All-order Resurgence from Complexified Path Integral in a Quantum Mechanical System with Integrability

We discuss all-order transseries in one of the simplest quantum mechanical systems: a U(1) symmetric single-degree-of-freedom system with a first-order time derivative term. Following the procedure of the Lefschetz thimble method, we explicitly evaluate the path integral for the generating function of the Noether charge and derive its exact transseries expression. Using the conservation law, we find all the complex saddle points of the action, which are responsible for the non-perturbative effects and the resurgence structure of the model. The all-order power-series contributions around each saddle point are generated from the one-loop determinant with the help of the differential equations obeyed by the generating function. The transseries are constructed by summing up the contributions from all the relevant saddle points, which we identify by determining the intersection numbers between the dual thimbles and the original path integration contour. We confirm that the Borel ambiguities of the perturbation series are canceled by the non-perturbative ambiguities originating from the discontinuous jumps of the intersection numbers. The transseries computed in the path-integral formalism agrees with the exact generating function, whose explicit form can be obtained in the operator formalism thanks to the integrable nature of the model. This agreement indicates the non-perturbative completeness of the transseries obtained by the semi-classical expansion of the path integral based on the Lefschetz thimble method.


Conclusions and discussion 30
Acknowledgments 31 A. Lefschetz thimble method 31 B. Differential equation for generating function 33

INTRODUCTION
Path integral formalism is one of the fundamental tools to formulate quantum systems. It is based on integration over infinite-dimensional functional spaces of fields. Although it is a general and intuitive formulation, path integrals can rarely be evaluated exactly. One can use the perturbative expansion to approximate a path integral as a power series of a coupling constant. Although such a perturbation series gives a good approximation when the coupling constant is small, it is usually an asymptotic series with a zero radius of convergence. Hence, the perturbation series truncated at a finite order has limited accuracy, particularly for a large expansion parameter. A possible prescription for such a divergent series is Borel resummation. It applies to asymptotic series with factorially divergent expansion coefficients and gives a closed form for the series if it is Borel summable, i.e. its Borel transform is non-singular on the positive real axis of the Borel plane. However, in many physical systems, perturbation series are non-Borel summable and associated with ambiguities depending on the regularization.
The remedy for such an ill-defined series is to construct the so-called transseries by appropriately summing up the contributions of saddle points, that is, classical solutions of the action. According to the resurgence theory [1] (see e.g. [2][3][4][5][6] for reviews on the application of the resurgence theory to field theories), all the ambiguities from the perturbative and non-perturbative sectors cancel in the transseries. As in the steepest descent (stationary phase) method for ordinary finite-dimensional integrals, we have to take into account complex saddle points which can be found by analytically continuing the action as a holomorphic functional of the fields (see e.g. [7][8][9][10][11][12][13][14] for examples of systems where complex saddle points called "bions" play important roles). Not all the saddle points are relevant, but a specific subset can contribute. Such a subset can be determined by the Lefschetz thimble method, which states that the relevant saddle points are those with steepest-ascent flows (dual thimbles) intersecting with the original path integration contour (the original configuration space). The contribution from each saddle point is given by the integral over the associated thimble (steepest descent flows). Its ambiguity is related to a Stokes phenomenon, a sudden change in the shape of the (dual) thimble, which occurs when the argument of the coupling constant is varied. Although the contribution from each saddle point can be ambiguous due to such a Stokes phenomenon, the transseries constructed through the Lefschetz thimble method is unambiguous thanks to the cancellation mechanism of the ambiguities. Such a resurgence structure enables us to find a well-defined closed form with the correct asymptotic expansion. The procedure for evaluating path integrals based on the Lefschetz thimble method can be summarized as follows: 1. The first step is to find saddle points by solving the complexified equations of motion derived from the classical action analytically continued to the complexified configuration space.
tween the original configuration space and the dual thimbles. The transseries can be constructed by summing up the contributions of all the saddle points using the intersection numbers as coefficients.
The question is whether such a transseries is exact or not. It would not be so difficult to see that the Lefschetz thimble method gives exact results for finite-dimensional integrals [7]. On the other hand, path integrals cannot be explicitly evaluated in almost all cases, and hence there is less chance to check the exactness of the transseries. If there exists a model with the following properties, it can serve as a testing ground for the Lefschetz thimble method: 1. All saddle point solutions can be found by solving the complexified equations of motion.
2. Expansion coefficients around each saddle point can be determined to all orders.
3. All intersection numbers can be determined.

