Nonlinear Schr¨odinger Equations and N=1 Superconformal Algebra 6 0 0 H. T.O¨zer∗and S. Saliho˘glu† 2 n Physics Department, Faculty of Science and Letters, a J Istanbul Technical University, 5 34469, Maslak, Istanbul, 2 Turkey 1 v February 1, 2008 8 8 1 1 Abstract 0 6 By using AKNS scheme and soliton connection taking values in N=1 0 superconformalalgebraweobtainnewcoupledsuperNonlinearSchr¨odinger / h equations. t - p e h : v i X r a ∗e-mail : ozert @ itu.edu.tr †e-mail : salihogl @ itu.edu.tr 1 1 Introduction UsingAblowitz,Kaub,Newell,Segur(AKNS)scheme[1]onecanobtaincou- pled Nonlinear Schr¨odinger (NLS) equations.Extensions of coupled NLS equa- tions have been obtained using a simple Lie algebra[2], a Kac-Moody algebra [3],a Lie superalgebra[3,4],a Virasoro algebra [5]in the literature. In physics supersymmetry [6] unites bosons and fermions into a single mul- tiplet. UsingsupersymmetryonecancancelmanynormallydivergentFeynman graphs and one can solve the hierarchy problem in grand unified theories. Su- persymmetry also helps to understand the cosmological constant problem in gravity and it reduces the divergences of quantum gravity. Conformal invariance in two dimensions is a powerful symmetry.Two - di- mensionalquantumfieldtheoriesthatpossessconformalsymmetrycanbesolved exactlybyexploitingtheconformalsymmetry. Conformalsymmetryhavefound remarkableapplicationsinstring theoryandin the study ofcriticalphenomena instatisticalmechanics. N=1supersymmetricextensionofconformalsymmetry ,called superconformal symmetry promotes string to superstring.The extension of string theory to supersymmetric string theories identified N=1 supercon- formal algebra (Neveu-Schwarz type[7] and Ramond type [8]) as the symme- try algebras of closed superstrings [9]. The N=1 superconformal algebra with Neveu-Schwarz and Ramond types are two possible super - extensions of the Virasoro algebra [10] for the case of one fermionic current. In this paper we will obtain super - extensions of coupled NLS equations usingN=1superconformalalgebrawithNeveu-SchwarzandRamondtypes. In sec.2 we will discuss the osp(1,2)superalgebravalued soliton connection and we will obtain coupled super NLS equations. Sec.3 and sec.4 concern the soliton connection for the N=1 superconformal algebra with Neveu-Schwarz type and Ramond type, respectively and we will obtain in these sections two different types of super- extensions of coupled NLS equations. 