SCFTs and susy RG flows

Ken Intriligator (UC San Diego)

Bay Area Particle Theory Seminar

Two topics: (1) based on work with Francesco Sannino and (2) with Clay Cordova and Thomas Dumitrescu

Spectacular Collaborators

Clay Córdova

Thomas Dumitrescu

1506.03807: 6d conformal anomaly a from ’t Hooft anomalies. 6d a-thm. for N=(1,0) susy theories.

1602.01217: Classify susy-preserving deformations for d>2 SCFTs.

+ to appear & work in progress

“What is QFT?”perturbation theory

around free field Lagrangian theories CFTs + perturbations

non-Lagrangian

higher dim’l thy compactified

unexplored... something crucial for the future?

susy string, brane, AdS, M, F-thy realizations

RG flowsUV CFT (+relevant)

IR CFT (+irrelevant)

course graining

“# d.o.f.”

E.g. Higgs mass E.g. dim 6 BSM ops

“chutes”

“ladders”

RG flowsUV CFT (+relevant)

IR CFT (+irrelevant)

RG course graining

“# d.o.f.”

“�L” =�

i

giOi. (OK even if SCFT is non-Lagrangian)

. Move on the moduli space of (susy) vacua.

. Gauge a (e.g. UV or IR free) global symmetry.

. Will here focus on RG flows that preserve supersymmetry.

The “deformations” examples:

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, e.g. parity anomaly matching in 3d. . Reducing # of d.o.f. intuition. For d=2,4 (& d=6?) : a-theorem

hTµµ i ⇠ aEd +

X

i

ciIi

aUV � aIR a � 0 For any unitary theory

d=even:

(d=odd: conjectured analogs, from sphere partition function / entanglement entropy.). Additional power from supersymmetry.

UV asymptotic safety?

• Suppose theory has too much matter, so not asymptotically free in UV.

• IR theory in free electric phase

• UV safe, interacting CFT, completion?

gIR ! 0.

gUV ! g? �(g?) = 0?with

�(g)

gg??

UVIR

e.g.��4? no (lattice)

I.e. in same theory as opposed to some dual?

UV asymptotic safetySurprise: examples found in non-susy QCD, D.F. Litim and F. Sannino ’14.

�(g)

gUVIR

�(g)

g

UV IR

“Banks-Zaks” Litim-Sannino

g?g?

Theory: SU(Nc) QCD with Nf Dirac fermion flavors +Yukawa coupling to Nf 2 gauge singlet scalars, with also quartic self- interactions. Nf >(11/2)Nc so not asymp. free. Multiple couplings, each IR free. Interacting CFT via cancellation of 1-loop and higher-loop beta fn contributions. Can be made perturbative, taking Nf just above (11/2)Nc .

Susy examples ?Warmup: recall example of N=1 susy gauge theory with 3 adjoint matter chiral superfields, with superpotential W = y✏ijkTr(�i�j�k)

g

y

N=4: IRIndividually IR free couplings combine to give IR-attractive, interacting SCFT.

UV starting point of these RG flows? Needs a UV completion. Not UV safety.

“susy UV asyp. safety?”K.I. and F. Sannino ’15.

“No.” at least not nearby, in perturbation theory, in broad classes of non-asymptotically free theories.

Via imposing

• (s) CFT unitarity constraints. • a-theorem: aUV > aIR. }• apply a-maximization (KI, Wecht ’03), if needed.

�(g)

gUV=SCFTIR

for susy theories

Also J.Wells & S. Martin ‘00. We were originally unaware of this excellent, early paper..

4d, N=1 SQFTs

Dimension of chiral fields ~ their running R-charges.

U(1)R ABJ anomaly.

Exact beta functions ~ linear combinations of R-charges of fields:

W's R-violation.

Energy-momentum tensor supermultiplet, contains U(1)R current

Energy-momentum conservation. X=chiral superfield,

supermultiplet of anomalies (useful!).

4d N=1 SCFTs

Can use the power of ’t Hooft anomaly matching.

Curved background

Anselmi, Freedman, Grisaru, Johansen.

Weyl2

Euler

The correct R-symmetry locally maximizes a(R). KI, B. Wecht

Now vary find

SQCD examplesSU(Nc) SQCD with Nf Dirac flavors, non AF: Nf > 3Nc

Superconf’l U(1)R determined by anomaly free + symm

W = SQ eQ

aUV?R

R* a(R) = 3TrR3 � TrR

R? ⌘ R(Q) = R( eQ) =Nf �Nc

Nf

aIR

2/3

Also, no asymp safe UV SCFT is possible for

�(S) =3

2R(S) = 3

Nc

Nf

Would violate unitarity for Nf > 3Nc �(S) � 1

Would violate the a-theorem: (Instead UV- completes to asymp. free

Seiberg dual.)

