I knew very early of
Stanley Mandelstam
I started physics
as a t
Prisoner of the s u Mandelstam Triangle I escaped from the Mandelstam Triangle
only to be ensnared in Light-Cone Superspace A Note of Personal Gratitude
1971 NAL Visit SuperConformal Theories
P. Ramond
(with S. Ananth, D. Belyaev, L. Brink and S.-S. Kim) Light-Cone Superspaces
N=4 Super Yang-Mills
N=8 SuperConformal
N=8 SuperGravity and E7(7)
N=16 SuperGravity and E8(8) N=8 Light-Cone Superspace houses
D=11: N=1 SuperGravity SO(9); F4/SO(9)
D=4: N=8 SuperGravity SO(2)x E7(7)
D=3: N=16 SuperGravity E8(8)
D=2: N=16 Theory E9(9) N=4 Light-Cone Superspace habitat for
D=10: N=1 Super Yang-Mills SO(8)
D=4: N=4 Super Yang-Mills PSU(2,2|4)
D=3: N=8 Super Conformal OSp(2,2|8) 1 i 1 ϕ (y) = 1 A (y) + i θm θn C (y) + 1θm θn θp θq $ ∂+ A¯ (y) + m n mn m n p mnpqq + ¯ ϕ (y) = +∂A (y) + √2θ θ Cmn (y) +12 θ θ θ θ $mnpq ∂ A (y) ∂ √2 √ 12 1 i 1 i m 2 m n p q m n m n p q + i+ +¯ θ χ¯m(y) + √2θ θ θ $mnpq χ (y) ϕ (y) = + A (y) + θ θ Cmn (y) + θ θ θ θ $mnpq ∂ ∂Am(y) 6 m n p q ∂ √2 12 + θ χ¯m(y) + θ θ θ $mnpq χ (y) 1 i m n 1 m n p ∂q+ + ¯ 6 ϕ (y) = A (y) + √θ θ C L(C2y) + Fθorθ maθ θ $lmnpqism∂ A (y) ∂+i m √ 2 m nmnp q12 + θ χ¯m(y) +2 θ θ θ $mnpq χ (y) 1 0 3 ∂+ 6 x± = (x x ) i √2 √ m m n p q 12 0± 3 + + θ χ¯m(y) + θ θ θ $mnpq χ (y) x± = (x x ) ∂ 6 √2 ± Light-Cone Co1ordina0 3tes: 1 0 3 x± = (x x ) ∂± = ( ∂ ∂ ) √2 ± √2 − ± 1 0 3 1 0 3 x± = (x x ) ∂± = ( ∂ ∂ ) 1 0√ 3 ± 1 1 2√2 − ±1 1 2 ∂± = ( ∂ 2∂ ) x = (x + ix ) x = (x ix ) √2 − ± √2 √2 − m m + 1 1 2 1 1 2 1 q+ , q¯+ n =1 i√2δ n ∂ x = 1(x + ix ) x = 1 (x ix ) x = (x1 +{ix2) x =} (x1 ix2) ∂ =√2 (∂1 + i∂2) ∂ = √(2∂1 i−∂2) √2 √2 − √2 √2 − 1 1 2 1 1 2 ¯ 1 1 2 1 1 2 ∂ = (∂qm+, qi¯∂ ) = ∂i√=2δm∂+(∂ i∂ ) ∂∂ ∂ = (∂ ( + i∂ ) +∂ =α¯ˆδ β(δˆ∂ +mi∂ ) α¯ˆδ βδˆ +m√2 { + + n } √n 2 − δ h = h + √2 κ 1 + 2m)∂ (e √(e2 ∂ − he− (e− ∂ h) 2m+1 m H ∂+ α β m=0 ! m m + m √ m + ¯ 1 q , q¯ = i√2δ ∂ q+ , q¯+ n = i 2δ n ∂ +∂∂ 0 ( +3 ++n α¯ˆδ βδˆ +m n α¯ˆδ βδˆ +m Light-Co{ ne Gaug} e:1 δH hA= =h + (A κ+1{+A2m) =)∂ 0(e} (e ∂ he− (e− ∂ h)α2m+1βm p q ∂+ √ dm dn ϕ = $mnpq d d ϕ 2m=0 2 ! ¯ ¯ 1 ∂∂ ( 1 + α¯ˆδ βδˆ +m α¯ˆδ β∂δˆ∂+m ( d d ϕ += α¯ˆδ$ βδˆ d+pmdq ϕ α¯ˆδ βδˆ +m δH h = h + κ 1 + 2m)∂0 (e 3(e ∂ heδ− h(e=− ∂ h +h)α2m+1κβm1 +m2mn)∂ (e (mnpqe ∂ he− (e− ∂ h) 2m+1 m ∂+ A− = ( A A ) is rHepla∂c+ed through eqs. of2 motions α β m=0 √2 − m=0 ! ! 1 p q dm dn ϕ = $mnpq d d ϕ 1 p q 2 dm dn ϕ = $mnpq d d ϕ 1 2 1 Physical Fields: A¯ = (A1 + iA2) A = (A1 iA2) √2 √2 −
4
4
4
4 The number of the Grassmann variables is the same as that of supersymmetry of the theory. For example, m run form 1 to 4 for the = 4 Super Yang-Mills theory and the N GrassmannThe numvariablesber of theθmGrassmanntransformvariablesas the 4isandthe sameθ¯ asasthethat4¯ ofundersupersymmetrySU(4). Forofthethe = 8 m N m Supergratheoryvit. yForin example,four dimension,m run formthey1 transformto 4 for theas θ=m 4 Sup8 erandYang-Millsθ 8¯ theoryunder andSU(8)the N ∼ ∼ Grassmann variables θm transform as the 4 and θ¯ as the 4¯ under SU(4). For the = 8 (m = 1 , 8). m N · · · m Supergravity in four dimension, they transform as θm 8 and θ 8¯ under SU(8) In Superspace, one can define functions that contain∼ the physical∼ fields together (m = 1 , 8). with the Grassmann· · · variables, which are called superfields. The supersymmetric In Superspace, one can define functions that contain the physical fields together transformations on superfields are manifest because they are represented as rotations with the Grassmann variables, which are called superfields. The supersymmetric and translations in Superspace. InFoorderur Coto fitmpltheexph ysical fields into superfields, one may transformations on superfieldsGrassmannare manifest Vbaecauseriabltheyesare represented as rotations constrainand translationsthe superfieldsin Supwitherspace.the cInhiralorderderivto fitativthees physical fields into superfields, one may constrain the superfields withChtheircahirall dederivrivativatiesves ∂ i ∂ i dm θm∂+ , d¯ + θ¯ ∂+ , (2–8) ¯ √ m ∂θm √ m ≡ − ∂θm∂− 2i ≡ ∂ i2 dm θm∂+ , d¯ + θ¯ ∂+ , (2–8) ≡ − ∂θ¯ − √ m ≡ ∂θm √ m which satisfy canonical anticommm utation2 relations 2 which satisfy canonical anticommutation relations
dm , d¯ = i√2δm ∂+ . (2–9) n − n dm , d¯ = i√2δm ∂+ . (2–9) ! "n − n The form of the constraints depends! on the" number of supersymmetries and the detail is The form of the constraints depends on the number of supersymmetries and the detail is discussed in the following subsections. discussed in the following subsections. 2.1.1 N = 4 Superspace 2.1.1 N = 4 Superspace
The TheN =N4 =Sup4 Superspaceerspaceis theis theSupSuperspaceerspacethatthatdescribdescribeses the maximallymaximallysupsupersymmetricersymmetric N theoriestheorieswithwith16(=16(=2 )2phN )ysicalphysicaldegreesdegreesofoffreedomfreedomininvvariousarious dimeions.dimeions.InInthisthisthesis,thesis,wewe restrictrestrictN =N 4=Sup4 Superspaceerspaceto tothetheoneoneininfourfourdimensions,dimensions, andand ththususititdescribdescribesesthethe maximallymaximallysupersymmetricsupersymmetrictheorytheorywithwith1616phphysicalysicaldegreesdegrees of freedom:freedom: ==4 4SupSuperer NN Yang-Mills.Yang-Mills. The Thephysicalphysicaldegreesdegreesof freedomof freedomofof ==44SupSupererYYang-Millsang-Mills theorytheory, ,thethetwtowogaugegauge NN fields defined as fields defined as
10 10 1 1 Aa = (Aa + i Aa) , A¯a = (Aa i Aa) , (2–10) √2 1 2 √2 1 − 2 the six real scalar fields satisfying
a 1 C = ! Ca pq , with ! = 1 mn 2 mnpq 1234
1 N=4: Hoa m me ofa the and finally four complex spin- 2 fermions χ and χ¯m (m = 1, . . . , 4), are captured in a 1 i m n 1 m n p q + ϕ (y) = A (y) + θ θ C (y) + θ θ θ θ $mnpq ∂ A¯ (y) complex superfield [8] ∂+ √2 mn 12 Constrainedi Chiral Sup√e2rfield + θm χ¯ (y) + θm θn θp $ χq(y) ∂+ m 6 mnpq a 1 a i m n a 1 m n p q + a 1 i a 1 ¯ aϕ (y) = + Aa (y) + θa m θn Cmn (y) + θm θn θp qθ $mnpq+∂ ¯aA (y) φ (y) = +∂ A (y) + √2θ θ Cmn (y) + 12θ θ θ θ !mnpq ∂ A (y) ∂ √2 12 1 i m a √2 ¯m ¯n p q a p q ¯ i+ θ χ¯ (√y)2+ dθm θdnθφ $= χ$mnpq(y) d d φ + θm∂+χ¯a (y) +m mθm θ6n θp ! χamnpqq(y)2, (2–11) ∂+ m d6 ϕ(y , θ ,mnpqθ) = 0
1 am p q √ m + where the light-cone coordinate yd¯ared¯givφen=by $q+ , dq¯+dn φ¯ = i 2δ n ∂ Chiral Constraint: m n dm ϕ2({ymnpq) = 0 }
m √ m + ¯ q+ , q¯+ n = i 2iδ n ∂ ∂∂ {ϕ(x+ , θ , θ(}) ϕ(y+,mθ¯,α¯ˆθδ ) βδˆ +m α¯ˆδ βδˆ +m δ yh== ( x, hx,¯ +x , y−κ 1x+− 2m)∂θ (eθm )(e, ∂ he− (e− ∂(2–12)h) 2m+1 m H ∂+ ≡ →− √2 α β m=0 ! 1 ∂∂¯ where + is interpreted as a ( 1 + α¯ˆδab βδˆ +abcm +α¯ˆδ c βδˆ +mb ∂ δH h = h +δ ϕ κ =1 + 2m)∂ ((∂¯eδ (e ∂gf ahe∂− ϕ(e)−δ ∂ ϕ h)α2ma+1βm ∂+ q¯ + 1 q¯+ p q m− =0 ∂ d −d ϕ = $ d d ϕ Inside-out Cons!traint: ! m n 2 mnpq " 1 1 φa(x−) =¯ ab dξ !(ξabc x+−) φc(ξ) , b ∂+ m = (∂ δ2 gf −∂ ϕ ) q¯[+ Brink,m ϕ Lindgren and Nilsson ’82 ] W ! − with !(z) being 1 for z > 0 and 1 for z < 0. The superfield also satisfies the chiral − δ ϕc = q¯ ϕc constraint q¯+ m + m
¯ ab ¯ ab abc + c ab ∂ δ m a (∂ δ gf a ∂ ϕ ) d →φ = 0 ; −d¯m φ¯ = 0 , ≡ D (2–13) as well as the inside-out constraints ∂ δab ∂ δab + →
