Aspects of SCFTs and their susy deformations

Transcription

1 Aspects of SCFTs and their susy deformations Ken Intriligator (UC San Diego) 20 years of F-theory workshop, Caltech Thank the organizers, Caltech, and F-theory Would also like to thank my

2 Spectacular Collaborators Clay Córdova Thomas Dumitrescu : 6d conformal anomaly a from t Hooft anomalies 6d a-thm for N=(1,0) susy theories : Classify susy-preserving deformations for d>2 SCFTs + to appear & work in progress

3 What is QFT? perturbation theory around free field Lagrangian theories CFTs + perturbations susy F-theory realizations non-lagrangian higher dim l thy compactified unexplored something crucial for the future?

4 # dof RG flows UV CFT (+relevant) RG course graining The deformations examples: IR CFT (+irrelevant) L = (OK even if SCFT is non-lagrangian) g i O i i Move on the moduli space of (susy) vacua Gauge a (eg UV or IR free) global symmetry Will here focus on RG flows that preserve supersymmetry

5 RG flow constraints d=even: t Hooft anomaly matching for all global symmetries (including NGBs + WZW terms for spont broken ones + Green-Schwarz contributions for reducible ones) Weaker d=odd analogs, eg parity anomaly matching in 3d Reducing # of dof intuition For d=2,4 (& d=6?) : a-theorem d=even: a UV a IR a 0 For any ht µ µ i ae d + X i c i I i unitary theory (d=odd: conjectured analogs, from sphere partition function / entanglement entropy) Additional power from supersymmetry

6 6d a-theorem? For spontaneous conf l symm breaking: dilaton has derivative interactions to give a anom matching Schwimmer, Theisen; Komargodski, Schwimmer 6d case: L dilaton = 1 2 ( )2 b ( )4 + a ( )6 3 6 (schematic) Maxfield, Sethi; Elvang, Freedman, Hung, Kiermaier, Myers, Theisen Can show that b>0 (b=0 iff free) but b s physical interpretation was unclear; no conclusive restriction on sign of Clue: observed that, for case of (2,0) on Coulomb branch, a b 2 >0 Cordova, Dumitrescu, KI: this is a general req t of N=(1,0) susy, and b is related to an t Hooft anomaly matching term a

7 Longstanding hunch eg Harvey Minasian, Moore 98 Susy multiplet of anomalies: should be able to relate a-anomaly to R-symmetry t Hooft-type anomalies in 6d, as in 2d and 4d T µ g µ J µ,a R A a R,µ Stress-tensor supermultiplet Sources = bkgrd SUGRA supermultiplet T T µ a? T T 4-point fn with too many indices Hard to get a, and hard to compute susy? J µ,a,t µ J,a,T I 8 J,a,T J,a,T Easier to isolate anomaly term, and enjoys anomaly matching

8 (1,0) t Hooft anomalies I origin 8 = 1 4! c 2 2(R)+ c 2 (R)p 1 (T )+ p 2 1(T )+ p 2 (T ) c 2 (R) tr(f SU(2) R ^ F SU(2)R ) p 1 (T ) 1 tr(r ^ R) 8 2 Background gauge fields and metric ( ~ background SUGRA) Computed for (2,0) SCFTs + many (1,0) SCFTs Harvey, Minasian, Moore; KI; Ohmori, Shimizu, Tachikawa; Ohmori, Shimizu, Tachikawa, Yonekura; Del Zotto, Heckman, Tomasiello, Vafa; Heckman, Morrison, Rudelius, Vafa Eg for theory of N small E8 instantons: E N : (,,, ) =(N(N 2 +6N + 3), N 2 (6N + 5), 7 8 N, N 2 ) Ohmori, Shimizu, Tachikawa (Leading N 3 coeff can be anticipated from Z2 orbifold of AN-1 (2,0) case)

