Topological String Geometry

Perturbative string amplitudes are correctly derived from the string geometry theory, which is one of the candidates of a non-perturbative formulation of string theory. In order to derive non-perturbative effects rather easily, we formulate topological string geometry theory. We derive the perturbative partition function of the topological string theory from fluctuations around a classical solution in the topological string geometry theory.


Introduction
String geometry theory is one of the candidates of non-perturbative formulation of string theory [1].Actually, the theory possesses appropriate properties as a non-perturbative formulation as follows.First, we can derive the all-order perturbative scattering amplitudes that possess the super moduli in IIA, IIB and SO(32) I superstring theories from the single theory by considering fluctuations around fixed perturbative IIA, IIB and SO(32) I vacua, respectively.Second, the theory is background independent.Third, the theory unifies particles and the space-time.
Next task is to derive non-perturbative effects from the theory.In order to derive nonperturbative effects rather easily, we formulate a topological string geometry theory in this paper.It is worthy to study the topological string geometry theory, since non-perturbative partition functions of the topological string theory on a certain class of the Calabi-Yau manifolds are proposed as we will explain below.
Non-perturbative partition functions in topological string theory on non-compact toric Calabi-Yau manifolds were conjectured by using dualities [2,3].In [3], a non-perturbative free energy is given by the combination of the unrefined free energy and Nekrasov-Shatashvili limit of the refined free energy [4].The authors in [5] proposed that the non-perturbative free energy in [3] closely relates to the quantization of the mirror curve [6,7], based on the duality between ABJM matrix model and the topological string theory on local P 1 × P 1 [8,9,10].This relation is sometimes called as Topological String/Spectral Theory correspondence.
(The overview of the correspondence is given in e.g.[11]).In addition, the authors in [12] show that the non-perturbative free energy in [3] agrees with that obtained by applying the resurgence technique, developed in [13,14], to the perturbative topological string theory in case of local P 2 .In spite of such progresses, the first principle calculations of the nonperturbative partition functions are still not known.We expect to provide an answer to this issue from topological string geometry theory.
The rest of the paper is organized as follows.In section 2, we perform a topological twist of string geometry theory and define topological string geometry theory.In section 3, we derive perturbative topological string theory in the flat background by considering fluctuations around a classical solution of the topological string geometry theory.In Appendix A, we develop a superfield formalism of the topological string since the string geometry theory is defined in a superfield formalism.

