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, October 31, 2011
Monday, October 31, 2011
w 3 w 4 w 5 e w 1 w 5 f d w 2 = ace bd f w 3 w 2 a c w 4 w 3 b w 5 Line bundle on graph = edge weights modulo gauge = monodromies around faces ( w i ) and homology generators of torus ( z 1 , z 2 ) subject to one condition: � w i = 1. Monday, October 31, 2011
Define a Poisson structure on the moduli space of line bundles by the formula { w i , w j } = ε ij w i w j (extend using Leibniz rule) where ε is a skew-symmetric form ε ij = − 1 if ε ij = 1 if w j w i w j w i ε ij = 0 else. genus 2 in the example A similar rule for { w i , z j } and { z i , z j } . ε is the intersection form on cycles on the “conjugate” surface obtained by reversing the cyclic orientation at black vertices. Monday, October 31, 2011
w 3 w 4 w 5 w 1 w 5 w 3 w 2 w 4 w 3 w 5 0 1 2 − 2 − 1 1 − 2 w 1 − 1 0 1 2 − 2 0 0 w 2 − 2 − 1 0 1 2 − 2 4 w 3 2 − 2 − 1 0 1 0 0 w 4 ε = 1 2 − 2 − 1 0 1 − 2 w 5 − 1 0 2 0 − 1 0 0 z 1 2 0 − 4 0 2 0 0 z 2 Monday, October 31, 2011
Goal: define commuting Hamiltonians. H 1 = z − 1 1 (1 + w 1 + w 1 w 2 + w 1 w 2 w 3 + w 1 w 2 w 3 w 4 ) H 2 = z − 1 1 ( w 2 1 w 2 w 3 + w 2 1 w 2 2 w 3 + w 2 1 w 2 2 w 3 w 4 + w 3 1 w 2 2 w 3 w 4 + w 3 1 w 2 2 w 2 3 w 4 ) C 1 = z 2 1 z 2 (commute with everything) Casimirs C 2 = w 4 1 w 3 2 w 2 3 w 4 z 1 z 3 2 Monday, October 31, 2011
w 3 w 4 w 5 w 1 w 5 w 3 w 2 w 4 w 3 w 5 A “zig-zag” path is in the kernel of ε (and these generate the kernel). “Casimirs” Monday, October 31, 2011
The Hamiltonians are normalized sums of weighted dimer covers A dimer cover has a weight = product of edge weights. Fix a “base point” dimer cover. 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 . Define the partition function � 1 z j ν ( m ) z i 2 ( − 1) ij P ( z 1 , z 2 ) = dimer covers m The normalized coefficients of P ( z 1 , z 2 ) are the Hamiltonians. 1 z j � H i,j = z i ν ( m ) 2 (divide by weight of a zig-zag path) m ∈ Ω i,j Monday, October 31, 2011
w 4 1 w 3 2 w 2 3 w 4 z 2 2 Coefficients of the dimer partition function C 0 , 1 z 2 z − 1 1 C 0 , 0 z 1 z − 1 2 C 0 , 0 = 1 + w 1 + w 1 w 2 + w 1 w 2 w 3 + w 1 w 2 w 3 w 4 C 0 , 1 = w 2 1 w 2 w 3 + w 2 1 w 2 2 w 3 + w 2 1 w 2 2 w 3 w 4 + w 3 1 w 2 2 w 3 w 4 + w 3 1 w 2 2 w 2 3 w 4 Monday, October 31, 2011
Proof of commutativity of Hamiltonians. ε is a sum of local contributions at vertices: ε R,B ( v ) = 1 ε R,B ( v ) = 1 ε R,B ( v ) = 0 ε R,B ( v ) = 0 2 and reverse sign if reverse vertex color or any path orientation. − 1 2 1 2 ε R,B = 0 Monday, October 31, 2011
For a pair of dimer covers R, B , let R ∗ , B ∗ be obtained by reversing colors on all topologically trivial loops. { R, B } + { R ∗ , B ∗ } = ε R,B RB + ε R ∗ ,B ∗ RB = ( ε R,B + ε R ∗ ,B ∗ ) RB = 0 also use Lemma : topologically nontrivial loops give net contribution zero. � Monday, October 31, 2011
