Tilings: Dessins, K3 Surfaces & Modular Group
YANG-HUI HE
Dept of Mathematics, City University, London;
School of Physics, NanKai University, China;
Merton College, University of Oxford, UK
ICMS, Edinburgh, May, 2014
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Acknowledgements
1201.3633 YHH, J. McKay and 1211.1931 YHH, J. McKay J. Read
1308.5233 YHH, J. McKay and 1309.2326 YHH, J. Read
1402.3846 YHH, M. van Loon
1402.xxxx S. Bose, J. Gundry, YHH
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Part I
Dramatis Personæ
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Platonic Solids
5 regular solids: (4,6,8,12,20)-sided (cf. McKay’s ADE Correspondence)
Proof: Schläfli Symbol:
{p, q} = (#edges/vertices per face, #faces/edges per vertice)
Each edge has 2 adjacent faces and vertices: pF = 2E = qV
Euler: V − E + F = 2
∴
1
p
+
1
q
=
1
2
+
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1
E
; 5 solutions (3, 3), (4, 3), (3, 4), (5, 3), (3, 5)
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Archimedean Solids
Each vertex has identical types of regular faces meeting X = (p1 , p2 , . . . , pq )
Proof: Let mp = multiplicity of p in X; so Fp = mp V /p and
q
q
P
P
P
1
1
F = p Fp = V
;
still
2E
=
qV
and
Euler;
∴
pi
pi =
i=1
i=1
q
2
−1
Three Types of solutions: (I) The 5 Platonics (II) Two (trivial) infinite series:
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(III) The 13+1 Archimedean Solids:
RM K :
+1 since pseudo.
no global isometry
Egyptian Fractions:
P 1
i pi
Erdös-Straus Conjecture
4
n
=
1
x
+
1
y
+
1
z
∃x, y, z ∈ N ∀ n ≥ 2
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Tessellations of Riemann Surfaces
Platonic/Archimedean solids ; Regular/Semi-Regular “tilings” or
tessellations of the sphere
In general, embed a graph into Riemann surface Σ, with symetry G
semi-regular tessellation if G acts transitively on vertices
regular tessellations if G acts transitively on {vertex, edges incident on vertex,
tile/face incident on vertex} (called flag)
Proof of finiteness as before, replace 2 − 2gΣ in Euler
g = 1 doubly-periodic tessellation of plane:
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Regular
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Semi-Regular
(cf. wiki)
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Modular Group & Cayley Graphs
Modular Group: Γ := P SL(2; Z) ' hS, T S 2 = (ST )3 = Ii
free product C2 ? C3 , C2 = hx|x2 = Ii and C3 = hy|y 3 = Ii.
Cayley Graph: nodes = group elements, arrows = group multiplication ;
free trivalent tree with nodes replaced by directed triangles
Finite index subgroups of Γ
Finite number of cosets
each coset ; node, arrows = group multiplication ⇒
coset graphs (finite directed trivalent graph): Schreier-Cayley Graphs
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An Index = 6
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Example
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Congruence Subgroups
Principal congruence subgroups:
Γ (m) := {A ∈ SL(2; Z) ; A ≡ ±I mod m} / {±I} ;
Congruence subgroups of level m: subgroups of Γ containing Γ (m) but not
any Γ (n) for n < m;
Unipotent matrices:



1
Γ1 (m) := A ∈ SL(2; Z) ; A ≡ ± 

0


 mod m / {±I} ;

1
b

Upper triangular matrices:



 a b

 ∈ Γ ; c ≡ 0 mod m / {±I} .
Γ0 (m) := 
 c d

Have: Γ (m) ⊆ Γ1 (m) ⊆ Γ0 (m) ⊆ Γ
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Finite Index, Genus 0, Torsion Free Subgroups
Torsion Free: nothing except I of finite order
Genus Zero: upper half plane H can quotient G ⊂ Γ ; Modular Curve ΣC
Γ/H ' P1 (upto cusps)
so what subgroup also gives P1 ? i.e. genus(ΣC ) = 0?
RARE & relevant to Moonshine (dim of Monster ∼ Klein-j;
McKay-Thompson series ∼ Hauptmoduln)
Complete classification by Sebbar (2003): torsion-free, genus 0: only 33
The 33 genus 0 torsion free subgroups of Γ, all are index 6, 12, 24, 36, 48, 60
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Hecke Groups
Generalize the modular group (Γ = H3 )
Hn := x, y|x2 = y n = I = C2 ? Cn