Topological string geometry
In this section, we will define topological string geometry by performing a topological twist of string geometry [1].
As in [1], on the topological super Riemannian surfaces Σ, there exists an unique Abelian differential dp that has simple poles with residues f i at P i where i f i = 0, if it is normalized to have purely imaginary periods with respect to all contours to fix ambiguity of adding holomorphic differentials.A global time is defined by z = τ + iσ := P dp at any point P on Σ [15,16].τ takes the same value at the same point even if different contours are chosen in P dp, because the real parts of the periods are zero by definition of the normalization.
In particular, τ = −∞ at P i with negative f i and τ = ∞ at P i with positive f i .A contour integral on τ constant line around P i : i∆σ = dp = 2πif i indicates that the σ region around P i is 2πf i .This means that Σ around P i represents a semi-infinite supercylinder with radius f i .The condition i f i = 0 means that the total σ region of incoming supercylinders equals to that of outgoing ones if we choose the outgoing direction as positive.That is, the total σ region is conserved.In order to define the above global time uniquely, we fix the σ regions 2πf i around P i .We divide N P i 's to arbitrary two sets consist of N − and N + P i 's, respectively (N − + N + = N ), then we divide equally −1 to f i = −1 N − , and 1 to f i = 1 N + .Thus, under a superconformal transformation, one obtains a topological super Riemann surface Σ that has coordinates composed of the global time τ and the position σ.Because Σ can be a supermoduli of super Riemann surfaces [17], any two-dimensional topological super Riemannian manifold Σ can be obtained by Σ = ψ( Σ) where ψ is a superdiffeomorphism times super Weyl transformation.
Next, we will define a model space E. We consider a state ( Σ, Φ T (τ s ), ΦT (τ s ), τs ) determined by Σ, a τ = τs constant hypersurface and an arbitrary map (Φ T (τ s ), ΦT (τ s )) from Σ| τs to the Euclidean space R d .Φ T (τ ) and ΦT (τ ) are defined by restricting (A.2) to the τ constant hypersurface whose conditions are given by hτm ∂ m φ + 2iχ z * z ρ z * = 0, hτm ∂ m φ + 2iχ z z * ρ z = 0, ( for T = A and hτm ∂ m φ = 0, hτm ∂ m φ + 2iχ z * z ρ z * + 2iχ z z * ρ z = 0, ( for T = B where hmn is the metric of the worldsheet, and χ z * z and χ z z * is the gravitino of the worldsheet. Σ is a union of N ± supercylinders with radii f i at τ ±∞.Thus, we define a string state as an equivalence class [ Σ, Φ T (τ s ±∞), ΦT (τ s ±∞), τs ±∞] by a relation ( Σ, Φ T (τ s ±∞), ΦT (τ s ±∞), τs ±∞) ∼ ( Σ , Φ T (τ s ±∞), Φ T (τ s ±∞), τs ±∞) , and ΦT (τ s ±∞) = Φ T (τ s ±∞) as in Fig. 1.Because Σ| τs S 1 ∪ S 1 ∪ • • • ∪ S 1 where Σ is the reduced space of Σ, and (Φ T (τ s ), ΦT (τ s )) : Σ| τs → M , [ Σ, Φ T (τ s ), ΦT (τ s ), τs ] represent many-body states of strings in R d as in Fig. 2. The model space E is defined by a collection of all the string states as Here, we will define topologies of where H q e d X J u + 0 x 3 V 9 o P X 3 9 + K y T W U v H m 3 P s m r 2 Q / y v W Z g / 0 A 7 P + q t 2 k e P r 8 D z 8 q e W l R e + T v z f g J c o s L M u H U U i y x 6 z c q i B n M Y p 6 6 s Y I E t p F E l q r b O M U F L q W w t C y t S x u 9 V C n g a 6 b w J a S t D 5 c D l K Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " + N I 2 r 4 i k y s A r q g l f H q e d X J u + 0 x 3 V 9 o P X 3 9 + K y T W U v H m 3 P s m r 2 Q / y v W Z g / 0 A 7 P + q t 2 k e P r 8 D z 8 q e W l R e + T v z f g J c o s L M u H U U i y x 6 z c q i B n M Y p 6 6 s Y I E t p F E l q r b O M U F L q W w t C y t S x u 9 V C n g a 6 b w J a S t D 5 c D l K Y = < / l a t e x i t >

