A Lagrangian for Argyres-Douglas theory and superconformal index - - PowerPoint PPT Presentation

A ‘Lagrangian’ for Argyres-Douglas theory and superconformal index

Kazunobu Maruyoshi

w/ Jaewon Song, 1606.05632, 1607.04281 w/ Prarit Agarwal and Jaewon Song, 160X.XXXXX

@YITP workshop “Strings and Fields”, August 11, 2016

(Seikei University)

• riginally has been found at the special locus on the Coulomb branch of

N=2 SU(3) pure SYM theory, where mutually non-local massless particles appear [Argyres-Douglas].

Argyres-Douglas theory

is strongly coupled N=2 SCFT with central charges [Aharony-Tachikawa] has one-dimensional Coulomb branch parametrized by the chiral

• perator u of scaling dimension 6/5.

a = 43 120, c = 11 30

• riginally has been found at the special locus on the Coulomb branch of

N=2 SU(3) pure SYM theory, where mutually non-local massless particles appear [Argyres-Douglas].

Argyres-Douglas theory

is strongly coupled N=2 SCFT with central charges [Aharony-Tachikawa] has one-dimensional Coulomb branch parametrized by the chiral

• perator u of scaling dimension 6/5.

a = 43 120, c = 11 30

The AD theory is the minimal nontrivial SCFT which saturates the central charge bound [Liendo-Ramirez-Seo].

We don’t know much about the AD theory. The full superconformal index? (in a limit [Cordova-Shao]) Other partition functions?

However….

We don’t know much about the AD theory. The full superconformal index? (in a limit [Cordova-Shao]) Other partition functions?

However….

This is (partly!) because of lack of the Lagrangian description…

We don’t know much about the AD theory. The full superconformal index? (in a limit [Cordova-Shao]) Other partition functions?

However….

This is (partly!) because of lack of the Lagrangian description…

In this talk, I present a Lagrangian which flows to the AD theory in the IR. This gives a new handle to study the strongly- interacting AD theory. Even more, we find a very general way to produce Lagrangians, in this sense, of many other N=2 SCFTs.

Gauge invariant chiral operators: Let us consider the following N=1 theory with SU(2) vector multiplets and with the following chiral multiplets:

An N=1 gauge theory

with the superpotential

trφ2, M1, M3, M 0

3, M5, . . .

W = φqq + M1φ2qq0 + M3qq0 + M5φq0q0 + M 0

3φ3q0q0,

q q’ 𝜚 M1 M3 M5 M3’ SU(2) ⃞ ⃞ adj 1 1 1 1 U(1)R0 1/2

• 5/2

1 2 4 6 4 U(1)𝓖 1/2 7/2

• 1
• 2
• 4
• 6
• 4

Consider the R charge and maximize central charge a [Intriligator-Wecht]

a-maximization and decoupling

• f chiral multiplets

a = 3 32

• trRIR(✏)3 − trRIR(✏)
• RIR(✏) = R0 + ✏F

Consider the R charge and maximize central charge a [Intriligator-Wecht]

a-maximization and decoupling

• f chiral multiplets

a = 3 32

• trRIR(✏)3 − trRIR(✏)
• RIR(✏) = R0 + ✏F

A caveat is that we have to check the chiral operators have dimension greater than one. If it is less than one, it is decoupled. Thus we subtract its contribution from central charge, and re-a-maximize

trφ2, M1, M3, M 0

3, M5

Consider the R charge and maximize central charge a [Intriligator-Wecht]

a-maximization and decoupling

• f chiral multiplets

a = 3 32

• trRIR(✏)3 − trRIR(✏)
• RIR(✏) = R0 + ✏F

A caveat is that we have to check the chiral operators have dimension greater than one. If it is less than one, it is decoupled. Thus we subtract its contribution from central charge, and re-a-maximize

trφ2, M1, M3, M 0

3, M5

Consider the R charge and maximize central charge a [Intriligator-Wecht]

a-maximization and decoupling

• f chiral multiplets

a = 3 32

• trRIR(✏)3 − trRIR(✏)
• RIR(✏) = R0 + ✏F

A caveat is that we have to check the chiral operators have dimension greater than one. If it is less than one, it is decoupled. Thus we subtract its contribution from central charge, and re-a-maximize

trφ2, M1, M3, M 0

3, M5

Consider the R charge and maximize central charge a [Intriligator-Wecht]

