an introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...an...
TRANSCRIPT
![Page 1: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/1.jpg)
An introduction to quantum hyperbolic geometry
From joint works with S. Baseilhac - Workshop-PRIN, Pisa -Febbraio 2013
![Page 2: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/2.jpg)
Two approaches to 3D hyperbolic geometry as a
classical field theory
W compact closed oriented 3-manifold.
(1) The fields are the Riemannian metrics g on W ; the hyperbolicstructures on W (if any) are the solutions gh of the 3D Einsteinequation for the Riemannian signature and with (normalized)negative cosmological constant Λ.
![Page 3: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/3.jpg)
(2)(Gauge theory)
PSL(2,C) = Isom+(H3)
![Page 4: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/4.jpg)
(2)(Gauge theory)
PSL(2,C) = Isom+(H3)
• The fields are the connections on PSL(2,C)-principal bundles overW (which can be trivialized as topological bundles).
![Page 5: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/5.jpg)
(2)(Gauge theory)
PSL(2,C) = Isom+(H3)
• The fields are the connections on PSL(2,C)-principal bundles overW (which can be trivialized as topological bundles).
•We consider the Chern-Simons action.
![Page 6: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/6.jpg)
(2)(Gauge theory)
PSL(2,C) = Isom+(H3)
• The fields are the connections on PSL(2,C)-principal bundles overW (which can be trivialized as topological bundles).
•We consider the Chern-Simons action.
•The critical points of the action are the flat connections (up togauge equivalence); equivalently the PSL(2,C)-charcters of W (i.e.of π1(W )) (admitting a lifting to SL(2,C)).
![Page 7: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/7.jpg)
•For every character ρ the Chern-Simons number:
CS(ρ) ∈ C/Z
is a well defined invariant.
![Page 8: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/8.jpg)
•For every character ρ the Chern-Simons number:
CS(ρ) ∈ C/Z
is a well defined invariant.
•Vol(ρ) := −π2
Im(CS(ρ)) ≥ 0
and the holonomy character ρh of a hyperbolic structure on W (ifany) maximises Vol(ρ).
![Page 9: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/9.jpg)
Relation between the two field theories
For every Riemannian metric g on W we can define:
Vol(W , g);
The Chern-Simons number CS(W , g) ∈ R/Z, via gauge theorywith group SO(3,R) over the tangent bundle of W .
![Page 10: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/10.jpg)
Relation between the two field theories
For every Riemannian metric g on W we can define:
Vol(W , g);
The Chern-Simons number CS(W , g) ∈ R/Z, via gauge theorywith group SO(3,R) over the tangent bundle of W .
We have:Vol(ρh) = Vol(W , gh)
CS(ρh) = CS(W , gh)−i
π2Vol(W , gh) .
![Page 11: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/11.jpg)
Simplicial formulas problem
Given:
(W , ρ), ρ being any PSL(2,C)-charcter of W ;
(T , b) any simplicial complex over W , which carries a simplicialfundamental class of W :
[W ] =∑
(∆,b)∈(T ,b)
∗b(∆, b), ∗b = ±1 .
[Here T refers to a “naked” triangulation of W by orientedtetrahedra, while b refers to the additional combinatorial structurethat converts every tetrahedron into a 3-symplex.]
![Page 12: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/12.jpg)
Determine:
1 A suitable enhancement T = (T , b, d) of (T , b) so that dencodes ρ.
2 A 3-cochain c(d) ∈ C 3(T , b;C/Z)
![Page 13: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/13.jpg)
Determine:
1 A suitable enhancement T = (T , b, d) of (T , b) so that dencodes ρ.
2 A 3-cochain c(d) ∈ C 3(T , b;C/Z)
So thatS(W , ρ) := c(d)([W ]) ∈ C/Z
is a well defined invariant, and
S(W , ρh) = CS(ρh) .
(Long story: Bloch, Dupont, Sah, Neumann, Yang, Zagier ...)
![Page 14: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/14.jpg)
ρ-Encoding via parallel transport along the edges
Every character ρ of W can be represented by (non-commutative)1-cocycles z with coefficients in PSL(2,C).
![Page 15: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/15.jpg)
ρ-Encoding via parallel transport along the edges
Every character ρ of W can be represented by (non-commutative)1-cocycles z with coefficients in PSL(2,C).