1 d¯ d¯ φa = ! dp dq φ¯a , (2–14) m n 2 mnpq a 1 ab abcd ijkl ∂ c + d b δq¯ φ = + ∂ δ gf & E φ E∂ φ η=0 & q¯+ φ − ∂ − ∂η · # $ %ijkl ( & ' ∂ ∂ δab ∂ δab gf abcd11&ijkl E φcE∂+ φd → − ∂η η=0 $ %ijkl & ' d d E = exp (η ) E = exp ( η ) · ∂+ − · ∂+
ϕa d¯1234 ϕb ∼
H = ( a , a ) W W a m 1 = i d4x d4θ d4θ a 4 W ∂+ 3 W m ) 4
E7(7)
1 2.2 SuperPoincar´e Algebra
There are two types of symmetry2.2 Supgenerators:erPoincarkinematical´e Algebranda dynamical ones. While
the kinematicalThere are generatorstwo types doof symmetrynot involvegenerators:the time derivkinematicalatives and areandrealizeddynamicallinearlyones.onWhile fields,the kinematicalthe dynamicalgeneratorsgeneratorsdo donotininvolvvolve timee thederivtimeativderives and,ativesinandgeneral,are realizedare realizedlinearly on non-linefields, thearlydynamicalon fields. Igeneratorsn order for thedo indynamicalvolve timegeneratorsderivativtoes band,e realizedin general,linearlyare, onerealized needsnon-lineto inarlytroduceon fields.auxiliaryIn orderfields.forUnfortunatelythe dynamical, findinggeneratorssuch auxiliaryto be realizedfields islinearlynot , one
anneedseasytotaskintroandducethereauxiliaryis no guidingfields.principleUnfortunatelyfor their, findingexistencesuc[4h].auxiliaryIn LC2 formalism,fields is not 1 i 1 therefore all the dynamical generators are non-linearly realized because therem aren no m n p q + ¯ an easy task and there is no guiding principleϕ (y) =for theirAexistence(y) + [4]. θIn LCθ Cformalism,mn (y) + θ θ θ θ $mnpq ∂ A (y) ∂+ √2 2 12 auxiliary fields. therefore all the dynamical generators are non-linearly realizedi because there√2are no + θm χ¯ (y) + θm θn θp $ χq(y) 2.2.1 Kinematical Super Poincar´e Generators ∂+ m 6 mnpq auxiliary fields. In this section, we review how the kinematical generators represented on light-cone, 2.2.11 Kinematicali Super Poincar1 ´e Generators 1 ϕ (y) = A (y) + θm θn C (y) + θm θn θp θq $ ∂+ A¯ (y) 0 3 whic+ h act linearly on the chiralmn superfield mnpq x± = (x x ) ∂ In this section,√2 we review how the12 kinematical generators represented on√ligh2 t-cone,± √ i m 2 m n p q i + θ χ¯m(y) + θ θ θ $mnpq χ (y) p p which act+ linearly on the chiral superfield λ = ( d d¯p d¯p d ) , (2–23) ∂ 6 + 1 0 3 4√2i∂ p − ∂± p= ( ∂ ∂ ) δϕ = iωKλϕ=, ( d d¯p d¯p d ) , (2–23) 4√2 ∂+ − √2 − ± measures the1 helicit0 y3of the superfield. For example, λ = + 1 for a chiral superfield and where K is a kinematicalmeasuresx± = Supthe(erxhelicitPoincarxy)of´eδgeneratortheϕ =supiωerfield.Kandϕ ,ωFisortheexample,transformation1 λ = + 1parameter.for a chiral 1superfield and 1 for an√an2ti-chiral± superfield. The actionx of=this (transvx1 + erseix2)rotationx =generator(x1 onixthe2) All the transformations− 1 forensurean anti-cthehiralchiralitsupyerfield. The action of this√2 transverse rotation generator√2 on− the where K is a kinematical− 1 Super Poincar´e generator and ω is the transformation parameter. chiral∂ sup= erfield( ∂is0 then∂3)expressed as δ φ = i ω j’φ. It is worth noting that under this chiral± superfield is then expressed as δ φ = i ω j φ. It is worth noting that under this Sup√2e− r±-Poincare1 1Gr2 oup1 1 2 All the transformationsSO(2) rotation,ensurethethetranslationchirality generator p∂and= p¯ transform(∂ + i∂ di)fferen∂tly=as (∂ i∂ ) dm δϕ = 0 . √ √ 1 SO(2) rotation, the 1translation generator p and p¯2transform differently as 2 − x = (x1 + ix2) x = (x1 ix2) There are three√2 kinematical momenta√2 − dm δϕ = 0 . m √ m + Kinematical Gen1era[ jt, op ]r=si p , [ j , p¯]q= , q¯np¯ ,1 = i 2 δ n ∂ 1 1 [ j , p ] = p ,m n[ j , p¯{] = p¯}, m n p q + ¯ 1 ϕ2(y) = + A1 (y) 2+ θ θ Cmn (y) −+ θ θ θ θ $mnpq ∂ A (y) ∂ = (∂ + i∂ ) 1 ∂ = ∂ (∂i i∂ ) √2 1 − 12 There are three√2 kinematical+ momen+ √ta2 − m n ¯ m n p q + ¯ ϕand(y)thpus=one= +caniA∂ (alsoy,) +assignp = θhelicitθi ∂ Cy, ,mnin a(p¯ysense)=+ √thati ∂θ, theθ helicitθ θ $ymnpqindicate(2–21)∂ Aho(yw) an operator Translatioandnsth, usSO(2)one∂− can also roassign√ta2ti−helicitionsm y, in a sense−12that2 mthenhelicitp y indicateq how an operator + θ ¯χ¯m(y) + θ θ θ $mnpq χ (y) m m + + ∂√∂ ( + α¯ˆδ βδˆ +m α¯ˆδ βδˆ +m transform under √SOi (2),mon these∂ generators2 m n (1p for6 p andq 1 for p¯). 2m+1 m The kinematical qtransformLoren, q¯tzn generatorsunder=+i 2SδOincludesnθ(2),∂ χ¯onδHthe(hthesey)=transv+ generators+ herseθ+ θSOθ(2)(1κ$forrotation1 +p and2χm()y∂)1 for(e p¯).(e ∂ he− (e− ∂ h)α β { +} ++ m ∂ mnpq − p = i∂∂ , p = i6∂ , m=0p¯ = i ∂¯ , − (2–21) TheTheremainingremaining− kinematicalkinematicalgeneratorsgenerators− !areare − 1 0 3 1 ∂ x∂± = (x x ) 1 ¯The kinematical Lorenj tz= generatorsx ∂¯ x¯ ∂ +includes( θm the transv1θ¯ 0erse) S3Oλ(2), rotation (2–22) p q ∂∂ ( + α¯ˆδ βδˆ +m α¯ˆδx±mβ=δˆ +mm(x x )√2d d ±ϕ = $ d d ϕ δ h = h + κ 1 + 2m)∂ (e (e− ∂ he2 − (∂eθ− −∂ h∂)θ¯ 2m+1− m m n mnpq H ∂+ √2 αm± β 2 m=0 i where! + + + + +¯+ ++ ++ 1 + i pp¯¯ ¯¯ pp j j ==i xi∂x ∂ , , ¯j j 1==imix¯x¯∂∂∂ ,, jj −−∂ == i x− ∂0 3((θθ ∂∂p ++θθp ∂∂ ))++i i. . (2–24)(2–24) j = x ∂¯ x¯ ∂ + ( θ 1 ∂θ¯±0 = 3) ( λ ∂, −kin∂2 ) p (2–22)p 1 ∂± = m ( ∂m ¯∂ ) − 2 − p q 2 ∂θ − ∂θm√2−− ±δ ϕ = K ϕ dm dn ϕ = $mnpq d d ϕ √2 − ± 2 ++ where WeWme ustmustnotenotethatthatthethej j−−generatorgeneratoris,is,inin fact,fact, aa dynamical generatorgeneratorsincesinceititconcontainstains 1 1 1 21 1 1 2 x = (xx1 +=ix2) (x x+=ix ) (x1m xix=2) ∂(x iix m) + + Kinemattheitheclighalighδlkint-cone Sut-coneϕ =sytimetimeK ϕderivderivativative;e;hohowwevever,er,ititbbecomesecomesqkinematical= ififoneone+ wworksorksθ at∂atxx+==0 0 √2 14 √2 √2 − −√2∂θ¯ −√2 + plane,plane,andandththususherehereititisisassumedassumedthatthatwweewworkork atat x+ = 0.0. 1 1 12 1 1 2 1 m ∂ ∂i = m +(∂ + i∂ ) 1 ∂ = 2 (∂ i∂m ) ∂ 1 1i ¯ 2 + q =Finally, +the kinematicalθ √∂2 ∂ =supersymmetry(∂ + i∂generators√)2 q¯−m∂ q=+=m and (q¯+∂m 1 iθ∂m ∂) Finally− ∂, θ¯the kinematical√2 sup14ersymmetry√2 generators q+ and√∂2θ q¯−+ m√−2 ∂ i m √ m + ¯m + q ∂, q¯n i=mmi +2 δ n ∂ ∂ m +i + q¯m = θqm ∂= { +} θ ∂ , q¯ =√ θ¯ ∂ , (2–25) m+ ∂¯ i qm ,+q¯n + m= i 2∂mδ n ∂ i m + ∂θ − √2q = − ∂θm+ √2θ ∂ , q¯ = ∂θ − √2 θ¯ ∂ , (2–25) + ¯ { }+ m m m − ∂θm √2 ∂θ − √2 which satisfy anticommutation relationdm , q¯ = qm , d¯ = 0 (respect ch¯ irality) n n which satisfy∂∂ anticomm(utation relation+{ α¯ˆδ βδˆ }+m {α¯ˆδ βδˆ }+m temperature? δ h = h + κ 1 + 2m)∂ (e (e ∂ he− (e− ∂ h) 2m+1 m H ∂+ α β m=0 ! m m + m + q+ , q¯+ n = i√2 δ n ∂ = √2 δ n p , (2–26) ∂∂¯ {m } m + −ˆ mm + ˆ m q+ , q¯+ n ( = 1i√2 δ n+∂p αq¯ˆδ= βδ √+2 δ n p α¯ˆδ, βδ + (2–26) δH h = h{ +d d ϕκ}=1 + 2$m)∂ d(ed ϕ(e − ∂ he− (e− ∂ h)α2m+1βm and anticommute∂with+ them cnhiral deriv2 ativmnpqes m=0 and anticommute with the !chiral derivatives δkin ϕ = K ϕ 1 d d ϕ = $ dp dq ϕ 1 m n m mnpq 4 m We may neglect “+” if it is obvious, and use q2 and q¯m instead of q+ and q¯+ m 1 m m We may 4neglect “+” if it is obvious, and use q and q¯m instead of q+ and q¯+ m kin δ 15 ϕ = ϕ O O 15 kin kin kin [ δ , δ ] ϕ = δ[ , ] ϕ O O! O O!
