Brane Tilings and New Horizons Beyond Them

Calabi-Yau Manifolds, Quivers and Graphs

Sebastián Franco

Durham University

Lecture 2

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2

Outline: Lecture 2

Brane Tilings as Physical Brane Configurations

Graphical QFT Dynamics

Orbifolds

Scale Dependence in QFT

Partial Resolution of Singularities and Higgsing

From Geometry to Brane Tilings

Orientifolds

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Brane Tilings as Physical Brane Configurations

Seb

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coBrane Intervals

An alternative approach for engineering gauge theories using branes (dual to branes at singularities)

4

TD-brane ~ 1/gs

TNS5 ~ 1/gs2

The field theory lives in the common dimensions. In this case: 4d

The relative orientation of the branes controls the amount of SUSY

NS NS NS NS NS

NS’

NS

NS’

0 1 2 3 4 5 6 7 8 9

D4 × × × × ×

NS5 × × × × × ×

NS5’ × × × × × ×

4,5

7,86

D4-branes

NS5-branes

N=2 SUSY N=1 SUSY

Hanany, Witten

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Brane tilings are a higher dimensional generalizations of this type of brane setups

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The NS5-brane wraps a holomorphic curve S given by:

Where x and y are complex variables that combine the x4, x5, x6 and x7 directions

0 1 2 3 4 5 6 7 8 9

D5 × × × × × ×

NS5 × × × × S

x4

x6

D5-branes NS5-brane

Field theory dimensions

P(x,y) = 0

P(x,y) is the characteristic polynomial coming from the toric diagram

Franco, Hanany, Kennaway, Vegh, Wecht

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QFT Dynamics, Tilings and Geometry

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Graphical Gauge Theory Dynamics

P1(Xi) P2(Xi)

X1 X2

P1(Xi) × P2(Xi)

W=X 1 P1 (X i )+X 2 P2 ( X i )− X1 X2+⋯

𝜕𝑋 1W=0

⇔X 2=P1 ( X i )

𝜕𝑋 2W=0

⇔X1=P2 ( X i )

W=P1 (X i )P2 ( X i )+⋯

2-valent nodes map to mass terms in the gauge theory. Integrating out the corresponding massive fields results in the condensation of the two nearest nodes

The equations of motion of the massive fields become:

Massive Fields

We are mainly interested in the low energy (IR) limit of these theories

Gauge Theory Dynamics

Graph Transformations

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coGeometry and Seiberg Duality

D3s

This is a purely geometric manifestation of Seiberg duality of the quivers! Full equivalence of the gauge theories in the low energy limit

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Brane Tiling(Gauge Theory)Calabi-Yau 3-fold

What happens if this map is not unique?

Quiver 1 Quiver 2

F0

Feng, Franco, Hanany, He Franco, Hanany, Kennaway, Vegh, Wecht

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For the F0 example, the two previous quivers theories correspond to the following brane tilings

Seiberg duality corresponds to a local transformation of the graph: Urban Renewal

Theory 2 Theory 1

Seiberg duality is a fascinating property of SUSY quantum field theories. Sometimes, it allows us to trade a strongly coupled one for a weakly coupled, and hence computable, dual

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Franco, Hanany, Kennaway, Vegh, Wecht

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Seiberg dualizing twice, takes us back to the original theory

The Calabi-Yau geometry is automatically invariant under this transformation

CY Invariance Cluster Transformation

Seiberg Duality

From the perspective of the dual quiver, this corresponds to a quiver mutation

SD 1 SD 2

We have generated massive fields and can integrate them out

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Seb

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1

2

2

22

3

3

3

4

4

5

5

5

6

6

1

1

4 6

1

46

1 1

1

1

1

1

1

2

2

2

2

3

3

3

34

4

4

4

5

5

5

56

6

6

6

3

31

1

4

4

4

4

6

6 6

6

2

2 2

2

5 5

5

5 5

5

56

6

2

2

2

2

3

3 3

3

41

41 41

41 41

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Generate new geometries and gauge theories from known ones

