Integrability of the symmetry reduced bosonic dynamics and soliton generating transformations in the low energy heterotic string effective theory

In tegrabilit y of the symmetry reduced b osonic dynamics and soliton generating tr ansformations in the low energy heterotic str ing effective theory G.A. Alekseev ∗ Steklov Mathematic a l Institute of the Russian A c ademy of Scienc es, Gubkina str. 8, 1 19991, Mosc ow, Russia Integrable structure of t h e sy mmetry redu ced dynamics of massless b osonic sector of the heterotic string effective action is presented. F or string background equ ations th at gov ern in the space-time of D dimensions ( D ≥ 4) th e dy n amics of interacting gravitati onal, dilaton, an tisymmetric tensor and any n umber n ≥ 0 of Ab elian vector gauge fields, all dep ending only on tw o coordinates, w e construct an e quivalent (2 d + n ) × (2 d + n ) matrix spectral p rob lem ( d = D − 2 ). This spectral problem provides the base for the development o f v arious solution constru cting pro cedures (d ressing transformations, integra l equation method s). F o r the case of the absence of Ab elian gauge fields, w e presen t the soliton generating transformations of any background wi th in teracting gra vitational, dilaton and the second rank antisymmetric tensor fields. This new soliton generating pro cedure is a v ailable for constructing of vari ous types of field configurations including stationary axisymmetric fields, interacting plane, cylindrical or some other types of wa ves and cosmolo gical solutions. P A CS num b ers: 04.20.Jb, 04.50.-h, 04.65.+e, 05.45.Yv In troduction In recent decades , dev elopment of the (super )string theory toward a consistent description of all in teractions suggested fundamental c ha nges of our view of the pic- ture of spa ce-time and field dynamics there [1, 2]. Many int eresting features o f this picture hav e b een discovered using the solutions of the corre s p o nding low-energy ef- fective theor ie s. Thes e solutions play an impor ta nt role in the analysis of v arious nonp erturbative a sp ects of the string theory (see [3, 4, 5] and the references therein). How ever, most o f these so lutions w ere found using some very particula r a ns ¨ atze, or global symmetry transforma- tions. Mor e systematic approa ches and flex ible metho ds may arise for symmetry reduced field equatio ns, provided these o ccur to b e in tegrable. This integrability may lead to constr ucting of large v arieties of multiparametric so- lutions for physically differen t types of interacting fields. The symmetry reduced equations o f pure v acuum Ein- stein gravity in D dimensio ns are clearly integrable and fo r their solution one can use (witho ut substantial changes) the inv er s e scattering approach dev elop ed for D = 4 space-times thirt y years a g o by Belinski and Za- kharov [6]. Many authors used this approa ch to co n- struct soliton solutions in four a nd five dimensions (e.g., [7, 8, 9, 10]). In D = 4 space-times with tw o commut ing isometries, the dynamics of dilato n and axion fields coupled to grav- it y a ls o was describ ed using the Belins k i and Zakha rov inv ers e scattering appro a ch [11]. In higher dimensio ns or/and in the presence of vector ga uge fields, this dyna m- ∗ Electronic ad dress: G.A.Alekseev@mi.ras.ru ics