Superstring Backgrounds in String Geometry
aa r X i v : . [ h e p - t h ] F e b Superstring Backgrounds in StringGeometry
Masaki Honda , ∗ , Matsuo Sato , † and Taiki Tohshima , ‡ Department of Physics, Waseda University, Totsuka-cho 1-104,Shinjuku-ku, Tokyo 169-8555, Japan Graduate School of Science and Technology, Hirosaki University,Bunkyo-cho 3, Hirosaki, Aomori 036-8561, Japan
Abstract
String geometry theory is a candidate of the non-perturbative formulation of string theory.In order to determine the string vacuum, we need to clarify how superstring backgroundsare described in string geometry theory. In this paper, we show that the superstringbackgrounds are embedded in configurations of the fields of a string geometry model.Especially, we show that the configurations satisfy the equations of motion of the stringgeometry model in α ′ → ∗ E-mail address : yakkuru [email protected] † E-mail address : [email protected] ‡ E-mail address : [email protected] ontents
1. Introduction
Superstring theory is a promising candidate of a unified theory including gravity. However,superstring theory is established at only the perturbative level as of this moment. Theperturbative superstring theory lacks predictability because it has many perturbativelystable vacua.String geometry theory is a candidate of non-perturbative formulation of superstringtheory [1], which can determine a non-perturbatively stable vacuum. In string geometrytheory, the path-integral of the perturbative superstring theory on the flat string back-ground is derived by taking a Newtonian limit of fluctuations around a fixed flat back-ground in an Einstein-Hilbert action coupled with any field on string manifolds [1, 2] .That is, the spectrum and all order scattering amplitudes in superstring theory on aflat background are derived from string geometry theory. However, perturbative stringtheory describes only propagation and interactions of strings in a fixed classical stringbackground, and cannot describe dynamics of the classical string background itself. Onlythe consistency with the Weyl invariance requires that the string background satisfies theequations of motion of supergravity. That is, string backgrounds are treated as externalfields in the perturbative string theory. In order to determine a string background, a non-perturbative string theory needs to be able to describe dynamics of the string backgroundsnot in consequence of consistency. A perturbative topological string theory is also derived from the topological sector of string geometrytheory [3].
1n paper [4], as a first step to determine the string vacuum, the authors studied howarbitrary configurations of the bosonic string backgrounds are embedded in configurationsof the fields of a bosonic string geometry model. Especially, the authors showed that theaction of the string backgrounds is obtained by a consistent truncation of the action ofthe string geometry model; the configurations of the fields of string geometry model sat-isfy their equations of motion if and only if the embedded configurations of the stringbackgrounds satisfy their equations of motion. This means that classical dynamics of thestring backgrounds are described as a part of classical dynamics in string geometry the-ory. This fact supports the conjecture that string geometry theory is a non-perturbativeformulation of string theory.This truncation is valid without taking α ′ → X (¯ σ ) → x .This fact will be important to derive the path-integral of the non-linear sigma model