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. su(2)! g ADE ii. 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 Atoms Radicals n 3 2 2 3 2 3 2 2
What is F-theory? Vafa 96 IIB: R 9,1 with position-dependent coupling = C 0 + ie T 2 R 9,1
All known 6D theories have F-theory avatar
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Dirac Pairing for String Charge Lattice Intersection Matrix! Dirac Pairing 4 1 4 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, SU(2) (1,0) SCFT ) U(2) *Bases related by blow-downs/blow-ups have same U(2)
Coarse Classification of Bases Heckman, Morrison, Vafa 13 x 1 x 2 x 3 x r p q = x 1 1 x 2... 1 x r :(z 1,z 2 ) 7! (e 2 i/p z 1,e 2 iq/p z 2 ) B 2 = C 2 /
Complete Classification of Bases
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Minimal Gauge Algebras su 3 1 2 3 4 5 6 8 12 f 4 e 8
Fiber Enhancements 3 + 8v, 8s, 8c 4 + 2 fundamentals 3 + 2 spinors su 3 3 4 + 2 fundamentals 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 Hom(su 2, g ADE ) Hom( ADE,E 8 ) M5-brane theories T 2 -fibered CY3
M5-Branes Probing C 2 / ADE N M5s C 2 / ADE 10 g g g g g 2 2 2 2 {z } N 1 g
M5-Branes Probing C 2 / ADE A k 1 : su k su k su k su k su k 2 2 2 D k : so 2k so 2k so 2k 2 2 so 2k sp k 4 so 2k sp k 4 so 2k sp k 4 so 2k 1 4 1 4 1 so 2k E 6 : e 6 e 6 2 e 6 su 3 su 3 1 3 1 6 1 3 1 e 6 e 6
Nilpotent Deformations Matrix of normal deformations characterizes positions of 7-branes View intersection points of CP 1 in base as marked points Let adjoint field have singular behavior at marked points ) Hitchin system coupled to defects: @ A = X p µ (p) C (p) F +[, ]= X p µ (p) R (p)
Nilpotent Deformations Split µ C = µ s + µ n, consider nilpotent part, get su 2 algebra:!! J + = µ C J = µ C Adjoint vevs µ C dz z J 3 = µ R ) Classified by Hom(su(2), g) (equivalently, by nilpotent orbits J + ) µ n
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 su 4 2 2 2 su 2 su 3 su 4 su 3 su 4 su 4 2 2 2 su 2 2 2 2 su 2 su 3 su 2 su 4 su 4 2 2 2 2 2 2 su 2
Partial Ordering of Nilpotent Orbits O µ O, Ōµ O, µ, P m i=1 µt i P m i=1 T i 8m > > > >
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 2, 9 flow T 1 T 2
Nilpotent Orbit Ordering Matches RG Ordering! su 4 su 4 2 su 4 su 4 2 2 su 2 su 3 2 su 4 su 4 2 2 su 2 2 su 4 su 4 2 2 su 2 su 2 2 su 3 su 4 2 2 2 su 2 su 3 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 2 g Let F (µ) be subgroup of G commuting with nilpotent element Claim: F (µ) is the global symmetry of the 6D SCFT associated with µ E.g. su 3 su 2 4 1 3 1 6 so 7 su 2 2 1 6
6D SCFTs and Hom( ADE,E 8 ) Consider M5-branes probing Horava-Witten wall and singularity!!! N! M5s E 8 Wall!! C 2 / ADE C 2 / ADE Boundary data ' flat E 8 connections on S 3 / ADE 10
6D SCFTs and Hom( ADE,E 8 ) For trivial boundary data, get 6D SCFT:!!!! e 8 g g g g 1 2 2 2 {z } N For non-trivial boundary data, global symmetry is broken to a subgroup g g g g g L 1 2 2 2 {z } N g g
6D SCFTs and Hom( ADE,E 8 ) Flat E 8 connections on S 3 / ADE, Hom( ADE,E 8 ) E.g. A 2, Hom(Z 3,E 8 ): su 2 su 3 su 3 e 8 1 2 2 2 2 su 2 su 3 su 3 su 3 e 7 1 2 2 2 2 su 2 su 3 su 3 su 3 su 3 so 14 1 2 2 2 2 su 3 su 3 su 3 su 3 e 6 1 2 2 2 2 su 3 su 9 su 3 su 3 su 3 su 3 su 3 1 2 2 2 2
6D SCFTs and Homomorphisms Large classes of 6D SCFTs have connections to structures in group theory The correspondence has been verified explicitly g L g g g g 2 2 2 2 g g L g g g g 1 2 2 2 g Hom(su(2), g) Hom( g,e 8 )
6D SCFTs and Automorphism Groups
6D SCFTs and Automorphism Groups The Dirac pairing of a 6D SCFT has an associated automorphism group Aut( ), which is calculable Elements of Aut( ) Green-Schwarz Couplings Geometric Phases of B2
Automorphism Groups Given 2 GL(n, Z), define Aut( ) by Aut( ) ={µ 2 GL(n, Z) µ T µ = }.
Automorphism Groups of 6D SCFTs For 6D SCFT, Dirac pairing, Aut( ) = Aut( end ) Aut(I k ) Dirac pairing after blowing down all -1 curves Dirac pairing associated with k blow-downs
E.g. Automorphism Groups of 6D SCFTs Aut( 4 1 4 ) = Aut( 3 3 ) Aut(I 1 ) Aut( 1 2 2 ) = Aut( 1 3 1 ) = Aut(I 3 )
Automorphism Groups of 6D SCFTs In general, n 1 n 2 n 3 Aut( 2 2 2 2 2 ) {z } {z } m 1 m 2 = Z 2 o S m1 +1 S m2 +1... Aut(I k )=S k o Z k 2
Outer Automorphisms For a symmetric endpoint, Aut( ) contains an additional factor associated with the quiver symmetry: Aut( 2 32 2 ) = Z 2 o (Z 2 o (S 2 S 2 )) Symmetry of quiver from left -2 curve from right -2 curve
Green-Schwarz Couplings Group elements label distinct choices!!!!! for Green-Schwarz coupling L 6 Z µ IJ B I ^ Tr(F J ^ F J ) I GS Tr(F I ^ F I )µ JI 1 JK µ KLTr(F L ^ F L ) µ IJ 2Aut( ), Dirac Quantization
(2,0) Automorphism Groups For a (2,0) SCFT, Aut( g ) = Aut(g) Group elements Permutations of M5-branes 1 2 3 4 1 3 4 2
Phases of 6D SCFTs For a general (1,0) SCFT, group!!! elements label tensor branch phases: X µ T IJ I > 0 These in turn correspond to geometric! phases of the base. I 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?