Geometry

Gauge Theory

At the level of the quiver, it basically amounts to adding images for gauge groups and fields and projecting the superpotential onto invariant terms

X

Y

Z

𝑊=∑𝑖

(𝑋 𝑖 ,𝑖+1𝑌 𝑖+1 , 𝑖+ 2−𝑌 𝑖 , 𝑖+ 1 𝑋 𝑖+ 1, 𝑖+ 2)𝑍 𝑖+2 , 𝑖𝑊=[ 𝑋 ,𝑌 ]𝑍

: N=4 SYM 5

Example:N orbifolds of correspond to identifications under rotations by multiples of 2p/N on each plane

3

Quotienting by a discrete group such as N or N × M

2p/3ℂ

Orbifolds

4

1

2

3

5

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From a brane tiling perspective, the M × N orbifold corresponds to enlarging the unit cell to include M × N copies of the original one

13

7 8 9 7

1 2 3 1

4 5 6 4

7 8 9 7

1 1

1 1

3

3

33

3

3

21

The explicit action of the orbifold group maps to the choice of periodicity on the torus

We can orbifold arbitrary geometries, by taking the corresponding brane tilings as starting points

3 × 3)ℂ3

X

Y

Z

W = [X,Y] Z

N=4 super Yang-Mills1

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5

8

6

7

3

4

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Replacing points by 2-spheres and sending their size to infinity

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Eliminating points in the toric diagramPartial Resolution

Cone over dP2 Cone over dP1

U(N) × U(N) U(N)d

In the brane tiling, it corresponds to removing edges and merging faces

12

3

45 12

3

4512

3

45

12

3

45

12

3

4/5 12

3

4/512

3

4/5

12

3

4/5

Removing and edge corresponds to giving a non-zero vacuum expectation value to a bifundamental field Higgs Mechanism

Example:

Franco, Hanany, Kennaway, Vegh, Wecht

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sufficiently large M and N

1

12

23

3

2

23

3

p1 p2 p3 p4 p5 p6

X11 1 1

X12 1 1

X21 1 1

X23 1

X32 1

X31 1 1

X13 1 1

P =

1

12/3

2/3

2/3

2/31

12

23

3

2

23

3

p1 p2, p3 p4

p5 p6

Suspended Pinch Point

p1 p2, p3 p4

p5

1) 2 × Remove p6

Possible partial resolutions = possible sub-toric diagrams

Remove X23

1 2/3

2) Conifold Remove p1 Also remove e.g. p2 Remove e.g. X12

1/2 31

12

23

3

2

23

31/23

3

3

3

1/2

1/2

1/2

1/2p2 p4

p5 p6

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Scale Dependence in QFT

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in QFT, coupling constants generically depend on the energy scale L (they run)

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Standard Model

log10 L (GeV)

ai-1

5 10 15 200

60

80

40

20

0

U(1)

SU(2)

SU(3)

𝛼 𝑖❑−1=

4𝜋𝑔𝑖2

Remarkably, in SUSY field theories we know exact expressions for the b-functions:

Gauge couplings (NSVZ): Ri: superconformal R-charge of chiral multiplets

The models we will study are strongly coupled Superconformal Field Theories (SCFTs). This implies they are independent of the energy scale

Renormalization Group

Superpotential couplings:

The running of any coupling l is controlled by its b-function:

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In a SCFT, the beta functions for all superpotential and gauge couplings must vanish. When all ranks are equal:

Conformal invariance constraints the geometry of the tiling embedding

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Superpotential couplings For every node: ∑𝑖∈node

𝑅𝑖=2

Gauge couplings For every face: ∑𝑖∈ face

(1−𝑅𝑖 )=2

Nfaces + Nnodes - Nedges = () = 0

We will focus on the torus. It would be interesting to investigate whether bipartite graphs on the Klein bottle have any significance in String Theory

