N =8 & 6 Superspace Constraints for 3D Gauge Theories. Robert Wimmer, ENS Lyon

Transcription

1 N =8 & 6 Superspace Constraints for 3D Gauge Theories Robert Wimmer, ENS Lyon H. Samtleben, R.W. JHEP 2010 & work in progress

2 Introduction & Motivation M-theory & AdS 4 /CF T 3 Correspondence M-theory (11D SUGRA) provides extended objects. M2-branes and M5- branes (Berman 07). Gain understanding of M-theory DOF: effective world volume theory for M2 branes 3D SCFT M2 branes (flat space) AdS4 x (S7) SUGRA bkg. AdS4/CFT3 Two Proposals: BLG ( N =8) & ABJM ( N =6) {Q αa,q βb } =2δ AB P αβ A,B =1,..., N R-symmetry SO(N ) BLG 07/08 ABJM 08 closure of susy e.o.m Enhance N =2 N =6 based on 3-algebra [Ta,Tb,Tc] = fabc d Td U(N)xU(N) N M2 branes on R 8 /Zk only SU(2)xSU(2) 2 M2 branes Lambert Tong 08 k=1,2 should be N =8 2-loop conf. sym.: Akerblom Saemann Wolf 09 Superspace formulation: susy manifest, broader classification, α corrections...

3 N = 8 Super Space R 2,1 16 Coordinates: x αβ... real symmetric 2x2 matrices θ αa= Majorana (3D) spinors in the 8s of SO(8): [ ] MW-Rep: ˆΓ I =, with I, A, Ḃ =1,...,8 Triality Derivatives: Q αa = αa iθ β A αβ αa θ β B = δ ABδα β {Q αa,q βb } =2δ AB ( i αβ ) D αa = αa + iθ β A αβ 0 Γ I Γ I T 0 Γ I AḂ {Q αa,d βb } =0 Connections: Eventually we will gauge global symmetries -> Superspace Connections: αa = D αa + A αa αβ = αβ + A αβ F αa,βb = { αa, βb } 2iδ AB αβ, F αβ,γδ =[ αβ, γδ ], F αβ,γc =[ αβ, γc ]

4 1. Free matter multiplet Component fields: φ I (x)... 8 v φ I =0 ψ α Ȧ... 8 c αβ ψ β Ȧ =0 δφ I = iɛγ I ψ δψ β Ȧ = ɛαa Γ I AȦ αβφ I Superfields: want Φ I (x, θ) =φ I (x) }{{} +θαa χ I αa(x) }{{} v 8 v 8 s = 8 c 56 c αa Φ I θ=0 = 8 c... susy covariant... too many DOF Constraint: D αa Φ I 56c =0 D αa Φ I = iγ I AȦΨ αȧ (global symmetry : ) GL(N,R) T(8N) δφ Ia =Λ Φ Ia + C Ia Integrability condition: D αa Ψ β Ȧ =ΓI AȦ αβφ I Superfield e.o.m.: Φ I =0, αβ Ψ β Ȧ =0 Superfield Expansion Component e.o.m. +?more? & susy transformations Equivalence with component multiplet

5 1. Free matter multiplet Component fields: φ I (x)... 8 v φ I =0 ψ α Ȧ... 8 c αβ ψ β Ȧ =0 δφ I = iɛγ I ψ δψ β Ȧ = ɛαa Γ I AȦ αβφ I Superfields: want Φ I (x, θ) =φ I (x) }{{} +θαa χ I αa(x) }{{} v 8 v 8 s = 8 c 56 c αa Φ I θ=0 = 8 c... susy covariant... too many DOF Constraint: D αa Φ I 56c =0 D αa Φ I = iγ I AȦΨ αȧ (global symmetry : ) GL(N,R) T(8N) δφ Ia =Λ Φ Ia + C Ia Integrability condition: D αa Ψ β Ȧ =ΓI AȦ αβφ I Superfield e.o.m.: Φ I =0, αβ Ψ β Ȧ =0 Superfield Expansion Component e.o.m. +?more? & susy transformations Equivalence with component multiplet

