Non-BPS exact solutions and their relation to bions in ${\mathbb C}P^{N-1}$ models

We investigate non-BPS exact solutions in ${\mathbb C}P^{N-1}$ sigma models on ${\mathbb R}^1 \times S^{1}$ with twisted boundary conditions, by using the Din-Zakrzewski projection method. We focus on the relation of the non-BPS solutions to the ansatz of multi-instanton (bion) configurations and discuss their significance in the context of the resurgence theory. We find that the transition between seemingly distinct configurations of multi-instantons occur as moduli changes in the non-BPS solutions, and the simplest non-BPS exact solution corresponds to multi-bion configurations with fully-compressed double fractional instantons in the middle. It indicates that the non-BPS solutions make small but nonzero contribution to the resurgent trans-series as special cases of the multi-bion configurations. We observe a generic pattern of transitions between distinct multi-bion configurations (flipping partners), leading to the three essential properties of the non-BPS exact solution: (i) opposite sign for terms corresponding to the left and right infinities, (ii) symmetric location of fractional instantons, and (iii) the transition between distinct bion configurations. By studying the balance of forces, we show that the relative phases between the instanton constituents play decisive roles in stability and instability of the muli-instanton configurations. We discuss local and global instabilities of the solutions such as negative modes and the flow to the other saddle points, by considering the deformations of the non-BPS exact solutions within our multi-instanton ansatz. We also briefly discuss some classes of the non-BPS exact solutions in Grassmann sigma models.