1. Convex integer polygon N triple crossing diagram 2. Minimal bipartite graph on T 2 line bundles 3. Cluster integrable system. Monday, October 31, 2011
Theorem [Goncharov-K] This Poisson bracket defines a completely integrable system of dimension 2 + 2Area( N ), with symplectic leaves of dimension 2int( N ), (twice the number of interior vertices). A basis for the Casimir elements is given by (ratios of) boundary coefficients of P . The commuting Hamiltonians are the normalized interior coefficients of P . A quantum integrable system can be defined using q -commuting variables: w i w j = q 2 ε ij w j w i . Monday, October 31, 2011
∂z i ∂t = { z i , H } ∂w i ∂t = { w i , H } Monday, October 31, 2011
Start: a convex polygon with vertices in Z 2 . Monday, October 31, 2011
Geodesics on the torus, one for each primitive edge of N . Monday, October 31, 2011
no parallel double crossings respect circular order Isotope to a “triple-crossing diagram” [D. Thurston] Monday, October 31, 2011
Monday, October 31, 2011
Monday, October 31, 2011
Obtain a bipartite graph Lemma: | white vertices | = | black vertices | = | faces | = 2Area( N ). Monday, October 31, 2011
Modding out by Casimirs, there is a change of variables { w i } ↔ { p 1 , . . . , p k , q 1 , . . . , q k } changing the symplectic form to the standard one k dp i ∧ dq i � p i q i i =1 � � � � it has the form: det A 1 det A 3 w i = � � � � A 2 A 4 det det where the A i are “generalized Vandermonde” matrices in p j , q j . Monday, October 31, 2011
Consider for example the following graph Monday, October 31, 2011
first define “A” variables A i,j = 1 1 1 1 p 1 p 2 p 3 p 4 det q 1 q 2 q 3 q 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4 “generalized vandermonde” Monday, October 31, 2011
the w variables (monodromies) are cross ratios of these: � � � � det A 1 det A 3 A 1 w i = � � � � A 2 A 4 det det A 4 A 2 A 3 Monday, October 31, 2011
p 2 p 2 p 2 p 2 1 1 1 1 1 q 1 2 q 2 3 q 3 4 q 4 p 1 p 2 p 3 p 4 p 1 p 2 p 3 p 4 det det p 2 p 2 p 2 p 2 q 1 q 2 q 3 q 4 1 2 3 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4 w i = p 2 p 2 p 2 p 2 1 1 1 1 1 q 1 2 q 2 3 q 3 4 q 4 p 1 p 2 p 3 p 4 p 1 p 2 p 3 p 4 det det p 2 p 2 p 2 p 2 q 1 q 2 q 3 q 4 1 2 3 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4 Monday, October 31, 2011
∂z i ∂t = { z i , H } ∂w i ∂t = { w i , 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 on a torus 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 of a pants decomposition 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
Triangle flip Λ 4 Λ 4 � λ − 1 λ � = 0 0 λ � = λ 1 (1 + λ 0 ) 1 � Λ 0 Λ 0 � � λ 2 (1 + λ − 1 Λ 1 Λ 3 Λ 1 Λ 3 0 ) − 1 λ � = 2 λ � = λ 3 (1 + λ 0 ) 3 λ 4 (1 + λ − 1 0 ) − 1 λ � = 4 � Λ 2 Λ 2 Urban renewal w 4 w 4 � w − 1 w � = 0 0 w � = w 1 (1 + w 0 ) 1 w 2 (1 + w − 1 0 ) − 1 w � = w 1 w 0 w 3 w 1 w 0 w 3 � � � 2 w � = w 3 (1 + w 0 ) 3 w 4 (1 + w − 1 0 ) − 1 w � = 4 w 2 w 2 � Monday, October 31, 2011
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