2 AKNS Scheme with osp(1,2) Superalgebra In AKNS scheme in 1+1 dimension the connection is defined as Ω= iλH1+Q+1E+1+Q−1E−1+P+12F+12 +P−12F−21 dx+ (1) (cid:16) −AH1+B+1E+1+B−1E−1+C+12F+21 +C−12F−(cid:17)12 dt (cid:16) (cid:17) where H1, E+1, E−1are bosonic generators and F+12 ,F−21 are fermionic gen- erators of osp(1,2) superalgebra. These generators have matrix representations 2 as 1 0 0 0 1 0 0 0 0 2 H = 0 −1 0 ;E = 0 0 0 ;E = 1 0 0 1  2  +1   −1   0 0 0 0 0 0 0 0 0 (2)  0 0 1  0 0 0  2 F+12 = 00 01 00 ;F−21 = 01 00 −021  2 2     Also, these generators satisfy the following commutation and anticommutation relations [H ,E ] = ±E 1 ±1 ±1 [E ,E ] = 2H +1 −1 1 H1,F±21 = ±12F±12 hF+12,F−21i = 12H1 (3) nE±1,F∓21o = −F±21 hF±12,F±21i = ±21E±1 In Eq.(1) λ is the spectranl parametero, Q±1 and P±12 are fields depending on space and time, namely x and t, and functions A,B±1 and C±21 are x,t and λ dependent. The integrability condition is given by d Ω + Ω ∧ Ω=0 (4) By using Eqs.(1) and (4) one can obtain following equations: Q+1t = B+1x+iλB+1+Q+1A+ 12P+21C+12 (5) Q−1t = B−1x−iλB−1+Q−1A− 12P−12C−21 (6) P+12t = C+21x+ 2iλC+21 + 12P+21A+B+1P−21 −C−12Q+1 (7) P−12t = C−21x− 2iλC−21 − 12P−12A+B−1P+21 −C+21Q−1 (8) 0=Ax+2B+1Q−1−2B−1Q+1− 12P+21C−21 − 21P−12C+12 (9) In AKNS scheme we expand A,B±1 and C±12 in terms of positive powersof λ as 2 2 2 A= λnan; B±1 = λnb±n1; C±21 = λnc±n12 (10) n=0 n=0 n=0 X X X Inserting Eq.(10) into Eqs.(5-9)gives 15 relations in terms of a ,b±1 and c±12 . n n n By solving these relations we get a0 =−2iQ+1Q−1−iP+21P−21; a1 =0; a2 =−2i; 3 b±1 =±iQ±1 ; b±1 =Q±1; b±1 =0 (11) 0 x 1 2 c±0 12 =±2iP±21x; c±1 12 =P±12; c2±21 =0 By using the relations given by Eq.(11) from Eqs.(5-8) we obtain the coupled super NLS equations as −iQ+1t =Q+1xx−2(Q+1)2Q−1−P+12P−21Q+1+P+12Px+21 iQ−1t =Q−1xx−2(Q−1)2Q+1−P+12P−21Q−1−P−12Px−12 (12) −iP+12t =2P+21xx+P−21Q+1x+2Q+1Px−21 −Q+1Q−1P+12 iP−21t =2P−12xx+P+21Q−1x+2Q−1Px+21 −Q+1Q−1P−12 3 AKNS Scheme with N=1 Superconformal Al- gebra (Neveu-Schwarz Type) We generalize the connection given by Eq.(1) as Ω= iλL0+Q+mL+m+Q−mL−m+P+m2 G+m2 +P−m2 G−m2 dx+ (cid:16) −AL0+B+mL+m+B−mL−m+C+m2G+m2 +C−m2G−m(cid:17)2 dt (cid:16) (cid:17) (13) where L0 , L±m are bosonic generators and G±m2 are fermionic generators of centerless N=1 superconformal algebra of Neveu-Schwarz type , namely they satisfy the following commutation and anticommutation relations [6] [L ,L ] = (r−s) L r s r+s {G ,G } = 2 L (14) r s r+s [L ,G ] = (r −s) G r s 2 r+s Here, L±m are generators with positive(negative) integer indices and G±m2 are generators with positive(negative) half integer indices. In Eq.