N=1 Lagrangians for
N=2 “non-Lagrangian” theories
Jaewon Song(Korea Institute for Advanced Study)
Based on work with
Prarit Agarwal (SNU), Kazunobu Maruyoshi (Seikei),
Emily Nardoni (UCSD), Antonio Sciarappa (KIAS)
1606.05632, 1607.04281, 1610.05311, 1707.04751, 1806.08353

Strings 2018, Okinawa, Japan

What is the “simplest”
interacting 4d N=2 SCFT?
A) Argyres-Douglas theory (very likely)

Argyres-Douglas theory•

Theory describing a special point in the Coulomb branch of N=2SU(3) SYM or N=2 SU(2) SQCD.
[Argyres-Douglas 95] [Argyres-Plesser-Seiberg-Witten 95]•At this special point, mutually non-local electromagnetically charged
particles become massless.

•It is a strongly-coupled N=2 SCFT with no tunable coupling.
Commonly referred to as a “non-Lagrangian theory”•Many generalizations. [Cecotti-Neitzke-Vafa 10][Cecotti-del Zotto 12][Xie 12]…

•Its Coulomb phase is well-understood, but the conformal phase isless-understood.
[Cornagliotto, Lemos, Liendo 17][Talk by Lemos]Properties of theH0 Argyres-Douglas theory•

There is a chiral operator of dimension 6/5
parametrizing the Coulomb branch.
43a=,
120

11c=
30

[Aharony-Tachikawa 07][Shapere-Tachikawa 08]•Central charges:•The central charge c above is the minimal value of any
interacting N=2 SCFT! [Liendo-Ramirez-Seo 15]
c

•

1130

The 2d chiral algebra corresponding to the AD theory isgiven by a non-unitary Virasoro minimal model.
[Beem-Lemos-Liendo-Rastelli-van Rees 13][Cordova-Shao 15]

Is it possible to write a Lagrangianfor the ‘simplest 4d N=2 SCFT’?
Challenges:••

Mutually non-local particles.
Chiral operators of fractional dimensions.

A) Sacrifice manifest (super)symmetryN=1 gauge theory flowing
to the H0=(A1, A2) SCFT
[Maruyoshi-JS 16]

Matter content

SuperpotentialSU(2)

qq’

22adj
𝜙0 0MX11

W = qq + M q q + X

2This theory has an anomaly free U(1) global symmetry that can bemixed with R-symmetry. R-charges fixed by a-maximization.
[Intriligator-Wecht 03]

gA
Figure 1 :A and B are saddlepoints.
The plop
C is stable, and there bothA is stgroups
are interactingis dri[Maruyoshi-JS][Maruyoshi-Nardoni-JS]

RG Flow to the H0 theory
W = gϕqq + g′Mϕq′q′ + λXϕg’

2g’

H0C

W=Xϕ2 fixed point@C, we get:

N=1 fixed pointgAFigure43
11 3 :
6of fig. 2.
a=,Thecopposite=(M)=120 g ′ is IR free 30
5for g = 0 butg̸
= 0 drives g ′ IR interacting.

Agrees with that of the Argyres-Douglas theory!
2

Two separa
combinN = 4 SY

N=1 gauge theory flowing tothe H1=(A1, A3) theory•Matter contents:
SU(2)•Interaction:

qq’

22adj
𝜙0

W = M qq + X

MX112This theory has SU(2)xU(1) flavor symmetry.
The U(1) symmetry can be mixed with R-symmetry.
@IR, we get:

11a=,
24

1c= ,
24(M ) =3

Checks & applications of the
N=1 Lagrangian for the AD theory•Moduli space, chiral ring agree with the known results.

•The full superconformal index of the AD theory. Can becompared against the Schur/Macdonald limits of theindex computed in [Cordova-Shao][Buican-Nishinaka][JS]•3d Argyres-Douglas theory: 3d N=2→N=4•More SUSY partition functions for the AD theory[Benvenuti-Giacomelli][Fredrickson-Pei-Yan-Ye][Gukov][Fluder-JS]

Where are these ‘Lagrangians’
coming from?
Is there any organizing principle?

N=1 Deformations of N=2
SCFT with global symmetry•

Consider N=2 SCFT TUV with non-abelian global symmetry F.

•

It has a moment map operator μ valued in the adjoint of F.

