1. 1 Sebastián Franco SLAC Theory Group IPPP Durham University

2. 2  Gauge group:  SU(N)  F-terms:  monomial = monomial  On the worldvolume of D3-branes, N =1 superconformal field theory with: Every field appears exactly twice in W with opposite signs (Toric Condition) CY3 N D3-branes  Torus fibrations over base spaces  Described by specifying shrinking cycles and their relations Toric Geometry 2-sphere compact 4-cycle  Encoded by web or toric diagrams Toric Diagram

3. 3 The dimer model is a physical configuration of NS5 and D5-branes  All the information defining the gauge theory can be encoded in a dimer model on T 2 3 1 1 1 1 2 2 4 4 12 34  Example:  Example: complex cone over F 0 Gauge TheoryDimer SU(N) gauge groupface bifundamental (or adjoint) edge superpotential termnode Franco, Hanany, Kennaway, Vegh, Wecht

4. 4  Perfect matching:  Perfect matching: configurations of edges such that every vertex in the graph is an endpoint of precisely one edge  Moduli Space:  Moduli Space: perfect matchings are the natural variables solving F-term equations Franco, Vegh Franco, Hanany, Kennaway, Vegh, Wecht p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p9p9 p8p8 (n1,n2) crossings of (z1,z2) directions

5. 5  This correspondence trivialized formerly complicated problems such as the computation of the moduli space of the SCFT, which reduces to calculating the determinant of an adjacency matrix of the dimer model (Kasteleyn matrix)  There is a one to one correspondence between perfect matchings and GLSM fields describing the toric singularity (points in the toric diagram) Franco, Vegh Franco, Hanany, Kennaway, Vegh, Wecht p 1, p 2, p 3, p 4, p 5 p8p8 p6p6 p9p9 p7p7 K = white nodes black nodes Kasteleyn Matrix Toric Diagram det K = P(z 1,z 2 ) =  n ij z 1 i z 2 j Example: F 0

6. 6  Local constructions of MSSM + CKM  Dynamical SUSY breaking  AdS/CFT correspondence in 3+1 and 2+1 dimensions BPS invariants of CYs (e.g. DT)  Mirror symmetry  Toric/Seiberg duality  D-brane instantons   Define an infinite class of interesting objects: largest classification of 4d, N=1 SCFTs  Make previous complicated calculations trivial: determination of their moduli space  The power of dimer models: Can they do it again? YES!  Define an infinite class of quantum integrable systems  Constructing all integrals of motion becomes straightforward Eager, SF SF, Hanany, Kennaway, Vegh, Wecht SF, Hanany, Krefl, Park, Uranga SF, Uranga SF, Hanany, Martelli, Sparks, Vegh, Wecht SF, Hanany, Park, Rodriguez-Gomez SF, Klebanov, Rodriguez-Gomez

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8. 8 1 3 33 32 2 44  One w i variable per gauge group:  Two 2-torus directions: w1w1 z1z1 Example: F 0  {w i,w j } = I ij w i w j I ij : intersection matrix  Idem for {w i,z j } and {z 1,z 2 } e.g: {w 1,w 3 } = 4 w 1 w 3 {w 1,w 2 } = -2 w 1 w 2 12 3 4 z2z2  exponential in p and q

9. 1 3 33 32 2 44 1 3 33 32 2 441 3 33 32 2 44  Every perfect matching defines a closed path on the tiling by taking the difference with respect to a reference perfect matching == w 1 w 4 The commutators define a 0+1d quantum integrable system of dimension 2 + 2 Area (toric diagram), with symplectic leaves of dimension 2 N interior  Casimirs: ratios of boundary points (commute with everything)  Hamiltonians: internal points (commute with each other)  Every perfect matching can be expressed in term of loops variables Goncharov, Kenyon Eager, Franco, Schaeffer

10. 10  This theory  This theory has 9 perfect matchings 1 3 33 32 2 44 1 + w 1 + w 1 w 4 + w 1 w 2 + w 3 -1 z2z2 z 1 -1 w 1 -1 w 2 -1 z 2 -1 w 1 w 4 z 1  Casimirs:  Hamiltonian: C 1 = z 1 z 2 C 2 = w 1 w 2 z 2 / z 1 C 3 = 1/(w 1 2 w 2 2 z 1 z 2 ) H = 1 + w 1 + w 1 w 4 + w 1 w 2 + w 3 -1

11.  Fully constructive prescription for building an integrable system given a spectral curve e.g.: relativistic periodic Toda chain (Conifold/Z n ) quiver/dimer model mirror manifold Feng, He, Kennaway, Vafa Hamiltonians Casimirs  Characteristic polynomial: P(z 1,z 2 ) coeficients and their ratios give Hamiltonians and Casimirs Spectral curve  P(z 1,z 2 ) = 0 Mirror manifold P(z 1, z 2 ) = Wu v = W Eager, Franco, Schaeffer Franco Brane configuration for: 5d, N=1, pure SU(n) gauge theory on S 1

