Topological terms of (2+1)d flag-manifold sigma models

We examine topological terms of $(2+1)$d sigma models and their consequences in the light of classifications of invertible quantum field theories utilizing bordism groups. In particular, we study the possible topological terms for the $U(N)/U(1)^N$ flag-manifold sigma model in detail. We argue that the Hopf-like term is absent, contrary to the expectation from a nontrivial homotopy group $\pi_3(U(N)/U(1)^N)=\mathbb{Z}$, and thus skyrmions cannot become anyons with arbitrary statistics. Instead, we find that there exist ${N(N-1)\over 2}-1$ types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians.


Introduction and Summary
Nonlinear sigma models appear as low-energy effective descriptions in broad areas of modern physics, ranging from high-energy to condensed-matter physics. For example, they have allowed systematic analyses of strong-interaction phenomena [1,2] even before the establishment of quantum chromodynamics (QCD), while they have also provided useful tools to study the long-range physics of anti-ferromagnetic spin systems [3].
In nonlinear sigma models, Lagrangian consists not only of the usual kinetic term but also of topological terms. Schematically, the kinetic term can be written as where σ : M d → X is a map from the spacetime M d to the target space X, and g ab (σ) is the metric of X, which defines the effective coupling constant. In practical use, we can add higher-derivative corrections to this Lagrangian, but (1.1) is often sufficient for our purpose.
Let us consider what kind of topological terms can be added to this Lagrangian. According to the standard lore, they are classified by the homotopy of the target space π * (X) [4]. In the path integral, we are interested in the configuration with the finite Boltzmann weight exp(−L kin [σ]) < ∞. Taking the infinite-volume limit M d → R d , in order to have a finite action, the sigma-model field must be constant as it approaches the infinities, lim |x|→∞ σ(x) = σ * ∈ X. (1.2) As a result, we can identify the infinities of the spacetime with a single point, and we may regard σ : S d → X. By definition, the topological nature of such maps is classified by π d (X). Based on this kind of argument, it has been considered that there is a theta term corresponding to nontrivial π d (X), and also that there is a Wess-Zumino term [5,6] when π d+1 (X) = Z while π d (X) = 0, as it allows us to extend the sigma-model configuration on S d to a disk D d+1 to define the well-defined phase factor mod 2π.
On the other hand, recent developments have led to more refined understanding of topological terms. Assuming quantum field theories (QFTs) can be defined on any spacetime manifolds (with given structures), the principle of locality requires their partition functions to be consistent under cutting and gluing of spacetime manifolds. Along this line together with the unitarity, (exponentiated) topological terms should be regarded as partition functions of the so-called invertible QFTs [7]. As a result, for the case at hand where the sigma-model target space is X, topological terms are classified up to continuous ones by a group Inv d (X) [8,9], which sits in the middle of the short exact sequence In addition, ordinary theta angles invisible in Inv d (X) are classified by Hom(Ω d (X), U (1)) along with discrete theta angles [10]. An important point is that these classifications use bordism groups Ω * (X) 1 rather than homotopy groups π * (X), and correspondingly the traditional arguments relying on homotopy groups might miss some subtleties. This indeed turns out to be the case, and one of the examples regarding the Hopf term in 3d CP 1 sigma model was recently pointed out by Freed, Komargodski, and Seiberg [11]. In this paper, we will further explore 3d sigma models with target spaces CP N −1 and flag manifolds U (N )/U (1) N , both of which can be viewed as a generalization of the simplest CP 1 case. 23 In passing, we mention another motivation to consider the flag-manifold sigma models. In (1 + 1)d, the sigma models with flag-manifold target space have appeared as effective descriptions of the SU (N )-analogue of antiferromagnetic Heisenberg-like chains [15][16][17][18], and are known to show a rich phase diagram thanks to various theta terms [19][20][21][22] (see [23] for a review). Similarly, the (2 + 1)d flag-manifold sigma models describe the 2d quantum SU (3) Heisenberg model on several lattices [24,25], and some of them are equivalent to 1 In this paper, we consider spin bordism groups Ω spin * (X) since the underlying microscopic theories are assumed to contain fermions.
2 Generalized (co)homologies including bordism have been also effectively used to analyze the nonlinear sigma model descriptions of 4d QCD-like theories [8,12,13]. 3 Analyses on the sigma-model dynamics from the bordism perspective have also been carried out in [14].
the Tsunetsugu-Arikawa point [26] of the SU (2) spin-1 model. It is expected that the SU (N ) anti-ferromagnetic Heisenberg systems may be realized with the ultracold atoms using alkaline earth metals [27][28][29][30][31][32][33], and thus theoretical predictions on these systems may be able to be verified by such experiments. Therefore, it is an interesting question to ask what kind of topological terms can enrich those quantum spin systems. 4 Let us summarize our result on the U (N )/U (1) N flag-manifold sigma model, which shall be discussed in detail in Section 4. Although the homotopy argument suggests the presence of Hopf-like term because of the nontrivial homotopy π 3 (U (N )/U (1) N ) = Z, we conclude that such a term does not exist since for N ≥ 3, assuming that the spacetime manifold is equipped with spin structure. We also find that there are N (N −1) 2 − 1 types of Chern-Simons-like terms according to the bordism computation, and we try to write down their explicit forms using the gauged sigma-model description. In such a description, we introduce U (1) gauge fields a 1 , . . . , a N −1 , which indeed allows us to write down the Chern-Simons coupling, but there still exists a puzzle. When naively writing down possible Chern-Simons term, we get and there are N (N −1) 2 of them, which overcounts the number of independent Chern-Simons couplings compared to the bordism computation. It turns out that there is a nontrivial relation, which allows us to eliminate one of the off-diagonal Chern-Simons couplings, and we find there exist globally well-defined 3-forms, which can be added to the effective Lagrangian, as a byproduct. For example for N = 3, the topological terms of the effective Lagrangian are given as where the first two terms are diagonal Chern-Simons terms, and the last one is a nonquantized topological coupling with a ij defined as (4.20), while the off-diagonal Chern-Simons term is absent. In particular, when the diagonal Chern-Simons terms have odd levels, skyrmions are transformed as fermions under spacetime rotations. The rest of the paper is organized as follows. In Section 2, we review the case of 3d CP 1 sigma model. We first directly compute bordism groups to identify the topological ("Hopf") term, and then discuss its consequence for the statistics of skyrmions from the physics point of view, using gauged realization of the model. In Sections 3 and 4, we further go on to the case of 3d sigma model with target spaces CP N −1 and flag manifolds U (N )/U (1) N , respectively. Appendix A serves as a supplement involving some subtleties arising in the flag manifold case. In Appendix B, various relevant cohomology groups are computed via Leray-Serre spectral sequence (LSSS). In Appendix C, we review the realization of the Hopf term of the CP 1 model as a partition function of a fermionic invertible phase [36].
2 Review on the topological term of 3d CP 1 model 3d CP 1 sigma model gives a low-energy description of the 2d anti-ferromagnetic quantum spin systems. Because of π 3 (CP 1 ) = Z which represents the Hopf fibration, it has been suspected for more than three decades that the model accommodates a topological theta term, which can be schematically written as iθ Hopf[n]. (2.1) The paper by Wilczek and Zee [37] has shown that the (magnetic) skyrmion acquires the fractional spin θ/2π and becomes an anyon due to this Hopf term. The Hopf term does not have a local expression in the original spin fields, but can be written as a Chern-Simons form using the U (1) gauge field in the gauged sigma-model description [38]. Meanwhile, starting from the (2 + 1)d SU (2) anti-ferromagnetic quantum Heisenberg spin system, it is known that the Hopf term does not appear in the low-energy theory [39]. Instead, the discrete value of the theta term θ = π arises when the spin variable is coupled to fermions by Yukawa interaction, after integrating out fermions [36].
In a recent paper [11], it has been elucidated that only these values, θ = 0 and θ = π, are consistent as local and unitary QFTs. Indeed, k = θ/π behaves as the level of U (1) spin Chern-Simons term, and the magnetic skyrmion becomes fermion if k is odd.
In the following of this section, we will give a review on these facts, as it is a basic ingredient in order to extend these results for general flag-manifold sigma models.

