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.

H 0 AD theory • There is a chiral operator of dimension 6/5 parametrizing the Coulomb branch. a = 43 c = 11 • Central charges: [Shapere-Tachikawa ‘08] 120 , 30 • The central charge c above is the minimal value of any interacting N=2 SCFT! [Liendo-Ramirez-Seo ‘15] 
 c ≥ 11 30 • The 2d chiral algebra corresponding to the AD theory is given by a non-unitary Virasoro minimal model. [Beem-Lemos-Liendo-Rastelli-van Rees ’13] 
 [Cordova-Shao ’15]

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 H 0 =(A 1 , A 2 ) SCFT q q’ M X ϕ Matter content SU(2) 2 2 adj 1 1 W = φ qq + M φ q 0 q 0 + X φ 2 Superpotential This theory has an anomaly free U(1) global symmetry that can be mixed with R-symmetry. (R-charges are not fixed)


 a-maximization [Intriligator-Wecht ‘03] • UV R-symmetry may not be the same as the IR R-symmetry when there is a non-baryonic U(1) symmetry. X R IR = R UV + ✏ i F i i • 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: 
 [Anselmi-Freedman-Grisaru-Johansen ‘97] a ( ✏ ) = 3 32(3Tr R 3 ✏ − Tr R ✏ ) • Upon determining exact R-charge, we can determine the operator dimension via Δ = 3/2 R.

RG Flow to the H 0 theory W = g Tr φ qq + g 0 M Tr φ q 0 q 0 + λ X Tr φ 2 Adjoint SQCD SU(2), Nf=1 g’ H 0 C Tr ϕ 2 decouples N=1 fixed point W=0 fixed point g A a = 43 c = 11 ∆ ( M ) = 6 @C, we get: 120 , 30 5 Agrees with that of the Argyres-Douglas theory!

N=1 Lagrangian for the 
 N=2 AD theory H 0 =(A 1 , A 2 ) • 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 of the singlet field M, not Tr ϕ 2 . • 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 H 1 =(A 1 , A 3 ) theory q q’ M X ϕ • Matter contents: SU(2) 2 2 adj 1 1 • Interaction: W = Mqq 0 + X φ 2 This theory has a SU(2)xU(1) global symmetry. The U(1) symmetry can be mixed with R-symmetry. a = 11 c = 1 ∆ ( M ) = 4 @IR, we get: 24 , 2 , 3

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 T UV 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: 
 W = Tr( Mµ ) • SU(2)xU(1) R-symmetry broken to U(1) R xU(1) F

N=1 Deformation 
 - Nilpotent Higgsing • Now, we give a nilpotent vev to M. [Gadde-Maruyoshi-Tachikawa-Yan] • The deformation triggers a flow to a new N=1 SCFT T IR [ T UV , ρ ]. 
 T UV T IR [ T UV , ⇢ ] • Nilpotent elements are classified by the SU(2) embeddings 
 ρ : SU(2) → F. 
 h M i = ρ ( σ + ) • Commutant becomes the global symmetry of the IR SCFT. • It preserves the U(1) F symmetry that can be mixed with R- symmetry.

Nilpotent Higgsing - Lagrangian theory • For a Lagrangian theory, giving a vev to M gives a nilpotent mass to the quarks. [Agarwal-Bah-Maruyoshi-JS] 
 [Agarwal-Intriligator-JS] • 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)

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=1 SUSY N=2 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. 
 cf) [Cordova-Shao][Buican-Nishinaka][JS][JS-Xie-Yan] • One can use this “Lagrangian description” to compute any RG invariant quantities. \

Deformations of SU(N) N f =2N 4d N = 2 SUSY SU (2 N ) ⇢ : SU (2) , → SU (2 N ) a c [1 4 ] 23 7 Yes; N c = 2 , N f = 4 24 6 7 2 SU (4) [3 , 1] Yes; ( A 1 , D 4 ) AD th. Here we list some of 12 3 11 1 [4] Yes; ( A 1 , A 3 ) AD th. 24 2 the deformations that 
 29 17 [1 6 ] Yes; N c = 3 , N f = 6 12 6 gives rational central SU (6) 13 7 [5 , 1] Yes; ( A 1 , D 6 ) AD th. 12 6 11 23 [6] Yes; ( A 1 , A 5 ) AD th. charges. 12 24 107 31 [1 8 ] Yes; N c = 4 , N f = 8 24 6 [2 , 1 6 ] 73801 43121 ? 17424 8712 Those with “?" have 9097 5129 SU (8) [4 , 4] ? 3888 1944 19 5 N=1 SUSY. [Evtikhiev] [7 , 1] Yes; ( A 1 , D 8 ) AD th. 12 3 167 43 [8] Yes; ( A 1 , A 7 ) AD th. 120 30 [1 10 ] 247 71 Yes; N c = 5 , N f = 10 24 6 Other deformations 5553943 6257387 [5 , 1 5 ] ? 1383123 1383123 give irrational central SU (10) [5 , 3 , 1 2 ] 92540867 52091009 ? 24401712 12200856 25 13 [9 , 1] Yes; ( A 1 , D 10 ) AD th. charges, therefore they 12 6 15 23 [10] Yes; ( A 1 , A 9 ) AD th. 8 12 flow to N=1 theories. 247 71 [1 12 ] Yes; N c = 6 , N f = 12 24 6 [4 3 ] 754501 424727 ? 138384 69192 SU (12) 31 8 [11 , 1] Yes; ( A 1 , D 12 ) AD th. 12 3 397 101 [12] Yes; ( A 1 , A 11 ) AD th. 168 42