∂ i qm = + θm ∂+ − ∂θ¯ √2
∂ i q¯ = θ¯ ∂+ m ∂θ − √2 m
4
4 1 i m n 1 m n p q + ϕ (y) = A (y) + θ θ C (y) + θ θ θ θ $mnpq ∂ A¯ (y) ∂+ √2 mn 12 1 i m i m √n 2 m n p 1 m qn p q + ϕ (y) = + A (yθ) +χ¯ (yθ) +θ Cθ (θy)θ+$ θ χθ (yθ) θ $mnpq ∂ A¯ (y) ∂+ ∂+ m√2 6 mn mnpq12 i √2 + θm χ¯ (y) + θm θn θp $ χq(y) ∂+ m 6 mnpq 1 0 3 x± = (x x ) √2 ± 1 0 3 x± = (x x ) 1√2 0 ± 3 ∂± = ( ∂ ∂ ) √2 − ± 1 0 3 ∂± = ( ∂ ∂ ) 1 √2 − ± 1 x = (x1 + ix2) x = (x1 ix2) √ √ − 21 12 x = (x1 + ix2) x = (x1 ix2) 1√2 √1 2 − ∂ = (∂1 + i∂2) ∂ = (∂1 i∂2) √21 √12 − ∂ = (∂1 + i∂2) ∂ = (∂1 i∂2) √2 √2 − qm , q¯ = i√2 δm ∂+ { n } n qm , q¯ = i√2 δm ∂+ { n } n ¯ ∂∂ ( m+ α¯ˆδ βδˆ +m α¯ˆδ βδˆ +m d ϕ(y , θ , θ) = 0 m 2m+1 m δH h = + h¯+ κ 1 + 2m)∂ (e (e ∂ hed− ϕ((ye,−θ , θ∂) = h0)α β ∂ ∂∂ ( + α¯ˆδ βδˆ +m α¯ˆδ βδˆ +m δ h = h m+=0 κ 1 + 2m)∂ (e (e ∂ he− (e− ∂ h) 2m+1 m H ∂+ ! α β dm ϕ(my=0, θ , θ) = 0 ! dm ϕ (y) = 0 m Kinematic1al Genperqadtoϕrs(y ) = 0 dm dn ϕ = 1$mnpq d d ϕ acdt ldineaϕ =rl2y $on thedp d qfieldsϕ dm ϕ (y) = m0 n 2 mnpq ϕ(x , θ , θ) ϕ(y ,ϕθ(x, θ, )θ , θ) ϕ(y , θ , θ) → → δkin ϕ = ϕ kin ϕ(x , θ , θ) ϕ(y , θ ,Oθδ) ϕ =O ϕ 1 1 δ ϕa = (∂¯δab gf abc∂+ ϕc ) δ ϕb →a O¯ abq¯ Oabc ++ c b q¯+ δ ϕ = (∂ δ − gf ∂ ϕ ) δ ϕ q¯ + ∂ q¯+ − − ∂ − kin kin kin ! " a 1 ab [ δ abc ,+δ! c ] ϕ∂ =b δi ϕ " δ ϕ = (∂¯δ gf m∂ ϕ ! ) δ ϕ a[ , m!] + ab abc + c b q¯ + qO =O q¯+ + Oθ=O (∂∂¯δ gf ∂ ϕ ) q¯ ϕ − ∂ −a = (∂¯δab− ∂gθ¯f abcW√∂m+2 ϕc ) q¯ −ϕb + m ! W m − " + m m ∂ i m + a = (∂¯δab gf abcq ∂+ =ϕc ) q¯ ∂ϕb+ i θ ∂ kin m m example:+ mkinematic¯al Su+δssyϕ = % q¯m ϕ W − q¯m kin=− ∂θ¯ m√2θm ∂ δs ϕ∂θ=−%√q¯2m ϕ δkin ϕ = %mq¯ ϕ ∂ i kin m s q¯m = θ¯ ∂+δs¯ ϕ = %¯mq ϕ m kin mm δs¯ ∂ϕθ =− √%¯m2q ϕ kin m ¯ ab ¯ ab abc + c ab δs¯ ϕ = %¯mq ϕ ∂ δ (∂ δ gf ∂ ϕ ) i → i − ≡ D [ δkin , δkin ] ϕ = %m%¯ ∂+ϕ = δkin ϕ s s¯ √2 m √2 t− ∂¯δab (∂¯δab gf abc∂+ ϕc ) ab ∂ δab ∂ δab + → − ≡ D → ∂¯δab (∂¯δab gf abc∂+ ϕc ) ab → − ≡ D ∂ δab ∂ δab + → a 1 ab abcd ijkl ∂ c + d b δq¯ φ = + ∂ δ gf % E φ E∂ φ η=0 % q¯+ φ − ab ∂ ab− ∂η · ∂ δ # ∂ δ + $ %ijkl ( → & ' 44 a 1 ab abcd ijkl ∂ abc + d ab abcdb ijkl ∂ c + d δq¯ φ = + ∂ δ gf % E∂ δφ E∂ φ ∂η=0δ %gfq¯+ φ % E φ E∂ φ − ∂ − ∂η → − · ∂η η=0 # $ %ijkl ( $ %ijkl a 1 ab abcd& ijkl ∂ ' c + d & b ' δq¯ φ = + ∂ δ gf % E φ E∂ φ η=0 % q¯+ φ − ∂ − ∂η d · d ab ab abcd# ijkl ∂ c E += dexpijkl(η ) E = exp( ( η ) ∂ δ ∂ δ gf % E φ $E∂ φ% · ∂+ − · ∂+ → − ∂η η=0& ' $ %ijkl ab ab abcd& ijkl ∂ ' c + d a 1234 b ∂ δ d ∂ δ gf % d ϕE φ Ed¯∂ φϕ η=0 E = exp (η →) E =−exp ( η ) ∂η + +$ %ijkl ∼ · ∂ − · ∂ & ' d d aE = 12exp34 (bη + ) E = exp ( η + ) ϕ d¯ ϕ · ∂ H = ( a−, · a∂) ∼ W W a m 1 = i d4x d4θ d4θ a ϕa d¯1234 ϕb W ∂+ 3 W m ∼ ) H = ( a , a ) W W 4 4 4 a m 1 a = i d x d θ d θ m H =W ( ∂a+, 3 Wa ) 1 ) W W a m 1 = i d4x d4θ d4θ a W ∂+ 3 W m ) 1 1 1 i m n 1 m n p q + ϕ (y) = A (y) + θ θ C (y) + θ θ θ θ $mnpq ∂ A¯ (y) ∂+ √2 mn 12 i √2 + θm χ¯ (y) + θm θn θp $ χq(y) ∂+ m 6 mnpq
1 0 3 x± = (x x ) √2 ±
1 0 3 ∂± = ( ∂ ∂ ) √2 − ± 1 1 x = (x1 + ix2) x = (x1 ix2) √q2m , d¯ = dm , q¯ √2 = −0 . (2–27) { + n } { + n } 1 1 2 1 1 2 ∂ = m(∂ + i∂ ) m∂ = (∂ i∂ ) 2.2.2 Dynamical Poincar√q ´e2 ,Generd¯n =atorsd , q¯+√n2 =−0 . (2–27) { + } { } m m + As we discussed earlier, all theq ,dynamicalq¯n = i√generators2 δ n ∂ are realized non-linearly on the 2.2.2 Dynamical Poincar´e {Generators} chiral superfield. Therefore, theirdm , actionq¯ =on theqm ,supd¯ erfield= 0is nontrivial. However, the form { n } { n } As we discussed earlier, all the dynamical generators are realized non-linearly on the of the dynamical generators without interactions are as simple as the kinematical ones. We ¯ chiral superfield. Therefore,∂∂ their( action+ onα¯ˆδ theβδˆ sup+merfieldα¯ˆδ is βnonδˆ +trivial.m However, the form here list thoseδH h =withouth + interactions.κ 1 + 2m)The∂ (eligh(et-cone∂ Hamiltonianhe− (e− ∂ is givh)αen2m+1byβm ∂+ m=0 of the dynamical generators! without interactions are as simple as the kinematical ones. We 1 ¯p q Dynamhere list thoseicawithoutl Genineteractions.ratdomrdsn Theϕ (l=inlighea$mnpqt-coner o∂dnl∂ Hamiltoniandy ϕfor freeis givtheoen bryies) p− =2 i . (2–28) − ∂+ The dynamical Lorentz generators areδkintheϕ b=oosts,ϕ O O ∂∂¯ dyn dyn,free −dyn,int Hamiltonianδ ϕ = δ ϕ +pδ = ϕ i= + .ϕ + non linear (2–28) O O O − ∂ O − kin kin kin [ δ , δ ] ϕ = δ[ , ] ϕ The dynamical Lorentz generatorsO¯are Othe! boosts,O O! ∂∂ p ∂ j− = i x i x− ∂ + i θ ∂¯ λ 1 , Boosts + p + ∂kin −dyn dyn − − ∂ [ δ ¯ , δ ] ϕ = δ[ !, ] ϕ " ¯ ∂O∂ O! O O! p ∂ ¯j− = i x¯ i x− ∂¯ + i θ¯ ∂ + λ 1 . (2–29) ∂¯+ − p − ∂+ ∂∂ p ∂ − m −∂ i !m +¯ " j = i x q + = i x ∂+ + iθ θ∂∂p λ 1 + , ∂ − − ∂θ¯ √2 − − ∂ They also preserve chirality because¯ ! " ¯ ∂∂ p ∂ Dynamical Su¯j− sy= i x¯ (squarei x r−o∂¯ot+ ofi theθ¯ ∂ Ham+ λ ilt1onian). (2–29) + ∂ i ¯ +p + ∂q¯m −= θm ∂ − ∂ ∂θ − √2 ! " m i m ∂ i ∂ [ j− , d ] = d , [ j− , d¯ ] = d¯ , (2–30) They also preserve chirality because2 ∂+ m 2 m ∂+ ¯ m ¯ m ∂ m ∂ Noticeq thati [thesej− , qare] the= “square-roq , ots”q¯ ofm the lighi [ j−t-cone, q¯m ]Hamiltonian,= q¯m . in (1)the sense that and satisfy− the≡ Poincar´e algebra.∂+ In particular− ≡ ∂+
m i m ∂ i ∂ [ j− , d ] = d , [ j− , d¯ ] = d¯ , (2–30) 2 ∂+ ∂∂¯ m 2 m ∂+ m √ m √ m ¯+ q , q¯ n += 4i 2 δ n =+ 2δ n p− , (2–33) [ j− , j ] −= − i j − j , [∂j+− , j − ] = i j− . (2–31) and satisfy the Poincar´e algebra.{ In− particular} − − The dynamicalimplying thatsupersymmetrythe light-conegeneratorsHamiltonianare isthena derivobtainedable quanby btitoostingy fromthethekinematicaldynamical supersymmetries. supersymmetries + + + [ j− , ¯j ] = i j − j , [ j− , j − ] = i j− . (2–31) − − The dynamical supersymmetry generators are then obtained by boosting the kinematical ¯ m ¯ m ∂ m ∂ q i [ j− , q + ] = q + , q¯ m i [ j− , q¯+ m ] = q¯+ m . (2–32) supersymmetries− ≡ ∂+ − ≡ ∂+
¯ m ¯ m ∂ m 16 ∂ q i [ j− , q + ] = q + , q¯ m i [ j− , q¯+ m ] = q¯+ m . (2–32) − ≡ ∂+ − ≡ ∂+
16
17 1 i m n 1 m n p q + ϕ (y) = A (y) + θ θ C (y) + θ θ θ θ $mnpq ∂ A¯ (y) ∂+ √2 mn 12 i √2 + θm χ¯ (y) + θm θn θp $ χq(y) ∂+ m 6 mnpq
1 0 3 1 i m xn± = (x 1 mx )n p q + ϕ (y) = A (y) + θ θ C (y) + θ θ θ θ $mnpq ∂ A¯ (y) ∂+ √2 mn √2 12± i √2 + θm χ¯ (y) + θm θn θp $ χq(y) ∂+ m 6 1 mnpq0 3 ∂± = ( ∂ ∂ ) √2 − ± 1 0 3 x± = (x x ) 1 √2 ± 1 x = (x1 + ix2) x = (x1 ix2) √2 √2 − 1 0 3 ∂± = ( ∂ ∂ ) √ 1 2 − ± 1 ∂ = (∂1 + i∂2) ∂ = (∂1 i∂2) 1 1 x = √(x21 + ix2) x = (x1 √2ix2) − √2 √2 − 1 qm , q¯ = 1i√2 δm ∂+ ∂ = (∂1 + i∂2) n ∂ = (∂1 ni∂2) √2 { } √2 − dm , q¯ = qm , d¯ = 0 { n } { n } qm , q¯ = i√2 δm ∂+ { n } n dm , q¯ = qm , d¯ = 0 ¯ { n } { n } ∂∂ ( + α¯ˆδ βδˆ +m α¯ˆδ βδˆ +m δ h = h + κ 1 + 2m)∂ (e (e ∂ he− (e− ∂ h) 2m+1 m H ∂+ α β ¯ m=0 ∂∂ !( + α¯ˆδ βδˆ +m α¯ˆδ βδˆ +m δ h = h + κ 1 + 2m)∂ (e (e ∂ he− (e− ∂ h) 2m+1 m H ∂+ α β mDynam=0 ical Genera1tors act non-linearly ! p q dm d(lninϕear= for fr$mnpqee theod drieϕs) 1 2 d d ϕ = $ dp dq ϕ m n 2 mnpq δkin ϕ = ϕ δkin ϕO= ϕ O dyn dyn,freeO dynO,int δ ϕ dyn= δ dyn,freeϕ + δ dyn,intϕ = ϕ + non linear O δ ϕ O= δ ϕ +Oδ ϕ = Oϕ+ − O O O O kin kin kin kin[ δ kin, δ ] ϕ kin= δ ϕ [ δ , δ ] ϕ ! = δ[ , ] ϕ[ , !] O O O! O O O! O O
kin kindyn dyn dyn dyn [ δ , δ ] ϕ = δ[ , ] ϕ O[ δ O,! δ ] ϕ O=O! δ[ , ] ϕ O O! O O!