Formal and explicit descriptions of topological terms of CP 1
Here, we first give a formal description on the possible topological terms in the CP 1 sigma model, and then write down the explicit forms for the topological term. Unless explicitly stated, we always assume that the spacetime manifolds are equipped with spin structure.
These groups can be computed using the Atiyah-Hirzebruch spectral sequence (AHSS) where the E 2 -page on the left hand side converges to the desired group on the right hand side. The explicit form of the E 2 -page is given as follows: It is known that, if the space X is connected, bordism groups split as and therefore differentials going into the p = 0 column (e.g. the dashed arrow) are all trivial. As a result, E 2 2,1 = H 2 (CP 1 ; Ω spin 1 (pt)) = Z 2 survives to the E ∞ -page, and one has Free(Ω spin 3 (CP 1 )) = 0, Tors(Ω spin 3 (CP 1 )) = Z 2 , Free(Ω spin 4 (CP 1 )) = Z. (2.5) The free part of Ω spin 4 (CP 1 ) comes from Ω spin 4 (pt) describing the gravitational Chern-Simons term, which is not of our interest; our interest will only be in Ω spin 4 (CP 1 ). The Hopf term with continuous θ ∈ R/2πZ parameter is in fact absent as the free part of Ω spin 3 (CP 1 ) is trivial. We identify the torsion part of Ω spin 3 (CP 1 ) as the "Hopf" term of 3d CP 1 model, but its theta angle should be discretized as it is Z 2 -valued.
Let us move on to explicit computations in order to write down the discrete theta term. We can realize the CP 1 sigma model as a U (1) gauge theory according to the fibration U (1) → SU (2) → CP 1 , that is, we introduce the SU (2)-valued scalar field U (x) and the U (1) gauge field a = a µ dx µ . We note that SU (2) matrix U can be written as with |z 1 | 2 + |z 2 | 2 = 1, and thus the SU (2) scalar field is equivalent to the two-component complex scalar field z = (z α ) α=1,2 with a unit norm. Assigning U (1) gauge charge 1 to this scalar z, the kinetic term in the Lagrangian is given by Since this is quadratic in a µ , the equation of motion can be solved as a = iz † dz. (2.8) With this U (1) gauge field, we define the Hopf term as the Chern-Simons term, following the idea of [38]. That is, we extend the spacetime manifold M 3 to the 4d spin manifold W 4 with ∂W 4 = M 3 and define whereã is the 4d extension of a as U (1) gauge fields.
We shall give concrete computations to show that this gives the correct discrete theta term in the next subsection. Before that, let us make several remarks, which would be confusing at first sight.
In the definition (2.9), we only extend the U (1) gauge field a, while the CP 1 field needs not be extended to W 4 , and therefore (2.8) holds only on the boundary M 3 but not on the 4d bulk. 6 We note that Ω spin 3 (CP 1 ) = Z 2 shows that the CP 1 field may not have a 4d extension when the spin structure is taken into account: Only after taking the double copy, M 3 M 3 , we can take the 4d bounding manifold M 4 such that the extension of the CP 1 field is possible.
Next, let us prove that the Hopf term defined in (2.9) is a bordism invariant of 3d spin manifolds equipped with a map σ = (z, a) to CP 1 . To see this, suppose the pair (M 3 , σ) is bordant to (M 3 , σ ), i.e., there exists a 4d spin manifold Y 4 and a map σ : Then we have Here, unlike in the definition (2.9), the 4d extension to Y 4 is taken as a CP 1 field, and thus the U (1) gauge field is subject to the constraint (2.8),ã = iz † dz. As a result, dã ∧ dã = 0 identically. Hence, we can see that which means that the Hopf term (2.9) gives a bordism invariant. 6 In the work of Abanov [42], the similar idea has been used to find the Hopf term by integrating out the fermion when the spin variable n = z † σz couples to it. Instead of directly evaluating the fermion determinant, they consider an infinitesimal variation of the spin variable, and try to integrate it to obtain the result. Within the CP 1 model, however, the Hopf term is a total derivative [38], and the infinitesimal variation identically vanishes. They circumvented this issue by embedding CP 1 ⊂ CP N −1 with N > 2. As BU (1) = CP ∞ , our prescription (2.9) can be formally regarded as the N → ∞ limit of it.