I. INTRODUCTION
Recent intensive study on the resurgence theory and the complexified path integral in quantum field theories and quantum mechanics has revealed the significance of topological soliton molecules, which are composed of (fractional) instantons and anti-instantons . Imaginary ambiguities arising in amplitudes of topologically neutral configurations of composite solitons can cancel out those arising in non-Borel-summable perturbative series in certain quantum theories [7-16, 18-21, 23, 24, 35-37]. In particular, in field theories on compactified spacetime, these objects are termed as "bions" [7][8][9][10]. It is expected that a full semi-classical expansion in perturbative and non-perturbative sectors as bions, which is called a "resurgent" trans-series , leads to unambiguous and self-consistent definition of field theories in the same manner as the conjecture in quantum mechanics [65][66][67][68][69][70][71][72]. It is also discussed that the complexification of variables and parameters in path integrals play significant roles in the resurgence theory [29,30].
In order to reach deeper understanding on bions and the related physics, it is of great importance to study examples in the field theory models in low-dimensions such as CP N −1 models [9, 10, 19-21, 23, 24], principal chiral models [13][14][15] and quantum mechanics [15][16][17]21]. In particular, the CP N −1 model in 1+1 dimensions has been studied for a long time as a toy model of the Yang-Mills theory in 3+1 dimensions, because of similarities between them such as dynamical mass gap, asymptotic freedom and the existence of instantons. The CP N −1 model on R 1 × S 1 with Z N -twisted boundary conditions admits fractional instantons (domain-wall instantons) as Bogomol'nyi-Prasad-Sommerfield (BPS) solutions with the minimal topological charge (Q = 1/N ) [73,74] (see also Refs. [75]). A part of the nonperturbative contributions in resurgence theory comes from bion configurations, consisting of a fractional instanton and anti-instanton. They are most conveniently obtained by means of an ansatz which reduces to solutions of field equations asymptotically at large separations of constituent fractional (anti-)instantons [19][20][21]24]. It has been found that the two-body forces change its character from attractive to repulsive as the relative phase of these fractional (anti-)instantons varies [19].
Non-BPS exact solutions in the CP N −1 model on R 1 ×S 1 with a Z N -twisted boundary condition are found [11], by means of the Din-Zakrzewski projection method [76][77][78] generating a tower of non-BPS solutions from a BPS solution. The simplest non-BPS solution that they found corresponds to the configuration composed of one doubly-compressed fractional instanton (Q = 2/N ) and two fractional anti-instantons (Q = −1/N ) for N ≥ 3. It is not yet clear how the balance of forces is achieved in the non-BPS exact solutions, in contrast to the naive configuration composed of the double fractional instanton and two fractional anti-instantons, where the attractive force exists among the nearest constituents [19]. On the other hand, the non-BPS exact solutions have been known to have unstable directions (negative mode) [76]. However, the local and global instabilities such as negative modes and the flow to the other saddle points have not been explicitly worked out for the non-BPS exact solutions. It is also observed recently that the non-BPS exact solutions exhibit unexpected properties with respect to the clustering [18].
The purpose of this work is to investigate the properties of non-BPS exact solutions [11] by using the ansatz of multi-instanton configurations, which is used to obtain the nonperturbative contributions needed for the resurgence [19][20][21]24]. We find that the non-BPS exact solutions are contained as subspaces of the parameter space of the ansatz, giving (a part of) the multi-instanton contributions, and hence they play a role in resurgence. Relative phases between constituent fractional instantons are found to play a decisive role to achieve the balance of forces. We also find that the transition between seemingly distinct configurations occurs as moduli parameters change in the non-BPS solutions, and the simplest non-BPS solution contains a multi-bion configuration with fully-compressed double fractional instantons. This fully-compressed double fractional instanton eliminates the strong phase dependence of two-body forces between a fractional instanton and anti-instanton [19], and helps to achieve the balance of forces in the non-BPS exact solution. We obtain a generic pattern of transitions among distinct multi-bion configurations (flipping partners), which helps to list various flipping partners in the non-BPS exact solutions. We observe the three essential properties of the non-BPS exact solutions : (i) opposite sign for terms corresponding to the left and right infinities, (ii) the symmetric location of instantons, and (iii) the transition between flipping partners. The balance of forces in the non-BPS exact solution is examined step by step, and many-body forces related to the relative phases are found to be important. We study the local and global instabilities of the non-BPS exact solution and find physical meaning of the negative modes and the flow to the other saddle points. We also briefly discuss some classes of the non-BPS exact solutions in Grassmann sigma models. This paper is organized as follows. In Sec. II, we review the multi-bion ansatz and introduce the simplest non-BPS solution in the CP N −1 model on R 1 × S 1 with a Z N -twisted boundary condition. In Sec. III we discuss the transition between seemingly distinct configurations in the non-BPS solutions, and conjecture a generic structure of non-BPS solutions diagrammatically. In Sec. IV we show how the balance of force exists in the non-BPS solutions, and discuss essential properties of the solutions. In Sec. V we investigate local and global instabilities of the non-BPS solutions by deforming them within the multi-bion ansatz, and discuss the number of negative modes. In Sec. VI we briefly discuss the simplest classes of non-BPS solutions in Grassmann sigma models. Section VII is devoted to a summary and discussion.
The unnormalized vector field ω is also called a moduli matrix, since it contains entire informations of moduli for BPS solutions [79]. We denote the total action as S and the total topological charge as Q. The covariant derivative D µ = ∂ µ + iA µ is defined in terms of the composite gauge field A µ . One should note that the unnormalized vector field ω(x) multiplied by arbitrary nonsingular functions should be identified, since it gives the same physical field n(x). We compactify the x 2 direction as x 2 + L ∼ x 2 , and impose the Z N -twisted boundary conditions on n and ω ω(x 1 , x 2 + L) = Ω ω(x 1 , x 2 ), Ω = diag. 1, e 2πi/N , e 4πi/N , · · · , e 2(N −1)πi/N .
Whenever a single component dominates, the normalized vector n becomes a vacuum configuration as (0, · · · , 1, · · · , 0). Consequently the action density s(x) and topological charge density q(x) are as depicted in Fig. 1. This kinky configuration has been understood as a D-brane configuration [80]. The condition ae − 2π NL x 1 = 1 gives the transition point at x 1 = N L 2π log a, which is defined as the position of the fractional instanton. Because of the translational invariance, integrated total action does not depend on the position of the fractional instanton. Moreover, the action density s(x) and topological charge density q(x) do not depend on the phase angle θ a .