m ∂ i m + q = ¯ + ∂ θ ∂i qm −= ∂θ √2+ θm ∂+ − ∂θ¯ √2 ∂ i ¯ + q¯m = θm ∂ ∂θ − ∂√2 i q¯ = θ¯ ∂+ m ∂θ − √2 m
4
4 N=4 Super Yang-Mills
• Maximally Supersymmetric Theory with max spin 1 • 16 massless states: 8 Bosonic + 8 Fermionic
• helicity: 1 1/2 0 -1/2 -1 # states: 1 4 6 4 1
SuperConformal Symmetry PSU(2,2|4)
(Conformal group SO(4,2) X SU(4) R-symmetry) 1 i m n 1 m n p q + ¯ ϕ (y) = A (y) + θ θ Cmn (y) + θ θ θ θ $mnpq ∂ A (y) ∂+ √2 ¯ 12 m ¯ m ∂ m ∂ q i mi [ j− , q + ] √=2 m nq +p, q¯ qm i [ j− , q¯+ m ] = q¯+ m . (3.14) + − ≡θ χ¯ (y) + θ∂+θ θ $ χ− (y)≡ ∂+ ∂+ m 6 mnpq These are the “square-roots” of the light-cone Hamiltonian, in the sense that 1 x = (x0 x3) ± ¯ √2 ±m m ∂∂ q , q¯ n = i √2 δ n . (3.15) { − − } ∂+ 1 0 3 ∂± = ( ∂ ∂ ) All operators have √been2 −construc± ted so as to be Hermitian with respect to the ¯ quadratic form m ¯ m ∂ m ∂ q i [ j− , q + ] = q + , q¯ m 1 i [1j− , q¯+2 m ] = q¯1+ m . 1(3.14)2 − ≡ ∂+ x −= ≡ (x + ix ) x∂=+ (x ix ) √2 √2 − 1 ( φ , ξ ) 2i d4x d4θ d4 θ¯ φ¯ ξ , (3.16) These are the “square-roots” of the light-cone Hamiltonian, in the≡sense that ∂+ 1 1 2 1 !1 2 where∂ = φ and(∂ ξ+arei¯∂ c)hiral ∂sup=erfields.(∂ Ini∂the) fully interacting theory, the kine- m √2 m ∂∂ √2 − q , q¯ n = i √2 δ n . (3.15) { − −matical} superP∂oincar+ ´e generators do not change, still acting linearly on the superfields, butm the dynamicalmgenerators+ act non-linearly, as we shall discuss All operators have been constructed so as to bqe Hermitian, q¯n = withi√2respδ n ect∂ to the in section 4.{ } quadratic form dm , q¯ = qm , d¯ = 0 { n } { n } 1 ( φ , ξ ) 3.22i d4xSupd4θ d4erθ¯ φconfor¯ ξ , mal Algebra(3.16) ¯≡ ∂+ ∂∂ ! ( + α¯ˆδ βδˆ +m α¯ˆδ βδˆ +m where φ and ξ are δcHhiralh =superfields.h +In thisInκsection,the1 +fully2mw)in∂eteractingcomplete(e (e theorythe∂ construction,hethe− kine-(e− ∂of theh)Pα2SmU+1(2βm, 2 4) generators. Our ∂+ | promcedure=0 is to start from the simplest kinematical generator in the conformal matical superPoincar´e generators do!not change, still acting linearly on the superfields, but the dynamical generatorsalgebra, andact non-linearlygenerate the, asrestwe shallby commdiscussutation. 1 p q in section 4. d d ϕ = $ d d ϕ The easiestm n starting2 pmnpqoint is the “plus” component of the conformal trans- formations, 3.2 Superconformal Algebra δkin ϕ = ϕ O O In this section, we complete thedynconstructiondynof,freethe P SUdyn(2,,2int4) generators.+ Our + δ ϕ = δ ϕ + δ | ϕ K= =ϕ 2+i xnonx¯ ∂ linear. (3.17) procedure is to start from theOsimplest kinematicalO generatorO in theOconformal − kin kin kin algebra, and generate the rest by commutation.[ δ , δ ] ϕ = δ ϕ + + For ease of algebra,! we contin[ ue, !to] work at x = 0, where K is kinematical. SuperConfoO rmaO l D=4O O Kinematics The easiest starting point is the “plus” component of the+ conformal trans- ¯ Since we already know j −, the commutation relation m formations,m ∂ m ∂ ¯ kin dyn dyn q i [ j− , q ] = + q , q¯ m i [ j− , q¯m ] = + q¯m . − ≡ ∂ − ≡ [ δ , δ ] ϕ =∂ δ[ , ] ϕ O O! O O! + + + + K = 2i x x¯ ∂ . [ K ,∂¯p− ] = (3.17)2i D + 2i j − , ∂ (3.18) m m ¯ m∂ i mm +− m m +q qi [ j=− ,mq ] =+ θq ∂ , q¯ m i [ j− , q¯m ] = q¯m . q , s¯ n = iδn (D + j − + ij)++ 2J n ¯ + + + For ease{of algebra,− }we −continue toyieldswork−theat≡ xdilatation= 0, −where∂generator,θ K √∂2is kinematical. − ≡ ∂ Since we already know j+ , the commutation relation − ∂ i q¯ = θ¯ ∂+ SupsmerCo= i [ nfj ,osrmma] l Sum sym m m + m − + ¯ q∂θ, −s¯ √n+2 = ¯iδn (D + j1 −∂+ ij)1+¯2∂J n −m+ ¯ m D+ ∂={im x−−∂ } x−∂ x¯∂ θ ∂ θ , (3.19) [ Kq , p− ]i=[ j−2,iqD +] 2=i j − , q , q¯−m −i(3.18)[ j−−, q¯2m ]∂θ=− 2 q¯∂mθ¯. − ≡ − ∂+ − ≡ ∂+ ∂¯ " ∂ # yields the dilatation generator,m+ ¯ m m [ Kq , q¯ in[]j−=whic,√q 2hs¯]+satisfies=n , + q , q¯ m m i [ j− , q¯m ]m= + q¯m . − ≡− ∂ − s≡ = i [ j− , s+ ] ∂ m m +− m q , s¯ n = iδn (D + j − + ij) + 2J n { − } − + ¯ m 1 ∂ 1 ¯m∂ + m 1 D = i x−∂ x∂ qx¯∂, s¯ nθ = iδθn (D +, j − + ij)(3.19)+m2J n m − −{ − −2 }∂θ −− 2[x ,∂Dθ¯ ] = i x ; [d , D] = i d . (3.20) " m # m − 2 s = i [ j− , s+ ] which satisfies + − ¯ By boosting K , we obtain4 the kinematicalSU(4)generators R-symK andmetrKy, + m [ K1 ,mq¯ n ] = √2 s¯+ n , [x , D] = i x ; [d , D] = i d .− (3.20) 2 7 m p m− p p m m p p m [ J n , J q ] = i δ q J n i δ q J n i δ n J q + i δ n J q By boosting K+, we obtain the kinematical generators K− and K¯ , −
7
5
5 5 ¯ m ¯ m ∂ m ∂ q i [ j− , q ] = q , q¯ m i [ j− , q¯m ] = q¯m . − ≡ ∂+ − ≡ ∂+
m m + m q , s¯ n = iδn (D + j − + ij) + 2J n { − } − ¯ m m ∂ m ∂ q i [ ¯j− , q ] = qm , q¯ m i [ j− , q¯ ] = q¯ . +s = i [ j− , ms+ ] m + m − ≡ ∂ − − ≡ ∂
m + m √+ m q , s¯ n[ K= , iq¯δnn(D] =+ j 2−s¯++nij,) + 2J n { − } − − m p m p p m m p p m [ J n , J q ] = i δ q J n i δ q J n i δ n J q + i δ n J q m − m− s = i [ j− , s+ ] − dyn a 1 ¯ ab abc + c m c δq¯ ϕ = + (∂ δ gf ∂ ϕ ) $ q¯m ϕ − ∂ + − [ K , q¯ n ] = √2 s¯+ n , ! − " m p dynm ap 1p abm m mc p p m [ J n , J q ] = iδδ qϕJ n= i δ q J n$ q¯i δϕ n J q + i δ n J q SuperConfs¯ orma− l∂ +D=4D − mDynamics 1 dyn a kin dyn¯ aba abc + dync am c Dynamδq¯ ϕica[=δl Su,+syδ (g∂]enδϕ e=ragtfes2 ∂ALLi δ¯ϕ )ϕd$ynamq¯m ϕics − K ∂ p− − − j− ! " 1 dynδdyn ϕdyna = a ab $mq¯ dynϕc a [ δs q¯ , δs¯ ] ϕ += √2 mδ ϕ − ∂ D − p−
kin dyn a dyn a [ δ , δ ] ϕ = 2 i δ¯ ϕ K p− − j−
dyn dyn a dyn a δs , δs¯ ϕ = √2 δ ϕ Com{ mutation} relatio−ns dep−termine unique dynamical Susy
5
5 dm ϕ = 0
∂¯ ∂ qm i [ ¯j , q m ] = q m , q¯ i [ j , q¯ ] = q¯ . − ϕ(x+, θ , θ) ϕ(ym, θ , θ) − m + m − ≡ ∂ → − ≡ ∂