6 1. Free matter multiplet Component fields: φ I (x)... 8 v φ I =0 ψ α Ȧ... 8 c αβ ψ β Ȧ =0 δφ I = iɛγ I ψ δψ β Ȧ = ɛαa Γ I AȦ αβφ I Superfields: want Φ I (x, θ) =φ I (x) }{{} +θαa χ I αa(x) }{{} v 8 v 8 s = 8 c 56 c αa Φ I θ=0 = 8 c... susy covariant... too many DOF Constraint: D αa Φ I 56c =0 D αa Φ I = iγ I AȦΨ αȧ (global symmetry : ) GL(N,R) T(8N) δφ Ia =Λ Φ Ia + C Ia Integrability condition: D αa Ψ β Ȧ =ΓI AȦ αβφ I Superfield e.o.m.: Φ I =0, αβ Ψ β Ȧ =0 Superfield Expansion Component e.o.m. +?more? & susy transformations Equivalence with component multiplet

7 2. Gauge Field Constraints The superconnection ( A αa, A αβ ) contains too many DOF (2 vector fields). It has turned out to be promising to impose partial flatness conditions on the bi-spinor field strength (twistors, pure spinors...): F αa,βb ((2, 8 s ) (2, 8 s )) sym = (3, 1) (1, 28) (3, 35) (3, 1) =0 conventional constraint, (3, 35) = 0 (no higher derivatives) (1, 28) 0 Deformation Potential ( = W AB (Φ) soon) W AB Constraint: { αa, βb } =2i (δ AB αβ + ε αβ W AB ) Bianchi Identities: consistency of the constraint B1 B2 [ αa, { βb, γc }] 0 cyclic { αa, [ βb, γδ ]} 0 B3 B4 cyclic cyclic [ ρa, [ αβ, γδ ]] 0 cyclic [ αβ, [ γδ, ρσ ]] 0

8 B1 : αa W BC 8 s 28 = 8 s 56 s 160 s αa W BC 160s =0 αa W BC = δ A[B λ C]α + ρ αabc Given a deformation Potential W AB (Φ) satisfying this condition -> integrability conditions thereof give identities for the composite SF s λ αc, ρ αabc. B2 B3 B4 are satisfyed and imply the SF- e.o.m: ( F) αβ = i 8 A (α λ β)a = i 28 A (α B β) W AB Chern-Simons SF - e.o.m. Therefore, the above W-constraint is the only consistency condition for the choice of the deformation Potential W AB (Φ). The coupling to the matter sector will give an additional condition.

9 B1 : αa W BC 8 s 28 = 8 s 56 s 160 s αa W BC 160s =0 αa W BC = δ A[B λ C]α + ρ αabc Given a deformation Potential W AB (Φ) satisfying this condition -> integrability conditions thereof give identities for the composite SF s λ αc, ρ αabc. B2 B3 B4 are satisfyed and imply the SF- e.o.m: ( F) αβ = i 8 A (α λ β)a = i 28 A (α B β) W AB Chern-Simons SF - e.o.m. Therefore, the above W-constraint is the only consistency condition for the choice of the deformation Potential W AB (Φ). The coupling to the matter sector will give an additional condition.

10 2. Gauge - Matter Coupling We discussed for the free case how to constrain the representation content (component fields) in a susy covariant way. -> + gauge covariance: Constraint: αa Φ I = iγ I AȦ Ψ αȧ Using the gauge field constraint one finds that the integrability condition determines, but: W AB Φ I αa Ψ β Ȧ 28 8 v = 8 v 56 v 160 v WIJ Φ K 160v =0 algebraic W-constraint W AB! = W AB (Φ) αβ Ψ β = 3 Ȧ 14 W ȦḂ Ψ αḃ 3i 16 ΓI AȦ λ αa Φ I i 336 ΓABC IȦ ρ αabc Φ I 2 Φ I ( = 1 8 3Γ I AȦ λ αa Ψ α Ȧ 1 21 ΓABC IȦ ρ αabc Ψ α ) Ḃ 3 14 V IJ Φ J W IJ (W JK Φ K ) W JK (W JK Φ I ) V λ SF-expansion, component susy transformations & e.o.m, Equivalence

