New soliton generating transformations in the bosonic sector of heterotic string effective theory

September 23, 2018 16:5 WSPC - Pro ceedings T rim Size: 9.75i n x 6.5in MG12˙2 1 NEW SOLITON GENERA TING TRANSF ORMA TIONS IN THE BOSONIC SECTOR OF HETEROTIC STRING EFFECTIVE THEOR Y George A. ALEKSEEV Steklov M athematic al Institute RAS, Gubkina 8, 119991, Mosc ow Russia E-mail: G.A.A lek se ev@mi.r as.ru In the author’s pap er Ref. 1, the integ rable structure of the symmetry r educed b osonic dynamics in the low energy heterotic string effectiv e theory was presen ted. In that pap er, for a complete system of massless b osonic fields which includes metric, dilaton field, an tis ymmetric tensor and an y nu mber of Ab elian vector gauge fields, considered in the space-time of D dimensions with D − 2 commuting isometries, the sp ectral problem equiv alen t to the symmetry reduced dynamical equations w as construct ed. How ev er, the soliton generating transformations w er e described in that paper only for the case in whic h all v ector gauge fields v anish. In this paper, we recall the integrabilit y structure of these equations and describ e some new type of s oliton generating transformations in which the v ector gauge fields can also en ter the bac kground (seed) solution as well as these can be generated eve n on v acuum background by an appropriate c hoice of soli ton parameters. Keywor ds : heterotic string; grav ity ; b osonic dynamics; symmetries; int egrability; solitons Massless b osoni c sector of the low-energy heterotic s tring theory The massless bos onic part o f heterotic string effectiv e ac tion in the string frame is S = Z e − b Φ ( b R ( D ) + ∇ M b Φ ∇ M b Φ − 1 12 H M N P H M N P − 1 2 n X p =1 F M N ( p ) F M N ( p ) ) q − b G d D x where M , N , . . . = 1 , 2 , . . . , D and p = 1 , . . . n , ( D is the space-time dimension and n is a num b er of Ab elian g a uge fields); b G M N po ssesses the “most p ositive” Lorentz signature. The components o f a three-form H and t wo-forms F ( p ) are determined in terms of antisymmetric tensor field B M N and Ab elian gaug e field p otentials A M ( p ) : H M N P = 3  ∂ [ M B N P ] − n X p =1 A [ M ( p ) F N P ] ( p )  , F M N ( p ) = 2 ∂ [ M A N ] ( p ) , B M N = − B N M . Metric b G M N and dilaton field b Φ a r e related to the metric G M N and dilaton Φ in the Einstein frame a s b G M N = e 2Φ G M N and b Φ = ( D − 2)Φ. Symmetry reduced b osoni c dynamics In what follows, we as s ume that in the spac e - time of D dimensions with d = D − 2 commuting K illing vector fields, all ”none-dynamical” field comp o nent s v anish: G M N =  g µν 0 0 G ab  , B M N =  0 0 0 B ab  , A M ( p ) =  0 A a ( p )      µ, ν , . . . = 1 , 2 a, b, . . . = 3 , 4 , . . . D September 23, 2018 16:5 WSPC - Pro ceedings T rim Size: 9.75i n x 6.5in MG12˙2 2 and a ll field co mpone nts and p otentials depend only on t wo coo rdinates x 1 and x 2 (one of which ca n be time-like or b oth a re space-like). The co o rdinates x 1 , x 2 can be chosen s o that g µν takes a conformally flat form g µν = f η µν with f ( x 1 , x 2 ) > 0. As such co ordinates, we choose ”geometric ally defined” functions α ( x 1 , x 2 ) and β ( x 1 , x 2 ) (gener alized W eyl co o rdinates) and use their linear co m binations ξ and η : ( ξ = β + j α, η = β − j α,      α : det k G ab k = ǫα 2 , β : ∂ µ β = ǫε µ ν ∂ ν α, η µν =  ǫ 1 0 0 ǫ 2  , ε µν =  0 1 − 1 0  , where ǫ = − ǫ 1 ǫ 2 and ǫ 1 = ± 1, ǫ 2 = ± 1 are the sign symbols whic h allow to consider v a rious types o f fields. The matrix η µν is in verse to η µν and ε µ ν = η µγ ε γ ν . The field equations imply that the function α ≥ 0 is “ha r monic”: η µν ∂ µ ∂ ν α = 0 and β ( x µ ) is defined as its “harmonic a lly” conjugated. The parameter j = 1 for ǫ = 1 (the hyperb olic cas e) and j = i for ǫ = − 1 (the elliptic case). T her efore the co ordinates ξ and η r esp e ctively b oth are r e a l or complex conjugated to eac h other. The sp ectral problem equiv alen t to dynamical equations As it was shown in Ref. 1, the symmetry reduced dynamics