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 d d ϕ = $ d d ϕ 1 p q m n 2 mnpq d d ϕ = $ d d ϕ m n 2 mnpq 1 1 1 2 Physical Fields: A¯ = (A1 + iA2) A = (A iA ) √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].