Classifying bions in Grassmann sigma models and non-Abelian gauge theories by D-branes

We classify bions in the Grassmann $Gr_{N_{\rm F},N_{\rm C}}$ sigma model (including the ${\mathbb C}P^{N_{\rm F}-1}$ model) on ${\mathbb R}^{1}\times S^{1}$ with twisted boundary conditions. We formulate these models as $U(N_{\rm C})$ gauge theories with $N_{\rm F}$ flavors in the fundamental representations. These theories can be promoted to supersymmetric gauge theories and further can be embedded into D-brane configurations in type II superstring theories. We focus on specific configurations composed of multiple fractional instantons, termed neutral bions and charged bions, which are identified as perturbative infrared renormalons by \"{U}nsal and his collaborators. We show that D-brane configurations as well as the moduli matrix offer a very useful tool to classify all possible bion configurations in these models. Contrary to the ${\mathbb C}P^{N_{\rm F}-1}$ model, there exist Bogomol'nyi-Prasad-Sommerfield (BPS) fractional instantons with topological charge greater than unity (of order $N_{\rm C}$) that cannot be reduced to a composite of an instanton and fractional instantons. As a consequence, we find that the Grassmann sigma model admits neutral bions made of BPS and anti-BPS fractional instantons each of which has topological charge greater (less) than one (minus one), that are not decomposable into instanton anti-instanton and the rests. The ${\mathbb C}P^{N_{\rm F}-1}$ model is found to have no charged bions. In contrast, we find that the Grassmann sigma model admits charged bions, for which we construct exact non-BPS solutions of the field equations.

Bions and the resurgence in the low-dimensional models have been extensively investigated for the CP N −1 model [8-10, 16, 17], principal chiral models [12,15], and quantum mechanics [11,13,14]. In Refs. [8,9], generic arguments on bion configurations were given in the CP N −1 model on R 1 × S 1 with Z N twisted boundary conditions, which is a corresponding situation to U(1) N −1 center-symmetric phase in QCD(adj.), based on the independent instanton description taking account of interactions between far-separated fractional instantons and anti-instantons. According to the study, the imaginary ambiguity in the amplitude of neutral bions has the same magnitude with an opposite sign as the leading ambiguity (∼ ∓iπe −2S I /N ) arising from the non-Borel-summable series expanded around the perturbative vacuum. The ambiguities at higher orders are cancelled by amplitudes of bion molecules (2-bion, 3-bion,...), and the full trans-series expansion around the perturbative and non-perturbative vacua results in unambiguous semiclassical definition of field theories.
Among other things, the two dimensional CP N −1 model enjoys common features with four-dimensional Yang-Mills theory [37] such as asymptotic freedom, dynamical mass gen-eration, and the presence of instantons [38,39]. Fractional instantons in the CP N −1 model on R 1 × S 1 with twisted boundary conditions were found in Ref. [40] (see also Refs. [41]).
Fractional instantons in the Grasssmann sigma model were also found in Ref. [42]. Explicit solutions or ansatze corresponding to bion configurations in the CP N −1 model have been investigated recently [10,16,17]. Although fractional instantons are Bogomol'nyi-Prasad-Sommerfield (BPS) solutions [43,44], bions are non-BPS as composite of fractional instantons and anti-instantons. In Ref. [10], such non-BPS solutions were found out in the CP N −1 model on R 1 × S 1 with the Z N twisted boundary condition by using the method of Ref. [39], which can be saddle points for the trans-series expansion. In our previous work [17] we have studied an ansatz corresponding to neutral bions in the CP N −1 model beyond exact solutions, and have shown that our ansatz is consistent with the result from the far-separated instanton gas calculus [8,9] even from short to large separations.
The purpose of our present work is to classify ansatze corresponding to all possible bion configurations in the CP N −1 and Grassmann sigma models on R 1 × S 1 with twisted boundary conditions. We study mainly the Z N twisted boundary condition for simplicity, although we can easily extend our study to more general twisted boundary conditions. In our study, we introduce a new viewpoint based on D-brane configurations to study bion configurations: The CP N −1 and Grassmann sigma models are formulated as a U(N C ) gauge theories with N F flavors in the fundamental representations [45][46][47][48][49][50][51], which can be embedded into supersymmetric gauge theories by adding fermions (and scalar fields) appropriately. Sigma model instantons (lumps) in the Grassmann sigma model are promoted to non-Abelian vortices [52][53][54][55][56] (see Refs. [57][58][59] as a review) in gauge theories, especially of semi-local type [60,61]. By doing so, the moduli space of BPS vortices (lumps) can be clarified completely in terms of the moduli matrix [58]. These theories can be further embedded into Hanany-Witten type D-brane configurations in type II string theories [62,63], where vortices can be identified with certain D-branes [52]. The T-duality transformation along S 1 maps vortices to domain walls [64], which can be described by kinky D-branes [65,66]. These D-brane configurations were used to study moduli space of non-Abelian vortices before [42].
In this paper, we show that these D-brane configurations as well as the moduli matrix offer very useful tools to classify all possible bion configurations in the Grassmann sigma model including the CP N −1 model, and the corresponding non-Abelian gauge theories. We unexpectedly find that the Grassmann sigma model admits neutral bions made of BPS and anti-BPS fractional instantons each of which has a topological charge greater (less) than one (minus one), but it cannot be decomposed into instanton anti-instanton and the rests. We find that the Grassmann sigma model admits charged bions, while the CP N −1 model does not. There are many different species of fractional instantons in the Grassmann sigma model.
Among them, we can choose species of BPS fractional instantons and anti-BPS fractional instantons that are noninteracting and can coexist stably. In such cases, we obtain exact non-BPS solutions representing charged bions. We also calculate the energy density and topological charge density of the bion configurations in these models numerically to obtain their interaction energies, which give valuable informations on the interactions between constituent fractional instantons, such as the sign and magnitude of the strength, and the dependence on the separations between constituent fractional instantons.
In Sec. II, we formulate the Grassmann sigma model Gr N F ,N C including the CP N F −1 model as U(N C ) gauge theory with N F Higgs scalar fields in the fundamental representation.
