[PPT] - Continuum Branching Observable in Higher Genus (Based on Discussion PowerPoint Presentation

SLIDE 1

Continuum Branching Observable in Higher Genus

(Based on Discussion with Nicolai Reshetikhin)

Matthew Bernard

mattb@berkeley.edu Advanced Computational Biology Center, Berkeley

Supported by The Lynn Bit Foundation, State of California

SLIDE 2

Abstract For all fjxed suffjciently large genus g=0, 1, ⩾2, multiedge connected, dual graph, we give a uniform bipartite observable of Grassmann kernel transfer matrices. On special hexagonal domain, we prove discriminant steepest descent of Grassmann kernel logarithmic asymptotics, and free Dirac Fermion convergence Ψ12×(1+O(1)). We conjecture: In large deviation functional, the Green’s function G for Dirichlet problem of variational principle minimizer is observable in the kernel asymptotics. Keywords: Continuum-branching, higher-genus, observable

2

SLIDE 3

1 Characterizations

Bipartite implies no adjacent-black (-white) vertices for all V

X = V

X ⊔ V
X :

V

X =
i•

ξ∩D: ξ=η(i• ξ, i• η, D)=∅

; D=
ξ={

iξ, jξ}: i=j, |iξ∩D|=1

.

Instance.

1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

. Non-instance.

1 2 3 4 5 6 7 8 9 10 11 12

no bipartite structure for triangular grids

.

3

SLIDE 4

Derivation. Rn r.v.X=(Xiξjξ; iξ, jξ∈ V

X) is Gaussian G ifg X=µ a.s., for

E[exp(it′X)] = exp

it′µ − 1

2t′Σt

µξ = E[Xiξjξ], Σξη = cov[Xiξjξ, Xiηjη]

⇐ ⇒ P{X ∈ dx} = 1

(2π)n det Σ

exp −(X − µ)T Σ−1 (X − µ) 2

dx

ifg t′X =

ξ tξXiξjξ is R-G, ∀ t ∈ Rn, where X is R-G ifg X=µ a.s., for

E[exp(itX)] = exp

iµt − Σt2

2

µ = E[X], Σ = var[X], ∀ t ∈ R

⇐ ⇒ P{X ∈ dx} = 1 √ 2πΣ exp

− (X − µ)2

2Σ

dx

ifg Xiξjξ ∈X is independent (Σξ,η=ξ = 0) ⇐ ⇒ X is Rn-G; where X is absolutely continuous ifg Σ is non-singular.

4

SLIDE 5

Derivation. UX d =X ifg centered X ∈Rn, Hermitian H |U =e

√−1H, UTU =UUT =I;

ifg

X X = X n

ξ=1

X2

iξjξ

is uniformly distributed on Sn−1 =

x ∈ Rn: x =1
where X is standard ifg centered (µ=0) and Σ=I | Xiξjξ ∼ N(0, 1).
Derivation. For n centered Gaussian, resp. Maxwell, particle velocity X,

E[itX] =

n

ξ=1

Φ(tξ) = Φ

t2

1 + · · · + t2 n

Φ(t) = exp
− 1

2Σt2

resp. E[itX] =

3

ξ=1

Φ(tξ) = Φ

t2

1 + t2 2 + t2 3

Φ(t) = exp
− 1

2Σt2 , Σ⩾0.

Remark. Taking n −

→ +∞, for support S, continuous density f(x): − 1 n log f ⊗n(Xi1j1, . . . , Xinjn)

a.s.

− → E [− log f(X)] = −

S

f(x) log f(x) = 1 2 log(2πeΣ).

5

SLIDE 6

1.1 Partition

Defjnition. An embedding X⊂Mg | V

X =

iξ∩D:

ξ=η(iξ, iη, D)=∅

is

partition σ∈Aut(D) ifg perfect-matching D=

ξ={

iξ, jξ}: i=j, |iξ∩D|=1

;

D={D, ∀ ξ}, Mg orientable compact, X closed, g ≫.

1 2 3 4 5 6 7 8

That is, σ implies 1 2

∂D= V

X

=

|Aut(D)| · (|{ σ}|)−1 exp

n ln 2 + n−1

k=1 ln k

=

ξ

σiξjξ

σiξjξ =

1 if ξ ∩ D 0 otherwise

V

X =(iξ∩D: |iξ|=2n); σ=(σ1, . . . , σ2n),

σ = σ|σ2ξ>σ2ξ−1; ξ, n∈N+ where X ⊂ Mg is CW cell-complex i.e. face ≈ topological disk i.e. no hole.

6

SLIDE 7

Derivation. (i) {[σ]} ∼

= (Sn×Sn

2 )(Aut(D)/(Sn×Sn

2 )) ∼

= [σ] ∼ = { σ}

(ii) |{ σ}|1/|D| ⩽

(2n)! · 2−((1/ε) mod c(X)) · e ln (a(X) · b(X))1/(2|D|)

where min(deg(X)) ⩾ n! · a(X) · b(X) ⌊2n−3⌋!! = ⌊n⌋−2

k=0

2k+1

a, b, c ∈ R+; n ⩾ 2;

Xiξjξ ≡ Xσ2ξ−1σ2ξ Sn∼ ={(σ1, . . . ,σ2n), . . . , (σ2n−1,σ2n, . . . ,σ1,σ2)}; σ ∈ S2n− →Im(Aut(D)) Sn

2 ∼

= {(σ1, . . . , σ2n), . . . , (σ2, σ1, . . . , σ2n, σ2n−1)}; [σ]={σ

Aut(D)}

(iii) (2n)(2n−1) 2 = 2 (2n)! n!2n

= |E

X| holds for complete graph X =K2n

n objects:

1 2 4 5 6 9 10 11 13 14 15 16 19 29 30 66

(regular) hexagonal grid domains.

1 2 3 4 5 6 7 8 9 10 11 12

square grid domains.

7

SLIDE 8

By E[σiξjξ σiηjη] = E[σiξjξ] ifg ξ = η, resp. zero if

ξ=η(iξ, iη, D) = ∅ i.e.

dimers of D sharing vertex: The local observable is dimer-dimer correlation i.e., for (Boltzmann) weights ωη, the conditional probability

k
η=1

σiηjη

def

= = Prob(i1∩D1, j1∩D1, . . . , ik∩Dk, jk∩Dk) = E

k
η=1

σiηjη

which equals
Dη

k

η=1

σiηjη × Prob(Dη) =

Dη

k

η=1

σiηjη

ξ ∩ Dη

ωξ

D
ξ ∩ D

ωξ = 1 Z × Z(η | η=1,...,k) = 0 ⇐ ⇒ 1 Z

Dη ∩ (i1 = i2,...ik,j1,...,jk)

ωDη = Prob(D) ⇐ ⇒ D =

Dη
η

{iη, Dη}

ωD=
ξ ∩ D

ωξ =

ξ ∩ D

e−

Ξξ K T = e− ΞD K T

ΞD=

ξ ∩ D

Ξξ, Z def = =

D
ξ ∩ D

ωξ for strict-sense positive partition function Z on dimer energy Ξ : E

X −

→ R+ | (iξ∩D, jξ∩D) − → Ξξ.

8

SLIDE 9

Defjnition. The space HX of height function hD or h is the whole of Z:

HX

def

= = { { {hD : F

X −

→ Z Z Z} } } | D ← − Bipartite surfaces h(F

i) =

h(F

i−1) + 1/3 if i• ξ is left on crossing ξ ∩ D

h(F

i−1) − 1/3 if i◦ ξ is left on crossing ξ ∩ D ; h(F 0) = 0.

Derivation. HX is given on the bipartite hexagonal X⊂R2 by:

h + 1

3

h + 2

3

9 5 13 10 16 14

h h + 1

3 ,

h h + 1

3

9 5 13 10 16 14

h + 2

3

h , h + 2

3

h + 1

3

9 5 13 10 16 14

h h + 2

3

for any perfect-matching D ∈ D, and base-face normalization hD(F

0)=0.

Theorem. For all HX: (i) hD =h i.e. hX is path (T∗) and D independent.

(ii) Curl sum dX =

F

dF =

F
ξ: ξ ∩ ∂F

ωiξjξ = 0 ifg F

X is all co-cycles.