It is important to note that both s(x) and q(x) for multi-fractional instantons (|Q| < 1) are independent of x 2 . Even if we have many fractional instantons at varius locations, x 2 appears only as the phase factor e −i 2π NL x 2 common to all terms in the same component due to the twisted boundary conditions (4). Then s(x) and q(x) do not depend on x 2 . In such a situation, we can take the compactification limit L → 0, and obtain the domain wall situation that has been studied extensively [81,82]. If and only if (more than) N fractional instantons coexist (not far apart), an additional term with a distinct x 2 dependent phase (by an additional amount e −i 2π L x 2 ) coexits in the same component. In such a situation with |Q| ≥ 1, x 2 dependence of s(x) and q(x) emerges.
From now on, we take the compactification scale L = 1 and the coupling constant g 2 = 1, for simplicity. Before discussing the non-BPS exact solutions in CP N −1 models with the Z Ntwisted boundary conditions (4), we introduce an ansatz [19,23,24] for multi fractional instanton configurations that reduces to solutions of field equations for asymptotically large separations of constituent fractional instantons, and carries the correct number of moduli for each individual constituent: its phase and position in x 1 . Therefore the ansatz serves as a basis to obtain nonperturbative contributions by integrating over the (quasi)-moduli of the multi fractional instantons.
Among them, we have bions (one-fractional-instanton + one-fractional-anti-instanton) that play a vital role to achieve a resurgent trans-series.
Let us first construct the ansatz for a bion as a non-BPS configuration as depicted in Fig. 2, by adding a fractional anti-instanton to the left of the fractional instanton (in Fig. 1). To return to the second vacuum for large negative x 1 , we need to add a term proportional to e − 2π N (z+z) in the second component in order to satisfy the twisted boundary condition (4). Thus the ansatz ω(x) for a bion in CP N −1 models is given by with a, b ∈ R, −π < θ a , θ b ≤ π. As x 1 varies from negative to positive values, the normalized vector n makes transitions between two different vacua as The first transition point gives the position of the left fractional anti-instanton and the second transition point gives the position of the right fractional instanton Thus, the separation R between these instanton constituents is whereas the center of mass position is given by x cm = N 4π log b. The configuration associated with ω in Eq.(7) for R > 1 is depicted in Fig. 2.
We note that action density s(x) and topological charge density q(x) depend on the (relative) phase θ b of two terms in a single component, but are independent of the phase θ a . Because of translational invariance, the total energy is independent of the center of mass position x cm . Therefore we consider densities s(x) and q(x) as functions of two relevant parameters : the separation R between the fractional instanton and anti-instanton and the relative phase θ b between them. We also note that densities s(x) and q(x) of this configuration are independent of the coordinate x 2 of the compactified direction, since no instantons (|Q| ≥ 1) is invloved in any region of x 1 .
As shown in [19,23,24], by decreasing this separation R from positive to negative values, the value of the total action changes from S = 2/N to S = 0. We can define the effective interaction potential between the two constituents as V eff (R, θ b ) = S − S I − SĪ = S − 2 N , representing the potential for the static two-body force between the consitituent fractional instanton and fractional anti-instanton. For large separations R, we have found [19,23,24] that The notable point is that the interaction is attractive (repulsive) for |θ b | ≤ π/2 (π/2 ≤ |θ b |). Based on the extended Bogomolnyi-Zinn-Justin prescription [65,66], it has been shown that the bion configuration in the CP N −1 model produces the imaginary ambiguity [21], which is expected to cancel the imaginary ambiguity in the non-Borel-summable perturbative contributions similarly to the case in the quantum mechanics [65,66].
Let us next introduce an ansatz for two bions. For simplicity, we assume N ≥ 3 [87]. In order to add one more bion to the left of the single bion in Eq.(7) as in Fig. 3(a), we need to add a term ce − 2π N (2z+z) (with a complex constant c) in the first component, and another term de − 4π N (z+z) to the second component. We will call this diagram in Fig. 3(a) asĪIĪI. If we wish to consider the diagram with orderingĪIIĪ in Fig. 3(b), we need to have a term f e − 2π Nz in the third component.
If we wish to consider the diagram with ordering IĪĪI in Fig. 3(c), we need to have a term general configuration for two bions (with no (anti-)instanton with |Q| = 1 in any region of with seven complex parameters a, b, c, d, f, g, ∈ C. One should note that one more diagram with the ordering IĪĪI in Fig Let us first consider the case with |g| ≫ |c| and |a| ≫ |f |, where the terms with c and f are superficially negligible. Then, as x 1 varies from negative infinity to positive infinity, the dominant term changes as de − 4π for a choice of the parameters |g| 2 ≫ |b||d|, |b| 2 ≫ |a||g|, |a| 2 ≫ |b|. It means that the vacua undergo the transition 0, 1, 0 → 0, 0, 1 → 0, 1, 0 → 1, 0, 0 → 0, 1, 0 , as shown in Fig. 3(c). Using a similar reasoning as the bion case, we find the four fractional instanton constituents show up successively as x 1 increases from negative infinity to positive infinity : the first fractional instanton at R 1 = N 2π log |d| |g| , the first fractional anti-instanton at R 2 = N 2π log |g| |b| , the second fractional anti-instanton at R 3 = N 2π log |b| |a| , and the second fractional instanton at R 4 = N 2π log |a|. The three separations between these instanton constituents are The configuration in Fig. 3(c) is visible in the parameter region |d/g| ≪ |g/b| ≪ |b/a| ≪ |a|.
On the other hand, we can recognize the configuration in Fig. 3(b) in another region of the parameter space: the case with |c| ≫ |g| and |f | ≫ |a|, where the terms with g and a are superficially negligible. Then we obtain transitions of vacua as instead of Eq. (14). As x 1 increases from negative infinity to positive infinity, four fractional instanton constituents show up successively : the first fractional anti-instanton at R ′ 1 = N 2π log |d| |c| , the first fractional instanton at R ′ 2 = N 2π log |c| |b| , the second fractional instanton at R ′ 3 = N 2π log |b| |f | , and the second fractional anti-instanton at R ′ 4 = N 2π log |f |. Thus, the three separations between these instanton constituents are The configuration in Fig. 3(b) is visible in the parameter region |d/c| ≪ |c/b| ≪ |b/f | ≪ |f |.
We note that there is a relation among six separations , leading to five independent separation variables. In both of these parameter regions corresponding It is interesting to note that the most general ansatz for two bions automatically contains two other possible diagrams containing the orderingĪIĪI in Fig. 3(a), or IĪIĪ in Fig. 3(d). The configuration ofĪIĪI in Fig. 3(a) is visible in the parameter region |d/c| ≪ |c/b| ≪ |b/a| ≪ |a|.
In this parameter region, the relative phases (a, c), and phases of b and d are relevant among the four phase parameters. The configuration of IĪIĪ in Fig. 3

