On exact-WKB analysis, resurgent structure, and quantization conditions

There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: Saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schr\"{o}dinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the"topology"of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller's ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact results. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

In large variety of quantum mechanical systems, it is now well-understood that perturbative (P) and nonperturbative (NP) physics are related in a deep way. This connection can be understood by multiple means. These are, • Semi-classical analysis based on saddle-point (e.g. instantons) and steepest descent (Lefschetz thimble) methods.
• Exact WKB method based on Schrödinger equation.
• Exact quantization methods based on generalization of Bohr-Sommerfeld quantization.
In each one of these constructions, to see the connection between P/NP physics, the most prominent role is played by resurgence theory, and Stokes phenomena. Despite the fact that much is known about each one of these methods, the precise relation between them is not yet completely clear.
In the resurgent asymptotic analysis, see [1][2][3][4][5][6][7][8] for mathematical background, the large order growth of the perturbative coefficients of fluctuations about a given sector (eg. perturbative sector), is related to low-order perturbative coefficients of fluctuations about other non-perturbative sectors (eg. the instanton-anti-instanton sector) in a precise way. Many signs of resurgent structure in the perturbative and instanton analysis are already hinted in the old physics literature [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. The renewed interest is due to the precise understanding of the connection between resurgence theory and physical problems in quantum mechanics , matrix models and string theory , and quantum field theory . 1 The power of formalism comes from the fact that the resurgent structure can provide a complete non-perturbative definition of quantum theories, reveal new saddles that are not obvious at all, and may provide at least a partial solution to the important renormalon problem in QFTs [80,100] In the semi-classical saddle point approach, we express physical quantities such as ground state energy or the partition function as series containing perturbative and non-perturbative contributions, 2 Z( ) = n a n n + e − S 1 n b n n + e − S 2 n c n n + ...
This type of series is called as a trans-series. The perturbative and non-perturbative parts are usually calculated independently, and typically, all series appearing in the transseries are asymptotic expansions. One can turn the divergent asymptotic series into something finite by Borel resummation, but the price one pays is that the result may be multi-fold ambiguous. For example, the Borel resummation of the perturbative part n a n n induces an ambiguity which has a non-perturbative factor e −S1/ (where S 1 is an information about another saddle in the problem, associated with the instanton antiinstanton action) is expressed as Here S ± stands for the operation of lateral (left/right) Borel summation, in which the sign ± means how the Laplace integral contour in the Borel resummation avoids the Borel singularity. This phenomenon is considered to be equivalent to the Stokes phenomena in the Picard-Lefschetz theory, where the structure of the Lefschetz thimble decomposition changes discontinuously [83,[115][116][117][118][119][120][121][122][123][124][125][126][127]. From a physical point of view, it means that the perturbative part, even after Borel resummation, is not well-defined by itself. But crucially, the way it is not well-defined carries non-perturbative information encoded into it. For stable quantum mechanical systems, the energy eigenvalues must be real and unambiguous. Fortunately, there are also non-perturbative contributions, instanton-anti-instanton correlated events [IĪ] ± , whose contributions are also two-fold ambiguous and cancels the ambiguity of Borel resummed perturbative series. This is first calculated in Refs. [13,14], but rigorous explanation is given in [50] by using the concept of critical points at infinity and Picard-Lefschetz theory. This type of resurgent cancellations encodes an intricate network of "large-order/low order" relations between different perturbative and non-perturbative sectors, and is partially proven.
A constructive version of resurgence, based on "low-order/low order" relations is discussed in [28,29,[52][53][54]. 2 Instead of partition function, one may also consider generalized partition function with the insertion of an operator leading to transition between different vacua, such as tr e −βH P in parity invariant systems. In such cases, the transseries will begin with an instanton factor instead of perturbative vacuum. 3 Of course, the usual WKB approximation is a text-book material. But it turns out that standard WKB can be made can be made exact as we review here. The exact-WKB, for which resurgent analysis is the fundamental tool, is more recent (a clarified understanding began with the works of Voros and Silverstone in the 80's [129,130]) and there is a very interesting body of mathematical works around it. One of our goals here is to make it more accessible, and use it as a tool to explore the connections between various non-perturbative methods.
application to the Schrödinger equation, Here ψ(x) correspond to the wave function and E stands for the energy eigenvalue. In exact-WKB, one maps the Schrödinger equation into a non-linear Ricatti equation, whose asymptotic solution provide the building block of WKB-wave function, which is by itself an asymptotic series. In exact-WKB, the position is first elevated to a complex variable, and the classical potential is used to turn the x ∈ C plane into Stokes graph and regions. 4 Once one considers the analytic continuation of the WKB-solution of ψ(x) from one Stokes region to an adjacent one in complex x plane, its asymptotic behavior sometimes changes discontinuously.
This is another type of the Stokes phenomenon, and this change can be examined by studying the Borel summation of the WKB wave functions. Starting with a decaying WKB wave function in the x → −∞, the existence of Stokes jumps and connection formula induce both exponentially decaying and exponentially increasing components as x → +∞. But the WKB wave function must vanish as x → +∞ for normalizability. This simple fact implies that the prefactor of the exponentially increasing WKB-wave function must vanish. This is the statement of the exact quantization condition. Exact quantization conditions necessarily involves both perturbative and non-perturbative cycles, and forces precise relations between perturbative and non-perturbative contributions. We investigate implications of this construction, and interpret it in terms of Euclidean path integral formulation. This reveals how the two different kinds of Stokes phenomena in the two "seemingly" different formulations are related.
Let us briefly summarize our findings: 1. Unified understanding of the two Stokes phenomena: As described above, in the semiclassical analysis of the double-well quantum system, the Stokes phenomenon occurs in such a way that the imaginary ambiguities are canceled out between the perturbative Borel resummation and the non-perturbative bion contributions. In the exact-WKB analysis, we find the same Stokes phenomenon takes place as the change of the "topology" of the Stoke curve. This correspondence is clearly incorporated in the Delabaere-Dillinger-Pham (DDP) formula [131], which expresses the resurgent relation as a relation between the different cycles crossing the Stokes curves.
2. Generalizing Gutzwiller trace formula: The Gutzwiller trace formula is a semi-classical method that ties spectrum of quantum theory to classical mechanical concepts, to periodic orbits calculations, actions, geometric phases [146]. It is almost exclusively used in systems without tunneling phenomena (instantons). Although it is sometimes suggested that by making time complex, one should be able to capture the non-perturbative effects, there is no systematic study capturing how this is done precisely. In fact, in 2018, Nekrasov [147] suggested that Gutzwiller's formula can be improved by the contributions of what he calls m,n-solutions (concrete examples of which are bion configurations as he points out), such that it can produce an exact formula. Our findings proves this proposal at least for a number of polynomial potentials.
3. Novel meaning of quasi-moduli integral: The above findings give new physical meaning to the quasi-moduli integral (QMI) in the semiclassical analysis of path integral. Using the perspective of Gutzwiller's quantization condition, the non-perturbative contribution obtained from QMI is shown to have a nontrivial relation with the perturbative contribution around the classical vacuum.

