Fractional instantons and bions in the O(N) model with twisted boundary conditions

Recently, multiple fractional instanton configurations with zero instanton charge, called bions, have been revealed to play important roles in quantum field theories on compactified spacetime. In two dimensions, fractional instantons and bions have been extensively studied in the ${\mathbb C}P^{N-1}$ model and the Grassmann sigma model on ${\mathbb R}^1 \times S^1$ with the ${\mathbb Z}_N$ symmetric twisted boundary condition. Fractional instantons in these models are domain walls with a localized $U(1)$ modulus twisted half along their world volume. In this paper, we classify fractional instantons and bions in the $O(N)$ nonlinear sigma model on ${\mathbb R}^{N-2} \times S^1$ with more general twisted boundary conditions in which arbitrary number of fields change sign. We find that fractional instantons have more general composite structures, that is, a global vortex with an Ising spin (or a half-lump vortex), a half sine-Gordon kink on a domain wall, or a half lump on a"space-filling brane"in the $O(3)$ model (${\mathbb C}P^1$ model) on ${\mathbb R}^{1} \times S^1$, and a global monopole with an Ising spin (or a half-Skyrmion monopole), a half sine-Gordon kink on a global vortex, a half lump on a domain wall, or a half Skyrmion on a"space-filling brane"in the $O(4)$ model (principal chiral model or Skyrme model) on ${\mathbb R}^{2} \times S^1$. We also construct bion configurations in these models.


I. INTRODUCTION
Instantons have been known for long time to play significant roles in non-perturbative dynamics of quantum field theories such as supersymmetric QCD. Recently, multiple fractional instanton configurations with zero instanton charge, called bions, have been revealed to play important roles in quantum field theories on compactified spacetime . The prime example which has been studied extensively is QCD with adjoint fermions (adj.) on R 3 × S 1 . Bions can be classified into two classes, magnetic (charged) bions carrying a magnetic charge, and neutral bions carrying no magnetic charge. Magnetic bions are conjectured to lead semiclassical confinement in QCD (adj.) on R 3 × S 1 [22][23][24][25][26][27][28][29][30][31][32]. On the other hand, neutral bions are identified as the infrared renormalons in field theory [6][7][8][9][10][11][12][13][14][15][33][34][35], and play an essential role in unambiguous and self-consistent semiclassical definition of quantum field theories in a process known as the resurgence; Imaginary ambiguities called renormalon ambiguities arising in non-Borel-summable perturbative series exactly cancel out with those arising in neutral bion's amplitude in the small compactification-scale regime of QCD (adj.) on R 3 × S 1 . It indicates that the full semi-classical expansion, referred as a resurgent expansion [36], that includes both perturbative and non-perturbative sectors, leads to unambiguous and self-consistent definition of field theories. In quantum mechanics this is known as the Bogomol'nyi-Zinn-Justin prescription [37][38][39].
On the other hand, two dimensional nonlinear sigma model enjoys a lot of common features with four-dimensional Yang-Mills theory [40] such as asymptotic freedom, dynamical mass generation, and instantons [41,42]. We can further expect a similar correspondence between fractional instantons and bions in nonlinear sigma models on R 1 × S 1 and those in Yang-Mills theory on R 3 × S 1 . Fractional instantons in the CP N −1 model [43] (see also Refs. [44]) and the Grassmann sigma model [45] were constructed on R 1 × S 1 with twisted boundary conditions by using the moduli matrix technique [46][47][48][49] (see Ref. [51] as a review) and D-brane configurations [45,52].
Bions and the resurgence have been extensively studied in the CP N −1 model [8-10, 16, 17, 20] and the Grassmann sigma model [18] on R 1 × S 1 . In particular in Refs. [8,9], bion configurations in the CP N −1 model were studied based on the dilute instanton description with taking account of interactions between well-separated fractional instantons and anti-instantons, to show explicitly that the imaginary ambiguity in the amplitude of neutral bions has the same magnitude with an opposite sign as the leading ambiguity arising from the non-Borel-summable series expanded around the perturbative vacuum. The ambiguities at higher orders are canceled by amplitudes of 2 bion molecules and the full trans-series expansion around the perturbative and non-perturbative vacua results in unambiguous semiclassical definition of field theories. Furthermore, neutral bion ansatz beyond exact solutions were found in the CP N −1 model [17] and the Grassmann model [18] in terms of the moduli matrix and was found to be consistent with the results from the wellseparated instanton gas calculus [8,9] from all ranges of separations. Bions and resurgence were also studied for principal chiral models [12,15] and quantum mechanics [11,13,14].
