Lagrangians for non-Lagrangian theories Jaewon Song (KIAS) in - - PowerPoint PPT Presentation

‘Lagrangians’ for 
 ‘non-Lagrangian’ theories

Jaewon Song 
 (KIAS)

in collaboration with 
 Prarit Agarwal (SNU), Kazunobu Maruyoshi (Seikei), 
 Emily Nardoni (UCSD), Antonio Sciarappa (KIAS)
 
 1606.05632, 1607.04281, 1610.05311, 1707.04751, 1711.xxxxx

@KEK - Nov. 13, 2017

What is the simplest interacting 4d N=2 SCFT?

Argyres-Douglas theory

• Originally discovered by looking at a special point in the

Coulomb branch of N=2 SU(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. “non-Lagrangian theory”
• Its Coulomb phase is well-understood, but the conformal

phase is rather poorly-understood.

H0 AD theory

• There is a chiral operator of dimension 6/5

parametrizing the Coulomb branch.

• Central charges:
• The central charge c above is the minimal value of any

interacting N=2 SCFT! [Liendo-Ramirez-Seo ‘15]


• The 2d chiral algebra corresponding to the AD theory is

given by a non-unitary Virasoro minimal model.

a = 43 120 , c = 11 30

[Beem-Lemos-Liendo-Rastelli-van Rees ’13]
 [Cordova-Shao ’15]

c ≥ 11 30

[Shapere-Tachikawa ‘08]

Is it possible to write a Lagrangian for the ‘simplest 4d N=2 SCFT’?

Is it possible to write a Lagrangian for the ‘simplest 4d N=2 SCFT’?

• It has been a long-standing problem to write a

Lagrangian for the QED with electron and monopole.

Is it possible to write a Lagrangian for the ‘simplest 4d N=2 SCFT’?

• It has been a long-standing problem to write a

Lagrangian for the QED with electron and monopole.

• Not possible if you want N=2 SUSY manifest.. 


eg) No way to get dim=6/5 Coulomb branch operator.

Is it possible to write a Lagrangian for the ‘simplest 4d N=2 SCFT’?

• It has been a long-standing problem to write a

Lagrangian for the QED with electron and monopole.

• Not possible if you want N=2 SUSY manifest.. 


eg) No way to get dim=6/5 Coulomb branch operator.

• Sometimes, sacrificing manifest symmetry can help. 


eg) ABJM theory, N=(4, 4) sigma model on K3.

N=1 gauge theory flowing to the H0=(A1, A2) SCFT

q q’ ϕ M X SU(2) 2 2 adj 1 1

This theory has an anomaly free U(1) global symmetry that can be mixed with R-symmetry. (R-charges are not fixed) Matter content Superpotential W = φqq + Mφq0q0 + Xφ2

a-maximization

• UV R-symmetry may not be the same as the IR R-symmetry

when there is a non-baryonic U(1) symmetry.

• The exact R-charge at the fixed point is determined by

maximizing the trial a-function, which is given in terms of the U(1)R ’t Hooft anomalies: 
 


• Upon determining exact R-charge, we can determine the
• perator dimension via Δ = 3/2 R.

[Intriligator-Wecht ‘03]

RIR = RUV + X

i

✏iFi

a(✏) = 3 32(3TrR3

✏ − TrR✏)

[Anselmi-Freedman-Grisaru-Johansen ‘97]

RG Flow to the H0 theory

Adjoint SQCD SU(2), Nf=1 W=0 fixed point Trϕ2 decouples

C g g’ A

H0

N=1 fixed point

@C, we get:

a = 43 120 , c = 11 30

∆(M) = 6 5 W = gTrφqq + g0MTrφq0q0 + λXTrφ2

Agrees with that of the Argyres-Douglas theory!

N=1 Lagrangian for the 
 N=2 AD theory H0=(A1, A2)

• This enables us to compute the full superconformal index of

the AD theory and compare against the simplification limits computed in [Cordova-Shao][JS].

• Pseudomoduli parametrized by X gets lifted via dynamically

generated superpotential.

• Coulomb branch emerges in the IR, parametrized via a vev
• f the singlet field M, not Trϕ2.
• Can further deform to a ‘simple N=1 SCFT’ via giving a mass

to M. We tested the claims of [Xie-Yonekura][Buican-Nishinaka].