9 (1,0) on tensor branch I origin 8 = 1 4! c 2 2(R)+ c 2 (R)p 1 (T )+ p 2 1(T )+ p 2 (T ) t Hooft anomaly matching requires I 8 I origin tensor branch 8 I 8 X 4 X 4 must be a perfect square, match I8 via X4 sourcing B: L GSW S = ib X 4 KI ; Ohmori, Shimizu, Tachikawa, Yonekura X (xc 2 (R)+yp 1 (T )) for some real coefficients x, y Our classification of defs gives: L tensor = Q 8 (O) L dilaton + L GSW S Then b = 1 (y x) 2 Adapting a SUGRA analysis of Bergshoeff, Salam, Sezgin 86 (!) Upshot: a origin = 16 7 ( + )+6 7

10 Change gears ( ) Classify susy-preserving deformations of SCFTs L =Q N Q O long D-term eg Kaher potential in 4d N=1 SCFT unitarity, bound grows with dim d: ( L) > 1 2 N Q + min (O long ) Irrelevant Eg for 6d N=(1,0) such operators have > = 10 L =Q n top O short ( L) = 1 2 n top + (O short ) eg F-terms, W in 4d N=1 Constrained by SCFT unitarity Short reps classified, in terms of the superconf l primary operator at the bottom of the multiplet Theory independent, just using SCFT rep constraints We study the Q descendants, looking for Lorentz scalar top ops Some oddball susypreserving ops do exist, including in middle of multiplet(!) We had to be careful - it s risky to claim a complete classification (embarrassing if something is overlooked)! Much more subtle and sporadic zoo than we originally expected (especially in 3d)

11 Some of our results: 6d (2,0): all 16 susy preserving deformations are irrel least irrelevant operator has dim = 12 6d (1,0): all 8 susy preserving deformations are irrel least irrelevant operator has dim = 10 Also J Luis, S Lust 5d: all susy preserving deformations are irrel, except for real mass terms associated with global symmetries 4d, N=3: no relevant or marginal deformations Also O Aharony and M Evtikhiev 3d, N>3: all have universal, relevant, mass deformations from stress-tensor; the only relevant deformations, and no marginal For N=4, also flavor current masses, no others

12 CFTs, first w/o susy SO(d, 2) Operators form representations P µ K µ P µ (O R ) descendants= total derivatives, such deformations are trivial O R primary K µ (O R )=0 Unitarity: primary + all descendants must have + norm, eg P µ O 2 O [K µ,p µ ] O 0 [P µ,k ] µ D + M µ Zero norm, null states = set to zero Nulls = both primary and descendant

13 SCFT super-algebras complete classification d>6 no SCFTs can exist d =6 OSp(6, 2 N ) SO(6, 2) Sp(N ) R (N, 0) d =5 F (4) SO(5, 2) Sp(1) R 8Qs 8N Qs d =4 Su(2, 2 N = 4) SO(4, 2) SU(N ) R U(1) R d =4 PSU(2, 2 N = 4) SO(4, 2) SU(4) R 4N Qs d =3 OSp(4 N ) SO(3, 2) SO(N ) R 2N Qs d =2 OSp(2 N L ) OSp(2 N R ) N L Qs + N R Qs

14 SCFT operator reps Q`(O R ) P µ K µ descendants Q S Q(O R ) {Q, Q} =2P µ {S, S} =2K µ S(O R )=0 O R super-primary L = X i g i O i primary, modulo descendants {Q, Q} P µ 0 Clifford algebra Level Q `(O R ) =0 max apple N Q

15 Typical, long multiplets Q S O top R = Q N Q (O R ) Q(O top R ) 0 Q `(O R ) NQ d OR * modulo descendants conformal primary ops at level l, 2 N Q d OR total O R super-primary S(O R )=0 Can generate multiplet from bottom up, via Q,or from top down, via S Reflection symmetry Unique op at bottom, so unique op at the top Operator at top = susy preserving deformation No other susy preserving operators in long multiplets Easy cases D-terms Unitarity bounds at bottom of give bounds at top