hav e been studied by many authors, who asserted its int egra bilit y , how ever, understanding of the former ha s not b een e no ugh to g ive rise to so me solution g enerating metho ds b eyond globa l symmetry transforma tions [12]. In this pa p e r, a completely int egra ble structur e of the dynamics of massless b osonic sector of heterotic string ef- fective ac tion for space-times with D ≥ 4 dimensions and with d = D − 2 commut ing isometries is des crib ed. F or int eracting g ravitational, dilaton, second ra nk antisym- metric tensor a nd any num be r n ≥ 0 o f Abelian gauge vector fields we construct an e quivalent (2 d + n ) × (2 d + n )- matrix sp ectra l pro ble m which provides the base for de- velopmen t of v arious solutio n co nstructing pro cedures. F or v anishing gauge v ector fields, the soliton generat- ing transformations are constructed here using an appro- priate mo dification of the construction [13] of Einstein - Maxwell solitons. F or v acuum fields, the relations b e- t ween o ur sp ectral problem and Belinski-Za k harov one and corresp onding v acuum solitons is shown explicitly . Massless b osonic mo des of heterotic string theory The mass le ss b osonic part o f heterotic str ing effective action in space-time with D ≥ 4 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 P p =1 F M N ( p ) F M N ( p ) ) p − b G d D x (1) where M , N , . . . = 1 , 2 , . . . , D and p = 1 , . . . n ; b G M N po ssesses the “ most p ositive” Lorentz signature. Met- ric b G M N and dilaton field b Φ are related to the metric 2 G M N and dilaton Φ in the Einstein frame as b G M N = e 2Φ G M N , b Φ = ( D − 2)Φ . (2) The comp onents of a three-form H and t wo-forms F ( p ) are determined in terms of antisymmetric tenso r field B M N and Ab e lian gauge field p otentials A M ( p ) as H M N P = 3  ∂ [ M B N P ] − n P p =1 A [ M ( p ) F N P ] ( p )  , F M N ( p ) = 2 ∂ [ M A N ] ( p ) , B M N = − B N M . Space-time sy mmetry ansatz W e co nsider the space-times with D ≥ 4 dimensio ns which admit d = D − 2 commuting Killing vector fields. All field comp o nents and potentials are assumed to b e functions of only tw o co o rdinates x 1 and x 2 , o ne of which can be timelike or b oth a re spacelike co or dinates. W e assume also the following structure of metr ic c omp onents G M N =  g µν 0 0 G ab  µ, ν, . . . = 1 , 2 a, b, . . . = 3 , 4 , . . . D (3) while the comp onents of field p otentials ta ke the forms B M N =  0 0 0 B ab  , A M ( p ) =  0 A a ( p )  . (4) W e c hos e x 1 , x 2 so that g µν takes a conformally flat form g µν = f η µν , η µν =  ǫ 1 0 0 ǫ 2  , ǫ 1 = ± 1 ǫ 2 = ± 1 where f ( x µ ) > 0 a nd the sign sym b ols ǫ 1 and ǫ 2 allow to c o nsider v a rious types of fields. The field equations imply that the function α ( x 1 , x 2 ) > 0 is “harmonic” one: det k G ab k ≡ ǫα 2 , η µν ∂ µ ∂ ν α = 0 , ǫ = − ǫ 1 ǫ 2 . where η µν is inv erse to η µν , and therefor e, the function β ( x µ ) can be defined as “harmonica lly ” co njuga ted to α : ∂ µ β = ǫε µ ν ∂ ν α, ε µ ν = η µγ ε γ ν , ε µν =  0 1 − 1 0  . Using the functions ( α, β ), w e cons tr uct a pair ( ξ , η ) of real null co ordina tes in the h yp erb olic ca se or complex conjugated to each other co ordinates in the elliptic case: ( ξ = β + j α, η = β − j α, j = ( 1 , ǫ = 1 − hyperb olic case , i, ǫ = − 1 − elliptic