New Superconformal Chern-Simons Theories and M2-branes Sangmin Lee Seoul National University Kazuo Hosomichi, Ki-Myeong Lee, SL, Sungjay Lee, Jaemo Park, Piljin Yi [JHEP0807:091, JHEP0809:002, JHEP0811:058] 29 Nov. 2008, PAQFT08, Singapore
Holographic duality in various dimensions QFT gravity (3+1)-dim : Super-Yang-Mills IIB string theory (2+1)-dim : well-known e.g. O(N) vector?? gravity?? M-theory (1+1)-dim : techniues special to 2d CFTs available.
Super-Chern-Simons Theories Super-YM-CS cannot go beyond N = 3. [Kao-K.Lee 92] CS without YM can go beyond N = 3! - useful for N = 8 M2-brane theory? [Schwarz 04] Bagger-Lambert theory [Bagger-Lambert 06-07][Gustavsson 07] - Super-conformal N = 8 CS-matter based on a Lie-3-algebra. Gaiotto-Witten construction [Gaiotto-Witten 0804.2907] - Large class of new N = 4 super-conformal CS theories. - List not complete / Relation to M2-branes unclear
N = 4 and twisted multiplets N = 4 super-algebra includes SO(4) = SU(2) L x SU(2) R R-symmetry (ordinary) SU(2) L SU(2) R α 2 1 (hyper) ψ α 1 2 A µ 1 1 (vector) χ α α 2 2 s α β 1 3 SU(2) L SU(2) R (twisted) 1 2 α 2 1 ψ α 1 1 Ã µ 2 2 χ α α 3 1 s αβ Gaiotto-Witten : most general N = 4 with ordinary hypers only. Our contribution : 1) Added twisted hypers to GW : complete classification 2) Explained relation to M2-brane world-volume theories
Gaiotto-Witten construction [Gaiotto-Witten 0804.2907] Key steps : Begin with N = 1 SUSY theory with SU(2) diag SU(2) L SU(2) R global symmetry. Adjust the N = 1 super-potential to restore SU(2) L SU(2) R. The SO(4) R-symmetry together with N = 1 SUSY generate a full N = 4 structure. Rewrite the action and SUSY transformation rules in a manifestly N = 4 covariant way.
Field content Matter : A α, ψ Ȧ α, FA α Sp(2n) with ω AB reality condition : α A = (A α ) = ɛ αβ ω AB B β Gauge group G Sp(2n) (t m ) A B satisfy [t m, t n ] = f mn pt p, Tr(t m t n ) = k mn t m AB ω AC(t m ) C B = t m BA Gauge : (A m ) µ, χ m Conditions for N = 4 SUSY k mn t m (AB tn C)D = 0 ( fundamental identity )
Classification in terms of Lie super-algebra [Gaiotto-Witten] Fundamental identity implies existence of a Lie super-algebra : [M m, M n ]= f mn pm p, [M m, Q A ]=Q B (t m ) B A, {Q A, Q B } = t m AB M m. [{Q A, Q B }, Q C ]+(cyclic) = 0 k mn t m (AB tn C)D = 0 Lie super-algebras classifies Gaiotto-Witten theories completely! Typical examples : U(N M), OSp(N M) G 1 G 2 G 1 G 2
Adding twisted hypers G 1 Hypers and twisted hypers should independently form super-algebra structure. G 2 G 3 Generically, they form uivers with the two types of hypers alternating between gauge groups. The uiver can either have open ends or form a closed loop. G 4 G 1 G 2 G 3 G 4 Short loops exhibit enhanced (N > 4) SUSY. G 1 G 2 G 1 G 2
Summary L = εµνλ 4π + 1 2 ω AB ( k mn A m m ν A n λ + 1 ) 3 f mnpa m µ A n ν A p λ ( ɛ αβ Dα A D B β + iɛ α β ) ψα Ȧ D/ ψ Ḃ β + 1 2 ω AB ( ɛ α β ) D α Ȧ D Ḃ β + iɛαβ ψ α A D/ ψ β B iπk mn ɛ αβ ɛ γ δ jα m γ jn β δ iπk mnɛ α β ɛ γδ j αγ jṅ ṁ βδ + 4πik mnɛ αγ ɛ β δ j m α β jṅ δγ +iπk mn (ɛ α γ ɛ β δ µ ṁ α β ψȧ γ t n AB ψḃ δ + ɛαγ ɛ βδ µ m αβ ψ γ A t n AB ψ ) δ B π2 6 f mnp(µ m ) α β (µn ) β γ(µ p ) γ α π2 6 f mnp( µ m ) α β ( µn ) β γ( µ p ) γ α +π 2 ( µ mn ) γ γ (µ m) α β (µ n) β α + π 2 (µ mn ) γ γ( µ m ) α β ( µ n) β α, δα A =+iηα α ψα Ȧ, δ α Ȧ = iη α α ψ α A, δa m µ = 2πiη α α γ µ (jα m α j αα), ṁ [ δψα Ȧ =+ Dα A + 2π ] 3 (t m) A B B β (µm ) β α η α α 2π(t m ) A B B β ( µm ) β α ηβ β, [ δ ψ α A = D α Ȧ + 2π ] 3 ( t m ) A B Ḃ β ( µm ) β α ηα α + 2π( t m ) A B Ḃ β (µm ) β αη β β.