fromfluctuations around the string background configurations in the string geometry theory,since the string backgrounds in the non-linear sigma model depend not only on the stringzero modes x but also on the other modes of X ( σ ) [5–7]: S = 14 πα ′ Z d σ √ g (cid:2)(cid:0) g ab G µν ( X ( σ )) + iǫ ab B µν ( X ( σ )) (cid:1) ∂ a X µ ∂ b X ν + α ′ Rφ ( X ( σ )) (cid:3) . (1)In this paper, we generalize the results in [4] to the supersymmetric case, whichpossesses interesting problems as follows. In general, it is too difficult to define non-linearsigma models in R-R backgrounds in the NS-R formalism of string theory. On the otherhand, if one can perform a supersymmetric generalization of the above results as theyare, there is an apparent contradiction that we can derive non-linear sigma models inR-R backgrounds in the NS-R formalism from string geometry theory. We will see howthis contradiction is resolved. There is an another interesting problem: Chern-Simonsterms cannot be defined in string geometry models because they are infinite dimensional,although supergravities, which should be reproduced from the models, possess Chern-Simons terms. Nevertheless, we will see that the type IIA and IIB supergravities arereproduced from a string geometry model.The organization of this paper is as follows. In Sec. 2, we introduce a string geometrymodel. In Sec. 3, we identify type IIA and IIB string background configurations andobtain the equations of motions of the type IIA and IIB supergravities from the equations2f motions of the string geometry model by a consistent truncation in α ′ →
2. String geometry model
We study a string geometry model whose action is given by, S = 1 G N Z D E D ¯ τ D X ˆ D √ G e − (cid:18) R + 4 ∇ I Φ ∇ I Φ − | H | (cid:19) − X p =1 | ˜ F p | ! , (2)where G N is a constant, I = { d, ( µ ¯ σ ¯ θ ) } , | H | := G I J G I J G I J H I I I H J J J , andwe use the Einstein notation for the index I . The action (2) consists of a metric G I I , ascalar field Φ, a field strength H I I I of a two-form field B I I and ˜ F p . ˜ F p are defined by P p =1 ˜ F p = e − B ∧ P k =1 F k , where F k are field strengths of (k-1)-form fields A k − . Forexample, ˜ F = F − B ∧ F + B ∧ B ∧ F . They are defined on a Riemannian stringmanifold , whose definition is given in [1]. String manifold is constructed by patching opensets in string model space E , whose definition is summarized as follows. First, a globaltime ¯ τ is defined canonically and uniquely on a super Riemann surface ¯ Σ by the real partof the integral of an Abelian differential uniquely defined on ¯ Σ [8, 9]. We restrict ¯ Σ toa ¯ τ constant line and obtain ¯ Σ | ¯ τ . An embedding of ¯ Σ | ¯ τ to R d represents a many-bodystate of superstrings in R d , and is parametrized by coordinates ( ¯ E , X ˆ D T (¯ τ ) , ¯ τ ) where ¯ E is a super vierbein on ¯ Σ and X ˆ D T (¯ τ ) is a map from ¯ Σ | ¯ τ to R d . ˆ D T represents all thebackgrounds except for the target metric, that consist of the B-field, the dilaton andthe R-R fields. String model space E is defined by the collection of the string states byconsidering all the ¯ Σ , all the values of ¯ τ , and all the X ˆ D T (¯ τ ). How near the two string The action of string geometry theory is not determined as of this moment. On this stage, we shouldconsider various possible actions. Then, we call each action a string geometry model and call the wholeformulation string geometry theory . In [1], the perturbative superstring theory on the flat spacetimeis derived