16 Unitary bounds All Q-descendants must have non-negative norm Eg at Q-level one: 0 Q O 2 O SQ O O {S, Q} O {S, Q} D (M µ + R) c(lorentz) + c(r symmetry) + shift Saturated iff there is a null state: a Q-descendant that is also a superconformal primary: O V = Q(O ) and S(O V )=0 Set O V =0 along with all its Q-descendants

17 Long - null = short Specific operator dimensions, in terms of Lorentz + R- symmetry + shifts, to get null states Set null states to zero: a short multiplet Simplest cases also have the reflection symmetry, unique operator at bottom and top = susy preserving deformation: O top R Act on bottom op with all Q s, setting the null linear combinations to zero But can also act with R-symmetry raising and lowering Some subtle cases short O R null O V Q S

18 Multiple top op cases (Unique bottom operator, so no reflection symmetry) Eg T µ multiplet of 4d N=4, top ops= T µ, O, O Conserved! of 5d N=1, top ops= µ J a,global J a,global µ, O m a Many examples, especially with conserved currents; in such cases, setting {Q, Q} P µ 0 requires care, since current cons laws are null, both primary and descendant But also examples of multiple top operators without conserved currents, eg in 4d N=2, O bottom = A 2 Ā 2 [0; 0] R=1,r=0 =3 O top = Q 3 Q2 O bottom No conserved currents in this multiplet, yet 2 tops: and O top = Q 3 Q 2 O bottom

19 Mid-level susy tops(!) top null state or 0 from Q S top {Q, Q} 0 O 3d N 4 T µ multiplet: the stress-tensor is at top, at level 4 Another top, at level 2, Lorentz scalar Gives susy-preserving universal mass term relevant deformations First found in 3d N=8 (KI 98, Bena & Warner 04; Lin & Maldacena 05) Seems special to 3d Indeed, these examples give a deformed susy algebra with a non-central extension with R-symm gens Rij playing role of central term (=3d loophole to Haag- Lopuszanski-Sohnius theorem)

20 Classify susy preserving deformations of SCFTs Many multiplets have mid-level Lorentz scalars, in all dimensions We do many cross checks that we re not overlooking any exotic susy deformations (eg verify that Q can map to an operator at the next level, check Bose-Fermi degeneracy, recombination rules, etc)

21 Detailed tables Give all susy-preserving deformations, relevant, marginal, and all irrelevant deformations, for all N, d>2 Eg 3d, N=8: L=long, A,B,C, =short 33 Primary O Deformation δl Comments (0, 0, 2, 0) B 1 O =1 (0, 0, 0, 2) B 1 O =1 (0, 0,R3 +4, 0) B 1 O =2+ 1R 2 3 (0, 0, 0,R4 +4) B 1 O =2+ 1R 2 4 (0, 0,R3 +2,R B 4 +2) 1 O =2+ 1(R R 4 ) { } (0,R B 2 +2,R 3,R 4 ) 1 O =2+R (R R 4 ) Q 2 Q 2 (0, 0, 0, 2) =2 (0, 0, 2, 0) =2 Q 8 (0, 0,R3, 0) =6+ 1R 2 3 Q 8 (0, 0, 0,R4 ) =6+ 1R 2 4 { } Q 10 (0, 0,R 3,R 4 ) =7+ 1(R R 4 ) { } Q 12 (0,R 2,R 3,R 4 ) =8+R (R R 3 ) Stress Tensor (T ) Stress Tensor (T ) F -Term ( T ) F -Term ( T ) universal mass! all others irrelev { } (R B 1 +2,R 2,R 3,R 4 ) 1 O =2+R 1 + R 2 + 1(R R 4 ) { } (R L 1,R 2,R 3,R 4 ) O > 1+R 1 + R 2 + 1(R R 4 ) { } Q 14 (R 1,R 2,R 3,R 4 ) =9+R 1 + R 2 + 1(R R 4 ) { } Q 16 (R 1,R 2,R 3,R 4 ) > 9+R 1 + R 2 + 1(R R 4 ) D-Term Table 16: Deformations of three-dimensional N =8SCFTs TheR-charges of the deformation are denoted by the so(8) R Dynkin labels R 1,R 2,R 3,R 4 Z 0

