Brane Tilings and New Horizons Beyond Them - PowerPoint Presentation

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Brane Tilings and New Horizons Beyond Them

Calabi-Yau Manifolds, Quivers and Graphs

Sebastián Franco

Durham University

Lecture 3

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Sebastian Franco

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Recent Developments 1: Cluster Integrable Systems

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Sebastian Franco

Multiple Applications of Brane Tilings

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

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



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

Quevedo et. al.

Feng, He, Kennaway, Vafa

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 integrable systems

Constructing

all integrals of motion

becomes straightforward

Mathematics

Physics

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Sebastian Franco

Constructing the Integrable System: Phase Space

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1

3

3

3

3

2

2

4

4

One wi variable per face:

Two 2-torus directions:

w1

z1

Example

:

F0

{wi,wj} = Iij wi wj

Iij: quiver intersection matrix

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

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

1

2

3

4

z2

exponential in p and q

Poisson Structure

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Sebastian Franco

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1

3

3

3

3

2

2

4

4

1

3

3

3

3

2

2

4

4

1

3

3

3

3

2

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)

Every perfect matching can be expressed in term of

loop variables

Goncharov, Kenyon

Eager, Franco, Schaeffer

The Integrable System

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Sebastian Franco

An Example: Integrable System from F0

Let us explore the explicit form of these operators for F0

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Recall this brane tiling has 9 perfect matchings

1

3

3

3

3

2

2

4

4

1 + w1 + w1w4 + w1w2 + w3-1

z2

z1-1

w1-1w2-1z2-1

w1w2 z1

Casimirs

:

Hamiltonian

:

C1 = z1z2

C2 = w1w2z2 / z1

C3 = 1/(w12w22z1z2)

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

It is straightforward to verify these operators commute following the commutation relations of the basic variables

Slide7

The Integrable System and Mirror Symmetry

Fully constructive

prescription for building an

integrable system

given a spectral curve

Feng, He, Kennaway, Vafa

Hamiltonians

Casimirs

Characteristic polynomial

: P(z1,z2)

coefficients and their ratios are Hamiltonians and Casimirs

Spectral curve

S

P(z1,z2) = 0

Mirror manifold

P(z1z2) = W

u v = W

Franco - Eager, Franco, Schaeffer

Interesting

:

this is also the wrapped M5-brane for 5d, N=1, pure SU(n) gauge theory on S1!

quiver/brane tiling

mirror manifold

This information is often encoded in the

spectral curve

of the integrable system

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Sebastian Franco

Connection to 4d and 5d Gauge Theory

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Among other things, we systematically answer the question: what is the integrable system associated to an arbitrary

4d N=2 gauge theory

? (spectral curve as

Seiberg-Witten curve

)

5d N=1 gauge theory on S1

M5-brane wrapped on

S

M-theory on CY3

Relativistic Integrable System

Spectral curve

S

Brane Tiling

S

inside mirror

4d N=2 gauge theory

Seiberg-Witten curve

s

Non-Relativistic Integrable System

Spectral curve

s

R → 0

pi → 0

Interestingly, some of these integrable systems have also emerged in connection to

other 5d and 4d gauge theories

(the latter should

not to be confused

with the 4d quivers we have been discussing)

Main underlying reason:

different avatars

of the spectral curve

S

Eager, Franco, Schaeffer

Slide9

An example: Relativistic Periodic Toda Chain

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Spectral curve

S

1

2

3

p-1

p

p+1

p+2

p+3

2p-1

2p

p/2 + 1

Nekrasov

reference p.m.

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

Brane Tiling

Following our discussion in a previous lecture, orbifolding corresponds to

enlarging the unit cell

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Sebastian Franco

Relativistic Toda Chain: The Integrable System

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Basic cyles

:

wi (i = 1, …, 2p), z1 and z2

di

i=1,…,p

ci-1

even i

ci

even i

Two additional cycles fixed by Casimirs

{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)

Bruschi, Ragnisco

Eager, Franco, Schaeffer

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Sebastian Franco

The Lax Operator from the Kasteleyn Matrix

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The

Kasteleyn matrix

is the

adjacency matrix

of the tiling

This is precisely the

Lax operator of the non-relativistic periodic Toda chain!