We also present BPS equations for BPS vortices or lumps in these theories and the moduli matrix which exhaust moduli parameters of BPS solutions. In Sec. III, we introduce D-brane configurations in type II string theories, that realize our theory on certain D-brane worldvolumes. We then study fractional instantons in Grassmann sigma model Gr N F ,N C including the CP N F −1 model in terms of D-brane configurations and the moduli matrix. In Sec. IV, we classify neutral bions in the CP N F −1 model. In Sec. V, we classify neutral and charged bions in the Grassmann sigma model Gr N F ,N C . In Sec. VI, we discuss interaction energy for bions with changing the distance between fractional instanton constituents. Sec. VII is devoted to summary and discussion. In Appendix A, we discuss solutions of constraint of Grassmann sigma model. Target spaces of supersymmetric nonlinear sigma models must be Kähler for four supercharges [67] and hyper-Kähler for eight supercharges [68]. The CP N F −1 and Grassmann sigma models can be obtained from supersymmetric gauge theories with four supercharges [45][46][47] and eight supercharges [48][49][50][51]. In this subsection we consider two-dimensional eu-clidean gauge field theories in the flat x 1 -x 2 plane with U(N C ) gauge group and N F flavors of scalar fields in the fundamental representation denoted as an N C × N F matrix H. The Lagrangian is given as where g is the gauge coupling, v is a real positive parameter (Fayet-Iliopoulos parameter in the context of supersymmetry) [58]. The covariant derivative D µ with the gauge field W µ and field strength F µν are defined as D µ H = (∂ µ + iW µ )H, . We use a matrix notation such as W µ = W I µ T I , where T I (I = 0, 1, 2, · · · , N 2 C − 1) are matrix generators of the gauge group G in the fundamental representation satisfying Tr(T I T J ) = Since the scalar fields H are massless, the Lagrangian has a global symmetry SU(N F ). It can be embedded into a supersymmetric theory with eight supercharges [58]. Consequently it admits BPS solitons [43,44] which preserve a part of supercharges [69]. Vacuum in this model is characterized by the vanishing vacuum energy This condition necessitates some of scalar fields H to be non-vanishing (rankH = N C ), implying that the gauge symmetry is completely broken (Higgs phase). This vacuum is called the color-flavor locked vacuum where N C out of N F flavors should be chosen to be non-vanishing and leaves only a diagonal SU(N C ) of color SU(N C ) and SU(N G ) subgroup of flavor SU(N F ) group beside the remaining SU(N F − N C ) × U(1) as the global symmetry.
Since we consider two euclidean dimensions, instantons are the usual vortices with codimension two. The Bogomol'nyi completion [43] can be applied to the Lagrangian L to give a bound L = Tr 1 +Tr −v 2 B 3 + 2i∂ [1 HD 2] H † ≥ Tr −v 2 B 3 + 2i∂ [1 HD 2] H † (II. 3) with a magnetic field B 3 ≡ F 12 . The bound is saturated if the following BPS vortex equations hold [52,53] When these BPS equations are satisfied, the total energy T is given by T ≡ d 2 xL = 2πv 2 Q, (II. 6) with the topological charge Q (instanton number) defined by measuring the winding number of the U(1) part of the broken U(N C ) gauge symmetry [92].
Let us define S = S(z,z) ∈ GL(N C , C) using a complex coordinate z ≡ x 1 + ix 2 We can solve [56] the first of the BPS equations (II.4) in terms of S H = S −1 H 0 (z), (II. 9) where H 0 (z) is an arbitrary N C by N F matrix whose components are holomorphic with respect to z, which is called the moduli matrix of BPS solitons. By defining a gauge invariant quantity Ω(z,z) ≡ S(z,z)S † (z,z), (II. 10) the second BPS equations (II.5) can be rewritten as We call this the master equation for BPS solitons [93]. This equation is expected to give no additional moduli parameters. It was proved for N F = N C = 1 (the ANO vortices) [71] and is consistent with the index theorem [52] in general N C and N F . Moreover, this fact can be easily proved in the strong coupling limit where the gauge theories reduce to the Grassmann sigma model, as we show in the next subsection. Thus we assume that the moduli matrix H 0 describes the moduli space completely.
However we note that there exists a redundancy in the solution (II.9): physical quantities H and W 1,2 are invariant under the following V -transformations H 0 (z) → H ′ 0 (z) = V (z)H 0 (z), S(z,z) → S ′ (z,z) = V (z)S(z,z), (II. 12) with V (z) ∈ GL(N C , C) for ∀ z ∈ C, whose elements are holomorphic with respect to z. Let us note that Ω is invariant under U(N C ) gauge transformations, but is covariant under the V -transformations Ω → V ΩV † (II. 13) Incorporating all possible boundary conditions, we find that the total moduli space of BPS solitons M total N C ,N F is given by (II. 14) where M N,N ′ denotes a set of holomorphic N × N ′ matrices [56,58]. B. Grassmann sigma model as a strong coupling limit Gauge theories reduce to nonlinear sigma models with target spaces as Grassmann manifolds in the strong gauge coupling limit g 2 → ∞. When they are embedded into supersymmetric gauge theories with four (eight) supercharges, they become (hyper-)Kähler (HK) nonlinear sigma models [67,68] on the Higgs branch [50,72] of gauge theories as their target spaces. This construction of (hyper-)Kähler manifold is called a (hyper-)Kähler quotient [48,49]. In order to have finite energy configuration, it is necessary to be at the minimum of the potential leading to a constraint Since the gauge kinetic terms for W µ disappear in the limit of infinite coupling, gauge fields W µ become auxiliary fields which can be expressed in terms of scalar fields H through their field equations (II. 18) After eliminating W µ , the Lagrangian (II. 19) with the constraints (II.2) becomes a nonlinear sigma model, 19) with the complex Grassmann manifold Gr N F ,N C as a target space (1) .
(II. 20) This is the Grassmann sigma model which is the main focus of our study. Now one can see and forH at strong couplingHH † = v 2 1 N F −N C , (II. 28) these solutions must satisfy the following orthogonality constraints The boundary conditions for the BPS solution H and the anti-BPS solutionH should be chosen to be associated with complementary vacua [64,73].