1 δ mϕa = (∂¯mδab gf abc+ ∂+ ϕc ) δ ϕmb q¯q , s¯ n += iδn (D + j − + ij) +q¯+2J n {− − }∂ − − ! " a = (∂¯mδab gf abc∂+mϕc ) q¯ ϕb m s = i [ j− , s+ ] + m W SOLE −Effe−ct of Interactions
inδ+ N=4ϕ cSu=peqrY¯ anϕgc-Mills: [ K q¯+,mq¯ n ] = +√m2 s¯+ n , − m p m p p m m p p m [ J n , J q ] = i δ q J n i δ q J n i δ n J q + i δ n J q ∂¯δab (∂¯δ−ab gf abc∂+−ϕc ) ab → − ≡ D 1 δdyn ϕa = (∂¯δab gf abc∂+ ϕc ) $mq¯ ϕc q¯ Transv+erabse Covarianabt Derivative m − ∂∂ δ −∂ δ + ! → " 1 δdyn ϕa = ab $mq¯ ϕc s¯ ∂+ D m a 1 ab abcd ijkl ∂ c + d b δq¯ φ = + ∂ δ gf & E φ E∂ φ η=0 & q¯+ φ − ∂ # −kin dyn a ∂η ijkl dyn a ( · [ δ Incr, δ edibl] ϕ $y= Simpl% 2 i δe!¯ ϕ K p− − & j− ' ∂ ∂ δab ∂ δab gf abcd &ijkl E φcE∂+ φd dyn dyn η=0 → dyn− a ∂η ijkl a δs , δs¯ ϕ =$ %√2 δ ϕ { } − p&− ' d d E = exp (η ) E = exp ( η ) · ∂+ − · ∂+
ϕa d¯1234 ϕb ∼
H = ( a , a ) W W a m 1 = i d4x d4θ d4θ a W ∂+ 3 W m )
E7(7)
E8(8)
1
5 Apply Same Algebraic Techniques to find the Light-Cone Superspace Formulation of the
Super-Conformal Theory in D=3
Bagger, Lambert, Gustavsson Use the N=4 Superfield
a a a a 1 i m n 1 m n p q + ϕ (y) = A (y) + θ θ C (y) + θ θ θ θ $mnpq ∂ A¯ (y) ∂+ √2 mn 12 i a √2 a + θm χ¯ (y) + θm θn θp $ χq(y) ∂+ m 6 mnpq
1 d¯ d¯ φsinc= e$ dp dq φ¯ m n 2 mnpq
OSp(2,m 2|8) m + q+ , q¯+ n = i√2δ n ∂ closes lin{early} (free theory) ¯ ∂ ∂ ( on +!α¯ˆδ(y)βδˆ +m α¯ˆδ βδˆ +m δ h = h + κ 1 + 2m)∂ (e (e ∂ he− (e− ∂ h) 2m+1 m H ∂+ α β m=0 ! (A is some label)
4 SO(4,2) Sp(2,2) ~ SO(3,2)
SU(4) SO(8)
PSU(2,2|4) OSp(2,2|8) ¯ m ¯ m ∂ m ∂ q i [ j− , q ] = q , q¯ m i [ j− , q¯m ] = q¯m . − ≡ ∂¯∂+ − ≡ ∂ ∂+ qm i [ ¯j , q m ] = q m , q¯ i [ j , q¯ ] = q¯ . − +¯ m − m + m −m≡ ¯ m ∂ ∂ m − ≡ ∂ ∂ q i [ j− , q ] = q , q¯ m i [ j− , q¯m ] = q¯m . − m + m − + m + ≡ q , s¯ n∂ = iδn (D + j −≡+ ij) + 2J n ∂ { − } − m m + m q , s¯ n = iδn (D + j − + ij) + 2J n { m − } − m + m q , s¯ n = miδn (D + j −m+ ij) + 2J n { − } s− = i [ j− , s+ ] − m m s = i [ j− , s+ ] − m m s+ = i [ j− ,√s+ ] [ K −, q¯ n ] = 2 s¯+ n , − + m p [ Km , q¯p n ] =p√2ms¯+ n , m p p m [ J n , J q ] = i δ q J n i δ q J n i δ n J q + i δ n J q + − − − m p m[ Kp , q¯ np] =m√2 s¯+ nm, p p m [ J n , J q ] = i δ q J n −i δ q J n i δ n J q + i δ n J q m p m p− p m− m p p m [ J n , Jdynq ] a= i δ 1q J n ¯ abi δ q J abcn +i δ c n Jm q + icδ n J q δq¯ ϕ = + (∂−δ gf −∂ ϕ ) $ q¯m ϕ − ∂ − dyn a 1 ¯ ab abc + c m c δq¯ ϕ = + (!∂ δ gf ∂ ϕ ) $ q¯m ϕ " −dyn a ∂ 1 ab− abc + c m c dyn ¯ 1 δq¯ ϕ = + a(∂ δ gfab ∂m ϕ )c$ q¯m ϕ − δs¯∂ !ϕ = − $ q¯m ϕ " 1∂+ D dyn a ! ab m c " δs¯ ϕ = + $ q¯m ϕ dyn a ∂ 1D ab m c δkins¯ ϕdyn= a $ q¯mdynϕ a [ δ , δ ] ϕ∂+=D 2 i δ¯ ϕ SO(8)K ⊃p− SU(4)X− j− U(1) kin dyn a dyn a [ δ , δ ] ϕ = 2 i δ¯ ϕ K p− j− kin dyn a − dyn a dyn dyn [ δdyn, δp ] ϕ a = 2√i δ¯j ϕ a [ δsK , δs¯− ] ϕ = − 2 δ− ϕ − p− (was helicity) dyn dyn a dyn a U(1) generato[ rδs , δs¯ ] ϕ = √2 δ ϕ − p− dyn dyn i a √ dyn1 a [ δs , δs¯ ] ϕ =l 2 δlp ϕ J = ( q q¯l − q¯l q )− . −i4√2 − 1 ∂+ J = ( ql q¯ q¯ ql ) . l l + mn The SO(8) algebra is completed− 4√b2iy obtainingl − thelcoset∂ 1 generators J and J¯mn J = ( q q¯l q¯l q ) . through − 4√2 − ∂+ mn ¯ TheCoSOse(8)t algebrageneisrcompletedators by obtaining the coset generators J and Jmn mn throughThe SO(8) algebra is completed by obtaining the coset generators J and J¯mn mn m n i m n 1 through J = q , s+ = q q { − } √ ∂+ mn m n i 2m n 1 J = q , s+ = q q { − } √ ∂+ mn m n 2ii m n 1 1 ¯ J = q , s+ = q q Jmn = q¯{m−, s¯+n } = √2q¯+m q¯+∂n+ . { − } √2 ∂+ ¯ i 1 Jmn = q¯ m , s¯+n = q¯+m q¯+n . − √ ∂+ ¯ { } 2 i 1 Jmn = q¯ m , s¯+n = q¯m q¯n . { − } √2 ∂+
6 6 6 ¯ m ¯ m ∂ m ∂ q i [ j− , q ] = + q , q¯ m i [ j− , q¯m ] = + q¯m . − ≡ ∂ ∂¯ ¯ − ≡ ∂ ∂ m ¯ m m ¯ m m∂ m ∂ q i [ jq− , q i []j−=, q ] =q ,q , q¯ q¯mm ii [[jj−−, ,q¯mq¯]m=] = q¯m . q¯m . − ≡ − ≡ ∂+∂¯ ∂+ − − ≡≡ ∂+ ∂+∂ m ¯ mm mm + m q i [ j− , qq , ]s¯ =n = iqδn (D, + j q−¯ +mij) + 2iJ[ jn− , q¯m ] = q¯m . − + + − ≡ { }m ∂ − m − + ≡ m ∂ m q , s¯ n =m iδn (D ++j − + ij) + 2J nm q , s¯ {n =− }iδn−(D + j − + ij) + 2J n { − } m− m m s = i [mj− , s+ ] + m − m m q , s¯ n = iδsn (=Di+[ j−j, s− ]+ ij) + 2J n − + { } m− − m s = i [ j− , s+ ] + − [ K , q¯ n+] = √2 s¯+ n , [−K , q¯ n ] = √2 s¯+ n , m − m m p m p s =p i [mj− , s+m] p p m [ J , J ]m = ipδ J m i δp J p mi δ J m +pi δ Jp m n [qJ n , J q ] q= +inδ−q J nq i δn q J n ni δ qn J q +ni δ n qJ q [ K ,−q¯ n ]−= √−2 s¯+−n , − m p m p p m m p p m [ J , Jdyn] = dyni δ1 J+ 1 i δ J i δ J + i δ J n q a δ ϕa[qK= ¯n, abq¯ (n∂¯δ] abqabc= √g+fn2abcs¯c∂++nmϕc,) n$mq¯c qϕc n q δq¯ ϕ =q¯ + (∂ δ+− gf ∂ ϕ−) $ q¯m ϕ m − − ∂ ∂ −− − m p m! p ! p m m "p " p m [ J n , J q ] = i δ1 q Jdynn i δ q1 J n i δ n J q + i δ n J q dyn a dyn 1aab abcab m+ cc m c δ ϕ = a δs¯(∂¯ϕδ− = abg+fm ∂$ −cq¯mϕϕ ) $ q¯ ϕ q¯ δs¯ +ϕ = ∂ $ Dq¯m ϕ m − ∂ ∂+ D−
dyn a 1 !kin dynab a abc +dync a m c" δ ϕ = [ δ (, ∂¯δ δ ] ϕ g=f 2∂i δ¯ϕ ϕ) $ q¯ ϕ q¯ kindyn+dynKa ap− 1 ab −dynm aj− c m − [ δ δ ,∂δ ϕ ] ϕ= = − 2 i δ$¯ q¯ϕ ϕ Ks¯ p− ∂+−D j− m dyn! dyn a dyn a " [ δs , δs¯ ] ϕ = √2 δ ϕ dyn a 1 ab m p− c dynδ dynϕ a= √ −dyn$ q¯ a ϕ [ δs kins¯, δs¯ dyn] ϕ =a + 2 δ ϕdynm a [ δ , δ ] ϕ ∂= D 2p−i δ ϕ K p− − ¯j i l − l − 1 J = ( q q¯l q¯l q ) + . i − 4√2 − 1 ∂ kin dyn l a l dyn a J = ( q q¯l q¯l q ) . mn The SO(8) algebra[dynδis completed, δdyn ] ϕbay obtaining= 2thei+δcosetdyn¯ ϕgenea rators J and J¯mn [ δ K −, δ4√p−2] ϕ =− √∂2 δj− ϕ s s¯ − p− through − mn The SO(8) algebra is completed by obtaining the coset generators J and J¯mn through dyn dyn a i dyn 1a [ δ J mn, δ =i ]qϕm , lsn= = √2lqδm q1n ϕ J s= s¯ ( q +q¯ q¯ q )p +. √{ − l} − √l 2 ∂−+∂ − 4 2 i− 1 J mn = qm , sn = qm qn ¯ + i + 1 mn ¯ The SO(8) algebra1 is completedi Jmn{ =−byq¯iobtainingm} , s¯+n√2=the1 cosetq¯∂m q¯n1 gene. rators J and Jmn m n{ − l } √2ml n ∂p+ q + ¯ φ (y) = + A (y) + J =θ θ Cmn((qy)q¯l+ q¯lθq )θ +θ .θ $mnpq ∂ A (y) through ∂ √2 − 4√2 i−12 1∂ J¯ = q¯ , s¯ =a q¯ q¯a . i mn m √+δ2n ϕ = ωJmϕ n + The AScOt(8)ionalgebra on theis completedm su{ p− erfieldby Uobtaining(1}m) n√p2 the coset∂ q generators J mn and J¯ + θ χ¯m(y) + θ θ θ $mnpqi χ (y)1 mn ∂+ J mn = qm 6, sn = qm qn through + ω + δ{[ δ−ϕa, δ=free}ω] ϕJaϕa=√2 δfreeϕa ∂ U(1U) (1) s 2 s ¯ ¯ 1 p iiq ¯ 11 ¯ mndm dn φ =m $nmnpqi d d φ m1 n JmnJ = =δq¯ mqϕ,a s,¯2=+sn+ ==ωmnq¯ q¯q¯qm q¯qnϕa ++ . { coset−{ − 2}}√2 √√m22 n ∂+ ∂∂