In order to understand more precise structures of fractional instantons and bions in generic field theories, it is worth to remind that fractional instantons in the CP N −1 and Grassmann models on R 1 × S 1 with the Z N twisted boundary conditions have a composite soliton structure [43,45].
When the coordinate x 2 is a compact direction, fractional instantons are domain walls extending to the x 2 direction (perpendicular to the x 1 direction) whose world volume a U(1) modulus is localized on and twisted half along. Fractional instantons can be therefore regarded as half sine-Gordon kinks on a domain wall. Since a domain wall carries unit instanton (lump) charge when the U(1) modulus is twisted once (full sine-Gordon kink) [53][54][55][56], the above configuration carries half instanton charge [57]. In this paper, we refer the above domain wall and sine-Gordon kink as a host soliton and daughter soliton, respectively. The simplest among CP N −1 model and the Grassmann model is the CP 1 model, which is equivalent to the O(3) sigma model described by a unit three-vector of scalar fields n = {n A (x)} (A = 1, 2, 3) with n 2 = 1. The Z 2 symmetric boundary condition reduces to (n 1 , n 2 , n 3 )(x + R) = (−n 1 , −n 2 , +n 3 )(x) in this notation.
This paper is organized as follows. In Sec. II, we first give the O(N) model. In Sec. III, we provide a general framework to construct fractional instantons as composite solitons in the O(N) model with the twisted boundary conditions. In Secs. IV and V, we discuss fractional instantons and bions in the O(3) model on R 1 × S 1 and the O(4) model on R 2 × S 1 , respectively, with the twisted boundary conditions. Sec. VI is devoted to a summary and discussion. We present a list of modification of the models which may make fractional instantons to be local or BPS.

II. O(N ) MODEL
We consider an O(N) nonlinear sigma model, whose Lagrangian is given by with N-component scalar fields n = (n 1 (x), n 2 (x), · · · , n N (x)) T with a constraint n 2 = 1. We have to consider higher derivative (or the Skyrme) term L h.d. to stabilize (fractional) instantons in higher dimensions higher than two or three, or two dimensions with a potential term. In some cases, we also consider a potential term V (n) for the stability of fractional instantons. We compactify the x N −1 coordinate to S 1 with a period R.
The target space of the model is M ≃ S N −1 which admits topological textures, sine-Gordon kinks (N = 1), lumps [41] or baby Skyrmions [61,62] (N = 2), Skyrmions [58] (N = 3). The topological instanton charges π 2 (S 2 ), π 3 (S 3 ) can be written as respectively. The charge π 3 (S 3 ) is also called the baryon number in the context of the Skyrme model. In general, the instanton charge in π N −1 (S N −1 ) for the O(N) model is given by (see, e.g., Ref. [59]) The O(3) model is equivalent to the CP 1 model. Let φ be a normalized complex two vector (φ † φ = 1), and consider the Hopf map from S 3 to S 2 by with the Pauli matrices σ A (A = 1, 2, 3). Let us define the stereographic coordinate u of S 2 (projective coordinate of the CP 1 ) by In terms of u, the Lagrangian can be rewritten as In this notation, the topological instanton charge can be rewritten as The boundary condition (−, −, +) can be expressed in terms of φ and u as The O(4) model is equivalent to a principal chiral model with a group SU(2) or the Skyrme model if four derivative term is considered. We define an SU(2)-valued field U(x) ∈ SU(2) in terms of four reals scalar fields n A (x) (A = 1, 2, 3, 4): where σ a are the Pauli matrices and n · n = 1 is equivalent to U † U = 1 2 . In terms of U(x), the Lagrangian can be rewritten as The symmetry of the Lagrangian isG = SU(2) L × SU(2) R acting on U as U → U ′ = g L Ug † R . This symmetry is spontaneously broken down toH ≃ SU(2) L+R , which in turn acts as U → U ′ = gUg † so that the target space isG/H ≃ SU(2) L−R ≃ S 3 . The baryon number (the Skyrme charge) of Q 3 ∈ π 3 (S 3 ) can be rewritten as The boundary condition (−, −, +, +) can be expressed in terms of U as so that the vacuum is center symmetric.