Exact results can be obtained via another method.
It is natural to imagine that the set of the properties described above implies integrability. Resurgence structure of integrable fields theories has been discussed in [15][16][17][18][19][20][21][22][23] and it has been shown that integrability is a powerful tool for studying resurgence structure. In this paper, we focus on the case of quantum mechanics that is exactly solvable due to integrability. An integrable quantum mechanical system is a model with a finite number of degrees of freedom possessing a maximal set of commuting conserved charges. In such a system, Hamiltonian can be written by using the action-angle variables (ν, θ) as a function depending only on the conserved charge H = V (ν). The simplest class of such integrable quantum mechanical models is the single variable U (1) symmetric first-order time-derivative system. This model can be viewed as a system of a particle on a 2D plane with a large rotationally invariant potential and a constant magnetic field. 1 Compared to quantum mechanics with quadratic kinetic terms, where resurgence has been extensively discussed [8][9][10][11][12][13][14], the first-order time derivative system has half the degrees of freedom and hence a single variable system is integrable if there is a conserved charge. Therefore, it provides a good playground where we can test the completeness of the Lefschetz thimble method. Another important property, which enables us to evaluate the perturbation series around each saddle point, is that the generating function Z for the conserved charge obeys a partial differential equation of the form where g is the coupling constant (expansion parameter), µ is the external source (imaginary chemical potential) for the conserved charge and X(g, ∂ µ ) is a differential operator which depends on the Hamiltonian of the system. By using the power series ansatz on top of the saddle point value e −S saddle /g 2 , the differential equation (1.1) can be rewritten into a recursion relation for the expansion coefficients.
Starting from the initial term corresponding to the one-loop determinant around the saddle point, we can solve the recursion relation and determine the all-order power series around each saddle point. Another convenient property of our model is that the intersection numbers are accessible in a simple way. In particular, we will explicitly determine the intersection numbers by solving the gradient flow equation.
Although the gradient flow equation is originally defined in the complexified configuration space, it is reduced to a finite-dimensional problem using a symmetry argument. Using these special properties, we will show that the transseries obtained in the path integral formalism agrees with the exact partition function obtained in the operator formalism. The organization of this paper is as follows. In section 2, we discuss the resurgence structure in the first-order time derivative system with a U (1) symmetric quartic potential. After defining the generating function for the conserved charge Z(g) in section 2, we discuss the perturbation series for Z(g) in section 2.2. All the coefficients of Z(g) are determined by perturbatively solving the differential equation for Z(g). We see that the perturbation series is non-Borel summable due to some singularities of its Borel transform. In section 2.3, we calculate the contributions of complex saddle point solutions and determine the relevant saddle points by examining the intersection numbers based on the Lefschetz thimble method in section 2.4. We see that the ambiguities of the saddle point contributions cancel those of the perturbative part. In section 2.5, we compare the generating function obtained in the path integral formalism with that calculated in the operator formalism. In secion 3, we discuss the generalization to the case of generic U (1) symmetric potential. Section 4 outlines a generalization to more general integrable quantum mechanical systems. Section 5 is devoted to conclusions and discussion. Appendix A is a brief review of the Lefschetz thimble method, and Appendix B is a supplement on the properties of the differential equation for the generating function.

FIRST-ORDER SYSTEM WITH A U (1) SYMMETRIC QUARTIC POTENTIAL
In this section, we discuss the resurgence structure of the first order time derivative system with a U (1) symmetric quartic potential. This quantum mechanical system is one of the simplest example of the models in which transseries for some quantities such as partition function can be exactly obtained in the path integral formalism.