•Add a chiral multiplet M in the adjoint of F and the following
superpotential:
W = Tr(M µ)•SU(2)xU(1) R-symmetry broken to U(1)RxU(1)Ats) via

N=1 Deformationvia
W = TrM µ ,Nilpotent
Higgsing
ving a nilpotent vev to M . A nilpotent element in F is classified b[Gadde-Maruyoshi-Tachikawa-Yan]+ ). to[Agarwal-Bah-Maruyoshi-JS]
Now,give givena nilpotent
M.
⇢ : SU• (2),! weF andby ⇢( vev
Thereforeone
can associate an N
[Agarwal-Intriligator-JS]

N = 2 SCFT and the SU (2) embedding ⇢ as:•The deformation triggers a flow to a new N=1 SCFT.

TU V

TIR [TU V , ⇢]Nilpotent elements are classified by the SU(2) embeddingss of this kind were previously considered in [3–6].
ρ: SU(2) → F.
+•

hM i =paper⇢( )demonstrated that the four
the last two authors of the current
ries obtainedin thishave richcharacterizedby• It preserves
themanner,
U(1)A symmetry
thatdynamics
can be mixed
with Rsymmetry.
d appearanceof accidental symmetries along their RG flow. The
such RG flows was the a-maximization [8] and its modification [9]t surprising, was the fact that many of these theories flow to IR fix
is an enhancement of supersymmetry from N = 1 to N = 2. By

Results[Maruyoshi-JS][Agarwal-Maruyoshi-JS]•For a number of cases, Supersymmetry enhances
to N=2 at the fixed point.
•

N=1 RG flows between (known) N=2 SCFTs•

N=1 deformed Lagrangian N=2 SQCD flows to the“non-Lagrangian” Argyres-Douglas (AD) theory!
N=1 SUSY
N=2 SUSY

B

Central charges corresponding to various nilpotent embedding

Deforming SU(N) Nf=2N: F=SU(2N)
Table 6: Nilpotent embeddings of SU (2N ) leading to rational values for a and c. Thepartition corresponding to 2N ! [12N ], reduces to 4d N = 2 SU (N ) gauge theory with 2Nhypers. The partitions [2N 1, 1] and [2N ] reduce to AD theories listed in the last columnof the above table.
SU (2N )SU (4)SU (6)SU (8)SU (10)SU (12)⇢ : SU (2) ,! SU (2N )ac4d N = 2 SUSY

[14 ]

2324

76Yes; Nc = 2, Nf = 4[3, 1]

71223Yes; (A1 , D4 ) AD th.

[4]

1124

12Yes; (A1 , A3 ) AD th.

[16 ]2912176Yes; Nc = 3, Nf = 6[5, 1]

131276Yes; (A1 , D6 ) AD th.

[6]

11122324Yes; (A1 , A5 ) AD th.

[18 ]10724316Yes; Nc = 4, Nf = 8[2, 16 ]

73801
17424

43121
8712

?

[4, 4]

9097
3888

5129
1944

?

[7, 1]

191253Yes; (A1 , D8 ) AD th.

[8]

167
120

4330Yes; (A1 , A7 ) AD th.

[110 ]

247
24716Yes; Nc = 5, Nf = 10

[5, 15 ]5553943
13831236257387
1383123?

[5, 3, 12 ]92540867
24401712

52091009
12200856?

[9, 1]

2512136Yes; (A1 , D10 ) AD th.

[10]

1582312Yes; (A1 , A9 ) AD th.

[112 ]

247
2471
6Yes; Nc = 6, Nf = 12[43 ]75450113838442472769192?

[11, 1]311283Yes; (A1 , D12 ) AD th.

[12]

397
168

10142Yes; (A1 , A11 ) AD th.

Here we list some ofthe deformations thatgives rational central
charges.
Those with “?" have
N=1 SUSY. [Evtikhiev]
Other deformations
give irrational central
charges, therefore theyflow to N=1 theories.

partition [14N +4 ] reduces to four-dimensional N = 2 Sp(N ) gauge theory with 4N + 4 halfhypermultiplets plus decouped fields. The partitions [4N + 1, 13 ] and [4N + 4] reduce toAD theories listed in the last column of the above table. Here Nc denotes the rank of thecorresponding symplectic gauge group and Nf denotes the number of half-hypermultipletstransforming in the fundamental representation of the gauge group.

Deforming Sp(N),Nf=2N+2: F=SO(4N+4)SO(4N + 4)

SO(8)SO(12)

SO(16)

SO(20)

ρ : SU (2) "→ SO(4N + 4)

ac4d N = 2 SUSY

[18 ]

2324

76Yes; Nc = 1, Nf = 8[32 , 12 ]71223Yes; (A1 , D4 ) AD th.