L(w) =

K =

~

~

~

~

~

~

~

P(z1,z2) = det K

Non-relativistic limit

:

linear orden in pi and z and define

L(w) - z ≡ K

Vi = Vi ≡ eq

i

-q

i+1

Hi = -Hi ≡ e(-1) p

i

/2

z1 ≡ e-z

z2 ≡ w

~

~

Rows:

Columns:

It controls

conserved quantities

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Sebastian Franco

Quiver Impurities = Spin Chain Impurities

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It is a member of a

family

, which is constructed by adding

impurities

to the chain. The spectral curves are given by:

Toric diagrams for the

cones over Y

p,q!

We constructed their quivers (equivalently

tilings

and

integrable systems) iteratively starting for Y

p,p and adding

(p-q) impurities

s

s

s

s

t

t

t

t

The

quiver impurities

are indeed in one-to-one with impurities in the

spin chains

Benvenuti, Franco, Hanany, Martelli, Sparks

Y4,0

Y4,1

Y4,2

Y4,3

Y4,4

Some of this integrable systems can also be realized as

spin chains

The relativistic, periodic Toda chain is one such example

(-1,p-q)

(0,p)

(0,0)

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Higgsing as a Generator of Integrable Systems

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In a previous lecture, we saw how brane tilings, and hence integrable systems, can be

systematically

constructed from any toric diagram

An alternative simple way to generate

new integrable systems from existing ones

is via the

Higgs mechanism (geometrically,

partial resolution)

1)

Remove loops containing an edge with a non-zero vev

2)

Re-express surviving loops with the replacement (wi wj) → wi/j

wj

wi

wj

wi

Start from the integrable system for the parent theory and turn on a vev for Xij

vev for Xij

Feng, He, Kennaway, Vafa

Hanany, Vegh

Eager, Franco, Schaeffer

Slide14

Conclusions

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Brane tilings define an infinite class of

integrable systems

The computation of

all integrals of motion

becomes straightforward

Many of these integrable systems are also associated to

5d N=1

and

4d N=2 gauge theories

Brane tilings provide a systematic procedure for constructing the integrable system for an arbitrary gauge theory of this type

Integrability

appears in multiple places in

QFT

and

String Theory

: Seiberg-Witten, vacua of supersymmetric theories, calculation of the spectrum of anomalous dimensions and scattering amplitudes in super-Yang-Mills. Interesting to explore the implications/applications of the new correspondence.

It is possible to extend these ideas to the continuous (1+1)-dimensional

integrable field theory

limit

Franco, Galloni, He

Sebastian Franco

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Sebastian Franco

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Recent Developments 2: Bipartite Field Theories

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Sebastian Franco

Bipartite Field Theories

a 4d N=1 gauge theory whose Lagrangian is defined by a bipartite graph on a Riemann surface (with boundaries)

Bipartite graph

Edge

:

chiral bifundamental

Face

:

U(N) group

Riemann surface

No superpotential term

Bipartite Field Theory

(BFT)

Franco

See also: Yamazaki, Xie

Node

:

superpotential term

Additional motivations:

on-shell scattering diagrams, ideal triangulations of Riemann surfaces and BPS quivers, cluster algebras from Riemann surfaces, etc

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Sebastian Franco

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Internal

boundary

External

Internal faces:

automatically

anomaly free

This is a rather natural choice in cases in which the graph has a

brane interpretation

Node color

:

External faces

Gauged

Global

There are

two types of faces

in the graph:

Sign

of corresponding superpotential term

Chirality

of bifundamental fields

Gauge and Global Symmetries

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Sebastian Franco

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1

3

4

2

5

6

9

8

7

10

9

1

3

4

2

6

8

7

10

The BFT is given by a quiver

dual

to the bipartite graph

The Dictionary

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Sebastian Franco

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Graphical Gauge Theory Dynamics