9

SLIDE 10

Proof Follows by divergence-free notion on X, that is, d∗

iD1D2= d∗ iD1− d∗ iD2= 0 ifg F X is all co-cycles;

d∗

iD = d∗ i =

j
ξ

ωiξjξ

fmow ωiξjξ = −ωjξiξ =

+1 if i: i•

ξ ∩ D

−1 if i: i◦

ξ ∩ D

0 otherwise.

Defjnition. Skew plane partition is sequence {λ | λ⊃µ} of diagonal slices

λ(t) =

πi, i+t ∈ N | i ⩾ max(0, −t), ∀ t ∈ Z
for fjnite monotone (πij ⩾ πi+r, j+s; r, s ⩾ 0) array in the generalized 3D

partition array π=(πij : (i, j)∈N2|πij =0, ∀ i+j ≫0) of X∗ cubes πij.

Remark. π is uniquely determined by X∗ bijection (projection) map

R3 − → R2 ⊃ {(t, h)}: t = y − x, h = z − (y + x)/2, ∀ (x, y, z) ∈ R3 for all cubes mod Z3

⩾0 projection, with boundary (base) condition (0, 0, 0).

The centers of the horizontal hexagonal tiling is given by. πC =

i − j, πij − (i + j − 1)/2
⊂ Z × 1

2 Z.

10

SLIDE 11

Cubes: 2D mixing algorithm

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 39 41 42 43 44 45 46 47 48 51 52 57 59 60 62 64 65 66 68 69 70 71 72 74 76 78 79 80 81 83 84 85 90 91 100 102 103 105 108 113 123 125 126 127 136 137 138 139

11

SLIDE 12

1 2 3 4 5 6

D (left)

HX (right)                       0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0                      

12

SLIDE 13

Proposition (combinatorial correspondence).

Dimers on X
∼

= ∼ = ∼ = bijection

height functions
.
Proof. For all D ←

− Discrete surfaces, with spanning dual trees T∗, family (Dimers) ← → family (Tilings). In particular, if X ⊂ R2 = planar (no intersected edge) orientable, then (i) 2D cell complex XR2 =X ⊂ R2: 0-cells, 1-cells, 2-cells = vertices, edges, faces, resp.

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Disjoint interiors. ∂X(k) = ((k−1)mod2)-cell ∂X(k) = boundary of two k-cells, k = 0, 1, 2.

Remark. XMg : 1-skeleton CW complex (orientable compact decomposition).

13

SLIDE 14

(ii) 2D dual cell complex X∗: 0-cells,1-cells,2-cells = resp.“centers”of 2-cells,1-cells,0-cells of X.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 25 26 27

X∗ = dual cell complex to X. (iii) For a dimer on X : Unique pair of 2-cells on X∗ share:

1 2 3 4 7 10 11 1 2 3 4 7 10 11 15 21 22 23 25

(iv) Therefore, the global bijection: (Dimers on X) ← → Tilings of X∗ by unique pair of 2-cells

.

□

14

SLIDE 15

Remark. On bipartite graph, two-color tiles are admissible:

1 2 3 4 7 10 11 15 21 22 23 25 1 2 3 4 7 10 11 15 21 22 23 25

(Below: one-color tiles to the left, and two-color tiles to the right)

43 35 44 31 14 15 31 32 35 34 10 13 35 34 44 45 16 19

.

15

SLIDE 16

Cubes: 2D rhombus tiling 3D projection π =

2 1

1 0

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 39 41 42 43 44 45 46 47 48 51 52 57 59 60 62 64 65 66 68 69 70 71 72 74 76 78 79 80 81 83 84 85 90 91 100 102 103 105 108 113 123 125 126 127 136 137 138 139

16

SLIDE 17

Lemma. Prob(D) = 1 Z

F

qhD(F

)

F

Z =
D
F

ξ ∩ ∂F

ω

εK

ξ (F

) ξ

hD(F

)

where qF=“essential” invariant parameter; εK

ξ , ε∂F= ε∂X, ε− ε∂F are edge,

fjxed (counterclockwise) boundary, counter orientation, ∀ F∈ F

X | iξ = jξ;

εK

ξ (F

) = −1 ≡ ξւ(F ) (or +1 ≡ ξր(F )) if εK

ξ ∈ ε− ε∂F (resp. εK ξ ∈ ε∂F).

Proof. Prob(D) is well-defjned.

□ Theorem. D ∼ h: F

X −

→Z

Prob(h) = 1

Z

F

qh(F

)

F

, Z =

h
F

ξ ∩ ∂F

ω

εK

ξ (F

) ξ

h(F

)

.

Proof. Follows by prior proposition and lemma.

□

Remark. Prob(D) = “gauge” invariant measure: ωξ −

→ s(ξ+) ωξ s(ξ−). Cases. (i) Uniform distribution: qF = 1 = = a−1b c−1a b−1c.

2 4 9 13 10 5

b c

a

b c

a

17

SLIDE 18

(ii) Prob(h) ∝

t

q|π(t)|

t

, qF = qt, |π| =

i
j

πij. Here, π =   3 2 1 2 1 0 1 0 0  .

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 87 90 91 92 93 94 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 123 124 125 126 127 128 129 130 131 132 136 137 138 139 140 141 142 145 146

x t

18

SLIDE 19

Remark. For 0<qt<1,

Prob(π) ∝

t

q|π(t)|

t

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 123 124 125 126 127 128 129 130 131 132 136 137 138 139 140 141 142 145 146

x t

19

SLIDE 20

Remark. For 0<qt<1,

the unbounded stack: Prob(π) ∝

t

q|π(t)|

t

.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 123 124 125 126 127 128 129 130 131 132 136 137 138 139 140 141 142 145 146

x t

20

SLIDE 21

1.2 What is known 1.2.1 Order of + + + and − − − Pfaffjans in Z Z Z for fjxed g⩾0 g⩾0 g⩾0

Kasteleyn (1963). For g=0, Z = ± Pfaffjan of Kasteleyn matrix. Kasteleyn (1963). For g=1, Z = linear in 4 Pfaffjans; 3“+”, 1“−”. Kasteleyn (1963). For g ⩾ 2, Z = conjecture: 22g Pfaffjans, appearing mysteriously i.e. proof was not given, at least not published.

1.2.2 Combinatorial representation of + + + and − − − in Z Z Z

Gallucio & Loebl (1999). Z := ±1; Mg compact orientable. Tesla (2000). Z :=√−1 and ±1; Mg non-orientable. Cimazoni & R. (2004, 2005). Z := ±1 by spin structure. Cimasoni (2006). Z := √−1 by pin-minus structure for double-cover; Mg non-orientable; a Tesla (2000) topological model ∼ = spin structure’s ±1.

21

SLIDE 22

1.2.3 Asymptotics of bipartite observable (Pfaffjans)

R. et al. (2006). For height functions h ∈ Z, face-weights qF, ∀ g⩾2,

Z(bipartite) = Const. ×

h
F

qh(F

)

F

Z = 1

2g

T ∈ S(Mg)

Arf(qK

T ) · Pf(XK T ).

And, as |X|− →∞, qF− →1, in Seiberg-Witten conjecture (Gaussian fjeld theory) entropy, Z is scaling-limit path integral: Z =

exp
− 1

2

Mg

(∂Φ)2 d2x +

Mg

λ(x) Φ(x)

where the term qh(F

) F

contributes to R.H.S R.H.S R.H.S linear multiple λ(x) Φ(x) by: qx = ξ−ε · λ(x)

ε = lattice step; λ = logarithmic scale, as ε −

→ 0. Moreover, in Alvarez-Gaumé, Moore, Nelson & Vafa (1986), studying Fermi and Bose partition correspondence on Riemann surfaces, R.H.S. R.H.S. R.H.S. ∼

T ∈ S(Mg)

Arf(T) × |Θ(z |T)|2

ω determines z.

22

SLIDE 23

Remark. Conjecture (critical-weight): In large thermodynamic scaling limit

asymptotics, the observable decaying linearly goes to eVolume × the free energy eVolume × the free energy eVolume × the free energy where the next leading term is sum of Θ functions, such that the square of each Θ function is next leading asymptotics of each of the Pfaffjans. The conjecture was confjrmed by: (i) Ferdinand (1967). On square-grid torus. (ii) Costa-Santos & McCoy (2002). Numerically: Arf(T) × |Θ(z |T)|2 , ∀ g⩾2 . That is, the conjecture works, but no proof yet i.e. still a conjecture.