Every b-oriented edge e of (T , b) is labelled by z(e) ∈ PSL(2,C) sothat they verify the four 2-facet relations
zi ,jzj ,kz−1k,i , i < j < k
on every 3-symplex (∆, b).[Note: This defines also a simplicial 3-cycle of BPSL(2,C)δ.]
![Page 16: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/16.jpg)
0 3
1 2
z
z
z
z
z
0,1
0,3
2,3
1,2
1,3
z0,1
z1,3
z0,3
−1=1
z0,2
![Page 17: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/17.jpg)
Idealization: ρ-encoding via cross-ratio systems
Fix a base point p0 ∈ S2∞ = ∂H3.
![Page 18: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/18.jpg)
Idealization: ρ-encoding via cross-ratio systems
Fix a base point p0 ∈ S2∞ = ∂H3.
If a 1-cocycle z representing ρ is “generic”, then for every 3-symplexthe points of S2
∞ :
(p0, p1 = z0,1(p0), p2 = z0,1z1,2(p0), p3 = z0,3(p0))
are distinct and span an hyperbolic ideal tetrahedron. Up toorientation preserving isometries of H3, this is encoded by a systemof cross-ratios {u[i ,j ]} which label the edges of (∆, b).
We stipulate that all edge decorations of a tetrahedron verify theproperty that opposite edges share the same decoration instance.
![Page 19: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/19.jpg)
Any cross-ratio system on (∆, b):
u[i ,j ] ∈ C \ {0, 1}
is determined by
u0 = u[0,1], u1 = u[1,2], u2 = u[0,2]
and verifiesuj+1 = 1/(1− uj), mod(3)
so that2∏
j=0
uj = −1 .
Hence it is eventually determined by u0. The final cross-ratioenhancement of (∆, b) takes into account the sign ∗b:
wj := u∗bj , j = 0, 1, 2 .
![Page 20: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/20.jpg)
The gluing variety G (T , b)
•Let n be the number of 3-symplexes of (T , b), and order them.
![Page 21: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/21.jpg)
The gluing variety G (T , b)
•Let n be the number of 3-symplexes of (T , b), and order them.
•(C \ {0, 1})n
represents the set of arbitrary system of cross-ratiosw = (w0,1, . . .w0,n), on every (∆, b) of (T , b).
![Page 22: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/22.jpg)
The gluing variety G (T , b)
•Let n be the number of 3-symplexes of (T , b), and order them.
•(C \ {0, 1})n
represents the set of arbitrary system of cross-ratiosw = (w0,1, . . .w0,n), on every (∆, b) of (T , b).
•The gluing variety G (T , b) is the algebraic subvariety of(C \ {0, 1})n defined by the system of edge equations, one for eachedge e of T
W (e) :=∏
E→e
w(E )∗b(E) = 1
W (e) is called the total cross ratio around e.
![Page 23: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/23.jpg)
Facts:•Every point w ∈ G (T , b) represents a PSL(2,C)-character ρ(w) ofW .
![Page 24: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/24.jpg)
Facts:•Every point w ∈ G (T , b) represents a PSL(2,C)-character ρ(w) ofW .
•The system of cross-ratios obtained via the idealization of anygeneric 1-cocycle belongs to G (T , b) and they represent the samecharacter.
![Page 25: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/25.jpg)
Facts:•Every point w ∈ G (T , b) represents a PSL(2,C)-character ρ(w) ofW .
•The system of cross-ratios obtained via the idealization of anygeneric 1-cocycle belongs to G (T , b) and they represent the samecharacter.
•For every symplicial complex (T , b) over W , G (T , b) carries(possibly infinitely many times) all characters of W .
![Page 26: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/26.jpg)
The simplicial “Volume Function”
The Bloch-Wigner dilogarithm:
D2(x) := Im(Li2(x)) + arg(1− x) log |x | .
D2(x) is real analytic on C \ {0, 1}.
![Page 27: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/27.jpg)
The simplicial “Volume Function”
The Bloch-Wigner dilogarithm:
D2(x) := Im(Li2(x)) + arg(1− x) log |x | .
D2(x) is real analytic on C \ {0, 1}.