Duality between U(N C ) gauge theories and U(N F −N C ) gauge theories can be formulated in terms of the corresponding moduli matrices H 0 andH 0 as Together with the complementary boundary conditions, this relation determinesH 0 uniquely from H 0 up to the V -equivalence (II. 12). Although this duality is not exact for finite coupling there still exists a one-to-one dual map by the relation among the moduli matrix H 0 in the original gauge theory and the (N F − N C ) × N F moduli matrixH 0 of the dual gauge theory. C.

Z N F twisted boundary conditions and fractional instantons
In the present subsection, we introduce a Z N F twisted boundary condition in the U(N C ) gauge theory with N F flavors or the Grassmann sigma model as its strong coupling limit on The Z N F twisted boundary conditions in a compactified direction is expressed in terms of a twisting matrix B as [9,10] H( The Z N F twisted boundary condition breaks the global SU(N F ) symmetry down to Z N F .
Fractional instantons (kink instantons) carry integer multiple of the minimum topological charge 1/N F in the Grassmann sigma models on R 1 × S 1 with a Z N F twisted boundary condition [40,42].
When x 2 is compactified with the period L, the lowest mass of Kaluza-Klein modes is 2π/L. In the case of Z N F twisted boundary condition, the lowest mass is also fractionalized to give 2π/(N F L). The fractional instanton is in one-to-one correspondence with the kink [40,42] as a function of x 1 . The study of BPS equations for kinks in the strong coupling limit reveals that the size of the fractional instanton is given by the inverse of the mass difference associated to the adjacent vacua [58,74]. Therefore the size of elementary fractional instanton with the instanton charge 1/N F is given by N F L/(2π). When two fractional instantons are compressed together, they can form a compressed fractional instanton, whose size should be a half of the individual fractional instantons. By the same token, n fractional instanton can be compressed together to form a compressed n-fractional-instantons whose size should be N F L/(2πn).
From next subsection we make all the dimensionful quantities and parameters dimensionless by using the compact scale L (L → 1) unless we have a special reason to recover it.

A. D-brane configurations
The gauge theory introduced in the last section can be made N = 2 supersymmetric (with eight supercharges) by doubling the Higgs scalar fields H and adding fermionic superpartners (Higgsino and gaugino) and adjoint scalars (dimensionally reduced gauge fields) [58]. Then, the theory can be realized by a D-brane configuration [66]. We first consider the Hanany-Witten brane configuration [62,63]. We are interested in euclidean space R × S 1 , but we consider a brane configuration in 2+1 dimensions by adding "time" direction. In Table I, we summarize the directions in which the D-branes extend. In Fig. 1 the brane configuration is schematically drawn. The U(N C ) gauge theory is realized on the N C coincident D3- Fig. 1: Brane configuration for k vortices. As for separation of the two NS5-branes, ∆x 3 corresponds to 1/g 2 and (∆x 4 , ∆x 5 , ∆x 6 ) correspond to the triplet of the FI parameters.  brane world-volume which are stretched between two NS5 branes. The D3 brane worldvolume have the finite length ∆x 3 between two NS5 branes, and therefore the D3 brane world-volume theory is (2 + 1)-dimensional U(N C ) gauge theory with a gauge coupling , with the string coupling constant g (B) s in type IIB string theory and the D3-brane tension s l 4 s ). The positions of the N F D5 branes in the x 7 -, x 8 -and x 9 -directions coincide with those of the D3 branes. Strings which connect between D3 and D5 branes give rise to the N F hypermultiplets (the Higgs fields H and Higgsinos) in the D3 brane worldvolume theory. The two NS5 branes are separated into the x 4 -, x 5and x 6 -directions, which give the triplet of the FI parameters c a [52,55]. We choose it as , with the string length l s . Now, we consider BPS non-Abelian vortices in this setup [52,55]. k vortices are represented by k D1-branes stretched between D3-branes from the following reasons. = v 2 of each D1 brane coincides with that of a vortex. Therefore, one concludes that the k D1 branes correspond to k vortices in the D3 brane world-volume theory. When N F = N C , the vortices are called local vortices, while forN F > N C the vortices are called semi-local vortices.
Next, we compactify the x 2 -direction on S 1 with the period L for our purpose. First, we turn on a constant background gauge field on the D5 brane worldvolume as a non-trivial Wilson loop around S 1 on the D5 branes. This precisely gives a twisted boundary condition in our context. In this paper, we consider the Z N symmetric twisted boundary condition corresponding to m n = 2πn/(LN F ), n = 1, · · · , N F .
We then take T-duality along the x 2 direction. Table II These separations give hypermultiplet masses in the D2 brane worldvolume theory. The worldvolume theory of the D2 branes is (1 + 1)-dimensional gauge theory with a gauge is the string coupling constant in type IIA string theory.  First, let us consider vacua without vortices (k = 0). As shown in Fig. 2, each D2 brane ends on one of the D4 branes, on each of which at most one D2 brane can end, which is known as the s-rule [62]. There are Let us consider vortices (k = 0). The D1 branes representing vortices are mapped by the T-duality to D2 branes, which we denote as D2 ′ , stretched between the D4 branes, as shown in the middle figure in Fig. 3, where the position of the D2 ′ brane in the x 1 coordinate is denoted as x 0 . The D2 branes are attached to different D4 branes at x 1 = −∞ and x 1 = +∞, and there must exist D2 branes which connect the D2 branes ending on different D4 branes at some point in the x 1 -coordinate. These D2 branes correspond to D2 ′ in Fig. 3.
Since the D2 ′ branes do not end in the x 1 -direction, they must be bent to the x 3 -direction Fig. 3: T-dualized configurations for k = 1: the D2 ′ brane is assumed to be located at  Fig. 4: Brane configuration for a wall.
to end on the NS5-branes. We denote these D2-branes by D2 * in Fig. 3. In Fig. 4, the brane configuration in the x 1 , x 2 , x 3 -coordinates is shown. This is nothing but a brane configuration [66] of a BPS kink (domain wall) [64] in the D2 brane theory. The energy of the kinks can be calculated from this brane configuration in the strong coupling limit. Since the gauge coupling 1 g 2 is proportional to ∆x 3 , the D2 * branes disappear in the limitĝ → ∞. The D2 ′ branes have the energy τ 2 ∆x 4 l 2 s ∆m = ∆x 4 ∆m s ls =ĉ ∆m coinciding with the energy of a kink. A set of D2 ′ +D2 * -branes between two D4-branes is a kink as a fractional vortex (or a lump), which corresponds to a fractional instanton in euclidean R × S 1 space in our context.