Remark. (i) Z is glueable (summable) on boundary spins of bounded surface.

(ii) “Higher” spin-structure is unknown, perhaps a para-polynomial theory. (iii) Observable, unlike d log(ω-system), is non-deterministic sophistication.

23

SLIDE 24

Goal

1. Operators

(i) Prove Z invariant for all multiedge connected, genus g bipartite T∗ (ii) Prove the O(n3) observable for all fjxed suffjcient-large genus g⩾0

2. Vertex algebras

(i) Prove Grassmann kernel convergence for special genus g domain T∗ (ii) Obtain the R logarithmic scaling asymptotics by variational principle (iii) State conjecture for the Green’s function · in large-deviation

24

SLIDE 25

1.3 Orientation

Defjnition. 1-skeleton CW complex (oriented compact cell-decomposition)

X ⊂ Mg is Kasteleyn XK if ∀ F ∈ F

X orientation εK

ξ | iξ = jξ, fjxed

counter (counterclockwise) ε−

ε∂F= ε− ε∂X boundary orientation ε∂F,

dd parity ρ− = 1F
ξւ

(mod 2) i.e. εK

F =

ξ ∈ ∂F

εK

ξ (F

) = −1

εK

ξ (F

) = −1 ≡ ξւ(F ) if εK

ξ ∈ ε− ε∂F

+1 ≡ ξր(F ) if εK

ξ ∈ ε∂F.

4 3 2 1 6 1 8 4 5 7 3 2 9 10

Given XK for ωξ trivial otherwise, ∀ ξ connecting iξ and jξ, XK

ij =

ξ

εK

iξjξ ωξ = −XK ji

XK

ij = 0

i = j ,

εK

iξjξ =

−1 if εK

ξ is jξ to iξ

+1 if εK

ξ is iξ to jξ.

Remark. Bipartite Kasteleyn orientation is well-defjned in hexagonal lattice.

25

SLIDE 26

Derivation. If εK

iξjξ = εK jξiξ = 1, then (XK ij ) is called: adjacency matrix

(resp. weighted adjacency matrix) ∀ ωξ=1 (resp. ∀ ωξ>1).

Derivation. Let XK⊂ Mg be bipartite, then

XK

ij = −XK ji =

ωξ if iξ jξ

r

iξ jξ −ωξ if iξ jξ

r

iξ jξ if i = j

r

ξ = η ∀ iξ , jη

r

XK

ij = −XK ji =

ωξ if iξ jξ

r

iξ jξ −ωξ if iξ jξ

r

iξ jξ if i = j

r

ξ = η ∀ iξ , jη.

26

SLIDE 27

The transition subgraph is symmetry D1∆D2=D1∪D2\D1∩D2 of 1-chain complex C1(XK ; Z2); 1-cycle homology H1(XK ; Z2)=H1(Mg; Z2) class of all ordered, even-length η=

α σCα ∩ D1∆D2 simple closed transition paths

Cα =

σnα−1+1, . . . , σnα
, ∀ α∈N+ | 1⩽α⩽η, n0=0,

traversing σnα−1+1, (σnα−1+1, σnα−1+2), . . . , σnα, (σnα, σnα−1+1) given by:

(σnα−1+1, σnα−1+2), . . . , (σnα−1, σnα)
⊆ D1
(σnα−1+2, σnα−1+3), . . . , (σnα, σnα−1+1)
⊆ D2 .

1 2 3 4 5 6 7 8

Remark. D1, D2 are equivalent if |D1∆D2| = 0 ∈ H1(Mg; Z2) −

→ Z2; D1, D2 = 1-chain in cell-complex C1(Mg; Z2); ∂D1, ∂D2 = C0(Mg; Z2).

27

SLIDE 28

Lemma (sign). The monomial sign for fjxed suffjcient large genus g, εK

D = (−1)t(σ)

ξ ∈ D

εK

σ2ξ−1σ2ξ

t(σ):= (σ1, . . . , σ2n) −

→ (1, . . . , 2n) is invariant of Aut(D).

Proof. εK

D is Aut(D) invariant by transposition of σ2ξ−1σ2ξ, with (−1)t(σ).

Now, let D1, D2 ∈ D orient from σ2ξ−1 to σ2ξ, resp. τ2η−1 to τ2η, in cyclic order of σ, resp. τ, ∀ Cα (transition even cycles). Then, exactly one εK

ξ∗∨ η∗(Cα) is + (resp. −) in clockwise (resp. counterclockwise) ∀ α. Hence,

for all composition γ = σ ◦ τ | σ2ν−1σ2ν = τ2ν−1τ2ν, +1 = εK

D1 εK D2 =

α
ξ ∈ Cα
η ∈ Cα

εK

σ2ξ−1σ2ξ εK τ2η−1τ2η

ν
εK

σ2ν−1σ2ν = εK τ2ν−1τ2ν

2

=

α
ξ ∨ ξ∗ ∈ Cα
η ∨ η∗ ∈ Cα

εK

σ2(ξ ∨ ξ∗) − 1 σ2(ξ ∨ ξ∗) εK τ2(η ∨ η∗) − 1 τ2(η ∨ η∗)

= ⇒ εK

D1 = εK D2, for 1 Cα

ξւ∨ξ∗ ւ∨ηւ∨η∗ ւ

= 1 (mod 2), ∀ α, by ξ∗∨ η∗ i.e. εK

D1 = εK D2, ∀ ρ− = 1 Cα

(·)ւ

≡ 1

Cα

(·)ր

= ρ+ through Aut(D1) invariance, resp. Aut(D2) invariance, ∀ D1, D2 ∈ D. □

28

SLIDE 29

Defjnition. Two orientations are equivalent if there exists reversing-map:

147

− →

148

Theorem. All Kasteleyn orientations of XK⊂ R2 are equivalent.

Proof. Given two Kasteleyn orientations K

−, K + marked by K − (resp.

K

+) on ith end (resp. jth end) of ξ, ∀ F

, ε∂F= ε∂X, then εK

−

ξ

= εK

+

ξ

· σK

−K +

ξ

, εK

+

ξ

= εK

−

ξ

· σK

−K +

ξ

σK

−K +

ξ

= εK

−

ξ

· εK

+

ξ

i.e. K

−−

→K

+ (resp. K +−

→K

−) by σK

−K +

ξ

multiplying K

− (resp. K +)

at every vertex; and, K

− ←

→ K

+ ←

→ equivalence class [K] in simple reversal of orientations around vertices by −1 = σK

−K +

ξ

:= ±1. □

Corollary. Equivalence class [K] is unique for XK⊂ R2.
Proof. ∃ one homotopy class of loops i.e. R2 trivial fundamental group. □

29

SLIDE 30

Theorem. Kasteleyn orientation equivalence classes [K] are exactly 22g.
Proof. The isomorphisms {[K]} are in characteristic-2 fjeld κ affjne closure

Sym2

κ(V ∧) of non-degenerate, skew-symmetric quadratic bilinear form

q(α + β) = q(α) + q(β) + α·β

q: V ⊗V −

→κ , ∀ α, β ∈ H1 = V ⊗V in fjrst homology space H1 ∋ α, for 1

|H1|
q ∈ (H1,·)

(−1)Arf (q)+q(α) = 1

Arf (q)=
{ξ, η}

q(ξ)q(η) ∈ κ/ f(κ) ⊂ Z2 where {ξ, η} are symplectic basis pairs for symplectomorphisms V − → V, Lang’s isogeny f : κ− →κ | x− →x2−x ∈ Gal/F2 (2-element Galois fjeld). By continuity ψ : XK− → Mg, every Mg\ψ(XK) connected-components (ψ-faces F ) ≈ open disk, i.e. χ(XK) = χ(Mg) in Euler-Poincaré bound |V

XK|−|E XK|+|F XK| = χ(XK) ⩾ χ(Mg). Vanishing composition ∂1◦∂2

f boundary operators ∂2:C2 −

→C1, ∂1:C1 − →C0 for basis C0, C1, C2 of 2D cell-complex V

XK, E XK, F XK, resp. implies 1-cycle space superset Ker(∂1)

f 1-boundary space ∂2(C2). Hence, independent of XK but depending only
n genus g: |H1(Mg; Z2)| = |H1(XK; Z2)| = |Ker(∂1)/∂2(C2)| = 22g. □

30

SLIDE 31

Theorem (existence). Kasteleyn orientation exists ⇐ ⇒ |V

XK| = even.

Proof. Following a rooted spanning dual tree T∗:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 29 30 31 32 33 34 35 36 37 39 42 43 44 45 46 47 48 57 59 60 62 64 65 66 68 69 70 71 72 74 76 78 79 80 81 83 84 85 90 91 100 102 103 105 108 113 136 137 138 139

Reduce X to ≪ by n×n − → exp(αn2); and, arbitrarily orient every ξ not crossing T∗. Then, deleting ξ∗ from leaves starting at root, make εK

F, ∀ F

.

31

SLIDE 32

Now,

F∈ F

XK

(−1)

ւ

ξւ(F

)

= (−1)
F∈ F

XK

ւ

ξւ(F

)

= (−1)|E

XK| =

⇒ |V

XK| = even

by the Euler-Poincaré equality: |V

XK| mod 2

= = = = |E

XK−F XK|.

□

Remark. Deleted-vertex changes Kasteleyn to non-Kasteleyn at “hole”:

1 2

4

3

5 6 9 10 11 13 14 15 16 19 29 30 66

≡

1

−1

2

−1 −1 −1

5 10 13 14

h0, h1, h2, h3 = K

1

2 4

3

6 9 10 11 13 14 15 16 19 29 30 66

012

h012 = non-K, h3 = K.

32

SLIDE 33

Remark. To convert the non-Kasteleyn orientation back to Kasteleyn:

h0 = h1 = · · · = h11 = −1.

33

SLIDE 34

Theorem. Let XK⊂ Mg | g=0 be multiedge embedding, then

|Pf(XK)| = Z

def

= =

D
ξ ∩ D

ωξ where Quot(K[D]) ∋ Pf(XK) = 1 n! 1 2n

σ ∈ S2n

sgn(σ) XK

σ1σ2 · · · XK σ2n−1σ2n

sgn(σ) = (−1)t(σ) | t(σ):= (σ1, . . . , σ2n) − → (1, . . . , 2n) XK

σ2ξ−1σ2ξ =

(σ2η−1, σ2η) = (σ2ξ−1, σ2ξ)

εK

σ2η−1σ2η ωσ2η−1σ2η

∀ η, n ∈ N+; ξ = 1, . . . , n.

34

SLIDE 35

Proof. XK=m×m =

⇒ det XK=det(−(XK)T)=(−1)m det XK=0 ⇐ ⇒ m=odd; but det XK=0 = ⇒ det XK= positive-defjnite, square of rational function of XK

ij | XK= 2n×2n.

Precisely, XK

i πi = −XK

πi i | i⩽πi =

⇒ sum of monomials in two partitions:                           

π

∈ S2n

(Sn × Sn

2 )

(−1)t(π)

2n

i=1

XK

i πi

j =π−1(πi) ←

→ i=j ∈{1, . . . , n} = ⇒ XK

i πi ≡ XK

π2ξ−1π2ξ, ∀ ξ=1, . . . , n;

t(π)= even (odd), for even n (otherwise) t(π) := (π1, . . . , π2n) − → (1, . . . , 2n)

+

π

∈ 2 ·

S(2n)!

n! 2n

S2 × S(2n)!

n! 2n − 2

(−1)t(π)

2n

i=1

XK

i πi

j =π−1(πi) ←

→ i=j ∈{1, . . . , n} = ⇒ XK

i πi ≡XK

π2ξ−1π2ξ, ∀ ξ=1, . . . , n;

t(π)= odd (even), for even n (otherwise) by Leibniz’s second-index permutations.

35

SLIDE 36

And, t(σ):= (σ1, . . . , σ2n)− →(1, . . . , 2n) implies the quadratic:                                     

σ =

σ

∈ S2n

(Sn × Sn

2 )

(−1)t(π) + n + t(σ) n

ξ=1

XK

σ2ξ−1σ2ξ

2
t(π)= even (odd),

for even n (otherwise)

+

2 ×



  

σ = σ = τ = τ

∈ S2n

(Sn × Sn

2 )

    ∼ =

S(2n)!

n! 2n

S2 × S(2n)!

n! 2n − 2

(−1)t(σ) + t(τ)

n

ξ=1

XK

σ2ξ−1σ2ξ

n

η=1

XK

τ2η−1τ2η

=

σ = σ

(−1)t(σ)

n

ξ=1

XK

σ2ξ−1σ2ξ

2

= Pf 2(XK)

t(σ):= (σ1, . . . , σ2n)

− → (1, . . . , 2n) ∀ min(deg( XK))⩾n!a( XK)b( XK)

⌊2n−3⌋!!; a,b∈R+; n⩾2; Aut(D

)⊆S2n.

36

SLIDE 37

Now, ∀ η connecting σ2ξ−1 and σ2ξ, write: Pf(XK) =

σ = σ

sgn(σ)

n

ξ=1
(σ2η−1, σ2η) = (σ2ξ−1, σ2ξ)

εK

σ2η−1σ2η ωσ2η−1σ2η

and by εK

D invariant of Aut(D),

Pf(XK) =

σ

Aut(D)
D

sgn(σ)

n

ξ=1

εK

σ2ξ−1σ2ξ

fjxed, ∀ σ ∈ Aut(D)
ξ ∩ D

ωξ = 1 n! 1 2n

σ

∈

Aut(D)

D

εK

D

fjxed, ∀ σ ∈ Aut(D)
ξ ∩ D

ωξ = sgn(σ)

n

ξ=1

εK

σ2ξ−1σ2ξ ·

D
ξ ∩ D

ωξ = (±)

D
ξ ∩ D

ωξ = ± Z.

37

SLIDE 38

That is, Pf(XK) =

σ
Aut(D)

sgn(σ)

n

ξ=1

XK

σ2ξ−1σ2ξ

|Pf(XK)| = Z

and, for all S2n\Aut(D) monomials vanishing, Pf(XK) = 1 n! 1 2n

σ ∈ S2n

sgn(σ) XK

σ1σ2· · ·XK σ2n−1σ2n

|Pf(XK)| = Z

difgering only in orientation, independent of σ ∈ Aut(D). □

Theorem. Observable is absolutely continuous ifg XKis non-singular.

Proof. k

ℓ=1

σiℓjℓ

= Pf
(XK)−1

ξη

D ∋ (i1j1), . . . , (ikjk); ξ, η= 1, . . . , k

|Pf(XK)| = partition function. □

Theorem. Combinatorials i.e. exponentials reduce to cubic complexity.
Proof. Pf(

AXK AT)=det( A)Pf( XK)− →O(n3) in diagonalization by skew symmetric Gaussian elimination, for spectrum analysis.

Remark. Recall mini-max contour deformation, critical point universality.

38

SLIDE 39

1.4 Grassmann (graded) integral

Defjnition. Grassmann (graded) algebra
XK

, ∀ XK basis (x1, . . . , x2n) is given by 22n=2n

k=0(dim kXK) =2n k=0

2n

k

dimensional basis vectors:

x0=1; xσk< = xσ1⊗ · · · ⊗ xσk | xσξ⊗ xση + xση⊗ xσξ = 0; σk< =(σ1 · · · σk) | σ1<· · ·<σk, ∀ σ1, . . . , σk, k = 1, . . . , 2n

.

Element is graded by

XK ∋ y(x) = y(0) ⊕

2n

i=1

y(i)xi ⊕

2n

k=2
τ ∈ Sσk<

(−1)t(τ) y(τ1···τk) xσk< =

2n

k=0
τ ∈ Sσk<

y(τ1···τk)

k

i=1

xτi

y([k=0])=y(0)

xσ0 =x0=1. Multiplication y1(x) y2(x) is given by y(0)

1 y(0) 2

⊕

2n

i=1
y(0)

1 y(i) 2 + y(i) 1 y(0) 2

xi ⊕ 1

2

σ ∈ Sσ<
y(0)

1 y(σ1 σ2) 2

+ + y(σ1)

1

y(σ2)

2

− y(σ2)

1

y(σ1)

2

+ y(σ1 σ2)

1

y(0)

2

xσ1⊗xσ2 ⊕ · · ·

39

SLIDE 40

Derivation.

2XK∋w= ijXK ij xi⊗xj =

⇒

2nXK∋wn=Pf(XK)xσ2n<.

kXK− →kXK: (−1)t(σ)wσ1∧ · · · ∧ wσk = 1 k!