m a √ m ia+ 1 ¯ q+ , δq¯+ n ϕ= =i ω2δJnϕ∂ Jmn {= q¯U(1m)}, s¯+n = q¯m q¯n . Constraints on Su{sy− transf} 6 or√ma2 tions∂+
dyn i ∂ a ma aω inta a a ! a δs¯ ϕ [ =δ , δ +6]$aϕ q¯m=ϕ +δaδsϕ¯ ϕ [ δ , δ ] ϕ = δ ϕ U(1δ√)U2(1∂)sϕ = ωJ2ϕs coset s s¯
a i mn 1 a a δ ϕ = ωa q¯ωq¯ a ϕ [ δ , δs ] ϕ = 0 coset [ δ , δ ] ϕ = m δn ϕ+ coset U(1) 2s√2 2 s ∂ a ! a [ δ , δ ] ϕ = δ ϕ m! mn coset s a s¯ " a [ δ , δ i] ϕ = δ ϕ 1 # = ω #n δ ϕcoseta = s ωmnq¯ s¯q¯ ϕa SO(8) vector: coset 2√2 m n ∂+ a [ δcoset , δs ] ϕ = 0 6 6 mn #m! = ω #n
4
7
7 1 i m n 1 m n p q + φ (y) = A (y) + θ θ C (y) + θ θ θ θ $mnpq ∂ A¯ (y) ∂+ √2 mn 12 i √2 + θm χ¯ (y) + θm θn θp $ χq(y) ∂+ m 6 mnpq
1 d¯ d¯ φ = $ dp dq φ¯ m n 2 mnpq
qm , q¯ = i√2δm∂+ { + + n } n
dyn aDespi ∂ermatela y intSeekina g Susy δs¯ ϕ = $ q¯ ϕ + δ ϕ √2 ∂+ m s¯ int a dyn a i ∂ m a δs¯ ϕ = + $ q¯m ϕ √2 ∂ + "su sy! ? (free theory)
In D=3 Dim ! is half-odd integer
int a b c d " ! ! ! ! susy ~ abcd f Dynamical Susy
4 dm ϕ(y , θ , θ) = 0
m d ϕ (y) =dm0ϕ(y , θ , θ) = 0
ϕ(x , θ , θ) ϕ(y ,dmθ ,ϕθ(y)) = 0 →
[ δ 1 , δ ] ϕa =ϕ(δx!, ϕθ a, θ) ϕ(y , θ , θ) δ ϕa =coset s(∂¯δab gfs¯abc∂+ ϕ→c ) δ ϕb q¯ + q¯+ − ∂ − m ! 1 d ϕ(y , θ , θ) "= 0 δ ϕa = (∂¯δab gf abc∂+ ϕc ) δ ϕb q¯ a + q¯+ a [ δcoset¯ ,ab−δs ] ϕ abc=∂ +0 c − b = (∂ δ gf ∂ ϕ ) q¯ m ϕ W m − ! +d mϕ (y) = 0 " a ¯ ab abc + c b m = (∂ δ gf ∂ ϕ ) q¯+ m ϕ m! Wmn ϕ(−x , θ , θ) ϕ(y , θ , θ) # kin= ω #mn → δs ϕ = % q¯m ϕ kin 1 m δ ϕδa s= ϕ = (%∂¯δabq¯m ϕgf abc∂+ ϕc ) δ ϕb q¯ + q¯+ − ∂ − kin m ! " a free a int a δs¯ ϕ = %¯mq ϕ a ¯ ab abc + c b δq¯ ϕ = δq¯ ϕ + δq¯ ϕ kinm = (∂ δ gmf ∂ ϕ ) q¯+ m ϕ − − − Wδs¯ ϕ = %¯m−q ϕ 1 ∂ a = α qkin¯ϕ kin i m + kin i kinm + [ δ , δ ] ϕ = % %¯ ∂ ϕ i=δs ϕ =δ % q¯ϕm ϕ i √2 ∂ · s s¯ kin kin m m +t− kin [ δs√2, δs¯ ] ϕ = % √%¯m2∂ ϕ = δt ϕ √2 √2 − abcd ijkl ∂ 1 +B b 1 +C c +D d δkin ϕ = %¯ q m ϕ + f # +A 1∂ ϕ +s¯ M 2∂m ϕ 3∂ ϕ ab ∂η ∂ ab E abc + ∂c E ab E ∂¯δ ! "ijk(l∂¯δ∂¯δab!gf ∂(∂¯δϕab ) gf abc∂+ ϕc ) ab " η=0 , α i# i $ → − →[ δkin , δkin ] ϕ−= ≡ %Dm%¯ ∂+ϕ =≡ D δkin ϕ s s¯ √2 m √2 t− free a 1 ∂ a δs ϕ = + # q¯ϕ √2 ∂ ∂¯δ·ab (∂¯δab gf abc∂+ ϕc ) ab 1 a 1 ab ∂ abcd →ijkl ∂ − c ≡+ Dd b a δq¯ abφ = abcd+ ij∂kδl gf % c + dE φ E∂ φ η=0 b % q¯+ φ δq¯ φ = + ∂ δ− gf ∂ #% − E φ∂Eη ∂ijklφ η=0 % q¯+ φ( · − ∂ − ∂η m $ m% · # d Eijη kϕl = i√2 η ϕ & ( ' abcd $1 Te%chnology∂ f δ φa = 1 ∂ δab g&f abcd %ijkl ' E φcE∂+ φd % q¯ φb a (#η¯ ,ζ) +Bα q¯b 1 + +Cα ∂c 1 +Dα d η=0 + ∆α = E#¯η∂ abϕ −E−#¯η ∂ ab# Eabcd−ζ ∂ 1ijkl ϕ E∂ηζ− ij∂kl c ϕ + d ( · +Aα ∂ δ ∂ δ+Mα gf % $ % E φ E∂ φ η=0 ∂ab ab abcd ij∂kl ∂ + ∂cη + d& ' % → − O → ∂ O ijkl ( ∂ δ ∂ δ gf % & E$φ E%∂ φ η=0 ' → − ab ∂η ab abcd ijkl ∂ & c + 'd coherent stat∂eδs ∂!δ ijklgf % E φ E∂ φ η=0 $→m % − √ m ¯ˆ ∂η ijkl q E" ϕ = i 2 $η dϕ $ % m (#η,ζ) Eη −=& e · '& ' d ∆ ˆ = 0 α η d¯ " q¯ˆ η d¯ˆ " q¯ˆ E = e · E =Eηe =· e · E" = e · η "ϕa d¯1234 ϕb ∼ 1 η d¯ˆ Eη− = e − · z ∂ˆ #¯ qˆ Ez = e E #η¯ = e · Exη∂ˆ E H== e( a , a ) x Wm W m √" q¯ˆ ˆ d 4Eη4ϕ 4= ia1m2 η1 ϕηad¯ = i d x Ed "θ d=θE−e ·= e − · Wη ∂+ 3 W m d- & q-eigenstates " 1 η d¯ˆ 1 E− = e − · ˆ = " q¯ˆ η m E E+" = em· d EηOϕ7(7)= ∂i√O2 η ϕ
m E m "q q¯ˆ E" ϕ8(8=) i√2 $ ϕ E" = e · − 1 SO(9)1 ϕa d¯1234 ϕb ∼ SO(8) 1 a a H = ( P SU, (2, 2 4)) W W | a m 1 = i d4x d4θ d4θ a W ∂+ 3 W m 2 1 ijkl !κ klmn pqij = ∂ Cijkl ∂µC + Cijkl C ∂ Cmnpq ∂µC + (κ4) L − 48 µ 2 µ O # $ E7(7)
1 mnpq C ijkl = $ijklmnpq C 4! E8(8)
1 = ij µν ij + c.c. 7 LV −8 Fµν GSO(9) written in terms of the self-dual complex field strengths
SO(8) 2
P SU(2, 2 4) |
2 1 ijkl κ klmn pqij = ∂ Cijkl ∂µC + Cijkl C ∂ Cmnpq ∂µC + (κ4) L − 48 µ 2 µ O " #
1 mnpq C ijkl = $ijklmnpq C 4!
1 = ij µν ij + c.c. LV −8 Fµν G written in terms of the self-dual complex field strengths
2 a ! a [ δcoset , δs ] ϕ = δs¯ ϕ
[ δ , δ ] ϕa = δ! ϕa coset s s¯a [ δ , δ ] ϕa = 0! a [ δcosetcoset , δss ] ϕ = δs¯ ϕ
a [ δcoset , δs ] ϕ = 0 #m! = ωmn a# [ δcoset , δs ] ϕ n= 0
m! mn # = ω #n m! mn # = ω #n δ ϕa = δfree ϕa + δint ϕa q¯ q¯ q¯ a ! a − − − [ δcoset , δs ] ϕ = δs¯ ϕ a free1 a ∂ int a δq¯ ϕ = δq¯ ϕ + δq¯ ϕ a = − α q¯ϕ − a √2 ∂f+ree −· a int a δq¯ ϕ 1 =∂ δq¯ ϕa + δq¯ ϕ a =− α −q¯ϕ − [ δcoset , δs ] ϕ = 0 + abcd ijkl ∂ 1 +B b 1 +C c +D d √2 ∂ 1· ∂ a + f # +A 1∂ ϕ +M 2∂ ϕ 3∂ ϕ = + α q¯ϕ∂η ∂ E ∂ E E abcd√2ij∂kl ∂· ! "ij1kl !+B b 1 +C c +D d " η=0 , α + f # chiral en#m!g1=in∂ ωeemnϕ # ring ∂ ϕ ∂ ϕ ∂η ∂+A E ∂n+M E2 # E3 $ abcd! ij"klijkl∂ ! 1 +B b 1 +C c "+ηD=0 ,dα + f # 1 ∂ 1∂ ϕ 2∂ ϕ 3∂ ϕ δfree∂ϕηa = ∂+A E# q¯ϕa# ∂+M E E$ s ! "ijkl !+ " η=0 , α two chiral ffieldsree a 1 ∂√2 ∂ a · # $ a freeδs a ϕ int= a # q¯ϕ δq¯ ϕ = δq¯ ϕ + δq¯ ϕ √2 ∂+ ·1 ∂ − − δ−fr1ee ϕa = # q¯ϕa 1bc(∂η) sa +B b+ 1 +C c = Z α=q¯ϕ +A Eη∂ √2ϕ∂ Eη−· ∂ ϕ bc(√η)2 ∂+ ·1 ∂ +B b 1 +C c Z = Eη∂ ϕ Eη− ∂ ϕ ∂+A m% bc(η) & abcdbc(η)ijkl d∂1Z 1 =+B 0 b +B 1b +1C c +C c +D d + fZ # = E ∂ ϕ1∂E ϕ ∂ ϕ ∂ ϕ ∂ ϕ m% bc∂η+(ηA) ∂η+A E η− &∂+M E2 E3 d Z! ∂ "ijk=l 0 ! " η=0 , α % # & $ fmree a(#η¯ ,ζ) 1 ∂ a δd ∆ϕα = = 0 # q¯ϕ f abcd m s(#η¯ ,ζ) √ 1+ a (thr#η¯ ,ζ)ee chiral fieldsd ∆+α: Bnαesb=ted01 c2o∂nstr· uct+ioCnα c 1 +Dα d ∆α = E#η¯ ∂ ϕ E#η¯− Eζ ∂ ϕ E− ∂ ϕ ∂+Aα ∂+Mα ζ ' * #¯ qˆ ( ) abcd E = e · E a (#η¯ ,ζ) f +#η¯Bα#¯ qˆb 1 1η +Cα c 1 +Dα d ∆α = EE#η¯ #¯η=∂ e ·ϕ EEη−#¯η Eζ ∂ ϕ E− ∂ ϕ ∂+Aα dm Zbc(η) ∂=+M0α ζ % & '(