General N=1 casesNo N=1 susy theories with non-asymptotically free matter content and W=0 can have a UV-safe SCFT. By a-maximization all such cases would violate the a-theorem:

aUV?

(W=0), R*

a(R) = 3TrR3 � TrR

aIR

2/3

Can satisfy a-thm only if some fields have large R charge, far from perturbative limit. Some possible examples via W terms - see Martin & Wells. Various other constraints to check. All satisfied? Do these exist? TBD.

Change gearsDiscuss work with C. Cordova and T. Dumitrescu

Study, and largely classify, possible susy-preserving deformations of SCFTs in various spacetime dims. !

!

Especially consider 6d susy and SCFT constraints. 6 = the maximal spacetime dim for SCFTs. A growing list of interacting, 6d SCFTs. Yield many new QFTs in lower d, via compactification.

6d a-theorem?For spontaneous conf’l symm breaking: dilaton has derivative interactions to give anom matching Schwimmer, Theisen; Komargodski, Schwimmer

�a

6d case:

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 .

Ldilaton =12(��)2 � b

(��)4

�3+ �a

(��)6

�6

(schematic)

�a

Clue: observed that, for case of (2,0) on Coulomb branch,

�a � b2

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.

>0.

Longstanding hunchSusy multiplet of anomalies: should be able to relate a-anomaly to R-symmetry ’t Hooft-type anomalies in 6d, as in 2d and 4d.

Tµ� � Jµ,aR

gµ� � AaR,µ

Stress-tensor supermultiplet

Sources = bkgrd SUGRA supermultiplet

T ��

T�� T ��

a?Jµ,a, Tµ� J�,a, T ��

J�,a, T�� J�,a, T ��I8

susy?

Easier to isolate anomaly term, and enjoys anomaly matching

Tµ�

4-point fn with too many indices. Hard to get a, and hard to compute.

e.g. Harvey Minasian, Moore ’98

6d (1,0) ‘t Hooft anomalies

Computed for (2,0) SCFTs + many (1,0) SCFTsHarvey, Minasian, Moore; KI; Ohmori, Shimizu, Tachikawa; Ohmori, Shimizu, Tachikawa, Yonekura; Del Zotto, Heckman, Tomasiello, Vafa; Heckman, Morrison, Rudelius, Vafa.

E.g. for theory of N small E8 instantons: Ohmori, Shimizu, Tachikawa

Iorigin

8

=1

4!

��c2

2

(R) + ⇥c2

(R)p1

(T ) + ⇤p21

(T ) + ⌅p2

(T )�

EN : (�,⇥, ⇤, ⌅) = (N(N2 + 6N + 3),�N

2(6N + 5),

7

8N,�N

2)

c2(R) ⌘ 1

8�2tr(FSU(2)R ^ FSU(2)R)

p1(T ) ⌘1

8�2tr(R ^R)

Background gauge fields and metric ( ~ background SUGRA)

(Leading N3 coeff. can be anticipated from Z2 orbifold of AN-1 (2,0) case.)

(1,0) on tensor branch

‘t Hooft anomaly matching requires

KI ; Ohmori, Shimizu, Tachikawa, Yonekura

for some real coefficients x, y

Our classification of defs. gives:

must be a perfect square, match I8 via X4 sourcing B:

LGSWS = �iB ⇥X4

X4 ⌘ 16�2(xc2(R) + yp1(T ))

b =1

2(y � x)

�I8

⇥ Iorigin

8

� Itensor branch

8

⇤ X4

⌅X4

Adapting a SUGRA analysis of Bergshoeff, Salam, Sezgin ’86 (!).

Ltensor

= Q8(O) � Ldilaton

+ LGSWS

Then

Iorigin

8

=1

4!

��c2

2

(R) + ⇥c2

(R)p1

(T ) + ⇤p21

(T ) + ⌅p2

(T )�

aorigin =16

7(�� ⇥ + ⇤) +

6

7⌅Upshot:

Change gears (1602.01217)

Classify susy-preserving deformations of SCFTs

• “�L” = QNQOlong

“D-term” e.g. Kaher potential in 4d N=1.

�(�L) > 1

2NQ +�min(Olong

)SCFT unitarity, bound grows with dim d:

Irrelevant. E.g. for 6d N=(1,0) such operators have � >1

28 + 6 = 10.

• “�L” = QntopO

short

�(�L) = 1

2ntop

+�(Oshort

)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 susy- preserving 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).

e.g. F-terms, W in 4d N=1.

