Ten-dimensional Dirac Algebra

We use the following representation of the 10d Dirac matrices (

Γ^A,Γ^B = 2η^AB):

Γ⁰=iγ⁰⊗I⊗I⊗I, Γ¹=iγ¹⊗I⊗I⊗I Γ²=iγ²⊗I⊗I⊗I, Γ³=iγ³⊗I⊗I⊗I, Γ⁴=γ⁵⊗σ2⊗I⊗σ1, Γ⁵=γ⁵⊗σ2⊗I⊗σ3, Γ⁶=γ⁵⊗σ1⊗σ2⊗I, Γ⁷=γ⁵⊗σ3⊗σ2⊗I, Γ⁸=γ⁵⊗I⊗σ1⊗σ2, Γ⁹=γ⁵⊗I⊗σ3⊗σ2,

whereI is the2×2 identity matrix, theγ⁰sare 4d Dirac matrices given by

γ⁰=σ1⊗I, γ¹=iσ2⊗σ1, γ²=iσ₂⊗σ₂, γ³=iσ₂⊗σ₃, γ⁵=iγ⁰γ¹γ²γ³,

and the Pauli matrices are given by equation (A.1).

Finally, we dene the 10d chirality operator as

Γ11= Γ⁰Γ¹Γ²Γ³Γ⁴Γ⁵Γ⁶Γ⁷Γ⁸Γ⁹.

Appendix B

Poincaré and Conformal

Supersymmetries of the BL SO(4) Theory

In this appendix we veried in complete detail the Poincaré and conformal supersymmetries of the BLSO(4)theory.

The BLSO(4)Lagrangian is given by

L=−1

2 Dµφ^I

a D^µφ^I

a+ i

2ψ^A_aγ^µ Dµψ^A

a

+ic1^abcdψ^A_a Γ^IJ

ABψ_b^Bφ^I_cφ^J_d −c2^{abcdef gd}φ^I_aφ^J_bφ^K_c φ^K_g φ^I_eφ^J_f +c3^µνλabcd

Aµab∂νAλcd+2

3AµabAνcgAλgd

,

where theci,i= 1,2,3,4 will be xed by supersymmetry. The eld content and the index notation are given in tables 2.1 and 2.2. The supersymmetry transformations are

δφ^I_a =iψ^A_aΓ^I_AA˙^A˙ =i^A˙Γ^I_AA˙ ψ^Aa, (B.1) δψAa =−γµ Dµφ^I

aΓ^I_AA˙^A˙ +c4^abcd Γ^{IJ K}

AA˙^A˙φ^I_bφ^J_cφ^K_d −φ^I_aΓ^I_AA˙η^A˙, (B.2) δA^µ_ab=i^abcdψ^A_cγµΓ^I_AA˙^A˙φ^I_d, (B.3) and

^A˙(x) =^A₀^˙ +γ^µx_µη^A˙,

where ^A₀^˙ is a constant 8c-spinor that correspond to the Poincaré supersymmetry parameter while η^A˙ correspond to the superconformal parameter.

Variations

Now, we will take the supersymmetry variations of the Lagrangian by separating it in ve terms: A, B, C, D and the Chern-Simons term.

• Term A=−¹₂ D_µφ^I

a D^µφ^I

a

First, we can write A up to boundary terms in the following way

A =−1

2 D_µφ^I

a D^µφ^I

a= 1

2 D^µD_µφ^I

aφ^I_a. Then variation is

δ(A) = 1

2δ D^µDµφ^I

aφ^I_a

=D^µDµφ^I_aδφ^I_a+1

2δA^µ_ab Dµφ^I

bφ^I_a+1

2Dµ δA^µ_abφ^I_b φ^I_a

=D^µDµφ^I_aδφ^I_a+δA^µ_ab Dµφ^I

bφ^I_a

=iD^µD_µφ^I_a ψ_aΓ^I

+i^abcd ψ_cγ_µΓ^J

φ^J_d D_µφ^I

bφ^I_a.