int a dm ∆(#ηα,ζ) =∂0 ∂ a (#η¯ ,ζ) intδ a ϕ = +c m α( a1)(α#η¯ ∂,ζ) ∂ a (#η¯ ,ζ∆) δ ϕs¯ = +c ( d1)∆ =[2 02α] ∆[2+2α] α η=ζ=0 s¯ −α [2 ∂2αη] −[2+2∂αζ] α η=ζ=0 | α=1/2, 1/2 α= 1−,0,1 ∂η − ∂ζ | α=1/2, '1/2− α= 1,0,'1− '− '− #¯ qˆ E #¯ = e · #¯ qˆ int Eδain#η¯ t ϕ=a e=· Eη δs¯ ϕ s¯ =
α α∂ ∂ ∂ ∂a (#η¯ ,ζ)a (#η¯ ,ζ) int a ( 1)( 1) α∆ ∂∆ ∂ a (#η¯ ,ζ) δ ϕ = +c[2 2α[2] 2[2α(]+21)α][2+2α α] αη=ζ=0 η=ζ=0∆ s¯ − −∂η −∂η ∂−ζ ∂ζ [2 2α| ] [2|+2α] α η=ζ=0 α=1/α2=1, /1/22, 1/2 − ∂η − ∂ζ | '−α=1−/2, 1/2 α= 1,0,1 '+− +− ∂ ∂ ∂ ∂ ( 1)α α ∆a (#η¯ ,ζ)a (#η¯ ,ζ) ( 1)[2 2α] [2in+2t α]a α ∆αη=ζ=0 η=ζ=0 − −∂η −∂η[2∂ζ2δα] ∂ϕζ[2+2=α] | | α= α1=,0,11,0,1 − s¯ '− '− i1 i2 2α i4 ∂ ∂ # ··· −i1 ···i2 2α i4 ∂ ∂ ∂ ∂ # ···∂ − ··· ∂ ∂ ∂ [2 2α] [2+2α] (2+2αα)!(2 2α)! i1 i2 2α a (i#η¯3 ,2ζα) i4 ∂η − [2 ∂2αζ ] [2+2≡α] ( 1) ∂η ··· −i1 ∆∂iαζ2 2−α ···ηi=3ζ=02α i4 ∂η − ∂ζ −≡ (2∂+2η−[2α)!(22α] 2∂αζ)![2∂+2ηα] ··· − ∂|ζ − ··· α=1/2, 1/2 − − +− 7 7 7
7 after some elementary manipulations... a ! a [ δ , δ ] ϕa = !δ aϕ a ! a [ δcosetcoset, δs s] ϕ = δs¯ ϕs¯ [ δcoset , δs ] ϕ = δs¯ ϕ
a a [ [δδcoset, ,δsδs] ϕ] ϕ ==0 0 a coset [ δcoset , δs ] ϕ = 0
mm! ! mnmn ## ==ωω #n#n m! mn # = ω #n
aa ffrreeee aa intint aa δδq¯ ϕϕ == δδq¯ ϕϕ ++δδq¯ ϕϕ q¯ q¯− q¯ −− − −− a free a int a 11 ∂∂ a δ ϕ = δ ϕ + δ ϕ == αα q¯q¯ϕϕa q¯ q¯ q¯ √ ∂++ · − − − √22 ∂ · 1 ∂ a abcd ijkl ∂ 1 +B b 1 +C c +D d = α q¯ϕ abcd ijkl ∂ 1 +B b 1 +C c +D d + + f # +A 1∂ ϕ +M 2∂ ϕ 3∂ ϕ √ ∂ · + f # ∂η ∂ +A E 1∂ ϕ∂ +M E 2∂ ϕE 3∂ ϕ 2 ! ∂η"ijkl ∂ ! E ∂ E E " η=0 , α ! "ijkl ! # $ " η=0 , α abcd ijkl ∂ 1 +B b 1 +C c +D d # $ + f # ∂ ϕ ∂ ϕa 1!∂ a ∂ ϕ +A 1 +M [ δfcoset2ree, δsa] ϕ = 3δs¯ ϕ1 ∂ a ∂η ijkl ∂ E ∂ δEsfreeϕ a =E + # q¯ϕ ηa=0 , α ! " ! δs ϕ = √2 ∂ + #· "q¯ϕ # a √2 ∂ $· [ δcoset , δs ] ϕ = 0 free a 1 ∂ a abcd δs ϕ = # q¯ϕf m! mn 1 a (#η¯ ,+ζ) abcd +# Bα= ωb #n 1 +Cα c 1 +Dα d ∆√α2 ∂ =· f E#¯η∂ ϕ E−#¯η 1 Eζ ∂ ϕ E− ∂1 ϕ a (#η¯ ,ζ) ∂+Aα +Bα b 1∂+Mα +Cα c ζ +Dα d ∆α = +A E#¯η∂ ϕ E−#¯η +M Eζ ∂ ϕ Eζ− ∂ ϕ α% a ! a α ( ∂ [ δcoset , δs ] ϕ = δs¯ ϕ ∂ & ' a free a %int a & '( δq¯ ϕ = δq¯ ϕ + δq¯ ϕ abcd − − − a ! a 1 ∂ [ δcoset , δs ] ϕm= δ(s¯#η¯ϕ ,ζ) a (#η¯ ,ζ) f +B b 1 1 +C a [cδ ,dδ1 ] ϕ+∆a D= 0 d = 0 α = α qα¯ϕ coset s m α(α#η¯ ,ζ) ∆ = E ∂ ϕ E− E√2∂∂+ · ϕ E− ∂ ϕ α +Aα #¯η #¯η +Mα ζ ζ d ∆α = 0 ∂ ∂ ∂ 1a 1 abcd ijkl [ δcoset , δs ] ϕ = 0 +B b +C c +D d % + f # m! mn ∂ ϕ ( ∂ ϕ ∂ ϕ & ∂η# =∂+ωA #E1 '∂+M E2 E3 ! "ijkl ! n #¯ qˆ " η=0 , α # $ m! Emn#η¯ = e · Eη # = ω #n #¯ qˆ m (#η¯ ,ζ) free a E 1 ∂= e a· E δs ϕ = #η¯ # q¯ϕ η d ∆α =a 0 free a int a √2 ∂+ · δq¯ ϕ = δq¯ ϕ + δq¯ ϕ − − − a f1ree∂ a int a δq¯ ϕ == δq¯ ϕ α+ δq¯q¯ϕaϕ − − + abcd − ∂ ∂ int a a (#η¯ ,ζ)√2 ∂ f · +B b 1 1 α +C c 1 +D d a (#η¯ ,ζ) twδ oϕ inde= p1 en∂ dena t+ cansaα ¨tz(e1) α α ∆ s¯ ∆α = = α+Aq¯ϕ E#¯η∂ ϕ E−#¯η +M Eζ ∂ ϕ Eζ− ∂ ϕ α η=ζ=0 #¯ qˆ +∂ α ∂ 1 ∂ α 1 [2 ∂2α] [2+2∂α] int a √2 ∂abcd· ijk%l −+B b α∂η − +C ∂c ζ +D (d a (#η¯ ,ζ|) E #η¯ = δe ·ϕ Eη= α+=1f /2, # 1/2 +cα= +1A,0,11∂( &ϕ1) +M 2∂ ϕ 3∂ 'ϕ ∆ s¯ − ∂∂η 1∂− E 1∂ E[2 2α] E [2+2α] α η=ζ=0 both transf+ f abcd)or#ijmkl ! a"sij k8l ) of! ∂ +SO(8)Bϕ−b ∂η∂+C−ϕc ∂+∂Dϕζd " η=0 , α | α=1/2, 1/∂2η αm∂+=A(#η¯1,Eζ,10) ,1 ∂+M E2# E3 $ )− ! "ijkdl ~∆)−α! = 0 " η=0 , α free a 1 in∂ t a a # $ δs ϕ = + # q¯ϕ free a 1√2δ∂∂s¯ ϕ· a = δs ϕ = + #¯ qˆq¯ϕ E √=2 ∂ e ·in· Et a int a α ∂ ∂ a#η¯ (#η¯ ,ζδ)s¯ ηϕ = δ ϕ = +c ( 1) abcd ∆ s¯ a (#η¯[2,ζ) 2αf] [2+2α+]B bα 1 1 η=ζ=0+C c 1 +D d abcd α ∂ ∂α α − ∆∂αη = f +∂A ζ E#¯η∂ ϕα E−#¯η 1+M | Eζ ∂ ϕ Eζ− ∂a (#η¯ ϕ,ζ) a (#η¯ ,ζ−) ∂ α +Bα b 1 ∂ α +Cα c 1 +Dα d α=1/2, 1/2 α= 1,0,1 ∆α = E#¯η∂( 1)ϕ E−#¯η Eζ∂∂ ϕ E∂ζ− ∂∆αϕ η=ζ=0 − int a ∂+Aα % ∂+[2Mα 2α&] [2+2α] a (#η¯ ,ζ) '( )− ) δs¯ ϕ = +−c ∂(η 1) ∂ ∂ζ ∂ ∆α η=ζ|=0 α=1/2,% 1/2 α − −&∂η[2 2α] ∂ζ[2+2α] a'((#η¯| ,ζ) α=1/2, 1/2 ( α=1)1,0,1 − ∆ ))−− m)− (#η¯ ,ζ) [2 2α] [2+2α] α η=ζ=0 −md ∆(#η¯α,ζ∂) η =−0 ∂ζ | int a α=1/2, 1/2 d ∆α ∂= 0 ∂ δ ϕ = )− α δint ϕa = a (#η¯ ,ζ) triality s¯ ( 1) s¯ ∆α η=ζ=0 [2#¯ qˆ2α] [2+2α] − E ∂=η #¯eqˆ−· E ∂ζ | α= 1,0,1 E #η¯α= e · E∂ η ∂ a (#η¯ ,ζ) − ( 1)#η¯ η ∆ and/or ) α ∂ [2 2∂α] a[2(#η¯ +2,ζ) α] α η=ζ=0 ( −1) ∂η ∂∆ζα η=ζ=0 | α= 1,0,1 − ∂η[2 2αi]1∂−ζ[2i+22 α2]α i4 | α ∂ ∂ ∂ aα=1(#η¯/2, ζ1∂)/2 − # ··· − ··· ∂ ∂ )−)− ∂ ∂ ∂ ∂ ( 1) inintt aa ∆ ∂ ∂ (#η¯ (,#η¯ζ),ζ) ∂ ∂ (#η¯ ,ζ) (#η¯ ,ζ) δδ ϕϕ == [2 2αα] ccα[2+2ηα=] ζ=0 ∆∆ + + c" c" i1 i2 2α ∆ i∆3 2α i4 [2 2α] s¯s¯ [2+2α] α [2[2 2α2]α] [2+2[2∂+2α](2α] α+2iα1α∂ )!(2i2 2α2α)!αi4 α[2 2[2α] 2α[2] +2α[2] +2αα] α − ∂η ∂ζ ∂η − ∂ζ∂∂ηη |−− α∂ζ∂≡ζ a (#η¯ ,ζ)∂∂η η∂−η ···−∂ζ −∂ζ ∂ζ − ··· − αα=1∂/2,, 11//22 ∂( 1) # ···α=−∆α−=1α,0,1···,0,1 η=ζ=0 ∂ ∂ α=1/2, 1/2 )−− − ∂η[2 2α] ∂ζ[2+2α])− )− | − α= 1,0,1 − ) [2 2α] )− [2+2α] i1 i2 2α i3 2α i4 ∂η − ∂ζ ≡ (2+2α)!(2 2α)! ∂η ··· − ∂ζ − ··· i1 i2 2α i4 i1 i2+2αj1 j2 2α − ∂ ∂∂ ∂# i1 i2+2αj#1 −···j2 2−α ∂··· ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ # ······ ······ − ∂ ∂ ∂ ∂ α a (#η¯ ,ζ) i i i i [2 2α] [2[2+2α2]α] [2+2α] (2+2α)!(2 i21α)!7 1i2 22α 2α j1 3 2α j2+24 α ∂η[2 −2α] ∂∂ζη[2+2−α] ≡∂ζ (2 + 2α≡)!(2 2α)! ∂η ·i1· ·∂∂ηη ···− i2−∂2ζα ∂ζ· ·j1·−∂ζ··· j2+2α ( 1) ∂η − ∆∂ζα ≡ (2η=+ 2ζα=0)!(2 2α)!−∂η · · · ∂η − ∂ζ · · · ∂ζ − ∂η[2 2α] ∂ζ[2+2α] | −− α= 1,0,1 − 7 )− Aα 1 = Aα + 1 Bα 1 = Bα7+ 1 Mα 1 = Mα 2 Aα−1 = Aα + 1 B−α 1 = Bα + 1 M− α 1 = M−α 2 i1 i2 2α i4 − − − − ∂ ∂ # ··· − ··· ∂ ∂ Cα 1 = Cα 1 Dα 1 = Dα 1 [2 2α] [2+2α] i1 i2 2−α i−3 2α i−4 − ∂η ∂ζ ≡ (2+2α)!(2 2α)! ∂η Cα 1 ∂=ζCα 1 Dα 1 = Dα 1 − − ··· − − −− ··· − −