The unit vortex corresponds to the D2 brane winding around the S 1 of the cylinder with exhibiting a kink as in Fig. 5 (a). The size of the kink in the x 1 -direction is that 1/g √ c of an Abrikosov-Nielsen-Olesen (ANO) vortex. Note that the scalar fieldΣ(x 1 ) has period 1/R. This vortex can be decomposed into two walls by changing the size of the vortex. In this configuration, the D2 brane is attached to the same D4 brane at x 1 → ±∞ with exhibiting kinks twice as in Fig. 5 (b). The relative distance between the two kinks can be interpreted as the size moduli of the single semi-local vortex (or lumps). The small size limit of the configuration reduces to the ANO vortex with the ANO size 1/(gv). In other words, the small lump singularity in the strong coupling limit g → ∞ is resolved by the size of the ANO vortex for finite g. By using the brane picture presented here, the moduli space of multiple non-Abelian vortices was classified in Ref. [42]. In this paper, we use this kinky D-brane picture to classify all possible bion configurations. We can visualize the kink exhibited in Σ( where P is the path-ordering and the gauge field W 2 is given in Eq. (II.18) in the Grassmann sigma model. Since this is a matrix, the intuitive meaning of the brane picture can be best visualized when the matrix H 0 H † 0 and Σ are nearly diagonal. Before doing that, we make a comment on a brane picture of the Seiberg-like duality, which exchanges the gauge group as U(N C ) ↔ U(N F − N C ) in Eq. (II.26). In the Hanany-Witten setup, this is achieved by the exchange of the positions in x 3 of the two NS5-branes.
When an NS5-brane passes through a D5-brane, a D3-branes is created (annihilated) if it is (not) stretched between the NS5-brane and the D5-brane before the crossing, due to the Hanany-Witten effect [62,63]. This changes the number of the D3-branes from N C to N F − N C . This exchange flips the sign of the FI parameters c a ↔ −c a . The Seiberg dual in the presence of vortices was studied in Ref. [61]. In the T-dual configuration, the presence dual In terms of the complex coordinate z = x 1 + ix 2 on R 1 × S 1 with 0 ≤ x 2 < L = 1, the fractional instantons for the CP 1 model satisfying the Z 2 twisted boundary condition can be parameterized by the following moduli matrices with real moduli parameters λ L , λ R > 0 and θ L , θ R One should note that the twisted boundary condition automatically introduces nontrivial From the T-duality, this is precisely the location of the BPS fractional instanton. Therefore, the fractional (anti-)instanton H 0L (H * 0L ) is situated at x 1 = 1 π log λ L , and the fractional (anti-)instanton H 0R (H * 0R ) is at x 1 = 1 π log 1 λ R . Since the moduli matrices H 0L and H 0R are holomorphic (depend on z only), they give BPS solutions with instanton charge Q = +1/2.  identified with 1) is parameterized as H 0n = 0, · · · , 0, λ n e iθn e −2πz/N F , 1, 0, · · · , 0 , H * 0n = 0, · · · , 0, λ n e −iθn e −2πz/N F , 1, 0, · · · , 0 , (III.6) where the value 1 corresponds to the (n + 1)-th flavor [94] One should note that the twisted boundary condition automatically introduces nontrivial x 1 dependence in H 0 . The BPS solution given by H 0n carries an instanton charge 1/N F , and its conjugate H * 0n carries an instanton charge −1/N F . We call these BPS solutions as the elementary fractional instantons.
The fractional instantons for H 0n and H * 0n are located at N F 2π log λ n . The other moduli θ n represents the relative phase of the n-th and (n+1)-th vacua. All these elementary fractional instantons are physically distinct, and are needed to form an instanton with unit charge as a composite of fractional instantons, as shown in Fig. 5. In the particular case of the Z N F twisted boundary condition, they are distinct only by the vacuum label n (n = 1, · · · , N F ) and have identical properties. In that sense, we will exhibit only one of them as the repre-sentative in the following.
For each topological charge n/N F , n = 1, · · · , N F −1, there are N F distinct BPS composite fractional instantons, but we will exhibit only one of them as a kink connecting the first flavor to the n + 1-th flavor, since all other solutions starting from other vacua have identical properties. When the topological charge reaches unity, it becomes a genuine instanton.
Those BPS solitons containing at least one instanton are not counted as fractional instantons here. The BPS composite fractional instanton with the maximal topological charge (N F − 1)/N F is given by the moduli matrix with 2N F − 2 moduli parameters. Constituent fractional instantons are located at If any one of these inequalities are not satisfied, for instance, N F 2π log λn λ n+1 ≥ N F 2π log λ n+1 λ n+2 , two fractional instantons are merged into one. In the limit of negative infinite relative separation, λ n+1 → 0, the solution becomes a compressed fractional instantons located at the common center N F 4π log λn λ n+2 . In the limit, the size of the compressed fractional instantons becomes half of that of the individual fractional instanton.  Fractional instantons are located at 3 2π log λ 1 λ 2 and 3 2π log λ 2 in x 1 , when λ 2 ≫ 1, as shown in Fig. 8(a). Keeping the center of mass position 3 4π log λ 1 fixed, we can decrease the relative separation 3 2π log λ 1 λ 2 2 . When λ 2 ≈ 1, two fractional instantons are touching and begin to merge as shown in Fig. 8(b). When λ 2 → 0, moduli matrix becomes H 0 = λ 1 e iθ 1 e −4πz/3 , 0, 1 and two fractional instantons are compressed completely to become a single compressed fractional instanton with a width of the half of individual fractional instanton.