• Term B=₂ⁱψ^A_aγ^µ Dµψ^A

a

δ(B) =δ i

2ψ_aγ^µ(Dµψ)_a

=iψ_aγ^µ∂_µδψ_a+iA_µabψ_aγ^µδψ_b+ i

2δA_µabψ_aγ^µψ_b

=iψ_aγ^µD_µδψ_a+ i

2δA_µabψ_aγ^µψ_b

=−i ψ_aγ^µγvΓ^I

DµDvφ^I

a+ 3ic4^abcd ψ_aγ^µΓ^{IJ K}

Dµφ^I_bφ^J_cφ^K_d

−i ψ_aγ^µΓ^Iη

Dµφ^I_a−i ψ_aγ^µγvΓ^IDµ

Dvφ^I

a

+ic4^abcd ψ_aγ^µΓ^{IJ K}Dµ

φ^I_bφ^J_cφ^K_d + i

2δAµabψ_aγ^µψb

−2ic4^abcd ψ_eγ^µΓ^{IJ K}

A^µ_aeφ^I_bφ^J_cφ^K_d

=−i ψ_aγ^µγ_vΓ^I

D_µD_vφ^I

a+ 3ic₄^abcd ψ_aγ^µΓ^{IJ K}

D_µφ^I_bφ^J_cφ^K_d + 3ic4^abcd ψ_aΓ^{IJ K}η

φ^I_bφ^J_cφ^K_d −2ic4^becd ψ_aγ^µΓ^{IJ K}

A^µ_abφ^I_eφ^J_cφ^K_d

−1

2^abcd ψ_cγµΓ^I

ψ_aγ^µψb

φ^I_d

=−i ψ_aΓ^I

D_µD^µφ^I

a−i1

2 ψ_aγ^µγ^vΓ^I

F_µvabφ^I_b + 3ic4^abcd ψ_aγ^µΓ^{IJ K}

Dµφ^I

bφ^J_cφ^K_d + 3ic4^abcd ψ_aΓ^{IJ K}η

φ^I_bφ^J_cφ^K_d −2ic4^becd ψ_aγ^µΓ^{IJ K}

A^µ_abφ^I_eφ^J_cφ^K_d

−1

2^abcd ψ_cγµΓ^I

ψ_aγ^µψb

φ^I_d.

• Term C=ic₁^abcdψ^A_a Γ^IJ

ABψ_b^Bφ^I_cφ^J_d δ(C) = 2ic1^abcdψ^A_a Γ^IJ

ABδψ_b^Bφ^I_cφ^J_d+ 2ic1^abcdψ^A_a Γ^IJ

ABψ^B_b δφ^I_cφ^J_d,

= 2ic₁^abcdψ^A_a Γ^IJ

AB[−γ_µ D_µφ^K

bΓ^K+c₄^{bef g}Γ^{LM N}φ^L_eφ^M_f φ^N_g

−φ^K_b Γ^Kη]φ^I_cφ^J_d −2c1^abcdψ_aΓ^IJψbψ_cΓ^Iφ^J_d,

=−2ic1^abcd ψ_aΓ^IJγµΓ^K

Dµφ^K

bφ^I_cφ^J_d + 2ic₁c₄^{abcdbef g} ψ_aΓ^IJΓ^KLM

φ^I_cφ^J_dφ^K_e φ^L_fφ^M_g

−2ic1^abcd ψ_aΓ^IJΓ^Kη

φ^K_b φ^I_cφ^J_d

−2c₁^abcd ψ_aΓ^IJψ_b

ψ_cΓ^I φ^J_d.

• Term D=−c2^{abcdef gd}φ^I_aφ^J_bφ^K_c φ^I_eφ^J_fφ^K_g

δ(D) =−3c2^{abcdef gd}δ φ^I_aφ^I_e

φ^J_bφ^J_fφ^K_c φ^K_g

=−3ic2^{abcdef gd} ψ¯aφ^I_e+ ¯ψeφ^I_a

Γ^Iφ^J_bφ^J_fφ^K_c φ^K_g . After relabeling dummy indices we get

δ(D) =−6ic2^{abcdbef g} ψ¯aΓ^K

φ^I_cφ^J_dφ^K_c φ^J_fφ^I_g.

• Chern-Simons term=c3^µνλabcd Aµab∂νAλcd+²₃AµabAνcgAλgd

δ(CS) = 2c3^µνλabcd(δAµab∂νAλcd+δAµabAνcgAλgd)

=c3^µνλabcdFνλδAµab

=ic₃^{µνλabcdabef} ψ_eγ_µΓ^I φ^I_fF_νλ

=ic34^µνλ δ^cd_ef

ψ_eγµΓ^I

φ^I_fFνλcd

=ic₃4^µνλ ψ_cγ_µΓ^I

φ^I_dF_νλcd, where we used the fact that

^µνλAeµcdδAνcgAλgd=^µνλδAeµcdAνcgAλgd.