Rediscoverig Maslov index and the intersection number of Lefschetz thimble:
The resolvent G(E) obtained from the quantization condition D(E) = 0 in the exact-WKB analysis can be rewritten in the Gutzwillertype representation and it can be compared to G(E) derived in the Gutzwiller's quantization condition. We then find that (−1) n appearing in the non-perturbative sector is interpreted as the Maslov index. Furthermore, we show the Maslov index turns out to be the intersection number of Lefschetz thimbles by expressing the Fredholm determinant and resolvent in a convenient form.

5.
Equivalence of the different quantization conditions: Based on above observations, we clarify the equivalence and the nontrivial relations among the different quantization conditions, based on path integral, exact-WKB and Gutzwiller methods.
6. Generalization to symmetric multi-well potential: We generalize the exact-WKB analysis and the DDP formula to the quantum systems with generic symmetric multi-well potentials. Again, we show that the Stokes phenomena occur as the topological change of Stokes curve in the exact-WKB analysis, and the resurgent structure in the semiclassical analysis is completely incorporated in the DDP formula.
The paper is constructed as follows. In Sec. II A we review path integral, Lefschetz thimble decompositions, resolvent methods and Gutzwiller' quantization in quantum theories. In Sec. III and Sec. III D we review the exact-WKB method, and explain Stokes curves for potential problems in simple examples. In Sec. IV we apply the exact-WKB analysis to double-well potential quantum mechanics by studying the associated Stoke curves, where we find the equivalence of the two Stokes phenomena and show the equivalence among several quantization conditions. In Sec. V we extend our investigation to the systems with generic multi-well potentials and discuss outcomes. Sec. VI is devoted to summary and discussion.

II. PREPARATION
Before starting our journey, we introduce the tools other than the exact-WKB analysis as prerequisite knowledge. These include saddle point decomposition of path integrals (Lefschetz thimbles), its relation to resolvent, Gutzwiller's quantization and Maslov index. Our discussion is basic and is streamlined according to what we need later.

A. Lefschetz thimble decomposition and Resolvent method
We start with the Lefschetz-thimble decomposition of path integral and the resolvent method in quantum mechanics. By use of the asymptotic series expansion and the trans-series expansion, the Euclidean partition function in quantum mechanical systems with field x(τ ) is expressed as = n a n n + e − S 1 n b n n + e − S 2 n c n n + ... , where β is Euclidean time period. From the viewpoint of Picard-Lefschetz theory, the Borel summation of a perturbative series around each saddle point corresponds to performing exact thimble integration associated with the corresponding saddle: 5 Here σ labels the saddle points and J σ are the corresponding Lefschetz thimbles. Here, we have to determine the index, n σ (0 or ±1), called the intersection number, to obtain the correct result of the path integral. However, there is no efficient method for calculating this index except for calculating the thimble numerically and plotting it explicitly, which is a hard task. Therefore, we have no reliable way to determine "which are relevant saddles" in the generic cases from the integration itself. In this paper, we propose a certain solution to this problem by using exact-WKB and simple Stokes graphs, and give a physical interpretation of the index in quantum mechanics. We now review the resolvent method [22] for quantum theories. In the latter part of this paper, this method will enable us to obtain the partition function of the system directly from the quantization conditions obtained from the exact-WKB analysis and to interpret the resurgent structure of the partition function in terms of the exact-WKB analysis.
First, we write down the partition function formally as a sum over saddle points where S[·] denotes the Borel summation of series expansions and x σ stands for saddle points. We then consider the Laplace transform of Z(β), which gives the trace of resolvent G(E). Since this transform is linear, we obtain the expression as tr 1 It is notable that the poles of G(E) give the eigenvalues and G σ (E) stands for the trace of resolvent for each sector where D σ (E) stands for the Fredholm determinant for each thimble. We note that the zeros of D(E) give the exact energy eigenvalues. 6 The main point of our analysis is following. From the exact exact-WKB analysis, we will obtain an exact quantization condition, D(E) = 0. This formula will be expressed in terms of perturbative and non-perturbative cycles, which involve perturbative as well as non-perturbative instanton/bion data. By reexpressing the condition D(E) = 0 as a sum over P/NP cycles, we will be able extract the index n σ and Maslov index from the exact-WKB analysis. Since we can go back to Z(β) by inverse Laplace transform it will enable us to obtain the partition function from the exact-WKB method.