N=1 gauge theory flowing to the H1=(A1, A3) theory

• Matter contents:
• Interaction:

q q’ ϕ M X SU(2) 2 2 adj 1 1

This theory has a SU(2)xU(1) global symmetry. The U(1) symmetry can be mixed with R-symmetry. W = Mqq0 + Xφ2

a = 11 24, c = 1 2, ∆(M) = 4 3

@IR, we get:

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

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

• Consider an N=2 SCFT TUV with non-abelian global

symmetry.

• For an N=2 SCFT with global symmetry F it has a moment

map operator μ transforms as the adjoint of F.

• Add a chiral multiplet M transforming as the adjoint of F and

the following superpotential:


• SU(2)xU(1) R-symmetry broken to U(1)RxU(1)F

W = Tr(Mµ)

N=1 Deformation 


• Nilpotent Higgsing
• Now, we give a nilpotent vev to M.
• The deformation triggers a flow to a new N=1 SCFT TIR[TUV ,ρ].

• Nilpotent elements are classified by the SU(2) embeddings 


ρ: SU(2) → F. 


• Commutant becomes the global symmetry of the IR SCFT.
• It preserves the U(1)F symmetry that can be mixed with R-

symmetry.

hMi = ρ(σ+)

[Gadde-Maruyoshi-Tachikawa-Yan]

TUV TIR[TUV , ⇢]

Nilpotent Higgsing - Lagrangian theory

• For a Lagrangian theory, giving a vev to M gives a nilpotent mass

to the quarks.

• Massive quarks can be integrated out.
• Some components of M remain coupled.
• Write all possible superpotential terms consistent with the
• symmetry. (many of them are dangerously irrelevant)

[Agarwal-Bah-Maruyoshi-JS]
 [Agarwal-Intriligator-JS]

Results - Surprise!

• Emergent N=2 supersymmetry:
• For a number of cases, SUSY enhances to N=2 at the

fixed point.

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

N=2 SUSY N=1 SUSY

Results - Surprise!

• N=1 deformation of the Lagrangian N=2 SQCD flows to

the “non-Lagrangian” Argyres-Douglas (AD) theory!

• This enables us to compute the full superconformal

indices of the AD theories. 


• One can use this “Lagrangian description” to compute

any RG invariant quantities.

\ cf) [Cordova-Shao][Buican-Nishinaka][JS][JS-Xie-Yan]

Deformations of SU(N) Nf=2N

SU(2N) ⇢ : SU(2) , → SU(2N) a c 4d N = 2 SUSY SU(4) [14]

23 24 7 6

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

7 12 2 3

Yes; (A1, D4) AD th. [4]

11 24 1 2

Yes; (A1, A3) AD th. SU(6) [16]

29 12 17 6

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

13 12 7 6

Yes; (A1, D6) AD th. [6]

11 12 23 24

Yes; (A1, A5) AD th. SU(8) [18]

107 24 31 6

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

73801 17424 43121 8712

? [4, 4]

9097 3888 5129 1944

? [7, 1]

19 12 5 3

Yes; (A1, D8) AD th. [8]

167 120 43 30

Yes; (A1, A7) AD th. SU(10) [110]

247 24 71 6

Yes; Nc = 5, Nf = 10 [5, 15]

5553943 1383123 6257387 1383123

? [5, 3, 12]

92540867 24401712 52091009 12200856

? [9, 1]

25 12 13 6

Yes; (A1, D10) AD th. [10]

15 8 23 12

Yes; (A1, A9) AD th. SU(12) [112]

247 24 71 6

Yes; Nc = 6, Nf = 12 [43]

754501 138384 424727 69192

? [11, 1]

31 12 8 3

Yes; (A1, D12) AD th. [12]

397 168 101 42

Yes; (A1, A11) AD th.

Here we list some of the deformations that
 gives rational central charges. Those with “?" have N=1 SUSY. [Evtikhiev] Other deformations give irrational central charges, therefore they flow to N=1 theories.

