Higgsing M2 worldvolume theories

Transcription

1 Higgsing M2 worldvolume theories Costis Papageorgakis Tata Institute of Fundamental Research Imperial College 19 November 2008

2 Based on: M2 to D2, Sunil Mukhi and CP, arxiv: [hep-th], JHEP 0805:085 (2008) M2-branes on M-folds, Jacques Distler, Sunil Mukhi, CP and Mark van Raamsdonk arxiv: [hep-th], JHEP 0805:038 (2008) D2 to D2, Bobby Ezhuthachan, Sunil Mukhi and CP arxiv: [hep-th], JHEP 0807:041 (2008) Bobby Ezhuthachan, Sunil Mukhi and CP in progress

3 In the Beginning... Recently, Bagger-Lambert and Gustavsson proposed a candidate theory for describing multiple M2-branes. Achieved via introduction of CS gauge fields for closure of N = 8 supersymmetry algebra and 3-algebras [Filipov] [T A, T B, T C ] = f ABC DT D with h AB = Tr(T A T B ) satisfying f ABCD = f [ABCD] and f [ABC G f E]F G D = 0 (F.I.)

4 With this write a 3d action L = 1 2 DµXA(I) D µ X (I) A 1 f ABCDX A(I) X B(J) X C(K) f ɛµνλ f ABCDAµ AB νa CD + i 2 ΨA Γ µ D µψ A + i 4 fabcdψb Γ IJ X C(I) X D(J) Ψ A D EF G X E(I) X F (J) X G(K) λ f G AEF f BCDG A AB µ Aν CD «Aλ EF This has the following exciting features Manifest SO(8) R-symmetry Maximally (N = 8) superconformal

5 However only one example with positive definite metric: A 4 -theory with f ABCD = fɛ ABCD SO(4) [Gauntlett-Gutowski, Papadopoulos] a discrete free parameter coming from the quantised CS level f = 2π k Intriguing but can be used as expansion parameter A 4 can be re-cast as SU(2) SU(2) CS-matter theory [van Raamsdonk] M-theory interpretation? Role of 3-algebra?

6 Moduli space analysis suggests 2 M2s on generalised orbifold ( M-fold ). [Lambert-Tong, Distler-CP-Mukhi-van Raamsdonk] Moduli space topology: R8 R 8 D 2k R8 R 8 Z 2 Z 2k Preserves N = 8 for any value of k Not conventional M-theory orbifold (R8 /Z k ) N S N - but agrees for k = 2 Precise spacetime interpretation remains unknown. Possible mysterious enhancement from N = 6?

7 ABJM generalised the above by relaxing manifest N = 8 to N = 6. They found a U(N) U(N) CS-matter theory describing N M2-branes on C 4 /Z k. [Aharony-Bergman-Jafferis-Maldacena] Use λ = N/k as t Hooft expansion parameter: Gauge theory now weakly coupled for λ 1. Good gravity description for λ 1 as AdS 4 S 7 /Z k or AdS 4 CP 3 AdS 4 /CFT 3 correspondence. ABJM can also be thought as relaxed 3-algebra: not totally antisymmetric structure constants modified version of FI. Leads to a classification of N = 6 CS-matter theories. [Bagger-Lambert, Schnabl-Tachikawa]

8 In both BLG and ABJM, there exists a novel version of the Higgs mechanism corresponding to giving a diagonal vev to a scalar, i.e. simultaneously taking the M2-branes away from orbifold. For sufficiently large vevs one of the CS gauge fields can be integrated out while the other one becomes dynamical. Resulting theory is SYM in 3d and can be thought as low energy theory on equal number of D2-branes in flat space.