B. Gutzwiller's quantization
The Gutzwiller's quantization, which is also known as Gutzwiller's trace formula [146], is a semi-classical construction that express the quantum mechanical density of states, (the spectral information of quantum theory), in terms of classical mechanics concepts. The formalism uses path integral in Minkowski space formulation (in real time), hence one is dealing with amplitudes in real time Dx e iS . It is used primarily in quantum chaos, in questions such as quantum billiards. As we describe below, the method uses the stationary phase approximation, the semi-classical methods in the → 0 limit. Once this is made, (since we are in real time formalism), what follows is classical, namely, 6 Mathematically, the definition of Fredholm determinant (or resolvent) needs a regularization, e.g. periodic orbit calculations, actions, geometric phases are manifestly classical. In certain sense, Gutzwiller method can be interpreted as the intermediate quantization method between the path integral and the Bohr-Sommerfeld quantizations, because the paths that are periodic in the whole phase space and not only in configurations space.
This method is almost exclusively used in systems without tunneling phenomena (instantons) [146]. In retrospect, this is not surprising because usual instantons in real time correspond to imaginary singular configurations [148], and it is not so obvious how to deal with it. It is sometimes loosely suggested that by making time complex t → te iα , one should add complex time periodic orbits to quantum mechanical cycle expansions, see eg. [149]. This suggestion is morally correct, but it does not address the important issue on how to systematically capture these effects. Indeed, at finite energy E, all solutions between the minimum and top of the barrier, see Fig.8, are doubly-periodic solutions, e.g. Weierstrass ℘-function is doubly periodic, with both purely real and imaginary periods. It is clear that in the more precise version of the Gutzwiller formula, doubly-periodic solutions must play a prominent role.
Recently, Nekrasov [147] suggested that Gutzwiller's formula can be improved by the contributions of what he calls m,n-solutions (concrete examples of which are bion configurations), such that it can produce an exact formula. Our findings proves this proposal at least for a number of polynomial potentials.
In the older literature on Gutzwiller formulas, it suffices to use classical mechanics in the usual textbook sense. However, in recent years, especially in the discussion of the Euclidean path integral formulation, it became quite clear that the moment we start to work with "semi-classics", action must be viewed as holomorphic. Therefore, the classical mechanics that one needs to deal with, in any semi-classical approach, is an holomorphic classical mechanics that captures general complex solutions [33,34].
Let us first review the standard discussion of the trace formula. Later, we will re-derive it from exact quantization conditions, and show the effects of complex bion paths. We first express the "Lorentzian" partition function as The resolvent, which has quantum spectral data, is given by where lim 0 is taken after integration. The resolvent can also be expressed as where Γ = S + Et. We also note that the action S is written as We here evaluate the t integral in (14) by the stationary phase method with considering the t derivative of Γ Since pdx is independent of time and only depend on the trajectory, we have d dt pdx = 0. Therefore, we obtain It means the leading contributions of the t integral are periodic classical solutions whose energy is E. When a periodic orbit is a solution, the configuration rotating n times is also a solution. By taking this fact into account, we find that the contribution is obtained just by the replacement as pdx → n pdx. Then, we obtain Γ = S + Et = n pdx − Et + Et = n pdx , (n = 1, 2, 3...) . After all, the contribution of the classical solutions to G(E) is expressed as where p.p.o. stands for a "prime periodic orbit", which is a topologically distinguishable orbit among the countless periodic orbits, i.e. periodic in phase space.
If we consider the sub-leading terms in stationary phase approximation, it gives where T (E) is the period of each cycle. Here, det δ 2 S δxδx is a functional determinant taking into account the fluctuation operator around the saddle point. Evaluation of this part requires care, as described below, this operator has negative eigenvalues when considering the expansion around a periodic orbit in general. The number of negative eigenvalues is called "Maslov index", which plays an important role in Gutzwiller's quantization.
The operator M has a zero eigenvalue if x cl depends on t, and The operator M has 2n-1 negative eigenvalues for n-cycle orbit. We below give a brief proof of this fact: Proof. Consider classical EoM: Take t differential for this equation. Then we get This expression is nothing but an eigenvalue equation for the zero eigenvalue of the fluctuation operator, Mψ 0 (t) = 0, and the eigenfunction is proportional toψ 0 (t) = dx cl dt . Next, let us consider a periodic classical solution x cl . When it is a one-cycle solution, the derivative dx cl dt typically has a behavior depicted in Fig. 1. The operator M is a Schrödinger-type operator, thus the level of the eigenfunction is determined by the number of zero points 7 (nodes). In the case of Fig. 1, M has two nodes since the periodic b.c. is imposed, and the endpoints are identical and regarded as a single node. The reason why it has one negative eigenvalues is that dx cl dt is the first excited state, but at the same time, dx cl dt is also an eigenfunction of the zero eigenvalue. Similarly, a n-cycle classical solution has 2n turning points, M has 2n − 1 negative eigenvalues because dx cl dt is the 2n − 1-th excited state.
If we consider an analogy with the Morse theory, the operator M corresponds to (the diagonal part of) Hessian where the action is viewed as a Morse function, and its negative eigenvalue corresponds to the Morse index. Thus, the Maslov index is essentially regarded as the Morse index in the functional integral. To rephrase this, we now express the contribution of functional determinants as Here, α is called the Maslov index. Note that only this exponential factor e iαπ appears in the latter calculation so we may call e iαπ as Maslov index instead of α. Here, ν is the number of negative eigenvalues of M . The determinant of the n-cycle is given by Therefore, the final form of G(E) (up to higher order quantum corrections) is given as Working of Maslov index in simple harmonic oscillator: As an example, we now consider the harmonic oscillator system. In this case, there is only one type of p.p.o. with constant T (E) and | det δ 2 S δxiδxj |, We then obtain Therefore the poles of G(E) are given by This is the Bohr-Sommerfeld quantization of harmonic oscillator. It should be emphasized that the contribution of the Maslov index is important in order to obtain the correct energy eigenvalues including the vacuum energy.
We set Q(x) = 2(V (x) − E) then rewrite the equation as In the WKB analysis, we consider the ansatz given by where S(x, ) is a formal power series expansion in expansion parameter , and S n (x) are functions of x. Substituting Eq. (31) into Eq. (30), leads to the non-linear Riccati equation By substituting Eq. (32) into Eq. (33), we obtain the recursive relation We note S −1 = ± Q(x). Since S n is recursively determined from S −1 , S n has two independent solutions: The first several terms are given by From Eq. (34), one finds the relation S − n = (−1) n S + n . Therefore, we reach the simple expression = ±S odd + S even .
Based on Eq. (41), Eq. (33) is rewritten as These two equations give Therefore the WKB wave function can be expressed as with a being an integral constant. For later calculations, we choose it as a turning point, which is a solution of Q(x) = 0. At the leading order, WKB wave function is evaluated as which is nothing but the solution in the text-book level WKB approximation. Since we have derived the WKB wave function recursively, it is regarded as a formal series in Note that the factor 1 2 in n+ 1 2 comes from 1 Here, both of these series turn out to be asymptotic expansions with respect to . In other words, the all orders WKB wave function is an divergent asymptotic expansion with respect to . In order to give it a precise meaning, we need another technology, the Borel resummation, applied to series for which the divergent coefficients ψ ± a,n (x) are x-dependent.