We have the Volume function defined on the gluing variety:
vol : G (T , b) → R
vol(w) :=∑
n
(∗b)nD2(w0,n)
![Page 28: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/28.jpg)
Facts:(1) Let ρ = ρ(w), then
vol(ρ) := vol(ρ(w))
is a well defined invariant.
![Page 29: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/29.jpg)
Facts:(1) Let ρ = ρ(w), then
vol(ρ) := vol(ρ(w))
is a well defined invariant.
(2) If ρ lifts to SL(2,C), then
Vol(ρ) = vol(ρ) .
![Page 30: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/30.jpg)
Facts:(1) Let ρ = ρ(w), then
vol(ρ) := vol(ρ(w))
is a well defined invariant.
(2) If ρ lifts to SL(2,C), then
Vol(ρ) = vol(ρ) .
(3) In particular: Vol(W , gh) = vol(ρh).
![Page 31: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/31.jpg)
Facts:(1) Let ρ = ρ(w), then
vol(ρ) := vol(ρ(w))
is a well defined invariant.
(2) If ρ lifts to SL(2,C), then
Vol(ρ) = vol(ρ) .
(3) In particular: Vol(W , gh) = vol(ρh).
[Note: This computes the volume of the scissors coungruence class of(W , gh); recall the third Hilbert problem. The Bloch group, thatorganizes these classes, is generated by the ideal tetrahedra.]
![Page 32: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/32.jpg)
Why does D2(x) work so well?
•It is completely invariant with respect to the tetrahedral symmetries:
D2(x−1) = −D2(x), D2(x) = D2(1/(1− x))
![Page 33: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/33.jpg)
Why does D2(x) work so well?
•It is completely invariant with respect to the tetrahedral symmetries:
D2(x−1) = −D2(x), D2(x) = D2(1/(1− x))
•It verifies all instances of functional 5-terms identities supported bythe enhanced versions
(T , b,w) ↔ (T ′, b′,w ′)
of the basic 2 ↔ 3 triangulation move. These define rationalrelations between gluing varieties supported by different simplicialcomplexes over W .
![Page 34: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/34.jpg)
a
b
c=ab
t=rs r s
![Page 35: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/35.jpg)
The Rogers dilogarithm
L(z) := −π2
6−
1
2
∫ z
0
(log(t)
1− t+
log(1− t)
t)dt
![Page 36: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/36.jpg)
The Rogers dilogarithm
L(z) := −π2
6−
1
2
∫ z
0
(log(t)
1− t+
log(1− t)
t)dt
It is complex analytic on C \ {(−∞; 0) ∪ (1; +∞)} and
L(z) = −π2
6−
1
2log(z) log(1− z) + Li2(z)
when |z | < 1.
![Page 37: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/37.jpg)
Defects of L(z)
•It is not defined on the whole of C \ {0, 1}.
![Page 38: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/38.jpg)
Defects of L(z)
•It is not defined on the whole of C \ {0, 1}.
•It is not invariant for the tetrahedral symmetries (there are “logdefects”).
![Page 39: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/39.jpg)
Defects of L(z)
•It is not defined on the whole of C \ {0, 1}.
•It is not invariant for the tetrahedral symmetries (there are “logdefects”).
•It verifies a special instance of decorated 5-terms identity:All the signs ∗b are equal to 1, and the cross ratios verify thegeometric constraint of representing two subdivisions of a positivelyoriented ideal convex octahedron.
![Page 40: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/40.jpg)
Defects of L(z)
•It is not defined on the whole of C \ {0, 1}.
•It is not invariant for the tetrahedral symmetries (there are “logdefects”).
•It verifies a special instance of decorated 5-terms identity:All the signs ∗b are equal to 1, and the cross ratios verify thegeometric constraint of representing two subdivisions of a positivelyoriented ideal convex octahedron.
We have to perform a clever analytic continuation that fixes thesedefects.
![Page 41: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/41.jpg)
W. Neumann uniformization mod (π2Z)
Facts:Consider the maximal Abelian covering
p0 : C → C \ {0, 1}
so that:
•Every point of C can be encoded in the form[w0; f0, f1, f2] ∈ C \ {0, 1} × Z2 so that
2∑
j=0
log(wj) + iπfj = 0 .