(d) is visible in the parameter region
|d/g| ≪ |g/b| ≪ |b/f | ≪ |f |, where only three phases are relevant : the relative phase (g, f ) and phases b and d. It is interesting to note that a different partial set of four relative phases are relevant in each parameter region corresponding to four different diagrams in Fig. 3. One should also note that there are five independent length parameters, but only three different combinations of them emerge as separations of constituent fractional (anti-)instantons in different parameter regions. In the next section, we will demonstrate that this ansatz (13) contains the simplest non-BPS exact solution. This fact implies that the non-BPS exact solution contributes to the resurgent expansion as a special configuration of two-bion configurations.

B. The simplest non-BPS exact solution
Based on the procedure of projection operations [76], the non-BPS exact solutions in CP N −1 model on R 1 ×S 1 with Z N -twisted boundary conditions are constructed in [11]. We here discuss the properties of the solutions. The non-BPS solutions are obtained through the following projection applied to any of the BPS solutions ω, with S = 2/3, Q = 2/3 shown in Fig. 4, the projection produces the non-BPS exact solution The total action of this solution is S = 4/3 while the total topological charge is Q = 0. This ω is obviously a special case of the generic two-bion configuration (13). We note that the action s(x) and topological q(x) charge densities are independent of overall phase variables θ 1 , θ 2 in each component. The exact solution (20) is included as a subspace of parameters in our most general ansatz in Eq. (13) as : What is special in this solution is that this solution contains in different corners of moduli space the two seemingly distinct configurations, each of which can be seen as a compressed case of the two fractional (anti-)instanton configuration in the middle, because of b = 0. As depicted in Fig Let us call these two configurations as the flipping partners. We show how densities of action s(x) and topological charge q(x) vary as 2l 1 /l 2 2 varies in Fig. 6, although the value of the integrated total action remains constant. We find the action densities of flipping partners are related by the We will find that the flipping partners contained in the non-BPS exact solution is important to achieve the balance of forces between the BPS and anti-BPS constituent fractional instantons.
One should also note that the two separations R 1 (R ′ 1 ) and R 2 (R ′ 2 ) from the middle compressed fractional instantons to the left and right fractional instantons are identical : for 2l 1 ≫ l 2 2 and for 2l 1 ≪ l 2 2 .

A. Flipping partners in the non-BPS exact solution
Let us analyze how transitions between different configurations occur in the non-BPS exact solution in the CP 2 model as shown in Fig. 5 (the extension to CP N −1 models is straightforward).
We can re-express the BPS solution with two fractional-instantons in Eq. (19) as with α = 2π/3. Then, the projection operation in Rq. (18) gives a non-BPS exact solution in Eq. (20), which is now re-expressed as where the difference by an overall factor from Eq.

(a)Fractional anti-instantons
When ABB in the first component and −BAĀ in the second component are dominant, the unnormalized vector field ω is equivalent to a simpler one as because the division of ω by a common factor AB to all components gives identical physical vector n. Since |B| = l 2 e −αx 1 and |Ā| = l 1 e −2αx 1 , the vacua undergo transition as (0, 1, 0) → (1, 0, 0), implying a fractional anti-instanton at x 1 = (1/α) log(l 1 /l 2 ). In the same manner, when B in the second component and BB in the third component are dominant, we obtain which implies the vacuum transition (0, 0, 1) → (0, 1, 0) and a fractional anti-instanton at implying the vacuum transition (0, 1, 0) → (0, 0, 1) and a fractional instanton at x 1 = (1/α) log(l 2 /2). These two fractional instantons correspond to the constituents of the original BPS solution in Fig. 4.

(d)Double fractional anti-instantons
When 2A in the first component and 2AĀ in the third component re dominant, we obtain implying the vacuum transition (0, 0, 1) → (1, 0, 0), and a compressed double fractional antiinstanton at In Figs. 9 and 10, we show the case for three fractional instanton constituents in the starting the instanton localizes in the two-dimensional x 1 − x 2 space when they get closer. Such a situation was discussed in [11].
If two fractional instantons are compressed into a single double fractional instanton in the starting BPS solution as in the left panel of Fig. 11, we find vanishing instanton charge for the non-BPS exact solution, which has only two types of configurations as in Fig. 12. It is interesting to observe that the compression of the two fractional instanton in the starting BPS solution (32) corresponds to unexpected movement of various constituents in the non-BPS solution (33) : the left-side fractional instanton moves to negative infinity. This is the reason why the topological charge in the visible (finite) region of x 1 changes from unity to zero, resulting in Fig. 12. This is a manifestation of the unusual clustering property of the non-BPS exact solutions compared to the starting BPS solutions, that was observed previously [18].

IV. BALANCE OF FORCES IN THE NON-BPS EXACT SOLUTION
To understand the reason why no force is acting on various constituents of the non-BPS exact solution, we first analyze in terms of two-body forces between solitons in various configurations and will reveal the presence of three-body forces. We consider the CP 2 model for simplicity.