III. GENERAL FRAMEWORK FOR FRACTIONAL INSTANTONS IN THE O(N ) MODEL
Here, we provide a general framework to construct fractional instantons in the O(N) model with the boundary condition (1). In general, the boundary condition (1) defines a fixed manifold as the fixed points of the action at the boundary. This is nothing but the moduli space of vacua, since the boundary condition does not induce the gradient energy for the fields n A (A = s + 1, · · · , N) while it does for that of the rests n A (A = 1, · · · , s) . From the homotopy group of N , one finds the existence of a host soliton (defect) in the bulk. Here, we have formally defined π −1 for a space-filling brane in the case of n = −1 (s = N) for the situation that there is no localized defects.
At the core of the defect, the nonzero fields in the bulk must vanish, and the relation This is nothing but the moduli localized on the host soliton's world volume (collective coordinates of the host soliton). This has a non-trivial homotopy group The host soliton has world volume along the compact direction and the rests. Therefore, the moduli M must be twisted along the world volume in the compact direction with the twisted boundary condition. It inevitably introduces a daughter soliton, which, we find, belongs to a "half" element of the homotopy group in Eq. (20). In other words, a homotopy group in Eq. (20) is modified by the boundary condition to take a value in a half integer. While this should be explained by a relative homotopy group more rigorously, we do not do that in this paper. We denote it symbolically by We thus have a composite soliton. Each composite soliton consists of a daughter soliton, belonging to a half element of the homotopy group π b.c. m (M) in Eq. (21) modified by the boundary condition, on a host soliton, belonging to the unit element of the homotopy group π n (N ) in Eq. (18). Consequently, the total homotopy group π N −1 (M) in Eq. (3) is a product of the elements in π n (N ) in Eq. (18) and π b.c. m (M) in Eq. (21), and so it belongs to a half element of the total homotopy group π N −1 (M) in Eq. (3), that is, a half instanton.
The sum of codimensions of a host soliton and of a daughter soliton is N − 1, which is 2 or 3 exists a certain relation between the dimensionality of the homotopy groups: which is n + m + 1 = 2 for the O(3) model and n + m + 1 = 3 for the O(4) model.
In the following sections, we discuss fractional instantons and bions in more detail for each boundary condition in the O(3) and O(4) models.

A. (−, +, +): global vortex with an Ising spin or half lump-vortex
The fixed manifold is characterized by n 1 = 0, equivalently (n 2 ) 2 + (n 3 ) 2 = 1, which is N ≃ S 1 . This is the moduli space of vacua as explained in the last section. It has a nontrivial homotopy π 1 (S 1 ) ≃ Z, allowing a global vortex having the winding in n 2 + in 3 . In the vortex core, the two fields constituting the vortex must vanish n 2 = n 3 = 0 and the rest field n 1 appears taking a value n 1 = ±1, giving an Ising spin degree of freedom to the vortex. Therefore In order to write down explicit configurations, it is useful to define a complex coordinate by Then, asymptotic forms near fractional instantons located at z = 0 can be given by model with the boundary condition (−, +, +). ⊙ and ⊗ correspond to n 1 = +1 and n 1 = −1, respectively. Black arrows represent (n 2 , n 3 ) with n 2 2 + n 2 3 = 1 (n 1 = 0) parameterizing the moduli space of vacua N ≃ S 1 : ←, →, ↑, ↓ correspond to n 3 = +1, n 3 = −1, n 2 = +1, n 2 = −1, respectively. We chose the vacuum n 3 = +1 at the boundary. Topological charges ( * , * , * ) denote a host vortex charge π 1 , an Ising spin π 0 in its core, and the total instanton charge π 2 , respectively. (a) An instanton is split into two fractional instantons (+1, + 1 2 , + This expression is also good for large compactification radius R. The topological charges of fractional (anti-)instantons with the boundary condition (−, +, +) are summarized in Table II.