Deforming Sp(N),4N+4 half-hypers SO (4 N + 4) ⇢ : SU (2) , → SU (4 N + 4) a c 4d N = 2 SUSY [1 8 ] 23 7 Yes; N c = 1 , N f = 8 24 6 [3 2 , 1 2 ] 7 2 ? 12 3 11 1 SO (8) [4 , 4] ≡ [5 , 1 3 ] Yes; ( A 1 , D 3 ) AD th. 24 2 6349 3523 [5 , 3] ? 13872 6936 43 11 [7 , 1] Yes; ( A 1 , A 2 ) AD th. 120 30 37 11 [1 12 ] Yes; N c = 2 , N f = 12 12 3 [4 2 , 2 2 ] 105027 61145 ? 59536 29768 SO (12) [9 , 1 3 ] 19 1 Yes; ( A 1 , D 5 ) AD th. 20 67 17 [11 , 1] Yes; ( A 1 , A 4 ) AD th. 84 21 [1 16 ] 51 15 Yes; N c = 3 , N f = 16 8 2 [5 , 1 11 ] 109031 123889 ? 27744 27744 18250741 10440877 SO (16) [5 , 3 3 , 1 2 ] ? 5195568 2597784 81 3 [13 , 1 3 ] Yes; ( A 1 , D 7 ) AD th. 56 2 91 23 [15 , 1] Yes; ( A 1 , A 6 ) AD th. 72 18 65 38 [1 20 ] Yes; N c = 4 , N f = 20 6 3 [2 2 , 1 16 ] 4181 2463 ? 400 200 [3 4 , 2 4 ] 29 133 ? 4 16 [4 4 , 2 2 ] 28361329 16338643 ? 4702512 2351256 737 817 SO (20) [9 , 5 , 3 , 1 3 ] ? 192 192 [11 , 1 9 ] 6638927 3700169 ? 1976856 988428 [11 , 2 2 , 1 5 ] 106413731 59339969 ? 31795224 15897612 [11 , 2 4 , 1] ≡ [11 , 3 , 1 6 ] 26650955 14869241 ? 7990296 3995148 106793099 59613689 [11 , 3 , 2 2 , 1 2 ] ? 32127576 16063788

Deformations of SO(N) SQCD, 1/2 Nf=N-2 Sp ( N � 2) ⇢ : SU (2) , ! Sp ( N � 2) 4d N = 2 SUSY a c [1 4 ] 19 5 Yes; N c = 4 , N f = 4 12 3 Sp (2) 10111 5381 [2 , 1 2 ] ? 7056 3528 65 35 [1 6 ] Yes; N c = 5 , N f = 6 24 12 Sp (3) [4 , 1 2 ] 325 341 ? 192 192 33 9 Sp (4) [1 8 ] Yes; N c = 6 , N f = 8 8 2 35 77 No non-trivial 
 Sp (5) [1 10 ] Yes; N c = 7 , N f = 10 6 12 47 26 [1 12 ] Yes; N c = 8 , N f = 12 N=2 fixed point! 6 3 Sp (6) [2 2 , 1 8 ] 589093 329335 ? 80688 40344 13065 7085 [4 , 1 8 ] ? 2312 1156 [1 14 ] 81 45 Yes; N c = 9 , N f = 14 8 4 59094550 129141025 Sp (7) [5 2 , 1 4 ] ? 10978707 21957414 375975613 406255085 [6 , 3 2 , 2] ? 72745944 72745944 [1 16 ] 305 85 Yes; N c = 10 , N f = 16 24 6 389 53 [4 2 , 2 2 , 1 4 ] ? 48 6 Sp (8) [5 2 , 3 2 ] 30593927 16735805 ? 4642608 2321304 28118905 3828919 [5 2 , 4 , 1 2 ] ? 4348848 543606