Statistics of skyrmions
Let us illustrate how the level k of the Hopf term is quantized. Since the 3d spin bordism group equipped with a map to CP 1 is given by Ω spin 3 (CP 1 ) = Z 2 , the Hopf term must take a Z 2 valued phase, exp iπ Hopf[M 3 , σ] = ±1. Therefore, we should identify . This shows that we can verify the definition (2.9) of the Hopf term by constructing a CP 1 configuration σ on a 3-manifold We can construct such a configuration by taking the spacetime M 3 to be S 2 × S 1 , introducing a magnetic skyrmion on the space S 2 , and then performing the adiabatic 2π rotation along the imaginary time S 1 . This is exactly the way to measure the statistics of the skyrmion in a semi-classical manner [37,38]. A skyrmion can be put on S 2 by setting where (θ, φ) is the spherical coordinate of S 2 with 0 ≤ θ ≤ π and 0 ≤ φ < 2π. We can perform the adiabatic 2π rotation along the imaginary time 0 ≤ τ < 2π as Here, exp i τ 2 σ 3 describes the adiabatic rotation, and we further multiply the overall U (1) phase e iτ /2 so that the configurations at τ = 0 and τ = 2π are in the same gauge. We then obtain the U (1) gauge field a on S 2 × S 1 from (2.15) as In order to evaluate (2.9), we need to extend (2.16) from S 2 × S 1 to a 4-manifold. We can take W 4 = S 2 × D 2 as an extension, where the polar coordinate of D 2 is given by (ρ, τ ) and the extension of a isã = − cos 2 θ 2 We can readily check that This means that the skyrmion becomes fermionic due to the Hopf term when we set k = 1. One should be able to confirm the statistics of the skyrmion also by checking the statistics of its creation/annihilation operator. Since a single skyrmion in the space S 2 introduces the configuration of the U (1) gauge field (2.16), we identify its creation/annihilation operator as the monopole operator m(x) for the U (1) gauge field. The monopole operator m(x) is defined as a defect operator. Roughly speaking, we remove a point x from the spacetime 3-manifold M 3 , and require the U (1) gauge field a to behave as where S 2 x is a sufficiently small two-sphere surrounding the removed point x. When the monopole operator m(x) is inserted, we assert that the path integral is performed over the gauge fields with this boundary condition.
We note that the monopole operator is not U (1) gauge invariant when the Chern-Simons term exists [43,44]: Under the gauge transformation, a → a + dε and z → e −iε z, (2.20) As a result, the gauge-invariant monopole operator should be defined as The quantum number of this operator should be identical with that of the corresponding skyrimon. In order to find the spin of the gauge-invariant monopole operator, note that the spherical coordinate dependence of the scalar field z is described by the monopole harmonics, Y q, ,m , near the monopole singularity [45]. For the minimal monopole charge, q = 1/2, and the allowed values of are ∈ |q| + Z ≥0 [46,47]. Therefore, the spin of the monopole operator is given by k/2 mod Z, which is consistent with the above computations.
3 Topological term of 3d CP N −1 sigma model with N ≥ 3 Let us move on to the 3d CP N −1 nonlinear sigma model, and consider its topological terms. Even though the expressions for the topological terms of CP 1 and CP N −1 are superficially similar, we shall emphasize the difference between them in this section. This understanding turns out to be useful to find the correct description of topological terms for the flag-manifold model in the next section.