9 Outline Introduction and Motivation Higgsing at lowest order Higgsing at four-derivative order Conclusions

10 Higgsing the A 4 -theory The Lagrangean for the A 4 -theory is in van Raamsdonk s formulation S A4 = k Z» d 3 x Tr ( 2π D µ X I ) DµX I 8 3 XIJK X KJI + fermions + 1 ɛµνλ A (1) 2 µ νa (1) λ A(1) µ A (1) ν A (1) λ A(2) µ νa (2) λ 2 3 A(2) µ A (2) ν A (2) λ where X IJK = X [I X J X K] The matter fields are complex-valued but obey the reality condition X a ḃ = ɛ ab ɛḃȧ X ȧb

11 They transform in the bi-fundamental representation (2, 2) D µ X I = µ X I + A (1) µ X I X I A (2) µ As a next step create linear combinations of the gauge fields The covariant derivative is now ( A µ = 1 2 A (1) µ + A (2) ) µ ( B µ = 1 2 A (1) µ A (2) ) µ D µ X I = D µ X I {B µ, X I } with D µ X I = X I + [A µ, X I ] F µν = µ A ν ν A µ + [A µ, A ν ]

12 The bosonic part of the action becomes S B = k [ d 3 x Tr (D µ X I ) D µ X I 8 2π 3 XIJK X KJI +{B µ, X I }{B µ, X I } + D µ X I {B µ, X I } {B µ, X I } D µ X I +ɛ µνλ( B µ F νλ B µb ν B λ ) ] Note that the new gauge field A µ is in the diagonal subgroup of SU(2) SU(2) and has an adjoint action on the Xs.

13 Expand the scalars in suitable basis and give one a vev v, e.g. X 8 where X 8 = 1 2 (v + x8 ) 1l + x 8 X i = 1 2 xi 1l + x i x I Ia σa = i x 2 with σ a the Pauli matrices. Have performed decomposition of bi-fundamental scalars into trace and traceless part, We will be interested in the limit of large vev, v.

14 Substitute back into the action to get S B = k { ( d 3 x 1 2π 2 µ x I µ x I + Tr D µ x I D µ x I + v2 2 [xi, x j ][x i, x j ] + 2vB µ D µ x 8 + v 2 B µ B µ + ɛ µνλ B µ F νλ ) } + higher-order The higher-order terms will be suppressed in powers of 1 v. The gauge field B µ has acquired a mass term by the Higgs mechanism. Since it has no kinetic term, it can be integrated out.

15 This will render A µ dynamical! The corresponding Goldstone boson is x 8, as is evident by grouping terms depending on x 8 and B µ v 2 ( B µ + 1 v D µx 8) 2 + ɛ µνλ ( B µ + 1 v D µx 8) F νλ and shifting B µ B µ 1 v D µx 8. The equation of motion for B µ is B µ = 1 2v 2 ɛµνλ F νλ Eliminating B µ from the action and rescaling x 1 v x, x 1 v x S B = k { ( 2πv 2 d 3 x 1 2 µ x I µ x I 1 + Tr 2 F µν F µν ) } + D µ x i D µ x i [xi, x j ][x i, x j ]

16 As last step, combine seven singlet scalars x i with SU(2) adjoints x i to make U(2) adjoints ˆX i = i 2 xi 1l + x i Left with singlet scalar x 8, to be dualised into an abelian gauge field ( ) d 3 x 1 2 µ x 8 µ x 8 d 3 x 1 4 F µν U(1) F µν U(1) ɛµνλ µ x 8 F U(1) νλ Integrating out x 8 on RHS gives Bianchi for F U(1) µν and the new abelian gauge field combines with SU(2) gauge field to form U(2) gauge field  µ = i 2 AU(1) µ 1l + A µ ˆF µν = i 2 F µν U(1) 1l + F µν

17 We have ended up with S B = k ( 2πv 2 d 3 1 x Tr ˆF µν 2 ˆFµν + D µ ˆXi D ˆXi µ [ ˆX i, ˆX j ][ ˆX i, ˆX ) j ] The higher-order terms have decoupled for k, v with k v 2 fixed as they are of higher order 1 v but have same k-dependence as leading terms. Fermions follow and we recover maximally supersymmetric U(2) YM with coupling constant g 2 YM = v2 /k, the low-energy theory on 2 D2s. For finite k the SYM theory becomes strongly coupled. By definition theory on 2 M2s proof that A 4 is related to membrane physics.