Franco

Yamazaki, Xie

When an Nf = Nc gauge group

confines

, the corresponding face is eliminated

Nf = Nc gauge group

Confinement

This process is often called

bubble reduction

Many transformations of the BFT work exactly as for brane tilings:

Integrating out massive fields

Seiberg duality

Higgsing

Condensation of 2-valent nodes

Urban renewal

Edge deletion

A

2-sided face

corresponds to a node in the quiver with a single incoming and a single outgoing arrows. This is an Nf = Nc gauge group.

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Sebastian Franco

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Reduced Graphs

Reduced Graph

:

a graph with the

minimal

number of

internal faces

within a class connected by moves and reductions

Only defined up to equivalence moves

not unique

3)

Confine Nf = Nc gauge groups

2)

Integrate out massive fields

1)

Seiberg dualize an Nf = 2Nc gauge group

Reduced graphs play a central role in scattering amplitudes

Two important questions

:

How to determine whether two graphs are connected by mergers and moves? And reductions?

How to identify reduced graphs

Example

:

Arkani-Hamed, Bourjaily, Cachazo, Goncharov, Postnikov, Trnka

Reduced Graph

:

a BFT with

minimal

gauge symmetry

within an IR equivalence class

Leading Singularities

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Sebastian Franco

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Perfect Matchings

Perfect matchings play a central role in connecting BFTs to

geometry

(moduli spaces)

(Almost) Perfect Matching:

p is a subset of the edges in G such that:

Every internal node is the endpoint of exactly on edge in p

Every external node belongs to either one or zero edges in p

p1

p2

p3

p4

p5

p6

p7

Once again, the

automatic solution of F-term equations

motivates the following

map

between chiral fields in the quiver Xi (edges) and perfect matchings p

m

:

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Sebastian Franco

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Generalized Kasteleyn Technology

Perfect matchings of bipartite graphs with boundaries can be efficently determined using a

generalization of the Kasteleyn matrix

technique

Franco

The starting point is the

master Kasteleyn matrix

of the graph:

All perfect matchings are then given by the following polynomial:

Slide23

Sebastian Franco

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Master space

space of solutions to F-term equations

parametrized in terms of

perfect matchings (GLSM fields) and

toric

1

2

3

4

5

Forcella, Hanany, He, Zaffaroni

7 perfect matchings

5d toric CY

Example

:

Master and Moduli Spaces

The

master and moduli spaces

of any BFT are

toric

CYs

(generically not 3-folds) and perfect matchings can be identified them with

GLSM fields

in their toric description

BFTs are naturally associated to certain geometries via their

master and moduli spaces

The Master Space

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Sebastian Franco

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The Moduli Space

Moduli Space

Space of solutions to vanishing F and D-terms

Projection

of the Master Space onto vanishing D-terms

One D-term contribution for every

gauge group

internal face

of the graph

2

The moduli space is

invariant

under all equivalence moves (integrating out massive fields, Seiberg duality, etc)

Ideal diagnostic for identifying graphs related by them

Perfect matchings whose difference is an internal loop

Example

:

1

2

3

4

5

Q1 =

4d toric CY

It applies to completely

general BFTs

. Other methods, e.g. permutations, exist for planar graphs

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Sebastian Franco

Stringy Embedding of BFTs

D3s

D7-D7



D3-D7



D3-D3



2-cycle

D7

D7’

4-cycle

It is possible to engineer generic planar BFTs using D-branes over toric CY 3-folds

Internal faces

Fractional D3-branes

External faces

Flavor D7-branes

Franco, Uranga

(to appear)

Heckman, Vafa, Yamazaki, Xie

(sub-classes)

Using mirror symmetry and QFT Higgsing, it is possible to determine the

spectrum

and

superpotential interactions

for a general D-brane configuration over toric singularities

D3-branes on toric CY 3-folds correspond to bipartite graphs on T2

RR Tadpole cancellation

Anomaly cancellation on every possible D-brane probe

D7

D3

Slide26

Sebastian Franco

Conclusions

We introduced BFTs, a new class of 4d, N = 1 gauge theories defined by

bipartite graphs

on

Riemann surfaces. We also developed efficient tools for studying them.