(13) we assume summation over the repeated indices. The fields Q±m and P±m2 are x,t depen- dent and also functions A, B±m and C±m2 are x,t and λ dependent. In N=1 superconformal algebra if we restrict L to have only L , L ±m 0 ±1 componentsandG±m2 tohaveonlyG±21 componentsweobtainosp(1,2)algebra given by Eq.(3) with the following definitions: H =−L ; E =L ; E =−L 1 0 +1 +1 −1 −1 (15) F+12 = 21G+21; F−12 =−21G−12 From the integrability condition given by Eq.(4) we obtain Q+m =B+m −imλB+m−mAQm t x 4 ∞ ∞ + (r−s)B+sQ+rδ + (r+s)B−sQ+rδ r−s,m r−s,m r,s=1 r,s=1 X Xr>s ∞ ∞ − (r+s)B+sQ−rδ−r+s,m+ 2P+r2C−2sδr−s,2m (16) r,s=1 r,s=1 Xr<s Xr>s ∞ ∞ + 2P−r2C+2sδ−r+s,2m+ 2P+r2C+2sδr+s,2m r,s=1 r,s=1 Xr<s X Q−m =B−m +imλB−m+mAQ−m t x ∞ ∞ − (r−s)B−sQ−rδ + (r+s)B−sQ+rδ −r−s,−m r−s,−m r,s=1 r,s=1 X Xr<s ∞ ∞ − (r+s)B+sQ−rδ−r+s,−m+ 2P+r2C−2sδr−s,−2m (17) r,s=1 r,s=1 Xr>s Xr<s ∞ ∞ + 2P−r2C+2sδ−r+s,−2m+ 2P−r2C−s2δ−r−s,−2m r,s=1 r,s=1 Xr<s X P+m2 t =C+m2x− imλC+m2 − mP+m2 A 2 2 ∞ ∞ + 1(r+s)Q+rC−s2δ2r−s,m− 1(r+s)Q−rC+s2δ−2r+s,m 2 2 r,s=1 r,s=1 X X 2r>s 2r<s ∞ ∞ + 1(r−s)Q+rC+2sδ2r+s,m+ 1(r+s)P+r2B−sδr−2s,m (18) 2 2 r,s=1 r,s=1 X rX>2s ∞ ∞ + 1(r−s)P+r2B+sδr+2s,m− 1(r+s)P−r2B+sδ−r+2s,m 2 2 r,s=1 r,s=1 X X r<2s im m P−m2 t =C−m2x+ λC−m2 4 P−m2 A 2 2 ∞ ∞ + 1(r+s)Q+rC−2sδ2r−s,−m− 1(r+s)Q−rC+s2δ−2r+s,−m 2 2 r,s=1 r,s=1 2Xr<s 2Xr>s ∞ ∞ − 1(r−s)Q−rC−s2δ−2r−s,−m+ 1(r+s)P+r2B−sδr−2s,−m (19) 2 2 r,s=1 r,s=1 X X r<2s 5 ∞ ∞ − 1(r−s)P−r2B−sδ−r−2s,−m− 1(r+s)P−r2B+sδ−r+2s,−m 2 2 r,s=1 r,s=1 X rX>2s and ∞ 0=−Ax+2 rB−rQ+r−rB+rQ−r+P+r2C−r2 +P−r2C+r2 (20) r=1 X(cid:0) (cid:1) In AKNS scheme we expand A,B±m and C±m2 in terms of the positive powers of λ as 2 2 2 A= λnan; B±m = λnb±nm; C±m2 = λncn±m2 (21) n=0 n=0 n=0 X X X InsertingEq.(21)intoEqs.(16-20)gives15relationsintermsofa ,b±m andc±m2 n n n (n=0,1,2) . By solving these relations we get ∞ ∞ a0 =2i Q+rQ−r+ m4i P+r2P−r2; a1 =const.=a10; a2 =−i; r=1 r=1 X X i b±m =∓ Q±m +a iQ±m; b±m =Q±m; b±m =0 (22) 0 m x 10 1 2 c0±m2 =a10iP±m2 ∓ m2iP±m2 x; c±1 m2 =P±m2; c±2 m2 =0 By usingthe relationsgivenby Eq.(22)fromEqs.