A. A bion configuration :ĪI
We begin with the bion configuration in Fig. 2 as the simplest non-BPS configuration. The effective interaction potential in Eq. (12) between the two constituents for large separation R is attractive for the relative phase |θ b | < π/2, but is repulsive for |θ b | > π/2. Consequently the total action at θ b = π/2 is flat along the R direction for large separations, satisfying a necessary condition to be an exact solution. However, the total action has a positive θ b derivative, and the bion cannot be a stationary point of the action. We need to remedy the strong θ b dependence to achieve the balance of force.
These features of the two body force is clearly visible in Fig. 15, where we depict the total

B. A fractional anti-instanton and a bion :ĪĪI
In order to eliminate the strong dependence on the relative phase, we next consider the addition of a fractional anti-instanton to the bion configuration from the small x 1 side as shown in Fig. 16. (34) is depicted for CP 2 model.

FIG. 16: The configuration of fractional anti-instanton + bion in
The configuration within our ansatz is given by with a, b, g ∈ R and −π < θ a , θ b , θ g ≤ π. The action and topological charge densities are independent of θ a and θ g . The separation of two fractional anti-instantons in the left is RĪĪ = 3 2π log b 2 ag and the separation between the middle fractional anti-instanton and the instanton is R bion = 3 2π log a 2 b . For large separations RĪĪ and R bion , the total action depends only on R bion and the phase θ b , and is independent of RĪĪ, reflecting the absence of static forces between fractional instantons.
Therefore we obtain no improvement by adding the fractional anti-instanton, as long as they are well separated.
If we let b → 0 in Eq. (34), the left-most fractional instanton is compressed into the middle fractional instanton. In the completely compressed limit b = 0, we obtain the unnormalized vector field ω as The configuration contains one fully-compressed double fractional anti-instanton (ĪĪ) and one fractional instanton I as shown in Fig. 17  (1, 0, 0) → (0, 1, 0), as x 1 varies from x 1 = −∞ to x 1 = ∞. The separation between the two constituents is given by R = (3/(4π)) log(a 3 /g)). We emphasize that this configuration is independent of all the phase parameters θ a , θ b , θ g , since the action density and topological charge density is independent of phases θ a , θ g for each component, and the phase θ b disappears in the limit of However, the approximation with two-body forces between the compressed double fractional anti-instanton and a fractional instanton is no longer valid, if we compress these two constituents further by letting a small with g (the center of mass position) fixed. In Fig. 18, we depict the action density (left) and topological charge density (right) for three values of the parameter a = 1/100, 1/10, 100 (from top to bottom) with g = 1 fixed. We can clearly see that the total action is not equal to an absolute value of the topological charge, even when the two constituents are compressed (a = 100).
In Fig. 19, we depict the total action as a function of the separation R = (3/4π) log(a 3 /g).  If we restrict the three parameters a, d, g in terms of two parameters l 1 , l 2 as a = 2l 1 /l 2 , d = l 2 1 , g = −2l 2 1 /l 2 , we obtain If the phase θ vanishes, the configuration becomes equivalent to dominant terms of the non-BPS exact solution (20) in a parameter region 2l 1 ≫ l 2 2 , representing one of the configurations in Fig. 5(a). We see that two separations are equal : R = R 1 = R 2 = (3/4π) log(4l 1 /l 2 2 ). Thus, the energy density is symmetric under the reflection around the middle compressed double fractional instanton. We call this symmetry as reflection symmetry. If the reflection symmetry is broken, the balance of force fails, as we will see later. Thus, the reflection symmetry is one of the essential properties of the non-BPS exact solutions.  Fig. 20 shows the total action of (37) as a function of the phase parameter θ and the parameter l 1 with l 2 = 1 fixed, which gives the separation R = (3/4π) log(4l 1 /l 2 2 ). For θ = π, in which the two terms in the second component in (37) have the same sign, the action decreases most rapidly as l 1 gets smaller (R gets smaller), thus the effective force is attractive. For θ = 0, in which the two terms in the second component have the opposite sign, the action increases as l 1 gets smaller (R gets smaller), thus the effective force is repulsive. If one wants to achieve the balance of force to obtain a solution of field equations, it is a fatal flaw to have an attractive interaction, since the configuration tends to decay into vacuum in our case of topologically trivial sector. This is in accordance with the sign of the two terms of the second component in the non-BPS exact solution (20).
It is important to realize that the force we have discussed here is a three-body force involving three constituent solitons : θ is the relative phase between the two terms corresponding to the left-most and right-most vacua. If one focuses on the two constituents, for example, the left anti-instanton and central double instanton, they have attractive force for any θ (only at small separations), as we have seen. Moreover, the reflection symmetry is also important, since the two-body force between constituents with shorter separation dominates to give the attractive force whenever the reflection symmetry is broken. This is why the reflection symmetry is also essential in the non-BPS exact solution. Now, the only question is how the non-BPS exact solution suppresses the repulsive force resulting in the total action S = 4/3 for any values of R at θ = 0 and the vanishing derivative with respect to θ. Compared to the present case, the non-BPS exact solution (20) has the other terms, which produce the flipping-partner configuration for l 2 2 > 2l 1 as shown in Fig. 5. We have to conclude that these terms for the flipping partner also work to avoid the increase of the total action for the parameter range 2l 1 > l 2 2 , even though these terms appear to be irrelevant at a glance. Thus, the existence of flipping partners is also essential in the non-BPS exact solution.