The CP N F −1 instanton with the unit instanton charge can be obtained with the moduli Fig. 9 shows the BPS instanton in the case of the CP 2 model.
The s-rule implies that there should be no crossing of color lines and To enumerate genuine fractional instantons, we identify BPS instanton with unit instanton charge as having the identical sets of flavors occupied by color lines for left and right vacua namely we consider f 1 = 1, f 2 = 2. There is only one elementary BPS fractional instanton with total instanton charge 1/4, specified by (k 1 , k 2 ) = (0, 1), as shown in Fig. 10(a). There are two BPS composite fractional instantons with total instanton charge 1/2, specified by (k 1 , k 2 ) = (1, 1) and (0, 2), as shown in Fig. 10(b). There is only one BPS composite fractional instantons with total instanton charge 3/4, specified by (k 1 , k 2 ) = (1, 2), as shown in Fig. 10(c). There are two BPS solutions with total instanton charge 1, specified by (2,2) and (1,3). It is surprising and interesting to find that the diagram (2, 2) shown in Fig. 10 (although physically distinct) and we do not list it. There is only one type of BPS composite fractional instantons with total instanton charge 1/2, specified by (k 1 , k 2 ) = (1, 1), as shown in Fig. 11(b). There is only one type of BPS composite fractional instantons with total instanton charge 3/4, specified by (k 1 , k 2 ) = (1, 2), as shown in Fig. 11(c). Since (2, 1) is of the same type as (1, 2) and is not listed. There is only one type of BPS solution with total instanton charge 1, specified by (2,2), as shown in Fig. 11(d). This (2, 2) solution is a genuine BPS instanton solution, since the left and right vacua are identical as a set of (a) (0, 1) The constituent fractional instantons are located at x 1 = 2 π log λ 3 λ 4 , 2 π log λ 4 λ 5 , 2 π log λ 1 , 2 π log λ 5 . This BPS configuration is nothing but an instanton with the unit instanton charge, as one can recognize from the fact that the set of flavor branes occupied by the color branes in the right infinity of this diagram are identical to the corresponding set at the left infinity. The total instanton charge is of course unity.
As another example, let us write down explicitly the moduli matrix for the BPS solution of (composite) fractional instantons with the set (2, 2), which is depicted in Fig. 10 is located at 2 π log λ 4 λ 5 , the fractional instanton connecting vacua (1, 3) → (2, 3) is located at 2 π log λ 1 λ 2 , the fractional instanton connecting vacua (2, 3) → (2, 4) is located at 2 π log λ 5 , and the fractional instanton connecting vacua (2, 4) → (3, 4) is located at 2 π log λ 2 . In the Fig. 10(e), the position of the kink on the first brane at 2 π log λ 1 λ 2 is placed to the right of the kink on the second brane at 2 π log λ 5 . One should note that their positions can be interchanged by taking λ 5 < λ 1 λ 2 . In that case, the character of fractional instantons at these positions change: the fractional instanton connecting vacua (1, 3) → (1, 4) is located at 2 π log λ 2 , and the fractional instanton connecting vacua (1, 4) → (2, 4) is located at 2 π log λ 1 λ 2 . Similarly to the CP N F −1 model, two BPS fractional instantons can be merged together and become a compressed fractional instantons with half of the size of the individual fractional instanton in the limit, as illustrated in Fig.12. This configuration of the compressed fractional instantons can be regarded as a boundary of the moduli space of separated fractional instantons. Since the compressed kink may be regarded as a reconnection of color lines, we call this phenomenon as BPS reconnection.  ansatz for the CP 1 model satisfying a Z 2 twisted boundary condition (II.31) as [17] H 0 = λ 1 e iθ 1 e −πz + λ 2 e iθ 2 e πz , 1 . (IV.1) As shown in Fig.14, the fractional instanton is located at 1 π log λ 1 , and the fractional antiinstanton is at 1 π log 1 λ 2 . As the separation becomes negative λ 1 λ 2 → ∞ (with λ 1 /λ 2 held fixed), they are compressed together and eventually becomes a vacuum H 0 → (1, 0).  Fig. 15(a)-(c) can be decomposed along the dotted line into two neutral bions, so that they are all reducible. We say a configuration to be reducible even when its subconfiguration can be decomposed into multiple neutral parts. An example of such a configuration is given in Fig. 15(d), where the two regions between the dotted lines are decomposed neutral bions in Fig. 15(a).
Irreducible neutral bions can be characterized in terms constituent BPS fractional instanton. We define the topological charge of neutral bions by the topological charge of constituent BPS fractional instanton, namely by the number of fractional instantons, which is the same with that of fractional anti-instantons from its neutrality. The total energy is proportional to the topological charge.
For instance, an irreducible bion in Fig. 14   In order to classify bions in the Grassmann sigma models systematically, we may consider two fundamental procedures to create neutral bions, as shown in Fig. 17; (a) a pair creation of fractional instanton and anti-instanton and (b) a crossing of two color branes, which may be called non-BPS crossing. The pair creation (a) in Fig.17 increases the numbers of  (16) can be all constructed by repeating the pair creations two and four times, respectively. As we see later, we can connect the right hand side of (a) in Fig.17 with finite energy continuously to left hand side of (a) representing the vacuum configuration with vanishing energy by changing parameters of field configuration.
On the other hand, we can write down a moduli matrix representing the non-BPS crossing shown in the right hand side of (b) in Fig.17 as We find the energy density to vanish identically, indicating that the right hand side of (b) is actually the same as the left-hand side, namely the vacuum itself [95]. Therefore we do not use the non-BPS crossing in our approach to generate bion configurations in Grassmann sigma model. It is conceivable that non-BPS crossing needs to be considered if we consider finite gauge coupling and/or quantum effects properly.
The definition of the reducibility is the same with the last subsection for the CP N F −1 model. In this paper, we classify irreducible neutral bions. In addition to the configurations in Fig. 15 (a) and (b) the configuration in Fig. 18 is a reducible neutral bion because each of them can be split into the left and right parts. The configuration in Fig. 18 is a Seiberg dual of that of Fig. 15 (b). We do not consider these reducible configurations when we classify irreducible neutral bions.
