6D SCFTs and Group Theory Tom Rudelius Harvard University Based On - - PowerPoint PPT Presentation

6D SCFTs and Group Theory

Tom Rudelius Harvard University

Based On

• 1502.05405/hep-th
• with Jonathan Heckman, David Morrison, and Cumrun Vafa
• 1506.06753/hep-th
• with Jonathan Heckman
• 1601.04078/hep-th
• with Jonathan Heckman, Alessandro Tomasiello
• 1605.08045/hep-th
• with David Morrison
• 1612.06399/hep-th
• with Noppadol Mekareeya, Alessandro Tomasiello
• work in progress
• with Fabio Apruzzi, Jonathan Heckman

Outline

I. Classification of 6D SCFTs i. Tensor Branches/Strings

• ii. Gauge Algebras/Particles
• II. 6D SCFTs and Homomorphisms

i. ii.

• III. 6D SCFTs and Automorphism Groups
• i. Automorphism Groups
• ii. Geometric Phases

ΓADE → E8

su(2) → gADE

Geometry Group Theory 6D SCFTs

The Big Picture

What is a 6D SCFT?

• S=supersymmetric (8 or 16 supercharges)
• C=conformal symmetry
• FT=Field theory in 5+1 dimensions

Classification of 6D SCFTs

Classification of 6D SCFTs

• 6D SCFTs can be classified via F-theory
• Nearly all F-theory conditions can be phrased

in field theory terms

• 6D SCFTs = Generalized Quivers

Classification of 6D SCFTs

• Looks like chemistry

“Atoms” “Radicals”

2 2 2 2 2 3 3 3 n

What is F-theory?

Vafa ’96

IIB: R9,1 with position-dependent coupling τ = C0 + ie−Φ

R9,1 T 2

All known 6D theories have F-theory avatar∗

n

Intersection Matrix ← → Dirac Pairing 4 4 1

Dirac Pairing for String Charge Lattice

ΩIJ negative definite ⇔ Curves contractible

Blow-down Operations

n

n − 1 n − 1

m

m − 1

n

Coarse Classification of Bases*

Heckman, Morrison, Vafa ’13

(2,0) SCFT ⇔ Γ ⊂ SU(2) (1,0) SCFT ⇒ Γ ⊂ U(2)

*Bases related by blow-downs/blow-ups have same Γ ⊂ U(2)

Coarse Classification of Bases

Heckman, Morrison, Vafa ’13

x1 x2 x3 p q = x1 − 1 x2 − . . . 1

xr

B2 = C2/Γ

Γ : (z1, z2) 7! (e2πi/pz1, e2πiq/pz2)

xr

Complete Classification of Bases

n

4 6 3

su3

2

Minimal Gauge Algebras

1 8 12 5 e8

f4

Fiber Enhancements

+ 2 fundamentals + 2 fundamentals

4 4 4

+ 8v, 8s, 8c + 2 spinors

3 3 3

su3

6D SCFTs and Homomorphisms

6D SCFTs and Homomorphisms

• Large classes of 6D SCFTs have connections to

structures in group theory

• The correspondence has been verified explicitly

M5-brane theories T2-fibered CY3

Hom(su2, gADE) Hom(ΓADE, E8)

M5-Branes Probing C2/ΓADE

2 2 2 2

g g g g g C2/ΓADE N M5s

| {z }

N−1

10

g

M5-Branes Probing C2/ΓADE

2 2 2

suk suk suk suk suk

2 2 2

so2kso2k so2k so2k

e6 e6 e6

so2k

so2k so2k

4 4 1 1 1

spk−4 spk−4 spk−4

so2k

1 1 6 3

su3

3

su3

1 1

e6 e6

Ak−1 :

Dk : E6 :

Nilpotent Deformations

• Matrix of normal deformations characterizes

positions of 7-branes

• View intersection points of in base as

marked points

• Let adjoint field have singular behavior at

marked points Hitchin system coupled to defects:

CP1

Φ ⇒

∂AΦ = X

p

µ(p)

C δ(p)

F + [Φ, Φ†] = X

p

µ(p)

R δ(p)

Φ

Nilpotent Deformations

• Split , consider nilpotent part ,

get algebra:

• Adjoint vevs

µn su2

µC = µs + µn

J+ = µC J− = µ†

C

J3 = µR Φ ∼ µC dz z Hom(su(2), g)

⇒ Classified by

(equivalently, by nilpotent orbits J+)

6D SCFTs and Hom(su(2), Ak−1)

2 2 2 su4 su4 su4 su4 2 2 2 su4 su4

su3

su2

2 2 2 su4 su4

su2 su2

2 2 2 su4 su3 2 2 2 su2 su3 su2

Hom(su(2), Ak−1) labeled by partitions of k:

⇔ µ ≥ ν ⇔ Pm

i=1 µT i ≥ Pm i=1 νT i ∀m

Oµ ≥ Oν ⇔ ¯ Oµ ⊃ Oν

Partial Ordering of Nilpotent Orbits

> > > >

Renormalization Group Flows

TIR

TUV

High Energy Short Distance Long Distance Low Energy

Partial Ordering of Theories

• Can define a partial ordering on theories using

RG flows:

T1 T2 T1 ≥ T2 ⇔ ∃ flow

Nilpotent Orbit Ordering Matches RG Ordering!