Deforming Sp(N),4N+4 half-hypers

SO(4N + 4) ⇢ : SU(2) , → SU(4N + 4) a c 4d N = 2 SUSY SO(8) [18]

23 24 7 6

Yes; Nc = 1, Nf = 8 [32, 12]

7 12 2 3

? [4, 4] ≡ [5, 13]

11 24 1 2

Yes; (A1, D3) AD th. [5, 3]

6349 13872 3523 6936

? [7, 1]

43 120 11 30

Yes; (A1, A2) AD th. SO(12) [112]

37 12 11 3

Yes; Nc = 2, Nf = 12 [42, 22]

105027 59536 61145 29768

? [9, 13]

19 20

1 Yes; (A1, D5) AD th. [11, 1]

67 84 17 21

Yes; (A1, A4) AD th. SO(16) [116]

51 8 15 2

Yes; Nc = 3, Nf = 16 [5, 111]

109031 27744 123889 27744

? [5, 33, 12]

18250741 5195568 10440877 2597784

? [13, 13]

81 56 3 2

Yes; (A1, D7) AD th. [15, 1]

91 72 23 18

Yes; (A1, A6) AD th. SO(20) [120]

65 6 38 3

Yes; Nc = 4, Nf = 20 [22, 116]

4181 400 2463 200

? [34, 24]

29 4 133 16

? [44, 22]

28361329 4702512 16338643 2351256

? [9, 5, 3, 13]

737 192 817 192

? [11, 19]

6638927 1976856 3700169 988428

? [11, 22, 15]

106413731 31795224 59339969 15897612

? [11, 24, 1] ≡ [11, 3, 16]

26650955 7990296 14869241 3995148

? [11, 3, 22, 12]

106793099 32127576 59613689 16063788

?

Deformations of SO(N) SQCD, 1/2 Nf=N-2

Sp(N 2) ⇢ : SU(2) , ! Sp(N 2) a c 4d N = 2 SUSY Sp(2) [14]

19 12 5 3

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

10111 7056 5381 3528

? Sp(3) [16]

65 24 35 12

Yes; Nc = 5, Nf = 6 [4, 12]

325 192 341 192

? Sp(4) [18]

33 8 9 2

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

35 6 77 12

Yes; Nc = 7, Nf = 10 Sp(6) [112]

47 6 26 3

Yes; Nc = 8, Nf = 12 [22, 18]

589093 80688 329335 40344

? [4, 18]

13065 2312 7085 1156

? Sp(7) [114]

81 8 45 4

Yes; Nc = 9, Nf = 14 [52, 14]

59094550 10978707 129141025 21957414

? [6, 32, 2]

375975613 72745944 406255085 72745944

? Sp(8) [116]

305 24 85 6

Yes; Nc = 10, Nf = 16 [42, 22, 14]

389 48 53 6

? [52, 32]

30593927 4642608 16735805 2321304

? [52, 4, 12]

28118905 4348848 3828919 543606

?

No non-trivial 
 N=2 fixed point!

Principal Deformations of 
 quiver theories

SO(2) − Sp(N) − SO(4N + 2) − Sp(3N)− . . . − Sp(2mN − N) − SO(4mN + 2) (A2m−1, D2Nm+1)

SO(N) − Sp(N − 2) − SO(3N − 4) − Sp(2N − 4)− . . . − Sp(m(N − 2)) − SO(2m(N − 2) + N) (A2m, Dm(N−2)+ N

2 )

Sp(N) − SO(4N + 4) − Sp(3N + 2) − SO(8N + 8) − . . . − Sp((m − 1)(2N + 2) + N) − SO(4m(N + 1)) Dm(2N+2)

m(2N+2)[m]

Non-principal deformations do not lead to SUSY enhancement.

See also: [Benvenutti-Giacomelli]

SU(N) − SU(2N) − . . . − SU(mN − N) − SU(mN) (Am−1, ANm−1)

U(1) − SU(k + 1) − . . . − SU(mk − k + 1) − SU(mk + 1) (Im,mk, S)

Deforming ‘non-Lagrangian’ theories

TUV ρ TIR[TUV , ρ] SU(N) with Nf = 2N [N] (A1, A2N−1) theory [N 1, 1] (A1, D2N) AD theory Sp(N) with Nf = 2N + 2 [4N + 4] (A1, A2N) theory [4N + 1, 13] (A1, D2N+1) AD theory (IN,k, F) [N] (AN−1, AN+k−1) theory (IN,−N+2, F) [N 1, 1] (A1, DN) theory E6 SCFT E6 H0 theory D5 H1 theory D4 H2 theory

Sp(n) SQCD with Nf = 2n + 2 $ (I2n+1,−2n+1, F) AD theory ρ = [4n + 1, 13] & . ρ = [2n, 1] (A1, D2n+1) AD theory

