Path integrals of perturbative strings on curved backgrounds from string geometry theory

String geometry theory is one of the candidates of the non-perturbative formulation of string theory. In this paper, from the closed bosonic sector of string geometry theory, we derive path integrals of all order perturbative strings on all the string backgrounds, $G_{\mu\nu}(x)$, $B_{\mu\nu}(x)$, and $\Phi (x)$, by considering fluctuations around the string background configurations, which are parametrized by the string backgrounds.


Introduction
String geometry theory is one of the candidates of non-perturbative formulation of string theory. It is formulated by a path integral of string manifolds, which belong to a class of infinite-dimensional manifolds, string geometry [1]. String manifolds are defined by patching open sets of the model space defined by introducing a topology to a set of strings. One of the remarkable facts concerning string geometry theory is that the path integral of perturbative superstrings on the flat background is derived including the moduli of super Riemann surfaces, by considering fluctuations around the flat background in the theory [1,2,3].
Moreover, configurations of fields in string geometry theory include all configurations of fields in the ten-dimensional supergravities, namely string backgrounds [4,5]. Especially, it is shown that an infinite number of equations of motion of string geometry theory are consistently truncated to finite numbers of equations of motion of the supergravities. That is, string geometry theory includes string backgrounds not as external fields like the perturbative string theories. Dynamics of string backgrounds are a part of dynamics of the fields in the theory. It is natural to expect to derive the path integral of perturbative strings on the sting backgrounds by considering fluctuations around the corresponding configurations in string geometry theory. Furthermore, a string background that minimizes the energy of the string background configurations, will be chosen spontaneously, because string geometry theory is formulated non-perturbatively [4,5].
For each background, one theory is formulated in case of a perturbative string theory, whereas perturbative string theories not only on the flat background but also on non-trivial backgrounds should be derived from a single theory in case of the non-perturbative formulation of string theory. In this paper, from the closed bosonic sector of string geometry theory, we derive the path integrals of perturbative strings on all the string backgrounds G µν (x), B µν (x), and Φ(x).
The organization of the paper is as follows. In section 2, we briefly review the closed bosonic sector in string geometry theory. In section 3, we set string background configurations parametrized by the string backgrounds G µν (x), B µν (x), and Φ(x), and set the classical part of fluctuations representing strings. In section 4, we consider two-point correlation functions of the quantum part of the fluctuations and derive the path integrals of the perturbative strings on the string backgrounds. In section 5, we conclude and discuss our results. In the appendix, we obtain a Green function on the flat string manifold.
2 Review of closed bosonic sector in string geometry theory In this paper, we discuss only the closed bosonic sector of string geometry theory. One can generalize the result in this paper to the full string geometry theory in the same way as in [1]. The closed bosonic sector [4,5] is described by a partition function where the action is given by The path integral is defined by integrating a metric G IJ , a scalar φ, and a two-form B IJ defined on an infinite dimensional manifold, so-called string manifold. String manifold is constructed by patching open sets in string model space E, whose definition is summarized as follows. First, a global timeτ is defined canonically and uniquely on a Riemann surfaceΣ by the real part of the integral of an Abelian differential uniquely defined onΣ [6,7]. We restrictΣ to aτ constant line and obtainΣ|τ . An embedding ofΣ|τ to R d represents a many-body state of strings in R d , and is parametrized by coordinates (h, X(τ ),τ ) 1 whereh is a metric onΣ and X(τ ) is a map fromΣ|τ to R d . String model space E is defined by the collection of the string states by considering all theΣ, all the values of τ , and all the X(τ ). How near the two string states is defined by how near the values ofτ and how near X(τ ).h is a discrete variable in the topology of string geometry, where an As a result, dh cannot be a part of basis that span the cotangent space in (2.4), whereas fields are functionals ofh as in (2.5). The precise definition of the string topology is given in the 1 "¯" represents a representative of the diffeomorphism and Weyl transformations on the worldsheet. Giving a Riemann surfaceΣ is equivalent to giving a metrich up to diffeomorphism and Weyl transformations. section 2 in [1]. By this definition, arbitrary two string states on a connected Riemann surface in E are connected continuously. Thus, there is an one-to-one correspondence between a Riemann surface in R d and a curve parametrized byτ from −∞ to ∞ on E. That is, curves that represent asymptotic processes on E reproduce the right moduli space of the Riemann surfaces in R d . Therefore, a string geometry model possesses all-order information of the perturbative string theory. Indeed, the path integral of perturbative strings on the flat spacetime is derived from the string geometry theory as in [1,3]. We use the Einstein notation for the index I, where I = {d, (µσ)}. The cotangent space is spanned by for µ = 0, 1, . . . , d − 1, while dh mn with m, n =τ ,σ cannot be a part of the basis becausē h mn is treated as a discrete valuable in the string topology. The summation overσ is defined by dσē(σ,τ ), whereē := hσσ . This summation is transformed as a scalar under τ →τ (τ , X(τ )), and invariant underσ →σ (σ).
From these definitions, we can write down the general form of the metric of the string geometry as follows.