Action, Hamiltonian and Generating Function
Let us consider the 1d system described by the action where φ stands for a complex scalar degree of freedom and g is a coupling constant. This model can be viewed as a system of a particle on the (x, y)-plane (φ ∝ √ B(x + iy)) with a large magnetic field B and a potential V ∝ B 2 |x + iy| 4 . Since the Lagrangian L is linear in the time derivative, the canonical conjugate of φ is identified with its complex conjugate iφ. The Hamiltonian of this system is given by This is the conserved quantity corresponding to the time translation invariance. Another conserved quantity is the Noether charge for the phase rotation symmetry φ → e iα φ In this section, we discuss the resurgence structure of this model by investigating the weak coupling expansion of the generating function for the expectation value of N where µ is the external source forN and can be interpreted as an imaginary chemical potential 2 . Since the canonical commutation relation in this system is given by 3 the operatorsφ andφ † do not commute with each other and hence we must specify the order of the operators to define the conserved charges. In this paper, we adopt the following ordering for the conserved chargesN With this convention, we can show that the generating function (2.4) satisfies the "heat equation" As we will see, this differential equation enables us to determine the perturbation series to all orders in the coupling constant g.
In the operator formalism, the generating function Z can be determined by using the number eigenstates. The Hamiltonian can be rewritten in terms of the number operatorN aŝ This implies that the energy eigenstates are the number eigenstates Therefore, the generating function can be written as We can confirm that this satisfies the differential equation (2.7). In the next section, we calculate the same quantity by applying the Lefschetz thimble method (see Appendix A for a briefly review of the Lefschetz thimble method) in the path integral formalism and check that the nontrivial resurgence structure obtained through the Lefschetz thimble method leads to the exact result (2.10) with no ambiguity.
(2.12) Therefore, in the path integral formalism, the generating function is given by where S E is the (Wick rotated t → −iτ ) action with the source term and S W is the part generated when the Hamiltonian is rewritten in terms of the Weyl ordered operators (2.14) Corresponding to the trace in Eq. (2.4), the path integral should be carried out over the configurations satisfying the periodic boundary condition It is convenient to rescale the variable as Then, S E and S W become Thus, identifying the coupling constant g as the Planck constant, we regard S E and S W as a "classical action" and an "operator insertion", respectively.

Perturbation Series
Let us first evaluate the path integral for the generating function Z in Eq. (2.13) by using the perturbative expansion. Although the standard diagrammatic perturbative expansion is possible, we can obtain the perturbation series to all orders in the coupling constant g more easily by using the differential equation for the generating function Z: This equation implies that the perturbation series can be obtained from the generating function of the free theory Z 0 = Z| g=0 as To compute the generating function of the free theory Z 0 , let us use the Fourier series expansion of the original variable (2.20) In terms of the Fourier coefficients c p , the free theory action can be rewritten as where we have defined Performing the Gaussian integral for each mode, we obtain (2.26) This perturbation series is a divergent asymptotic series. To show this, let us expand the generating function of the free theory as 5 Using this expanded form of Z 0 , we can rewrite the perturbation series in Eq. (2.26) as (2.28) Since (2n)!/n! = 4 n Γ(n + 1/2)/Γ(1/2), this perturbation series is factorially divergent. Rewriting the series as Note that the sign of the infinite product is ambiguous. Relabeling p = p + q with an arbitrary integer q, we find that the sign depends on the choice of q This ambiguity is related to the "anomaly" of the periodicity (large gauge transformation) µ ∼ µ+2π/β, which is canceled if SW is appropriately taken into account. 5 Here, the summation should be interpreted as we obtain the formal Borel resummation of the perturbation series Since there are singularities at t = βω 2 p /(2g), we have to regularize the integral to obtain a finite value.

