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

A ‘Lagrangian’ for Argyres-Douglas theory and superconformal index Kazunobu Maruyoshi (Seikei University) w/ Jaewon Song, 1606.05632, 1607.04281 w/ Prarit Agarwal and Jaewon Song, 160X.XXXXX @YITP workshop “Strings and Fields”, August 11, 2016

Argyres-Douglas theory originally 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] . is strongly coupled N=2 SCFT with central charges [Aharony-Tachikawa] a = 43 c = 11 120 , 30 has one-dimensional Coulomb branch parametrized by the chiral operator u of scaling dimension 6/5 .

Argyres-Douglas theory originally 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] . is strongly coupled N=2 SCFT with central charges [Aharony-Tachikawa] a = 43 c = 11 120 , 30 has one-dimensional Coulomb branch parametrized by the chiral operator u of scaling dimension 6/5 . The AD theory is the minimal nontrivial SCFT which saturates the central charge bound [Liendo-Ramirez-Seo] .

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

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

⃞ ⃞ An N=1 gauge theory Let us consider the following N=1 theory with SU(2) vector multiplets and with the following chiral multiplets : M 1 M 3 M 5 M 3 ’ q q’ 𝜚 SU(2) adj 1 1 1 1 U(1) R0 1/2 -5/2 1 2 4 6 4 1/2 7/2 -1 -2 -4 -6 -4 U(1) 𝓖 with the superpotential W = φ qq + M 1 φ 2 qq 0 + M 3 qq 0 + M 5 φ q 0 q 0 + M 0 3 φ 3 q 0 q 0 , Gauge invariant chiral operators: tr φ 2 , M 1 , M 3 , M 0 3 , M 5 , . . .

a-maximization and decoupling of chiral multiplets Consider the R charge R IR ( ✏ ) = R 0 + ✏ F and maximize central charge a [Intriligator-Wecht] a = 3 tr R IR ( ✏ ) 3 − tr R IR ( ✏ ) � � 32

a-maximization and decoupling of chiral multiplets Consider the R charge R IR ( ✏ ) = R 0 + ✏ F and maximize central charge a [Intriligator-Wecht] a = 3 tr R IR ( ✏ ) 3 − tr R IR ( ✏ ) � � 32 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 tr φ 2 , M 1 , M 3 , M 0 3 , M 5

a-maximization and decoupling of chiral multiplets Consider the R charge R IR ( ✏ ) = R 0 + ✏ F and maximize central charge a [Intriligator-Wecht] a = 3 tr R IR ( ✏ ) 3 − tr R IR ( ✏ ) � � 32 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 tr φ 2 , M 1 , M 3 , M 0 3 , M 5

a-maximization and decoupling of chiral multiplets Consider the R charge R IR ( ✏ ) = R 0 + ✏ F and maximize central charge a [Intriligator-Wecht] a = 3 tr R IR ( ✏ ) 3 − tr R IR ( ✏ ) � � 32 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 tr φ 2 , M 1 , M 3 , M 0 3 , M 5

a-maximization and decoupling of chiral multiplets Consider the R charge R IR ( ✏ ) = R 0 + ✏ F and maximize central charge a [Intriligator-Wecht] a = 3 tr R IR ( ✏ ) 3 − tr R IR ( ✏ ) � � 32 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 a = 43 c = 11 ✏ = 13 tr φ 2 , M 1 , M 3 , M 0 3 , M 5 120 , 15 , 30 dimension 6/5

N=1 deformation 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 couple N=1 chiral multiplet M in the adjoint rep of F by the superpotential W = tr µM

N=1 deformation Suppose we have an N=2 SCFT T with non-Abelian flavor symmetry F. Then let us couple N=1 chiral multiplet M in the adjoint rep of F by the superpotential W = tr µM give a nilpotent vev to M (which is specified by the embedding ρ : SU(2) → F), which breaks F.

N=1 deformation Suppose we have an N=2 SCFT T with non-Abelian flavor symmetry F. Then let us couple N=1 chiral multiplet M in the adjoint rep of F by the superpotential W = tr µM give a nilpotent vev to M (which is specified by the embedding ρ : SU(2) → F), which breaks F. This gives the IR theory T IR [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.

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 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 Other choices of vevs to SO(8): vev preserves SU(2) → H 1 theory (SU(2) flavor symmetry) vev preserves SU(2)xU(1) → H 2 theory (SU(3) flavor symmetry) others → possibly N=1 SCFTs

Partial list of results Let us consider the deformation of other N=2 theories which break F completely. Theories with the IR N=2 enhancement when T = rank-one theories H 1 , H 2 , D 4 , E 6 , E 7 , E 8 → H 0 SU(N) SQCD with 2N flavors → (A 1 , A 2N ) Sp(N) SQCD with 2N+2 flavors → (A 1 , A 2N+1 ) (A 1 , D k ) theory [Cecotti-Neitzke-Vafa] → (A 1 , A k-1 ) Theories with no IR N=2 enhancement when T = other rank-one theories [Argyres et al.] T N , and R 0,N theories of class S [Gaiotto, Chacaltana-Distler] N=4 SU(2) SYM theory

Superconformal index 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 1 2 ) Γ ( z ± ( pq ) − 5 1 7 1 2 ) Γ ( z ± 2 , 0 ( pq ) I = κ Γ (( pq ) 3 ξ − 6 ) Γ ( z ± ( pq ) 2 ξ − 1 ) 4 ξ 4 ξ I dz Γ ( z ± 2 ) Γ (( pq ) 1 ξ − 2 ) 2 π iz ξ : fugacity for U(1) F (We subtract the contributions of the decoupled operators!)

Superconformal index 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 1 1 2 ) Γ ( z ± ( pq ) − 5 7 1 2 ) Γ ( z ± 2 , 0 ( pq ) I = κ Γ (( pq ) 3 ξ − 6 ) Γ ( z ± ( pq ) 2 ξ − 1 ) 4 ξ 4 ξ I dz Γ ( z ± 2 ) Γ (( pq ) 1 ξ − 2 ) 2 π iz ξ : fugacity for U(1) F (We subtract the contributions of the decoupled operators!) 1 3 5 ( pq ) We substitute for the correct IR R symmetry. After that ξ → t 10 basically one can compute the integral 1 Coulomb index limit (pq/t=u, p,q,t → 0): I C = 6 1 − u 5 Macdonald limit (p → 0) agrees with the index by [Cordova-Shao]

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?