18 Summary and effective Higgs rules Summarising the Higgs process Give vev to a bi-fundamental scalar X 8 One non-dynamical gauge field integrated out while other becomes dynamical in diagonal of SU(2) SU(2) Decompose scalars into trace and traceless part X 8 traceless part is the Goldstone mode X 8 trace part becomes abelian gauge field after duality Seven remaining scalars become U(2) adjoint scalars Everything combines into U(2) N = 8 SYM in 3d

19 Net result can be captured by set of effective rules: For CS gauge fields For scalars L CS 2 v 2 f µ f µ with f µ = 1 2 ɛµνλ ˆFνλ D µ X 8 1 v f µ, X ij8 1 4v [ ˆX i, ˆX j ], D µ X i 1 v Dµ ˆXi, X ij8 1 4v [ ˆX i, ˆX j ] We have gone from M2 to D2! This has not involved conventional compactification to IIA but is reminiscent of similar effects in models of deconstruction. [Mukhi-CP, Distler-Mukhi-CP-van Raamsdonk]

20 Higgsing ABJM Higgs mechanism extends to ABJM with minor modifications. [Aharony-Bergman-Jafferis-Maldacena, Pang-Wang, Li-Liu-Xie] The fields are now in U(N) U(N) and decompose as Y A = Y A0 T 0 + iy Aa T a with Y A = X A + ix A+4 and the vev is Y 40 = kv. The dynamical gauge field is now in the U(N) diagonal subgroup. Need to cancel one more degree of freedom.

21 After implementing the Higgsing find that traceless part of X 4 and trace part of X 8 cancel out. The degrees of freedom work out right no need for abelian duality. Implies compactification of X 8 transverse to X 4 which developed the vev. [Ezhuthachan-Mukhi-CP in progress] Get U(N) theory of N D2-branes with YM coupling g 2 Y M = v2 /k. The theory is weakly coupled when Ngeff 2 1, implying k N.

22 Knowing the geometric orbifold action it is easy to see the effective compactification. Start with (Y 1, Y 2, Y 3, Y 4 ) e 2πi k (Y 1, Y 2, Y 3, Y 4 ) For large k, v we can expand around the vev In components X 4 X 4 Y 4 Y 4 + 2πi k ( kv) +... X 8 X 8 + 2πR with R = v k = g Y M l 3/2 pl

23 Many applications of this in ABJM context. Serves as check for consistency of proposals relating to weakly-coupled gauge theory. Aside: Alternative interesting recent use in attempt of field-theoretic derivation of S-duality. [Hashimoto-Tai-Terashima] Consider N M2s on C 4 /Z k Z n Give vevs to 2 scalars - hence M-theory on T 2 or IIA on S 1 Use field-theoretic version of T-duality (deconstruction - Taylor) to turn D2s into D3s Freedom in Higgsing leads to different values of gy 2 M related by S- and T- SL(2, Z) transformations Unfortunately these only give a one real-parameter subspace of possible couplings

24 Four-derivative corrections to A 4 Another application is the determination of four-derivative terms in the BLG/ABJM action. This result also applies to finite k. Same problem solved for Lorentzian BLG-theories. Lorentzian BLG theories have N = 8, no free parameter and indefinite metric. [Gomis et al., Benvenuti et al., Ho et al.] An appropriately gauged version of the above is on-shell equivalent to YM after performing a non-abelian duality in 3d. [Bandres et al., Gomis et al., Ezhuthachan-Mukhi-CP, de Wit-Nicolai-Samtleben] Usefulness of the un-gauged theory limited. [Verlinde]

