6D SCFTs and Group Theory. Tom Rudelius Harvard University

Transcription

1 6D SCFTs and Group Theory Tom Rudelius Harvard University

2 Based On /hep-th with Jonathan Heckman, David Morrison, and Cumrun Vafa /hep-th with Jonathan Heckman /hep-th with Jonathan Heckman, Alessandro Tomasiello /hep-th with David Morrison /hep-th with Noppadol Mekareeya, Alessandro Tomasiello work in progress with Fabio Apruzzi, Jonathan Heckman

3 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

4 The Big Picture Group Theory 6D SCFTs Geometry

5 What is a 6D SCFT? S=supersymmetric (8 or 16 supercharges)! C=conformal symmetry! FT=Field theory in 5+1 dimensions

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9 Classification of 6D SCFTs

10 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

11 Classification of 6D SCFTs Looks like chemistry Atoms Radicals n

12 What is F-theory? Vafa 96 IIB: R 9,1 with position-dependent coupling = C 0 + ie T 2 R 9,1

13 All known 6D theories have F-theory avatar

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18 Dirac Pairing for String Charge Lattice Intersection Matrix! Dirac Pairing IJ negative definite, Curves contractible

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20 Blow-down Operations m n n m 1 n 1 n 1

21 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)

22 Coarse Classification of Bases Heckman, Morrison, Vafa 13 x 1 x 2 x 3 x r p q = x 1 1 x x r :(z 1,z 2 ) 7! (e 2 i/p z 1,e 2 iq/p z 2 ) B 2 = C 2 /

23 Complete Classification of Bases

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25 Minimal Gauge Algebras su f 4 e 8

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

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28 6D SCFTs and Homomorphisms

29 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

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

31 M5-Branes Probing C 2 / ADE A k 1 : su k su k su k su k su k 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 so 2k E 6 : e 6 e 6 2 e 6 su 3 su e 6 e 6

32 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 A = X p µ (p) C (p) F +[, ]= X p µ (p) R (p)

33 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

34 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 su 2 su 3 su 4 su 3 su 4 su su su 2 su 3 su 2 su 4 su su 2

35 Partial Ordering of Nilpotent Orbits O µ O, Ōµ O, µ, P m i=1 µt i P m i=1 T i 8m > > > >

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

37 Partial Ordering of Theories Can define a partial ordering on theories using RG flows: T 1 T 2, 9 flow T 1 T 2

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

39 6D SCFTs and Hom(su(2),D k )

40 6D SCFTs and Hom(su(2),E 6 )

41 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 so 7 su

42 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

43 6D SCFTs and Hom( ADE,E 8 ) For trivial boundary data, get 6D SCFT:!!!! e 8 g g g g {z } N For non-trivial boundary data, global symmetry is broken to a subgroup g g g g g L {z } N g g

44 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 su 2 su 3 su 3 su 3 e su 2 su 3 su 3 su 3 su 3 so su 3 su 3 su 3 su 3 e su 3 su 9 su 3 su 3 su 3 su 3 su

45 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 g g L g g g g g Hom(su(2), g) Hom( g,e 8 )

46 6D SCFTs and Automorphism Groups

47 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

48 Automorphism Groups Given 2 GL(n, Z), define Aut( ) by Aut( ) ={µ 2 GL(n, Z) µ T µ = }.

49 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

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

51 Automorphism Groups of 6D SCFTs In general, n 1 n 2 n 3 Aut( ) {z } {z } m 1 m 2 = Z 2 o S m1 +1 S m Aut(I k )=S k o Z k 2

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

53 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

54 (2,0) Automorphism Groups For a (2,0) SCFT, Aut( g ) = Aut(g) Group elements Permutations of M5-branes

55 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

56 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

57 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?