22 4d, N=3 (all irrelevant) 40 Primary O Deformation δl Comments (R1 +4, 0; 2R B 1 B 1 +8) 1 Q 4 Q 2 (R1, 0; 2R 1 +6) O =4+R 1 =7+R 1 F -Term ( ) (0,R2 +4; 2R B 1 B 2 8) 1 Q 2 Q 4 (0,R2 ; 2R 2 6) O =4+R 2 =7+R 2 F -Term ( ) {( R1 +2,R 2 +2;2(R 1 R 2 ) ) } {( B 1 B 1 Q 4 Q 4 R1,R 2 ;2(R 1 R 2 ) ) } O =4+R 1 + R 2 =8+R 1 + R 2 (0, 0; r +6),r>0 LB 1 O =1+ 1r Q 6 (0, 0; r),r>0 =4+ 1r>4 F -term ( ) 6 6 (0, 0; r 6), r < 0 B 1 L O =1 1 r Q 6 (0, 0; r),r<0 =4 1 r>4 F -Term ( ) 6 6 (R1 +2, 0; r +4),r>2R LB O =2+ 2R r Q 6 Q 2 (R1, 0; r),r>2r 1 +6 =6+ 2R r>7+R ( ) 6 1 (0,R2 +2;r 4), r < 2R B 1 L 2 6 O =2+ 2 R r Q 2 Q 6 (0,R2 ; r),r< 2R 2 6 =6+ 2 R r>7+r ( ) 6 2 { } (R1,R LB 2 +2;r +2),r>2(R 1 R 2 ) 1 O =3+ 2(R R 2 )+ 1r Q 6 Q 4 (R 1,R 2 ; r),r>2(r 1 R 2 ) =8+ 2(R R 2 )+ 1r>8+R ( ) R 2 { } (R1 +2,R B 1 L 2 ; r 2),r<2(R 1 R 2 ) O =3+ 2(2R R 2 ) 1r Q 4 Q 6 (R 1,R 2 ; r),r<2(r 1 R 2 ) =8+ 2(2R R 2 ) 1r>8+R ( ) R 2 (R 1,R 2 ; r) (R 1,R 2 ; r) { LL 2 3 O > 2+max 1 + R 2 ) 1r } Q 6 Q 6 { R 2 )+ 1r > 8+max 1 + R 2 ) 1r } D-Term R 2 )+ 1r 6 Table 25: Deformations of four-dimensional N =3SCFTs Thesu(3) R Dynkin labels R 1,R 2 Z 0 and the u(1) R charge r R denote the R-symmetry representation of the deformation

23 d=5, 6 = simpler No exotic susy deformations (but not a 100% proof) 5d, N=1: Q 2 C 1 [0, 0] R=2 =[0, 0] R=0 4 (Eg gauge kinetic terms) Q 4 C 1 [0, 0] R+4 =[0, 0] R R Q 8 L 1 [0, 0] R =[0, 0] R > R mass terms via flavor symms irrel F-terms irrel D-terms 6d, N=(1,0): Q 4 D 1 [0, 0, 0] R+4 =[0, 0, 0] R =10+2R irrel F-terms Q 8 L[0, 0, 0] R =[0, 0, 0] R >10+2R irrel D-terms

24 Conclude QFT is vast, expect still much to be found susy QFTs and RG flows are rich, useful testing grounds for exploring QFT Strongly constrained: unitarity, a-thm, etc Can rule out some things Exact results for others Thank you! Happy birthdays, F-theory and Dave!