Cancellations

We can classify the terms we obtain from the supersymmetries variations in ve dierent types as shown in table B.1. Now we will show how these dierent types of contributions cancel by choosing the rightci's parameters.

Table B.1. Classication of the supersymmetry variations of the BLSO(4)Lagrangian DDφψ F φψ ψ(Dφ)φ² ψφ³η ψ³φ φ⁵ψ

A × 0 × 0 0 0

B × × × × × 0

C 0 0 × × × ×

D 0 0 0 0 0 ×

CS 0 × 0 0 0 0

• Term DDφψ

This cancelation is trivial

iD^µD_µφ^I_a ψ_aΓ^I

−i D_µD^µφ^I

a ψ_aΓ^I

= 0.

• Term F φψ

We rst simplify the contribution from the B term as

−i1

2 ψ_aγ^µγ^vΓ^I

Fµvabφ^I_b =−i1

2^µvρ ψ_aγρΓ^I

φ^I_bFµvab, then for the cancelation we need

−iκc34^µvρ ψ_aγρΓ^I

φ^I_bFµvab−i1

2^µvρ ψ_aγρΓ^I

φ^I_bFµvab = 0, therefore this implies that

c3=1 8.

• Term ψφ³η

First, let us simplify both contributions in the following way

3ic₄^abcd ψ_aΓ^{IJ K}η

φ^I_bφ^J_cφ^K_d = 3ic₄^abcd ψ_aΓ^IΓ^JΓ^Kη

φ^I_bφ^J_cφ^K_d ,

−2ic1^abcd ψ_aΓ^IJΓ^Kη

φ^K_b φ^I_cφ^J_d =−2ic1^abcd ψ_aΓ^IΓ^JΓ^Kη

φ^I_bφ^J_cφ^K_d, then we need

c4= 2 3c1.

• Term ψ(Dφ)φ² We want

i^abcd ψ_aγµΓ^J

Dµφ^I

bφ^I_dφ^J_c + 3ic4^abcd ψ_aγ^µΓ^{IJ K}

Dµφ^I

bφ^J_cφ^K_d

−2ic₁^abcd ψ_aγ^µΓ^IJΓ^K

D_µφ^K

bφ^I_cφ^J_d = 0, expanding the second and third contributions we get

i^abcd ψ_aγ_µΓ^J

φ^J_b D_µφ^I

dφ^I_c

+ic4^abcd ψ_aγ^µ ΓKΓ^T_IΓJ+ ΓIΓ^T_JΓK−ΓIΓ^T_KΓJ

Dµφ^K

bφ^I_cφ^J_d

−2ic1^abcd ψ_aγ^µΓ^IΓ^T_JΓK

Dµφ^K

bφ^I_cφ^J_d = 0, and using the following simplication of the second term

^abcd ψ_aγ^µ 2δ^IKΓ^J+ 3Γ_IΓ^T_JΓ_K−4Γ_Iδ^KJ

D_µφ^K

bφ^I_cφ^J_d = 3^abcd ψ_aγ^µΓIΓ^T_JΓK

Dµφ^K

bφ^I_cφ^J_d −6^abcd ψ_aγ^µΓ^J

Dµφ^I

dφ^I_cφ^J_b, we nd that the coecients are

c₁=1 4, c₄=1

6.

• Term ψ³φ

The required cancellation is

0 =−1

2^abcd ψ_aγ^µψb

ψ_cγµΓ^I

φ^I_d+ 2c1^abcd ψ_aΓ^IJψb

ψ_cΓ^J φ^I_d

=−1

2T1 + 2c1T2.

Then, we have two structures

T1 =^abcd ψ_aγ^µψ_b

ψ_cγ_µΓ^I φ^I_d, and

T2 =^abcd ψ_aΓ^IJψ_b

ψ_cΓ^J φ^I_d, and we want to prove they are equivalent.

We decompose the rst structure in the following way

T1 =^abcd ψ_aγ_b^µψb

ψ_cγµΓ^I φ^I_d= 1

3^abcd





ψ_aγ^µ ψ_bψ_c−ψ_cψ_b γ_µΓ^I + ψ_aγ^µψ_b

ψ_cγ_µΓ^I



φ^I_d.

