integrable systems in the dimer model R. Kenyon (Brown) A. - - PowerPoint PPT Presentation
integrable systems in the dimer model R. Kenyon (Brown) A. Goncharov (Yale) Monday, October 31, 2011 1. Convex integer polygon triple crossing diagram 2. Minimal bipartite graph on T 2 line bundles 3. Cluster integrable system. Monday,
Monday, October 31, 2011
line bundles triple crossing diagram
Monday, October 31, 2011
Monday, October 31, 2011
w1 w2 w3 w4 w4 w5 w5 w5 w3 w3
b c d e a f
w2 = ace bd f
subject to one condition: wi = 1. Line bundle on graph =
monodromies around faces (wi) and homology generators of torus (z1, z2)
edge weights modulo gauge =
Monday, October 31, 2011
{wi, wj} = εijwiwj where ε is a skew-symmetric form εij = 1 if εij = −1 if εij = 0 else. A similar rule for {wi, zj} and {zi, zj}. wi wj wi wj
by the formula
(extend using Leibniz rule) Define a Poisson structure on the moduli space of line bundles
ε is the intersection form on cycles on the “conjugate” surface
genus 2 in the example
Monday, October 31, 2011
1 2 −2 −1 1 −2 −1 1 2 −2 −2 −1 1 2 −2 4 2 −2 −1 1 1 2 −2 −1 1 −2 −1 2 −1 2 −4 2
w1 w2 w3 w4 w4 w5 w5 w5 w3 w3
ε = w1 w5 z1 z2 w2 w3 w4
Monday, October 31, 2011
Goal: define commuting Hamiltonians. (commute with everything) H1 = z−1
1 (1 + w1 + w1w2 + w1w2w3 + w1w2w3w4)
H2 = z−1
1 (w2 1w2w3 + w2 1w2 2w3 + w2 1w2 2w3w4 + w3 1w2 2w3w4 + w3 1w2 2w2 3w4)
C2 = w4
1w3 2w2 3w4z1z3 2
C1 = z2
1z2
Casimirs
Monday, October 31, 2011
w1 w2 w3 w4 w4 w5 w5 w5 w3 w3 A “zig-zag” path is in the kernel of ε (and these generate the kernel). “Casimirs”
Monday, October 31, 2011
Fix a “base point” dimer cover.
A dimer cover has a weight = product of edge weights. The Hamiltonians are normalized sums of weighted dimer covers
Monday, October 31, 2011
Combining with another cover gives a set of cycles. The ratio of weights is the product of the cycle monodromies.
(and so is independent of gauge)
Monday, October 31, 2011
Let M(G) be the set of dimer covers of G. The normalized coefficients of P(z1, z2) are the Hamiltonians. Define the partition function P(z1, z2) =
ν(m)zi
1zj 2(−1)ij
(divide by weight of a zig-zag path)
Hi,j = zi
1zj 2
ν(m)
Monday, October 31, 2011
C0,0 C0,1z2 z−1
1
Coefficients of the dimer partition function z1z−1
2
w4
1w3 2w2 3w4z2 2
C0,0 = 1 + w1 + w1w2 + w1w2w3 + w1w2w3w4 C0,1 = w2
1w2w3 + w2 1w2 2w3 + w2 1w2 2w3w4 + w3 1w2 2w3w4 + w3 1w2 2w2 3w4
Monday, October 31, 2011
εR,B = 0 and reverse sign if reverse vertex color or any path orientation.
1 2 −1 2
Proof of commutativity of Hamiltonians. ε is a sum of local contributions at vertices: εR,B(v) = 1
2
εR,B(v) = 0 εR,B(v) = 0 εR,B(v) = 1
Monday, October 31, 2011
{R, B} + {R∗, B∗} = εR,BRB + εR∗,B∗RB = (εR,B + εR∗,B∗)RB = 0 For a pair of dimer covers R, B, let R∗, B∗ be obtained by reversing colors on all topologically trivial loops. Lemma: topologically nontrivial loops give net contribution zero. also use
line bundles triple crossing diagram
N
Monday, October 31, 2011
is given by (ratios of) boundary coefficients of P. (twice the number of interior vertices). A basis for the Casimir elements
Theorem [Goncharov-K] This Poisson bracket defines a completely integrable system A quantum integrable system can be defined using wiwj = q2εijwjwi. q-commuting variables:
The commuting Hamiltonians are the normalized interior coefficients of P.