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.

CFTs, first w/o susy SO(d, 2) Operators form representations

OR

Pµ Kµ

Kµ(OR) = 0primary

Pµ(OR) descendants= total derivatives, such deformations are trivial.

[Pµ,K� ] � �µ�D + Mµ�

��Pµ|O���2 � �O|[Kµ, Pµ]|O� � 0

Unitarity: primary + all descendants must have + norm, e.g. Zero norm, null states = set to zero. Nulls = both primary and descendant.

.

...

...

SCFT super-algebras complete classificationd > 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

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

d = 3 OSp(4|N ) ⇥ SO(3, 2)� SO(N )R

d = 2 OSp(2|NL)�OSp(2|NR)

8Qs8NQs

4NQs

2NQs

NLQs +NRQs

SCFT operator reps

OR

Pµ Kµ

descendants Q S{Q,Q} = 2Pµ

{S, S} = 2Kµ S(OR) = 0super-primary

Q(OR)

Q`(OR)

“�L” =X

i

giOi primary, modulo descendants.

{Q,Q} � Pµ � 0 Grassmann algebra.

Level Q�`(OR) � = 0 . . . �max

NQ

Typical, long multiplets

OR S(OR) = 0super-primary

Q�`(OR)

Otop

R = Q�NQ(OR) Q(Otop

R ) � 0

modulo descendantsQ S

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.

*

✓NQ

�

◆dOR

conformal primary ops at level l, 2NQdOR total

Unitary boundsAll Q-descendants must have non-negative norm.

E.g. at Q-level one:

0 ���Q|O�

��2 � �O†|SQ|O� � �O†|{S, Q}|O�

{S, Q} � D � (Mµ� + R)

� � c(Lorentz) + c(R� symmetry) + shift

Saturated iff there is a null state: a Q-descendant that is also a superconformal primary:

OV = Q(O�) S(OV) = 0and

OV = 0Set along with all its Q-descendants.

Long - null = shortSpecific 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:

OVOR

Q S

null

short

.Otop

RAct 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.

Multiple top op. cases (Unique bottom operator, so no reflection symmetry.)

E.g. multiplet of 4d N=4, top ops= !

Conserved of 5d N=1, top ops=

Tµ⌫ Tµ⌫ , O⌧ , O⌧

Ja,globalµ Ja,global

µ , Oma

Many examples, especially with conserved currents; in such cases, setting requires care, since current cons. laws are null, both primary and descendant. But also examples of multiple top operators without conserved currents, e.g. in 4d N=2,

{Q,Q} � Pµ � 0

Obottom = A2

A2

[0; 0]R=1,r=0

�=3

Otop = Q3Q2Obottom

Otop’ = Q3Q2ObottomandNo conserved currents in this multiplet, yet 2 tops:

Mid-level susy tops(!)

O

Q S

null state or 0 from top

top

3d N � 4 Tµ⌫ 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).

{Q,Q} ⇠ 0

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 (e.g. verify that Q can map to an operator at the next level, check Bose-Fermi degeneracy, recombination rules, etc).

Detailed tablesGive all susy-preserving deformations, relevant,

marginal, and all irrelevant deformations, for all N, d>2

Primary O Deformation δL Comments

B1

!(0, 0, 2, 0)∆O = 1

"Q2O ∈

!(0, 0, 0, 2)∆ = 2

"Stress Tensor (T )

B1

!(0, 0, 0, 2)∆O = 1

"Q2O ∈

!(0, 0, 2, 0)∆ = 2

"Stress Tensor (T )

B1

!(0, 0, R3 + 4, 0)∆O = 2 + 1

2R3

"Q8O ∈

!(0, 0, R3, 0)∆ = 6 + 1

2R3

"F -Term (#T )

B1

!(0, 0, 0, R4 + 4)∆O = 2 + 1

2R4

"Q8O ∈

!(0, 0, 0, R4)∆ = 6 + 1

2R4

"F -Term (#T )

B1

!(0, 0, R3 + 2, R4 + 2)∆O = 2 + 1

2(R3 +R4)

"Q10O ∈

!(0, 0, R3, R4)

∆ = 7 + 12(R3 +R4)

"−

B1

!(0, R2 + 2, R3, R4)

∆O = 2 +R2 +12(R3 +R4)

"Q12O ∈

!(0, R2, R3, R4)

∆ = 8 +R2 +12(R2 +R3)

"−

B1

!(R1 + 2, R2, R3, R4)

∆O = 2 +R1 +R2 +12(R3 +R4)

"Q14O ∈

!(R1, R2, R3, R4)

∆ = 9 +R1 +R2 +12(R3 +R4)

"−

L

!(R1, R2, R3, R4)

∆O > 1 +R1 +R2 +12(R3 +R4)

"Q16O ∈

!(R1, R2, R3, R4)

∆ > 9 +R1 +R2 +12(R3 +R4)

"D-Term

Table 16: Deformations of three-dimensional N = 8 SCFTs. The R-charges of the deformation are denoted by the so(8)RDynkin labels R1, R2, R3, R4 ∈ Z≥0.