25 The four-derivative corrections obtained by applying the same procedure to α 2 -corrected D2-brane action. [Alishahiha-Mukhi] Corrections arranged themselves in terms of 3-brackets and it was conjectured that the same should hold for Euclidean BLG-theories. With this assumption, the Higgs mechanism determines the form of the four-derivative corrections to the BLG A 4 -theory uniquely. [Ezhuthachan-Mukhi-CP in progress] There is a limit in which ABJM reduces to Lorentzian BLG. This should also extend to higher-derivative corrections. [Honma et al., Antonyan-Tseytlin]

26 The non-abelian 3BI What is the form of the most general operator one can add? The non-linear form of the action for the single membrane points to l 3 pl -corrections. Thus the full non-abelian 3BI action will have an expansion S 3BI = S A4/ABJM + S l 3 p +... Look for dimension 6 operators. In the bosonic sector, take the most general gauge-invariant operator built out of 3-algebra building blocks.

27 No CS-terms for gauge fields (not gauge invariant) No F 2, F 4 terms for gauge fields (extra d.o.f. - break susy) Left with all legal index contractions for ( DX) ( 4 : k 2 l 3 pl STr a D µ X I Dµ X J Dν X J Dν X I +b D µ X I Dµ X I Dν X J Dν X J ) X IJK ( DX) ( ) 3 : k 2 l 3 plɛ µνλ STr c X IJK Dµ X I Dν X J Dλ X K + c h.c. (X IJK ) 2 ( DX) ( 2 : k 2 l 3 plstr d X IJK X IJK Dµ X L Dµ X L +e X IJK X IJL Dµ X K D µ X L ) ( (X IJK ) 4 : k 2 l 3 pl STr f X IJK X IJK X LMN X LMN +g X IJM X KLM X IKN X JLN +h X IJK X IJL X MNK X MNL )

28 Higgs and fix numerical coefficients by comparing with D2-brane effective action at O(α 2 ). Latter obtained using symmetrised trace in non-abelian DBI. [Tseytlin, Bergshoeff-Bilal-de Roo-Sevrin, Cederwall-Nielsson-Tsimpis] Sα B α 2 2 = gy 2 M [ d 3 1 x STr ˆF νρ 4 µν ˆF ˆFρσ ˆF σµ 1 ˆF µν ρσ 16 ˆFµν ˆF ˆFρσ 1 4 D µ ˆX i D µ ˆXi D ν ˆXj D ν ˆXj D µ ˆX i D ν ˆXi D ν ˆXj D µ ˆXj ˆX [ij] ˆX[jk] ˆX[kl] ˆX[li] 1 16 ˆX [ij] ˆX[ij] ˆX[kl] ˆX[kl] ˆF µν ˆF νρ D ρ ˆXi D µ ˆXi 1 4 ˆF µν ˆF µν D ρ ˆXi D ρ ˆXi 1 8 ˆF µν ˆF µν 1 4 D µ ˆX i D µ ˆXi ˆX[kl] ˆX[kl] ˆX [ij] ˆX[jk] D µ ˆXk D µ ˆXi ˆF µν D ν ˆXi D µ ˆXj ˆX[ij] ] ˆX[kl] ˆX[kl]

29 3BI to DBI Proceed with Higgsing as before. The e.o.m. for the gauge field B µ get corrected Two ways to get l 3 pl terms: B µ = 1 v 2 f µ l3 pl 2v δ B µ (L l 3 pl ) Plug in lowest-order e.o.m. to four-derivative action Plug in four-derivative correction to lowest-order action these vanish Higgs rules extend to this order.