Deforming ‘non-Lagrangian’ theories

TUV ρ TIR[TUV , ρ] SU(N) with Nf = 2N [N] (A1, A2N−1) theory [N 1, 1] (A1, D2N) AD theory Sp(N) with Nf = 2N + 2 [4N + 4] (A1, A2N) theory [4N + 1, 13] (A1, D2N+1) AD theory (IN,k, F) [N] (AN−1, AN+k−1) theory (IN,−N+2, F) [N 1, 1] (A1, DN) theory E6 SCFT E6 H0 theory D5 H1 theory D4 H2 theory

Sp(n) SQCD with Nf = 2n + 2 $ (I2n+1,−2n+1, F) AD theory ρ = [4n + 1, 13] & . ρ = [2n, 1] (A1, D2n+1) AD theory

Deforming ‘non-Lagrangian’ theories

IR duality between
 Lagrangian and 
 non-Lagrangian thy

TUV ρ TIR[TUV , ρ] SU(N) with Nf = 2N [N] (A1, A2N−1) theory [N 1, 1] (A1, D2N) AD theory Sp(N) with Nf = 2N + 2 [4N + 4] (A1, A2N) theory [4N + 1, 13] (A1, D2N+1) AD theory (IN,k, F) [N] (AN−1, AN+k−1) theory (IN,−N+2, F) [N 1, 1] (A1, DN) theory E6 SCFT E6 H0 theory D5 H1 theory D4 H2 theory

Sp(n) SQCD with Nf = 2n + 2 $ (I2n+1,−2n+1, F) AD theory ρ = [4n + 1, 13] & . ρ = [2n, 1] (A1, D2n+1) AD theory

Is there any pattern of SUSY enhancement?

2d Chiral Algebra associated to TUV

• For any 4d N=2 SCFT, there is a subsector described by

a two-dimensional chiral algebra


• If the chiral algebra is given by the affine Kac-Moody

algebra , the stress tensor is given by the Sugawara tensor with the central charge

c2d = −12c4d, k2d = −1 2k4d . cSugawara = k2ddimF k2d + h∨

TUV 7! χ2d[TUV ]

[Beem-Lemos-Liendo-Peelaers-Rastelli-van Rees]

b Fk2d

When does SUSY enhances?

TUV TIR[TUV , ⇢]

When does SUSY enhances?

• When does TIR exhibits SUSY enhancement?

TUV TIR[TUV , ⇢]

When does SUSY enhances?

• When does TIR exhibits SUSY enhancement?
• TUV satisfies c2d[TUV]= cSugawara, and F is of ADE type.

TUV TIR[TUV , ⇢]

When does SUSY enhances?

• When does TIR exhibits SUSY enhancement?
• TUV satisfies c2d[TUV]= cSugawara, and F is of ADE type.
• When ρ is the principal embedding.

TUV TIR[TUV , ⇢]

When does SUSY enhances?

• When does TIR exhibits SUSY enhancement?
• TUV satisfies c2d[TUV]= cSugawara, and F is of ADE type.
• When ρ is the principal embedding.
• Or ρ is the ‘next to principal’ and TUV saturates the flavor

central charge bound. [BLLPRvR][Lemos-Liendo]

TUV TIR[TUV , ⇢]

When does SUSY enhances?

• When does TIR exhibits SUSY enhancement?
• TUV satisfies c2d[TUV]= cSugawara, and F is of ADE type.
• When ρ is the principal embedding.
• Or ρ is the ‘next to principal’ and TUV saturates the flavor

central charge bound. [BLLPRvR][Lemos-Liendo]

WHY?

TUV TIR[TUV , ⇢]

Summary

Summary

• To a given N=2 SCFT T with non-abelian global symmetry

F, one can obtain N=1 SCFT TIR[T ,ρ] labelled by the SU(2) embedding ρ of F. 


• When TUV = SQCD or certain quiver gauge theory, and ρ

is (nearly) principal, 
 TIR[T ,ρ] = (generalized) Argyres-Douglas theory.

• N=1 Lagrangian theory flowing to the N=2 AD theory. It

can be used to compute various RG-invariant quantities.

TUV TIR[TUV , ⇢]

Future directions

• SUSY enhancement in other dimensions? 


3d N=2->N=4 [Benvenutti-Giacomelli]

• More supersymmetric partition functions for the AD theory. 


eg) Lens index [Fluder-JS]

• When and why N=2 enhancement happens?
• Can we find N=1 gauge theories flowing to other ‘non-

Lagrangian’ theories? Other AD theories, TN SCFTs, N=3 
 cf) E6 SCFT [Gadde-Razamat-Willett]

• New topological invariants of 4-manifold? [Gukov][Moore]

Thank you!