String background configurations and fluctuations representing strings
In this paper, we consider only static configurations, including quantum fluctuations: In this section, we will set classical backgrounds including string backgrounds and consider fluctuations that represent strings around them. The Einstein equation of the action (2.2) is given byR whereR,R M N ,R M N P Q , and∇ M denote the Ricci scalar, Ricci tensor, curvature tensor and covariant derivative constructed from the metricḠ M N , respectively. We consider a perturbation with respect to the metricḠ M N : whereh M N denotes a fluctuation around the 0-th order backgroundĜ M N .We raise and lower the indices byĜ M N in the following. We also consider a perturbation with respect to the 2-formB M N and the scalarφ around the 0-th order backgrounds 0.
First, we generalize the harmonic gauge to the one when we have the dilaton. If we definē We impose a generalization of the harmonic gauge: which reduces to the ordinary harmonic gauge if the dilaton is zero. Then, the Einstein Next, we set the 0-th order backgroundĜ M N as a flat background: where a d = 1 and a (µσ) =ē . Then, the gauge fixing condition (3.7) becomes and the components of (3.6) read Next, the equation of motion of the scalar of the action (2.2) is written asR up to the first order in the fields,h IJ ,B IJ , andφ. Furthermore, this can be written as which is written as We consider classical backgrounds corresponding to the string background configurations: where g µν (x) and B µν (x) satisfy gauge fixing conditions, which imply (3.10) and (3.21). Indeed, these are equivalent tō These are the string background configurations themselves [4,5]. If we impose that g µν (x), B µν (x) and Φ(x) satisfy Laplace equations, Thus, we also consider the scalar fluctuation ψ dd around the general perturbative vacua. We set the classical part of ψ dd as where Rh is the scalar curvature of the two-dimensional metrich mn and G(X; X ) is a Green function on the flat string manifold given by G(X; X ) = δ(X − X ), (3.40) where N is a normalizing constant. A derivation is given in the appendix. As a result,ψ dd is not on-shell but satisfies Furthermore, we consider the quantum part of ψ dd , (3.44) 4 Deriving the path integrals of the perturbative strings on curved backgrounds In this section, we will derive the path integrals of the perturbative strings up to any order from the tree-level two-point correlation functions of the quantum scalar fluctuations of the metric. In order to obtain a propagator, we add a gauge fixing term corresponding to (3.7) into the action (2.2) and obtain This is expressed as up to the first order in the classical fields and the second order inφ. Here, we take the regularization parameter D → ∞. Then, (4.2) becomes By shifting the fieldφ asφ =φ − 2 3 , the first order term inφ vanishes as where surface terms are dropped and the gauge fixing condition in (3.28) and a relation (3.4) are applied. By normalizing the leading part of the kinetic term asφ = 2(1 −ψ dd −