30 So use D µ X 8 1 v f µ, X ij8 1 4v [ ˆX i, ˆX j ], Dµ X i 1 v Dµ ˆXi The terms we get are weighted by: l 3 pl k2 /v 4 = g s l 3 s/g 4 Y M = α 2 /g 2 Y M which is exactly what we want. But 3 (X IJK ) 4 terms in 3BI while 2 ( ˆX [ij] ) 4 in DBI. Saved by SU(2) structure leading to identities for DBI and 3BI respectively ( ) STr ˆX[ij] ˆX[jk] ˆX[kl] ˆX[li] = 1 ( ) 2 STr ˆX[ij] ˆX[ij] ˆX[kl] ˆX[kl]

31 Also ( STr X IJK X IJL X MNK X MNL ) =2STr (X IJM X KLM X IKN X JLN ) = 1 ( 3 STr X IJK X IJK X LMN X LMN ) Can fix the coefficients uniquely a = 1 2, b = 1 4, c = 2 3 d = 4 3, e = 8, f = 16 9 Corrections including fermions work out in a similar way. We have obtained the full four-derivative correction to the A 4 -theory for any k.

32 The full answer is S A4 +l 3 = k Z» d 3 x Tr ( pl 2π D µ X I ) DµX I 8 3 XIJK X KJI + 1 ɛµνλ A (1) 2 µ νa (1) λ A(1) µ A (1) ν A (1) λ A(2) µ νa (2) λ 2 3 A(2) µ A (2) ν A (2) λ Z + k2 l 3 p 4π 2» d 3 1 x STr D µ X I DµX J 2 Dν X J DνX I 1 4 D µ X I DµX I Dν X J DνX J XIJK X KJI X LNP X P NL XIJK X KJI DµX L Dµ X L 8 X IJK X KJL DµX L Dµ X I 2 3 ɛµνλ DµX I DνX J Dλ X K X IJK + D µx I DνX J Dλ X K X IJK + fermions...which are far worse.

33 ...and look something like this S F l 3 = k2 l 3 Z p p 4π 2 d 3 x STrhÂ Ψ Γ IJ [X K, X L, Ψ] Ψ Γ KL [X I, X J, Ψ] + ˆB Ψ Γ µ Dν Ψ Ψ Γ ν DµΨ + Ĉ Ψ Γ µ [X I, X J, Ψ] Ψ Γ IJ DµΨ + ˆD Ψ Γ µγ IJ DνΨ D µ X I Dν X J + Ê Ψ Γ µ Dν Ψ D µ X I DνX I + ˆF Ψ Γ IJKL DνΨ X [IJK] Dν X L + Ĝ Ψ Γ IJ DνΨ X [IJK] Dν X K + Ĥ Ψ Γ IJ [X J, X K, Ψ] D µ X I DµX K + Î Ψ [X I, X J, Ψ] D µ X I DµX J + Ĵ Ψ Γ µν [X I, X J, Ψ] D µx I DνX J + ˆK Ψ Γ µνγ IJ [X J, X K, Ψ] D µ X I Dν X K + ˆL Ψ Γ µγ IJ [X K, X L, Ψ] D µ X I X [JKL] + ˆM Ψ Γ µ[x I, X J, Ψ] D µ X K X [IJK] + ˆN Ψ Γ µγ IJKL [X L, X M, Ψ]X [IJK] Dµ X M + Ô Ψ Γ µγ IJ [X K, X L, Ψ]X [IJK] Dµ X L + ˆP Ψ Γ IJKL [X M, X N, Ψ]X [IJL] X [KMN] + ˆQ Ψ Γ IJ [X K, X L, Ψ]X [IJM] X [KLM] + Hermitian conjugates with the same coefficients

34 Conclusions We have reviewed an interesting version of the Higgs mechanism in the context of M2 worldvolume theories For large vevs one gets effective compactification and the theory is equivalent to that of D2-branes This connection holds beyond linear order and helped us establish the form of the four-derivative correction to the A 4 -theory Expect the same to work for ABJM

35 Crucial test would be to show that these corrections are compatible with N = 8 supersymmetry In that case it would show that the novel Higgs mechanism can tell us something about these theories at finite k 3-algebra structure again evident Relation between 3-algebras and supersymmetry?