of mass less b osonic fields in heterotic string effective theory is integrable and the solution of the dynamical equations is equiv alent to so lution of the sp ectr al problem found there. This s p ectr al problem is formulated in terms of four (2 d + n ) × (2 d + n )-matrices Ψ ( ξ , η , w ), U ( ξ , η ), V ( ξ , η ), W ( ξ , η , w ) which should satisfy the linear system for Ψ with the algebraic conditions which determine the canonical J ordan forms o f its co efficients:    2( w − ξ ) ∂ ξ Ψ = U ( ξ , η ) Ψ 2( w − η ) ∂ η Ψ = V ( ξ , η ) Ψ       U · U = U , tr U = d, V · V = V , tr V = d, (1) and this system should a dmit a s ymmetric matr ix integral K ( w ) such tha t    Ψ T WΨ = K ( w ) K T ( w ) = K ( w )       ∂ W ∂ w = Ω , Ω =   0 I d 0 I d 0 0 0 0 0   (2) where w ∈ C is a spectr al para meter, Ω is (2 d + n ) × (2 d + n )-matrix, I d is a d × d unit matrix. Denoting by W (3)(3) the lower right n × n blo ck of W , we require a lso Ψ ( ξ , η , w ) = Ψ ( ξ , η , w ) , K ( w ) = K ( w ) , W (3)(3) = I n . (3) In accorda nce with Ref. 1, any s o lution { Ψ , U , V , W } of the the Eqs. (1)–(3) de- termines uniq ue ly some solution of the dyna mical equations and vice versa. Soliton generating transformation Given some solution a s background for so lito ns, we denote its (2 d + n ) × (2 d + n )- matrices by ” ◦ ”, a nd for one soliton on this background w e assume Ψ = χ · ◦ Ψ , χ = I + R ( ξ , η ) w − w o , χ − 1 = I + S ( ξ , η ) w − w o = ⇒ det χ ≡ 1 September 23, 2018 16:5 WSPC - Pro ceedings T rim Size: 9.75i n x 6.5in MG12˙2 3 where w o is a real consta n t a nd (2 d + n ) × (2 d + n )-matr ices R and S are rea l and depend on ξ , η only . W e also assume K ( w ) = ◦ K ( w ). Then the consistency conditions χ χ − 1 = χ − 1 χ = I imply S = − R and R · R = 0. This means that R is degenerate and for simplicity , w e consider R having the rank equa l to 1: R = n ⊗ m , ( m · n ) = 0 , where m ( ξ , η ) and n ( ξ , η ) are (2 d + n )-vector row and column r esp ectively . Substi- tution of the ab ov e expressions int o the Eqs. (1)–(3) leads to a set of relations a t the p ole w = w o and at w → ∞ which can b e solved explicitly . Thus w e obtain: U = ◦ U + 2 ∂ ξ R , V = ◦ V + 2 ∂ η R , W = ◦ W − Ω · R − R T · Ω The vector functions m ( ξ , η ) a nd n ( ξ , η ) ar e determined by the expres sions n = æ − 1 p , m = k · ◦ Ψ − 1 ( ξ , η , w o ) , p = ◦ Ψ ( ξ , η , w o ) · l , æ = 1 2 ( p T · Ω · p ) . where the re a l (2 d + n )-vectors k and l ar e constant; k is determined unequally in terms o f l , and the choice of l is arbitra r y provided it satisfies an algebr aic constra in t: k = l T · ◦ K ( w o ) , ( l T · ◦ K ( w o ) · l ) = 0 . Thu s, ch o osing any ba ckground solutio n and any rea l constant (2 d + n )-vector col- umn l which satisfy o nly the las t men tioned constraint, we ca n co nstruct the matr ix W whose components (in accordance with the expressions g iven in Ref. 1) allow us to calculate all field comp onents and p otentials of generated one solito n solution. It is necessary to ment ion, how ever, that for sta tionary ca se, generation of one soliton can le a d (simila rly to v acuum solitons of Belinski and Zakharov) to a change of signature of metric a nd therefore the nu mber of solitons in this case should be even. O n the other ha nd, the describ ed here one so liton generating transformation can b e g eneralized easily to the multisoliton cas e . Ac kno wledgeme n ts The autho r is thankful to the O rganizing Committee of MG12 for partial financial suppo rt for participation in the Meeting. This w ork was suppo rted in par ts by the Russian F oundation for Bas ic Research (gra n ts 08-01-0 0 501, 08 -01-0 0 618, 09-0 1- 92433 -CE) and the progr am “Ma thematical Metho ds of Nonlinear D ynamics” of the Russia n Academ y of Sciences. References 1. G.A. Alekseev, Integrabil ity of the symmetry reduced b osonic dy namics and soliton generating transformations in the low energy heterotic string effective theory , Phys. R ev. D80 , 0419 01(R) (2009); arXiv:hep-th/0811.1358.