We conclude that conformal invariance requires the tiling to live on either a torus or a Klein bottle

Summing over the entire tiling

Franco, Hanany, Kennaway, Vegh, Wecht

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The vanishing of the beta functions now becomes:

Let us introduce the following change of variables:

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Superpotential couplings For every node: ∑𝑖∈node

𝜃𝑖=2𝜋

Gauge couplings For every face: ∑𝑖∈ face

(𝜋−𝜃𝑖 )=2𝜋

𝜃𝑖

R-charges can be traded for angles in the isoradial embedding

Isoradial Embedding: every face of the brane tiling is inscribed in a circle of equal radius

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From Geometry to Brane Tilings

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They can be efficiently implemented using a double line notation (alternating strands)

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oriented paths on the tiling that turn maximally left at white nodes and maximally right at black nodes

Zig-Zag Paths

Feng, He, Kennaway, Vafa

Example: F0

clockwise/counterclockwise around white/black nodes

They provide an alternative way for connecting brane tilings to geometry

every intersection gives rise to an edge

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Question: given a toric diagram, how do we determine the corresponding brane tiling(s)?

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Answer: the vectors normal to the external faces of the toric diagram determine the homology of zig-zag paths in the brane tiling Hanany, Vegh

(1,1)

(1,-1)

(-2,1)

(0,-1)1 2

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Del Pezzo 1

Seiberg duality corresponds to relative motion of the zig-zag paths

1

23

41

23

4

1

23

41

23

4

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Applications and Extensions:Orientifolds

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Dimer models solve the problem of finding the gauge theory on D-branes probing an arbitrary toric Calabi-Yau 3-fold singularity

Quotient by the action of:

w: worldsheet orientation reversal (in the quiver, it conjugates the head or tail of arrows)

s: involution of the Calabi-Yau

FL: left-moving fermion number

At the level of the gauge theory, it adds new possibilities:

New representations for fields: e.g. symmetric and antisymmetric

New gauge groups: symplectic and orthogonal

Orientifold Projection w s (-1)FL

The correspondence can be extended to more general geometries

Orientifold Planes: fixed point loci of s. Closed cousins of D-branes

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2 identification in the dimer

There are two classes of orientifolds:

Fixed points Fixed lines

Fixed points: preserve U(1)2 mesonic flavor symmetry

Fixed lines: projects U(1)2 to a U(1) subgroup

Fixed points and lines correspond to orientifold planes and come with signs that determine their type

There is a global constraint on signs for orientifolds with fixed points

1

1

1

1

2 2

2 2

1

1

1

1

2 2

2 2

1

1

1

1

2 2

2 2

1

1

1

1

2 2

2 2

OrientifoldingFranco, Hanany, Krefl, Park, Vegh

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Gauge group Matter

Signs: (+,+,+,-)

SO(N) + +

1

1

1

1a

bc

d

1

1

1

1a

bc

d

+1

1

1

1a

bc

d

+

+1

1

1

1a

bc

d

+

+

+

1

1

1

1a

bc

d

+

+

+

-

Superpotential: project parent superpotential

Supersymmetry constrains sign parity to be (-1)k for dimers with 2k nodes

O+/O- on face projects gauge group to SO(N)/Sp(N/2)

O+/O- on edge project bifundamental to /

Assign a sign to every orientifold point O+/O-

Orientifold of

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1

1

1

1

2

1

3

2

2

33

2

1

1

All these theories contain gauge anomalies unless the ranks of the gauge groups are restricted or (anti)fundamental matter is added.

Orientifolds of

Orientifolds of 3

For (-,+,+,+):

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1

6

5

4

2

1

2 1

2

1

2

3

6

5

Orientifolds of L1,5,2

2

1 2

1

3

3

2

1 2

1

Orientifolds of SPP