D. Essential properties of non-BPS exact solutions
To sum up, we list essential properties of the non-BPS exact solution (20), which give the constant total action without annihilation into vacuum in the entire moduli space.

Relative sign
The terms in the solution have the appropriate relative signs : these signs serve to suppress the dominant two-body attractive forces, and to provide many-body repulsive forces suppressing a decay into vacuum.

Reflection symmetry
The reflection symmetry around the middle compressed double fractional instanton prevents the dominance of attractive two-body forces.

Flipping partner
The non-BPS exact solution makes a transition between two seemingly different configurations (flipping partners), as moduli parameters are varied across the point where all the constituents get closer. They never annihilate each other even though no topological quantum number guarantees the flatness of the total action. This transition is essential to provide the many-body forces to cancel the two-body and three-body forces in configurations without the flipping partners.
All the configurations we have discussed including the non-BPS exact solutions correspond to some special cases of the generic two-bion ansatz in (13). We emphasize that this ansatz provides the multi-instanton computation needed to achieve the resurgence. Therefore, the non-BPS exact solutions contribute to a part of the multi-instanton moduli integral relevant to the resurgence theory.

V. LOCAL AND GLOBAL STABILITY
It has been pointed out that the non-BPS exact solutions have unstable modes [76]. In this section, we analyze the local stability of the non-BPS exact solution in Eq. (20) to understand the physical meaning of unstable negative modes within the parametrization of our general ansatz in Eq. (13). We focus only on the simplest non-BPS solution in the CP 2 model. As we see in We here omit the common phases θ 1 , θ 2 of the two terms in the first and second components which are included in (20), since the above configurations are independent of them. The argument in Sec.IV C suggests that the total action decreases due to the attractive force between the constituents if we flip the relative sign (θ = 0 → π) between the second component in (20). In Fig. 21, we show the total action as a function of the phase θ and l 2 with l 1 = 1 fixed (we note R = (3/4π) log(l 2 2 /l 1 ) for l 2 2 > 2l 1 ). We also display the total action as a function of θ with the fixed separation l 1 = 1, l 2 = 1 in Fig. 22. We find that the total action decreases when we change θ from 0 to π, indicating a negative mode along theta direction. For the large separation 2l 1 ≫ l 2 2 or 2l 1 ≪ l 2 2 , the total action dependence on θ becomes small (but exists), since the magnitude of the two-body force decreases for small separations.

B. Negative modes for asymmetric separation
In this subsection we violate the second of the essential properties (reflection symmetry) of the non-BPS exact solution by introducing two multiplicative factors γ, γ ′ ∈ R in order to change the left and right separations For γ, γ ′ = 1, the configurations are no longer solutions.
In the parameter region 2l 1 > l 2 2 , we have a configuration similar to that in Fig. 5(a). However, the separation between the left instanton and the middle compressed double anti-instanton is decreased by γ > 1.
Similarly, the separation between the middle compressed double anti-instanton and the right instanton is decreased by In the parameter region 2l 1 < l 2 2 , we have another configuration similar to that in Fig. 5(b). The separation between the left anti-instanton and the middle compressed double instanton is decreased by γ > 1, and the separation between the middle compressed double instanton and the right anti-instanton is decreased by for the γ → ∞ limit, where the left fractional anti-instanton is fully compressed with the double fractional instanton. The value 5/9 of the action comes from the compressed part of fractional anti-instanton + double fractional instanton as found in Fig. 19. Similarly, increasing γ ′ with γ fixed is found to be a negative mode, leading to a configuration of fully compressed fractional double fractional anti-instanton + instanton together with a (almost) noninteracting fractional instanton to their left.
We next consider the deformation combining (38) and (39), As we have discussed in the previous section, this configuration behaves differently with θ = 0 and θ = π. In both cases, the increase of γ > 1 with γ ′ = 1 or the increase of γ ′ > 1 with γ = 1 decreases the total action from S = 4/3 to S = 8/9 since the double fractional instanton and the fractional anti-instanton at each side have attractive interaction. However, things change if one increases both γ and γ ′ with γ = γ ′ .
For θ = 0, the three instanton constituents have effectively repulsive interaction as we have seen in Fig. 20. Thus the total action increases as one increases γ = γ ′ > 1. In Fig. 25, we depict the total action as a function of R ′ R = 3 4π log In the calculation, we fix l 1 = 1 and l 2 = 1000, and vary γ, γ ′ . Along R ′ R = R ′ L → 0, the action increases. For θ = π, the three instanton constituents have effectively attractive interaction. Thus the total action decreases as one increases γ = γ ′ > 1. In Fig. 26, we depict the total action as a function of R ′ R and R ′ L , where we fix l 1 = 1 and l 2 = 1000, and vary γ, γ ′ . In any direction of decreasing R ′ R , R ′ L (increasing γ, γ ′ ) from the original configuration, the action decreases. Figure  27 shows the total action as a function of R = R ′ R = R ′ L , showing clearly that the total action decreases to S = 0 towards R = R ′ R = R ′ L → −∞. (This corresponds to the curve obtained as a section along R ′ R = R ′ L in Fig. 26 although we change the parameter value in the two figures.) In Fig. 28, corresponding to (45) with 2l 1 < l 2 2 , we depict how the instanton constituents meet and are compressed when γ = γ ′ increases.