case . In particula r, for stationar y axisymmetric fields ξ = z + iρ , η = z − iρ , whereas for plane wav es or for c osmolog ic a l solutions ξ = − x + t , η = − x − t or these may have more complicate expressio ns in terms of x 1 , x 2 . Dynamical e quations The sy mmetr y r educed dyna mical e q uations can b e presented in the form of a matrix analog ue of the known Ernst equations [1 8] expressed in terms of the string frame v ar iables – a symmetric d × d -matrix G = e 2Φ k G ab k , an tisy mmetric d × d -matrix B = k B ab k , a rect- angular d × n - matrix A = k A a ( p ) k and a scalar b Φ:      η µν ∂ µ ( α∂ ν E ) − α η µν ( ∂ µ E − 2 ∂ µ AA T ) G − 1 ∂ ν E = 0 , η µν ∂ µ ( α∂ ν A ) − α η µν ( ∂ µ E − 2 ∂ µ AA T ) G − 1 ∂ ν A = 0 , η µν ∂ µ ∂ ν α = 0 , (5) where T means a matrix transp osition and E = G + B + AA T , det G = ǫα 2 e 2 b Φ . (6) Equations (5) imply the exis tence of tw o other matrix po tent ials. These are the antisymmetric d × d -matrix po tent ial e B and d × n -matrix p otential e A defined as ∂ µ e B = − ǫαε µ ν G − 1 ( ∂ ν B − ∂ ν AA T + A ∂ ν A T ) G − 1 , ∂ µ e A = − ǫαε µ ν G − 1 ∂ ν A + e B ∂ µ A . (7) The remaining part of the field equations do e s not ha ve a dynamical character and it allows to calc ula te the confor- mal factor f in quadr atures (see the expres s ions b elow), provided the solution o f dyna mical equations w as found. “Null curv ature” representation It is rema rk able that the dynamical equations (5) can be transformed in to a pair of first order matrix equations for t wo (2 d + n ) × (2 d + n )-matrix functions U and V whic h are rea l (in the hyper bo lic cas e) or complex conjugated to each other (in the elliptic case): ∂ η U + ∂ ξ V + [ U , V ] ξ − η = 0 , ∂ η U − ∂ ξ V = 0 , (8) where U and V possess the 3 × 3 blo c k-matrix structures U =    I d 0 0 B + I d 0 C + 0 I n       I d −E ξ − 2 A ξ 0 0 0 0 0 0       I d 0 0 − B + I d 0 − C + 0 I n    V =    I d 0 0 B − I d 0 C − 0 I n       I d −E η − 2 A η 0 0 0 0 0 0       I d 0 0 − B − I d 0 − C − 0 I n    in which the s ubscripts ξ and η denote par tial deriv a tives, I d and I n mean d × d and n × n unit matr ic es r esp ectively . The expressions for d × d -blo cks B ± and n × d -blo cks C ± include the po tentials defined in (7): B + = e B − j α G − 1 C + = − ( e A T + A T B + ) B − = e B + j α G − 1 C − = − ( e A T + A T B − ) (9) 3 In these coor dinates a nd notations, the seco nd relation in (7) is equiv alent to ∂ ξ e A = B + ∂ ξ A , ∂ η e A = B − ∂ η A . Spe ctral problem T o o btain a s pe ctral pr oblem equiv alent to the dynam- ical equa tions, we construct a linea r system with a free complex (“sp e ctral”) parameter w ∈ C and with the in te- grability conditions (8) and supply it with the cons tr aints providing U a nd V to ha ve the ab ov e structures and sat- isfy (6) – (9). In our sp ectra l problem it is requir e d to find four (2 d + n ) × (2 d + n )-matrix functions Ψ ( ξ , η , w ) , U ( ξ , η ) , V ( ξ , η ) , W ( ξ , η , w ) (10) which should satisfy the following linea r sy stem with al- gebraic constraints on its matrix 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 (11) Besides that, it is r equired that the s ystem (11) posses ses a symmetric matrix integral K ( w ) such that ( 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   (12) where Ω is (2 d + n ) × (2 d + n ) matrix. W e require also Ψ ( ξ , η , w ) = Ψ ( ξ, η , w ) , K ( w ) = K ( w ) , W (3)(3) = I n . (13) where W (3)(3) is the lower right n × n blo ck o f W . In accorda nce with (10)–(12), W (3)(3) is a constant matr ix and therefore, the co ndition W (3)(3) = I n can b e alwa ys achiev e d by a n a ppropria te gaug e trans fo rmation. The e quiv alence of the sp ectr a l problem (10)–(13) to (5) can be s een from a direct calculation similar to [14]. Field v ariables and p otent ials The conditions (10)–(13) imply , that W has the for m W = ( w − β ) Ω + G , G =     ǫα 2 G − 1 − e B G e B + e A e A T e B G + e AA T e A −G e B + A e A T G + AA T A e A T A T I n     (14) where (2 d + n ) × (2 d + n ) matrix G is real and symmetric, that its d × d matr ix blocks G , e B ar e symmetric and anti- symmetric resp ectively and that these matrix v a riables, together with d × n matrices A and e A , satisfy (5) – (7). The conformal factor b f = e 2Φ f in the W eyl form of conformally flat part of string frame metric b f ( dα 2 − ǫ dβ 2 ) can be ca lculated in quadratures (“tr ” denotes a trace):      ∂ ξ log  α d/ 2 e − b Φ b f  = − 1 8 tr  ( G − 1 + E ) U T ΩU  ∂ η log  α d/ 2 e − b Φ b f  = − 1 8 tr  ( G − 1 + E ) V T ΩV  where (2 d + n ) × (2 d + n ) matrix E = I − Ω 2 . Global sym metries The spectra l problem constructed above admits global symmetry transforma tio ns (including the discr ete ones) U → A UA − 1 , V → A V A − 1 Ψ → AΨ , W → ( A T ) − 1 W A − 1      A T ΩA = Ω W (3)(3) ≡ I n (15) where the real constant matr ix A is determined by tw o inv arianc e co nditio ns shown just ab ov e on the right. Some of these symmetries are not pur e ga uge and gener- ate ph ysically different solutions from a given one. Soliton generating transformations with A ≡ 0 F or v a nishing v ec tor gauge fields ( A ≡ 0), the problem (10)–(13) reduces to 2 d × 2 d matrix form which a dmits the so liton generating transformations . Given a so lution of (5) with A ≡ 0 , we denote its 2 d × 2 d matrices b y ” ◦ ”. F or the one-soliton solution on this background we assume Ψ = χ ◦ Ψ , χ = I + R ( ξ , η ) w − w 1 , χ − 1 = I + S ( ξ , η ) w − e w 1 where w 1 , e w 1 are real consta nt s and 2 d × 2 d - matrices R , S dep end on ξ , η only . F or cons istency w e also assume K ( w ) =  w − e w 1 w − w 1  ◦ K ( w ) . Then the conditions (10)–(13) imply that for this so liton solution U , V and W take the forms W = ◦ W − Ω · R − R T · Ω + ( w 1 − e w 1 ) Ω , U = ◦ U + 2 ∂ ξ R , V = ◦ V + 2 ∂ η R , R = ( w 1 − e w 1 ) p · ( m · p ) − 1 · m in which ( d × 2 d ) matrix m and (2 d × d ) matrix p are determined in terms of the background solution as m = k · ◦ Ψ − 1 ( ξ , η , w 1 ) p = ◦ Ψ ( ξ , η , e w 1 ) · l      k · ◦ K − 1 ( w 1 ) · k T = 0 , l T · ◦ K ( e w 1 ) · l = 0 . (16) with the ”integration constants ” – the real ( d × 2 d ) ma- trix k and (2 d × d ) matr ix l , which must hav e the ra nks 4 equal d a nd satisfy the algebr aic constra int s (16). T o solve (16), w e no te tha t R and W remain unchanged if w e m ultiply k from the left and l fro m the righ t b y some constant nondegenerate d × d matr ic es. Therefore, without lo ss of g enerality , w e ca n put so me d columns of k and d rows of l eq ual to d × d unit matrices. Then, for example, a transformation ◦ Ψ → ◦ Ψ · C ( w ), s uch that ◦ K → C T · ◦ K · C = Ω , linearizes the constra int s (16). This pro cedure can b e generaliz ed to any num b er of solito ns. On the Beli nski-Zakharov v acuum solitons F or v acuum fields in D dimensions, the fundamental solution Ψ ( ξ , η , w ) of our sp ectral pro blem takes the form Ψ = 1 λ 2 − ǫα 2 λ I d −G − ǫα 2 G − 1 λ I d ! ψ BZ 0 0 λ ( ψ − 1 BZ ) T ! where 2 w = λ + 2 β + ǫα 2 /λ and ψ BZ ( ξ , η , λ ) is a fun- damental solution of Belinski-Zak harov v acuum d × d - matrix s pe ctral proble m. The po les λ = µ k ( ξ , η ) o f ψ BZ on the λ pla ne corresp ond to p o les w = w k of Ψ ( ξ , η , w ) on the w plane (2 w k = µ k + 2 β + ǫα 2 /µ k ) and ther efore, the known v acuum so litons are a lso solito n solutions for our sp ectral pr oblem (sometimes, howev er , of a mor e gen- eral t yp e). F or any D ≥ 4 and for different backgrounds, our so liton generating pr o cedure lea ds to large families of solutio ns for in teracting fields whose physical and ge- ometrical prop erties need further inv estig a tion. Monodromy transform approach for A 6 = 0 The sp ectral problem (10)–(13) p o ssesses an impor - tant mono dromy preserving prop erty providing the base for application of the so called mono dr omy trans fo rm ap- proach sug gested in [15, 16] (see a ls o [17]) for E instein- Maxwell fields . It allows to reduce o ur sp ectral pr oblem to the equiv alent system o f linear s ing ular integral equa- tions, which admits an explicit calculation of infinite hier- archies of solutions for a n y (a nalytically matc hed) ratio- nal mono dr o my data. How ever, such developmen ts a re exp ected to be the s ub ject of subsequent publications. Ackno wle dgements This work was supp or ted in parts by the Russian F oun- dation for Basic Resear ch (grants 08- 01-00 501, 08-01 - 00618 , and 09-01-92 433-CE ) and the programs ”Math- ematical Metho ds of Nonlinear Dynamics” of Russian Academy of Sciences and ”Le a ding Scientific Sc ho ols” of Russian F ederation, (grant NSh-1959 .2008 .1). [1] M. B. Green, J. H. Sc hw arz, and E. Witten, Sup erstring the ory (Cambridge Un ivers ity Press, Cambridge, Eng- land, 1987), V ols. 1,2. [2] G.T. Horowitz, New J.Ph ys. 7, 201 (2005). [3] D. Y oum, Phys.Rept. 316 (1999) 1-232; [4] T. Mohaupt, Class. Quantum Gra v. 17 (2000) 3429. [5] R. Emparan and H. Reall, Living Rev. R elativit y , 11 , (2008), 6. [6] V.A. Belinski and V .E. Zakharov, Sov. Phys. JETP 4 8 , 985 (1978); 50 , 1 (1979). [7] V. Belinski and R. Ruffin i, Ph ys.Lett. 89B 195 (1980). [8] A.A. Pomeransky , Phys.Rev. D73 , 044004 (2006). [9] H. Elv ang and P . Figueras, J. High Energy Phys. 05 (2007) 050. [10] H . Elv ang and M. J. Ro driguez, J. High Energy Ph ys. 04 (2008) 045. [11] I.Bak as, Phys. Rev. D 54, 6424 (1996). [12] O.V . Kec hkin, Phys. Particl es an d Nu clei 35 , 383 (2004). [13] G.A. Alekseev, JETP Lett., 32 , 277 (1980). [14] G.A. Alekseev, Theor. Ma th. Phys. 144 (2), 1065 (2005). [15] G.A. Alekseev, Sov. Phys. Dokl. 30 , 565 (1985). [16] G. A. A lekseev, Pro c St eklo v Inst Math / T rudy Matem- atic heskogo institu t a imeni V A Steklov a ), 3, 215 (198 8). [17] G.A.Alekseev, Physica D (Amsterdam) 152 , 97 (2001). [18] F or stationary fields, in different notations the matrix Ernst equations were found earlier (see the su rvey [12]).