τ ∈ Sσk<

(−1)t(τ)

k

i=1

wτi.

Defjnition. With respect to orientation θ ∈

2nXK ∼

= R,

2nX

f = fθ

f = fθ θ + · · ·
lower
rder terms

by formal rule

2n
i=1

xi ⊗

2n

i=1

dxi = (−1) 2n−1

i=1

i

2n
i=1

(xi ⊗ dxi) = (−1)n(2n−1).

Derivation. For degenerate integral =

⇒ deg(x) < deg(dx),

k
i=1

xσi ⊗ dx = (−1)t(σ) if k=2n if k<2n

dx = (−1)n(2n−1) 2n

i=1 dxi

t(σ):= (σ1, . . . , σ2n) − → (1, . . . , 2n).

Derivation. θ = x1⊗· · ·⊗x2n if (xi) is basis of XK.

40

SLIDE 41

Theorem. Let f(x) =

XKexp
λ0 + 1

2

x, XKx
dx satisfy Pf(XK)

constraints, then f uniquely maximizes −

XK(1/|f|) log(1/|f|) dx; and,

(i) Pf(XK) =

XK

exp 1 2

ij

xiXK

ij xj

dx

(ii) Pf

XK

−(XK)

T

= det(XK)

(iii) (Pf(XK))2 = det(XK) (iv) ∂ ∂XK

i1j1

· · · ∂ ∂XK

ikjk

Pf(XK) = Pf(XK) · Pf((XK−1)xy)

x=i1, . . . , ik

y=j1, . . . , jk.

41

SLIDE 42

Proof. (i). Since all exponents except n vanish,

XK

exp 1 2

x, XKx
dx

= 1 n! 1 2n

XK
x, XKx

n dx where, precisely, x, XKx n dx =

σ ∈ Sn×Sn

2

XK

i1j1· · ·XK injn(xi1⊗xj1)⊗· · ·⊗(xin⊗xjn) dx =

=

σ ∈ Sn×Sn

2

(−1)t(σ)XK

i1j1· · ·XK injn

t(σ) := (i1, j1, . . . , in, jn)

− → (1, . . . , 2n). Therefore, with “equality” of permutations σ ∈ Sn×Sn

2 ,

V

exp 1 2

x, XKx
dx

= Pf(XK).

Remark. II, III and IV follow from the latter integral formula.

42

SLIDE 43

(ii). Choosing splitting XK= W K⊕W K for block structure, where XK is isomorphic to algebra (tensor product) generated by ui, vi|i=1, . . . , n with relations uiuj =−ujui, uivj =−vjui, and vivj =−vjvi: (x1, . . . , x2n) = =

u1, . . . , un
basis in W K

, v1, . . . , vn

basis in W K
.

As a result,

x ,
XK

−(XK)

T

x
=

2

u, XKv
i.e. need to prove
n(W K⊕W K)

exp (

u, XKv
) du dv

= det(XK). (iii). Similar.

43

SLIDE 44

(iv).

exp

1 2

x, XKx
+ x, η
dx

= =

exp

1 2

x+XK−1η , XK(x+XK−1η)
− 1

2

η, XK−1η
dx

= exp

− 1

2

η, XK−1η
Pf(XK).

∂ ∂XKi1j1 · · · ∂ ∂XK

ikjk

Pf(XK) = =

exp

1 2

x, XKx
xi1xj1 · · · xikxjk dx

= ∂ ∂η

2k

exp 1 2

x, XKx
+ η, x
dx.

44

SLIDE 45

Then, by Kullback-Leibler distance D(··) and Jensen’s inequality for any U, − D

U
f
=
XK

1 |U| log (1/|f|) (1/|U|) ⩽log

XK

1 |U| (1/|f|) (1/|U|) = log

XK

(1/|f|) = log 1 i.e. −

XK

1 |U| log 1 |U|

= −
XK

1 |U| log |f| |U| · 1 |f|

= − D
U
f
−
XK

1 |U| log 1 |f|

⩽−
XK

1 |U| log 1 |f|

= −
XK

1 |U| log

1
XK

eλ0+1

2x,XKxdx

= −
XK

1 |f| log 1 |f|

where the inequality is equality ifg U(x)=f(x) almost everywhere.

□ Lemma.

XKgraded identity, up to tensors on superalgebra Ma,b minimal

subfjeld, is isomorphic to kernel of either Q or prime-ordered fjeld Fq = pm.

Proof. ♥.
Theorem. The ideal of Mpr+qs, ps+qr is contained in ideal of Mp,q⊗Mr,s.
Proof. Follows from the prior lemma.

□

45

SLIDE 46

Theorem. Let XK⊂ Mg | g=0 be bipartite multiedge embedding, then

(i) Z = |det(CXK)|

CXK = RV◦

XK←

֓ , RV (XK) = RV•

XK ⊕ RV◦ XK←

֓ where ← ֓ = ⇒ nested. (ii)

σb1w1 · · · σbkwk
= det
(CXK)−1
b w
b =

b1, . . . , bk w = w1, . . . , wk where b = white-vertex identifjed with b.

46

SLIDE 47

Proof. (i). XK⊂ Mg | g=0 implies Z = εK

X

exp

1 2

ij

xi (XK

ij ) xj

dx
εK

X = (−1)σ εK

σ1σ2 · · · εK σ2n−1σ2n

2n = |V (XK)|. XK⊂ Mg | g=0 bipartite V

XK= V

XK ⊔ V
XK implies

XK =

BXK

−(BXK)T

BXK : RV◦

XK −

→ RV•

XK

RV (XK) = RV•

XK ⊕ RV◦ XK

dim(RV•

XK) = dim(RV◦ XK) = n

|V (XK)| = 2n. Identifying V•(XK), V◦(XK) via a diagram {b} ∼ {w} with “hole” XK =

CXK

−(CXK)T

RV (XK) = RV•

XK ⊕ RV◦ XK←

֓ CXK = RV◦

XK←

֓ where ← ֓ = ⇒ nested i.e. Z = |det(CXK)|. □

47

SLIDE 48

(ii). Write

σb1w1 · · · σbkwk
=

∂ ∂ w(b1w1) · · · ∂ ∂ w(bkwk) ln Z = det

(CXK)−1
b w
b =

b1, . . . , bk w = w1, . . . , wk where b = white-vertex identifjed with b. □

Remark. The “physical” meaning:
σb1w1 · · · σbkwk
=

=

ψ∗

b1ψw1· · · ψ∗ bkψwk exp

ψ∗CXKψ
dψ∗dψ ·
exp
ψ∗CXKψ
dψ∗dψ

which corresponds to the free Fermionic observable.

48

SLIDE 49

Corollary (dimer-monomer problem). Let monomers ← → dimers:

X ⊂ Mg

remove vertices and adjacent edges

49

SLIDE 50

then taking monomer cover

X ⊂ Mg

b1 b2 w1 w2

the monomer-monomer observable is given by Mb1···bnw1···wn = Z(XK

b1···bnw1···wn)

Z(XK)

50

SLIDE 51

such that: adjacent monomers (ibξ, iwξ) = ⇒ dimer (ibξ iwξ), ∀ ξ ∈ D: X ⊂ Mg . and, Mb1···bnw1···wn for all |{[K]}| = 22g+2n−1, 2n = |vertices|, is a special dimer case for nontrivial fundamental-group surfaces:

Proof. ♥.

51

SLIDE 52

1.5 Generating function for equivalence class

The order of an equivalence class of D, |σ

Aut(D)| = |{

σ}| = |D| ⩽ |{σ

Aut(D)}| =
exp
n ln 2 +

n

k=2

ln k |D| is given by two-variable generating function:

σ =

σ

ξ ∈ D(σ)
η ∈ (σ(2ξ−1), σ(2ξ))

1 =

D(N1, ..., Nk) | (k

ν=1 Nν) = n

(±)

k

ν = 1

ωNν

ν

∀ η connecting σ(2ξ−1) and σ(2ξ); Nν=|ν-class dimers|; ω1=1=ω2, k=2. Derivation I. Let X⊂Mg =planar M×N square grid, where ∂X =open. |{ σ(X; M, N)}| = 2(MN

2 )

M

i=1

N 2

j=1
cos2 πi

M +1

+ cos2 πj

N +1

N =

= even = |{ σ(X; N, M)}|

M = even

= 0

MN = odd.
Show. ♥.