Here, we have defined the value of π 0 for the Ising spin to be ±1/2 to be consistent with the other boundary conditions discussed below.  Let us discuss the interaction between fractional instantons. When constituent fractional instantons are well separated at distance r in a large compactification radius R, the interaction between them is E int ∼ ± log r (the force is F ∼ ±1/r) because they are global vortices. Here, the interaction is repulsive for a pair of (anti-)vortices, and attractive for a pair of a vortex and an anti-vortex.
Therefore, it is attractive for a pair of fractional (anti-)instantons constituting an (anti-)instanton.
On the other hand, when the compactification radius R is as the same as the size of fractional instantons as in Fig. 5, a sine-Gordon (anti-)kink connects a fractional instanton and anti-instanton so that they are confined by a linear interaction energy E int ∼ r with distance r and the force between them is constant.
Next let us discuss bion configurations. Configurations near a bion can be written as This is good for large compactification radius R. Bions for small compactification radius R are schematically drawn in Fig. 5. Each of Fig. 5 shows a sine-Gordon (anti-)kink connecting two fractional (anti-)instantons for small compactification radius R, and consequently they are confined by a linear potential E int ∼ r with distance r for large separation.
(a) (+1, Before going to the other boundary conditions, let us make a comment on fractional instantons in related models. There exist topologically the same fractional instantons on R 2 without twisted boundary condition. One is baby Skyrmions [61,62] in an O(3) sigma model with a potential term V = m 2 n 2 1 and a four derivative (baby Skyrme) term [63][64][65]. The other is a vortex in a U(1) gauged O(3) sigma model with a potential term V = m 2 n 2 1 , in which the U(1) acting on n 2 + in 3 is gauged [66][67][68][69][70]. Vortices in this case are local, that is, of Abrikosov-Nielsen-Olesen (ANO) type [71]. In both cases, the potential term plays an alternative role of the twisted boundary condition. Interactions between fractional instantons are rather different from our case of the twisted boundary condition. In the former, the interaction between fractional instantons constituting an instanton is attractive at large distance and repulsive at short distance, resulting in a stable molecule [63][64][65]. In the latter, the interaction between them is exponentially suppressed which is either repulsive or attractive for type-II or type-I superconductor, and non-interactive for the critical limit, which is BPS [69]. modulus is twisted [53,55,72]. When the size of the domain wall ring is that of the compactification radius R, the top and bottom of the domain wall ring touch each other through the compact direction x 2 with the twisted boundary condition. Then, a reconnection of two parts of the ring occurs, and it can be split into two domain wall lines separated into the x 1 direction, as shown in Fig. 6. The U(1) modulus is twisted half along the domain lines extending to the x 2 direction, resulting in fractional (anti-)instantons. We have two pairs for instanton and anti-instanton respectively as seen in Fig. 7. We thus find four kinds of fractional (anti-)instantons shown in Fig. 1 (2a)-(2d). Each fractional instanton wraps a half sphere of the target space S 2 . For instance, the left half of Fig. 7(a) wraps a half sphere as in Fig. 4(b).
These ansatz are different from Ref. [17], but their asymptotic behaviors are the same. The interactions between fractional (anti-)instantons are exponentially suppressed so that the total action becomes a sum of each action when they are well separated.
Before going to the next case, let us give a brief comment on a relation to dimensional reduction in this case. In the zero radius limit of the compact direction, the theory is dimensionally reduced.
By assuming the dependence of the fields on the compact direction x 2 as (n 1 , n 2 ) = n 1 (x 1 ) cos π R x 2 ,n 2 ( in the presence of the twisted boundary condition, we see that a potential term is effectively in-17 duced from the gradient term of the fields: This is known as the Scherk-Schwarz dimensional reduction in the context of supersymmetric theories in which the induced mass is called a twisted mass. The dimensionally reduced CP 1 model is often called as the massive CP 1 model in the context of supersymmetry. Lumps (instantons) are reduced to (a pair of) domain walls in the massive CP 1 model [73,74]. This case was generalized to the CP N −1 and Grassmann sigma models, for which domain walls [46,52], instantons (lumps) [75,76], fractional instantons [45], bions [18] were studied. An (anti-)instanton is separated into two fractional (anti-)instantons with the boundary condition (−, −, −) as shown in Fig. 9. The ansatz for an isolated fractional (anti-)instanton can be given as Each fractional (anti-)instanton wraps a half sphere of the target space S 2 . For instance, the left half of Fig. 9(a) wraps a half sphere as in Fig. 4(c). The topological charges of fractional (anti-)instantons are summarized in Table IV. Here, we formally use π −1 for space-filling solitons of codimension zero, to be consistent with the other cases.