C. Number and directions of negative modes
Before discussing the directions associated to the remaining parameters, we quantify the mass squared matrix of fluctuations around the non-BPS exact solution to find the number and directions of negative modes described by γ, γ ′ , θ in Eq. (45). As a natural coordinates of the parameter space, we take R ′ R , R ′ L , θ, since they are associated to the flat metric for kinetic terms at least for large separations. For R = 2.20 (l 2 = 100, l 1 = 1), the separation is relatively large. The second-derivative matrix at γ = 1, γ ′ = 1, θ = 0 is numerically calculated as where the label R ′ R , R ′ L , θ stands for the ordering of rows and columns. By denoting the unit vectors as e R , e L , e θ respectively, we approximately obtain the eigenvalues and the eigenvectors as We find that there are two negative modes in e R − e L and e θ directions in this case.
For R = 1.10 (l 2 = 10, l 1 = 1), the separation is relatively small and the constituents are about to crash. The second-derivative matrix at γ = 1, γ ′ = 1, θ = 0 is numerically calculated as Then the eigenvalues and the eigenvectors are approximately given by In this case too, we find that there are two negative modes in e R − e L and e θ directions.
For R = 0 (l 2 = 1, l 1 = 1), the separation is zero and the constituents crash. The secondderivative matrix at γ = 1, γ ′ = 1, θ = 0 is numerically calculated as Then the eigenvalues and the eigenvectors are approximately given by ∼ 0.025 : e R − e L (56) In this case, there is one negative mode in e θ directions. We should mention that coordinates R ′ R and R ′ L do not have a simple physical meaning as separations at small R region such as R = 0. These results show that the number of negative modes depends on the parameter region. However, the relative phase fluctuation always gives a negative mode.

D.
Splitting of two bions To analyze the stability of the non-BPS exact solution, we still need to consider two more parameters : a separation parameter corresponding to the splitting of the middle compressed double fractional (anti-)instanton, and its associated phase. The splitting can be described by the following ansatz containing a new term with the parameter δe iθ ′ in the second component with δ ≥ 0, 0 ≤ θ ′ < 2π. For δ = 0, the configuration is no longer a solution. In the limit of the exact solution (δ = 0), the phase θ ′ obviously is ill-defined and loses a physical meaning. Therefore we consider here only deformations due to these new parameters δ, θ ′ and postpone the analysis combined with other parameters to the next subsection.
A nonzero δ splits the double fractional (anti-)instanton in the middle of Fig. 5 into two fractional (anti-)instantons, and leads to the two bion configurations as shown in Fig. 3(b) and (c).
We first study the case of large separations between all consitituents, where the two-body forces [19] in Eq. (12) between fractional instanton and anti-instanton is applicable. Since two-body forces are attractive (repulsive) for the relative phase smaller (larger) than π/2, we find that the effective forces between the fractional constituents in both two bions in Fig. 3(b) and (c) become attractive for any values of δ, only at θ ′ = π/2. This is because some of the terms in (58) have relative signs, in such a way that one of the constituent pairs has an attractive interaction while the other has a repulsive interaction for θ ′ = π/2. One should note that two fractional (anti-)instantons experience no static force because they are mutually BPS.
For example, let us consider the parameter region 2l 1 < l 2 2 , corresponding to Fig. 3(b). The fractional anti-instanton in the left end is located at 3 2π log l 1 l 2 , whereas the left fractional instanton emerging from the middle double instanton is located at 3 2π log l 1 l 2 δ . They have a relative phase e iπ /e iθ ′ = e i(π−θ ′ ) , and compose a left bion. Thus the constituents have an attractive force for |θ ′ | > π/2, and repulsive for |θ ′ | < π/2. Similarly, the fractional instanton and anti-instanton in the right bion have a relative phase e iθ ′ , and exhibit an attractive force for |θ ′ | < π/2, and repulsive for |θ ′ | > π/2. If |θ ′ | = π/2 and the separation is small, which is beyond the scope of the two-body force approximation, the strong attractive force between the constituents of bions emerges as shown in Fig. 15. Therefore attractive forces are the strongest around θ ′ = ±π/2 since constituents of both bions are attractive for small separation only at these parameters.
We note that, for 2l 1 > l 2 2 , the situation is similar. (left). The total action as a function of δ with θ ′ = π/2 fixed is also depicted (right).
In Fig 29 and Fig. 30, the total action is depicted as a function of δ and θ ′ for fixed l 1 and l 2 .
One can see that, for θ ′ = 0, π, the total action decreases at first as δ gets nonzero, then at some point it takes a turn and increases. It means that, for these values of θ ′ , the effective force between the fractional instanton and anti-instanton is changed from attractive to repulsive ones at some value of δ. Here we used the parameter δ instead of R to describe the length of the middle vacuum region between the two fractional instantons, since the metric for the two BPS solitons is typically cigar-like and δ is more appropriate at small separations [83]. (left). The total action as a function of δ with θ ′ = π/2 fixed is also depicted (right).
Our observation shows that, for example, if we fix θ ′ = π/2, the parameter δ is identified as one of the unstable modes, which connects the non-BPS solution and the two bion configuration. We show how the action density changes with δ for θ = π/2 in Fig. 31. In both left and right bions, the fractional constituents gradually annihilate into vacuum as they approach each other.