12. 12  Multiple avatars of the Riemann surface   Among other things, we systematically address the question: what is the integrable system associated to an arbitrary 4d N=2 gauge theory? (spectral curve as SW curve) 5d N=1 gauge theory on S 1  M5-brane wrapped on   M-theory on CY3 Relativistic Integrable System  Spectral curve  Dimer Model   inside mirror 4d N=2 gauge theory  Seiberg-Witten curve  Non-Relativistic Integrable System  Spectral curve  R → 0p i → 0 Eager, Franco, Schaeffer

13. 13  Spectral curve  1 2 3p-1p p+1 p+2 p+32p-12p p/2 + 1 Nekrasov  It corresponds to Y p,0 (Z p orbifold of the conifold)  Dimer model: reference p.m. 5d, N=1, pure SU(p) gauge theory on S 1

14. 14  Basic cyles: w i (i = 1, …, 2p), z 1 and z 2 didi i=1,…,p cici even i C i-1 even i  Two additional cycles fixed by Casimirs {c k,d k } = c k d k {c k,d k-1 } = c k d k-1 {c k,c k+1 } = - c k c k+1 H k =   c i d j  Hamiltonians in terms of non-intersecting paths: k factors  A more convenient basis: H 1 =   c i + d i ) Eager, Franco, Schaeffer Bruschi, Ragnisco

15. 15  The Kasteleyn matrix is the adjacency matrix of the dimer  This is precisely the Lax operator of the non-relativistic periodic Toda chain! p1p1 e q 1 -q 2 e q p -q 1 w e q 1 -q 2 p2p2 e q 2 -q 3 e q p- 1 -q p e q p -q 1 w -1 e q 2 -q 3p L(w) = -H 1 -H 1 z 1 V1V1 V p z 2 V1V1 H 2 +H 2 z 1 -1 V2V2 V2V2 V p-1 V p z 2 -1 V p-1 H p +H p z 1 -1 K = ~ ~ ~ ~ ~ ~ ~ P(z 1,z 2 ) = det K  Non-relativistic limit: linear orden in p i and z and defineL(w) - z ≡ K V i = V i ≡ e q i -q i+1 H i = -H i ≡ e (-1) p i /2 z 1 ≡ e -z z 2 ≡ w ~~  Rows:  Columns: It controls conserved quantities

16. 16  Relativistic, periodic Toda chain  5d, N=1, pure SU(p) on S 1  Quantized cubic coupling in prepotential: c cl = 0, …, p (disappears in 4d limit)  These are the toric diagrams for Y p,q manifolds  Y p,p : C 3 /(Z 2 ×Z p )  Y p,0 : conifold/Z p  Quivers constructed iteratively starting for Y p,p and adding (p-q) impurities   The quiver impurities are indeed impurities in XXZ spin chains  c cl = q Benvenuti, Franco, Hanany, Martelli, Sparks (0,1) (0,0) (0,p) (-1,p-q) Y 4,0 Y 4,1 Y 4,2 Y 4,3 Y 4,4 Eager, Franco, Schaeffer

17. 17  In addition, they define an infinite class of quantum integrable systems  The computation of all integrals of motion becomes straightforward  These integrable systems are also associated to 5d N=1 and 4d N=2 gauge theories  Dimer models provide a systematic procedure for constructing the integrable system for an arbitrary gauge theory of this type  Dimer models are brane configurations in String Theory connecting Calabi-Yau’s and quantum field theories in various dimensions  Quantum Teichmüller Space: one-to-one correspondence between edges in dimer models and Fock coordinates in the Teichmüller space of . The commutation relations required by integrability imply Chekhov-Fock quantization. Franco

18.  Study the continuous (1+1)-dimensional integrable field theory limit  Classification of possible integrable impurities and interfaces in integrable field theories Franco, Galloni, He, and in progress 18  Applications to 3d-3d generalizations of the Alday-Gaiotto-Tachikawa (AGT) correspondence M 3 =  × I Z 3d SL(2,R) CS = Z 3d N=2 theory Terashima, Yamazaki  Connection to quivers encoding the BPS spectrum of N=2 gauge theories, obtained from ideal triangulations of the SW curve. Alim, Cecotti, Cordova, Espahbodi, Rastogi, Vafa Also Gaiotto, Moore, Neitzke

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20. 20 Integrabe Systems 5d, N=2 Gauge Theory 4d, N=1 Gauge Theory 4d, N=1, SCFT quivers N=2 BPS states 3d-3d AGT Calabi- Yaus  Dimer models provide natural, systematic bridges connecting integrable systems to several physical systems.