[4, 4] ≡ [5, 13 ]1124

12Yes; (A1 , D3 ) AD th.

[5, 3]

6349
13872

3523
6936

?

[7, 1]

43120

1130Yes; (A1 , A2 ) AD th.

[112 ]

3712113Yes; Nc = 2, Nf = 12

[42 , 22 ]

10502759536

61145
29768?

[9, 13 ]1920

1Yes; (A1 , D5 ) AD th.

[11, 1]67841721Yes; (A1 , A4 ) AD th.

[116 ]

518

152Yes; Nc = 3, Nf = 16

[5, 111 ]10903127744

12388927744?

[5, 33 , 12 ]18250741
5195568

10440877
2597784?

[13, 13 ]8156

32Yes; (A1 , D7 ) AD th.

[15, 1]

9172

2318Yes; (A1 , A6 ) AD th.

[120 ]

65
6383Yes; Nc = 4, Nf = 20

[22 , 116 ]

4181
400

2463
200?

[34 , 24 ]

29413316?

8.

Table 8: Nilpotent embe-cddings of Sp(Nc 2) (s.t. Sp(1) ' SU (2)) leading to rational
values for a and c. The partition corresponding to 2Nc 4 ! [12Nc 4 ], reduces to 4d N = 2SO(Nc ) gauge theory with Nf = 2Nc 4 half-hypers.

Deforming SO(N) Nf=N-2: F=Sp(N-2)
ac4d N = 2 SUSY

[14 ]191253Yes; Nc = 4, Nf = 4[2, 12 ]

10111
7056

5381
3528

?

[16 ]

65
243512Yes; Nc = 5, Nf = 6[4, 12 ]325
192

341192?

Sp(4)

[18 ]

33892Yes; Nc = 6, Nf = 8Sp(5)[110 ]

3567712Yes; Nc = 7, Nf = 10[112 ]

476263Yes; Nc = 8, Nf = 12

[22 , 18 ]58909380688

32933540344?

[4, 18 ]

13065
2312

7085
1156?

[114 ]

818454Yes; Nc = 9, Nf = 14

[52 , 14 ]

59094550
10978707

129141025
21957414?

[6, 32 , 2]375975613
72745944

406255085
72745944?

[116 ]

3052485
6Yes; Nc = 10, Nf = 16

[42 , 22 , 14 ]389
48536?

[52 , 32 ]

30593927
4642608

16735805
2321304?

[52 , 4, 12 ]28118905
43488483828919543606?

Sp(NSp(2)

Sp(3)

Sp(6)

Sp(7)

Sp(8)

2)

⇢ : SU (2) ,! Sp(N2)

No non-trivial
N=2 fixed point!

Is there any pattern in
the SUSY enhancement?

the central charges of the chiral algebra [65] and whether the flavor central charge satura
the bound of [65, 66].

7

Chiral Algebra associated to TUVDiscussion[Beem-Lemos-Liendo-Peelaers-Rastelli-vanRees]NIn this paper, we considered the N = 1 deformationof N = 2 SCFTs. Among variousSCFTs, we found the deformation of a particular class of theories flow to the IR fixed poT!
7[T]UV2dUVwith the enhanced N = 2 supersymmetry. We list the summary of our result in the table
To any N = 2 SCFT T , there is an associated two-dimensional chiral algebra χ[T• For any 4d N=2 SCFT, there is a subsector described bydiscussed in [65]. The central charges for the chiral algebra are given asa chiral algebra withc2d = −12c4d ,1k2d = − k4d .
2(7If the two-dimensional
algebra is given by the Sugawara construction of the aﬃ• If the chiralVirasoro
algebrais given by the affine Kac-Moody
Lie algebra, the 2d central
charge has to be given by c2d = cSugawara , wherêalgebra ℱk2d , the stress tensor is given by the Sugawara
k2d dimF
tensor with the central
chargecSugawara =,(7∨k2d + hk2ddimℱcSugawara =k2d + h ∨– 36 –

W = TrM µ ,giving a nilpotent vev to M . A nilpotent element in F is classified byngs ⇢ : SU (2) ,! F and given by ⇢( + ). Therefore one can associate an N
[Agarwal-Maruyoshi-JS][Agarwal-Sciarappa-JS]
by N = 2 SCFT and the SU (2) embedding⇢ as:

When does SUSY enhances?
[Giacomelli]