⌧2
< l a t e x i t s h a 1 _ b a s e 6 4 = " D g j g U C s 0 B y M y F 6 6 4 m R E L d 8 q J / T w = " , and is small enough, because the τs ±∞ constant hypersurface traverses only supercylinders overlapped by Σ and Σ .
U is defined to be an open set of E if there exists such that U ([ Σ, Φ T (τ s ), ΦT (τ s ), τs ], ) ⊂ U for an arbitrary point [ Σ, Φ T (τ s ), ΦT (τ s ), τs ] ∈ U .The topology of E satisfies the axiom of topology.The proof is the same as in [1].
Although the model space is defined by using the coordinates [ Σ, Φ T (τ s ), ΦT (τ s ), τs ], the model space does not depend on the coordinates, because the model space is a topological space.does not transform to σ and θ and vice versa, because the string states are defined by τ constant hypersurfaces.Under these restrictions, the most general coordinate transformation is given by
Riemannian topological string manifold is obtained by defining a metric, which is a section of an inner product on the tangent space.The general form of a metric is given by (2.6) We summarize the vectors as dΦ I T (τ ) (I = d, (I σ θ), ( J σ θ)), where dΦ d T (τ ) := dτ , dΦ Then, the components of the metric are summa-rized as G IJ ( Ē, Φ T (τ ), ΦT (τ ), τ ).The inverse of the metric G IJ ( Ē, Φ T (τ ), ΦT (τ ), τ ) is defined by The components of the Riemannian curvature tensor are given by R I JKL in the basis The volume is √ G, where G = det(G IJ ).
By using these geometrical objects, we formulate topological string theory non-perturbatively as 45 where As an example of sets of fields on the topological string manifolds, we consider the metric and an u(1) gauge field A I whose field strength is given by F IJ .The path integral is canonically defined by summing over metrics and gauge fields on M. By definition, the theory is Under G IJ ( Ē, Φ T (τ ), ΦT (τ ), τ ) and A I ( Ē, Φ T (τ ), ΦT (τ ), τ ) are transformed as a symmetric tensor and a vector, respectively and the action is manifestly invariant.
3 Perturbative topological string from topological string geometry In this section, from the topological string geometry theory, we will derive the partition function of the A model for topological strings in all-order string coupling constant.The partition function of the B model can be derived in the same way.
The background that represents a perturbative vacuum for the A model is given by where we fix charts by choosing T = A on E. ρ( h) := 1 4π dσē Rh, where Rh is the scalar curvature of hmn .D is a volume of the index (I σ θ) and ( J σ θ): D := dσd 4 θ Êδ Then, v(Φ A (τ ), ΦA (τ )) satisfies where we have used (2.1), and e σ z is the vierbein.The inverse of the metric is given by because From the metric, we obtain By using these quantities, one can show that the background (3.1) is a classical solution6 to the equations of motion of (2.8).We also need to use the fact that v(Φ A (τ ), ΦA (τ )) is a harmonic function with respect to Φ A (τ ) and ΦA (τ ), η I J ∂ (I σ θ) ∂ ( J σ θ) v = 0.In these calculations, we should note that Ē A M , Φ I A (τ ), Φ Let us consider fluctuations around the background (3.1), G IJ = ḠIJ + GIJ and A I = ĀI + ÃI .The action (2.8) up to the quadratic order is given by, where There is no first order term because the background satisfies the equations of motion.If we take G N → 0, we obtain where the fluctuation of the gauge field is suppressed.In order to fix the gauge symmetry (2.9), we take the harmonic gauge.If we add the gauge fixing term we obtain (3.9) In order to obtain perturbative topological string amplitudes, we perform a derivative and take α → 0, (3.11)where α is an arbitrary constant in the solution (3.1).
We normalize the fields as HIJ := Z IJ GIJ , where . āI represent the background metric as ḠIJ = āI δ IJ , where ād = 2λρ and ā(Iσ θ) = ē2 Ê √ h .Then, (3.9) reduces to where and In the same way as in [1], a part of the action with decouples from the other modes.
In the following, we consider a sector that consists of local fluctuations in a sense of strings as H KL = dσd 4 θ Êf KL ( Ē, Φ A (τ ), ΦA (τ ), τ ). (3.17) In the same way as in [1], we have because the leading term of Φ I A is φ I and covariant derivatives with respect to σ apply to all the other terms including φ I in Φ where n and nσ are components of h in the ADM formalism, for example summarized in [1], we obtain (3.15) with where we have taken D → ∞.
Because (3.23) means that ∆ F is an inverse of H, ∆ F can be expressed by a matrix element of the operator Ĥ−1 as In order to define two-point correlation functions that is invariant under the general coordinate transformations in the topological string geometry, we define in and out states as where and E i and E f represent the topological super vierbein of the super cylinders at τ = ∓∞, respectively.When we insert asymptotic states, we integrate out Φ Af , ΦAf , Φ Ai , ΦAi , E f , and E i in the two-point correlation function for these states, (3.28) In the same way as in [1], by inserting completeness relations of the eigen states, we obtain If we integrate out p τ , p φ , p φ, p F , and p F by using the relation of the ADM formalism, we obtain , where b(t) and c(t) are bc ghosts, we obtain where we have redefined as c(t) → T (t)c(t).Z 0 represents an overall constant factor, and we will rename it Z 1 , Z 2 , • • • when the factor changes in the following.This path integral is obtained if which has a manifest one-dimensional diffeomorphism symmetry with respect to t, where T (t) is transformed as an einbein [20].
Under dτ dτ = T (t), T (t) disappears in (3.33) as in [1], and we obtain where Φ A and Φ A are given by replacing ∂ ∂ τ with ∂ ∂t in Φ A and ΦA , respectively.This action is still invariant under the diffeomorphism with respect to t if τ transforms in the same way as t.
If we choose a different gauge In the second equality, we have redefined as c(t)(1 − dτ (t) dt ) → c(t) and integrated out the ghosts.The path integral is defined over all possible two-dimensional topological super Riemannian manifolds with fixed punctures in R d as in Fig. 4. By using the two-dimensional where χ is the Euler number of the two-dimensional Riemannian manifold.This is the all order perturbative partition function of the A model for topological strings in the flat target space itself [26,17,27,28].