0 −1
0 −1
 , y=
 ,
x=
1 0
1 λn
λn =
q
2 + 2 cos
2π
n
λn are algebraic numbers
Congruence Subgroups I is an ideal of Z[λn ]
Hn (I)
1
Hn
(I)
0
Hn
(I)

 a

 c

 a
= Hn ∩ 
 c

 a
= Hn ∩ 
 c
= Hn ∩
b
d
b
d
b
d

 ∈ PSL (2, Z [λn ]) ; a − 1, b, c, d − 1 ∈ I




 ∈ PSL (2, Z [λn ]) ; a − 1, c, d − 1 ∈ I




 ∈ PSL (2, Z [λn ]) ; c ∈ I



Have: Hn (I) ⊆ Hn1 (I) ⊆ Hn0 (I) ⊆ Hn
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Grothendieck’s Dessin d’Enfant
Belyı̌ Map: rational map β : Σ −→ P1 ramified only at (0, 1, ∞)
Theorem [Belyı̌]: β exists ⇔ Σ can be defined over Q
(β, Σ) Belyı̌ Pair
Dessin d’Enfants = β −1 ([0, 1] ∈ P1 ) is a bi-partite graph on Σ: label β −1 (0)
black and β −1 (1) white,
then β −1 (∞) lives one
per face


Ramification data:


 r0 (1), r0 (2), . . . , r0 (B)
r1 (1), r1 (2), . . . , r1 (W )



r∞ (1), r∞ (2), . . . , r∞ (I)






equivalently, Permutation Triple: σB , σW and σB σW σ∞ = I (encodes how
the sheets are permuted at the ramification points)
σB = (. . .)r0 (1) (. . .)r0 (2) . . . (. . .)r0 (B),
σW = (. . .)r1 (1) (. . .)r1 (2) . . . (. . .)r1 (W )
Cartographic group: hσB , σW i
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Elliptic Surfaces
Consider complex surface S → P1 with elliptic fibration:
{y 2 = 4x3 − f (s)x − g(s)} ⊂ C[x, y] ,
Klein J-invariant J(s) =
f (s)3
1
1728 f (s)3 −27g(s)2
s ∈ P1
is rational map
When is J(s) : P1 → P1 Belyi?
Whenit is ; trivalent,
clean dessin (black 3 and white 2) with ramification