52

SLIDE 53

Derivation II. Let X⊂Mg =cylindrical M ×N square grid. |{ σ(X; M, N)}| = = 2(MN

2 )

M

i=1

N 2

j=1
sin2π(2i−1)

M

+ cos2 πj

N +1

N = even

= 2(MN

2

− M

2 + 1)

M

i=1

N 2

j=1
sin2π(2i−1)

M

+ cos2 πj

N +1

N = odd

= 0

MN = odd.
Show. ♥.

53

SLIDE 54

Derivation III. Let X⊂Mg =toroidal M ×N square grid. |{ σ(X; M, N)}| = = 2(MN

2

− 1)

               

M

i=1

N 2

j=1
sin2π(2i−1)

M

+ sin22πj

N

+

M

i=1

N 2

j=1
sin22πi

M

+ sin2π(2j−1)

N

+

M

i=1

N 2

j=1
sin2π(2i−1)

M

+ sin2π(2j−1)

N



              

N =even

= |{ σ(X; N, M)}|

M = even

= 0

MN = odd.
Show. ♥.

54

SLIDE 55

Derivation IV. Let X⊂Mg =planar 6×8 square grid, where ∂X =open. |{ σ(X; M, N)}| = = 16777216 1 4 + cos2(π 7)

cos2(π

9) + cos2(π 7)

cos2(π

7) + cos2(2π 9 )

×

×

cos2(π

7) + sin2( π 18) 1 4 + sin2( π 14)

cos2(π

9) + sin2( π 14)

×

×

cos2(2π

9 ) + sin2( π 14)

sin2( π

18) + sin2( π 14) 1 4 + sin2(3π 14)

×

×

cos2(π

9) + sin2(3π 14)

cos2(2π

9 ) + sin2(3π 14)

sin2( π

18) + sin2(3π 14)

.
Show. ♥.

55

SLIDE 56

Derivation V. Let X⊂Mg =cylindrical 6×8 square grid. |{ σ(X; M, N)}| = = 5242880 1 4 + cos2(π 9) 2 1 + cos2(π 9) 1 4 + cos2(2π 9 ) 2 × ×

1 + cos2(2π

9 ) 1 4 + sin2( π 18) 2 1 + sin2( π 18)

.
Show. ♥.

Derivation VI. Let X⊂Mg =toroidal 6×8 square grid. |{ σ(X; M, N)}| = = 8388608 18225 131072 + cos4(π 8) 3 4 + cos2(π 8) 4 sin4(π 8) 3 4 + sin2(π 8) 4 + + 1 4 + cos2(π 8) 4 1 + cos2(π 8) 21 4 + sin2(π 8) 4 1 + sin2(π 8) 2 .

Show. ♥.

56

SLIDE 57

1.6 Partition as sum of Pfaffjans

Lemma. Z = 1 2g

[K]

Arf(qK

D0) · εK

(D0) · Pf(XK)

±1 = Arf(q) = 1

2g

α ∈ H1

(−1)q(α) 2g = |H1(XK; Z2)| where [K] = all equivalence classes of Kasteleyn orientations, 22g in total qK

D0 = quadratic form on H1(Mg; Z2), corresponding to Kasteleyn

rientation with respect to reference perfect matching D0

εK (D0) = (−1)σ εK

σ1σ2· · · εK σ2n−1σ2n

σ ∈ Aut(D0) ⊆ Aut(D)

σ

Aut(D ∈ D) ∼

= Aut(D)

(Sn×Sn

2 ).

Proof. ♥.

57

SLIDE 58

Theorem. Z = 1 2g

T ∈ S(Mg)

Arf(qK

T ) · Pf(XK T )

±1 = Arf(q) = 1

2g

α ∈ H1

(−1)q(α) 2g = |H1(XK; Z2)| where Arf(qK

T ) := quadratic form qK T on H1(Mg; Z2) for spin structure T

XK

T = Kasteleyn matrix corresponding to spin structure T

S(Mg) = set of all spin structures on Mg.

Proof. ♥.

58

SLIDE 59

Theorem. Let X ⊂ Mg be bipartite, such that

height function = = section of the non-trivial Z-bundle. then Z

Hx1, . . . , Hxg, Hy1, . . . , Hyg
=

=

D
ℓ ∈ D

ω(ℓ)

g

i=1

exp

i

Hxi∆xih + +

i

Hyi∆yih

where (x1, . . . , xg, y1, . . . , yg) are fundamental cycles, and

∆Ch = change in height function along Mg noncontractible cycle C.

Proof. ♥.

59

SLIDE 60

1.7 Limits

N N N uniform measure Prob(h) = 1 |HX| N − →∞ .

60

SLIDE 61

Theorem (Schur process; Okounkov & R). Let ϕε:Z2֒ →R2|D⊂R2; ε ε

D

such that: ε− →0, |Dε|− →∞ Dε = ϕε

Z2

∩ D . Then, for cube-stack with measure Prob(π) =

t

qπ(t)

t

π
t

qπ(t)

t

π ∈ HX

π ∼ = D, there is existence of: Thermodynamic limit (|Dε|− →∞) + + Scaling limit (q=e−ε, ε− →+0).

Proof. ♥.

61

SLIDE 62

x1N y1N x2N y2N

|

uN | vN

|

where u + v = = x1 + x2 + y1 + y2; N = ε−1, q = e−ε.

62

SLIDE 63

2 Vertex algebras

Points: (i) Prove Grassmann kernel convergence for special genus g domain T∗ (ii) Obtain the R logarithmic scaling asymptotics by variational principle (iii) State conjecture for the Green’s function · in large-deviation

63

SLIDE 64

2.1 Grassmann (graded) kernel

Pairing

XK∗⊗
XK−

→R: σ(k)> =(σ(1), . . . , σ(k))

σ(1)>· · ·>σ(k),
ϕ(x∗), ψ(x)

def = = ϕ0 ψ0 +

2n

k=1

ϕkψk +

2n

k=1
σ(k)<

ϕσ(k)···σ(1) ψσ(1)···σ(k) = = |ψ0|2 +

2n

k=1
σ(k)<

|ψσ(1)···σ(k)|2 d 2nx, ∀ |ψ|2 ∝ |ϕ|2 ∈ R such that for the dual space, graded basis x∗

σ(k)>,

X ∋ψ(x) = ψ0 +

2n

k=1
σ(k)<

ψσ(k)< xσ(k)<

kXK∋
ψσ(k)< xσ(k)<
XK∗∋ϕ(x∗) = ϕ0 +

2n

k=1
σ(k)>

ϕσ(k)> x∗

σ(k)>

kX∗∋
ϕσ(k)< x∗

σ(k)>

where

XK∗ is the dual graded algebra to
XK generated by

x0=1; xσk< = xσ1⊗ · · · ⊗ xσk | xσξ⊗ xση + xση⊗ xσξ = 0; σk< =(σ1 · · · σk) | σ1<· · ·<σk, ∀ σ1, . . . , σk, k = 1, . . . , 2n

.

64

SLIDE 65

Fixing integrals on

XK,
XK∗,
(XK∗⊗XK) by choosing

x1, . . . , x2n ∈

2nXK, x∗ 2n, . . . , x∗ 1 ∈ 2nXK∗

and x∗

2n, . . . , x∗ 1 , x1, . . . , x2n ∈ 2nXK∗⊗ 2nXK

then

η
i=1

x∗

σ(i) η

i=1

xτ(i) dx∗ dx = , η = 2n (−1)(σ + τ + n(2n−1)) , η = 2n σ : (σ(1), . . . , σ(2n)) − → (1, . . . , 2n) τ : (τ(1), . . . , τ(2n)) − → (1, . . . , 2n) . Lemma.

ϕ(x∗), ψ(x)
=
exp
i

x∗

i xi

ϕ(x∗) ψ(x) dx∗ dx .
Proof. ♥.

65

SLIDE 66

Lemma. Let Y K: XK−

→XK by ψY K(x) =

{i}< , {j}<

x{i}< Y{i}<{j}< ψ{j}< = ψ0 ⊕ Y ψ1 ⊕ Y ⊗2 ψ2 ⊕ · · · then ψY K(w) = =

exp
−x∗Y Kw
exp(

−x∗x) ψ(x)dx∗ dx.

Proof. ♥.