from a gravitational model coupled with a u (1) field on a Riemannian string manifold,whereas in [2], it is derived from gravitational models coupled with arbitrary fields on a Riemannianstring manifold. Thus, the perturbative superstring theory on the flat spacetime is derived from thismodel. ¯ represents a representative of the super diffeomorphism and super Weyl transformation on theworldsheet. Giving a super Riemann surface ¯ Σ is equivalent to giving a supervierbein ¯ E up to superdiffeomorphism and super Weyl transformations. τ and how near X ˆ D T (¯ τ ) . By this definition,arbitrary two string states on a connected super Riemann surface in E are connectedcontinuously. Thus, there is an one-to-one correspondence between a super Riemannsurface in R d and a curve parametrized by ¯ τ from ¯ τ = −∞ to ¯ τ = ∞ on E . That is,curves that represent asymptotic processes on E reproduce the right moduli space of thesuper Riemann surfaces in R d . Therefore, a string geometry model possesses all-orderinformation of superstring theory. The cotangent space is spanned by d X d ˆ D T := d ¯ τd X ( µ ¯ σ ¯ θ )ˆ D T := d X µ ˆ D T (cid:0) ¯ σ, ¯ τ , ¯ θ (cid:1) , (4)where µ = 1 , . . . ,
10. The summation over (¯ σ, ¯ θ ) is defined by R d ¯ σd ¯ θ ˆ E (¯ σ, ¯ τ , ¯ θ ). ˆ E (¯ σ, ¯ τ , ¯ θ ) := n ¯ E (¯ σ, ¯ τ , ¯ θ ), where ¯ n is the lapse function of the two-dimensional metric. This summa-tion is transformed as a scalar under ¯ τ ¯ τ ′ (¯ τ , X ˆ D T (¯ τ )) and invariant under (¯ σ, ¯ θ ) (¯ σ ′ (¯ σ, ¯ θ ) , ¯ θ (¯ σ, ¯ θ )). For example, an explicit form of the line element is given by ds ( ¯ E , X ˆ D T (¯ τ ) , ¯ τ )= G ( ¯ E , X ˆ D T (¯ τ ) , ¯ τ ) dd ( d ¯ τ ) +2 d ¯ τ Z d ¯ σd ¯ θ ˆ E X µ G ( ¯ E , X ˆ D T (¯ τ ) , ¯ τ ) d ( µ ¯ σ ¯ θ ) d X µ ˆ D T (¯ σ, ¯ τ , ¯ θ )+ Z d ¯ σd ¯ θ ˆ E Z d ¯ σ ′ d ¯ θ ′ ˆ E ′ X µ,µ ′ G ( ¯ E , X ˆ D T (¯ τ ) , ¯ τ ) ( µ ¯ σ ¯ θ ) ( µ ′ ¯ σ ′ ¯ θ ′ ) d X µ ˆ D T (¯ σ, ¯ τ , ¯ θ ) d X µ ′ ˆ D T (¯ σ ′ , ¯ τ , ¯ θ ′ ) . (5)The inverse metric G IJ ( ¯ E , X ˆ D T (¯ τ ) , ¯ τ ) is defined by G IJ G JK = G KJ G JI = δ KI , where δ dd = 1 and δ µ ′ ¯ σ ′ ¯ θ ′ µ ¯ σ ¯ θ = δ µ ′ µ δ ¯ σ ′ ¯ θ ′ ¯ σ ¯ θ , where δ ¯ σ ′ ¯ θ ′ ¯ σ ¯ θ = δ (¯ σ ¯ θ )(¯ σ ′ ¯ θ ′ ) = E δ (¯ σ − ¯ σ ′ ) δ (¯ θ − ¯ θ ′ ). ¯ E is a discrete variable in the topology of string geometry, where an ǫ -open neighborhood of[ ¯ Σ , X ˆ D T s (¯ τ s ) , ¯ τ s ] is defined by U ([ ¯ E , X ˆ D T s (¯ τ s ) , ¯ τ s ] , ǫ ) := n [ ¯ E , X ˆ D T (¯ τ ) , ¯ τ ] (cid:12)(cid:12) q | ¯ τ − ¯ τ s | + k X ˆ D T (¯ τ ) − X ˆ D T s (¯ τ s ) k < ǫ o , (3)As a result, d ¯ E cannot be a part of basis that span the cotangent space in (4), whereas fields arefunctionals of ¯ E as in (5). The precise definition of the string topology is given in the section 2 in [1]. . Consistent truncation In this section, we will show that we can consistently truncate the string geometry modeleq. (2) to the type IIA or IIB supergravities if we apply appropriate configurations to themodel, respectively and take α ′ → G IJ , Φ, B L L and A L ··· L p − are derived as R IJ + 2 ∇ I ∇ J Φ − H IL L H L L J − G IJ ( R + 4 ∇ I ∇ I Φ − ∂ I Φ ∂ I Φ − | H | ) − e