Fig. 1: Contours on Borel plane
Giving a small imaginary part to the coupling constant g (arg g = ), or equivalently, performing the Borel resummation along the contours C ± shown in Fig. 1, we can avoid the singularity and obtain a finite value. However, the Borel resummation gives different answers depending on the sign of Im β/2g ω p where erf(z) is the error function defined by The discontinuity at argg = 0 is given by pert are the perturbation series for arg g > 0 and arg g < 0, respectively. This discontinuity has non-perturbative factors, and hence it is expected to be related to non-perturbative effects. In the next section, we show that there are complex saddle points of the Euclidean action whose non-perturbative contributions cancel these ambiguities of the perturbation series.
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Complex Saddle Points
In this section, we look for the saddle points responsible for the non-perturbative effects in this model. In the following, we use the rescaled variable ϕ = √ g φ so that the action takes the form given in (2.17).
In addition to the classical vacuum solution ϕ = 0, the classical action S E in Eq. (2.17) has non-trivial complex saddle point solutions. Such solutions can be found by complexifying the degree of freedom where S E [ϕ,φ] is interpreted as a holomorphic functional of two independent complex variables ϕ and ϕ. The saddle points can be found by solving the complexified equations of motion We can show by using the conservation laws that besides ϕ =φ = 0, there are infinitely many complex saddle points labeled by an integer p ∈ Z where θ is an integration constant (moduli parameter) and ω p = µ + 2πp/β as in the previous section. Note thatφ is not the complex conjugate of ϕ (see Fig. 2.3) and hence these solutions are complex saddle points which are not contained in the original configuration space before the complexification. The values of the action at these saddle points are given by We can confirm that these values agree with the non-perturbative exponents of the discontinuity of the perturbative part (2.33). To compute the contributions from these complex saddle points, let us consider the integration over the thimble J p associated with the p-th complex saddle point We first focus on the leading order contribution in the weak coupling limit g → 0. Let c q andc q be the Fourier coefficients of ϕ andφ The p-th saddle point corresponds to the configuration with Now let us consider the change of the integration variables from (c q ,c q ) to a set of coordinates parameterizing the neighborhood of the saddle point in the configuration space. Choosing the new integration variables b q ,b q (q = ±1, ±2, · · · ), a and θ around the saddle point as 6 we find that around the saddle point, the action S E and S W take the forms of In the weak coupling limit, the integration measure takes the form of where the normalization factor N is the same as the one used to compute the perturbation series in Eq. (2.24). Using this integration measure, one can evaluate the leading order contribution from the p-th saddle point as where we have determined the integration contours by the steepest descent method (Lefschetz thimble method). To evaluate the infinite product, let us consider the ratio between Z p and the leading order contribution around the perturbative vacuum Z 0 where we have used the fact that Z 0 in (2.23) can be rewritten as Since Z 0 = (1 − e −iβµ ) −1 , we find that the leading order contribution of the p-th saddle point is given by The higher order part can be determined by using the differential equation (2.18). Since the leading order part takes the form of the "heat kernel", it satisfies the differential equation (2.18). As shown in Appendix B, this is the unique solution that is regular at ω p = 0. Therefore, there is no higher order correction, i.e. Z p is one-loop exact (2.52)

Intersection Numbers
Although we have determined the integral along the thimbles associated with the non-perturbative saddle points, not all of them contribute to the generating function Z. In the Lefschetz thimble method, the generating function can be constructed by combining the perturbation series and the non-perturbative contributions from the complex saddle points as The reason why the intersection is the fixed point is because the p-th saddle point is a fixed point of this symmetry and hence if the intersection point were not invariant under the symmetry, we have a continuous family of flow lines that intersects with the original contour along the orbit of the symmetry action (see Fig. 3). This contradicts the assumption that the dual thimble intersects with the original integration contour at a single point. Therefore, we assume the following invariant ansatz for the flow connecting the p-th saddle point and the intersection point where S p (ν) is the value of the action obtained by substituting the ansatz (2.58) withν = ν into the the model so that the intersection becomes an isolated point. Therefore, this flow is a straight line on the complex ν-plane (see Fig. 4). Since Im ν = 0 and Re ν > 0 on the original integration contour (φ =φ), an intersection point exists only when the line (2.67) intersects the positive real axis on the complex ν-plane, that is, if the parameters satisfy the condition Re ν | Im ν=0 = |ω p | sin cos(arg g/2) > 0, (2.68) there is an intersection between the original contour and the dual thimble. Therefore, the intersection number is given by where we have assumed that arg g is small and hence cos(arg g/2) > 0.