Although we do not consider non-BPS crossing in this work, we here present a case that the non-BPS crossing and reconnection reduce to the other procedure that we use, namely pair annihilation, as shown in Fig. 19. Instead of considering the reconnection from the non-BPS crossing (b) in Fig. 19 to the vacuum (d) (in the two lower color lines), the non-BPS crossing (b) can be safely deformed by deformations keeping BPS properties to (a) and (c), and then finally (c) to (d) by pair-annihilation process. Therefore we find that the non-BPS crossing and reconnection can be reduced to the pair-annihilation process in some region of parameter space by deformations keeping BPS conditions. This result supports our strategy not to use non-BPS crossing to enumerate bion configurations.
When we repeat to insert pair creations in Fig. 17 (a), we do not insert the second pair outside the first pair, resulting in a zigzag configuration as in Fig. 15(a), which is reducible. Instead, we allow inserting the second pair creations between the first pair of fractional instanton and anti-instanton, as in Fig. 16 (a). If we insert the second pair with the direction opposite to the first one, we again have a zigzag configuration as in Fig. 15 (a), which is reducible.
For each color brane, k BPS fractional instantons are placed on the left (right) and k fractional anti-instantons are placed on the right (left) to cancel the instanton charge in total. We label it by k (−k). Therefore, irreducible neutral bions in G N F ,N C can be labeled by a set of N C integers: where (k 1 , · · · , k N C ) and Q give constituent fractional instanton numbers and the total instanton charge of the BPS fractional instantons corresponding to the left half of the bion.
The BPS reconnection in Fig. 12 is always possible, but a configuration after the reconnection is a compressed limit of the configuration before the reconnection. The moduli We will write down the moduli matrix of these neutral bions explicitly for a more general Similarly to the CP N F −1 case, fractional instanton is situated where the magnitude of two neighboring elements in each row (each color) become equal. In Eq. (V.3), the fractional instanton is located at x 1 = N F 2π log λ 1 , and fractional anti-instanton is at x 1 = N F 2π log 1 λ 3 . By creating a pair of fractional instanton and anti-instanton between the pair of fractional instanton and anti-instanton on the second brane in Fig.21(a), we obtain Fig.21(b) with a BPS fractional instantons (0, 2) for the left half of the diagram. The moduli matrix for this (0, 2) neutral bion for the model Gr N F ,2 is given by The fractional instantons are located at x 1 = N F 2π log λ 1 , N F 2π log 1 λ 2 , and fractional antiinstantons are at x 1 = N F 2π log 1 λ 4 , N F 2π log λ 4 λ 5 . By creating a pair of fractional instanton and anti-instanton on the first color brane in Fig.21(a), we obtain Fig.21(c) with a BPS fractional instantons (1, 1) for the left half of the diagram. The moduli matrix for this (1, 1) neutral bion for the model Gr N F ,2 is given by The fractional instantons are located at x 1 = N F 2π log λ 3 , N F 2π log λ 1 , and fractional antiinstantons are at To visualize the brane diagram for the bion (1, 1) given by the moduli matrix ansatz (V.5), we computed the relative weight of absolute value square of each flavor components of moduli matrix for each row corresponding to the parameter set λ 1 = 10 −2 , λ 2 = 10 −2 , λ 3 = 10 −5 , λ 4 = 10 −5 , N F = 4, and plotted in Fig. 22.
Since Σ is almost diagonal when fractional instantons are far apart, the relative weight becomes indistinguishable with the diagonal elements of Σ given in Eq. (III.3). This result nicely agrees with the schematic picture in Fig. 21(c). By further creating a pair of fractional instanton and anti-instanton between the pair of fractional instanton and anti-instanton in the second color brane in Fig. 21(c), we obtain  (1,2) neutral bion for the model Gr N F ,2 is given by The fractional instantons are located at x 1 = N F 2π log λ 4 λ 5 , N F 2π log λ 5 , N F 2π log λ 1 , and fractional anti-instantons are at If we further create a pair of fractional instanton and anti-instanton between the innermost pair of fractional instanton and anti-instanton in the first color brane in Fig.21(d), we obtain Fig.21(e) with a BPS fractional instantons (2, 2) for the left half of the diagram. The moduli matrix for this (2,2) neutral bion for the model Gr N F ,2 is given by The fractional instantons are located at x 1 = N F 2π log λ 6 λ 7 , N F 2π log λ 1 λ 2 , N F 2π log λ 7 , N F 2π log λ 2 , and fractional anti-instantons are at x 1 = N F 2π log 1 λ 4 , N F 2π log 1 λ 9 , N F 2π log λ 4 λ 5 , N F 2π log λ 9 λ 10 . There are other dagrams as a composite of BPS solutions and anti-BPS solutions with the set of higher fractional instanton numbers (k 1 , k 2 ), but they contain at least one instanton and are not bions anymore, or reducible diagrams. For instance, if we create a pair of fractional instanton and anti-instanton between the innermost pair of fractional instanton and anti-instanton in the second color brane of (1, 2) in Fig. 21(d), we obtain Fig. 23(a) with a BPS configuration (1, 3) for the left half of the diagram. Since this left half of the brane diagram is the BPS instanton (1, 3) in Fig. 10(f), Fig. 23 (1, 3) instanton-anti-instanton pair for the model Gr N F ,2 is given by The fractional instantons are located at x 1 = N F 2π log λ 6 λ 7 , N F 2π log λ 1 λ 2 , N F 2π log λ 7 , N F 2π log λ 2 , and fractional anti-instantons are at x 1 = N F 2π log 1 λ 4 , N F 2π log 1 λ 9 , N F 2π log λ 4 λ 5 , N F 2π log λ 9 λ 10 . The brane diagram in Fig. 23(b) is characterized by (2, 3), and is a reducible neutral bion diagram because of the decomposition (2, 3) = (1, 3) + (1, 0), namely it is a composite of an instanton-anti-instanton pair (1, 3) and a feactional instanton (1, 0). Fig. 23(c) is characterized by (−1, 1), and is constructed by placing in the left half of the diagram the exact non-BPS charged bion solution in Fig. 21(g). This is a reducible neutral bion diagram. In general, all the exact non-BPS solutions that we can construct in moduli matrix formalism is the case of composite of non-interacting BPS and anti-BPS fractional instantons. Therefore neutral bions constructible from these non-BPS exact solutions are always reducible. The brane diagram in Fig. 23 (d) and (e) are characterized by (−1, 2), and (−2, 1), respectively.