E. Combined effects of deformation parameters
The final step of analyzing the (global) stability of the non-BPS exact solution is to consider the five relevant deformation parameters at the same time. As a most significant effect, we consider the case where the strong phase dependence of two-body forces become most visible. Let us consider the following modification of the non-BPS exact solution with δ ≥ 0, 0 ≤ θ < 2π. We have chosen the phase of the the second term of the second component as −δe − 2π 3 (z+z) , so that the strong phase dependence of two-body forces in Eq. (12) of both bions can collaborate together to act as attractive forces when θ = π.
We depict the total action as a function of the symmetric separation R = 3 2π log l 3 2 δ of the bions at both right and left sides with θ = π fixed in Fig. 32, and compare it to the sum of the total action of each individual left and right bion. Up to numerical errors, these two plots are consistent.
It indicates that the configuration (59) is seen as almost two-bion configurations, and there is no interaction between the two bions.
We note that the bion parts have the attractive interaction. It means that, if we calculate their amplitudes by integrating moduli integrals, we find the imaginary ambiguities, which will be cancelled out combined with the perturbative calculations around the appropriate background. Again, we conclude that the non-BPS solutions can be seen as a special limit of the configurations relevant to the resurgence theory.

VI. NON-BPS SOLUTIONS IN GRASSMANN SIGMA MODELS
In this section, we briefly discussextensions to the Grassmann sigma models Gr N F ,N C ≃ SU (N )/SU (N F − N C ) × SU (N C ) × U (1) with twisted boundary conditions. For this case, we have several ways of constructing non-BPS solutions [77,78]. We introduce two simple classes of non-BPS solutions in the Grassmann sigma models.

A. Simple Extension of CP N −1 Projection
We introduce one simple class of non-BPS solutions in the Grassmann model [77]. which is given by the following projection, with and 0 ≤ k 1 < k 2 < · · · < k Nc ≤ N F − 1 .
ω(z) is a arbitrary holomorphic N F vector. This projection makes N c N F -vectors orthogonal. H 0 is a moduli matrix, which is related to the physical scalar H by H = (H 0 H † 0 ) −1/2 H 0 for this case since H 0 H † 0 is diagonal. We apply this projection to Gr 4,2 . We consider the holomorphic vector ω = l 1 e iθ 1 e −πz , l 2 e iθ 2 e − π 2 z , 1, 0 .
We here drop the phase variables θ 1 , θ 2 since they do not contribute to the action and topological charge densities. This solution is shown in Fig. 33. The total action of this solution is S = 3/2 while the total topological charge is Q = 1/2. It is notable that positions of constituent instantons in different color lines coincide in the configurations of the solution.
This solution is shown in Fig. 34 for l 1 ≪ l 2 2 , l 3 ≫ l 2 4 . The total action of this solution is S = 4/3 while the total topological charge is Q = 0. More generic forms of non-BPS solutions in Grassmann sigma model are discussed in [78]. We will discuss the properties of these non-BPS solutions and their relation to bions in Grassmann sigma model in the future works.

VII. SUMMARY AND DISCUSSION
We can summarize the present work as follows. Firstly, we have studied the non-BPS exact solutions in terms of ansatz for arbitrarily many fractional instantons, which gives solutions of field equations for asymptotically large separations of constituent fractional instantons. Since the ansatz serves as a basis to obtain all the multi-instanton contributions as integrals over the (quasi-)moduli, we find that the non-BPS exact solutions are also included as a part of multi-instanton contributions, which play an important role in the resurgence theory. Secondly, we have studied the balance of forces that assures the non-BPS configuration to be an exact solution. The interaction between a fractional instanton and a fractional anti-instanton is attractive (repulsive) depending on the value of the relative phase modulus θ. From this strong dependence of two-body forces, we found three essential properties of non-BPS exact solutions: (i) the appropriate relative sign in two