Conflict between homotopy argument and the gauged-sigma model
The CP N −1 sigma model can be realized using a C N -valued scalar field z = (z α ) α=1,...,N with a unit norm, with a U (1) gauge field a. This corresponds to the fibration which is identical with the Hopf fibration for N = 2. In the following, we consider the case with N ≥ 3. As a consequence of the homotopy exact sequence, for k ≥ 3, as π k (S 1 ) = π k−1 (S 1 ) = 0, and we immediately find that Therefore, in the conventional view of the homotopy, we do not expect the 3d CP N −1 model to have any topological terms, neither theta terms nor Wess-Zumino terms. However, if we regard the CP N −1 model as an example of U (1) gauge-Higgs systems, we can write down a Chern-Simons coupling for the U (1) gauge field, where W 4 is a four-dimensional extension of the spacetime 3-manifold along with the spin structure. We would like to resolve the discrepancy between these two arguments, and also find the physical consequence of the topological terms.

Spin bordism of CP N −1 and the topological term
To find possible topological terms, we again consider AHSS The E 2 page is given by the following figure: where the homology of CP N −1 is (see Appendix B.1 to confirm this result from the Leray-Serre spectral sequence (LSSS)). Important difference from the N = 2 case is that H 4 (CP N −1 ) = Z for N ≥ 3 and this gives rise to a possibly-nontrivial differential d 2 : E 2 4,0 → E 2 2,1 . This map is known to be given by a dual of Steenrod square composed with mod 2 reduction [48], and here the generator Therefore, (not only the ordinary ones but also) the discrete theta angles do not exist in the 3d CP N −1 sigma model for N ≥ 3. Instead, we have which is nothing but the Chern-Simons coupling (3.4) in our description. Here, the map is a multiplication by two, 7 since the latter is the kernel of d 2 : E 2 4,0 → E 2 2,1 . To be more explicit, 8 we can take CP 2 ⊂ CP N −1 as the generator of H 4 (CP N −1 , Z), which satisfies CP 2 da 2π 2 = 1. On spin 4-manifolds X, however, the Atiyah-Singer index theorem tells that X da 2π 2 is always an even integer, so there is a factor-2 difference between the integral homology and the spin bordism of degree 4.
Let us make several remarks. Although the definition (3.4) looks very similar to the one (2.9) in the CP 1 model, there are important differences. As Ω spin 3 (CP N −1 ) = 0 for N ≥ 3, the CP N −1 configurations in 3 dimensions are always null-bordant within the spin manifold. Therefore, unlike in the case of CP 1 , we can take the 4d extension of the fields (z,ã) as CP N −1 fields in the definition (3.4). A related remark is that, for CP N −1 models, we do not have any identifications on k, although the Hopf tem for the CP 1 model has (2.13). For such an identification to exist, the phase factor exp i 4π ada needs to be quantized. This indeed occurs for the CP 1 model because dã ∧ dã = 0 whenã is restricted toã = iz † dz. However, for CP N −1 with N ≥ 3, one can easily check that dã ∧ dã = 0 even with the constraintã = iz † dz, so such quantization of phases does not occur. This explains the fact that the topological term (3.4) comes from Ω spin 4 (CP N −1 ), not from Ω spin 3 (CP N −1 ), in an intuitive way.