A. Borel summation
Let us consider the following formal series (not necessarily asymptotic) with respect to .
The Borel transform of this series is defined as The directional Borel resummation S[Z] is defined as where θ denotes the direction of integration. The resurgence property tells us that the Borel transform admits an analytic continuation in z plane, which allows via Borel resummation the reconstruction of the exact value of the result. If the series is convergent, this procedure just return the original series due to the identity For an asymptotic divergent series, however, it gives one analytic functions which have the series as its asymptotic series. The Borel summation is a homomorphism, so that the following algebraic properties hold.
Now, we apply the Borel summation procedure to the WKB wave function, then we obtain Because the coefficient ψ ± n (x) depends on x, the position of Borel singularity also depends on x. The position of singularities of integrand is z = ± x a Q(x) when the Stokes curve is not degenerate. From now on, we will express the Borel-summed wave function S[ψ ± a ]( ) as just ψ ± a ( ) unless otherwise noted. We now look into details of the Borel singularities. One of them is always the endpoint of the integration path, and 1 √ x is regular at x = 0. Therefore the other one contributes to Stokes phenomena. These two singularities are point-symmetric. Hence, Stokes phenomena occur due to the situations that one of the singularities is on the integration path as shown in Fig. 2. This condition can be expressed as Note that 1 x a Q(x) = 1 S −1 is the leading term of WKB expansion. Therefore, in order to understand when Stokes phenomena occur, we do not have to calculate Borel summation of ψ(x) explicitly, but just evaluate this term. The path derived from Eq. (57) is called a Stokes curve, which is part of a structure that is called Stokes graph. In principle, the exact energy spectrum of the theory can be calculated just from the Stokes curve data. We denote by S ± the lateral Borel resummation with a positive/negative (small) angle θ.

B. Stokes curves and Stokes phenomena
Let a be a turning point (a solution of Q(x) = 0). In this case, the Stokes curve associated with a is defined as Also, each segment of the Stokes curve has an index, ±. This index indicates which on of the ψ + and ψ − pair increases exponentially when moving from the point a to infinity (more precisely, a to ∞e iθ * where θ * is the phase of corresponding segment) along the Stokes curve. The parts between the Stokes lines is called Stokes regions or just regions. When the index of the corresponding Stokes curve is +, ψ + increases exponentially 9 and When the index is −, then ψ − increases exponentially in the case The Stokes curve indicates where the Stokes phenomena occur, when ψ(x) is analytically connected between adjacent Stokes regions.

C. Connection formula and monodromy matrix
For a generic potential and at a typical value of the energy, the turning points are non-degenerate. To each such turning point, one attaches an Airy-type Stokes graph. Therefore, a Stokes graph for a general potential is a composite of elementary building blocks of Airy-type Stokes graphs.
We now give a connection formula for the Airy-type Stokes graph. When one considers the Borel-resummed wave functions for a given potential and its analytic continuation in terms of a complex x, one has to take into account the effect of the Borel singularity on the Borel plane, i.e., the Stoke phenomenon.
Roughly speaking, in order to compute the effect, we decompose a global Stokes graph into the Airy-type Stokes graphs locally and then consider the effect of crossing the Stokes line by using a connection formulas. As we emphasize, the Airy-type Stokes graph is a building block of any given graph, hence, it is important to understand it fully in simple examples. Apart from the connection formula, one also pay attention to the change of normalization point of the wave function corresponding to the change of turning points. We describe both below.
We suppose the wave function is normalized at a simple turning point a and consider analytic continuation from the region I to II as shown in Fig. 3. When x crosses a Stokes line, the relation between wave function can be expressed by The monodromy matrix M multiplies the wave function according to following rules: Furthermore, if it crosses the branch cut emerging from the simple turning point a, it moves between the first and second Riemann sheets as S odd (x) → −S odd (x). Thus, we have for clockwise crossing of a cut (63) A complete cycle around a turning point gives identity map: In order to consider the analytic continuation globally beyond the Airy-type Stokes curve, one has to incorporate the change of normalization point and then employ the connection formula due to curves or cuts emerging from other simple turning points. Two wave functions normalized at different turning points a 1 , a 2 are are related by the equation The quantity, a2 a1 S odd is called the Voros multiplier. One may be tempted to think that the Voros multiplier is an asymptotic function, because S odd is defined by the recursive relation and asymptotic itself. However, the Voros multiplier appearing here is Borel resummed because the ψ ± a1 (x) are already Borel resummed wave functions. We then write down the normalization matrix as The orientation of N a1a2 is flipped before/after crossing a branch cut.
D. Warm-up: Harmonic oscillator with Airy-type Stokes graph For a general potential with multiple degenerate harmonic minima, we associate a Stokes graph which is a combination of the Stokes graph of harmonic oscillator. The Stokes graph for harmonic oscillator is a combination of two Airy-type graphs as shown in Fig.4. Therefore, one can quickly learn how the formalism works in practice in this simple, but essential example. Therefore, we first review this example, and then move to more interesting examples 4. The Stokes graph for harmonic oscillator. In order to obtain the quantization condition, we take the orbit from the left to the right below the real axis by taking into account the Stokes phenomena (Left panel). The cycle A is defined as an oriented cycle enclosing two turning points a1 and a2 (Right panel).
of double-well, triple-well and N -ple well examples. The harmonic potential is given by V (x) = 1 2 ω 2 x 2 . Its Stokes curve is depicted in Fig. 4 assuming E > 0. There are two turning points, The blue arrow is a trajectory of the analytic continuation. If we start with a decaying solution at the beginning of blue line, we will demand a decaying solution at the end of journey, for the full WKB solution to be normalizable. However, the Stokes phenomena will induce terms that will be exponentially growing at the end. In order to have a physical answer, we will demand that the pre-factor of the exponentially growing part to vanish. That will give us the quantization condition that will determine the spectrum of the theory.
First, let us take wave function normalized at a 1 and consider analytic continuation from the region I to II. Then the wave function changes as Second, consider analytic continuation from the region II to III. In this case, the Stokes curve to be crossed is starting from the other turning point, a 2 . Therefore we have to change the normalization as Then we can multiply the monodromy matrix as As a result, 10 , we obtain the connection formula where the cycle A = e A S odd = e 2 a 2 a 1 S odd is depicted in Fig. 4. As x → −∞ in the region I, ψ + is normalizable, it decays as x → −∞. (This is true on first Riemann sheet which we stick through this argument.) Therefore we take ψ + in the region I and we find that it changes to ψ + a1,III (x) + i(1 + A)ψ − a1,III (x) in the region III. In region III, ψ + a1,III (x) is decaying as x → −∞ while ψ − a1,III (x) is blowing up. Therefore, in order to satisfy the normalization condition, the coefficient of ψ − must be zero. This is the quantization condition: where e a 2 a 1 S odd is the Voros multiplier connecting two turning points. This is equivalent to For the harmonic oscillator, where ω = 2π T and T is the classical period, T = A (2(V (x) − E)) −1/2 dx . Therefore, the quantization condition D(E) = 0 obtained from the exact-WKB analysis gives The Stokes curve in Fig.4 correspond to E > 0 and the turning points are real. This puts a restriction that n = 0, 1, 2..., which is just the spectrum of simple harmonic oscillator.

