Brane Tilings, CSTheories, M2 Branes
Amihay Hanany

In collaboration withAlberto Zaffaroni

Brane Tilings, CS Theories, and M2 Branes

David Vegh

Yang-Hui He

Motivation

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Motivation• How many CFT’s do we know in 2+1
dimensions?

6Motivation• How many CFT’s do we know in 2+1
dimensions?

• What is the world volume theory of a stackof M2 branes in M theory?

6Motivation• How many CFT’s do we know in 2+1
dimensions?

• What is the world volume theory of a stackof M2 branes in M theory?

• Understand Chern Simons (CS) theories
better6

Motivation: AdS/CFT

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Motivation: AdS/CFT• Long standing problem:

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Motivation: AdS/CFT• Long standing problem:• What is the theory dual to AdS x H , H
4

Sasaki Einstein

777Motivation: AdS/CFT• Long standing problem:• What is the theory dual to AdS x H , H
4

Sasaki Einstein•M2 probing CY4 = Cone over H7777

What is special in 2+1d?

8What is special in 2+1d?
• Gauge coupling of YM has dimension 1/2

8What is special in 2+1d?
• Gauge coupling of YM has dimension 1/2• All IR theories are strongly coupled

8

What is special in 2+1d?
• Gauge coupling of YM has dimension 1/2• All IR theories are strongly coupled• CS terms exactly marginal

8

What is special in 2+1d?
• Gauge coupling of YM has dimension 1/2• All IR theories are strongly coupled• CS terms exactly marginal• CS levels have dimension 08

What is special in 2+1d?
• Gauge coupling of YM has dimension 1/2• All IR theories are strongly coupled• CS terms exactly marginal• CS levels have dimension 0• Integer coefficients
8

What is special in 2+1d?
• Gauge coupling of YM has dimension 1/2• All IR theories are strongly coupled• CS terms exactly marginal• CS levels have dimension 0• Integer coefficients• Scale invariant
8

Simple observations in
2+1d CS theories

9Simple observations in
2+1d CS theories• No beta function for CS levels

9Simple observations in
2+1d CS theories• No beta function for CS levels• Finite renormalization - typically at 1 loop

9Simple observations in
2+1d CS theories• No beta function for CS levels• Finite renormalization - typically at 1 loop
• N=2 supersymmetry (4 supercharges): nocorrections

9

Simple observations in2+1d CS theories• No beta function for CS levels• Finite renormalization - typically at 1 loop
• N=2 supersymmetry (4 supercharges): nocorrections• Infinite family of SCFT’s parametrized by CSterms

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A lattice of SCFT’s

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A lattice of SCFT’s• For one gauge group - a 1d lattice of SCFT’s

10

A lattice of SCFT’s• For one gauge group - a 1d lattice of SCFT’s• For a product of G gauge groups a G
dimensional lattice of SCFT’s

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A lattice of SCFT’s• For one gauge group - a 1d lattice of SCFT’s• For a product of G gauge groups a G
dimensional lattice of SCFT’s• If put c conditions on CS levels G-cdimensional sub - lattice of SCFT’s

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Back to 3+1 dimensions

AdS/CFT12

AdS/CFT• Have a good understanding for the case of
D3 branes probing CY312

AdS/CFT• Have a good understanding for the case of
D3 branes probing CY3•

AdS5 x H5, H5 Sasaki Einstein base of CY312

AdS/CFT• Have a good understanding for the case ofD3 branes probing CY3•• Best description in terms of “Brane Tilings”
AdS5 x H5, H5 Sasaki Einstein base of CY312

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Periodic bipartite tiling13

Tiling - Quiverdictionary

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Tiling - Quiverdictionary• 2n sided face - U(N) Gauge group with nNflavors

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Tiling - Quiverdictionary• 2n sided face - U(N) Gauge group with nNflavors• Edge - A bi-fundamental chiral multipletcharged under the two gauge groups
corresponding to the faces it separates.

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Tiling - Quiverdictionary• 2n sided face - U(N) Gauge group with nNflavors• Edge - A bi-fundamental chiral multiplet

charged under the two gauge groups
corresponding to the faces it separates.

• k valent node - A k-th order interaction
term in the superpotential

14

Z3

3
orbifold of C
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Example: Conifold
16Comments

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Comments• Arrows are oriented in an alternating fashion

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Comments• Arrows are oriented in an alternating fashion• Graph is bi-partite: Nodes alternate between
clockwise (white) and counterclockwise(black) orientations of arrows

17Comments• Arrows are oriented in an alternating fashion• Graph is bi-partite: Nodes alternate between
clockwise (white) and counterclockwise(black) orientations of arrows• black (white) nodes connected to white(black) only

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Comments• odd sided faces are forbidden by anomaly
cancellation condition

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Comments• odd sided faces are forbidden by anomaly
cancellation condition• white nodes with + sign in thesuperpotential