Lemma.

exp
−x∗Y Kw
exp(

−x∗x) exp

−W K∗W Kx
dx∗dx =

= exp

−w∗W KXKw
.
Proof. ♥.
Remark. Hence, exp(

−w∗Y Kw) is Y K “integral kernel” acting on

2nXK.

66

SLIDE 67

2.2 Vertex operators

(i). The Fermionic Fock space F i.e. XK

m

∈ C Z +1

2

is given by F =

XK

m1 ∧ XK m2 ∧ · · ·

mi ∈ Z + 1

2

mi+1 = mi−1 i ≫ 1

.

(ii). The Clifgord algebra is given by ClZ =

ψm , ψ∗

m

m ∈ Z + 1

2

ψm ψm′ + ψm′ ψm = ψ∗

m ψ∗ m′ + ψ∗ m′ ψ∗ m = 0

ψm ψ∗

m′ + ψ∗ m′ ψm = δm m′ .

(iii). The Clifgord algebra acting on the Fock space F : ψm xm1 ∧ xm2 ∧ · · · = xm ∧ xm1 ∧ xm2 ∧ · · · ψ∗

m xm1 ∧ xm2 ∧ · · ·

=

∞

i=1

(−1)i δmi, m xm1 ∧ · · · ∧ xm1 ∧ · · ·

67

SLIDE 68

(iv). The Heisenberg algebra is given by

αn
n ∈ Z\{0}

[αn, αn′] = −n δn, −n′ . (v). The Heisenberg algebra acting on the Fock space F :

As part of Bose-Fermi correspondence in 1D:

αn − →

m ∈ Z +1

2

ψm+n ψ∗

m .

As operator in F:
αn , ψξ
= ψξ+n ,
αn , ψ∗

ξ

= −ψ∗

ξ−n .

(vi). The vertex operators in F are given by XK

± (x) = exp

∞

n=1

xn n α±n

XK

− (x)v, w

=

=

v, XK

+ (x)w

=
XK

+ (x)w, v

.

68

SLIDE 69

(vii). The commutation relations are given by XK

+ (x) XK − (y)

= (1−x) · XK

− (y) XK + (x)

XK

+ (x) ψ(z)

= (1−z−1 x)−1 · ψ(z) XK

+ (x)

XK

− (x) ψ(z)

= (1−x z)−1 · ψ(z) XK

− (x)

XK

+ (x) ψ∗(z)

= (1−z−1 x) · ψ∗(z) XK

+ (x)

XK

− (x) ψ∗(z)

= (1−z x) · ψ∗(z) XK

− (x).

(viii). The eigenvectors are given by XK

− (x)

i

ψ∗(wi)

j

ψ∗(zj) v(n) = =

i

(1−x zi)−1

j

(1−x wj)

i

ψ∗(wi)

j

ψ∗(zj) v(n) where v(n) = vn−1

2 ∧ vn−3 2 ∧ · · ·

69

SLIDE 70

2.3 Fermionic Kasteleyn operators

For the one cube X∗ of two-color tiles on bipartite hexagonal lattice X :

2 3 4 5 7 8 9 10 12 13 14 15 16 17 18 19 22 23 25 26 62 68 70 71

x t let the general parameterization for bipartite hexagonal lattice be given by b(h, t) = (h, t− 1

2),

w(h, t) = (h, t+ 1

2).

70

SLIDE 71

Kasteleyn matrix by the above-given b ∼ w diagram is then given by K(h, t) = (h, t) −

h+ 1

2, t+1

+ yh,t
h− 1

2, t+1

.

Placing Fermions x∗

h, t , xh, t respectively at b(h, t) and w(h, t):

4 8 9 10 12 13 16 17

(h, t) (h− 1

2, t+1)

(h+ 1

2, t+1)

x∗Kx =

h, t

x∗

h,t xh,t −

h, t

x∗

h+1

2, t+1 xh,t +

h, t

x∗

h−1

2, t+1 xh,t yh,t

=

t
x∗

t xt + xt V x∗ t+1 + xt V −1 xt x∗ t+1

.

71

SLIDE 72

Theorem. Assuming xh,t = xt, analogous to the notation qh,t = qt,

[Diagram]

Prob(π)

∝

t

q |π(t)|

t

the boundary conditions imply Z =

exp
x∗Y Kx
dx∗ dx =

=

XK

−

x−1

2

· · · XK

−

xu0+1

2

XK

+

x1

2

· · · XK

+

xu1+1

2

v(0)

0 , v(0)

.

72

SLIDE 73

Proof (outline).

· · · exp
x∗

t−1 xt−1

· exp
xt−1
V −V −1XK

t

x∗

t

·

· exp

x∗

t xt

· exp
xt
V −V −1XK

t

x∗

t+1

· · ·

= = · · ·

V −V −1XK

t−1

∼−1
XK

+ (xt)

·

V −V −1XK

t

∼−1
XK

− (xt)

· · · where XK

+ (xt) and XK − (xt) each depends on t such that

Y K = Y K, where V ←

֓ is lifted to

∞ 2 V

V =
m ∈ Z+1

2

C vh under boundary conditions, etc. □

Remark. Direct proof exists combinatorially besides the Kasteleyn way.

Corollary. Z =

u1 − 1

2

m = 1

2

−1

2

m′ = u0 + 1

2

1 − x−

m′ x+ m

−1 .

73

SLIDE 74

Theorem. (Okounkov & R., 2005).
σ(h1t1) · · · σ(hktk)
= det(K((ti, hi), (tj, hj)))1 ⩽ i, j ⩽ k

K((ti, hi), (tj, hj)) = = 1 (2πi)2

|z| < R(t1)
|z| <

R(t2) Φ−(z, t1) Φ+(w, t2) Φ+(z, t1) Φ−(w, t2) · · 1 z − w · z

− h1−B(t1)−1

2

· w
h2−B(t2)−1

2

dz dw

where |w|<|z|, t1⩾t2 |w|>|z|, t1<t2

R(t)= min

m > t((x+ m)−1),

R(t)=max

m < t(x− m), B(t)= |t| 2 − |t−u0| 2

Φ+(z, t)=

m > max(t, 1

2)

(1 − z x+

m), Φ−(z, t)= m < max(t, −1

2)

(1 − z−1 x−

m).

Proof. ♥.

74

SLIDE 75

2.4 Thermodynamic limit with scaling

[Diagram] x+

m = aqm

x−

m = a−1qm

assumed

corresponding to Prob(π) ∝ q |π| .

75

SLIDE 76

Consider limit ε− →0, for q=e−ε, u1=ε−1v1, u0=ε−1v0; fjxed v1, v0: Z =

u0 < n < 0

0 < m < u1

(1 − x−

m x+ n )−1

=

u0 < n < 0

0 < m < u1

(1 − qm−n)−1 |π| = q ∂ ∂q ln Z = ε−3 u1

u0

s−t 1−e t−s 3D volume function ds dt + · · · where ln Z = ε−2 u1

u0

ln ( 1−e−s+t 2D partition function ) ds dt + · · ·

76

SLIDE 77

2.5 Graded (Grassmann) kernel asymptotics

Consider the limit ε− →0 where ti = ε−1τi, h1 = ε−1χi, for fjxed τi, χi: [Diagram] (τi, χi) in the bulk K((t1, h1), (t2, h2)) − → − → 1 (2πi)2

Cz
Cw

exp

ε−1(S(z, t1, χ1) − S(z, t2, χ2))
·

· (zw)1/2 (z−w)−1 dz dw where S(z, t, χ) = = − (χ + τ 2 − u0) ln Z + Li2(ze−v0) + Li2(ze−v1) − Li2(z) − Li2(ze−τ) and Li2(z) = z

t−1 ln(1−t) dt.

77

SLIDE 78

2.6 Critical point discriminants

exp

χ+τ

2

=

(1−ze−v0)(1−ze−v1) (1−z)(1−ze−τ) gives quadratic equation, implying a discriminant for two real solutions or two complex-conjugate solutions, or a zero-discriminant. [Diagram] ∂χ h0(τ, χ) = 1 π arg(z0)

σ(h, t)
= K((t, h), (t, h)) −

→ ε ∂χ h0(τ, χ).

78

SLIDE 79

2.7 Steepest descent

K((t1, h1), (t2, h2)) = − ε 2π ·

exp{ε−1(S1(z1) − S2(w2))}

(z1 − w2)

−w2S′′

2(w2)

z1S′′

1(z1)

− − exp{ε−1(S1(z1) − S2(w2))} (z1 − w2)

−w2S′′

2(w2)

z1S′′

1(z1)

+ c. c.