![Page 42: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/42.jpg)
W. Neumann uniformization mod (π2Z)
Facts:Consider the maximal Abelian covering
p0 : C → C \ {0, 1}
so that:
•Every point of C can be encoded in the form[w0; f0, f1, f2] ∈ C \ {0, 1} × Z2 so that
2∑
j=0
log(wj) + iπfj = 0 .
The integer triple f := (f0, f1, f0) represents a further edge decorationcalled a flattening of (w0,w1,w2), with associated log-branches
lj := log(wj) + iπfj .
![Page 43: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/43.jpg)
•The formula
L([w0; f ]) := L(w0) +iπ
2(f0 log(1− w0) + f1 log(w0))
defines an analytic function
L : C → C/π2Z
Setp0 : C
n → (C \ {0, 1})n .
![Page 44: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/44.jpg)
The induced infinite covering over the Gluing
variety
G (T , b) is the complex analytic subset of p−10 (G (T , b)) ⊂ Cn defined
by the system of edge equations, one for each edge e of T
L(e) :=∑
E→e
∗b(E )(log(w(E )) + iπf (E )) = 0
![Page 45: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/45.jpg)
The induced infinite covering over the Gluing
variety
G (T , b) is the complex analytic subset of p−10 (G (T , b)) ⊂ Cn defined
by the system of edge equations, one for each edge e of T
L(e) :=∑
E→e
∗b(E )(log(w(E )) + iπf (E )) = 0
L(e) is called the total log-branch around e.
![Page 46: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/46.jpg)
The induced infinite covering over the Gluing
variety
G (T , b) is the complex analytic subset of p−10 (G (T , b)) ⊂ Cn defined
by the system of edge equations, one for each edge e of T
L(e) :=∑
E→e
∗b(E )(log(w(E )) + iπf (E )) = 0
L(e) is called the total log-branch around e.
By restriction we get an infinite analytic covering
p0 : G (T , b) → G (T , b)
![Page 47: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/47.jpg)
The induced infinite covering over the Gluing
variety
G (T , b) is the complex analytic subset of p−10 (G (T , b)) ⊂ Cn defined
by the system of edge equations, one for each edge e of T
L(e) :=∑
E→e
∗b(E )(log(w(E )) + iπf (E )) = 0
L(e) is called the total log-branch around e.
By restriction we get an infinite analytic covering
p0 : G (T , b) → G (T , b)
We define the analytic function
L : G (T , b) → C/π2Z
L([w ; f ]) =∑
n
∗bnL([w0,n; fn] .
![Page 48: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/48.jpg)
The simplicial “Chern-Simons Function”
Facts:•Every point [w ; f ] ∈ G (T , b) represents a couple (ρ(w), h(f )) whereh(f ) ∈ H1(W ;Z/2Z).
![Page 49: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/49.jpg)
The simplicial “Chern-Simons Function”
Facts:•Every point [w ; f ] ∈ G (T , b) represents a couple (ρ(w), h(f )) whereh(f ) ∈ H1(W ;Z/2Z).
•For every (ρ, h) there is [w ; f ] ∈ G (T , b) such that(ρ, h) = (ρ(w), h(f )).
![Page 50: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/50.jpg)
The simplicial “Chern-Simons Function”
Facts:•Every point [w ; f ] ∈ G (T , b) represents a couple (ρ(w), h(f )) whereh(f ) ∈ H1(W ;Z/2Z).
•For every (ρ, h) there is [w ; f ] ∈ G (T , b) such that(ρ, h) = (ρ(w), h(f )).
•If (ρ, h) = (ρ(w), h(f )), then
L(W , ρ, h) := L([w ; f ])
is a well defined invariant.
![Page 51: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/51.jpg)
The simplicial “Chern-Simons Function”
Facts:•Every point [w ; f ] ∈ G (T , b) represents a couple (ρ(w), h(f )) whereh(f ) ∈ H1(W ;Z/2Z).
•For every (ρ, h) there is [w ; f ] ∈ G (T , b) such that(ρ, h) = (ρ(w), h(f )).
•If (ρ, h) = (ρ(w), h(f )), then
L(W , ρ, h) := L([w ; f ])
is a well defined invariant.
•If ρ lifts to SL(2,C) and h = 0, then
L(W , ρ) = CS(ρ)
this holds in particular for ρh.