11 2. Gauge - Matter Coupling We discussed for the free case how to constrain the representation content (component fields) in a susy covariant way. -> + gauge covariance: Constraint: αa Φ I = iγ I AȦ Ψ αȧ Using the gauge field constraint one finds that the integrability condition determines, but: W AB Φ I αa Ψ β Ȧ 28 8 v = 8 v 56 v 160 v W IJ Φ K 160v =0 algebraic W-constraint W AB! = W AB (Φ) αβ Ψ β = 3 Ȧ 14 W ȦḂ Ψ αḃ 3i 16 ΓI AȦ λ αa Φ I i 336 ΓABC IȦ ρ αabc Φ I 2 Φ I ( = 1 8 3Γ I AȦ λ αa Ψ α Ȧ 1 21 ΓABC IȦ ρ αabc Ψ α ) Ḃ 3 14 V IJ Φ J W IJ (W JK Φ K ) W JK (W JK Φ I ) V λ SF-expansion, component susy transformations & e.o.m, Equivalence

12 3. Solutions for the W-constraints The dynamics is finally determined by a choice for W AB (Φ) functional form, gauge group G & representation R of the matter fields. Conformal theories: G GL(N,R) [W ]=1, [Φ] = 1 2 (W IJ ) a b := f a b,cd Φ c IΦ d J... g (polynomial) 1. αa W BC Ψ Φ 8 s 56 c 160 v 2. W IJ Φ K 160 =0 f a [bcd] Note: 160 v but also 8 v The group G and the representation R are restricted by W IJ g : f g ab f cde =3f g [cd f e]ab... Fundamental Identity 3-Algebras G = SO(4), R= fund. Superfield e.o.m.: 2 Φ Ka = 1 4 f a bc f def Φ b I Φ c J Φ d I Φ e J Φ Kf + Yukawa BLG

13 3. Solutions for the W-constraints The dynamics is finally determined by a choice for W AB (Φ) functional form, gauge group G & representation R of the matter fields. Conformal theories: G GL(N,R) [W ]=1, [Φ] = 1 2 (W IJ ) a b := f a b,cd Φ c IΦ d J... g (polynomial) 1. αa W BC Ψ Φ 8 s 56 c 160 v 2. W IJ Φ K 160 =0 f a [bcd] Note: 160 v but also 8 v The group G and the representation R are restricted by W IJ g : f g ab f cde =3f g [cd f e]ab... Fundamental Identity 3-Algebras G = SO(4), R= fund. Superfield e.o.m.: 2 Φ Ka = 1 4 f a bc f def Φ b I Φ c J Φ d I Φ e J Φ Kf + Yukawa BLG

14 3. Solutions for the W-constraints The dynamics is finally determined by a choice for W AB (Φ) functional form, gauge group G & representation R of the matter fields. Conformal theories: G GL(N,R) [W ]=1, [Φ] = 1 2 (W IJ ) a b := f a b,cd Φ c IΦ d J... g (polynomial) 1. αa W BC Ψ Φ 8 s 56 c 160 v 2. W IJ Φ K 160 =0 f a [bcd] Note: 160 v but also 8 v The group G and the representation R are restricted by W IJ g : f g ab f cde =3f g [cd f e]ab... Fundamental Identity 3-Algebras G = SO(4), R= fund. Superfield e.o.m.: 2 Φ Ka = 1 4 f a bc f def Φ b I Φ c J Φ d I Φ e J Φ Kf + Yukawa BLG