Using the following Fierz transformation

ψbψ_c−ψcψ_b=1

8γ^µψ_bγµψc− 1

16Γ^IJψ_bΓ^IJψc+ 1

384Γ^{IJ KL}γ^µψ_bΓ^{IJ KL}γµψc, we get

T1 = 1 3^abcd











−¹₈ ψaΓ^Iγ^ν

ψ_bγνψc

−₁₆³ ψ_aΓ^LMΓ^I

ψ_bΓ^LMψc

−₃₈₄¹ ψ_aΓ^{LM N O}Γ^Iγ^ν

ψ_bΓ^{LM N O}γ_νψ_c + ψ_aΓ^Iγ_µ

ψ_bγ^µψ_c









 φ^I_d,

=1 3^abcd





7

8 ψ_aΓ^Iγ^ν

ψ_bγ_νψ_c

−₁₆³ ψ_aΓ^LMΓ^I

ψ_bΓ^LMψ_c

−₃₈₄¹ ψ_aΓ^{LM N O}Γ^Iγ^ν

ψ_bΓ^{LM N O}γ_νψ_c



φ^I_d.

For the second structure we get

T2 =^abcd ψ_aΓ^IJψb

ψ_cΓ^J φ^I_d=1

3



^abcd





ψ_aΓ^IJ ψbψ_c−ψcψ_b Γ^J + ψ_aΓ^IJψb

ψ_cΓ^J



φ^I_d



. Using the Fierz transformation and the following identities

Γ^IJΓ^J= 7Γ^I,

Γ^IJΓ^LMΓ^J= 3Γ^LMΓ^I −8Γ^Lδ^IM+ 8Γ^Mδ^IL, Γ^IJΓ^{LM N O}Γ^J=−Γ^{LM N O}Γ^I,

we get

T2 = 1 3









 ^abcd











7

8 ψ_aΓ^Iγ^ν ψ¯_bγ_νψ_c

− ψ_aΓ^J

ψ_bΓ^IJψ_c

−₃₈₄¹ ψ_aΓ^{LM N O}Γ^Iγ^ν

ψ_bΓ^{LM N O}γ_νψ_c

−₁₆³ ψ_aΓ^LMΓ^I

ψ_bΓ^LMψc

+ ψ_cΓ^J

ψ_aΓ^IJψb









 φ^I_d











= 1 3^abcd





7

8 ψ_aΓ^Iγ^ν ψ¯bγνψc

−₁₆³ ψ_aΓ^LMΓ^I

ψ_bΓ^LMψc

−₃₈₄¹ ψ_aΓ^{LM N O}Γ^Iγ^ν

ψ_bΓ^{LM N O}γνψc



φ^I_d.

ThereforeT1 = T2,and

c1=1 4.

• Term φ⁵ψ We want

2ic1c4^{abcdbef g}ψ^aΓ^IJΓ^KLMφ^I_cφ^J_dφ^K_eφ^L_fφ^M_g −6ic2^{abcdbef g}ψ^aΓ^Kφ^I_cφ^J_dφ^K_eφ^J_fφ^I_g= 0,

then we would like to show that

3c2^{abcdbef g}Γ^Kφ^I_cφ^J_dφ^K_e φ^J_fφ^I_g=c1c4^{abcdbef g}Γ^IJΓ^KLMφ^I_cφ^J_dφ^K_eφ^L_fφ^M_g . The right-hand side can be simplied by noting thatΓ^IJΓ^KLM can be written as

Γ^IJΓ^KLM = Γ^{IJ KLM}+ 6Γ^{N OP}δ_{N Q}^[IJ]δ^[KLM]_QOP + 6Γ^Nδ_{[J I]N}^[KLM]. and^{bacdbef g}= 6δ^[acd]_{ef g} then

^{abcdbef g}Γ^IJΓ^KLMφ^I_cφ^J_dφ^K_e φ^L_fφ^M_g

=−6 ¯ψ^a

6Γ^{N OP}δ_{N Q}^[IJ]δ_QOP^[KLM]+ 6Γ^Nδ_{[J I]N}^[KLM]

φ^I_cφ^J_dφ^K_a φ^L_cφ^M_d

=−36ψ^aΓ^M0δ^[KIJ]_{[M L]M}0φ^I_cφ^J_dφ^K_aφ^L_cφ^M_d

=−36ψ^aΓ^M0δ^[KIJ]_{M LM}0φ^I_cφ^J_dφ^K_aφ^L_cφ^M_d

= 36ψ^aΓ^M0δ_{LM M}^{IJ K} 0δ_{ef g}^[acd]φ^K_e φ^J_fφ^I_gφ^L_cφ^M_d

= 36ψ^aδ^[acd]_{ef g} Γ^Kφ^K_eφ^J_fφ^I_gφ^I_cφ^J_d,

where the term Γ^{N OP}δ_{N Q}^[IJ]δ^[KLM]_QOP in the second line cancel since only its symmetric part in I←→LandJ ←→M contribute but this part is clearly zero if we rewrite this term as