![Page 52: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/52.jpg)
Tensor networks
If (ρ, h) = (ρ(w), h(f )), set:
H(W , ρ, h) := exp(2
iπL(W , ρ, h)) = exp(
2
iπL([w ; f ]))
and interprete the last term as the total contraction of a suitable“1-tensor network” carried by (T , b).
![Page 53: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/53.jpg)
•(T , b) is a network of 3-symplexes connected by a system oforiented arcs transverse to the 2-facets, and contained in the1-skeleton of the cell decomposition of W dual to (T , b). At each3-symplex, if ∗b = 1 the ordered couple of arcs at (F2, F0) areoutgoing, at (F3, F1) they are ingoing. Viceversa if ∗b = −1.
![Page 54: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/54.jpg)
•(T , b) is a network of 3-symplexes connected by a system oforiented arcs transverse to the 2-facets, and contained in the1-skeleton of the cell decomposition of W dual to (T , b). At each3-symplex, if ∗b = 1 the ordered couple of arcs at (F2, F0) areoutgoing, at (F3, F1) they are ingoing. Viceversa if ∗b = −1.
•For every N ≥ 1, (T , b) can be converted into a N-tensor networkvia the following procedure:(1) Associate to every 2-facet Fj , j = 0, 1, 2, 3, of every 3-symplex(∆, b) a copy Vj of C
N .
![Page 55: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/55.jpg)
•(T , b) is a network of 3-symplexes connected by a system oforiented arcs transverse to the 2-facets, and contained in the1-skeleton of the cell decomposition of W dual to (T , b). At each3-symplex, if ∗b = 1 the ordered couple of arcs at (F2, F0) areoutgoing, at (F3, F1) they are ingoing. Viceversa if ∗b = −1.
•For every N ≥ 1, (T , b) can be converted into a N-tensor networkvia the following procedure:(1) Associate to every 2-facet Fj , j = 0, 1, 2, 3, of every 3-symplex(∆, b) a copy Vj of C
N .
(2) Let every (∆, b) carry an operatorA = A(∆, b) ∈ End(CN ⊗CN), by encoding its matrix elements Ap,q
r ,s ,p, q, r , s ∈ {0, . . . ,N − 1}, as follows:
A =
{
(Ai ,jk,l ) : V3 ⊗ V1 → V2 ⊗ V0 if ∗b = 1
(Ak,li ,j ) : V2 ⊗ V0 → V3 ⊗ V1 if ∗b = −1 .
(1)
![Page 56: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/56.jpg)
1
0
3
23
2
1
0 k l
j i
![Page 57: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/57.jpg)
State sum
A state σ of a given N-tensor network A carried by (T , b) is alabelling of the connecting arcs by {0, . . . ,N − 1}. Every stateselects a matrix element A(∆, b)σ at each 3-symplex. The state sum
S(A) :=∑
σ
∏
(∆,b)
A(∆, b)σ
determines a scalar ∈ C and is the total contraction of A.
![Page 58: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/58.jpg)
Clearly:•H(W , ρ, h) is the total contraction of the 1-tensor network carriedby (T , b,w , f ), made by the tensors
R(∆, b,w , f ) := exp(2
iπL(∆, b,w , f ))∗b .
•These tensors verify multiplicative 5-terms identities.
![Page 59: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/59.jpg)
Non commutative invariant state sums
Problem:For N > 1, find N-tensor networks RN carried by (T , b,w , f ), sothat the state sums well define invariants HN(W , ρ, h).
![Page 60: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/60.jpg)
Non commutative invariant state sums
Problem:For N > 1, find N-tensor networks RN carried by (T , b,w , f ), sothat the state sums well define invariants HN(W , ρ, h).
One expects both a good behaviour with respect to the tetrahedralsymmetries and the verification of the multiplicative andnon-commutative 5-terms functional identities.