E. Resolvent and Spectral form
We derive the partition function starting with the quantization condition D(E) and resolvent G(E). The reason we are presenting this is because we will follow verbatim the same procedure in the theories with instantons and we will reach to fairly non-trivial results. It is therefore useful to recall this tool in a simple example.
The quantization condition D is written as 11 where we have used reflection formula: Γ(x)Γ(1 − x) = π sin(πx) . We then obtain the resolvent G(E) from the quantization condition as: The partition function is the inverse Laplace transform of resolvent: To calculate this quantity, we consider the contour C depicted in Fig. 5, where C is determined by the condition E > 0. It leads − 1 2 ω < < 1 2 ω and C is closing in the positive real region. 11 The zeta function regularization for Fredholm determinant D(E) = det(H − E) gives: It removes the irrelevant Gamma function, which does not contribute to the partition function defined through contour C. Inside the contour C, πi ω and ∂ ∂E log Γ 1 2 + E ω are holomorphic, hence do not contribute to integration. Furthermore, ∂ ∂E log Γ 1 2 − E ω has infinitely many poles in C and all the residues 12 are 1. Therefore we find the partition function of harmonic oscillator. We will use the same strategy in more general cases involving instantons to describe the partition functions of the systems from the exact quantization condition.

IV. SYMMETRIC DOUBLE-WELL POTENTIAL
We consider the exact-WKB analysis for the symmetric double-well potential. It is known that (1) the leading non-perturbative contribution to its ground state energy comes from the instanton configuration, and (2) the Borel ambiguity of the perturbation theory for the ground state is cancelled by that of the bion (correlated instanton-antiinstanton configuration) contribution. This pattern continue to higher states under the barrier. The exact form of the bion contribution can be obtained from the quasi-zero mode integration (quasi-moduli integral). We first review the resurgent structure of the partition function in this system. Then, we find an explicit mapping between this construction and Gutzwiller's quantization. In doing so, we figure out the relation between the phase ambiguity of quasi-moduli integral, the topological properties of the Stokes curve in terms of Gutzwiller's quantization.
For the symmetric double-well potential, where a i are turning points. Then the Stokes curve of this systems 13 is schematically depicted as shown in Fig. 6. 12 ∂ ∂x log Γ(x) ∼ 1 x in x → 0 13 This Stokes curve corresponds to the low energy region, to energies below the barrier height. Above the barrier height, two of the real turning points, a 2 and a 3 turns into complex conjugate turning points. Even in high energy region, we can show the topological structure of Stokes curve is corresponding to phase ambiguity. 6. The Stokes graph for the double-well potential with Im > 0 (Left panel) and Im < 0 (Right panel). We took the two branch cuts such that their end-points are turning points (a1, a2) and (a3, a4). We take the orbit for obtaining the quantization condition from the left to the right below the real contour.
Perturbative cycles (A, C) and a non-perturbative cycle (B) for the symmetric double-well potential. The perturbative cycles, A and C, are defined as oriented cycles enclosing (a1, a2) and (a3, a4), respectively, and the non-perturbative cycle, B, is an oriented cycle enclosing (a2, a3) and intersecting with the two branch cuts.
As we did for the harmonic oscillator in Sec. III D, we obtain the quantization condition from the normalized condition by performing the analytical continuation of the wave function from −∞ to ∞ using the connection formulas.
for Im > 0 for Im < 0 The quantization condition for this case is given by where the cycles are defined as 14 and C ± := S ± [C] for C ∈ {A, B, C} as shown in Fig. 7. The A and C cycles are perturbative and B cycle is non-perturbative, B ∝ e − S , where S corresponds to the single bion contribution. We here defined the notation, where C is used as series forms and C ± is used as Borel-summed forms. However, from now on, we would use a simplified notation where we use C instead of C ± for simplicity unless it causes a confusion. 15 To evaluate the non-perturbative contribution to the ground state energy and the phase ambiguity term, let us consider the asymptotic form of A, which does not include non-perturbative contribution, before being Borel-resummed. It is: This ω A (E, ) is an asymptotic expansion in . In the low energy limit, it can be regarded as a harmonic frequency of the classical (harmonic) vacuum: where x vac is a minimum of the potential. We emphasize that writing down this expression corresponds to taking the Borel-resummed A back to its asymptotic expansion form. This procedure helps us to see that the quantization condition D has the phase ambiguity. However, of course, this ambiguity disappears when we consider the Borelresummed form. We will show it in the next subsection. We now set E = ω A 1 2 + δ , where δ roughly stands for the energy deviation from that of the harmonic oscillator. The quantization condition D = 0 then becomes Or equivalently, Here ± in the latter form stands for parity. We emphasize that this result is obtained without any approximation.
In [14][22], [49], The quantization condition was calculated using path integral(QMI) method. The result is (90) is the contribution from quantum fluctuations, this part is included in B and ω A in (89). The extra Gamma function Γ(1 + δ) is coming from the negative energy part when we consider the argument in (79), so it can be ignored under the condition E > 0. Therefore, this result is regarded as the complete quantization condition with full quantum fluctuations.

A. Gutzwiller's quantization
Gutzwiller's quantization is based on prime-periodic orbit (p.p.o.) as a fundamental unit, but the way how to add up this p.p.o. has not been clearly known except for simple systems. We will see that one can exactly obtain the Gutzwiller's form from the quantization conditions in the exact-WKB analysis and it reveals a new physical meaning of the quasi-moduli integral in the path integral method.
First, let us rewrite the quantization condition Eq. (83), using C = 1/A, in terms of only A and B cycles where D A =1 + A −1 (Im > 0) or 1 + A (Im < 0). This rewriting allows us to write the trace of resolvent G(E) = − ∂ ∂E log D(E), derived from the quantization condition Eq. (83), in a useful form: The derivative term ∂ ∂E A produces the "period" and similarly, where ± corresponds to Im > 0 and Im < 0 respectively. Since the classical solutions in the lower part of the potential are doubly-periodic, and our definition of S −1 = √ Q where Q = 2(V − E), T A is purely imaginary and T B is real. Our construction instructs us that complex periodic paths are part of Gutzwiller formula. This seems to be the mechanism through which Gutzwiller formula is able to capture the tunneling (instanton) effects. The magnitudes of the quantities T A , T B can be called quantum periods and its leading term corresponds exactly to the period of the classical orbit. Using these quantities, G(E) can be expressed as the reversal of the direction of the A cycle of the non-perturbative term. As we show later, this transition can give the new perspective of the quasi-moduli integration.