Γ^{N OP}δ_{N Q}^[IJ]δ^[KLM]_QOP = 2Γ^ILMδ_J^K+ 2Γ^IKLδ^M_J

+ 2 Γ^{IM K}δ^L_J −Γ^{LJ K}δ^I_M

− 2Γ^{J LM}δ_I^K+ 2Γ^{J M K}δ_I^L .

Now, the left-hand side can be written like

3c₂^{abcdbef g}Γ^Kφ^I_cφ^J_dφ^K_e φ^J_fφ^I_g= 18c₂δ^[acd]_{ef g} Γ^Kφ^I_cφ^J_dφ^K_eφ^J_fφ^I_g. Finally, for the cancellation we need

c2= 2c1c4.

• Parameters of the Lagrangian

Finally the constraints on the parameters of the Lagrangian that come from supersymmetry are

c1=1 4, c2= 1

12,

c3=1 8, c4=1

6.

Note that they agree exactly with those of Bagger and Lambert up to minus signs.

Appendix C

Verication of Superconformal Symmetry

C.1 The U(1) × U(1) Theory

Let us check the supersymmetry of theU(1)×U(1)theory. We only analyze half of the terms, since the other half are just their adjoints. Omitting the factor of k/2π, the variation of the Lagrangian contains (dropping total derivatives)

∆1=−D^µX^ADµδXA=iD²X^Aε¯^IΓ^I_ABΨ^B, and

∆2=iδΨ¯Aγ·DΨ^A=−iΓ^I_ABε¯^Iγ·DX^Bγ·DΨ^A

=iΓ^I_ABε¯^ID²X^BΨ^A−1

2Γ^I_ABε¯^Iγ^ρµ(Fρµ−Fˆρµ)X^BΨ^A.

Note that the gauge elds only appear in the covariant derivatives in the combinationA−Aˆ, which has a vanishing supersymmetry variation. The variation of the Chern-Simons term, using the rst term in equation (4.6), contributes

∆3= 1

2ε^µνλε¯^IγµΨ^AΓ^I_ABX^B(Fνλ−Fˆνλ).

Usingε^µνλγ_µ =γ^νλ, we see that ∆₁+ ∆₂+ ∆₃= 0. The other half of the terms in the variation of the action, which are the adjoints of the ones considered here, cancel in the same way. The conserved supersymmetry current can be computed by the standard Noether procedure. This gives (aside from an arbitrary normalization)

Q^I_µ= Γ^I_ABγ·DX^Aγ_µΨ^B−Γ˜^IABγ·DX_Aγ_µΨ_B.

One can check this result by computing the divergence. This vanishes as a consequence of the equations of motionγ·DΨ^B = 0,D·DX^A= 0, andF_µν−Fˆ_µν= 0.

Let us now explore the conformal supersymmetry, with an innitesimal spinor parameter η^I, using the method explained in [27]. As a rst try, consider replacingε^I byγ·xη^I in the preceding equations, since this has the correct dimensions. Using∂µε(x) =γµηandγ^µγ^ργµ=−γ^ρ, this gives a variation of the action that almost cancels, except for a couple of terms. These remaining terms can be canceled by including an additional variation of the spinor elds. It has the form

δ⁰Ψ^A=−Γ˜^IABη^IXB and δ⁰ΨA= Γ^I_ABη^IX^B. Correspondingly, the conserved superconformal current is

S_µ^I =γ·x Q^I_µ+ Γ^I_ABX^AγµΨ^B−Γ˜^IABXAγµΨB.

As a check, one can compute the divergence using the conservation of Q^I_µ and the spinor eld equation of motion

∂^µS_µ^I =γ^µQ^I_µ+ Γ^I_ABγ·DX^AΨ^B−Γ˜^IABγ·DX_AΨ_B= 0.

The various bosonic OSp(6|4) symmetry transformations are obtained by commuting ε and η transformations. Of these only the conformal transformation, obtained as the commutator of two η transformations, is not a manifest symmetry of the action. It is often true that scale invariance implies conformal symmetry. However, this is not a general theorem, so it is a good idea to check the conformal symmetry (or the conformal supersymmetry) explicitly.