integrable systems in the dimer model R. Kenyon (Brown) A. Goncharov (Yale)

Transcription

1 integrable systems in the dimer model R. Kenyon (Brown) A. Goncharov (Yale)

2 1. Convex integer polygon triple crossing diagram 2. Minimal bipartite graph on T 2 line bundles 3. Cluster integrable system.

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4 w 3 w 5 w 4 w 3 w 1 f e w 2 d w 5 w 2 = ace bdf a c w 4 b w 3 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.

5 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 =1if w i w j ε ij = 1 if w i w j ε ij = 0 else. 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. genus 2 in the example

6 w 3 w 5 w 4 w 1 w 5 w 3 w 2 w 4 w 3 ε = w 1 w 2 w 3 w 4 w 5 z 1 z 2 w

7 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 (w2 1w 2 w 3 + w 2 1w 2 2w 3 + w 2 1w 2 2w 3 w 4 + w 3 1w 2 2w 3 w 4 + w 3 1w 2 2w 2 3w 4 ) C 1 = z1z 2 2 C 2 = w1w 4 2w 3 3w 2 4 z 1 z2 3 Casimirs (commute with everything)

8 w 3 w 5 w 4 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

9 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.

10 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)

11 Let M(G) be the set of dimer covers of G. Define the partition function P (z 1,z 2 )= dimer covers m ν(m)z i 1z j 2 ( 1)ij The normalized coefficients of P (z 1,z 2 ) are the Hamiltonians. (divide by weight of a zig-zag path) H i,j = z i 1z j 2 m Ω i,j ν(m)

12 w 4 1w 3 2w 2 3w 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 1w 2 w 3 + w 2 1w 2 2w 3 + w 2 1w 2 2w 3 w 4 + w 3 1w 2 2w 3 w 4 + w 3 1w 2 2w 2 3w 4

13 Proof of commutativity of Hamiltonians. ε is a sum of local contributions at vertices: ε R,B (v) = 1 ε 2 R,B (v) =0 ε R,B (v) =1 ε R,B (v) =0 and reverse sign if reverse vertex color or any path orientation ε R,B =0

14 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.

15 1. Convex integer polygon N triple crossing diagram 2. Minimal bipartite graph on T 2 line bundles 3. Cluster integrable system.

16 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 2ε ij w j w i.

17 z i t = {z i,h} w i t = {w i,h}

18 Start: a convex polygon with vertices in Z 2.

19 Geodesics on the torus, one for each primitive edge of N.

20 no parallel double crossings respect circular order Isotope to a triple-crossing diagram [D. Thurston]

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23 Obtain a bipartite graph Lemma: white vertices = black vertices = faces = 2Area(N).

24 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: w i = det det (A 1 ) (A 2 ) det det (A 3 ) (A 4 ) where the A i are generalized Vandermonde matrices in p j,q j.

25 Consider for example the following graph

26 first define A variables det A i,j = p 1 p 2 p 3 p 4 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

27 the w variables (monodromies) are cross ratios of these: A 1 w i = det det (A 1 ) (A 2 ) det det (A 3 ) (A 4 ) A 4 A 2 A 3

28 w i = det det p 2 1q 1 p 2 2q 2 p 2 3q 3 p 2 4q 4 p 1 p 2 p 3 p 4 q 1 q 2 q 3 q 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4 p 2 1q 1 p 2 2q 2 p 2 3q 3 p 2 4q 4 p 1 p 2 p 3 p 4 p 2 1 p 2 2 p 2 3 p 2 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4 det det p 1 p 2 p 3 p 4 p 2 1 p 2 2 p 2 3 p 2 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q p 1 p 2 p 3 p 4 q 1 q 2 q 3 q 4 p 1 q 1 p 2 q 2 p 3 q 3 p 4 q 4

29 z i t = {z i,h} w i t = {w i,h}

30 What distinguishes these two examples the dimer theory and the Teichmüller theory fromthegeneraltheoryofclusterpoissonvarietiesisthatineachofthemthesetof 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 Monday, October moduli 31, 2011space of spectral data means the space S/T. So combining results of [GK] with Teichmüller theory [FG] and theory of dimers. Here is a dictionary relating key objects in these three theories. Dimer Theory Teichmüller Theory Cluster varieties Convex integral polygon N Oriented surface S with n>0punctures 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 Teichmüller 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

31 Triangle flip Λ1 Λ 4 Λ 0 Λ 2 Λ 3 Λ1 Λ 4 Λ 0 Λ 2 Λ 3 λ 0 = λ 1 0 λ 1 = λ 1 (1 + λ 0 ) λ 2 = λ 2 (1 + λ 1 0 ) 1 λ 3 = λ 3 (1 + λ 0 ) λ 4 = λ 4 (1 + λ 1 0 ) 1 Urban renewal w 1 w 4 w 0 w 2 w 3 w 1 w 4 w 0 w 2 w 3 w 0 = w0 1 w 1 = w 1 (1 + w 0 ) w 2 = w 2 (1 + w0 1 w 3 = w 3 (1 + w 0 ) w 4 = w 4 (1 + w 1 0 ) 1