![Page 61: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/61.jpg)
Faddeev-Kashaev 6j-symbols
For every odd N ≥ 3, set ζ = exp(2iπ/N)
![Page 62: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/62.jpg)
Faddeev-Kashaev 6j-symbols
For every odd N ≥ 3, set ζ = exp(2iπ/N)
Via the idealization of the 6j-symbols of the cyclic representationstheory of the Borel quantum subalgebra Bζ of the quantum group
Uζ(sl(2,C))
![Page 63: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/63.jpg)
Faddeev-Kashaev 6j-symbols
For every odd N ≥ 3, set ζ = exp(2iπ/N)
Via the idealization of the 6j-symbols of the cyclic representationstheory of the Borel quantum subalgebra Bζ of the quantum group
Uζ(sl(2,C))
we get an explicit family of invertible tensors of the form
LN(∆, b, exp(1
Nlog(w))) ∈ End(CN ⊗ C
N)
![Page 64: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/64.jpg)
Faddeev-Kashaev 6j-symbols
For every odd N ≥ 3, set ζ = exp(2iπ/N)
Via the idealization of the 6j-symbols of the cyclic representationstheory of the Borel quantum subalgebra Bζ of the quantum group
Uζ(sl(2,C))
we get an explicit family of invertible tensors of the form
LN(∆, b, exp(1
Nlog(w))) ∈ End(CN ⊗ C
N)
and moreover, LN(∆, b, x) is a determined rational function of x .
[A conceptual explaination is based on De Concini-Kac-Procesi“quantum coadjoint action” theory]
![Page 65: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/65.jpg)
These tensors formally have the same properties and the samedefects of (the exponential of) the “classical” Rogers dilogarithm:
![Page 66: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/66.jpg)
These tensors formally have the same properties and the samedefects of (the exponential of) the “classical” Rogers dilogarithm:
•They are not defined over the whole of C \ {0, 1}.
•They are not invariant for the tetrahedral symmetries.
•They verify the same special instance of multiplicative 5-termsidentity.
![Page 67: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/67.jpg)
These tensors formally have the same properties and the samedefects of (the exponential of) the “classical” Rogers dilogarithm:
•They are not defined over the whole of C \ {0, 1}.
•They are not invariant for the tetrahedral symmetries.
•They verify the same special instance of multiplicative 5-termsidentity.
Once again we have to perform a clever analytic continuation thatfixes these defects.
![Page 68: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/68.jpg)
A tentative solution
The analytic continuation would lead to a solution having thefollowing features:
![Page 69: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/69.jpg)
A tentative solution
The analytic continuation would lead to a solution having thefollowing features:
•Let us enhance (T , b) with a further system of edge decorationcalled charges
(c0, c1, c2) ∈ Z3
one for each (∆, b), so that
c0 + c1 + c2 = 1 ,
and satisfying the global constraints: for every edge e of T
C (e) =∑
E→e
c(E ) = 2
C (e) is called the total charge around e.
![Page 70: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/70.jpg)
•Having fixed a charge c, then for every [w ; f ] ∈ G (T , b) we have asystem of Nth-roots
w ′
N := exp(1
N(log(w) + iπ(N + 1)(f − ∗bc)) .
![Page 71: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/71.jpg)
•Having fixed a charge c, then for every [w ; f ] ∈ G (T , b) we have asystem of Nth-roots
w ′
N := exp(1
N(log(w) + iπ(N + 1)(f − ∗bc)) .
Then the final N-tensor network RN is made by tensors of the form:
RN(∆, b,w , f ) = α(w ′
N , c)LN(∆, b,w ′
N)
where α(w ′N , c) is a “smart” simmetrizing scalar factor.
![Page 72: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/72.jpg)
Charge difficulty and actual solution
Unfortunately such charges c do not exist because of aGauss-Bonnet-like obstruction at the spherical combinatorial linkLink(T ,b)(v ) of every vertex of (T , b).
![Page 73: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/73.jpg)
Charge difficulty and actual solution
Unfortunately such charges c do not exist because of aGauss-Bonnet-like obstruction at the spherical combinatorial linkLink(T ,b)(v ) of every vertex of (T , b).
Making the theory consistent via a link fixing:•Let L be a knot or more generally a link in W and assume that(T ,H, b) is a distinguished simplicial complex over (W , L), so that His a Hamiltonian sub-complex of T (1) that realizes L.
![Page 74: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/74.jpg)
Charge difficulty and actual solution
Unfortunately such charges c do not exist because of aGauss-Bonnet-like obstruction at the spherical combinatorial linkLink(T ,b)(v ) of every vertex of (T , b).