2 2 2 su4 su4 su4 su4

su2

2 2 2 su4 su4

su2

2 2 2 su4 su4

su3

su2

2 2 2 su4 su3 su2 2 2 2 su3 su2

6D SCFTs and Hom(su(2), Dk)

6D SCFTs and Hom(su(2), E6)

Nilpotent Orbits and Global Symmetries

• Consider nilpotent orbit
• Let be subgroup of commuting with

nilpotent element

• Claim: is the global symmetry of the 6D

SCFT associated with

• E.g.

Oµ ∈ g F(µ) G F(µ) µ

1 6 3

su3

1 4

su2

1 6

2

su2

so7

• Consider M5-branes probing Horava-Witten wall

and singularity

• 6D SCFTs and Hom(ΓADE,E8)

C2/ΓADE N M5s E8 Wall

C2/ΓADE

Boundary data ' flat E8 connections on S3/ΓADE

10

• For trivial boundary data, get 6D SCFT:
• For non-trivial boundary data, global symmetry is

broken to a subgroup

6D SCFTs and Hom(ΓADE,E8)

2 2 2 1

g g g g g

| {z }

N

e8 2 2 2 1

g g g g g

| {z }

N

gL

6D SCFTs and Hom(ΓADE,E8)

Flat E8 connections on S3/ΓADE ⇔ Hom(ΓADE, E8)

E.g. ΓA2, Hom(Z3, E8):

2 2 2

1 e8

su3

2

su2

2 2 2

1

su3

2

su2

2 2 2

1 su3

2 2 2 2

1

2 2 2 2

1

2

e7

su3 su3 su3 su3 su3 su3 su2 so14 su3 su3 su3 su3

e6

su3 su3 su3 su3 su3 su3

su9

6D SCFTs and Homomorphisms

• Large classes of 6D SCFTs have connections to

structures in group theory

• The correspondence has been verified explicitly

2 2 2 1

g g g g g

gL

Hom(Γg, E8)

2 2 2 2

g g g g g

gL

Hom(su(2), g)

6D SCFTs and Automorphism Groups

6D SCFTs and Automorphism Groups

The Dirac pairing of a 6D SCFT has an associated automorphism group , which is calculable

Aut(Ω) Ω

Green-Schwarz Couplings Geometric Phases of B2

Aut(Ω)

Elements of

Automorphism Groups

Aut(Ω) = {µ ∈ GL(n, Z)|µT Ωµ = Ω}.

Given Ω ∈ GL(n, Z), define Aut(Ω) by

Automorphism Groups of 6D SCFTs

For 6D SCFT, Dirac pairing ,

Ω

Dirac pairing after blowing down all -1 curves Dirac pairing associated with k blow-downs

Aut(Ω) = Aut(Ωend) × Aut(Ik)

Automorphism Groups of 6D SCFTs

E.g.

4 4 1

Aut( )

3 3

Aut( )

=

Aut( ) =

2 2 1

Aut( )

1 3 1

= × Aut(I1) Aut(I3)

Automorphism Groups of 6D SCFTs

In general,

Aut( )

2 2 2 2 2

=

n1

n2 n3

| {z } | {z }

m1 m2

Z2 o Sm1+1 × Sm2+1 × ... Aut(Ik) = Sk o Zk

2

Outer Automorphisms

For a symmetric endpoint, contains an additional factor associated with the quiver symmetry:

Aut(Ω)

Aut( )

2 2 2 2

=

2 3

Z2 o (Z2 o (S2 × S2))

Symmetry of quiver from left -2 curve from right -2 curve

Green-Schwarz Couplings

• Group elements label distinct choices

for Green-Schwarz coupling

• L6 ⊃

Z µIJBI ∧ Tr(FJ ∧ FJ) µIJ ∈Aut(Ω) ⇔ Dirac Quantization

IGS ⊃ Tr(FI ∧ FI)µJIΩ−1

JKµKLTr(FL ∧ FL)

(2,0) Automorphism Groups

For a (2,0) SCFT, Aut(Ωg) = Aut(g) Group elements Permutations of M5-branes

1 2 4 3 1 2 4 3

Phases of 6D SCFTs

• For a general (1,0) SCFT, group

elements label tensor branch phases:

• These in turn correspond to geometric

phases of the base .

• X

I

µT

IJφI > 0

B2

• So far…
• 6D SCFTs have been classified
• There are remarkable relationships between 6D

SCFTs and two classes of homomorphisms

• Phases of 6D SCFTs are labeled by automorphism

groups of their Dirac pairing

Summary

• In the future…
• Can we classify full set of 6D RG Flows in terms
• f group theory data?
• Can we understand compactifications to lower

dimensions?

• Can we understand these algebraic/geometric

correspondences from a purely mathematical perspective?

Further Research