a-maximization and decoupling

• f chiral multiplets

a = 3 32

• trRIR(✏)3 − trRIR(✏)
• RIR(✏) = R0 + ✏F

A caveat is that we have to check the chiral operators have dimension greater than one. If it is less than one, it is decoupled. Thus we subtract its contribution from central charge, and re-a-maximize

trφ2, M1, M3, M 0

3, M5

dimension 6/5

a = 43 120, c = 11 30

✏ = 13 15,

Suppose we have an N=2 SCFT T with non-Abelian flavor symmetry F. Then let us

N=1 deformation

Suppose we have an N=2 SCFT T with non-Abelian flavor symmetry F. Then let us

N=1 deformation

couple N=1 chiral multiplet M in the adjoint rep of F by the superpotential

W = trµM

Suppose we have an N=2 SCFT T with non-Abelian flavor symmetry F. Then let us

N=1 deformation

give a nilpotent vev to M (which is specified by the embedding ρ: SU(2)→F), which breaks F. couple N=1 chiral multiplet M in the adjoint rep of F by the superpotential

W = trµM

Suppose we have an N=2 SCFT T with non-Abelian flavor symmetry F. Then let us

N=1 deformation

give a nilpotent vev to M (which is specified by the embedding ρ: SU(2)→F), which breaks F. This gives the IR theory TIR[T, ρ], which is generically N=1 supersymmetric and supposed to be conformal. However quite often a-maximization implies the IR supersymmetry gets enhanced to N=2. couple N=1 chiral multiplet M in the adjoint rep of F by the superpotential

W = trµM

In this case, F = SO(8) We consider the vev which breaks SO(8) completely (principal embedding) → after integrating out the massive fields, we get the Lagrangian in previous slide

T = SU(2) w/ 4 flavors

In this case, F = SO(8) We consider the vev which breaks SO(8) completely (principal embedding) → after integrating out the massive fields, we get the Lagrangian in previous slide

T = SU(2) w/ 4 flavors

Other choices of vevs to SO(8): vev preserves SU(2) → H1 theory (SU(2) flavor symmetry) vev preserves SU(2)xU(1) → H2 theory (SU(3) flavor symmetry)

• thers → possibly N=1 SCFTs

Theories with the IR N=2 enhancement when T = rank-one theories H1, H2, D4, E6, E7, E8 → H0 SU(N) SQCD with 2N flavors → (A1, A2N) Sp(N) SQCD with 2N+2 flavors → (A1, A2N+1) (A1, Dk) theory [Cecotti-Neitzke-Vafa] → (A1, Ak-1)

Partial list of results

Theories with no IR N=2 enhancement when T =

• ther rank-one theories [Argyres et al.]

TN, and R0,N theories of class S [Gaiotto, Chacaltana-Distler] N=4 SU(2) SYM theory Let us consider the deformation of other N=2 theories which break F completely.

Now we had a Lagrangian theory which flows to the AD theory in the IR. The superconformal index can be simply given from the matter content

Superconformal index

(We subtract the contributions of the decoupled operators!)

ξ : fugacity for U(1)F

I = κΓ((pq)3ξ−6) Γ((pq)1ξ−2) I dz 2πiz Γ(z±(pq)

1 4 ξ 1 2 )Γ(z±(pq)− 5 4 ξ 7 2 )Γ(z±2,0(pq) 1 2 ξ−1)

Γ(z±2)

Now we had a Lagrangian theory which flows to the AD theory in the IR. The superconformal index can be simply given from the matter content

Superconformal index

(We subtract the contributions of the decoupled operators!)

ξ : fugacity for U(1)F

We substitute for the correct IR R symmetry. After that

ξ → t

1 5 (pq) 3 10

basically one can compute the integral Coulomb index limit (pq/t=u, p,q,t→0): Macdonald limit (p→0) agrees with the index by [Cordova-Shao]

IC = 1 1 − u

6 5

I = κΓ((pq)3ξ−6) Γ((pq)1ξ−2) I dz 2πiz Γ(z±(pq)

1 4 ξ 1 2 )Γ(z±(pq)− 5 4 ξ 7 2 )Γ(z±2,0(pq) 1 2 ξ−1)

Γ(z±2)

Questions

✪ Complete the list: non-principal embedding [work in progress] ✪ What is the condition of UV theory T for the enhancement? ✪ Why the enhancement? ✪ The IR Coulomb branch comes from M, gauge-singlet in the UV… ✪ string/M-theory realization?