Making the theory consistent via a link fixing:•Let L be a knot or more generally a link in W and assume that(T ,H, b) is a distinguished simplicial complex over (W , L), so that His a Hamiltonian sub-complex of T (1) that realizes L.
•A charge c on (T ,H, b) is defined as above, provided that for everye ∈ H, the total charge C (e) = 0 (instead of C (e) = 2). Note that acharge also encodes the link L.
![Page 75: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/75.jpg)
Facts:(1) Assume that (T ,H, b) is as above. Every charge c on (T ,H, b)carries a class k(c) ∈ H1(W ;Z/2Z). For every k ∈ H1(W ;Z/2Z)there exists a charge c such that k = k(c).
![Page 76: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/76.jpg)
Facts:(1) Assume that (T ,H, b) is as above. Every charge c on (T ,H, b)carries a class k(c) ∈ H1(W ;Z/2Z). For every k ∈ H1(W ;Z/2Z)there exists a charge c such that k = k(c).
(2) For every couple (W , L) there exist distinguished simplicialcomplexes (T ,H, b).
![Page 77: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/77.jpg)
Facts:(1) Assume that (T ,H, b) is as above. Every charge c on (T ,H, b)carries a class k(c) ∈ H1(W ;Z/2Z). For every k ∈ H1(W ;Z/2Z)there exists a charge c such that k = k(c).
(2) For every couple (W , L) there exist distinguished simplicialcomplexes (T ,H, b).
(3) (like in the classical case) For every couple (ρ, h) there exist[w ; f ] ∈ G (T , b) so that (ρ, h) = (ρ(w), h(f )).
![Page 78: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/78.jpg)
(4) For every (W , L), (T ,H, b), k = k(c) and (ρ, h) = (ρ(w), h(f ))as above, let RN(T , b,w , f , c) be the corresponding N-tensornetwork.
![Page 79: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/79.jpg)
(4) For every (W , L), (T ,H, b), k = k(c) and (ρ, h) = (ρ(w), h(f ))as above, let RN(T , b,w , f , c) be the corresponding N-tensornetwork.
Then, up to multiplication by a 2N-root of unity (phase anomaly),
HN(W , L, ρ, h, k) := S(RN(T , b,w , f , c))
is a well defined quantum hyperbolic invariant at level N.
![Page 80: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/80.jpg)
(4) For every (W , L), (T ,H, b), k = k(c) and (ρ, h) = (ρ(w), h(f ))as above, let RN(T , b,w , f , c) be the corresponding N-tensornetwork.
Then, up to multiplication by a 2N-root of unity (phase anomaly),
HN(W , L, ρ, h, k) := S(RN(T , b,w , f , c))
is a well defined quantum hyperbolic invariant at level N.
(5) Via the above map w ′N = exp( 1
N(log(w) + iπ(N + 1)(f − ∗bc)),
the state sum function, which is defined on G (T , b), is factorizedthrough an algebraic finite covering (of degree N2)
GN(T , b) → G (T , b)
and a determined rational regular function defined on GN(T , b).
![Page 81: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/81.jpg)
A special instance: HN(S3, L)
The theory is non trivial even at the minimal level of topologicalcomplexity:
For every link L ⊂ S3, up to the phase ambiguity,
HN(S3, L) = JN(L)(exp(2iπ/N))
where JN(L)(q) ∈ Z[q±1] is the colored Jones polynomial normalizedby JN(KU)(q) = 1 on the unknot KU .
![Page 82: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/82.jpg)
Some challenging open questions
(1) Improve or even fix the phase anomaly, possibly by introducingfurther preserved structures on the 3-manifolds. At first one expectsto improve 2N to N, by fixing at least the sign ambiguity.
![Page 83: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/83.jpg)
(2) QH Asymptotic problem, when N → +∞. Let P be anypattern such the the QHI are defined (this is allusive to the fact thatQHI are defined for further patterns such as the the cuspedhyperbolic 3-manifolds), then
H∞(P) := lim supN→+∞
(log |HN(P)|/N)
is finite.
![Page 84: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/84.jpg)
(2) QH Asymptotic problem, when N → +∞. Let P be anypattern such the the QHI are defined (this is allusive to the fact thatQHI are defined for further patterns such as the the cuspedhyperbolic 3-manifolds), then
H∞(P) := lim supN→+∞
(log |HN(P)|/N)
is finite.