Gauge theory dynamics is captured by simple graph transformations

CY manifolds emerge as moduli spaces

Other Topics

We developed a full understanding of D3-D7 systems on toric CYs. This provides a D-brane embedding of BFTs but has many other applications

BFTs provide an alternative perspective on various equivalent systems: D-brane probes, integrable systems and on-shell diagrams

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For graphs on a

disk

, they are related to the classification of cells in the

positive Grassmannian

BFTs provide natural generalizations, based on standard N=1 gauge theory knowledge, of the Grassmannian objects

beyond the planar case

Slide27

Sebastian Franco

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The Future

BFTs generate

ideal triangulations

of Riemann surfaces (Seiberg-Witten and Gaiotto curves of 4d, N=2 theories)

N=2 BPS quivers

Explore the role of the master space and moduli space CYs for

scattering amplitudes

beyond the planar case

Geometric transitions and BFTs

Reducibility and Gauge Theory Dynamics

UV

IR

Multi-loop integrand

Leading singularity

Franco

Heckman, Vafa, Yamazaki, Xie

Alim, Cecotti, Cordova, Espahbodi, Rastogi, Vafa

Franco, Uranga

(to appear)

Slide28

Sebastian Franco

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The Future

RG flow interpretation of graph reductions?

Field theoretic criterion for graph reducibility?

If so, can we map the classification of leading singularities to a classification of IR fixed points?

Deconstruction

Two data points

:

the 6d (2,0) and little string theories on T2 are deconstructed by BFTs

on T2

BFTs might provide the natural framework for studying

6d gauge theories

via

deconstruction

. This could result in a more

physical understanding

of the emergence of certain mathematical structures such as the Grassmannian and cluster algebras

Arkani-Hamed, Cohen, Kaplan, Karch, Motl

Arkani-Hamed, Cohen, Georgi

Slide29

Sebastian Franco

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General Conclusions

Slide30

Sebastian Franco

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A Rich Web of Connections

There exists an

intricate web

connecting interesting objects and ideas

We understand some of them in great detail, but are just starting to investigate others. Exciting time!

Slide31

Sebastian Franco

S. Franco, A. Hanany, K. D. Kennaway, D. Vegh and B. Wecht, “Brane dimers and quiver gauge theories,” hep-th/0504110.

A Few References

K. D. Kennaway, “Brane Tilings,” arXiv:0706.1660 [hep-th].

M. Yamazaki, “Brane Tilings and Their Applications,” arXiv:0803.4474 [hep-th].

S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht, “Gauge theories from toric geometry and brane tilings,” hep-th/0505211.

A. Goncharov and R. Kenyon, “Dimers and cluster integrable systems,” arXiv:1107.5588 [math.AG]

R. Eager, S. Franco and K. Schaeffer, “Dimer Models and Integrable Systems,” arXiv:1107.1244 [hep-th].

S. Franco, “Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes,” arXiv:1207.0807 [hep-th].

D. Xie and M. Yamazaki, “Network and Seiberg Duality,” arXiv:1207.0811 [hep-th].

S. Franco, D. Galloni and R.-K. Seong, “New Directions in Bipartite Field Theories,” arXiv:1211.5139 [hep-th].

B. Feng, Y.-H. He, K. D. Kennaway and C. Vafa, “Dimer models from mirror symmetry and quivering amoebae,” hep-th/0511287.

Brane Tilings

Cluster Integrable Systems

Bipartite Field Theories

Slide32

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Sebastian Franco

Thank you Chiara and Susha!