6D SCFTs and Group Theory. Tom Rudelius IAS

Transcription

1 6D SCFTs and Group Theory Tom Rudelius IAS

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 Noppadol Mekareeya, Alessandro Tomasiello work in progress with Darrin Frey Thanks to Carl P. Feinberg for financial support of this research.

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. Implications for 6D SCFTs i. The a-theorem in 6D ii. Classification of RG Flows

4 Classification of 6D SCFTs

5 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

6 Classification of 6D SCFTs Looks like chemistry Atoms c.f. Morrison, Taylor 12 Radicals n

7 All known 6D theories have F-theory avatar *up to subtleties involving frozen singularities, see Alessandro s talk

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9 n

10 n

11 Minimal Gauge Algebras su 3 f e 8

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

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14 6D SCFTs and Homomorphisms

15 6D SCFTs and Group Theory 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 )

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

17 Nilpotent Deformations Matrix of normal deformations characterizes positions of 7-branes View intersection points of CP 1 in base as marked points Can 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)

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

19 6D SCFTs and Hom(su(2),A k 1 ) Hom(su(2),A k 1 ) labeled by partitions of k: su 2 su 3 su 3 su su su su 2 su su su 2

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

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

22 RG Flows in 6D SCFTs T UV su 3 T IR su

23 Nilpotent Hierarchy Matches RG Hierarchy! su 2 su su su 2 su 2 2 su su 2 su 3 2 2

24 6D SCFTs and Hom(su(2),D k )

25 6D SCFTs and Hom(su(2),E 6 )

26 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 O µ E.g. su 3 su so 7 su

27 6D SCFTs and Hom( ADE,E 8 ) Consider M5-branes probing Horava-Witten wall and singularity C 2 / ADE C 2 / ADE N M5s E 8 Wall Boundary data ' flat E 8 connections on S 3 / ADE ' Hom( ADE,E 8 ) 10

28 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

29 6D SCFTs and 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

30 Classification of Hom( ADE,E 8 ) A n case: done (Kac 83) E 8 case: done (Frey 98) D n case: open! E 6 case: open! E 7 case: open!

31 Classification of Hom( Dn,E 8 ) Hom( Dn ' Dic n 2,E 8 ) are uniquely labeled by a nilpotent orbit of D n together with a simple Lie algebra! E.g. : D 5! E 8 sp 2 so 10 sp 1 so 10 + so $ sp 2 sp 2 so 10 sp 1 so

32 6D SCFTs and Group Theory 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 )

33 Geometry and Group Theory Physics 6D SCFTs Elliptically-fibered CY3 s Homomorphisms Math

34 Implications for 6D SCFTs

35 Implications for 6D SCFTs There is significant evidence for the a-theorem (and an infinite collection of other c-theorems) in 6D SCFTs Connections to group theory provide a proof in certain classes of RG flows We speculate that a full classification of RG flows among 6D SCFTs is possible through these connections to group theory

36 t Hooft Anomalies in 6D SCFTs Anomaly polynomial calculable for any 6D SCFT Ohmori, Shimizu, Tachikawa, Yonekura 14 I = c 2 (R) 2 + c 2 (R)p 1 (T )+ p 1 (T ) 2 + p 2 (T )+... Trace anomaly related to 6D Euler density 3 1 ht µ µ i = ae Can be expressed in terms of anomaly polynomial: a = 8 ( + )+ 3 Cordova, Dumitrescu, Intriligator 15

37 Expand a curve in base to large size / Tensor Branch 6 e fundamental fundamental

38 Evidence for the a-theorem Tensor branch flows: a-theorem proven! Cordova, Dumitrescu, Intriligator 15 6 e fundamental Higgs branch flows: numerical sweep Heckman, T.R fundamental 6

39 Nilpotent Orbit SCFTs Can relate anomalies to data of nilpotent orbit su r1 su r2 su r3 su r su f1 su f2 su f3 su f4 ) = 12 X i,j = N 1 = = i,j r X ir j + 2(N 1) i 1 X ri 2 2! X i r i f i + 30(N 1) C 1 Cremonesi, Tomasiello 15 d H d O Allows for proof of a-theorem for these flows 7 2 i! X r i f i + 60(N 1) i r 2 i

40 Summary and Future Research So far Classified 6D SCFTs in terms of CY3 s Found relationships between 6D SCFTs and two classes of homomorphisms Found strong evidence for the a-theorem in 6D

41 Summary and Future Research In the future Can mathematics give deeper insight into the geometry-group theory correspondence? Can we classify full set of 6D RG Flows in terms of group theory data? Can we prove a-theorem in full generality?