They are reducible neutral bion diagrams because they contain a configuration in Fig. 18.
Lastly, let us consider the other case of left vacuum being two non-adjacent flavor branes occupied by color branes. We can obtain neutral bions by combining BPS fractional instantons in Fig. 11 The fractional instanton is located at x 1 = N F 2π log λ 1 , and fractional anti-instanton is at x 1 = N F 2π log 1 λ 3 . Two color lines of (1, 1) in Fig.24(b) do not share any common flavor lines. This is the case of composite of two non-interacting elementary fractional instantons whose moduli matrix can be given as The neutral bion in Fig.24(c) with a BPS fractional instantons (1,2) for the left half of the diagram is given by the moduli matrix The fractional instantons are located at One can think of another possibility of (2, 1), but this case is equivalent to (1, 2) case and is not listed here. In this smallest Grassmann sigma model Gr 4,2 , we encountered a neutral bion whose constituent fractional instanton has unit instanton charge but is different from genuine instanton, as shown in Fig. 21(e). For larger Grassmann sigma models, there exist neutral bions with the instanton charge even larger than one. In Fig. 25, we show a neutral bion in G 5,2 , half of which has the instanton charge 6/5 greater than one. This cannot be decomposed into an instanton and the rest. This kind of phenomena arises due to an increasingly large number of different species of fractional instantons as N F , N C increases.

B. Charged bions in the Grassmann sigma models
Charged bions have no instanton charge in total but non-zero vector of fractional instanton numbers for the whole field configuration. Although the CP N F −1 models do not admit lines are non-interacting, namely they do not exert any static force. In such circumstances, we have observed already that our moduli matrix formalism allows non-BPS exact solutions of field equations [66]. The moduli matrix of the exact solution can be given by (V.14) To visualize the brane picture more exactly, we compute Σ defined in Eq. (III.3) for this charged bion. Since Σ turned out to be diagonal (reflecting the fact that two fractional instantons on different color lines are noninteracting), we can exactly compute Σ from the moduli matrix (V.14). The plot of Σ in Fig. 27 with the parameter set λ 1 = 10 2 , λ 2 = 10 2 , N F = 4 nicely realizes the brane picture that is schematically drawn in Fig. 26 (a).
For larger Grassmann sigma models, there are more combinations. An example of charged bion in G 6,3 is shown in Fig. 26 (b).
Before discussing the results of numerical evaluation, we make some comments:First, we take the phase parameter θ i = 0 (i = 1, 2, 3, 4) for simplicity. Indeed, we find that nonzero θ i just increase the total energy as with the neutral bion in the CP N −1 model [17],   as shown in Fig. 29. It is notable that the size of the fractional (anti-)instantons becomes a half smaller (∼ 4 →∼ 2) when they are compressed with another (anti-)instantons, which is consistent with the argument in Sec.II C (size of instanton= 1/∆m = L/S). In this case, the total energy is unchanged. For λ 1, the instanton and the anti-instanton characterized  Next, we calculate the parameter dependence of the total energy for the neutral bion configuration (V.5). As a characteristic case, we again vary λ = λ 1 = λ 2 with λ 3 = λ 4 = 10 −5 fixed. The result is given in Fig. 31. τ ∼ 10 corresponds to the compressed-kink cases as Fig. 29 while τ = 1 corresponds to the pair-annihilation case between the instanton and anti-instanton as Fig. 30. The total energy is changed from S = 4 ×1/4 = 1 to S = 2 ×1/4 = 1/2 as τ gets smaller. Since it is known that BPS solitons do not exert any static force, this result shows that the there is an attractive static force between fractional instanton λ 1 and anti-instanton λ 2 .
In order to study mutual interactions between constituent fractional (anti-)instantons more quantitatively, we define the interaction energy density as the energy density s(x 1 ) minus the two fractional-instanton density and two fractional-anti-instanton density 2s ν=1/N F + 2s ν=−1/N F , s int (x 1 ) = s(x 1 ) − (2s ν=1/N F + 2s ν=−1/N F ) . (VI.5) The integrated interaction energy is then given by  where C(N F ) is a y-intercept. In Fig. 32 we simultaneously depict these analytic functions.
The slope 2π/N F of this line is equivalent to that of the elementary neutral bion in the CP N F −1 model [17], which contains only one fractional instanton and one fractional antiinstanton. This means that the interaction energy is dominated by the interaction between λ 1 and λ 2 , and not by the other interactions. In Fig. 33 we plot exp[C(N F )] as a function of N F for N F = 3, 4, 5, 6, 7. By fitting the N F -dependence of the constant C(N F ), we obtain the interaction energy formula identical to the CP N F −1 model [17], which means C N F ∼ 4/N F .
(We note that we have included the factor 1/(2π) in the definition of Lagrangian in Eq. (II. 19) in this paper compared to our previous paper [17].). Namely we find Next we study the interaction between fractional instanton λ 3 and anti-instanton λ 4 in outer pair. To isolate the interaction between the outer pair, we take the annihilation limit of inner pair, fixing the parameters λ 1 = λ 2 = 10 5 . Then practically no remnant is left from the inner pair. By varying the separation between λ 3 and λ 4 , we find that the interaction energy is given precisely by the same formula as in the case of inner pair λ 1 and λ 2 in Eq. (VI.8). By using this formula for outer pair, we can see that the contribution from outer pair becomes tiny in the geometrical configurations in Figs. 28, 29, and 30. This justifies to neglect outer pair interaction enrgy in analyzing the interaction energy of inner pair a posteriori.
With our level of numerical accuracy, we cannot obtain definite results for other possible interactions between fractional instanton and anti-fractional instanton residing on different color lines, such as λ 1 − λ 4 , and λ 3 − λ 2 , except that they are at least as small as interaction energy between outer pair at the same separation.