It may be worth to mention that the existence of a daughter soliton is not required from the (a) (+1, + 1 2 ,   The columns represent the homotopy groups of a host soliton π −1 , a daughter soliton π 2 , and the total instanton π 2 from left to right. π −1 is merely formal. boundary condition, because the x 1 -dependent rotation in Eq. (33) is not necessary and a configuration (n 1 , n 2 , n 3 ) = (− cos π R x 2 , sin π R x 2 , 0) is in fact a minimum energy state. This is in contrast to the other boundary conditions in which the existence of a daughter soliton is required in the presence of a host soliton.
In this case, the boundary condition is not enough to stabilize fractional instantons, unlike the (a) (+1, + 1 2 , the interaction is that of two sine-Gordon kinks, which is repulsive. Bions with the boundary condition (−, −, −) are shown in Fig. 10. The function f in Eq. (33) behaves as as f = 0 at x 1 −∞, f ∼ π in some intermediate region and back to f = 0 at x 1 → +∞.
The interaction energy between two fractional instantons constituting a bion would be suppressed exponentially because of the same reason with the above while the detailed form depends on the choice of the potential term. For the potential term in Eq. (35), the interaction is that between a sine-Gordon kink and an anti-kink.

A. (−, +, +, +): global monopole with an Ising spin or half Skyrmion-monopole
The fixed manifold is characterized by n 1 = 0, equivalently (n 2 ) 2 + (n 3 ) 2 + (n 4 ) 2 = 1, which is the moduli space of vacua N ≃ S 2 . Therefore, it has a nontrivial homotopy π 2 (S 2 ) ≃ Z, allowing a global monopole. In the monopole core n 2 = n 3 = n 4 = 0, the field n 1 appears taking a value  Table V. Here, we have defined the value of π 0 for the Ising spin to be ±1/2 to be consistent with the other boundary conditions discussed below.   . The columns represent the homotopy groups of a host soliton π 2 , a daughter soliton π 0 , and the total instanton π 3 from left to right.
We need higher derivative (Skyrme) term for the stability of fractional instantons (Skyrmions) [77], as is so for usual Skyrmions.
When the compactification radius R is large, the interaction between two well-separated fractional instantons is the same with that of global monopoles at large distance. For a small compactification radius R of the order of fractional instanton size, the interaction between two wellseparated fractional instantons at distance r is E int ∼ r because of a lump string connecting them.
(a) (+1, + 1 2 , Black arrows represent (n 1 , n 2 ) with n 2 1 + n 2 2 = 1 (n 3 = n 4 = 0) parameterizing the moduli space of vacua N ≃ S 1 , while red arrows represent (n 3 , n 4 ) with n 2 3 + n 2 4 = 1 (n 1 = n 2 = 0) parameterizing the moduli space of a vortex M ≃ S 1 . An instanton can be represented as a vorton, that is, a vortex ring along which the U (1) modulus is twisted once. Brackets ( * , * , * ) denote topological charges for a host global vortex characterized by π 1 , that for a sine-Gordon kink characterized by π 1 , and that for an instanton characterized by π 3 . a winding in n 3 + in 4 . In the vortex core, the winding field must vanish n 3 = n 4 = 0, and the other fields n 1 and n 2 appear with a constraint (n 1 ) 2 + (n 2 ) 2 = 1, giving a modulus M ≃ U(1) to the vortex. For a fractional instanton, this U(1) modulus is twisted half along the vortex string extending to the compactified direction, as described below.
An instanton (Skyrmion) can be represented by (a global analog of) a vorton [80], that is, a vortex ring along which a U(1) modulus is twisted. This fact was first found in the context of Bose-Einstein condensates (BEC) [81,82] (see also [83]), and stable solutions in a Skyrme model were also constructed in Refs. [84][85][86]. Configurations of Skyrmions as vortons are shown in Ref. [50]. By considering all possibilities of twisted vortex rings, we find four kinds of fractional (anti-)instantons, as summarized in Fig. 2 (2a)-(2d).