Exact Generating Function and Comparison with Operator Formalism
Having determined the perturbation series (2.26), all the non-perturbative contributions (2.52) and the intersection numbers (2.69), we can construct the transseries for the generating function by combining them as By applying the Borel resummation to the perturbation series as in Eq. (2.31) and taking into account the discontinuities of the intersection numbers (2.69), we can write down the unambiguous form of the full generating function as This shows that the ambiguities of the perturbative and non-perturbative sectors completely cancel out each other in the transseries obtained through the Lefschetz thimble method. We can show that the expression (2.71) is not only well-defined but also exact by comparing it with the generating function obtained in the operator formalism. By using the number eigenstates, the generating function can be written as This relation can be regarded as a variant of the Poisson resummation. Applying this resummation method, we can rewrite the generating function as Evaluating the integrals by using the definition of the error function (2.32), we find the complete agreement of the generating functions obtained through the Lefschetz thimble method (2.71) and the operator formalism (2.74). It is worth examining how the transseries for the generating function is obtained from the viewpoint of the operator formalism. To extract the perturbation series from (2.74), let us consider steepest ascent pathC p of S p (ν) starting from the origin in the complex ν-plane and decompose the integral along the positive real axis R ≥0 as where R + −C p is the path consisting of the positive real axis and the inverse path ofC p connected at the origin (see Fig. 5). We can show that the first term gives the perturbative part by changing the variable as Summing over p, we find that the collection of these terms and 1/2 in (2.74) correspond to the Borel resummation of the perturbation series (2.30). The second integral in (2.75) can be evaluated by applying the Lefschetz thimble method to this integral. The saddle point of S p (ν) is located at ν = −iω p , which is nothing but the value of ν = ϕφ for the p-th saddle point of the original action S E in Eq. (2.37). Evaluating the integral along the associated thimble, we find that the saddle contribution agrees with (2.52). The dual thimble is the path determined from Im S p (ν) = Im S p (−iω). This agrees with the flow determined by the equation (2.64) reduced from the original flow equations. Therefore, we obtain the same intersection numbers as Eq. (2.69). In this way, we can see the agreement of the transseries obtained from the path integral and operator formalism through the thimble analysis of the single variable functions S p (ν).

Preliminary
In the previous section, we have seen that the Lefschetz thimble method gives exact results in the case of the quartic potential. It is also possible to generalize the discussion to the case of an arbitrary U (1) symmetric potential In the following, we assume that the potential V (|ϕ| 2 ) has its minimum at ϕ = 0 and can be expanded as As in the previous case, we consider the generating function where the HamiltonianĤ and the conserved chargeN are given bŷ We can show that the generating function satisfies the differential equation We will use this differential equation to determine the perturbation series in the following. From the viewpoint of operator formalism, the generating function can be calculated by using the number eigenstatesN |n = n|n as On the other hand, the path integral expression for the generating function is given by where S E is the classical Wick-rotated action 8) and S W is the part generated when the original Hamiltonian is rewritten in terms of the Weyl ordered operators To derive this expression, we have used 10) where (N ) n W stands for the Weyl ordered operator defined in (2.12). We will not use the details of the higher order terms since their contributions can be determined through the differential equation.

Perturbation Series
Let us first consider perturbative expansion of the generating function Z with respect to the coupling constant g. Substituting the power series ansatz into the differential equation (3.5), we obtain to a recursion relation from which the coefficients C k can be determined order-by-order as where the initial term is given by the generating function in the free theory We can show that the Borel resummation of the perturbation series is given by where ω p = µ + 2πp β and ν p (gt) is the solution of the equation We can check that Eq. (3.14) gives the correct perturbation series by confirming that it satisfies the differential equation (3.5). For this purpose it is convenient to change the integration variable from t to ν as where the integration contour C p is the image of the positive real axis under the map from the t-plane to the ν-plane, that is, the ascending flow of S p (ν) emanating from the origin on the complex ν-plane (see the examples in Fig. 6). Substituting into (3.5), we find that (3.16) satisfies the differential equation Furthermore, (3.16) satisfies the initial condition (3.13) and hence Z pert in (3.14) gives the correct perturbation series. The formal Borel resummation (3.14) of the perturbation series is non-Borel summable if the integrand (Borel transform) has singularities along the positive real axis in the Borel plane (complex t-plane). This occurs when one of the contours C p in Eq. (3.16) connects the origin and a saddle point of S p (ν), that is, a point at which ν satisfies Although such singularities can be avoided by complexifying the coupling constant g → |g|e ±i0 , the Borel resummation of the perturbation series has ambiguities of the form where C (±) p are the ascending flows of S p (ν) for arg g = ±0. Noting that C is the thimble associated with the saddle point connected to the origin by the flowC p | arg g=0 (see Fig. 6-(b)), we can rewrite the discontinuity as where σ is the label of the saddle points of S p (ν), J p,σ is the thimble 8 associated with the saddle point 8 The orientation of the thimble is chosen so that Jp,σ dν g exp (−Sp(ν)) = 2π βgS p (νp,σ) exp (−Sp(νp,σ)) 1 + O(g) . ν p,σ and the coefficient m p,σ is given by Note that each (∆Z pert ) p,σ satisfies the differential equation (3.5). In the next section, we will see that these ambiguities are canceled by the contributions from complex saddle point solutions.