Statistics of skyrmions
Let us discuss the statistics of skyrmions. We add the level-1 Chern-Simons coupling to In order to discuss the quantum number of skyrmions, we can look at the statistics of corresponding monopole operators, as we have briefly explained in Sec. 2.2. Because of the Chern-Simons coupling (3.4), the naive monopole operator is not U (1) gauge invariant, and the gauge-invariant monopole operator is given as (3.10) Therefore, the skyrmion is in the defining representation of SU (N ) flavor symmetry. In order to discuss its spin, we again note that the z field in the vicinity of the monopole singularity should be decomposed with the monopole harmonics Y 1/2, ,m . Thus, the z field has a half-integer spin, and so does the monopole operator m(x)z α(x) . As a result, the corresponding skyrmion is a fermion. Let us confirm this result by directly evaluating the topological term (3.4) for the adiabatic 2π-rotation of a skyrmion. To be specific, let us set N = 3 and put a CP 1 7 We would like to note that this factor 2 appears due to the spin structure of the spacetime. When we only assume the orientation structure, we need to compute the oriented bordism, instead. As (reduced) oriented bordism is exactly identical to the homology at low degrees without this extra factor, we can classify the topological terms using ordinary (co)homology in such cases [49]. Use of generalized (co)homology allows us to deal with the subtle factor related to the spacetime structure. 8 We thank the anonymous referee for suggestion of giving the following explicit explanation.
skyrmion configuration in the first two components of CP 2 fields: Here, we take the spherical coordinate of the space S 2 , and make θ and φ run over [0, π] and [0, 2π) respectively. Then, we can realize the 2π rotation of this skyrmion by multiplying the following SU (3) matrix, (3.12) Here, α ∼ α + 2π is regarded as the imaginary-time circle at ρ = 1, and it is extended to the two-dimensional disk D 2 . Importantly, unlike the case of CP 1 model, we here take a 4d extension of both the U (1) gauge field and the scalar field. 9 Now, let us evaluate the Chern-Simons term. The U (1) gauge field is obtained as The phase factor acquired by the skyrmion is given by which shows that the skyrmion with the Chern-Simons coupling becomes a fermion.

Topological terms of 3d flag-manifold sigma model
We finally consider sigma models having flag manifolds as a target space, which are another type of generalizations of CP 1 = U (2)/U (1) 2 . 9 We note that the same is true for skyrmions of 4d N f -flavor QCD; for N f = 2, we cannot extend the skyrmion configuration with 2π rotation into 5d, but it becomes possible if we embed it into larger flavors N f ≥ 3 [50].

Spin bordism of flag manifolds
As before, the classification of topological terms is given in terms of Ω spin * (U (N )/U (1) N ). The AHSS we consider is and the E 2 page is as follows: 2 for the computation of integral cohomology of flag manifolds, which is necessary for filling in the above table. Here, note that the Z 2 cohomology ring of flag manifolds is known to be [51] where Λ Z (x 1 , . . . , x N ) is a ring of symmetric functions of generators x 1 , . . . , x N ∈ H 2 (BU (1) N ). This implies that the differential d 2 : E 2 4,0 → E 2 2,1 given by a dual Steenrod square composed with mod 2 reduction [48] is nontrivial, and none of the Z 2 's involved survive to the E ∞ -page. This shows that Ω spin 3 (U (N )/U (1) N ) = 0, (4.5) and thus both ordinary and discrete theta angles do not exist in the 3d U (N )/U (1) N sigma model for N ≥ 3. We also find that each factor corresponding to a Chern-Simons-like term, and there are N (N −1) 2 − 1 of them in total. 10 In this formula, we have already neglected the purely gravitational term by considering the reduced bordism. 10 The factor 2 in front of Z appears in the same way as we have discussed in (3.9). Here, it means that the diagonal Chern-Simons terms are quantized with the coefficient 1 4π , while the off-diagonal ones are quantized with the coefficient 1 2π , which is twice of that of diagonal ones. off-diagonal Chern-Simons terms, where f I = da I . However, this overcounts the number of independent Chern-Simons couplings according to the bordism computations (4.6). We shall resolve this issue in the next subsection by closely looking at the constraint on U (1) gauge fields.
In passing, we note that this is again in conflict with the homotopy argument, where the homotopy exact sequence on the fibration Thus, the homotopy argument would imply the existence of one continuous theta angle. However, the bordism suggests that the 3d theta terms (both ordinary and discrete) do not exist. Instead, there should to be many Chern-Simons terms, which are also invisible from the homotopy group.

Effective Lagrangian of 3d flag-manifold sigma models
Let us consider the form of the effective Lagrangian for 3d flag-manifold sigma models. We take an approach of the gauged nonlinear sigma model, by introducing an SU (N ) scalar field U and U (1) gauge fields a I=1,...,N −1 . We denote the SU (N )-valued scalar field U as where z i are N -component complex vectors with the constraint and also with det[z 1 , . . . , z N ] = 1. We can then express z N in terms of the other scalars We can write down the SU (N )-symmetric kinetic term as where we have set a N := −(a 1 + · · · + a N −1 ) to emphasize the Z N permutation symmetry of this Lagrangian. We can solve the equation of motion for the gauge fields to find the constraint a I = iz † I dz I , (4.17) for I = 1, . . . , N − 1. With this setup, we try to find the possible topological terms that can be added to this Lagrangian. We have observed that there are potentially N (N −1)