Conclusion and discussion
In this paper, we first defined topological string geometry theory by twisting string geometry theory.From the single theory, we derived both the A and B models of the topological strings in the flat target space, by considering fluctuations around the type A and B string manifolds covered by the A and B charts, respectively.This fact implies that we see the mirror symmetry in topological string geometry theory perturbatively.
The string coupling constant g is not a parameter of non-perturbatively formulated string theory, but an expectation value of the dilaton.Actually, g is not a parameter of the string geometry theory, too.g = e λ is a free parameter of the background solution (3.1) in string geometry theory when we derive the partition function of the perturbative topological string.
During the derivation, we take a limit of another free parameter of the solution: α → 0. Therefore, it is natural to identify α as e − 1 g .We expect to obtain right non-perturbative corrections to the partition function by evaluating the contribution of the α expansions to the propagator in the string geometry theory, corresponding to the perturbative partition function.
Concretely, we can calculate the non-perturbative corrections [21], as follows.In [22], perturbative string theories are derived in Newtonian limits of string geometry theory including arbitrary fields.The post Newtonian expansion is the α expansion and can be identified with the e − 1 g expansion.Thus, by moving to the first quantization formalism in topological string geometry theory as in this paper, we obtain e − n g corrections to the perturbative g m contributions of the partition function.This behaviour is consistent with the results of the resurgence on the asymptotic expansion in the string coupling of the partition function in the topological string theory.That is, the summation over m and n of the e − n g corrections to the g m contributions of the partition function equals to the summation over n and m of the perturbative g m expansion in the n-th instanton background, whose factor is given by e − n g .We can calculate these corrections by using the localization in the first quantization formalism of topological string geometry theory, and compare them with the non-perturbative partition function which are conjectured by using dualities in [2,3] and are coincident with the results of the resurgence in [23,24] .We also calculate them in a geometric approach as follows.We obtain corrections to the condition that curves are holomorphic as a result of the above localization.Then, we obtain corrections to the definition of the Gromov-Witten invariants and calculate corrections to Gromov-Witten potential, which corresponds to the partition function at the zero-th order in the genus expansion.We compare them with the result in Physics.In the case that the target space is C 3 /Z 2 , we can calculate the corrections because C 3 /Z 2 can be treated as an orbifold of C 3 in topological string geometry theory 7 , whereas it can be treated as a limit of a blow-up geometry in the conjecture, in the result of the resurgence, and in symplectic geometry.
Here, we discuss how to derive perturbative string theories on more general backgrounds.
First, the perturbative vacuum solution (3.1) is a solution even if v(Φ A (τ ), ΦA (τ )) is generalized to arbitrary holomorphic plus anti-holomorphic functions.It is an immediate task to clarify on which Kähler manifolds topological string theories are reproduced from the fluctuations around the generalized backgrounds.Second, M. Honda and M. S. found that general configurations of the fields of a supergravity are included in configurations of the fields of the string geometry in [25].Similarly, we expect that general Kähler manifolds are included in configurations of the fields of the topological string geometry and that we can reproduce topological string theories on various compact and non-compact Calabi-Yau manifolds.
Because string backgrounds are included in configurations of the fields of the string geometry [25], we expect that instantons of the string geometry reduce to instantons of the string backgrounds and instanton effects of string geometry give non-perturbative effects in string theory where a string background changes to another.which are the actions of the A and B models of topological strings [17,26,27,28], respectively.
Therefore, the new superfields E, Φ I T and ΦĪ T define superfield formalisms of the topological string theories.