2Φ 9 X p =1 h p − F IL ··· L p − ˜ F L ··· L p − J − G IJ | ˜ F p | i = 0 , (6) R + 4 ∇ I ∇ I Φ − ∂ I Φ ∂ I Φ − | H | = 0 , (7) X p =3 [ p − ] X n =0 n +1 · ( p − F I ··· I p − − n B J K · · · B J n K n ˜ F I ··· I p − − n J K ··· J n K n L L + ∇ I ( e − H IL L ) = 0 , (8) ∇ I ˜ F IL ··· L p − + (cid:18) (cid:19) n ∇ I h B J K · · · B J n K n ˜ F J K ··· J n K n IL ··· L p − i = 0 , (9)respectively.We consider the following ansatz, which we call the IIA or IIB string backgroundconfiguration,Metric: G (¯ τ , X ) = − G ( µ ¯ σ ¯ θ )( µ ¯ σ ¯ θ ) (¯ τ , X ) = G µ µ (cid:0) X (¯ σ ¯ θ ) (cid:1) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) the others = 0 , (10)Scalar field: Φ (¯ τ , X ) = Z d ¯ σd ¯ θ ˆ E δ (¯ σ, ¯ θ )(¯ σ, ¯ θ ) φ ( X (¯ σ, ¯ θ )) (11)5 field: B ( µ ¯ σ ¯ θ )( µ ¯ σ ¯ θ ) (¯ τ , X ) = B µ µ (cid:0) X (¯ σ ¯ θ ) (cid:1) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) the others = 0 , (12)p-form field: A ( µ ¯ σ ¯ θ ) ··· ( µ p ¯ σ p ¯ θ p ) (¯ τ , X ) = A µ ··· µ p (cid:0) X (¯ σ ¯ θ ) (cid:1) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) · · · δ (¯ σ p − ¯ θ p − )(¯ σ p ¯ θ p ) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) , the others = 0 , (13)where A µ ··· µ p = 0 (p: even)˜ F = − ∗ ˜ F , ˜ F = ∗ ˜ F ,A = C ,A = C + B ∧ C , (14)for IIA string background configuration, or A µ ··· µ p = 0 (p: odd)˜ F = ∗ ˜ F , ˜ F = − ∗ ˜ F , ˜ F = ∗ ˜ F ,A = C ,A = C + B C ,A = C + 12 B ∧ C + 12 B ∧ B C , (15)for IIB string background configuration. G µ µ ( x ) is a symmetric tensor field, φ ( x ) is ascalar field, B µ µ ( x ) is an B field and C µ ··· µ p ( x ) are p-form fields on a 10-dimensional6pacetime.We remark that the string background configuration has a non-trivial dependence onthe worldsheet. The consistent truncation will be ensured due to the relation between theworldsheet dependence of the fields and of the indices of the string geometry fields. Forexample, see (¯ σ ¯ θ ) dependence on the string background configuration for the metric. Inaddition, the factor δ (¯ σ ¯ θ )(¯ σ ¯ θ ) reflects that the point particle limit is a field theory.The α ′ → ∼ (9) with the IIA string backgroundconfiguration are equivalent to the equations of motion of the type IIA supergravity S IIA = 12 κ Z d x √ G (cid:18) e − φ (cid:18) R + 4 ∇ µ φ ∇ µ φ − | H | (cid:19) − | ˜ F | − | ˜ F | (cid:19) − Z B ∧ dC ∧ dC ! . (16)The α ′ → ∼ (9) with the IIB string backgroundconfiguration are also equivalent to the equations of motion of the type IIB supergravity, S IIB = 12 κ Z d x √ G (cid:18) e − φ (cid:18) R + 4 ∇ µ φ ∇ µ φ − | H | (cid:19) − | ˜ F | − | ˜ F | − | ˜ F | (cid:19) − Z C ∧ H ∧ dC ! , (17)and the self-dual condition, ˜ F = ∗ ˜ F . Here we display a mechanism how the α ′ → G ( µ ¯ σ ¯ θ )( ν ¯ σ ¯ θ ) with the string background configuration is equivalent to the equation ofmotion of G µν . By substituting the string background configuration, the left hand side ofthe Einstein equation becomes R ( µ ¯ σ ¯ θ )( ν ¯ σ ¯ θ ) − G ( µ ¯ σ ¯ θ )( ν ¯ σ ¯ θ ) R = δ (¯ σ ¯ θ )(¯ σ ¯ θ ) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) (cid:18) R µν (cid:0) X ˆ D T (¯ σ ¯ θ ) (cid:1) − G µν (cid:0) X ˆ D T (¯ σ ¯ θ ) (cid:1) Z d ¯ σd ¯ θ ˆ E (¯ σ ¯ θ ) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) R (cid:0) X ˆ D T (¯ σ ¯ θ ) (cid:1)(cid:19) . (18)7s one can see in this formula, if an equation of motion includes a trace (in R in this case),the reduced equation of motion includes an extra summation R d ¯ σd ¯ θ ˆ E (¯ σ ¯ θ ) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) againstthe equation of motion of the string backgrounds. Fortunately, the terms including theextra summation vanish by using the string background configuration and the equation ofmotion of the scalar, as one can see below. Actually, by substituting the string backgroundconfiguration into the equation of motion of G ( µ ¯ σ ¯ θ )( ν ¯ σ ¯ θ ) , we obtain0 = R ( µ ¯ σ ¯ θ )( ν ¯ σ ¯ θ ) − H ( µ ¯ σ ¯ θ ) I I H I I ( ν ¯ σ ¯ θ ) + 2 ∇ ( µ ¯ σ ¯ θ ) ∇ ( ν ¯ σ ¯ θ ) Φ − G ( µ ¯ σ ¯ θ )( ν ¯ σ ¯ θ ) ( R + 4 ∇ I ∇ I Φ − ∂ I Φ ∂ I Φ − | H | ) − e