15 YM - Theories: G = G YM T k Matter is in the adjoint of GYM, Φ I =Φ a I T a. We also gauge a subgroup of the translations, defined by a fixed vector ξ I SO(8) SO(7): Gauging subgroup T k T(8N) : τ a := ξ I Ta I τ a Φ b I = ξ I δa b A M αβ T M = Âαβ a T a + Bαβτ a a F αβ,γδ Φ I = ˆF αβ,γδ Φ I +2ξ I ˆ [αβ B γδ] Deformation Potential: W IJ = 2Φ a [I ξ J]T a + f bc a Φ b IΦ c Jτ a Superfield e.o.m.: W-constraints: 1. αa W AB 160 =0 2. W AB Φ K 56 v (no 8v!) F αβ = ξ I αβ Φ Ia T a f bc a ( Φ Ib αβ Φ Ic + iψ b αȧψc βȧ ) τ a... CS-eom Non-abelian dual photon formulation of N =8 SYM

16 Re-dualize: Set ξ I = g YM δ I8 and fix gauge Φ 8 =0 ~Ta : ˆF a αβ = g YM B a αβ... eliminate B αβ ~ τa : 1 g 2 YM ε γδ ˆ γ(α ˆFβ)δ = 1 2 [Φi, ˆ αβ Φ i ]+iψȧ(α ΨȦβ) Standard 2 nd order N =8 SYM equations

17 N = 6 Super Space R 2,1 12 The R-symmetry group is now SO(6) SU(4), so I =1,...,6 now. Otherwise the structure is very similar. We will employ an SU(4) notation: ˆΓ I = [ 0 Γ I Γ I 0 ], Γ I ij with i,j =1,... 4 θ αi θ α ij = 1 2 ɛijkl θα kl a.s.o. Another difference is that matter transforms now in the 4 and 4 of SU(4): (Φ ia ) =: Φ ia, (Ψ a αi) =: Ψ i αa... N = 6 matter multiplet 1. Gauge Field Constraints F α ij,β kl ((2, 6) (2, 6)) sym =(3, 1) (1, 15) (3, 20) Constraint: { α ij, β kl } = i ( ɛ ijkl αβ + ε αβ ɛ mij[k W m ) l] W-constraint: α ij W k l 64 =0

18 N = 6 Super Space R 2,1 12 The R-symmetry group is now SO(6) SU(4), so I =1,...,6 now. Otherwise the structure is very similar. We will employ an SU(4) notation: ˆΓ I = [ 0 Γ I Γ I 0 ], Γ I ij with i,j =1,... 4 θ αi θ α ij = 1 2 ɛijkl θα kl a.s.o. Another difference is that matter transforms now in the 4 and 4 of SU(4): (Φ ia ) =: Φ ia, (Ψ a αi) =: Ψ i αa... N = 6 matter multiplet 1. Gauge Field Constraints F α ij,β kl ((2, 6) (2, 6)) sym =(3, 1) (1, 15) (3, 20) Constraint: { α ij, β kl } = i ( ɛ ijkl αβ + ε αβ ɛ mij[k W m ) l] W-constraint: α ij W k l 64 =0

19 N = 6 Super Space R 2,1 12 The R-symmetry group is now SO(6) SU(4), so I =1,...,6 now. Otherwise the structure is very similar. We will employ an SU(4) notation: ˆΓ I = [ 0 Γ I Γ I 0 ], Γ I ij with i,j =1,... 4 θ αi θ α ij = 1 2 ɛijkl θα kl a.s.o. Another difference is that matter transforms now in the 4 and 4 of SU(4): (Φ ia ) =: Φ ia, (Ψ a αi) =: Ψ i αa... N = 6 matter multiplet 1. Gauge Field Constraints F α ij,β kl ((2, 6) (2, 6)) sym =(3, 1) (1, 15) (3, 20) Constraint: { α ij, β kl } = i ( ɛ ijkl αβ + ε αβ ɛ mij[k W m ) l] W-constraint: α ij W k l 64 =0

20 2. Gauge - Matter Coupling To appropriately restrict the representation content, the matter SF Φ i contains now O(θ) =χ i,jk α 4 20, we impose: Constraint: α ij Φ k 20 =0 α ij Φ k = iδ k [i Ψ j] α The integrability condition thereof again puts another restriction on the deformation potential: W i j Φ k 15 4 = algebraic W-constraint: W i j Φ k 36 =0 In same fashion as before we obtain from these relations superfield e.o.m. of CS-matter type in terms of composite SF s derived from the deformation potential W i j.