B. Charged bions
We next consider the charged bion in the Grassmann sigma model. For Gr 4,2 model, we found only one irreducible charged bion with the fractional instanton number (−1, 1) in Fig. 26(a), which is given by the moduli matrix in Eq. (V.14). This is an exact non-BPS solution, since BPS and anti-BPS sectors reside on color branes which do not share any common flavors, and are non-interacting.
To find out the properties of the solutin in detail, we study it numerically using our formula in Eq. II.24. We observe analytically that the energy and charge densities are independent of the moduli parameters θ 1 and θ 2 . For the symmetric case λ ≡ λ 1 = λ 2 , the separation between the fractional instanton (at −(N F /2π) log λ 1 ) and fractional antiinstanton (at (N F /2π) log λ 2 ) is given by τ = (N F /π) log λ. We depict energy and charge densities for three sets of parameters,    We find that, unlike the neutral bions, the energy density is still nonzero even in the no separation limit τ = 0 (λ 1 = λ 2 = 1) of the fraectional instanton and anti-instanton. As shown in Fig. 36, lumps of the topological charge density annihilates and disappear in this case. Although total topological charge happens to vanish, BPS fractional instanton and anti-instanton are not of the same species, and cannot annihilate each other, as anticipated.
To show details, we depict the separation (τ ) dependence of the total energy in Fig. 37. The total energy is independent of the separation, and keeps the constant value S = 2×1/4 = 1/2.  for N F = 4 fixed.

VII. SUMMARY AND DISCUSSION
In this paper, we have considered topologically trivial configurations in the Grassmann sigma model on R 1 × S 1 with the Z N F symmetric twisted boundary conditions, to study properties of bions composed of multiple fractional instantons. By formulating these models as gauge theories, we proposed to use the moduli matrix to classify bion configurations. By embedding these models to D-brane configurations in type-II string theories, we have found that D-brane configurations together with the moduli matrix are useful to classify all possible bion configurations in the CP N F −1 models and the Grassmann sigma models. We have found that the Grassmann sigma models admit neutral bions made of BPS and anti-BPS fractional instantons each of which has a topological charge greater (less) than one (minus one), nevertheless it cannot be decomposed into (anti-)instanton and the rests. We have found that Grassmann models admit charged bions, while the CP N F −1 models do not admit them. We have also constructed exact solutions of charged bions in the Grassmann model.
We have calculated the energy density and topological charge density of the bion configurations in these models numerically, and have obtained their interactions. The dependence of these interactions on the separations between fractional instanton constituents is studied explicitly.
We have studied the Grassmann sigma model without fermions. On the other hand, fermions can be coupled to the Grassmann sigma model. This is the case of the supersymmetric Grassmann sigma model, which can be formulated from supersymmetric gauge theories. In this case, fermions are localized at the fractional instantons and contribute to the interactions between bions.
In this paper, we have concentrated on the Z N unboken gauge symmetry occurs for gauge theory on R 3 × S 1 with partially degenerated twisted boundary conditions. We can prepare corresponding situations by putting boundary conditions of some flavors to coincide. In this case, kinks carry non-Abelian moduli and can be called non-Abelian kinks [78][79][80]. In the brane picture, the s-rule admits at most n color branes can sit on n coincident flavor branes, and there remains a U(n) gauge symmetry on the color branes which is a source of non-Abelian moduli for non-Abelian kinks. We will consider this situation with "non-Abelian bions" in a future publication.
One of future directions is to extend our method to bion configurations in other nonlinear sigma models. Since nonlinear sigma models on Hermitian symmetric space (including Grassmann and CP N F −1 ) can be formulated as gauge theories [47], the moduli matrix can be used for these cases, although embedding to brane configurations is not yet available. In gauge theory perspective, changing the gauge group from U(N C ) studied in this paper to other groups is also one possible direction. The (hyper-)Kähler quotients for G = SO and USp were obtained in Ref. [81], in which fractional instantons were also studied. We may study bions in these cases.
As for the relation between two and four dimensional theories, the effective theory on a non-Abelian vortex in four dimensions [52,53,56] is the two-dimensional CP N F −1 model, thereby the Yang-Mills instantons and monopoles are CP N F −1 instantons and kinks inside a vortex, respectively [40,54,55,58,82,83]. Therefore, bions in Yang-Mills theory can exist inside the vortex as the CP N F −1 bions when the vortex world-sheet is wrapped around S 1 .
With this regards, a non-Abelian vortex in gauge theories with gauge group G admits a G/H nonlinear sigma model on its world-sheet [84]. In particular, the cases of G = SO(N) and USp(2N) were studied extensively [81,85]. With twisted boundary conditions, it admits monopoles as kinks on G/H sigma models in the vortex theory [86], in which kinky brane-like picture was obtained. Quark matter in high density QCD also admits a non-Abelian vortex [87] whose effective theory is the CP 2 model [88] (see Ref. [89] as a review). A bound state of a kink and an anti-kink appears quantum mechanically inside a vortex, representing a meson of a monopole and an anti-monopole. The quark-hadron duality between the confining phase at low density and the Higgs phase at high density may be explained through a non-Abelian vortex [90]. Bions inside a non-Abelian vortex wrapped around S 1 in these cases are interesting to study in future.
We may compactify two or more directions in higher dimensional theories. For instance, the CP N F −1 model, Grassmann sigma model and corresponding gauge theories on R 2 × S 1 × S 1 admit not only fractional lumps (vortices) from one of S 1 's which are string-like, linearly extended structure of the fractional instantons studied in this paper but also their intersections with Yang-Mills instanton charge [83]. These configurations are called Amoebas in mathematics and reduce to domain wall junctions [91] for a small S 1 radii limit. Bions in this theory will have more varieties because fractional solitons have networks in two directions, and hopefully are useful for four dimensional gauge theories.
which satisfies the constraint However, V U is not unitary, and is different from U ′ . Even the eigenvalues are different Ω ′ d = Ω d . Therefore the resulting solution H ′ of the constraint obtained from the formula (II.24) is in general different from H. Although relations between U ′ , Ω ′ d and U, Ω are complicated, we find that the following matrixŨ is a unitary matrix