33

3d, N=8:

universal mass !

all others irrelev.

E.g.

L=long, A,B,C,.. =short.

Primary O Deformation δL Comments

B1B1

{(R1 + 4, 0 ; 2R1 + 8)

∆O = 4 +R1

}Q4Q

2O ∈

{(R1, 0 ; 2R1 + 6)∆ = 7 +R1

}F -Term (∗)

B1B1

{(0, R2 + 4 ; −2R2 − 8)

∆O = 4 +R2

}Q2Q

4O ∈

{(0, R2 ; −2R2 − 6)

∆ = 7 +R2

}F -Term (∗)

B1B1

{(R1 + 2, R2 + 2 ; 2(R1 −R2)

)

∆O = 4 +R1 +R2

}Q4Q

4O ∈

{(R1, R2 ; 2(R1 −R2)

)

∆ = 8 +R1 +R2

}−

LB1

{(0, 0 ; r + 6) , r > 0

∆O = 1 + 16r

}Q6O ∈

{(0, 0 ; r) , r > 0∆ = 4 + 1

6r > 4

}F -term (⋆)

B1L

{(0, 0 ; r − 6) , r < 0

∆O = 1− 16r

}Q

6O ∈

{(0, 0 ; r) , r < 0∆ = 4− 1

6r > 4

}F -Term (⋆)

LB1

{(R1 + 2, 0 ; r + 4) , r > 2R1 + 6

∆O = 2 + 23R1 +

16r

}Q6Q

2O ∈

{(R1, 0 ; r) , r > 2R1 + 6

∆ = 6 + 23R1 +

16r > 7 +R1

}(†)

B1L

{(0, R2 + 2 ; r − 4) , r < −2R2 − 6

∆O = 2 + 23R2 −

16r

}Q2Q

6O ∈

{(0, R2 ; r) , r < −2R2 − 6∆ = 6 + 2

3R2 −16r > 7 +R2

}(†)

LB1

{(R1, R2 + 2 ; r + 2) , r > 2(R1 −R2)

∆O = 3 + 23(R1 + 2R2) +

16r

}Q6Q

4O ∈

{(R1, R2 ; r) , r > 2(R1 − R2)

∆ = 8 + 23(R1 + 2R2) +

16r > 8 +R1 +R2

}(‡)

B1L

{(R1 + 2, R2 ; r − 2) , r < 2(R1 − R2)

∆O = 3 + 23(2R1 +R2)−

16r

}Q4Q

6O ∈

{(R1, R2 ; r) , r < 2(R1 − R2)

∆ = 8 + 23(2R1 +R2)−

16r > 8 +R1 +R2

}(‡)

LL

⎧⎪⎨

⎪⎩

(R1,R2 ; r)

∆O > 2 + max

{23(2R1 +R2)−

16r

23(R1 + 2R2) +

16r

}

⎫⎪⎬

⎪⎭Q6Q

6O ∈

⎧⎪⎨

⎪⎩

(R1,R2 ; r)

∆ > 8 + max

{23(2R1 +R2)−

16r

23(R1 + 2R2) +

16r

}

⎫⎪⎬

⎪⎭D-Term

Table 25: Deformations of four-dimensionalN = 3 SCFTs. The su(3)R Dynkin labels R1, R2 ∈ Z≥0 and the u(1)R charge r ∈ R

denote the R-symmetry representation of the deformation.

40

4d, N=3 (all irrelevant)

d=5, 6 = simpler

5d, N=1: Q2C1[0, 0]R=2 = [0, 0]R=0

4mass terms via flavor symms

irrel. F-termsQ4C1[0, 0]R+4 = [0, 0]R8+ 3

2R

Q8L1[0, 0]R = [0, 0]R�>8+ 3

2Rirrel. D-terms

6d, N=(1,0):Q4D1[0, 0, 0]

R+4 = [0, 0, 0]R�=10+2R irrel. F-terms

Q8L[0, 0, 0]R = [0, 0, 0]R�>10+2R irrel. D-terms

No exotic susy deformations (not yet a 100% proof).

(E.g. gauge kinetic terms)

Conclude

• QFT is vast, 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 !