(16-19)we obtainthe coupled super NLS equations as −i Q+m = Q+m +ia Q+m t m xx 10 x ∞ ∞ −2imQ+m Q+rQ−r −4imQ+m 1 P+r2P−r2 r r=1 ! r=1(cid:20) (cid:21) ! X X ∞ ∞ 2r−m 2r−m −i Q+rQ+(m−r)−i Q+rQ−(r−m) m−r x m−r x r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX<m rX>m ∞ ∞ +i 2r−r m Q−(r−m)Q+xr+4i r−12m P+r2Px−(r2−m) (23) r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>m rX>2m ∞ ∞ −4i 1 P−(r2−m)Px+r2 −4i 1 P+r2Px+(m−r2) r 2m−r r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>2m rX<2m 6 i Q−m = Q−m +ia Q−m t m xx 10 x ∞ ∞ +2imQ−m Q+rQ−r +4imQ−m 1 P+r2P−r2 r r=1 ! r=1(cid:20) (cid:21) ! X X ∞ ∞ 2r−m 2r−m +i Q−rQ+(r−m)+i Q−rQ+(r−m) r−m x r−m x r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX<m rX>m ∞ ∞ +i 2r−r m Q+(r−m)Q−xr+4i 1r P+(r2−m)Px−r2 (24) r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>m rX>2m ∞ ∞ −4i 1 P−r2Px+(r2−m)−4i 1 P−r2Px+(r2−m) r−2m r−2m r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>2m rX<2m P+m2 t = −m2iP+m2 xx+ia10P+m2 x ∞ ∞ −imP+m2 Q+rQ−r −2imP+m2 1 P+r2P−r2 r r=1 ! r=1(cid:20) (cid:21) ! X X ∞ ∞ +2i 3rr−−mm P+r2Qx−(r2−m2)+ 2i 3r−r m P−(r−m2)Q+xr r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>m rX>m 2 ∞ ∞ +i 3r−m P+r2Qx−(m2−r2)+i 3r−m Q+r2Px−(r−m2) (25) 2 r−m 2r−m r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX<m rX>m 2 ∞ ∞ +i 3r−m Q−(r2−m2)Px+r2 +i 3r−m Q+rPx−(m2−r) 2r 2r−m r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>m rX<m 2 and P−m2 t = m2iP−m2 xx+ia10P−m2 x ∞ ∞ +imP−m2 Q+rQ−r +2imP−m2 1 P+r2P−r2 r r=1 ! r=1(cid:20) (cid:21) ! X X ∞ ∞ +2i 3r−r m P+(r−m2)Q−xr+ 2i 3rr−−mm P−r2Q+x(r2−m2) r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>m rX>m 2 ∞ ∞ +i 3r−m P−r2Qx+(r2−m2)+i 3r−m Q+(r2−m2)Px−r2 (26) 2 r−m 2r r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX<m rX>m 7 ∞ ∞ +i 3r−m Q−rP+(r−m2)+i 3r−m Q−rP+(r−m2) x x 2r−m 2r−m r=1 (cid:20) (cid:21) r=1 (cid:20) (cid:21) rX>m rX<m 2 2 4 AKNS Scheme with N=1 Superconformal Al- gebra (Ramond Type) We take the soliton connection as Ω= iλL +Q+mL +Q−mL +P+mG +P−mG dx+ 0 +m −m +m −m (cid:16) AL +DG +B+mL +B−mL +C+mG +C−(cid:17)mG dt 0 0 +m −m +m −m (cid:16) (cid:17)(27) where L , L are bosonic generators and G are fermionic generators of 0 ±m ±m centerless N=1 superconformal algebra of Ramond type , namely they satisfy the following commutation and anticommutation relations [6] [L ,L ] = (r−s) L r s r+s {G ,G } = 2 L (28) r s r+s [L ,G ] = (r −s) G r s 2 r+s Here,L and G are generators with positive(negative) integer indices. In ±m ±m Eq.