∈ {0, 1, 2}, (wrj )N = ±zrj for some point z = (zrj ) ∈ Z ;
Q
about edges: ∀ E ∈ E (T ), j,r wrj (E ) = ζ −1 .
If non empty, ZN is a space of N-th roots of the shape parameters
in Z . Note that the edge equations are the same for all edges.
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18/26
Again, any point w = (wrj ) ∈ ZN determines:
I
a holonomy ρ(w) ∈ X ;
I
a class γN (w) ∈ H 1 (∂ M̄; C∗ );
I
a class γN,2 (w) ∈ H 1 (∂ M̄; {±1}).
Below we will assume that γN,2 (w) = 1 (for simplicity). Then,
setting
AN := {(u, v ) ∈ C2 | (u N , v N ) ∈ A}
we can put
hZ ,N : ZN
w
−→ AN
7−→ (γN (w)(l), γN (w)(m)).
We deduce a commutative diagram
Z∞
hZ ,∞
classical
/ A∞
1
e N (·)
ZN
(·)N
Z
hZ ,N
/(NZ)2
quantum
/ AN
hZ
/(Z/NZ)2
/A
We call hZ ,∞ (l) and hZ ,N (w) the weights of l and w.
But do we have (Z 6= ∅ ⇒ Z∞ , ZN 6= ∅) ?
What can be said about the image of hZ ,∞ and hZ ,N ?
19/26
Theorem BIG (follows from Neumann’s work) We have:
(1) The map hZ ,∞ : Z∞ → A∞ maps onto a dense open subset
(no lift is missed).
(2) The fibers of the covering Z∞ → Z are affine spaces over an
abelian group C that fits in an exact sequence
(γ,γ2 )
0 → Zn(edges) → C −→ H 1 (∂ M̄; Z) ⊕ H 1 (M̄; Z/2Z)
r −i ∗
−→ H 1 (∂V ; Z/2Z) → 0
(3) The isomorphism γ : C ⊗ Q → H 1 (∂ M̄; Q) is symplectic with
respect to Neumann-Zagier’s 2-form and the intersection product.
Rq. Given any two points l, l0 ∈ Z∞ , (γ(l) − γ(l0 ), γ2 (l) − γ2 (l0 ))
lies in Ker(r − i ∗ ). This helps proving (1) from (2).
20/26
1. Gluing varieties
2. The coverings
3. The Chern-Simons function
21/26
In 1992, Neumann introduced a function (a sum of dilogarithms)
that can be written as
H1 : Z∞ −→ C/2πiZ.
He showed:
Theorem (Neumann) Let Y be M or a closed hyperbolic Dehn
filling M 0 of M, and ρY the hyperbolic holonomy of Y . Then:
(1) H1 (l) is constant on points l ∈ Z∞ with holonomy ρ(l) = ρY
and weight hZ ,∞ (l) = 0.
(2) For such an l we have H1 (l) = exp π2 Vol(Y ) + 2πiCS(Y ) .
22/26
23/26
The proof is homological: Neumann produced an isomorphism
b
H3 (BPSL(2, C); Z) ∼
= B(C)
with an extended Bloch group of scissors congruences in H3 , and
used Dupont’s 3-cochain representing p̂1 in terms of dilogarithms.
Another, direct proof (knowing the answer), is due to J. Marché;
he computed the Chern-Simons functional of flat connections on
permutohedra with “nice” representatives of the connections.
For any compact oriented 3-manifold Y , Chern-Simons theory with
gauge group PSL(2, C) defines:
I
the Chern-Simons bundle L∂Y → X (∂Y ), which is a
C∗ -bundle with connection and inner product;
I
the parallel Chern-Simons section sY : X (Y ) → res ∗ L∂Y ,
where res : X (Y ) → X (∂Y ) is the restriction map.
When Y is closed, sY is just a function, and if Y is hyperbolic,
then Yoshida and Dupont proved independently that
2
Vol(Y ) + 2πiCS(Y )
sY (ρY ) = exp
π
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If Y = V ∪ϕ (D 2 × S 1 ) is a closed hyperbolic Dehn filling of M
with holonomy ρY close to ρM in X , then Kirk-Klassen proved:
I
Fixing the gauge on i ∗ L∂ M̄ by taking as local coordinates on
X the standard logarithms log(λ) and log(µ), we have
2
Vol(M) + 2πiCS(M) .
sM̄ (ρhyp ) = exp
π
I
If the Dehn filling instruction maps the meridian ∂(D 2 × ∗) to
l q mp and the longitude ∗ × S 1 to l s mr (so ps − qr = 1), then
sY (ρY ) = sM̄ (ρhyp )
!
Z ρY
1
× exp −
(log(λ)d log(µ) − log(µ)d log(λ))
2πi ρhyp
× exp (−(s log(λ) + r log(µ))) .
26/26
Putting together all these results gives a characterization of sM̄ as
a sol. of a differential problem. Using Neumann’s BIG Theorem,
one can deduce:
Proposition The function S(M) := exp(H1 (·)) : Z∞ → C∗ factors
through hZ ,∞ to define a function on A∞ , and it descends to a
parallel section s(M) of a flat trivial C∗ -bundle L(M) → A with
canonical connection 1-form and inner product, such that
h∗ (L(M), s(M)) ∼
= (i ∗ L∂ M̄ , sM̄ ).