7 7 7 i1 i2+2αj1 j2 2α ! ··· ··· − ∂ ∂ ∂ ∂ (2 + 2α)!(2 2α)! ∂ηi1 · · · ∂ηi2 2α ∂ζj1 · · · ∂ζj2+2α − − i1 i2+2αj1 j2α 2α ! − ∂ ∂ ∂ ∂ ··· ( ···1) i1 i2 2α i4 ∂ ∂ − ! ··· − ··· i1 i2i12αi2 2αj1 i3 2α ji42+2α (2 + 2α(2)!(2+2α)!(22α2α)!)! ∂η · · · ∂η∂η ··· ∂−ζ ∂ζ· · −· ∂···ζ − − − ( 1)α ∂ ∂ Aα 1 = Aα + 1i1 i2Bα2α 1 =i4 Bα + 1 Mα 1 = Mα 2 −− ! ··· − −··· − − (2+2α)!(2 2α)! ∂ηi1 i2 2α ∂ζi3 2α i4 −Light-Cone Ham··· −iltonian− ···
Cα 1 = Cα 1 Dα 1 = Dα 1 − − − − Aα 1 = Aα + 1 Bα 1 = Bα + 1 Mα 1 = Mα 2 − − − − [ δfree + δint , δfree + δint ]ϕa = δdyn ϕa s s s¯ s¯ p− Cα 1 = Cα 1 Dα 1 = Dα 1 − − − − free int a int free a [ δs , δs¯ ]ϕ + [ δs , δs¯ ]ϕ abcd terms freelinear inint f int free a dyn a [ δs + δs , δs¯ + δs¯ ]ϕ = δH ϕ + m ∂ ¯ ∂ K = 2i( x ∂ 2 x− ∂ + θ m + θm ) x − − ∂θ ∂θ¯m [ δfree , δint ]ϕa + [ δint , δfree ]ϕa s s¯[ K , p− ] = s 2i js¯− −
8
8 dm ϕ(y , θ , θ) = 0
[ δ , δ ] ϕa = δ! ϕa dm ϕ (y) = 0 coset s s¯
a [ δcoset , δs ] ϕ = 0 ϕ(x , θ , θ) ϕ(y , θ , θ) →
m! mn 1 # = ω #n δ ϕa = (∂¯δab gf abc∂+ ϕc ) δ ϕb q¯ + α ∂ q¯+ ∂ a ("η¯ ,ζ) − ∂ −( 1) ∆ ! − ∂η[2 2α] ∂ζ[2+2" α] α |η=ζ=0 α=1/2, 1/2 − a !fr−ee a int a δq¯ aϕ = =(∂¯δδabq¯ ϕgf abc+ δ∂q¯+ ϕϕc ) q¯ aϕb −m − − δ + ϕm = W 1 −∂ p− = α q¯ϕa √2 ∂+ · ∂ kin α ∂m ∂ a (zη,ζ) α ∂ ∂ a (η,zζ) δs ϕ = % q¯m ϕ i&¯ & ( 1) abcd[2 ij2kαl ] ∂[2+2α] ∆1α ++B b 1 ( 1)+C c[2 2α+]D d[2+2α] ∆ 1 +α · ∂z − + f ∂η #− ∂ζ 1∂ ϕ − ∂ ∂ϕη −∂ ∂ϕζ 2 α= 1 , 1 ∂η ∂+A E α= ∂1+,0M,1 E2 E3 !2 − 2 ! "ijkl ! !− " η=0 , α z=η=ζ=0 # $ δkin ϕ = %¯ q m ϕ s¯ m α ∂ ∂1 ∂ a ("η¯ ,ζ) ( 1) δfree ϕa = ∆# αq¯ϕa η=ζ=0 − ∂sη[2 2α] ∂ζ[2+2α+] | α= 1,0,1 − √2 ∂ · − kin kin !i m + i kin [ δs , δs¯ ] ϕ = % %¯m∂ ϕ =i1 i2 2δαt i4ϕ ∂ √2∂ bc(η) & 1···√2− ···− +B b ∂ 1 +C c ∂ Z = +A Eη∂ ϕ Eη− ∂ ϕ [2 2α] [2+2α] ∂ i1 i2 2α i3 2α i4 ∂η − ∂ζ ≡ (2+2α)!(2 2α)! ∂η ··· − ∂ζ − ··· i i j j − ab 1 2+2ab α 1 abc2 +2α c % ab & ¯ !i1 ···¯i2+2αj1 ···j2 2−α ∂ ∂ ∂ ∂ ∂ δ & (···∂ δ g···f − ∂ ϕ∂) ∂ ∂ ∂ → −abcd ≡i1 D i2 2α j1 j2+2α (2 + 2αf)!(2 2α)! ∂i1η · · ·i2∂η21α − j1 ∂ζ ·j2+2· · α∂ζ a (#η¯(2,ζ)+ 2α)!(2 2α)! ∂+ηBα · ·b· ∂η1 ∂ζ · +· ·C∂α ζ c 1 +Dα d ∆ = −E ∂ ϕ E− − E ∂ ϕ E− ∂ ϕ i1 i2+2αj1 j2 2α α +Aα− #η¯ #η¯ +Mα ζ ζ ! ··· ··· − ∂ ∂ ∂ ∂ ∂ ∂ αα ' ( )* i1 (( 1)1)i2 2α i ji1i i i j2+2i α ∂ ∂ ∂ ∂ (2a+ 2α)!(21 2αab)! ∂η abcd· · ·ij∂kηl −∂ ∂ζ1 1 2· ·22·αc∂2ζα4 + 4d b δ φ = ∂ δ gf %−− & !··· ···E− φ−···E∂··· φi i % q¯i φ i q¯ + − (2+2α)!(2 2α)! m bc∂(η) 1 η=02i1 2αi2∂ζ2+3α 2α i34 2α i4 − ∂ − (2+2α)!(2 ∂η2α)! ∂···η −··· ·− −∂ζ··· − ··· α# −− ijkl ∂ d Z∂ = 0 ( $( 1)%α ∆a ("η¯ ,ζ) ( 1) i1 i2 2α i4 ∂ [2& 2α] ∂ [2+2α] α' η=ζ=0 − a − ∂η ∂ζ | − ! ··· α···=1/2, 1/i2 i − i i (2+2α)!(2 2α)! δ !ϕ∂−η 1= 2 2α ∂ζ 3 2α 4 ab − Aabα 1p=−abcdAα ij+kl1··· −∂ Bα 1 −= B···c α ++1 d Mα 1 = Mα 2 ∂ δ ∂ δAα −1gf= Aα% + 1 δ B−ϕdαmaE∆=1φa=(E#η¯ ∂B,ζ)αφ=+01 − Mα 1 =− Mα 2 → − ∂η p− − α η=0 − − $ %ijkl & ' ∂ A = A + 1∂ ∂ B ∂= B α + 1∂ M∂ a=(zηM,ζ) 2 α ∂ ∂ ∂ a∂(η,zζ) α 1 α α i&¯ & α 1 ( 1)αˆ a (xη,ζ)α 1∆ α+ ( 1)α ∆ a (η,xζ) − − η dC¯ α [21 2=α] Cα[2+2"α−1]q¯ˆ α Dα#¯−q1ˆ = Dα 1 [2 2α] [2+2α] 1 +α i#¯ # ( 1) · ∂z − ∆∂η ∂ζ +E = e · E( 1)− ∂η ∂ζ 2 ∆ 1 [2 2α] Eη1 [2=1+2e α· ] −α−E" = −e · #η¯ −α= η1,0,1 − −[2 2α] [2+2α] +α α= 2 , 2 C = C 1 D = D 1 · ∂x − ∂η − ∂!ζ − α 1 α α!− −1 α∂η − ∂ζ z=η=2ζ=0 α= 1 , 1 − −α= 1,0,1− − 2 2 ∂ ∂ − x=η=ζ=0 $− α $a ("η¯ ,ζ) dyn Cα 1 = Cα 1 Dfreeα( 11)=ˆinDt α free1 int∆α a η=ζ=0 a − [ δs −−+zδ∂s∂η[2, δ2s¯α] ∂Eζ+[2+2δs¯α=] E]ϕE =| δ ϕ − α=Ez1,0,=1 e − − zη z η p− − dyn α ∂ ! free ∂ int afree($η¯ ,ζ) int a a [ δs + δs i1 ,i2δs¯2α i4+ δs¯ ]ϕ = δ ϕ ( 1) ∂ ∂ & ∆··· − ··· ∂ ∂ p− free int free[2 2α]int [2a+2α] dynα a η=ζ=0 − ∂η∂η[2 2α] ∂ζ∂[2+2ζ αfree] ≡ ¯ˆ(2+2intα)!(2a 2α)! ∂ηini|t1 i2 free2α ∂ζi3a 2α i4 [ δs + δs , δs¯ −−+ δ1s¯ ][ϕδ =η,dδδp ]ϕϕ−+ [ δ ···, δ− ]ϕ − ··· α= 1,0,1 int a Eη− = se − · s¯ − s α s¯ ∂ ∂ a (#η¯ ,ζ) − Confδ oϕrma=i1 i2+2l αgj1 enj2 2αe+rc ator ( 1) ∆ $ s¯ & ··· ··· − ∂ ∂ ∂ [2 2α∂] [2+2α] α η=ζ=0 free int a− ∂inηt − free∂ζ a | i iα=1/2, 1i/2 iα1= 1,0,1i2 2α j1 j2+2α 1(2 + 22α)!(22α[ δ2sα4)! ∂,ηδs¯· · · ]∂ϕη −+∂[ζδs· · ·,∂δζ s¯ ]ϕ ∂ ∂ # ··· − +−···− 1 +− ∂ ∂ ∂ ∂ [ δfree , δint K]ϕa=+ [ δin2ti,xδ"(freeq¯ˆ x]∂ϕa x ∂+ + θm + θ¯ ) s s¯ E(" 1)s=α e ·s¯ i −i ∂ i ∂m mi [2 2α] [2+2α] − i21 i2 2α−1i4 in2t 2aα ∂θ3 2α 4∂θ¯ ∂η − ∂ζ ≡ (2+2α)!(2− 2α&)!···∂−η ··· ···δ −ϕ =∂ζ − ··· m (2+2α)!(2 −2α)! ∂ηs¯i1 i2 2α ∂ζi3 2α i4 − ···+− m− ···∂ ¯ ∂ i1 i2+2αj1 j2 2α K = 2i(∂x ∂ 2 x−∂∂ + θ + θm ) x # − ∂ + m ∂ ∂ ∂ m ¯ ···K = 2···i( x ∂ 2 x− ∂ +−θ [ K+,−pθ¯−m] = ) x2i j− ∂θ ∂θm From AHamα 1 = Aαil+t1onimBanα 1 =tBoα +¯Bo1 oMsαt1 = Mα 2 − − −i ∂iθ − j∂θm − −j − (2 + 2α)!(2 2α)! ∂η 1 · · · 1∂η 2 2α ∂ζ 1 ·6· · ∂ζ 2+2α − − Cα 1 = Cα[ K[ j1 , ,pp−Dα]] 1=== D0α 21i j− α [ K , p− ] = −2i j− − − − − −− ( 1) i1 i2 2α i−4 ∂ ∂ − # ··· − ··· free ini1t freei2 2αint a i3 2dynα ia4 (2+2α)!(2 2α)! [ δs ∂+ηδs , δs¯ + δs¯ ∂]ϕζ = δp ϕ − ··· − − −··· [ j− , p− ] = 0 yet to be checked free int a int free a [ δs , δs¯ ]ϕ + [ δs , δs¯ ]ϕ Aα 1 = Aα + 1 Bα 1 = Bα + 1 Mα 1 = Mα 2 − − − − 1 + m ∂ ¯ ∂ K = 2i x ( x ∂ x− ∂ + θ m + θm ) − 2 − ∂θ ∂θ¯m
Cα 1 = Cα 1 Dα[ K ,1p−=] =Dα7 2i j−1 − − − − −
[ j− , p− ] = 0 [ δfree + δint , δfree + δint ]ϕa = δdyn ϕa s s s¯ s¯ p−
free int a int free a [ δs , δs¯ ]ϕ + [ δs , δs¯ ]ϕ
7 + m ∂ ¯ ∂ K = 2i( x ∂ 2 x− ∂ + θ m + θm ) x − − ∂θ ∂θ¯m
[ K , p− ] = 2i j− −
[ j− ,8p− ] = 0
8
7 Consistency being checked (with Belyaev, Brink and S-S Kim)
Work in Progress
Check boost and Hamiltonian commute
Compute f-f term
abcd Properties of f from commutation
Path Integral measure over ! f abcd
Happy Birthday
Stanley