B. Partition function
In this subsection, we calculate the partition function based on the resolvent method in Sec.III E. In particular, when evaluating the partition function using path integral, it is important to evaluate the contribution of the integral called quasi-moduli integral(QMI). It is shown that this can be evaluated explicitly by the calculation using exact WKB. We also show that the partition function itself is invariant under the Borel sum.

Comparison to quasi-moduli integral
Using the decomposition of resolvent given in (92), we can write the partition function as where and where we have used integration by parts moving to the second line. We now clarify the relation between the above quasi-moduli integral and our result on the non-perturbative contribution Using this asymptotic expansion in Eq.
E ω A (E, ) again. Then D A is given by For our purpose, we drop the irrelevant Gamma function factor , which corresponds to the negative eigenvalue and does not contributes to the integral in the case of harmonic oscillator.
Then, we rewrite Z np as Essentially √ 2π Γ(1+s) in 2 sin(πs) = 2π Γ(−s)Γ(1+s) corresponds to the negative eigenvalues, so if we define the integral path to take only positive eigenvalues, this integral can be approximated 16 as Here, the partition function obtained by calculating the path integral is as follows [49] [22], (Appendix A6): Comparing this with our results, we can see that we obtain the path integral representation and indeed each sector in Eq. (108) 16 The residues of 1 sin(πs) and Γ(−s) are different, so just removing Γ(1 + s), even though it essentially corresponds to the negative eigenvalues, would change the result. However if we only consider the residue around s = 0, which corresponds to the ground state energy, the factor Γ(1 + s) can be ignored. 17 There are two QMI for one bion and it gives two Gamma functions. topological angle (HTA) [34]) of n-bion configuration though, the index has very important role: this quantity can be regarded as the intersection number of Lefschetz thimble 18 .
Furthermore, comparing the QMI calculation and Gutzwiller's perspective, we can see the new physical meaning of QMI. The QMI calculation is based on the approximation that the cycle is sufficiently large, but from Gutzwiller's point of view, the B cycle is so short that it requires the A cycle to rotate infinite times in order to earn the sufficiently long cycle, and therefore it is considered to be represented in the form of D −2 A B. This perspective explains the riddle in the calculation of [49] [14]: The form of the Γ function derived from the vacuum and the form of the Γ function derived from QMI matched despite both were calculated entirely separately.
The first are from two vacua and the latter ones are from QMI. This miracle is easily explained by this Gutzwiller's representation. Both have essentially the same origin, the infinite number of A cycles, D −1

The intersection number of Lefschetz thimble
It is notable that the of quantization condition in Eqs. (83) determines the "relevant saddles" in the path integral and the intersection number of Lefschetz thimble (n σ ). Firstly, as we mentioned in Sec. II A, the Fredholm determinant can be expressed as Now, the quantization condition given by Eq. (83) can be rewritten as The first (1 + A) and 1 + A −1 are regarded as the Fredholm determinant coming from the vacuum saddle points and the latter ones are ones from n-bion saddle points.
This representation is a factorized form for each bion, and as explained in the previous section, the Z n obtained from this D n is indeed the partition function of n-bions. The power of each bracket [...] is nothing other than the Maslov index. Therefore, we can see the intersection number of Lefschetz thimble of non-perturbative contributions is exactly corresponding to the Maslov index. Strictly speaking, the Maslov index is attached to both A and B cycles. However, the former is not regarded as the intersection number of Lefschetz thimble but the only latter's is. This difference is related to the following In this subsection, we would like to briefly review the Delabaere-Dillinger-Pham (DDP) formula [131]. For the double-well potential, as we have seen the previous sections, the A, C-cycles are defined as asymptotic series. These series are non-Borel nonsummable if Im = 0. When the an asymptotic expansion is Borel nonsummable for Im = 0, the Borel transformed cycles have a singular point on the positive real axis of the Borel plane, in other words an imaginary ambiguity happens according to the choice of the sign of Arg( ) for the Borel resummation.
By employing this imaginary ambiguity the information of A, C-cycles can be carried into the B-cycle via the Stokes automorphism. This relationship is so called the resurgence relation. This type of resurgence relation connects high orders of the asymptotic expansion of the A, C-cycles to low orders perturbative expansion of B-cycle. In the physical sense, the A, C-cycles and B-cycle are now interpreted as a perturbative expansion(fluctuation) in terms of around (locally) bounded potential and the nonperturbative bion background, respectively, so that their asymptotic expansions can be related to each others by the resurgence relation. Instead of directly looking at the Borel plane, there exits a way to find the same relation from the Stokes graph of the exact WKB analysis, which is so called the Delabaere-Dillinger-Pham (DDP) formula. The DDP formula can be directly applied to any functions of the cycles, and it would also have the important role to see the cancellation of imaginary ambiguities for the partition function, discussed in the later section. From here, we would like to demonstrate some physical applications to (DDP) formula [131,142] to potential-well problems.
For simplicity, let us start with the simple setup which is the double-well potential with low energy shown in Fig. 9. The Stokes graph has four turning points on the real axis labelled by a 1 < · · · < a 4 , and we consider three anticlockwise cycles enclosing a pair of two turning points. These are (a 1 , a 2 ), (a 2 , a 3 ), (a 3 , a 4 ). We named these oriented loops as γ 12 , γ 23 , γ 34 in Fig.9, and the quantities A, B, C-cycles are defined along the each loops. Since the branch-cut lay on the complex x-plane with the endpoints at turning points, the A and C-cycles are now defined on the first Riemann have nontrivial value. In contrast, the B-cycle twice crosses the independently defined branch-cut, and the lower(upper) half contour lies on the first(second) Riemann sheet. This means that the B-cycle has the intersection with each A-and C-cycles once on the first sheet but does not on the second sheet 19 . Under this setup, the DDP 19 If the A(C)-cycle is defined on the second sheet, the relation with the cycle on the first sheet can be found as formula is obtained as follows. Here all symbols such as A, B used so far means Borel summed ones. In order to make this point clear, we will describe its asymptotic form asÃ,B and Borel summed ones as S ± [Ã] and S ± [B] in this sectiopn. The DDP formula is given as where S ± is the Borel resummation for sign(Im ) = ±1. The exponent of (1 + S[B]) is by intersection number (γ A , γ B ) = ±1 which is defined as follows: If the intersection between perturbative and non-perturbative cycles, A and B, is right(left)-handed, we say that (γ A , γ B ) = +1(−1). If a perturbative cycle does not have intersection with non-perturbative cycles, then it gives (γ A , γ B ) = 0. Fig. 10 shows how to determine the intersection number.
In the previous section we separately obtained the quantization condition D for Im > 0 and Im < 0, but one can see that these can be related to each other via the DDP formula: where D + and D − are given by eq.(83) for the positive and negative Im , respectively, but replaced A, B, C with A ± , B, C ± . It is important to mention that from eq.(119) one can indeed show the imaginary ambiguity cancellation for all order of bion sectors. We will summarize the proof in Appendix B.