21 2. Gauge - Matter Coupling To appropriately restrict the representation content, the matter SF Φ i contains now O(θ) =χ i,jk α 4 20, we impose: Constraint: α ij Φ k 20 =0 α ij Φ k = iδ[i k Ψ j] α The integrability condition thereof again puts another restriction on the deformation potential: W i j Φ k 15 4 = algebraic W-constraint: W i j Φ k 36 =0 In same fashion as before we obtain from these relations superfield e.o.m. of CS-matter type in terms of composite SF s derived from the deformation potential W i j.

22 2. Gauge - Matter Coupling To appropriately restrict the representation content, the matter SF Φ i contains now O(θ) =χ i,jk α 4 20, we impose: Constraint: α ij Φ k 20 =0 α ij Φ k = iδ[i k Ψ j] α The integrability condition thereof again puts another restriction on the deformation potential: W i j Φ k 15 4 = algebraic W-constraint: W i j Φ k 36 =0 In same fashion as before we obtain from these relations superfield e.o.m. of CS-matter type in terms of composite SF s derived from the deformation potential W i j.

23 3. Conformal solution for the W-constraints The most general polynomial ansatz is again bilinear w/ an inv. tensor: (W i j) a b := f ac bd ( Φ id Φj c 1 4 δi j Φ kd Φkc ) 1. ij W k l c.c. = c.c. 64 g 2. W i jφ k 36 =0 f [ac] [bd] Note: W i j Φ k W =W M TM f ac bd L l=1 1 g (l) κ MN (l) T (l) M a b T (l) c N d G =G1 x...x GL This expression with the above antisymmetry condition has been classified : U(N)xU(M), SU(N)xSU(N), U(N), Sp(N) and R=(N,M). Schnabl Tachikawa 08 The resulting SF-e.o.m. lead to the following Lagrangian: L ABJM = Tr{ Φ i Φ i i Ψ i 4 Ψ i + V bos + L Yuk k 4π (A da A3 Ã dã 2 3Ã3 )} ABJM

24 3. Conformal solution for the W-constraints The most general polynomial ansatz is again bilinear w/ an inv. tensor: (W i j) a b := f ac bd ( Φ id Φj c 1 4 δi j Φ kd Φkc ) 1. ij W k l c.c. = c.c. 64 g 2. W i jφ k 36 =0 f [ac] [bd] Note: W i j Φ k W =W M TM f ac bd L l=1 1 g (l) κ MN (l) T (l) M a b T (l) c N d G =G1 x...x GL This expression with the above antisymmetry condition has been classified : U(N)xU(M), SU(N)xSU(N), U(N), Sp(N) and R=(N,M). Schnabl Tachikawa 08 The resulting SF-e.o.m. lead to the following Lagrangian: L ABJM = Tr{ Φ i Φ i i Ψ i 4 Ψ i + V bos + L Yuk k 4π (A da A3 Ã dã 2 3Ã3 )} ABJM

25 3. Conformal solution for the W-constraints The most general polynomial ansatz is again bilinear w/ an inv. tensor: (W i j) a b := f ac bd ( Φ id Φj c 1 4 δi j Φ kd Φkc ) 1. ij W k l c.c. = c.c. 64 g 2. W i jφ k 36 =0 f [ac] [bd] Note: W i j Φ k W =W M TM f ac bd G =G1 x...x GL This expression with the above antisymmetry condition has been classified : U(N)xU(M), SU(N)xSU(N), U(N), Sp(N) and R=(N,M). Schnabl Tachikawa 08 The resulting SF-e.o.m. lead to the following Lagrangian: L ABJM = Tr{ Φ i Φ i i Ψ i 4 Ψ i + V bos + L Yuk k 4π (A da A3 Ã dã 2 3Ã3 )} ABJM