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Comments• odd sided faces are forbidden by anomaly
cancellation condition• white nodes with + sign in thesuperpotential• black nodes with - sign in the superpotential

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Comments• odd sided faces are forbidden by anomaly
cancellation condition• white nodes with + sign in thesuperpotential• black nodes with - sign in the superpotential• These rules define a unique Lagrangian in
3+1 dimensions18

3+1d: we know how to3+1d: we know how to• Compute the moduli space of vacua

3+1d: we know how to• Compute the moduli space of vacua• Spectrum of scaling dimensions

3+1d: we know how to• Compute the moduli space of vacua• Spectrum of scaling dimensions• Central charge and volume of SE manifold

3+1d: we know how to• Compute the moduli space of vacua• Spectrum of scaling dimensions• Central charge and volume of SE manifold• Master space - Baryonic & mesonic moduli
space of vacua

3+1d: we know how to• Compute the moduli space of vacua• Spectrum of scaling dimensions• Central charge and volume of SE manifold• Master space - Baryonic & mesonic moduli
space of vacua• Hilbert Series - partition function to countthe spectrum of the Chiral Ring

2+1d Lagrangians

2+1d Lagrangians• Given a 2d periodic, bipartite tiling with Gtiles, add G CS levels, 1 for each tile.

2+1d Lagrangians• Given a 2d periodic, bipartite tiling with Gtiles, add G CS levels, 1 for each tile.

• Largest known family of SCFT’s in 2+1d!

2+1d: current results

2+1d: current results• Mesonic moduli space of vacua - CY4

2+1d: current results• Mesonic moduli space of vacua - CY4• interacting SCFT’s in the IR2+1d: current results• Mesonic moduli space of vacua - CY4• interacting SCFT’s in the IR• Non-trivial scaling dimensions

2+1d: current results• Mesonic moduli space of vacua - CY4• interacting SCFT’s in the IR• Non-trivial scaling dimensions• Master space - partial baryonic & mesonicmoduli space2+1d: current results• Mesonic moduli space of vacua - CY4• interacting SCFT’s in the IR• Non-trivial scaling dimensions• Master space - partial baryonic & mesonicmoduli space• Hilbert Series

SummarySummary• All theories described are conjectured tolive on the world volume of M2 branes
probing the CY4 - mesonic moduli spaceSummary• All theories described are conjectured tolive on the world volume of M2 branes
probing the CY4 - mesonic moduli space• Infinite families of SCFT’s

Summary• All theories described are conjectured tolive on the world volume of M2 branes
probing the CY4 - mesonic moduli space• Infinite families of SCFT’s• Count how many?

Summary• All theories described are conjectured tolive on the world volume of M2 branes
probing the CY4 - mesonic moduli space• Infinite families of SCFT’s• Count how many?
• Know for 2 terms in W and arbitrary G

The 2+1d Lagrangian−!

+ i+!

!

4d θ"

† −Va
VbXab eXab e

Xab4d θG"

a=1ka!1dtVa D̄ (e0d θW (Xab ) + c.c.
2αtVa−tVaDα e)Choice of CS levels

Choice of CS levels
G!

a=1

ka = 0,gcd({ka }) = 1

Choice of CS levelsG!

ka = 0,a=1

C=!

1k1

1
k2gcd({ka }) = 1

1k3...
...

1kG

".

Vacuum Equations∂Xab Wµa (X) :=G!
b=1†
Xab Xab−G!

=††
XcaXca + [Xaa , Xaa] =04ka σac=1

σa Xab − Xab σb=0

Forward Algorithm

Classification of 2+1dtheories

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Classification of 2+1dtheories• “order parameters”

27Classification of 2+1dtheories• “order parameters”• Number of gauge groups G

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Classification of 2+1dtheories• “order parameters”• Number of gauge groups G• Number of fields in the quiver E27

4 fields in the quiver5 fields in the Quiver6 fields in the QuiverG=2, E=4, Model IG=2, E=4, Model IIG=3, E=5, Model IG=4, E=6, Model IVCounting Quivers
1 Hexagon
f1 (t) ==1(1 − t)(1 − t2 )(1 − t3 )1 + t + 2t2 + 3t3 + . . .

Counting Quivers
Chessboard Tilingf2 (t) ==1 − t6(1 − t)(1 − t2 )2 (1 − t3 )(1 − t4 )1 + t + 3t + 4t + 8t + . . .
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Summary37

Summary• Infinitely many quivers

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Summary• Infinitely many quivers• Each represents a lattice of SCFT’s in 2+1d

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Summary• Infinitely many quivers• Each represents a lattice of SCFT’s in 2+1d
• A variety of scaling dimensions

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Summary• Infinitely many quivers• Each represents a lattice of SCFT’s in 2+1d
• A variety of scaling dimensions• Toric Duality

37Summary• Infinitely many quivers• Each represents a lattice of SCFT’s in 2+1d
• A variety of scaling dimensions• Toric Duality• ...
37