Monday, October 31, 2011
∂wi ∂t = {wi, H} ∂zi ∂t = {zi, H}
Monday, October 31, 2011
Start: a convex polygon with vertices in Z2.
Monday, October 31, 2011
Geodesics on the torus, one for each primitive edge of N.
Monday, October 31, 2011
Isotope to a “triple-crossing diagram” [D. Thurston]
respect circular order
no parallel double crossings
Monday, October 31, 2011
Monday, October 31, 2011
Monday, October 31, 2011
Lemma: |white vertices| = |black vertices| = |faces| = 2Area(N).
Obtain a bipartite graph
Monday, October 31, 2011
{wi} ↔ {p1, . . . , pk, q1, . . . , qk}
k
dpi pi ∧ dqi qi wi = det
where the Ai are “generalized Vandermonde” matrices in pj, qj. changing the symplectic form to the standard one Modding out by Casimirs, there is a change of variables
Monday, October 31, 2011
Consider for example the following graph
Monday, October 31, 2011
“A” variables
“generalized vandermonde”
Ai,j = first define
1 1 1 1 p1 p2 p3 p4 q1 q2 q3 q4 p1q1 p2q2 p3q3 p4q4
det
Monday, October 31, 2011
A1 A2 A3 A4 wi = det
Monday, October 31, 2011
1 1 1 1 p1 p2 p3 p4 p2
1
p2
2
p2
3
p2
4
p1q1 p2q2 p3q3 p4q4 p2
1q1
p2
2q2
p2
3q3
p2
4q4
p1 p2 p3 p4 q1 q2 q3 q4 p1q1 p2q2 p3q3 p4q4 p2
1q1
p2
2q2
p2
3q3
p2
4q4
p1 p2 p3 p4 p2
1
p2
2
p2
3
p2
4
p1q1 p2q2 p3q3 p4q4 det det det det wi =
1 1 1 1 p1 p2 p3 p4 q1 q2 q3 q4 p1q1 p2q2 p3q3 p4q4
Monday, October 31, 2011
∂wi ∂t = {wi, H} ∂zi ∂t = {zi, H}
Monday, October 31, 2011
Teichm¨ uller theory [FG] and theory of dimers. Here is a dictionary relating key objects in these three theories. Dimer Theory Teichm¨ uller Theory Cluster varieties Convex integral polygon N Oriented surface S with n > 0 punctures Minimal bipartite graphs ideal triangulations of S seeds
spider moves of graphs flips of triangulations seed mutations Face weights cross-ratio coordinates Poisson cluster coordinates Moduli space of spectral Moduli space of framed cluster Poisson variety data on toric surface N PGL(2) local systems Harnack curve + divisor complex structure on S Moduli space of Teichm¨ uller space of S positive real points of Harnack curves + divisors cluster Poisson variety Tropical Harnak curve Measured lamination with divisor Moduli space of tropical space of measured real tropical points of Harnack curve + divisors laminations on S cluster Poisson variety Dimer integrable system Integrable system related to pants decomposition of S Hamiltonians Monodromies around loops
What distinguishes these two examples – the dimer theory and the Teichm¨ uller theory – from the general theory of cluster Poisson varieties is that in each of them the set of real / tropical real points of the relevant cluster variety has a meaningful and non-trivial interpretation as the moduli space of some geometric objects. Here by moduli space of certain objects related to the toric surface N we mean the space parametrizing the orbits of the torus T acting on the objects. For example, the moduli space of spectral data means the space S/T. So combining results of [GK] with
Monday, October 31, 2011
Λ0 Λ1 Λ2 Λ3 Λ4 Λ0 Λ1 Λ2 Λ3 Λ4 w0 w1 w2 w3 w4 w0 w1 w2 w3 w4
Triangle flip
= λ−1 λ
1
= λ1(1 + λ0) λ
2
= λ2(1 + λ−1
0 )−1
λ
3
= λ3(1 + λ0) λ
4
= λ4(1 + λ−1
0 )−1
w = w−1 w
1
= w1(1 + w0) w
2
= w2(1 + w−1
0 )−1
w
3
= w3(1 + w0) w
4
= w4(1 + w−1
0 )−1
Urban renewal
Monday, October 31, 2011