(a) Understand when H∞(P) = 0 (sub-exponential case) or whenH∞(P) 6= 0 (exponential case).
(b) Which “classical” information is carried by H∞(P)?
![Page 85: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/85.jpg)
Volume Conjecture for hyperbolic knots in S3
A famous particular instance is the Kashaev-Murakami-Murakamivolume conjecture for the hyperbolic knots K in S3:
2π lim(log |JN(K )(exp(2iπ/N)|/N) = Vol(S3 \ K ) .
This is proved for a few knots including the famous figure-8-knot.
![Page 86: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/86.jpg)
(3) Study the relations between different instances of 3-dimensionalquantum invariants arising from different sectors of the wholerepresentation theory of the quantum group Uζ(sl(2,C)).
![Page 87: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/87.jpg)
Some References
S. Freed, Classical Chern-Simons Theory, 1, Adv. in Math. 113(1995) 237–303
P. Kirk, E. Klassen, Chern-Simons invariants of 3-manifoldsdecomposed along tori and the circle bundle over therepresentation space of T 2, Comm. Math. Phys. 153 (3) (1993)521–557
W.P. Thurston, The geometry and topology of 3-manifolds,Princeton Univ. Press (1979)
R. Benedetti, C. Petronio, Lectures on Hyperbolic Geometry,Universitext, Springer Verlag (1992)
P.B. Shalen, Representations of 3–manifold groups, Handbookof geometric topology, North–Holland, Amsterdam (2002)955–1044
![Page 88: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/88.jpg)
S. Baseilhac, R. Benedetti, Quantum hyperbolic invariants of3-manifolds with PSL(2,C)-characters, Topology 43 (6) (2004)1373–1423
S. Baseilhac, R. Benedetti, Classical and quantum dilogarithmicinvariants of flat PSL(2,C)-bundles over 3-manifolds, Geom.Topol. 9 (2005) 493–570
S. Baseilhac, R. Benedetti, Quantum hyperbolic geometry, Alg.Geom. Topol. 7 (2007) 845–917
S. Baseilhac, R. Benedetti, The Kashaev and quantumhyperbolic invariants of links, J. Gokova Geom. Topol. 5 (2011)31–85
S. Baseilhac, R. Benedetti Analytic families of quantumhyperbolic invariants and their asymptotical behaviour, I,arXiv:1212.4261v1
![Page 89: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/89.jpg)
L. Lewin, Polylogarithms and associated functions,Elsevier(1981)
W. D. Neumann, Combinatorics of triangulations and theChern-Simons invariant for hyperbolic 3-manifolds, Topology’90(Columbus, OH, 1990), De Gruyter, Berlin (1992) 243–271
W.D. Neumann, Extended Bloch group and theCheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413–474
W.D. Neumann, D. Zagier, Volumes of hyperbolic 3-manifolds,Topology 24, 307–332 (1985)
![Page 90: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/90.jpg)
L.D. Faddeev, R.M. Kashaev, A.Yu. Volkov, Strongly coupledquantum discrete Liouville theory. I: Algebraic approach andduality, Comm. Math. Phys. 219 (1) (2001) 199–219
R.M. Kashaev, Quantum dilogarithm as a 6j-symbol, Mod.Phys. Lett. A Vol. 9 (40) (1994) 3757–3768
R.M. Kashaev, A link invariant from quantum dilogarithm, Mod.Phys. Lett. A Vol. 10 (40) (1995) 1409–1418
![Page 91: An introduction to quantum hyperbolic geometrymanifolds.sns.it/2013/posters/51345b5f958cetalk...An introduction to quantum hyperbolic geometry From joint works with S. Baseilhac -](https://reader033.vdocuments.mx/reader033/viewer/2022052720/5f08d6a57e708231d423f7c7/html5/thumbnails/91.jpg)
R. M. Kashaev, The hyperbolic volume of knots from thequantum dilogarithm, Lett. Math. Phys. 39 (1997) 269–275
H. Murakami, J. Murakami, The colored Jones Pulynomials andthe simplicial volume of a knot, Acta Math. 186 (2001), 85-104
F. Costantino, 6j–symbols, hyperbolic structures and the volumeconjecture, Geom. Topol. 11 (2007) 1831–1853