Complex Saddle Points
Let us look for the saddle points that cancel the ambiguities of the perturbation series in Eq. (3.20). The complexified equations of motion δS E /δϕ = δS E /δφ = 0 are given by Using the conservation law, we can show that the solution takes the form where ν is a constant satisfying the condition This is nothing but the condition in Eq. (3.18) that determines the locations of the singularities in the Borel plane for the perturbation series in Eq. (3.14). Suppose that ν p,σ is a solution of (3.25). Then, we can show that the value of action for the solution corresponding to ν p,σ is given by The leading order contributions from these saddle points can be calculated similarly to the previous case. For example, we can show that the one-loop determinant can be obtained by replacing g in the previous section to gV (ν p,σ ) .
The leading order part of S W is given by and hence the leading order part of the saddle point contribution takes the form of This leading order contribution is identical to that of the corresponding ambiguity of the perturbation series in Eq. (3.20). Since the higher order part can be uniquely determined from the leading part by the differential equation, the agreement of the leading order parts implies that the saddle point contribution Z p,σ and the corresponding ambiguity (∆Z pert ) p,σ in Eq. (3.20) agree to all orders in the coupling constant g Therefore, the transseries do not have ambiguities if the intersection numbers n p,σ have appropriate discontinuities ∆n p,σ = ±1 at arg g = 0. In the next section, we determine the intersection numbers n p,σ by using the flow equation.

Intersection numbers
To determine the intersection numbers, let us consider the flow equations By solving this conservation law, we can draw a flow line from each saddle point on the complex ν-plane and determine the intersection number by checking if the flow intersects the positive real axis in the complex ν-plane corresponding to the original integration contourφ =φ. We can also rephrase the condition for the intersection number as whereD p is the region in the ν-plane surrounded byC p and the positive real axis, that is, the orbit of the positive real axis under the ascending flow (see examples in Fig. 6). If the saddle point ν p,σ is on the boundary ofD p , that is,C p , the intersection number has discontinuity at arg g = 0. From the facts that • sign(Im S p (ν p,σ )) = ∓1 for arg g = ±0, (∵ S p (ν p,σ ) = β g [V (ν p,σ ) + iω p ν p,σ ] ∈ R ≥0 at arg g = 0 by assumption), • sign(Im S p (ν)) = sign(Im S p (0)) in the neighborhood ofC p inD p , we conclude that the discontinuity of the intersection number is given by This completely cancels the discontinuity of the perturbation series (3.20) and hence the transseries (3.31) obtained through the Lefschetz thimble method has no ambiguity.

Operator formalism
So far, we have seen from the viewpoint of the path integral formalism that the transseries expression for the generating function Z takes the form in Eq. (3.31) with the intersection numbers determined through Eq. (3.37). Here, we confirm that the transseries in Eq. (3.31) is consistent with that obtained from the viewpoint of operator formalism.
By using the number eigenstateN |n = n|n , the generating function (3.3) can be rewritten as This expression can be further rewritten by using the Poisson resummation (2.73) as This full generating function has the same form as the perturbative part (3.16) except for the integration contour: the ascending flow C p of S p (ν) for the perturbative part and the positive real axis for the full generating function. By the change of integration variable from ν to t = S p (ν) given in (3.15), the generating function can be rewritten as The integration contour C p on the complex t-plane is the image of the positive real axis on the complex ν-plane under the map ν → t = S p (ν). By deforming the integration contour C p , we can decompose the full generating funciton (3.42) into perturbative and non-perturbative parts as where C p,σ is the contour surrounding each singularity and associated branch cut. If t = t p,σ is a singularity, the corresponding intersection number is given by

