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 su (2) → g ADE i. ii. Γ ADE → E 8 III. 6D SCFTs and Automorphism Groups i. Automorphism Groups ii. Geometric Phases

The Big Picture Group Theory 6D SCFTs Geometry

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 “Radicals” “Atoms” n 2 3 3 2 2 3 2 2

What is F-theory? Vafa ’96 IIB: R 9 , 1 with position-dependent coupling τ = C 0 + ie − Φ T 2 R 9 , 1

All known 6D theories have F-theory avatar ∗

n

Dirac Pairing for String Charge Lattice Intersection Matrix ← → Dirac Pairing 4 1 4 Ω IJ negative definite ⇔ Curves contractible

Blow-down Operations m n n m − 1 n − 1 n − 1

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 x 1 x 2 x 3 x r 1 p q = x 1 − x 2 − . . . 1 x r Γ : ( z 1 , z 2 ) 7! ( e 2 π i/p z 1 , e 2 π iq/p z 2 ) B 2 = C 2 / Γ

Complete Classification of Bases

n

Minimal Gauge Algebras f 4 e 8 su 3 5 8 2 3 6 12 1 4

Fiber Enhancements 4 3 + 8 v , 8 s , 8 c + 2 fundamentals 3 4 + 2 spinors + 2 fundamentals su 3 3 4

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 T 2 -fibered Hom( su 2 , g ADE ) theories CY 3 Hom( Γ ADE , E 8 )

M5-Branes Probing C 2 / Γ ADE C 2 / Γ ADE 10 N M5s g g g g 2 2 2 2 g g | {z } N − 1

M5-Branes Probing C 2 / Γ ADE A k − 1 : su k su k su k 2 2 2 su k su k D k : sp k − 4 so 2 k so 2 k sp k − 4 so 2 k so 2 k sp k − 4 1 1 2 so 2 k 4 1 4 2 so 2 k so 2 k so 2 k E 6 : e 6 su 3 su 3 e 6 2 3 e 6 e 6 1 6 1 3 e 6 1 1

Nilpotent Deformations • Matrix of normal deformations characterizes Φ positions of 7-branes • View intersection points of in base as CP 1 marked points • Let adjoint field have singular behavior at Φ marked points Hitchin system coupled to ⇒ defects: µ ( p ) µ ( p ) X X F + [ Φ , Φ † ] = R δ ( p ) ∂ A Φ = C δ ( p ) p p

Nilpotent Deformations • Split , consider nilpotent part , µ C = µ s + µ n µ n get algebra: su 2 � J − = µ † J + = µ C J 3 = µ R C � dz • Adjoint vevs Φ ∼ µ C z ⇒ Classified by Hom( su (2) , g ) (equivalently, by nilpotent orbits J + )

6D SCFTs and Hom( su (2) , A k − 1 ) Hom( su (2) , A k − 1 ) labeled by partitions of k : su 4 su 4 su 4 2 2 2 su 4 su 2 su 3 su 4 2 2 2 su 4 su 4 su 3 2 2 2 su 2 su 2 su 3 2 2 2 su 4 su 4 su 2 2 2 2 su 2

Partial Ordering of Nilpotent Orbits O µ ≥ O ν ⇔ ¯ O µ ⊃ O ν ⇔ µ ≥ ν ⇔ P m i ≥ P m i =1 µ T i =1 ν T i ∀ m > > > >

Renormalization Group Flows High Energy Short Distance T UV T IR Low Energy Long Distance

Partial Ordering of Theories • Can define a partial ordering on theories using RG flows: T 1 T 1 ≥ T 2 ∃ flow ⇔ T 2

Nilpotent Orbit Ordering Matches RG Ordering! su 4 su 4 su 4 2 2 2 su 4 su 3 su 4 su 4 2 2 2 su 2 su 4 su 4 su 2 2 2 2 su 2 su 2 su 3 su 4 2 2 2 su 2 su 3 2 2 2

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

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

Nilpotent Orbits and Global Symmetries • Consider nilpotent orbit O µ ∈ g • Let be subgroup of commuting with F ( µ ) G nilpotent element • Claim: is the global symmetry of the 6D F ( µ ) SCFT associated with µ • E.g. su 3 3 6 1 1 4 su 2 su 2 6 2 1 so 7