(27) we assume summation over the repeated indices. The fields Q±m and P±mare x,t dependent and also functions A,D, B±m and C±m are x,t and λ dependent. From the integrability condition given by Eq.(4) we obtain Q+m =B+m −imλB+m−mAQm−2P+mD t x ∞ + 2P+sC+r +(s−r)Q+sB+r δ r+s,m r,s=1 X (cid:0) (cid:1) ∞ + 2P−rC+s−(r+s)Q−rB+s δ (29) −r+s,m r,s=1 Xr<s (cid:0) (cid:1) ∞ + 2P+rC−s+(r+s)Q+rB−s δ r−s,m r,s=1 Xr>s (cid:0) (cid:1) Q−m =B−m +imλB−m+mAQ−m−2P−mD t x ∞ + 2P+sC+r +(s−r)Q+sB+r δ r+s,m r,s=1 X (cid:0) (cid:1) 8 ∞ + 2P−rC+s−(r+s)Q−rB+s δ (30) −r+s,m r,s=1 Xr<s (cid:0) (cid:1) ∞ + 2P+rC−s+(r+s)Q+rB−s δ r−s,m r,s=1 Xr>s (cid:0) (cid:1) m P+m =C+m −imλC+m−mP+mA− Q+mD t x 2 ∞ 1 − (r−2s)P+rB+s+(2r−s)Q+sC+r δ r+s,m 2 r,s=1 X (cid:0) (cid:1) ∞ 1 − (2r+s)P−rB+s+(r+2s)Q−rC+s δ (31) −r+s,m 2 r,s=1 Xr<s (cid:0) (cid:1) ∞ 1 + (2r+s)P+rB−s+(r+2s)Q+rC−s δ r−s,m 2 r,s=1 Xr>s (cid:0) (cid:1) m P−m =C−m +imλC−m+mP−mA+ Q−mD t x 2 ∞ 1 (s−2r)P−rB−s+(s−2r)Q−rC−s δ −r−s,−m 2 r,s=1 X (cid:0) (cid:1) ∞ 1 − (r+2s)P−rB+s+(r+2s)Q−rC+s δ (32) −r+s,−m 2 r,s=1 Xr>s (cid:0) (cid:1) ∞ 1 + (2r+s)P+rB−s+(r+2s)Q+rC−s δ r−s,−m 2 r,s=1 Xr<s (cid:0) (cid:1) ∞ 0=−A +2 rB−rQ+r−rB+rQ−r+P+rC−r+P−rC+r (33) x r=1 X(cid:0) (cid:1) and ∞ 3 0=−D + r P+rB−r−P−rB+r−Q−rC+r +Q+rC−r (34) x 2 r=1 X (cid:0) (cid:1) In AKNS scheme we expand A,D,B±m and C±m in terms of the positive powers of λ as 9 2 2 2 2 A= λna ; D = λnd ; B±m = λnb±m; C±m = λnc±m (35) n n n n n=0 n=0 n=0 n=0 X X X X Inserting Eq.(35) into Eqs.(29-34)gives 18 relations in terms of a ,d ,b±m and n n n c±m (n=0,1,2) . By solving these relations we get n ∞ ∞ 2i a =2i Q+rQ−r+ P+rP−r; a =0; a =−i; 0 m 1 2 r=1 r=1 X X i b±m =∓ Q±m ; b±m =Q±m; b±m =0 (36) 0 m x 1 2 i c±m =∓ P±m ; c±m =P±m; c±m =0 0 m x 1 2 ∞ ∞ 3i 3i d = P−rQ+r+ Q+rP−r; d =0; d =0; 0 1 2 2 2 r=1 r=1 X X By usingthe relationsgivenby Eq.(36)fromEqs.(29-32)we obtainthe coupled super NLS equations as ∞ −i Q+m = Q+m −2imQ+m Q+rQ−r t m xx ! r=1 X ∞ ∞ −4imQ+m 1 P+r2P−r2 −3i Q+rP−r+Q−rP+r r r=1(cid:20) (cid:21) ! r=1 X X(cid:0) (cid:1) ∞ 2 2r−m −i P+(m−r)P+r− Q+(m−r)Q+r (37) r x r x r=1 (cid:18)(cid:20) (cid:21) (cid:20) (cid:21) (cid:19) rX<m ∞ 2 2r−m +i P+rP−(r−m)+ Q+rQ−(r−m) r−m x r−m x r=1 (cid:18)(cid:20) (cid:21) (cid:20) (cid:21) (cid:19) rX>m ∞ 2 2r−m −i P−(r−m)P+r− Q−(r−m)Q+r r x r x r=1 (cid:18)(cid:20) (cid:21) (cid:20) (cid:21) (cid:19) rX>m ∞ i Q−m = Q−m +2imQ−m Q+rQ−r t m xx ! r=1 X ∞ ∞ +4imQ−m 1 P+r2P−r2 +3i Q+rP−r+Q−rP+r r r=1(cid:20) (cid:21) ! r=1 X X(cid:0) (cid:1) 10