The ansatz for fractional (anti-)instanton configurations with the boundary condition (−, −, +, +) is given as n 3 + in 4 = sin g(r)e iθ , n 1 + in 2 = cos g(r)e iζ(z) , where (r, θ, z) are cylindrical coordinates. The topological instanton charge (Skyrmion charge or baryon number) can be calculated as General formula of the instanton (Skyrme) charge for a vortex string with the winding number Q, along which the U(1) modulus is twisted P times, was calculated to be P Q in Ref. [88] in the context of Hopfions and in Ref. [84] for Skyrmions. The topological charges of fractional (anti-)instantons with the boundary condition (−, −, +, +) are summarized in Table VI.  VI: Homotopy groups of fractional instantons in the O(4) model with the boundary condition (−, −, +, +). The columns represent the homotopy groups of a host soliton π 1 , a daughter soliton π 1 , and the total instanton π 3 from left to right.
Interestingly, we do not need higher derivative (Skyrme) term even though fractional instantons are Skyrmions. Indeed, stable configurations of (half) Skyrmions inside a vortex string was constructed without the Skyrme term in Ref. [84] on R 3 without twisted boundary condition.
Fractional instantons with the boundary condition (−, −, +, +) are global vortices in the x 1 -x 2 plane so that the interaction between them is E int ∼ ± log r with distance r for large separation (the force is F ∼ ±1/r), where positive sign is for a pair of (anti-)vortices and negative sign is for a pair of a vortex and anti-vortex.
Bions can be constructed by combining configurations in (2a) and (2c) in Fig. 2, or (2b) and (2d) in Fig. 2. In the both cases, instanton charges are canceled out. The interaction between fractional instantons constituting a bion is E int ∼ − log r with distance r for large separation and  The ansatz for fractional (anti-)instanton with the boundary condition (−, −, −, +) is given as The The topological charges of fractional (anti-)instantons with the boundary condition (−, −, −, +) are summarized in Table VII.  −, −, +). The columns represent the homotopy groups of a host soliton π 0 , a daughter soliton π 2 , and the total instanton π 3 from left to right.
Interestingly, we do not need higher derivative (Skyrme) term even though fractional instantons are Skyrmions, as in the case of the boundary condition (−, −, +, +). Indeed, stable configurations of a unit (not half) Skyrmion inside a domain wall was constructed in R 3 without twisted boundary condition [90,91]. We expect that the same holds for half instantons (Skyrmions).
Fractional instantons with the boundary condition (−, −, −, +) are domain walls perpendicular to the x 1 coordinate so that the interaction between them is E int ∼ −e −mr with distance r for large separation. The energy of domain walls are linearly divergent in the x 2 direction.
Bions can be constructed by combining configurations in (3a) and (3c) in Fig. 2, or (3b) and (3d) in Fig. 2, where the instanton charge is canceled out. The interaction between fractional instantons constituting a bion is attractive and exponentially suppressed E int ∼ −e −mr . One (anti-)instanton is separated into two fractional (anti-)instantons with the boundary condition (−, −, −, −), as summarized in Fig. 2 (4a)-(4b). Each fractional instanton wraps a half sphere of the target space S 3 .
We need higher derivative (Skyrme) term for the stability of fractional instantons (Skyrmions).   The columns represent the homotopy groups of a host soliton π −1 , a daughter soliton π 3 , and the total instanton π 3 from left to right. Here, π −1 is merely formal.

VI. SUMMARY AND DISCUSSION
We have found that a fractional instanton model with a potential V = m 2 n 2 1 on R 2 allows local vortices [66][67][68][69][70]. If we further choose the gauge coupling to be e 2 = m 2 , fractional (anti-)instantons become (anti-)BPS and the theory can be made supersymmetric [69]. In our case, the twisted boundary condition would play a role of the potential, so the gauge coupling should be correlated to the compactification radius for vortices to be BPS.  [93][94][95]. A generalization of the model to R 1 × S 1 with twisted boundary condition is expected to admit BPS fractional instantons.