Unambiguity of the partition function under the Borel resummation
In this section, we will show that the partition function itself does not have Borel ambiguity, that is, the partition function itself does not change due to the sign of the imaginary part of the . To show this statement, we now rewrite the quantization condition in Eq. (83) in a way that clearly shows the imaginary amgibuity part as Then we can show the quantization condition can be expressed as the form without the imaginary ambiguity term (See Appendix B.) We now have lim Arg( )→0± where the symbolĈ is called medianization and is defined as: The A-and C-cycles are related to each other,Â = 1/Ĉ =Ĉ * . S ± (Ω(L)) = −(ABC) −1 = 0 is an invariant quantity under the lateral Borel summation S ± , which does not contribute to the partition function itself.
Therefore, the partition function can be expressed by It is notable that the n-th bion sector can be identified from Eq. (125) by taking a form that where where | • | denotes the 1-norm and we assumed that |s|=n−m s∈N ⊗m−1 0 · · · = 1 when m = 1. Here, we usedD ± A = 1 +Â ±1 andÂ =Ĉ −1 .
By construction, Eq. (121) is invariant under lateral Borel summation so this form of the partition function is also invariant 20 under S ± .

V. GENERIC POTENTIALS (N PLE-WELL POTENTIAL)
The procedure to derive the quantization condition from the exact-WKB analysis and the construction of the pathintegral explained in Sec. IV can be extended to the cases with more generic forms of potentials. Since the extension is straightforward, we would show only the quantization condition and make some comments.
We now focus on a parity-symmetric real-bounded potential, V (x) ∈ R[[x 2 ]] with lim |x|→∞ V (x) = +∞. There are N perturbative and N − 1 non-perturbative cycles as shown in Fig. 11. By repeating the same argument as that in Sec. IV C, the DDP formula can be easily extended to these cases. When the N non-perturbative cycles (B i ) lays on the complex x-plane, the DDP formula for a perturbative cycle (Ã j ) can be expressed as [131] In this notation, we chose the orientation ofB i -cycles such thatB i is exponentially small. Then, by considering the path-orbit of the wave function from the left to the right and using the connection formula, the quantization condition for the positive/negative Arg( ) can be obtained as where n = (n 0 , n 1 , · · · , n N , n N +1 ) ∈ (Z/2Z) ⊗N +2 with n 0 = n N +1 = 0, |n| denotes the 1-norm of n,B 0 =B N = 0, and Φ (n) (Ã) := N k=1Ã n k k . 21 In the similar way to the double-well potential summarized in Appendix B, one finds S + [D + ] = S − [D − ] and shows that the imaginary ambiguity is cancelled for all order of bion sectors. After the cancellation, the quantization condition can be expressed in terms of unambiguous parts as: with the Stokes automorphism being expressed as S 0 . Notice that Ω(L) does not contribute to the path-integral. Because of parity symmetry and the reality condition for the potential, the different cycles have nontrivial relation Thus, eq.(136) is real. Furthermore, one can find the origin of the intersection number (index) of the thimble decomposition, i.e. the Maslov index, in terms of the quantization condition. For example, the quantization condition (133), (134) for N = 3 can be expressed as where We also use the fact that A 3 = A −1 1 and B 1 = B 2 =: B. Therefore, we obtain As shown in Eq. (112), the power (−1) n is the Maslov index of each nonperturbative cycle and is regarded as the intersection number of Lefschetz thimble. In the similar way to Eqs. (113)- (114), the partition function in the asymptotic limit can be expressed as with A1 B means dropping the factor , essentially the negative eigenvalue part. Now, the non-perturbative contribution of the partition function for the triple-well potential can be interpreted as follows. There are two types of bions as shown in Fig. 12: the one that reverses immediately at the turning point (blue) and the other passing through the barrier (green). The former gives B ) and the latter gives ). We can identify how to sum up the non-perturbative contributions and this result shows the structure of partition function as a trans-series 22 . The phase ambiguity is interpreted as the reversal of the direction of the cycle A 2 . We also note the Maslov index is not cancelled like the case of double-well potential because e ±πi E ω A 2 produces (±i) n , not (−1) n . The results in this section clearly indicate that the findings in the paper are generic and applicable at least to the symmetric multi-well potential quantum mechanics. The extension to the other systems is also possible, but it is left for future work.