data
dg


 3
2df



n1 , n 2 . . . , n t






Riemann-Hurwitz on Σg=0 → Σg=0 ;
d = 3df = 2dg =
X
ni ;
df + dg + t = d + 2
i
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Modular Elliptic Surfaces
Recall: γ ∈ Γ y H by τ 7→ γ · τ =
aτ +b
cτ +d ,
with γ =
a
c
, det γ = 1
d
b
Extend and twist (Shioda, 1970’s):
(γ, (m, n)) ∈ Γ o Z2 y H × C :
(τ, z) 7→
aτ + b z + mτ + n
,
cτ + d
cτ + d
Quotient: (H × C)/(Γ o Z2 ) is a complex surface which is fibred
Base: H/Γ = the modular curve ΣC
Fibre: (generically) C/Z2 ' a torus
Get a surface elliptically fibred over ΣC : modular elliptic surface with complex
parametre τ (torus T 2 ' C/(mτ + n))
Take finite index genus 0 subgroup of Γ: base is P1 ⇒ elliptic surfaces over
P1 of Euler number = index of group!
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Case 1: Genus 0, Torsion Free, Modular K3
Take euler number = 24 ; K3 surfaces elliptic over P1 (9 of these)
Fibration: Dependence on base s coord J(s) : P1C −→ P1 ; rational map
Ia
IIIa
Ib
IIIb
IIa
IIb
IIIc
IV
IIId
• YHH, McKay 1201.3633
• J(s) are Belyi: dessins on modular curve P1 = Coset graphs
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Case 2: Extremal Semi-Stable K3
Classified by Miranda-Persson (1989) 112 of these
extremal: max Picard number (= 20)
semistable: all singular fibres are of Kodaira type In
Each elliptic curve ; modular-invariant j (inspired by Moonshine)
[ni ] a partition of 24
Beukers-Montanus (2008): all ; trivalent clean dessins
YHH, McKay, Read, 1211.1931: find all corresponding modular subgroup G:
G = Cartographic Group = Finite index subgroup (via coset graph)
mostly non-congruence groups (difficult subject)
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Grothendieck, 1984
This discovery, which is technically so simple, made a very strong impression
on me, and it represents a decisive turning point in the course of my
reflections, a shift in particular of my centre of interest in mathematics, which
suddenly found itself strongly focussed. I do not believe that a mathematical
fact has ever struck me quite so strongly as this one, nor had a comparable
psychological impact.
This is surely because of the very familiar, non-technical nature of the objects
considered, of which any child’s drawing scrawled on a bit of paper (at least if
the drawing is made without lifting the pencil) gives a perfectly explicit
example. To such a dessin we find associated subtle arithmetic invariants,
which are completely turned topsy-turvy as soon as we add one more stroke.
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Part II
Bipartite Graphs, Quivers and
Gauge Theories
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Quiver Gauge Theories & Toric CY3
2000 Feng-Hanany-YHH, Observatio Curiosa∗ : Ng − Nf + Nw = 0
2005 Hanany-Kennaway, Franco-Hanany-Kennaway-Vegh-Wecht,
Benvenuti-Franco-Hanany-Martelli-Sparks, Feng-Kennaway-YHH-Vafa
D3 quiver theory / toric CY3 ' double-periodic bipartite tilings of the plane
dimer models or brane tilings
R-charge ; *; bipartite (each field appears twice with +/- sign) ; Toric
(binomial ideal)
2013 Franco et al: BFT (Bipartite field theories)
relation to Arkani-Hamed’s reducing N = 4 scattering amplitudes to bipartite
graphs on disk (via positive cells in the Grassmannian)?
genus other than g = 1
Cremonesi-Hanany-Seong g = 2
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Dimers as Dessins
Jejjala-Ramgoolam-Rodriguez-Gomez (1012.2351): represent toric quiver
theories (dimers) as permutation triples (i.e., dessins) on elliptic curve
Hanany-YHH-Jejjala-Ramgoolam-Rodriguez-Gomez (1104.5490, 1105.3471,
1107.4101): compute the complex structure τ and its Klein j-invariant J(τ )
Many interesting observations and puzzles
Like R-charges, τ are algebraic numbers (initial motivation of Grothendieck)
Elliptic curves of Seiberg duals are isogenous (equal j)
transcendence deg over Q inv
relations between τ (R-charge max), τ (dessin) and τ (mirror)??
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Revisiting Familiar Examples
Matter content and interaction of SUSY gauge theory with toric moduli
space is specified by Belyı̌ pair
Our most familiar example of N = 4 super-Yang-Mills:
Theory
Toric Diag
x
Belyı̌ Pair
Dessin on T 2 (dimer)
1