Example
To illustrate the discussion in this section, let us consider the monomial potential as an example In this case the generating function is given by We can easily verify that this generating function satisfies the differential equation (a) l = 4 (g = β = ω p = 1) (b) l = 6 (g = β = ω p = 1) Fig. 6: Lefschetz thimbles for S p (ν) = β g (ν 4 + iω p ν) (l = 4, left panel) and for S p (ν) = β g (ν 6 + iω p ν) (l = 6, right panel). For l = 4, the integration contour R + can be decomposed to the ascending pathC p from the origin and the thimble associated with the saddle point in the regionD p (forth quadrant). For l = 6, the the ascending pathC p depends on arg g. The differenceC + p −C − p is the thimble associated with the saddle point on the negative imaginary axis, whose intersection number is n = 1 for arg g < 0 and n = 0 for arg g > 0. By using the power series ansatz, we can determine the perturbative part as . (3.48) Since this perturbation series is factorially divergent, let us consider the Borel resummation (3.49) The Borel transform B(tg) takes the form of Examples of the thimble structure of S p (ν) for l = 4 and l = 6 are shown in Fig. 6. To see this, let us rewrite the exact generating function as where the functions B p (tg) are the same functions as those which appeared in the Borel transform (3.51).
The integration contours C p are the image of the positive real axis on ν-plane under the change of the variable t = β g (ν l + iω p ν) The saddle points (3.57) are enclosed by the curve C and the positive real axis on the Borel plane, and hence they have contributions to the generating function. In particular, for l = 2 mod 4, the singularity with q = l−1−(l−2)/4 is on the positive real axis and hence gives rise to an ambiguity of the perturbative part.

GENERALIZATION TO QUANTUM MECHANICS WITH INTEGRABILITY
The analysis in this paper can also be generalized to the multi-variable cases. In particular, it would be possible to obtain exact results if there exist the same number of conserved charges as degrees of freedom. For example, in the N -variable system described by the Lagrangian there are N conserved charges N = (|ϕ| 2 1 /g, · · · , |ϕ| 2 N /g) corresponding to the phase rotations ϕ i → e iα i ϕ i (i = 1, · · · , N ) and hence some exact results can be obtained. For example, the generating function satisfies the differential equation where µ = (µ 1 , · · · , µ N ) are chemical potentials for the conserved charges. The perturbation series can be determined from this differential equation with the initial condition Z g=0 = N i=1 (1 − e −iβµ i ) −1 . In general, the Borel transform of the perturbation series has singularities corresponding to non-perturbative saddle points. Such saddle point solutions of the Wick rotated equation of motion can be obtained by using the conservation laws where θ = (θ 1 , · · · , θ N ) are moduli parameters (integration constants) and we have defined The values of ν = (ν 1 , · · · , ν N ) are determined from the conditions The contribution from these saddle points can also be determined from the one-loop determinant by solving the differential equation. The intersection number can be determined by the flow equation.
Using the ansatz we can reduce the flow equation for (ϕ,φ) to that for ν i Combining the saddle point contributions and the intersection numbers, we can construct the transseries for the generating function.
In the operator formalism, the generating function is given by By applying the Poisson resummation formula (2.73) to each summation, the generating function can be rewritten as where Z m is given by integrals over m-face of the region ν i ≥ 0 (i = 1, · · · , N ) (4.12) Applying the Lefschetz thimble method to each integral, we can confirm the correspondence between the path integral and operator formalisms. For example, each solution of the saddle point conditions (4.6) in the path integral formalism is a saddle point of one of Z m . In this way, we can check the correspondence of saddle points, gradient flows, and turning points in the path integral and operator formalisms. Note that, in general, the saddle point configurations satisfying (4.6) are complex saddle points. This shows that the complexification of the path integral is indispensable for obtaining exact transseries. It would also be possible to generalize the discussions to general integrable systems, where the action can be rewritten by using the action-angle variables (ν, ϑ) as (4.13) Assuming that ϑ is on the invariant torus ϑ = 2πτ β p + θ (p ∈ Z N ), the saddle point condition and flow equation for S E reduces to those for the function (4.14) On the other hand, in the operator formalism, the generating function can be rewritten into a form similar to (4.12) depending on the details of the quantization conditions of the conserved charges ν. Then, applying the Lefschetz thimble method, we can confirm the correspondence between the path integral and operator formalisms. In this way, it would be possible to show that the Lefschetz thimble formalism gives exact results which are consistent with the operator formalism in general integrable systems.

