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Dimer Models, Integrable Systems and Gauge Theory

Maths of String and Gauge Theory, London

March 2012

Sebastián Franco

SLAC Theory Group

IPPP Durham University

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Gauge group: P 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)

The Quiver Gauge Theory

D-branes over Toric Singularities

CY3

N D3-branes

Torus fibrations over base spaces

Described by specifying shrinking cycles and their relations

Toric Geometry2-sphere

compact 4-cycle

Encoded by web or toric diagramsToric Diagram

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Brane Dimers

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 T2

3

11

1 1

2 2

4

4

1 2

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Example: complex cone over F0

Gauge Theory Dimer

SU(N) gauge group face

bifundamental (or adjoint) edge

superpotential term node

Franco, Hanany, Kennaway, Vegh, Wecht

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Perfect matching: configurations of edges such that every vertex in the graph is an endpoint of precisely one edge

Perfect Matchings and Geometry

Moduli Space: perfect matchings are the natural variables solving F-term equations

Franco, Vegh

Franco, Hanany, Kennaway, Vegh, Wecht

p1 p2 p3 p4 p5

p6 p7 p9p8

(n1,n2) crossings of (z1,z2) directions

Perfect Matchings and Geometry

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

p1, p2, p3, p4, p5

p8

p6

p9

p7

K =

white nodes

black nodes

Kasteleyn Matrix Toric Diagram

det K = P(z1,z2) = nij z1i z2

j Example: F0

Multiple Applications

Mathematics Physics

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

Dimers Models and Integrable Systems

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From a Dimer to Phase Space

1

3

3 3

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2

4 4

Poisson Structure

One wi variable per gauge group:

Two 2-torus directions:

w1

z1

Example: F0

{wi,wj} = Iij wi wj Iij: intersection matrix

Idem for {wi,zj} and {z1,z2}

e.g: {w1,w3} = 4 w1 w3 {w1,w2} = -2 w1 w2

1 2

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z2

exponential in p and q

1

3

3 3

32

2

4 4 1

3

3 3

32

2

4 4 1

3

3 3

32

2

4 4

Every perfect matching defines a closed path on the tiling by taking the difference with respect to a reference perfect matching

= = w1 w4

The commutators define a 0+1d quantum integrable system of dimension 2 + 2 Area (toric diagram), with symplectic leaves of dimension 2 Ninterior

Casimirs: ratios of boundary points (commute with everything)

Hamiltonians: internal points (commute with each other)

The Integrable System

Every perfect matching can be expressed in term of loops variables

Goncharov, Kenyon

Eager, Franco, Schaeffer

This theory has 9 perfect matchings

1

3

3 3

32

2

4 4

1 + w1 + w1w4 + w1w2 + w3-1

z2

z1-1

w1-1w2

-1z2-1

w1w4 z1

Casimirs:

Hamiltonian:

C1 = z1z2

C2 = w1w2z2 / z1

C3 = 1/(w12w2

2z1z2)

H = 1 + w1 + w1w4 + w1w2 + w3-1

An explicit example: F0

The Integrable System

Fully constructive prescription for building an integrable system given a spectral curve

e.g.: relativistic periodic Toda chain (Conifold/Zn)

quiver/dimer model mirror manifold

Feng, He, Kennaway, Vafa

Hamiltonians

Casimirs

Characteristic polynomial: P(z1,z2)coeficients and their ratios give

Hamiltonians and Casimirs

Spectral curve S

P(z1,z2) = 0

Mirror manifold

P(z1,z2) = W u v = W

Eager, Franco, Schaeffer

Franco

Brane configuration for: 5d, N=1, pure SU(n) gauge theory on S1

Connection to 4d and 5d Gauge Theory

Multiple avatars of the Riemann surface S

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 S1

M5-brane wrapped on S M-theory on CY3

Relativistic Integrable System

Spectral curve S

Dimer Model

S inside mirror

4d N=2 gauge theory

Seiberg-Witten curve s

Non-Relativistic Integrable System

Spectral curve s

R → 0 pi → 0

Eager, Franco, Schaeffer

An example: Relativistic Periodic Toda Chain

Spectral curve S

1 2 3 p-1 p

p+1 p+2 p+3 2p-1 2p

p/2 + 1

Nekrasov It corresponds to Yp,0 (Zp orbifold of the conifold)

Dimer model:

reference p.m.

5d, N=1, pure SU(p) gauge theory on S1

Basic cyles: wi (i = 1, …, 2p), z1 and z2

di

i=1,…,p

ci

even i

Ci-1

even i

Two additional cycles fixed by Casimirs

Relativistic Toda Chain: The Integrable System

{ck,dk} = ck dk

{ck,dk-1} = ck dk-1

{ck,ck+1} = - ck ck+1

Hk = S P ci dj Hamiltonians in terms of non-intersecting paths:

k factors

A more convenient basis:

H1 = S (ci + di)

Eager, Franco, Schaeffer

Bruschi, Ragnisco

The Kasteleyn matrix is the adjacency matrix of the dimer

This is precisely the Lax operator of the non-relativistic periodic Toda chain!

p1 eq1-q2 eqp-q1w

eq1-q2 p2 eq2-q3

eq2-q3

eqp-1-qp

eqp-q1w-1 eq2-q3 pp

L(w) =

-H1-H1z1V1 Vp z2

V1 H2+H2z1-1 V2

V2

Vp-1

Vp z2-1 Vp-1 Hp+Hpz1

-1

K =

~~

~

~~

~~

P(z1,z2) = det K

Non-relativistic limit: linear orden in pi and z and define

L(w) - z ≡ K

Vi = Vi ≡ eqi-qi+1 Hi = -Hi ≡ e(-1) pi/2 z1 ≡ e-z z2 ≡ w~ ~

Rows: Columns:

It controls conserved quantities

Quiver Impurities = Spin Chain Impurities

Relativistic, periodic Toda chain 5d, N=1, pure SU(p) on S1

Quantized cubic coupling in prepotential: ccl = 0, …, p (disappears in 4d limit)

These are the toric diagrams for Yp,q manifolds

Yp,p : C3/(Z2×Zp)

Yp,0 : conifold/Zp

Quivers constructed iteratively starting for Yp,p and adding (p-q) impurities

s s ss tttt The quiver impurities are indeed impurities in XXZ spin chains

ccl = q

Benvenuti, Franco, Hanany, Martelli, Sparks

(0,1)(0,0)

(0,p)

(-1,p-q)

Y4,0Y4,1Y4,2Y4,3Y4,4

Eager, Franco, Schaeffer

Conclusions

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 S. The commutation relations required by integrability imply Chekhov-Fock quantization. Franco

Future Directions

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

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Applications to 3d-3d generalizations of the Alday-Gaiotto-Tachikawa (AGT) correspondence

M3 = S × I Z3d SL(2,R) CS = Z3d 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, VafaAlso Gaiotto, Moore, Neitzke

Thank you!

Integrabe

Systems

5d, N=2 Gauge Theory

4d, N=1 Gauge Theory

4d, N=1, SCFT

quiversN=2 BPS

states

3d-3d AGT

Calabi-Yaus

Multiple Connections

Dimer models provide natural, systematic bridges connecting integrable systems to several physical systems.