2Φ 9 X p =1 (cid:16) p − F ( µ ¯ σ ¯ θ ) I ··· I p − ˜ F I ··· I p − ( ν ¯ σ ¯ θ ) − G ( µ ¯ σ ¯ θ )( ν ¯ σ ¯ θ ) | ˜ F p | (cid:17) = δ (¯ σ ¯ θ )(¯ σ ¯ θ ) δ (¯ σ ¯ θ )(¯ σ ¯ θ ) R µν (cid:0) X (¯ σ, ¯ θ ) (cid:1) − H µµ µ H µ µ ν + 2 ∇ µ ∇ ν φ − e R d ¯ σd ¯ θ ˆ E δ (¯ σ, ¯ θ )(¯ σ, ¯ θ ) φ ( X (¯ σ, ¯ θ )) 9 X p =1 p − F µµ ··· mu p − ˜ F µ ··· µ p − ν − G µν Z d ¯ σd ¯ θ ˆ E δ (¯ σ, ¯ θ )(¯ σ, ¯ θ ) 9 X p =1 | ˜ F p (cid:0) X (¯ σ, ¯ θ ) (cid:1) | ! . (19)In the second equality, we have used the equation of motion of the scalar (7) and theterms in the second line, which includes a part of the extra summation, vanishes. Byusing the common property in the IIA and IIB string background configurations,˜ F − n = ± ∗ ˜ F n we obtain | ˜ F − n | = −| ˜ F n | . (20)If we substitute this relation into (19), the last term, which includes the remaining partof the extra summation, vanishes. Furthermore, If we take α ′ → ,
8e obtain e R d ¯ σd ¯ θ ˆ E δ (¯ σ, ¯ θ )(¯ σ, ¯ θ ) φ ( X (¯ σ, ¯ θ )) → e φ ( x ) , (21)and thus, (19) gets to be equivalent to0 = R µν ( x ) − H µµ µ H µ µ ν + 2 ∇ µ ∇ ν φ − e φ X p =1 p − F µµ ··· mu p − ˜ F µ ··· µ p − ν . (22)This formula is equivalent to the equation of motion of the metric of S ′ = 12 κ Z d x √ G e − φ (cid:18) R + 4 ∇ µ φ ∇ µ φ − | H | (cid:19) − X p =1 | ˜ F p | !! (23)under the equation of motion of the scalar of this action and (20). The same appliesto the other fields. Furthermore, the equations of motion of (23) where the zero modepart of the IIA or IIB string background configurations (14), (15) are substituted areequivalent to the equations of motion of IIA or IIB supergravities (16), (17), as one cansee a proof in an appendix of [10]. Thus, the IIA and IIB supergravities, which possessthe Chern-Simons terms, are derived from the string geometry model, which does notpossess the Chern-Simons term. Therefore, we conclude that the string backgrounds canbe embedded into the string geometry model in the sense of the consistent truncation in α ′ → α ′ → α ′ → . We do not need the special mechanism that the extra summation vanishes as in (19) to derive thesupergravities because the extra summation automatically vanishes in α ′ → . Equations that determine a string back-ground Because the string background configuration (10) ∼ (15) is stationary with respect tothe string geometry time ¯ τ , the energy of it is defined as E = Z D E D X T = Z D E D X ( − ∇ ∇ Φ + 14 H L L H L L + 12 e
2Φ 9 X p =1 p − F I ··· I p − ˜ F I ··· I p − + G ( − ∇ I Φ ∇ I Φ + 2 ∇ I ∇ I Φ − | H | − e
2Φ 9 X p =1 | ˜ F p | ))= Z D X Z D E Z d ¯ σd ¯ θ ˆ E δ (¯ σ, ¯ θ )(¯ σ, ¯ θ ) (2 ∇ µ φ (cid:0) X (¯ σ, ¯ θ ) (cid:1) ∇ µ φ − ∇ µ ∇ µ φ + 14 | H | )= Z D X Z D e Z d ¯ σ ¯ e (¯ σ ) δ ¯ σ ¯ σ (2 ∇ µ φ ( X (¯ σ )) ∇ µ φ − ∇ µ ∇ µ φ + 14 | H | )= Z d x p − G ( x )(2 ∇ µ φ ( x ) ∇ µ φ − ∇ µ ∇ µ φ + 14 | H | ) (24)On the second and third line in the above formula, we have substituted (10) ∼ (15) andobtained the fourth line. On the fourth line, because δ (¯ σ, ¯ θ )(¯ σ, ¯ θ ) ∝ ¯ θ ¯¯ θ , if one integrates ¯ θ and ¯¯ θ , only the bosonic leading terms remain and we obtained the fifth line. On the fifthline, we have regularized the integral over the embedding function as Z D X = N Y j =1 Z d x j q − G ( x j ) Z d x j q − G ( x j ) = 1 