Brane Tilings and New Horizons Beyond Them
Calabi-Yau Manifolds, Quivers and Graphs
Sebastián Franco
Durham University
Lecture 2
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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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coBrane Tilings
Brane tilings are a higher dimensional generalizations of this type of brane setups
5
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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coGeometry and Seiberg Duality
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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coGeometry and Seiberg Duality
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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coDual Phases of del Pezzo 3
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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coOrbifolds
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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coOrbifolds and Brane Tilings
From a brane tiling perspective, the M × N orbifold corresponds to enlarging the unit cell to include M × N copies of the original one
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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
29
5
8
6
7
3
4
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coPartial Resolution and Higgsing
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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co Any toric geometry can be obtained by partial resolution of a M × N) orbifold for
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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coRunning Couplings in QFT
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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coGeometry of the Tiling and Conformal Invariance
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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coIsoradial Embedding and R-charges
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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coZig-Zag Paths
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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coBrane Tilings from Geometry
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
Feng, He, Kennaway, Vafa
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Applications and Extensions:Orientifolds
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coOrientifolds
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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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coOrientifolding Dimers
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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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coOrientifold Rules: Fixed Points
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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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coExamples
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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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coExamples
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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
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