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, 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