Embedding in String/M-theory [Gaiotto-Witten] [Hosomichi-Lee 3 -Park] IIB brane configuration 0 1 2 3 4 5 6 7 8 9 D3(O3) NS5 D5 NS5 D5 NS5 D3(O3) T-dual! M : Replace (C 2 /Z n ) 2 by Taub-NUT(Z n ) 2 N 4 Chern-Simons Theories = M2-branes on Orbifolds
[N = 4] [Hosomichi-Lee 3 -Park][Benna-Klebanov-Klose-Smedback][Terashima-Yagi] Linear uivers with U(N M) or OSp(N M) Circular uiver with (2m) gauge group factors and alternating hyper/twisted-hypers The theory describes N M2-branes on C 4 /(Z m Z mk ) or C 4 /(Z m D mk ) Relation to brane-crystal model [Kim-SL-Lee-Park] A i B i C i D i
[N = 5] [Hosomichi-Lee 3 -Park] Extended OSp(N M) R-symmetry enhancement from SU(2) x SU(2) to USp(4) = SO(5) Hypers and twisted hypers together form an N = 5 multiplet. The theory describes N M2-branes on D k orbifolds. A derivation of the field theory from M-IIB T-duality and brane construction is given.
[N = 6] [Aharony-Bergman-Jafferis-Maldacena] ABJM = Extended U(N M) R-symmetry enhancement from SU(2) x SU(2) to SU(4). Hypers and twisted hypers together form an N = 6 multiplet. When M = N, at CS level k, the theory is claimed to describe N M2-branes on C 4 /Z k orbifold. A derivation of the field theory from M-IIB T-duality and brane construction is given.
[N = 8] Bagger-Lambert ( Based on Lie-3-algebra f abcd, h ab) [Bagger-Lambert 06-07][Gustavsson 07] Positive h ab allows nothing but f abcd = ε abcd [Papadopoulos] SO(4) [Gauntlett-Gutowski] The SO(4) theory is an ordinary CS-matter theory [Raamsdonk] SO(4) Bagger-Lambert = Extended PSU(2 2) From N = 8 to N = 4 : SO(8) SO(4) 1 SO(4) 2 (SU(2) L SU(2) A ) (SU(2) B SU(2) R ) X : (2, 2, 1, 1) : (1, 1, 2, 2), Ψ ψ : (2, 1, 2, 1) ψ : (1, 2, 1, 2), ε η : (2, 1, 1, 2) η : (1, 2, 2, 1).
Instantons in 3-dim Yang-Mills Monopole in 4-dim Instanton in 3-dim D1 D0 D3 D3 D2 D2 Idea : compare world-volume theory and bulk theory A non-perturbative test of the ABJM proposal!
Instantons in ABJM theory [Hosomichi-Lee 3 -Park-Yi] ABJM model 11/10-dim bulk Classical instanton action Higgs mechanism Zero-instanton effective action One-instanton effective action D0 : DBI + RR Strings stretched between D2 s Super-graviton exchange D0 exchange subtleties due to imaginary value of Chern-Simons action, etc.
Discussions Superconformal index [Bhattacharya-Minwalla][Choi-SL-Song] Beta-deformation [Jeong-Jin-Kim-SL, work in progress] Vortex solitons, non-relativistic AdS/CFT Topological twisting of CS-matter theories : relation to pure-cs and Rozansky-Witten theories? Solution to the N 3/2 problem?