3. The O(4) model with (−, +, +, +). If one gauges the SO(3) symmetry action on (n 2 , n 3 , n 4 ), a half-Skyrmion monopole becomes local, that is, of 't Hooft-Polyakov type having finite energy, in which case the interaction between them is exponentially suppressed. This is because an SO(3) gauged Skyrme model with a potential V = m 2 n 2 1 on R 3 allows a local 't Hooft-Polyakov type monopole with finite energy [79]. A BPS limit is not known in this case. Again in our case, the twisted boundary condition would play a role of the potential. 4. The O(4) model with (−, −, +, +). If one gauges the U(1) symmetry acting on n 3 + in 4 , vortices as half instantons become local vortices having finite energy, in which case the interaction between them would be exponentially suppressed.

5.
The O(4) model on S 2 × S 1 . If we consider a geometry S 2 × S 1 instead of R 2 × S 1 , instantons (Skyrmions) are BPS for untwisted boundary condition [96]. An extension to a twisted boundary condition should be possible, in which case fractional (anti-)instantons may be also (anti-)BPS. 6. The O(4) model with only a six derivative term. If we consider Lagrangian containing only a six derivative term, which is baryon charge density squared, and a suitable potential term, instantons (Skyrmions) are BPS, which is indeed the case of R 3 [97]. It may be generalized to the case of R 2 × S 1 with a twisted boundary condition, in which case fractional (anti-)instantons may be also (anti-)BPS.
In these cases, fractional instantons will play a role in resurgence, which is indeed the case of the O(3) model with the boundary condition (−, −, +) [8,9,17] as denoted above.
When we compactify more than one directions, we can consider more general twisted boundary conditions. For instance, we may consider the O(3) model on R n × (S 1 ) 2 with a twisted boundary condition (−, −, +) for one direction and (+, −, −) for the other direction. A complete classification of these more general cases remain as an interesting problem.
A lattice of half Skyrmion appear in finite baryon density [98]. There may be certain relation with our half Skyrmions in the presence of a compact direction with twisted boundary conditions.
Hopfions are knot like solitons supported by the Hopf charge π 3 (S 2 ) ≃ Z in the O(3) model with four derivative (Faddeev-Skyrme) term [99]. Since Hopfions on R 3 are closed lump strings along which U(1) moduli are twisted (see, e.g. Ref. [100]), those on R 2 × S 1 with an untwisted boundary condition can be twisted closed lump strings wrapping around S 1 [88,101]. If we impose twisted boundary conditions, we will be able to obtain a fractional Hopfion as a halftwisted lump string wrapping around S 1 .
By applying our method to non-Abelian gauge theories, classification of fractional Yang-Mills instantons may be possible, which would be important toward the resurgence of gauge theories.
To this end, realizations of Yang-Mills instantons as various composite solitons summarized in Ref. [102] will be useful, as has been demonstrated for Skyrmions in this paper. Yang-Mills instantons are Skyrmions inside a domain wall [103], lumps inside a vortex [43,48,104,105], or sine-Gordon kinks on a monopole string [54]. Investigating boundary conditions realizing these would be an important first step toward the resurgence of gauge theories.
Finally, let us make a comment on duality. As seen in this paper, a CP 1 instanton with the boundary condition (−, −, +) is decomposed into a set of two fractional instantons which are half twisted domain walls, as seen in Fig. 6, and one of them becomes a domain wall in a small compactification radius limit in which the other is removed to infinity [43]. The same relation holds between a Yang-Mills instanton and a BPS monopole, which can be also understood as a Tduality acting on D-branes in type-II string theory [106]. In Ref. [43], CP N −1 fractional instantons were realized as fractional Yang-Mills instantons trapped inside a vortex in a U(N) gauge theory, which explains a relation between the above mentioned two T-dualities. Here, in this paper, we have added one more example, that is, a T-duality between a Skyrmion and a vortex. In the O(4) model with the boundary condition (−, −, +, +), equivalently Eq. (16), a Skyrmion is decomposed into a set of two fractional instantons which are half twisted vortex strings as seen in Fig. 14. One of them becomes a vortex in a small compactification radius limit, in which the other is removed to infinity. We think that a further T-duality maps this configuration to a domain wall through a domain wall Skyrmion.