4Ĝ
IJh IJ )φ , we have This can be written as where Here we have added terms which is true because of the gauge fixing condition (3.28).
The propagator forφ defined by In order to obtain a Schwinger representation of the propagator, we use the operator formalism (ĥ,X(τ )) of the first quantization, whereas the conjugate momentum is written as (ph,p X (τ )). The eigen state is given by |h, X(τ ) >.
Since (4.11) means that ∆ F is an inverse of H, ∆ F can be expressed by a matrix element of the operatorĤ −1 as This fact and (4.12) imply In order to define two-point correlation functions that is invariant under the general coordinate transformations in the string geometry, we define in and out states as integrate out X f , X i , h f and h i in the two-point correlation function for these states; The invariant measure is defined implicitly by the most general invariant norm without derivatives for elements δh mn of the tangent space of the metric, ||δh|| 2 = d 2 σ √ h(h mp h nq + Ch mn h pq )δh mn δh pq with C an arbitrary constant, and a normalization Dδh exp − 1 2 ||δh|| 2 = 1.
This can be written as in [1] 4 , By integrating out p X (τ (t), t), we move from the canonical formalism to the Lagrange formalism. Because the exponent of (4.18) is at most the second order in p X (τ (t), t), in-tegrating out p X (τ (t), t) is equivalent to substituting into (4.18), the solution p X (τ (t), t) which is obtained by differentiating the exponent of (4.18) with respect to p X (τ (t), t). The solution is given by , (4.20) up to the first order in the classical backgrounds g µν (X) and B µν (X). By substituting this, we obtain +h 01 ∂ t X µ (τ (t), t)∂σX ν (τ (t), t) + 1 2h 11 T (t)∂σX µ (τ (t), t)∂σX ν (τ (t), t)) where we use (3.41) and the ADM decomposition of the two-dimensional metric, In this way, the Green function can generate all the terms withoutτ derivatives in the string action as in (3.41), but cannot do those withτ derivatives, which need to be derived non-trivially, because the coordinates X µ (τ ) in string geometry theory are defined on theτ constant lines. We should note that the time derivative in (4.21) is in terms of t, notτ at this moment. In the following, we will see that t can be fixed toτ by using a reparametrization of t that parametrizes a trajectory.
By inserting DcDbe , where b(t) and c(t) are bc-ghost, we obtain where we redefine as c(t) → T (t)c(t), and Z 0 represents an overall constant factor. In the following, we will rename it Z 1 , Z 2 , · · · when the factor changes. The integrand variable p T (t) plays the role of the Lagrange multiplier providing the following condition, which can be understood as a gauge fixing condition. Indeed, by choosing this gauge in we obtain (4.23). The expression (4.25) has a manifest one-dimensional diffeomorphism symmetry with respect to t, where T (t) is transformed as an einbein [8].

Conclusion and Discussion
In this paper, in the closed bosonic sector of string geometry theory, we fix the classical part of the scalar fluctuation of the metric around the string background configurations, which are parametrized by the string backgrounds, G µν (x), B µν (x), and Φ(x). We showed that the two-point correlation functions of the quantum parts of the scalar fluctuation are path integrals of the perturbative strings on the string backgrounds. In this derivation, we move from the second quantization formalism to the first one, where the coordinates of the two fields in the correlation functions become the asymptotic fields that represent the initial state X µ (τ = −∞, σ) and the final state X µ (τ = ∞, σ), respectively. All the paths on the string manifolds from X µ (τ = −∞, σ) to X µ (τ = ∞, σ) are summed up in the first quantization representation of the two-point correlation functions. Because the paths on the string manifolds are world-sheets with genera as shown in the section two in [1], they reproduce the path integrals of the perturbative strings up to any order, although the correlation functions are at tree level.
Next task is a supersymmetric generalization of our result. It is known to be too difficult to describe the action of the perturbative strings on the R-R backgrounds in the NS-R formalism. Because string geometry theory is formulated in the NS-R formalism, we should derive the path integrals of the perturbative strings on the NS-NS backgrounds.