CONCLUSIONS AND DISCUSSION
In this paper, we have discussed the resurgence structure of the generating function for the conserved charge in the U (1) symmetric first-order time derivative systems. We have explicitly evaluated the path integral for the generating function by following the Lefschetz thimble method with the help of the differential equation which enables us to determine the all-order perturbation series around each saddle point. We have checked that the results obtained through the Lefschetz thimble method were consistent with the exact expressions obtained in the operator formalism. This fact indicates the non-perturbative completeness of the Lefschetz thimble method.
We have seen that the resurgence structure of the quantum mechanical system considered in this paper can be correctly captured by the Lefschetz thimble method. It would be interesting to generalize the discussion to the more general quantum mechanical systems with explicit analytic solutions. The key point that enables us to analyze exact results explicitly is integrability, i.e., the property that the number of degrees of freedom is the same as that of conserved charges. It would be possible to generalize our discussion to general integrable quantum mechanical systems. The explicit analysis of thimbles of the action written in terms of the action-angle variables (4.13) is important future work. Quantum mechanics with a single degree of freedom is one of the simplest classes of models where the action can be rewritten into the form (4.13) by using the conserved energy. Therefore, we can apply the analysis in this paper to such systems. It would be interesting to analyze the relationship between the method discussed in this paper and the exact WKB analysis.
It is also important to generalize the thimble analysis to the integrable quantum field theories. The non-linear Schrödinger system in two dimensions, whose 1d reduction is the model discussed in Sec. 2, is one of the examples of integrable field theories. It is more non-trivial to correctly determine the resurgence structure of field theories due to the existence of so-called renormalons [17], whose relation to saddle point configurations has not yet been well understood. Understanding the resurgence structure, in particular, the renormalons in the path integral formalism of exactly solvable models is important future work.
The contribution associated with each saddle point is given by the path integral over the Lefschetz thimble J σ : The coefficients n σ indicate how the original integration contour C is decomposed: They can also be defined as the intersection numbers between C R and "the dual thimble K σ " defined as the set of points which flows to the saddle point σ: Since the thimble J σ and its dual K σ are defined in terms of the flow, it follows that the real and imaginary parts of the complexified action satisfy Re S| Jσ ≥ Re S| sol,σ ≥ Re S| Kσ , Im S| Jσ = Im S| sol,σ = Im S| Kσ . (A.9) These properties imply that J σ and K σ intersect exactly once at the saddle point σ, and J σ cannot intersect with K σ (σ = σ) since Im S| Jσ = Im S| K σ for a generic action. Therefore, the intersection pairing of J σ and K σ , regarded as middle dimensional relative homology cycles, is given by Using this pairing, we can calculate the coefficients n σ as the intersection number of the original contour C R and the dual thimble K σ : The perturbative part of the partition function corresponds to Z 0 defined as the path integral over the thimble J 0 emanating from the trivial vacuum configuration. Non-perturbative contributions are given by the path integral over thimbles associated with non-trivial saddle points σ. It is often the case that the partition function for a real positive coupling constant g is on the Stokes line, i.e., the line on which the thimbles J σ and the coefficients n σ change discontinuously when we vary the coupling constant in the complex g plane. If J 0 jumps on the real axis (Im g = 0), the perturbative part Z 0 has an ambiguity depending on how we take the limit Im g → ±0. However, the original partition function Z has no ambiguity since it is defined independently of J σ and n σ . Therefore, the ambiguity of Z 0 has to be canceled by those associated with other non-trivial saddle points. In the case of CP 1 quantum mechanics, such saddle points correspond to the bion configurations [45][46][47][48], and their contributions have ambiguities as can also be seen in the result of the Gaussian approximation (B.1). We will see below that the ambiguity of the bion contribution originates from the discontinuous change of the intersection number n σ associated with the bion saddle points.
In general, the non-perturbative contributions of non-trivial saddle points take the form of Z p,σ = 2π βgV (ν p,σ ) exp − S p (ν p,σ ) g 1 + a 1 g + a 2 g 2 + · · · . (B.5) The coefficients a n can be determined by the differential equation (3.5), which reduces to the recursive differential equation of the form of (X∂ µ + n)a n = Y n (µ), where Y is a function of µ determined once the solutions a i (i = 1, · · · , n − 1) are given and X(ω p ) is a function of ω p such that lim ωp→0 X(ω p ) = 0. (B.7) This implies that the general solution of the homogeneous equation (X∂ µ + n)a n = 0 is singular in the limit ω p → 0. Therefore, we can uniquely fix the solution of the differential equation (B.6) by requiring that it is regular in the limit ω p → 0. Thus, all the coefficients around the saddle points can be uniquely determined by solving the differential equation (3.5).