6D SCFTs and Hom( Γ ADE , E 8 ) • Consider M5-branes probing Horava-Witten wall and singularity C 2 / Γ ADE � C 2 / Γ ADE � 10 � � N M5s E 8 Wall � � Boundary data ' flat E 8 connections on S 3 / Γ ADE

6D SCFTs and Hom( Γ ADE , E 8 ) • For trivial boundary data, get 6D SCFT: � g g g g � 1 2 2 2 e 8 g � | {z } N � • For non-trivial boundary data, global symmetry is broken to a subgroup g g g g 1 2 2 2 g g L | {z } N

6D SCFTs and Hom( Γ ADE , E 8 ) Flat E 8 connections on S 3 / Γ ADE ⇔ Hom( Γ ADE , E 8 ) E.g. Γ A 2 , Hom( Z 3 , E 8 ): su 2 su 3 su 3 1 2 2 e 8 2 2 su 2 su 3 su 3 su 3 e 7 1 2 2 2 2 su 3 su 2 su 3 su 3 su 3 2 2 1 2 2 so 14 su 3 su 3 su 3 su 3 1 2 2 e 6 2 2 su 3 su 3 su 3 su 3 su 3 su 3 2 1 2 2 su 9 2

6D SCFTs and Homomorphisms • Large classes of 6D SCFTs have connections to structures in group theory • The correspondence has been verified explicitly g g g g g g g g 2 2 2 2 g 1 2 2 2 g g L g L Hom( Γ g , E 8 ) Hom( su (2) , g )

6D SCFTs and Automorphism Groups

6D SCFTs and Automorphism Groups The Dirac pairing of a 6D SCFT has an associated Ω automorphism group , which is calculable Aut( Ω ) Elements of Green-Schwarz Geometric Couplings Phases of B 2 Aut( Ω )

Automorphism Groups Given Ω ∈ GL ( n, Z ), define Aut( Ω ) by Aut( Ω ) = { µ ∈ GL ( n, Z ) | µ T Ω µ = Ω } .

Automorphism Groups of 6D SCFTs For 6D SCFT, Dirac pairing , Ω Aut( Ω ) = Aut( Ω end ) × Aut( I k ) Dirac pairing after Dirac pairing associated blowing down all -1 with k blow-downs curves

Automorphism Groups of 6D SCFTs E.g. Aut( ) Aut( ) × Aut( I 1 ) 3 3 4 1 4 = Aut( ) = Aut( ) 1 2 2 1 3 1 Aut( I 3 ) =

Automorphism Groups of 6D SCFTs In general, Aut( ) 2 n 1 2 2 2 2 n 2 n 3 | {z } | {z } m 1 m 2 = Z 2 o S m 1 +1 × S m 2 +1 × ... Aut( I k ) = S k o Z k 2

Outer Automorphisms For a symmetric endpoint, Aut( Ω ) contains an additional factor associated with the quiver symmetry: Aut( ) Z 2 o ( Z 2 o ( S 2 × S 2 )) 2 2 3 2 2 2 = Symmetry of quiver from left -2 curve from right -2 curve

Green-Schwarz Couplings • Group elements label distinct choices for Green-Schwarz coupling � Z µ IJ B I ∧ Tr( F J ∧ F J ) L 6 ⊃ � � � I GS ⊃ Tr( F I ∧ F I ) µ JI Ω − 1 JK µ KL Tr( F L ∧ F L ) � µ IJ ∈ Aut( Ω ) ⇔ Dirac Quantization

(2,0) Automorphism Groups For a (2,0) SCFT, Aut( Ω g ) = Aut( g ) Permutations of Group elements M5-branes 1 3 4 2 1 2 3 4

Phases of 6D SCFTs • For a general (1,0) SCFT, group elements label tensor branch phases: � X µ T IJ φ I > 0 � I � • These in turn correspond to geometric phases of the base . B 2 �

Summary • 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

Further Research • In the future… • Can we classify full set of 6D RG Flows in terms of group theory data? • Can we understand compactifications to lower dimensions? • Can we understand these algebraic/geometric correspondences from a purely mathematical perspective?