TU V

TIR [TU V , ⇢]By studying a large set of Lagrangian/non-Lagrangian theories, we observe
tionsthatof thiskind
werepreviouslyconsideredin[3–6].
the SUSY is enhanced in the IR if and only if, 7], the last two authors of the current paper demonstrated that the four-diTUV satisfiesc2d[T& F dynamics
is of ADE type(can have U(1)’s)by op
UV]= cSugawara(+n)
heories •obtainedin thismanner,have richcharacterized
and appearance• Either of of
theaccidental
two cases: symmetries along their RG flow. The mafor such RG flows was the a-maximization [8] and its modification [9].
1. ρ is the principal embedding.
most surprising, was the fact that many of these theories flow to IR fixedρ is the ‘next toand TUV saturates
the1flavor
ere is an2.enhancementof principal’supersymmetry
from N =to Ncentral= 2. By inv
bound.
kF ≧ kdown
F,b [BLLPRvR][Lemos-Liendo]
flow, it was charge
possibleto write
N = 1 Lagrangians whose IR fixed p
be the Argyres-Douglas theory [10] and its generalization of (A1 , AN ) t
hem to obtain the full superconformal index of the these theories.

WHY?

What about the minimal
4d N=1 SCFT?

Minimal 4d N=1 SCFT?
•

There is no analytic bound on the value of a or c.

•

There is a candidate minimal SCFT suggested by
conformal bootstrap with c~0.11. But no explicit
construction of such theory. [Poland-Stergiou][Li-Meltzer-Stergiou]•We explored a large set of SCFTs with Lagrangian
descriptions by considering a simple setup.

•SU(2) adjoint SQCD with Nf=1 + gauge singlets with all
possible superpotential couplings

A Landscape of Simple SCFTs[Maruyoshi-Nardoni-JS]•We found 35 fixed points
having small centralcharges.

•They pass a number ofconsistency checks:
central charge bounds,
unitarity constraints, index•a/c lie in a narrow range
with mean value and stdc0.60
0.55
0.50
0.450.400.35
0.35

0.40

0.45

0.50

a0.8733 ± 0.0398

Some examples

W =X 2(a, c) = (0.453,0.499)•

H0* has the smallest value of a.

M q q̃

[Xie-Yonekura][Buican-Nishinaka]•qq

T0 and H1* have the smallest awith U(1) and SU(2) flavor.

•H1
qq + X̂M(0.345,0.349)•T̂

T0 has the smallest value of c.

M q̃ q̃[Benvenuti]

43 11
TM contains ‘unphysical’ operator 120, 30
that is not in the chiral ring.
Is it really ‘bad’ or just ‘ugly’?
Can it be a new “minimal” theory? 263 271
768 , 768

H0M

M2H1⇤

T0

11 124 , 2927 1023
2048 , 2048

M2 q q̃ + M3 qq + M2 M3qq + X̂ q̃ q̃
Tµ2H0⇤

TM417
449
2048 , 2048

711
807
2048 , 2048

Summaryor, µ (which is the scalar component in the N = 2 supermultipletts) via

Summary
W = TrM µ ,ving a nilpotent vev to M . A nilpotent element in F is classified b• To a given N=2 SCFT TUV with
non-abelian global symmetry F,+⇢ : SU (2) ,! F and given by ⇢( ). Therefore one can associate an N
one can obtain N=1 SCFT TIR[TUV ,ρ] labelled by SU(2)N = 2 SCFT and the SU (2) embedding ⇢ as:embedding ρ of F.

TU VTIR [TU V , ⇢]• For some special cases, TIR have enhanced N=2 SUSY.
s of thiskind were previously considered in [3–6].
the last
twoauthorsof
thecurrent
paper
demonstrated
that
thefour• N=1 Lagrangian theories flowing to the N=2 Argyres-Douglasries obtainedincan
thisrich dynamics characterized bytheories
be manner,realized inhavethis way.
d appearance of accidental symmetries along their RG flow. The• Many new “simple N=1 SCFTs” with small central charges cansuch RG flows was the a-maximization [8] and its modification [9]be constructed from a simple gauge theory setup.
t surprising, was the fact that many of these theories flow to IR fix
is an enhancement of supersymmetry from N = 1 to N = 2. By

Outlook•

When and why SUSY enhancement happens?

•Other ‘non-Lagrangian’ theories? general ADs, TN, N=3cf) E6, E7, R0, N SCFT [Gadde-Razamat-Willett][Agarwal-Maruyoshi-JS]•SUSY enhancements in other d?

•What is the minimal N=1 SCFT?

[Gaiotto-Komargodski-Wu][Benini-Benvenuti][Gang-Yamazaki]

Thank you!