The Superconformal Index and the Weyl Anomaly Great Lakes Strings - PowerPoint PPT Presentation

The Superconformal Index and the Weyl Anomaly Great Lakes Strings 2018, University of Chicago Brian McPeak Leinweber Center for Theoretical Physics University of Michigan Based on work presented in 1804.04155

Conformal Symmetry and Weyl Invariance Conformal or Weyl invariant theory is unchanged by g ab → e − 2 σ ( x ) g ab Invariance means the stress tensor satisfies T µ µ = 0 This symmetry might be anomalous – then � T � = g µν � T µν � � = 0 � T � can be given by curvature invariants. In 6D: � T � = − aE 6 + ( c 1 I 1 + c 2 I 2 + c 3 I 3 ) + D µ J µ , Euler density: E 6 = ǫǫ RRR Contractions of the Weyl tensor: I i = C 3

Holographic computations of the Weyl Anomaly c = c 2 − c 3 c ′ = c 1 − 4 c 2 c ′′ = c 1 − 2 c 2 + 6 c 3 , , . 32 192 192 Holographic computation at leading order Henningson, Skenderis (1998) In 6D, one-loop corrections are still an active area of research ◮ δ a was computed for all spins using AdS with a sphere boundary Beccaria, Tseytlin (2014) ◮ We computed δ ( c − a ) for spins ≤ 2 on Ricci-flat backgrounds Liu, McPeak (2017) ◮ With minimal (1,0) SUSY, c ′′ vanishes ◮ With (2,0) SUSY, c ′ and c ′′ vanish

Superconformal Index The 6D superconformal index : ˆ I ( p , q , s ) = Tr H ( − 1) j 1 + j 3 e − βδ q ∆ s j 1 p j 2 . ˆ ∆ = ∆ − 1 2 k δ = ˆ ∆ − 3 2 k − 1 2 ( j 1 + 2 j 2 + 3 j 3 ) Compute the index for short multiplets– general structure is: ∆ χ j 1 j 2 ( s , p ) ˆ I ( p , q , s ) ∼ q D ( p , q , s ) χ is the SU (3) character D comes from superconformal descendents

Group Theory Invariants The the anomaly and the index may both be written in terms of SU (3) invariants d ( j 1 , j 2 ) is the dimension I ( j 1 , j 2 ) are the Dynkin indices– e.g. Tr [ T R a T R b ] = 2 I 2 ( R ) δ ab For SU (3), d ( j 1 , j 2 ) = 1 2 ( j 1 + 1)( j 2 + 1)( j 1 + j 2 + 2) I 2 ( j 1 , j 2 ) = 1 12 d ( j 1 , j 2 )[ j 2 1 + 3 j 1 + j 1 j 2 + j 2 2 + 3 j 2 ] , I 3 ( j 1 , j 2 ) = 1 60 d ( j 1 , j 2 )( j 1 − j 2 )( j 1 + 2 j 2 + 3)(2 j 1 + j 2 + 3) , I 2 , 2 ( j 1 , j 2 ) = 3 � 8 I 2 ( j 1 , j 2 ) � 5 I 2 ( j 1 , j 2 ) d ( j 1 , j 2 ) − 1

Differential Operators ∆ χ j 1 j 2 ( s , p ) ˆ I ( p , q , s ) ∼ q D ( p , q , s ) We may write the Dynkin indices in terms of differential operators acting on the character: � d ( j 1 , j 2 ) = χ ( j 1 , j 2 ) ( s , p ) � s = p =1 ˆ I 2 ( j 1 , j 2 ) = 1 2 ( s ∂ s ) 2 χ ( j 1 , j 2 ) ( s , p ) � � s = p =1 I 3 ( j 1 , j 2 ) = ( p ∂ p )( s ∂ s ) 2 χ ( j 1 , j 2 ) ( s , p ) ˆ � � s = p =1 ˆ I 2 , 2 ( j 1 , j 2 ) = 1 2 ( s ∂ s ) 4 χ ( j 1 , j 2 ) ( s , p ) � � s = p =1

Sum Over Multiplets Fields in representations (∆ , j 1 , j 2 , j 3 ) k of U (1) × SU (4) × SU (2) R Short multiplets contribute to δ a , long multiplets give 0 For (1,0) theory, we find that δ a ∼ A ( j 1 , j 2 , ˆ ∆) � 4 � 4 A ( j 1 , j 2 , ˆ ˆ ∆) ∼ − 10 ∆ − 2 d ( j 1 , j 2 ) 3 � 2 � 4 [4 I 2 ( j 1 , j 2 ) + d ( j 1 , j 2 )] + 530 � 4 � ˆ ˆ + 20 ∆ − 2 ∆ − 2 I 3 ( j 1 , j 2 ) 3 9 3 − 80 9 [ I 2 , 2 ( j 1 , j 2 ) + 3 I 2 ( j 1 , j 2 )] − 11 3 d ( j 1 , j 2 ) , A depends on ( ˆ ∆ , j 1 , j 2 ) of lowest representation in the multiplet

Results We find the operator for δ a , � � δ a = O a D ( p , q , s ) I ( p , q , s ) � � p = q = s =1 where � 4 � 2 1 � � 4 � 4 (4ˆ O a = − 10 3 q ∂ q − 2 + 20 3 q ∂ q − 2 I 2 + 1) 2 5 · 6! � 4 � � + 530 I 3 − 80 I 2 ) − 11 ˆ 9 (ˆ I 2 , 2 + 3ˆ 3 q ∂ q − 2 9 3

Operator for ( c − a ) We only know δ ( c − a ) values for shortened multiplets starting at spin (0 , 0 , 0) However δ ( c − a ) is much simpler than a , which lets us almost determine the operator: 1 � � 4 � � ˆ I 3 + 1 + λ (ˆ I 2 , 2 − ˆ O ( c − a ) = − 90 3 q ∂ q − 2 I 2 ) 2 5 · 6!

Summary ◮ Superconformal index and Weyl anomaly both non-zero only for short multiplets ◮ Both can be written in terms of SU (3) invariants ◮ We find an operator which returns δ a when acting on the index ◮ δ ( c − a ) operator is determined up to one coefficient ◮ Fixing it requires anomaly for multiplets with spin > 2 Thank you!