NODE = Gauge Group 2
y = x3 + 1
ARROW = Bi−fundamental (Adj) Field y+1
β(x, y) = 2
y
z
M ' C3
W = Tr(X[Y, Z])
1
1
1
1
1
1
1
1
1
1
Rmk: Absolute Galois group Gal(Q/Q) acts faithfully over the dessin, even
on subsets like dessin on P1 or T 2 ...
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Klebanov-Witten’s Conifold Theory
Theory
Toric Diag
(0,1,1)
(1,1,1)
(1,1,1)
>>
A 1,2
>>
B 1,2
(0,0,1)
DIAGRAM
(1,0,1)
W = Tr(
QUIVER
il jk Ai Bj Al Bk )
M ' {uv − wz = 0} ⊂ C4
TORIC DIAGRAM
Belyı̌ Pair
Dessin on T 2 (dimer)
(1,0,1)
>>
A 1,2
1
1
y = x(x − 1)(x −
β(x, y) =
1
2)
B1 1
2
x
2x−1
1
1
2
A2
1
2
1
2
1B2
2
2
B 1,2
1
A12
2
2
1
2
1
2
>>
2
2
Q
QUIVER
1
2
Graph Dual
1
2
1
Dimer Model on Torus
DIMER
Fundamental Region
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Plethora of Non-Trivial Examples
e.g., Cone over F0 ' P1 × P1 (zeroth Hirzebruch surface);
Toric Diag
Theory
3
2
1
4
W
=
1 X4 X2 + X2 X2 X1 + X1 X1 X1 +
X14
43 31
14 43 31
24 43 32
2 X3 X2 − X1 X1 X1 − X2 X2 X1 −
+X24
43 32
43 31 14
43 32 24
3 X2 X2 − X4 X1 X2
−X43
43 32 24
31 14
M 'Hirzebruch 0
Dessin on T 2 (dimer)
Belyı̌ Pair
2
y 2 = x − x3
√
3
i(x2 − 3 −1)
β(x, y) = 3√3x2 (x2 −1)
2
2
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4
1
2
3
4
4
3
1
1
1
2
4
3
4
3
1
2
4
1
2
4
3
1
3
3
2
4
3
2
3
1
4
1
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Part III
Correspondences
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genus 0: dessins on the Sphere
Can immediately impose bipartite structure on some. e.g. cube
M'C
W = Tr(Φ1 Φ2 Φ3 + Φ4 Φ5 Φ6 + Φ7 Φ8 Φ9 + Φ10 Φ11 Φ12 − Φ1 Φ4 Φ12 − Φ2 Φ7 Φ6 − Φ3 Φ10 Φ9 − Φ5 Φ8 Φ11 )
can do so for all Platonic and Archimedeans which admit bipartite structure
; cube and truncated cube
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genus 0: clean dessins on the Sphere
Or impose clean dessin (insert bi-valent node) e.g.,
e.g., Cube ∼ Γ(4) ∼ K3 = {x(x2 + 2y + 1) +
j(s) =
YANG-HUI HE (London/Tianjin/Oxford)
s2 −1
2
s2 +1 (x
− y 2 ) = 0}
16(1 + 14s4 + s8 )3
s4 (s4 − 1)4
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Archimedean Clean Dessins
Name
V
Vertex type
F
E
Sym
Trivalent?
Hecke Group
I
Tetrahedron
4
(3, 3, 3)
4
6
Td
Yes
I
Cube
8
(4, 4, 4)
6
12
Oh
Yes
Γ ∼
= H3
Γ ∼
= H3
I
Octahedron
6
(3, 3, 3, 3)
8
12
Oh
No
I
Dodecahedron
20
(5, 5, 5)
12
30
Ih
Yes
I
Icosahedron
12
(3, 3, 3, 3, 3)
20
30
Ih
No
II
n-prism
2n
(4, 4, n)
n+2
3n
Dnh
Yes
II
n-antiprism
2n
(3, 3, 3, n)
2n + 2
4n
Dnd
No
III
Truncated tetrahedron
12
(3, 6, 6)
8
18
Td
Yes
III
Truncated cube
24
(3, 8, 8)
14
36
Oh
Yes
III
Truncated octahedron
24
(4, 6, 6)
14
36
Oh
Yes
III
Truncated isocahedron
60
(5, 6, 6)
32
90
Ih
Yes
III
Truncated dodecahedron
60
(3, 10, 10)
32
90
Ih
Yes
III
Truncated cuboctahedron
48
(4, 6, 8)
26
72
Oh
Yes
Γ ∼
= H3
Γ ∼
= H3
III
Truncated icosidodecahedron
120
(4, 6, 10)
62
180
Ih
Yes
Γ ∼
= H3
III
Cuboctahedron
12
(3, 4, 3, 4)
14
24
Oh
No
H4
III
Icosidodecahedron
30
(3, 5, 3, 5)
32
60
Ih
No
H4
III
Rhombicuboctahedron
24
(3, 4, 4, 4)
26
48
Oh
No
H4
III
Rhombicosidodecahedron
60
(3, 4, 5, 4)
62
120
Ih
No
H4
III
Pseudorhombicuboctahedron
24
(3, 4, 4, 4)
26
48
D4d
No
H4
III
Snub cube
24
(3, 3, 3, 3, 4)
38
60
O
No
H5
III
Snub dodecahedron
60
(3, 3, 3, 3, 5)
92
150
I
No
H5
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H4
Γ ∼
= H3
H5
Γ ∼
= H3
H4
Γ ∼
= H3
Γ ∼
= H3
Γ ∼
= H3
Γ ∼
= H3
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genus 1: brane tilings
Our familiar myriad of toric AdS4 /CF T5 theories, C3 , conifold, Y p,q , La,b,c ..
Of these, the (semi-)regular planar tessellations are
hexagonal
C3
square
Conifold
Trucated Square
HirzebruchII
Truncated trihexagonal
del Pezzo 3IV
Regular
Semi-Regular
Many more for higher genus Σ
moduli space = CY2g+1
Finite in number
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Summary
Physics
Combinatorics
Bipartite Graphs on Σ
g
Dessin
d’Enfants
Algebraic Geometry
encode
SUSY (Quiver)
Gauge Theory
moduli
space
(Dimer Model/Brane Tiling)
general
Toric CY2g+1
Hecke Group
semi−regular tessellations
an interesting family
trivalent
regular
Modular Group
includes N=4, Coni & abelian orb
& congruence subgp
g>1
g=1
Elliptic K3
j (τ) Belyi
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Bipartita
Modular
Surfaces etc
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