26 3. Conformal solution for the W-constraints The most general polynomial ansatz is again bilinear w/ an inv. tensor: (W i j) a b := f ac bd ( Φ id Φj c 1 4 δi j Φ kd Φkc ) 1. ij W k l c.c. = c.c. 64 g 2. W i jφ k 36 =0 f [ac] [bd] Note: W i j Φ k W =W M TM f ac bd 1 g (δa dδ c bδ bãδ d c δ a bδ c dδ b c δ dã) G =G1 x...x GL This expression with the above antisymmetry condition has been classified : U(N)xU(M), SU(N)xSU(N), U(N), Sp(N) and R=(N,M). Schnabl Tachikawa 08 The resulting SF-e.o.m. lead to the following Lagrangian: L ABJM = Tr{ Φ i Φ i i Ψ i 4 Ψ i + V bos + L Yuk k 4π (A da A3 Ã dã 2 3Ã3 )} ABJM

27 4. Enhancement N =6 N =8, Monopole operators ABJM is believed to describe N M2 branes in C 4 /Z k AdS 4 xs 7 /Z k For CS-level k=1,2 N =8 susy. Monopole operators play an important role. Create U(1)-flux through 2-sphere -> 1. Integrate fields w/ Dirac monopole singularities in PI (Kapustin Witten) 2. CFT operator vortex states (Kapustin Borokhov Wu): ρ : U(1) G, F d 1... GNO x mi H i M.O. with fluxes m i -> R with h.w. m i ( L G). With CS-term h.w. km i. N =8 susy by ferm. symmmetry of our constraints: δφ ia = ɛ α M ab Ψi αb δa α ij,δa αβ With M =0 this leads to M ae f bc ed = ˆf [abc] d! M ab ã b εab εã b Proposed additional currents for R-enhancement SU(4) SO(8): ( ) J ij αβ = M ab Φ a[i αβ Φ j]b + i 2 ɛijkl Ψ a k(α Ψb β)l Benna Klebanov Klose 09 Conservation? (dim.2). With M =0 we find same result.

28 U(2)xU(2) Kwon Oh Sohn 09 (unfinished U(N)xU(N) attempt fails). Gustavsson Rey 09 (inconsistent). Fake monopole operators, monopole operator (k=2) has to be in sym-sym representations. Real monopole operators : 3. thooft 78: Singular gauge transformations acting on fields, matter invariant under center Z(G) of gauge group. For pure CS, action not invariant -> boundary term from tube around insertion (Moore Seiberg, Gawedzki 89): M Tr R P exp ( i ) A Ã R =(N kq sym, N kq sym ) Problem: Matter not invariant under center Z(G)...

29 Conclusions and Outlook We gave a systematic analysis of the N = 8,6 superspace constraints. The possible models are encoded in a single quantity, the deformation potential W subjected to the W-constraints. Direct N = 8,6 classification, no reference to N = 2 needed Conformal and YM theories on same footing Susy covariant and efficient formalism (see e.g. N = 6 N = 8) SF e.o.m, etc. look formally the same as component expressions What more do you want? Close the 8v gap in the N = 8 case, are there more models? Determine quantum corrections in a non-perturbative way Find twistor space formulation for the constraints, integrability Tame down the monopole operators, mirror symmetry (Superspace) CFT N 3/2 scaling

30 Conclusions and Outlook We gave a systematic analysis of the N = 8,6 superspace constraints. The possible models are encoded in a single quantity, the deformation potential W subjected to the W-constraints. Direct N = 8,6 classification, no reference to N = 2 needed Conformal and YM theories on same footing Susy covariant and efficient formalism (see e.g. N = 6 N = 8) SF e.o.m, etc. look formally the same as component expressions What more do you want? Close the 8v gap in the N = 8 case, are there more models? Determine quantum corrections in a non-perturbative way Find twistor space formulation for the constraints, integrability Tame down the monopole operators, mirror symmetry (Superspace) CFT N 3/2 scaling Thank you!