Z D e Z d ¯ σ ¯ e (¯ σ ) δ ¯ σ ¯ σ = 1 N N X i =1 , (25)and obtained the sixth line.Therefore, in string geometry theory, a string background is determined by minimizingthe energy (24) of the solutions to the IIA or IIB equations of motions. In other words, inthe IIA case, by using the method of Lagrange multiplier, the equations that determine10tring backgrounds are obtained by differentiating˜ E = E + Z d x p − G ( x )( λ µνG ( x ) f Gµν ( x ) + λ φ ( x ) f φ ( x ) + λ µνB ( x ) f Bµν ( x )+ λ µC ( x ) f C µ ( x ) + λ µ µ µ C ( x ) f C µ µ µ ( x )) (26)with respect to the IIA string backgrounds G µν ( x ), φ ( x ) , B µν ( x ), C ( x ) and C ( x ) andthe Lagrange multipliers λ µνG ( x ), λ φ ( x ), λ µνB ( x ), λ µC ( x ) and λ µ µ µ C ( x ), where f Gµν ( x ) = 0, f φ ( x ) = 0, f Bµν ( x ) = 0, f C µ ( x ) = 0 and f C µ µ µ ( x ) = 0 represent the IIA equations ofmotions, respectively. The same applies to the IIB case.
5. Conclusion and Discussion
In this paper, we have shown that arbitrary configurations of the type IIA and IIB super-string backgrounds are embedded in configurations of fields of a string geometry modelas in (10) ∼ (15). Especially, the single action of the string geometry model is con-sistently truncated to the type IIA and IIB supergravity actions by applying the IIAand IIB string background configurations to the model, respectively and take α ′ → α ′ → ∼ (9) with the IIA and IIBstring background configurations are equivalent to the equations of motion of the typeIIA and IIB supergravities, respectively. This means that classical dynamics of both thetype IIA and IIB string backgrounds are described as a part of classical dynamics instring geometry theory. This fact supports the conjecture that string geometry theory isa non-perturbative formulation of string theory.The above results are consistent with the fact that one can derive both the type IIAand IIB perturbative string theories on the flat background from a single string geometrymodel as shown in [1]: in this case, the configurations of the backgrounds are formally thesame, whereas the charts that cover the backgrounds are different (IIA and IIB charts).These results strongly indicate that string geometry theory does not depend on stringbackgrounds. Here we comment on supersymmetry. Although the single action of theten-dimensional gravity (23) possesses both fields in type IIA and IIB supergravities,namely all the R-R filelds with odd and even degrees, the action cannot be generalized tobe supersymmetric even if fermions are coupled. However, in string geometry theory, anarbitrary action can be generalized to be supersymmetric, as one can see in [1]. Actually,11he string geometry model (2) is supersymmetric, although it possesses the tensor fieldsthat includes all the R-R fields with odd and even degrees.Furthermore, we have defined an energy of the superstring background configura-tions, because they are stationary with respect to the string geometry time ¯ τ . Thus, asuperstring background can be determined by minimizing the energy of the solutions tothe equations of motions of the superstring backgrounds. Therefore, we conclude thatstring geometry theory includes a non-perturbative effect that determines a superstringbackground. Acknowledgements
We would like to thank K. Hashimoto, Y. Hyakutake, Y. Nakayama T. Onogi, S. Sugi-moto, Y. Sugimoto, S. Yamaguchi, and especially H. Kawai and A. Tsuchiya for long andvaluable discussions.
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