Figure 1 :
Figure 1: An equivalence class of a string state.If the supercylinders and the embedding functions are the same at τ −∞, the states of strings at τ −∞ specified by the red lines s A m S G U p w = = < / l a t e x i t > ⌧3 < l a t e x i t s h a 1 _ b a s e 6 4 = " e yx W Q / W b q S 5 R F i k Z D B 3 + x s G e n f U = " > A A A C j X i c h V G 7 S g N B F D 1 Z 3 / G R q I 1 g I w b F S m Z 9 o I i I Y K F l Y o y J G A m 7 6 y Q u 2 Re 7 s 4 G 4 5 A f 8 A C 0 s f I C F 2 P s D N v 6 A h Z 8 g l g o 2 F t 5 s F k R F v c P M n D l z z 5 0 z X N U x d E 8 w 9 h i T 2 t o 7 O r u 6 e + K 9 f f 0 D i e T g 0 L Z n + 6 7 G c 5 p t 2 G 5 B V T x u 6 B b P l p P U P m z + U q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e yx W Q / W b q S 5 R F i k Z D B 3 + x s G e n f U = " > A A A C j X i c h V G 7 S g N B F D 1 Z 3 / G R q I 1 g I w b F S m Z 9 o I i I Y K F l Y o y J G A m 7 6 y Q u 2 Re 7 s 4 G 4 5 A f 8 A C 0 s f I C F 2 P s D N v 6 A h Z 8 g l g o 2 F t 5 s F k R F v c P M n D l z z 5 0 z X N U x d E 8 w 9 h i T 2 t o 7 O r u 6 e + K 9 f f 0 D i e T g 0 L Z n + 6 7 G c 5 p t 2 G 5 B V T x u 6 B b P l p P U P m z + U q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e yx W Q / W b q S 5 R F i k Z D B 3 + x s G e n f U = " > A A A C j X i c h V G 7 S g N B F D 1 Z 3 / G R q I 1 g I w b F S m Z 9 o I i I Y K F l Y o y J G A m 7 6 y Q u 2 Re 7 s 4 G 4 5 A f 8 A C 0 s f I C F 2 P s D N v 6 A h Z 8 g l g o 2 F t 5 s F k R F v c P M n D l z z 5 0 z X N U x d E 8 w 9 h i T 2 t o 7 O r u 6 e + K 9 f f 0 D i e T g 0 L Z n + 6 7 G c 5 p t 2 G 5 B V T x u 6 B b P l p P U P m z + U q A = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " e yx W Q / W b q S 5 R F i k Z D B 3 + x s G e n f U = " > A A A C j X i c h V G 7 S g N B F D 1 Z 3 / G R q I 1 g I w b F S m Z 9 o I i I Y K F l Y o y J G A m 7 6 y Q u 2 Re 7 s 4 G 4 5 A f 8 A C 0 s f I C F 2 P s D N v 6 A h Z 8 g l g o 2 F t 5 s F k R F v c P M n D l z z 5 0 z X N U x d E 8 w 9 h i T 2 t o 7 O r u 6 e + K 9 f f 0 D i e T g 0 L Z n + 6 7 G c 5 p t 2 G 5 B V T x u 6 B b P r X c 3 k 0 c j a 2 / / a s y a P e w / 6 n 6 0 7 O H G u Z D r 4 K 8 2 y E T / E J r 6 5 u H J y / r i / l J f 4 p d s m f y f 8 E e 2 R 3 9 w G y + a l c 5 n j / 9 w 4 9 K X l r U H v l 7 M 3 6 C z Z m M T D g 3 m 1 4 r X c 3 k 0 c j a 2 / / a s y a P e w / 6 n 6 0 7 O H G u Z D r 4 K 8 2 y E T / E J r 6 5 u H J y / r i / l J f 4 p d s m f y f 8 E e 2 R 3 9 w G y + a l c 5 n j / 9 w 4 9 K X l r U H v l 7 M 3 6 C z Z m M T D g 3 m 1 4 r X c 3 k 0 c j a 2 / / a s y a P e w / 6 n 6 0 7 O H G u Z D r 4 K 8 2 y E T / E J r 6 5 u H J y / r i / l J f 4 p d s m f y f 8 E e 2 R 3 9 w G y + a l c 5 n j / 9 w 4 9 K X l r U H v l 7 M 3 6 C z Z m M T D g 3 m 1 4 r X c 3 k 0 c j a 2 / / a s y a P e w / 6 n 6 0 7 O H G u Z D r 4 K 8 2 y E T / E J r 6 5 u H J y / r i / l J f 4 p d s m f y f 8 E e 2 R 3 9 w G y + a l c 5 n j / 9 w 4 9 K X l r U H v l 7 M 3 6 C z Z m M T D g 3 m 1 4

Figure 2 :
Figure 2: Various string states.The red and blue lines represent one-string and two-string states, respectively.
-open neighbourhood, arbitrary two string states on a connected topological super Riemann surface in R d are connected continuously.Thus, there is an one-to-one correspondence between a topological super Riemann surface with punctures 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 the topological super Riemann surfaces in R d .By a general curve parametrized by t on E, string states on different topological super Riemann surfaces that have even different numbers of genera, can be connected continuously, as in Fig. 3, whereas different topological super Riemann surfaces that have different numbers of genera cannot be connected continuously in the moduli space of the topological super Riemann surfaces.
background independent.DE is the invariant measure of the super vierbeins E A M on the two-dimensional topological super Riemannian manifolds Σ. E A M and Ē A M are related to each others by the super diffeomorphism and super Weyl transformations.

JA
(τ ), and τ are all independent.Because the equations of motion are differential equations with respect to Φ I A (τ ), Φ J A (τ ) and τ , Ē A M is a constant in the solution (3.1) to the differential equations.The dependence of Ē A M on the background (3.1) is uniquely determined by the consistency of the quantum theory of the fluctuations around the background.Actually, we will find that all the perturbative topological string amplitudes are derived.

Figure 4 :
Figure 4: A path and a topological super Riemann surface.The line on the left is a trajectory in the path integral.The trajectory parametrized by τ from τ = −∞ to τ = ∞, represents a topological super Riemann surface with fixed punctures in R d on the right.