2
Chern-Simons type terms listed in (4.7) for the 3d sigma model on the flag manifold. Meanwhile, due to the E 2 page of the AHSS (4.3) implies the existence of N (N −1) 2 − 1 of them. This suggests that one specific linear combination of Chern-Simons terms in (4.7) somehow becomes trivial, which enables us to identify N (N −1) 2 − 1 independent Chern-Simons terms modded out by such a relation. In Appendix A, we actually find that the U (1) gauge fields are subject to the relation given by where we have introduced a ij = z † i dz j (4.20) for i = j. Under the U (1) gauge transformations, they behave as where we have used z † j · z i = 0 for i = j. Therefore, although a ij are not U (1) gauge invariant, they transform covariantly. Accordingly, following combinations a ij ∧ a jk ∧ a ki , (4.22) for i = j = k, are gauge invariant 11 , which define globally well-defined 3-forms. This shows that the right-hand-side of (4.19) is an exact 4-form, and we have reproduced the result of the cohomology group by an explicit computation. We note that (a ij a jk a ki ) * = a ji a ik a kj , (4.23) so the right-hand-side of (4.19) is real-valued as it should be. According to this relation (4.19), the following combination involving Chern-Simons terms potentially provides a bordism invariant of 3d spin manifolds equipped with a map to U (N )/U (1) N , as in the case of the Hopf term in the CP 1 model. Here, M 3 is a 3d spacetime manifold where ∂W 4 = M 3 . However, since the spin bordism Ω spin 3 (U (N )/U (1) N ) = 0, we can take a 4d extension as a flag-manifold field, not as U (1) gauge fields. Hence, the above candidate term turns out to be trivial and we can simply eliminate one of the off-diagonal Chern-Simons term by using (4.19). 12 In the following, let us explicitly work on the case N = 3, i.e. on the 3d SU (3)/U (1) 2 flag-manifold sigma model. Topological terms of the effective Lagrangian can be written as Here, we note that (a 12 a 23 a 31 ) * = a 12 a 23 a 31 − d(a 12 a 21 ), (4.27) and thus it gives a real-valued 3-form up to a total derivative. Therefore, we do not need to introduce coupling constants separately for a 12 a 23 a 31 and (a 12 a 23 a 31 ) * = a 21 a 13 a 32 . The coupling constant h for a 12 a 23 a 31 does not need to be quantized unlike the Chern-Simons couplings k 1 , k 2 , and it does not have a periodicity unlike the theta terms, even though the term is topological. Even if we add the off-diagonal Chern-Simons term, k 12 2π dã 1 ∧ dã 2 , its effect can be taken into account with the above Lagrangian by rewriting the couplings Let us give a quick observation on how we could introduce the Chern-Simons coupling in view of symmetry. Our kinetic term (4.16) is constructed so that it has Z 3 permutation symmetry, z i → z i+1 mod 3 , which is often a consequence of the underlying lattice symmetry of quantum spin systems. However, the Chern-Simons coupling cannot be introduced in a Z 3 -symmetric way. Indeed, the only Z 3 symmetric combination of the diagonal Chern-Simons term is , and because of the relation (4.19), this is a trivial element of the cohomology. This would suggest that, if we try to construct a SU (3)/U (1) 2 model with nontrivial Chern-Simons terms from the SU (3) Heisenberg-like model on triangular lattices, we need to break the full wallpaper group, p6m, to some smaller subgroup explicitly. Lastly, we consider the properties of skyrmions for the SU (3)/U (1) 2 sigma model. As there are two independent U (1) gauge fields, there are two different skyrmions charges defined by Q I = 1 2π M 2 da I for I = 1, 2. Let us introduce a skyrmion for Q 1 and discuss its statistics as we have done in Sec. 3. Such a skyrmion can be constructed as We can further construct a 4d extension of the adiabatically 2π-rotated skyrmions by multiplying an SU (3) matrix from the left, exactly as we did in (3.12). One can readily check that the skyrmion becomes a fermion when k 1 is odd. Note that the non-quantized topological term, a 12 a 23 a 31 , does not affect the result since it vanishes identically for this configuration on S 2 × S 1 .