VI. DISCUSSION AND SUMMARY
In this work, we explored the connections between exact WKB method, saddle point analysis of Euclidean path integration, and Gutzwiller trace formula. Our main findings can be listed as follows: 1. Both exact WKB and saddle point method applied to path integrals take place in complex domain. In exact WKB, position x is complexified, and in path integrals, the space of paths is complexified. Stokes phenomena permeate through both constructions. We showed that the Stokes phenomena in exact-WKB expressed through exact quantization condition maps to the Stokes phenomena in the saddle-point method in path integration.
2. Traditionally, the instanton methods are commonly used to deduce ground state (or few excited state) properties by considering β → ∞ limit. If we wish to derive the instanton contribution to arbitrary energy level, we need to study the partition function at finite β. At this end, we found a new physical interpretation of the quasi-moduli integral (QMI) in the semiclassical analysis of path integral, which relates it to the fluctuation determinant around the harmonic minima. For the perturbative saddle, we are considering fluctuation operator around x cr (τ ) = 0, which ultimately produce harmonic state sums ∞ n=0 e −βn(ω+ 1 2 ) with the perturbative corrections. The QMI at finite-β for the bion configuration also produce the same factor, see 110. Despite apparent difference, this is physically reasonable, because the bion configurations is also almost x cr (τ ) = 0 everywhere, except for the instanton and anti-instanton cores, and this is natural from the perspective of Gutzwiller's quantization condition. This conspiracy has to take place in order the Euclidean path integral to reproduce correctly the level number dependence over the Hilbert space, and we believe this is important for future work.
3. The Maslov index that appears in the non-perturbative contribution is identified with the intersection number of corresponding Lefschetz thimble.
4. We showed the equivalence of Bohr-Sommerfeld, Gutzwiller, path-integral quantization conditions via the exact-WKB analysis. Exact-WKB and Stokes graphs are extremely powerful tools, and we believe that there is much to be learned by using them.
5. The resurgent structures of partition functions naturally continues to hold for generic symmetric multi-well potential, and ambiguity cancellations holds to all orders in semi-classical expansion. This is to a certain extend implicit i exact-WKB construction of exact quantization condition.
Our results not only uncover the unknown facts on quantization conditions and resurgent structures in quantum mechanics but also exhibit that the exact-WKB method and the Stokes curves can be powerful tools to study physical problems. Below, we describe a couple of examples to which we hope to apply our methods • In semi-classics approach to Euclidean path integral formulation, the action must be complexified at the beginning to determine the set of saddles that can possibly contribute. But it is not always easy to know which complex saddles contribute and which do not. Exact WKB method, via the use of basic Stokes graph, immediately answers this part of question. Furthermore, it can also be used to determine the Stokes multipliers of the corresponding saddles. There is clearly much to be learned in semi-classical approach to path integral by using the knowledge of exact (complex) WKB.
• A streamlined construction of the semi-classical expansion of Euclidean partition function at finite β, which is capable of addressing all the states in the Hilbert space, addressing both below and above the barrier in the spectrum, and not just restricted to few lowest lying states as usually done in literature.
• Analysis of graded partition functions in quantum mechanical systems coupled to Grassmann valued fields (e.g. supersymmetric or quasi-exactly solvable systems) or equivalently, Wess-Zumino terms, i.e, path integral of a particle x(τ ) with intrinsic spin. We hope to discuss both path integral as well as the exact-WKB construction for these systems.
• Quantum mechanics of particles on S 1 in the presence of a potentials − cos(N x), x ∼ x + 2π and topological theta angle. Study of such systems using exact WKB methods would be interesting.
• Detailed investigation of classical and quantum tilted multi-well systems by using exact-WKB.
• An outstanding question is whether the constructive (low-order/low order) resurgence between P/NP sectors e.g. [28,54], (eg. connecting pertubation theory around perturbative vacuum to perturbation theory around instanton) can be generalized to systems with higher genus with the help of exact WKB formula. We now give a brief review on the quasi-moduli integral in quantum mechanics. One of the methodologies for studying nonperturbative aspects of quantum theory in the Euclidean setup is the dilute instanton gas approximation (DIGA), where we ignore the interaction between the instantons. However, the recent study on quantum mechanics and low-dimensional quantum field theories uncovers that DIGA is not sufficient in terms of the nontrivial relation, called resurgent relation, between large-order growth of perturbative series and nonperturbative contribution from instanton-anti-instanton pair (bion) [32,37,43,48,80,81,84,85,94]. Here, both of the perturbative Borel resummation and the nonperturbative bion contribution are accompanied with imaginary ambiguities, which are cancelled out. This bion amplitude is calculated via quasi-moduli integral (QMI) of the interaction potential between instanton and anti-instanton. Quasi-moduli parameters are not genuine moduli but come to be moduli in the well-separated limit of the distance of instanton and anti-instanton [33,37,43,48,94].
One of the properties of the quasi-moduli integral is that the results includes gamma functions. Below, we show this fact briefly. For instance, the effective potential for a bion in the double-well quantum mechanics is given by where τ stands for the distance between instanton and anti-instanton, which is nothing but a quasi moduli in this case. The last term τ corresponds to the deformation term originating in quantum-mechanical fermionic degrees of freedom. We are interested only in the bosonic quantum mechanics, thus we take a → 0 limit in the end of calculation. It is shown in Ref. [33] that this integral is performed by complexifying τ and applying the Lefschetz thimble decomposition of the integration contour, which corresponds to the thimble decomposition of the complexified path integral associated with complex saddle points. We instead calculate it in a distinct but equivalent manner. In this method, since the integral is not convergent in a small region, we have to take a prescription called Bogomol'nyi-Zinn-Justin prescription [14], where we first regard − as positive-valued and take analytic continuation as − = e ±iπ in the end. The bion contribution to the partition function is calculated via the quasi-moduli integral as where S I = 1/(3 ) is the instanton action and β is an imaginary time period. One finds a gamma function emerges in this result. By taking analytic continuation as − = e ±iπ , where the sign corresnponds to the direction of its analytic continuation, and also it is regarded as the Stokes phenomena of this integral. The symmetric double well corresponds to taking → 0 limit but its gives 1/ pole, which is known as 1/ problem. In Ref. [49], the above calculation of the quasi-moduli integral for double-well quantum mechanics is extended to the n-bion contribution, which is obtained as We use the following relation, Here the ± is corresponded to the sign of Im( ), which comes from Stokes phenomenon of this quasi-moduli integral. the symmetric double well then, we obtain the final expression as QM I n ( → 0) = 1 2πin The partition function of symmetric double well is this form: The two Z 0 is from the two vacua 23 and B is a bion contribution (with its fluctuation). The rest of summation is from multi-bion and the linear β is from the translation symmetry of (imaginary) time dependent solutions. If we set V i (τ i ) = 0, which means the dilute gas approximation, the QMI becomes Then the partition function is This is usual dilute instanton gas approximation of symmetric double well potential.
Appendix B: Imaginary ambiguity cancellation for the double well potential In this appendix, we would show that for the symmetric double-well potential where S ± [D ± ] is given by eq.(119). In order to avoid confusion, we would like to employ the following notation for any cycles C( ) ∈ {A( ), B( ), C( )}: C( ) : Asymptotic expansion or transseries, e.g. n a n n or e −c/ n b n n , thenĈ( ) ∼C( ) for any cycles and the ambiguity δĈ( ) can be evaluated from the DDP formula, (116)- (118). Since 23 If we do the calculation with finite, we need to take into account the differences of the energy of each vacuum and QMI.See [49] the DDP formula gives We would rewrite the quantization condition in terms of A( ) and C( ). From the explicit calculation, one can make sure that the DDP formula for the quantization condition (119) Notice that a 1 = −a 4 and a 2 = −a 3 . Hence, the contour integration can be written as where 0 < δ 1,2 1 ∈ R + . Since S odd (x, ) = S odd (−x, ) for the symmetric double-well potential, one finds thatÂ( ) = 1/Ĉ( ). In addition, the complex conjugation for A-cycle gives [Â( )] * = 1/Â( ), which implies that [Â( )] * =Ĉ( ). Therefore, eq.(B11) is real, which gives the statement (B1).
Notice that this discussion is applicable to any bounded potentials preserving parity symmetry and reality condition.