· (1 + O(1))

That is, for H+ = {z ∈C, Im z >0} | z0(χ, τ) = inner process, such that z1 = z0(χ1, τ1) w2 = z0(χ, τ), K((t1, h1), (t2, h2)) = = ε 2π exp{ε−1(Re(S(z0(χ1, τ1))) − Re(S(z0(χ2, τ2))))} · · exp{i ε−1(Im(S′(z1)) − Im(S(w2)))} (z1−w2) + + exp{i ε−1(Im(S′(z1)) − Im(S(w2)))} (z1−w2) + c. c.

· (1 + O(1))

(∗).

79

SLIDE 80

Hence, (Kasteleyn-Fermions convergence to free Dirac-Fermions) solution: 1 √ε ψ x = exp{ε−1Re(S(z0))} ·

ψ+(z0) exp(iε−1Im(S(z0))) +

+ ψ−(z0) exp(iε−1Im(S(z0)))

· (1 + O(1))

1 √ε ψ∗

x = exp{ε−1Re(S(z0))} ·
ψ∗

+(z0) exp(iε−1Im(S(z0))) + + ψ∗ −(z0) exp(iε−1Im(S(z0)))

· (1 + O(1))

where E(ψ∗ ±(z) ψ±(w)) = 1 z − w E(ψ∗ ±(z) ψ∓(w)) = E(ψ∗ ψ∗) = E(ψ ψ) = 0 such that ψ∗ ±(z), ψ±(w) are spinors: ψ∗ ±(z) = ψ∗ ±(w)

∂w

∂z , ψ±(z) = ψ±(w)

∂w

∂z .

80

SLIDE 81

Remark. The observable is given by:

9 5 13 10 16 14

x
(σ

x1−σ x1) (σ x2−σ x2)

= K12K21 =

= ε2 (2π)2 ∂z1 ∂x1 ∂w2 ∂x2 (z1−w2)2 − ∂z1 ∂x1 ∂ w2 ∂x2 (z1−w2)2 + c. c.

×

× (1 + O(1)). In particular, σ x1−σ x1 = ε ∂x ϕ(z0(τ, x)) + · · ·

ϕ(z) = Gaussian free fjeld on H+

such that the Green’s function of Dirichlet problem on H+ is given by ϕ(z) ϕ(w) = 1 2π ln

z−w

z−w

and, the Bose-Fermi correspondence is given by

∂xϕ = : ψ(z, z) ψ(z, z) : · · · .

81

SLIDE 82

2.8 Scaling limit with Kasteleyn operator

Let X =Dε=ϕε(L) ∩ D, for arbitrary lattice L |AK

G = difgerence operator,

.

where ε − → 0 in the asymptotics of the equation for G

x,y given by

(AK

X)x · G x,y = δx,y

Cases. (i) Hexagonal lattice: Utilizes the weighted as above, for qt = e−ε f(t), t = τ ε , ε− →0.

Theorem. G

x,y = same as (∗), with difgerent z0(τ, x).

Proof. ♥.

(ii) Periodic lattice: Utilizes variational principle.

82

SLIDE 83

2.9 Variational principle

(i). For the N ×M torus [Diagram] Z(H, V ) =

D
ℓ ∩ D

ωℓ × exp(H∆ahD + V ∆bhD) = 1 2

Pf
AK1

+ Pf

AK2

+ Pf

AK3

− Pf

AK4

where N, M − →∞, for fjxed N

M.

And, ω(ℓ)=1 = ⇒ eigenvalues of Kasteleyn matrices by Fourier transform.

Theorem. (McCoy & Wu, 1969; Kenyon & Okounkov, 2005).

lim

N,M− →∞

1 NM ln ZNM = ln |1+zw| dz z dw w = f(H, V ) = |z| = eH |w| = eV .

83

SLIDE 84

(ii). Taking Legendre transform σ(s, t) = max

H,V (Hs + Vt − f(H, V ))

then

D

1 =

D
D

w (e) = exp(MN σ(s, t) · (1 + O(1))) where ∆ahD M = s, ∆bhD N = t, M, N → ∞, N M fjxed. (iii). For domain [Diagram] ∆ah = sM, ∆bh = tN.

Theorem. (Cohn, Kenyon, & Propp, 2000).
D

1 = exp(MN σ(s, t) · (1 + O(1))) with the boundary conditions of height function hD.

84

SLIDE 85

(iv). For domain [Diagram] Mi × Nj ZDε =



     

values of height functions

n

boundaries between rectangles

      

Z Mi Nj (h bound) =

{∆xh, ∆yh}ij

exp Mi Nj Mi Mj σ ∆xh Mi , ∆yh Nj

= exp
ε−2
D

σ(∂xh0, ∂yh0) dx dy (1 + O(1))

where h0 = minimizer for

S[h] =

D

σ(∂xh0, ∂yh0) dx dy.

85

SLIDE 86

Theorem. (Cohn, Kenyon, & Propp, 2000).

lim ε→0 ε2 ln ZDε =

D

σ − → ∇h0

dx dy

for 0 < ∂xh, ∂yh < 1 | h0 = minimizer h0

∂D = b, the boundary condition appearing in the limit ε −

→ 0 [Diagram] for height function h = ε−1h0 + ϕ = ε−1 (h0 + εϕ) with respect to h0 = limit shape, and ϕ = distribution (factor).

86

SLIDE 87

2.10 Physics way of the higher genus observable

S[h0 + εϕ] = S[h0] + ε2 2

D

aij(x)∂iϕ ∂jϕ d2x aij(x) = ∂i∂j ϕ(s, t)

 

s = ∂1h0 t = ∂2h0 such that:

Partition function equals

Z = exp(ε−2S(h0))

exp

1 2

D

aij(x)∂iϕ ∂jϕ d2x

Dϕ

where D = scalar fjeld with Riemannian metric induced by h0;

Correlation equals

ϕ(x) ϕ(y) = G (x, y) where G= Green’s function for ∆ = ∂i(aij∂j).

Conjecture. Gis same as obtained by asymptotics of Kasteleyn operators.
Remark. The conjecture is theorem in certain cases.

87

SLIDE 88

(Chebotarev, Guskov, Ogarkov & Bernard, 2019). For free-action

r interaction Gaussian theory,

S[g, ¯ ϕ] ≡ Z[g, j = ˆ G−1 ¯ ϕ] Z[j = ˆ G−1 ¯ ϕ] = =

∞

n=0

(−1)n n! n

a=1
dΓaeiλa ¯

ϕ(xa)

e

−1

2 n

a,b=1

λaλbG(xa−xb)

. (Bernard, Guskov, Kalugin, Ivanov & Ogarkov, 2019). In critical nonpolynomial phase theory, Z [g; dµ] =

dσt

dµ(x)

e f[ϕ(x);x]
1 =

= C1 [g; dµ]

1 + π(1 − η)

Γ 2(1

4)

dµ(x)
2g(x)

+ O 1 g

.

88

SLIDE 89

Conclusion: the higher genus observable yet

1. How to make (simulate) such pictures of perfect-matching mixture by:

(i). Monte Carlo for exp

∝10002

(ii). Sampling around most probable region by MCMC

2. How to describe such random surface invariant-limit analytically by:

(i). Equipartition Pfaffjan asymptotics with boundary conditions (ii). Variational principle: Minimizer functional in large deviation

89

SLIDE 90

References

[1]

M. Bernard, V.A. Guskov, M.G. Ivanov, A.E. Kalugin, and S.L.
Ogarkov. Nonlocal scalar quantum fjeld theory: functional integration,

basis functions representation and strong coupling expansion. Particles, 2(3):385–410, 2019. [2] I.V. Chebotarev, V.A. Guskov, S.L. Ogarkov, and M. Bernard. S-matrix

f nonlocal scalar quantum fjeld theory in basis functions representation.

Particles, 2(1):103–139, 2019. [3] P.W. Kasteleyn. Dimer statistics and phase transitions. J. Math. Phys., 4:287–293, 1963. [4]

A. Okounkov and N. Reshetikhin. Random skew plane partitions and

the Pearcey process. Comm. Math. Phys., 269:571–609, 2007.

90

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Thank you!

91