A Derivation of constraint relations for the flag sigma model
In this appendix, we prove the relation for the flag-manifold sigma model, where i =j<k means the summation over all 1 ≤ i, j, k ≤ N satisfying i = j and both i, j < k. Combined with the relation f N = −f 1 − · · · − f N −1 , we find that where capital I, J run over 1, . . . , N − 1, while i, j, k run over 1, . . . , N . Let U be a U (N )-valued scalar field and define the Maurer-Cartan form as Let P i be the projector to i-th component and denote We note that, in this appendix, the gauge fields, A and A i , are taken to be anti-Hermitian, and thus there is an extra fator of i to make the connection with the main text of the paper: Let us prove the following relations, The first relation (A.7) can be shown very easily. Since dA = −A 2 , we find that 13 Since Ai is a U (1) gauge field, A 2 i -term in the definition of Fi is not necessary, as it automatically vanishes. However, the proof in this appendix can be generalized to the case where Ai is a U (n) gauge field, which corresponds to more general flag manifolds, U (n1 + . . . + nN )/(U (n1) × · · · × U (nN )). Therefore, in the proof, we do not use any special properties for U (1) gauge fields, and keep the A 2 i -term in the field strength. Therefore, Here, we use the cyclic property of the trace, which gives tr(P i AP j A) = −tr(P j AP i A). Let us show the second relation (A.8).
In order to obtain the last expression, we replace the summations i, j, k over 1, . . . , N into the summations I, J, K over 1, . . . , N − 1 and N is treated separately, and we eliminate P N by substituting P N = 1 − P 1 − · · · − P N −1 . By cyclic property, tr((P I A) 4 ) = 0 and J,K,L,M tr(P J AP K AP L AP M A) = 0. We therefore find The first equation on the right-hand-side is obtained by replacing A 2 = −dA in (A.11) with the above note. As it turns out to be a total derivative, we extract the exterior derivative to find the second equality, and replace dA by −A 2 inside the overall differential. In order to go from the second line to the third line on the right-hand-side, we have replaced A 2 = A( L P L +P N )A and then consider the combinatorics for the domain of summations.
To obtain the last line from the third one, we note that tr(P J AP K AP L A) is cyclically symmetric in J, K, L, so we can replace J =K =L = 3 J =K<L in the first term. Then, we can combine the first and second terms in the third line by reintroducing the dummy indices i, j, k running over 1, . . . , N , which gives the last line, and this completes the proof.

B Leray-Serre spectral sequence for cohomologies of sigma models
Leray-Serre spectral sequence (LSSS) provides us a systematic way to find cohomologies from the fibration, When π 1 (B) acts trivially on the cohomology of the fiber H * (F ), LSSS gives that is, the E 2 page can be written down by knowing the cohomologies of the base space B and the fiber F , and it converges to the cohomology of the total space E. LSSS is a first-quadrant spectral sequence, so if B or F is finite dimensional, then LSSS collapses at some finite page. There are two typical ways to use LSSS. The first one is to obtain H * (E) when we already know about H * (B) and H * (F ). The another one is the "opposite" way: knowing H * (E) and one of cohomologies of B or F , we constrain the possible cohomologies of the last one.
In the context of the sigma model, we are interested in the cohomology of the target space X. In many cases, the target space is given by the homogeneous Riemannian space G/H, because this appears when a global symmetry G is spontaneously broken to its subgroup H by some condensates. Then, there is a principal H-bundle, When both G and H are continuous and connected Lie groups, and G is moreover simply connected, then the homotopy exact sequence tells where BH is the classifying space of H. When H is connected, π 1 (BH) π 0 (H) = 0, and thus LSSS again works nicely. We obtain Therefore, this spectral sequence makes a relationship between topological terms of H gauge fields and topological terms of G/H-valued sigma models. In the following of this appendix, we first study the cohomology of CP N −1 as an exercise, and then compute and reinterpret the cohomology of SU (N )/U (1) N −1 .

B.1 Cohomology of CP N −1
Let us apply the LSSS to the fibration (3.1), which gives We note that H * (S n ) = Z for * = 0, n and others vanish. Therefore, the cohomologies of the total space and the fiber is known, and thus we can constrain H * (CP N −1 ) from consistency. Let us explicitly work on the case N = 3, i.e. for the case of CP 2 . The E 2 page is given by Since CP N −1 is connected, H 0 (CP N −1 ) = Z, and then we can fill the 0th column. Since the total-space cohomology H * (S 5 ) does not have the nontrivial cohomology until * = 5, Z in (p, q) = (0, 1) must be eliminated by the d 2 differential. Let y ∈ H 1 (S 1 ) be the generator, and we denote d 2 (y) = x and H 2 (CP N −1 ) Z is generated by x. Then, (p, q) = (2, 1) is generated by their tensor products xy, which again should be eliminated by d 2 , and the Leibniz rule gives d 2 (xy) = d 2 (x)y + xd 2 (y) = x 2 . Thus, x 2 is the generator of H 4 (CP 2 ) Z. Since (p, q) = (4, 1) must not be eliminated by differentials because H 5 (S 5 ) = Z, we find that d 2 (x 2 y) = 0, and then we obtain H * (CP 2 ) Z[x]/(x 3 ) as a ring isomorphism. This reproduces (3.7).
We would like to identify this generator x as a U (1) field strength da/2π in the gauged sigma model description. Since the fibration (3.1) is principal, we can obtain another fibration out of it: Let us apply the LSSS to it. We note that π 1 (BU (1)) = 0, and thus the complication of local coefficients does not come up. To be specific, let us consider the case N = 3, while the discussion for general N is also straightforward. LSSS gives As CP 2 is a 4-manifold, H * (CP 2 ) must vanish for * ≥ 5. Just because of this trivial information, we find that the d 6 differentials should be identity maps, which are drawn with blue solid arrows. As a result, we find that, for N ≥ 3, This elucidates that the topological term of our interest can be described as the Chern-Simons coupling.
In this case, this argument may seem to be tautological, reminding the fact that BU (1) CP ∞ . This, however, turns out to be a good exercise for the flag-manifold sigma models, as we discuss in the next subsection.
Let us take the specific example, N = 3, in the following. The E 2 page looks as Here, we denote Z ⊕n = ⊕ n i=1 Z. As a result, we find that Although we did not identify the cohomology ring structure, it is consistent with (4.4) as a module. 14 With this information, we can start the computation of the spin bordism using AHSS, as we did in the main text. Here, instead, let us take a close look at the physical meaning of these nontrivial cohomologies. As in the case of CP N −1 model, it is convenient to describe the SU (N )/U (1) N −1 sigma model using the SU (N )-valued scalar field U with (N − 1) U (1) gauge fields, a 1 , . . . , a N −1 . Then, the natural question is whether the nontrivial cohomologies of SU (N )/U (1) N −1 are related to the nontrivial topological terms of these U (1) gauge fields.
The answer turns out to be affirmative. To see this, we consider the following fibration, We apply LSSS to this fibration. Let us set N = 3 again for simplicity, then the E 2 page 14 Indeed, LSSS is not necessarily useful to determine the cohomology ring structure. In particular, the cohomology ring of the total space cannot be determined from LSSS. For example, we can consider the fibration, CP 1 → U (3)/U (1) 3 → CP 2 , which can be obtained from U (2) × U (1) → U (3) → CP 2 by dividing the total space and the fiber by U (1) 3 . LSSS immediately reproduces (B.17) as a module, but it does not give the cohomology ring structure.
is given by Here, the d 4 differentials are drawn with blue solid arrows, and the d 6 differentials are drawn with the red dashed arrows. The spectral sequence clashes after the E 6 page. We then find that only the q = 0 components with H p (BU (1) 2 ) for p = 0, 2, 4, 6 can survive in the E ∞ page, and thus the nontrivial cohomology H * (SU (3)/U (1) 2 ) has a well-defined origin in H p (BU (1) 2 ). For general N , we get Thus, we can express these topological terms as f i ∧ f j , but there is a constraint compared with the U (1) N −1 gauge theory. We have computed the explicit relation among them in Appendix A.

C Realization of topological terms by a free massive fermion
In this section, we present an explicit realization of the Hopf term (2.9) as a partition function of a massive free fermion. Let us consider a (2 + 1)d Euclidean relativistic Dirac fermion coupled with spin fields n which take values on sphere S 2 , L = iψ( / D + mn · σ)ψ, n 2 = 1. (C.1) Here, / D = γ µ (∂ µ − iA µ ) is a covariant derivative with A µ a spin c gauge field, γ µ 's are the 2 by 2 gamma matrices, and σ are Pauli matrices for the internal spin space. It was shown that the large-scale dynamics of the spin fields n compared to the inverse of the mass, 1/m > 0, is described by the nonlinear sigma model with the Hopf term [36], The imaginary part of the effective action S eff is the Chern-Simons term corresponding to the spin c bordism groupΩ spin c 4 (CP 1 ) = Z, and it describes two topological responses: a skyrmion of the spin fields n has a unit U (1) charge, and the adiabatic 2π rotation of the skyrmion texture gives rise to the Berry phase (−1) on the ground state wave function. By restricting the background spin c fields A to a spin one, the third term of (C.4) vanishes and only the Z 2 -quantized Hopf term of (C.4) remains, which is nothing but the Hopf term (2.9) with k = 1.
thus it is expected that the ground state with a skyrmion indicates desired topological responses for an appropriate parameter region. Hereafter, we set m = 1 and focus on λ = 0, ±2, ±4 as representatives of different invertible phases. A skyrmion texture in a finite square lattice with sites {1, . . . , L x } × {1, . . . , L y } in the periodic boundary conditions is introduced, for example, by setting the unit vector n to n 1 (x, y, τ = 0) = sin 2πx L x − π cos πy L y − π 2 , (C.8) n 2 (x, y, τ = 0) = sin 2πx L x − π cos 2 πx L y − π 2 , (C.9) n 3 (x, y, τ = 0) = 1 − 2 cos 2 πx L x − π 2 cos 2 πy L y − π 2 . (C.10) This construction is based on the smash product S 1 ∧ S 1 ∼ = S 2 [54]. We numerically find that the number of electrons of the ground state |Ω (equivalently, the number of negative eigenvalues of one-particle Hamiltonian of (C. for sufficiently large L x and L y . Here, 2L x L y is the number of electrons of the ground state without skyrmions. Thus, we confirm that a skyrmion behaves as an electron for 1 < |λ| < 3. We also confirm that ∆N coincides with the second Chern number − 1 8π 2 T 2 ×S 2 trF(k, n) 2 of the semiclassical Hamiltonian H(k, n) over the Bloch-momentum space T 2 and the spin space S 2 with the identity map n : S 2 → S 2 .