Differential geometric invariants for time-reversal symmetric Bloch-bundles II: The low dimensional"Quaternionic"case

This paper is devoted to the construction of differential geometric invariants for the classification of"Quaternionic"vector bundles. Provided that the base space is a smooth manifold of dimension two or three endowed with an involution that leaves fixed only a finite number of points, it is possible to prove that the Wess-Zumino term and the Chern-Simons invariant yield topological quantities able to distinguish between inequivalent realization of"Quaternionic"structures.


Introduction
The present paper continues the study of the classification of "quaternionic" vector bundles started in [8; 10; 11]. The main novelty with respect to the previous papers consists of the use of differential geometric invariants to classify inequivalent isomorphism classes of "quaternionic" structures. In this sense, as expressed by the title, this paper represents a continuation of [9] where differential geometric techniques have been used to classify "real" vector bundles.
At a topological level, "quaternionic" vector bundles, or Q-bundles for short, are complex vector bundles defined over spaces with involution and endowed with a further structure at the level of the total space. An involution on a topological space X is a homeomorphism of period 2, ie 2 D Id X . The pair .X; / will be called an involutive space. The fixed point set of the involutive space .X; / is by definition X WD fx 2 X j .x/ D xg: A Q-bundle over .X; / is a pair .Ᏹ; ‚/, where Ᏹ ! X denotes the underlying complex vector bundle and ‚W Ᏹ ! Ᏹ is an antilinear map which covers the action of on the base space and such that ‚ 2 acts fiberwise as multiplication by 1. A more precise description is given in Definition 2.2. Q-bundles were introduced for the first time by J L Dupont in [12] (under the name of symplectic vector bundles). They form a category of topological objects which is significantly different from the category of complex vector bundles. For this reason the problem of the classification of Q-bundles over a given involutive space requires the use of tools which are structurally different from those typically used in the classification of complex vector bundles. The aim of the present work is to define differential geometric invariants able to distinguish the elements of Vec m Q .X; /, where the latter symbol denotes the set of isomorphism classes of rank m Q-bundles over .X; /.
The interest for the classification of Q-bundles has increased in the last years because of the connection with the study of topological insulators. Although this work does not focus on the theory of topological insulators -the interested reader is referred to the recent reviews by Ando and Fu [2] and Hasan and Kane [25] -it is worth mentioning that the first example of topological effects in condensed matter related to a "quaternionic" structure dates back to the seminal works by L Fu, C L Kane and E J Mele [18; 31]. The existence of distinguished topological phases for the so-called Kane-Mele model is the result of the simultaneous presence of two symmetries. The first symmetry is given by the invariance of the system under spatial translations. This fact allows the use of the Bloch-Floquet theory -see Kuchment [36] -for the analysis of the spectral properties of the system. As a result, a well-established procedure provides the construction of a vector bundle, usually known as Bloch bundle, from each gapped energy band of the system. Even though the details of the construction of the Bloch bundle will be omitted in this work -the interested reader is referred to Panati [42] or the authors [7, Section 2] -it is important to remark that the Bloch bundle is a complex vector bundle over the torus T d ' R d =.2 Z/ d . The integer d represents the dimensionality of the system and the physically relevant dimensions are d D 2; 3. The second crucial ingredient for the topology of the Kane-Mele model is the fermionic (or odd) time-reversal symmetry (TRS). In terms of the Bloch bundle the TRS translates into the involution TR W T d ! T d of the base space given by TR .k 1 ; : : : ; k d / WD . k 1 ; : : : ; k d / and into an antilinear map ‚ of the total space such that ‚ 2 D 1 fiberwise. Therefore, one concludes that the different topological phases of the Kane-Mele model are labeled Differential geometric invariants for time-reversal symmetric Bloch bundles, II 2927 by the inequivalent realization of Q-bundles over the torus T d with involution TR , namely by the distinct elements of Vec m Q .T d ; TR /. The classification of the topological phases of the Kane-Mele model given in [18; 31] is summarized by where Z 2 WD f˙1g is the cyclic group of order 2 presented in multiplicative notation. The topological classification (1-1) has been rigorously derived with the use of different techniques in various papers -see eg [8], Fiorenza, Monaco and Panati [14], and Graf and Porta [24] -and generalized to any (low-dimensional) involutive space .X; / by Lawson, Lima-Filho, Michelsohn and dos Santos [37; 45] and in [10; 11], independently. However, the topological classification based on the construction of homotopy invariants (such as characteristic classes) has the disadvantage of being difficult to compute. For this reason one is naturally inclined to look for different types of invariants.
A special role in the classification of complex vector bundles is played by the Chern classes. The latter, in view of the Chern-Weil homomorphism, can be represented via differential forms and integrated over suitable cocycles. The resulting Chern numbers are enough to provide a complete classification of complex vector bundles in several situations of interest in condensed matter. This is, for instance, the case of the quantum Hall effect and the related TKNN numbers; see Thouless, Kohmoto, Nightingale and den Nijs [46]. Using this observation as Ariadne's thread, one expects to find differential and integral invariants able to classify Q-bundles at least under some reasonable hypotheses. Indeed, "gauge-theoretic invariants" have already been used to reproduce the classification (1-1). The first pioneering works in this direction are Essin, Moore and Vanderbilt [13], Fu and Kane [17], and Qi, Hughes, Wang and Zhang [44; 47], where the Chern-Simons field theory has been used to relate the topological phases of the Kane-Mele model in 2 C 1 and 3 C 1 space-time dimensions with integral quantities like the (time-reversal) polarization. Afterwards, these results have been revisited and put in a solid mathematical background in various works like Carpentier, Delplace, Fruchart, Gawędzki, Monaco and Tauber [6; 5; 21; 22; 41], Freed and Moore [16], and Kaufmann, Li and Wehefritz-Kaufmann [32], just to mention some of them. If one ignores the differences due to the use of distinct mathematical techniques, it is possible to recognize a common outcome from all the papers listed above: the topological phases of the two-dimensional Kane-Mele model are governed by the Wess-Zumino term [15; 20; 21] while in the three-dimensional case the relevant object is the Chern-Simons invariant [15; 21; 28]. The present work is inspired by the latter consideration and it aims to provide a general and rigorous description of the relation between the classification of Q-bundle and the Wess-Zumino term, or the Chern-Simons invariant. The main achievements are presented below.
The two-dimensional case will be described first. In this case the relevant family of base spaces is restricted by the following: Definition 1.1 (oriented two-dimensional FKMM-manifold) An oriented two-dimensional FKMM-manifold is an involutive space . †; / subject to the following conditions: (a 0 ) † is an oriented two-dimensional compact Hausdorff manifold without boundary.
(b 0 ) The involution preserves the manifold structure and the orientation.
(c 0 ) The fixed point set † ¤ ∅ consists of a finite collection of points.
Let us point out that manifold structure in (b 0 ) shall be eventually assumed to be a smooth one as is stated at the beginning of Section 3. An example of oriented twodimensional FKMM-manifold is provided by the torus T 2 with the involution TR . The set of oriented two-dimensional FKMM-manifolds forms a subclass of the FKMMspaces defined in Definition 2.8 below. Q-bundles over these spaces are completely classified by a characteristic class called FKMM-invariant; see Theorem 2.9.
The crucial result for the classification of Q-bundles over two-dimensional FKMMmanifolds is expressed by the chain of isomorphisms The where X † is any compact three-dimensional oriented manifold whose boundary coincides with † and Q W X † ! SU .2/ is any smooth extension of ; see Definition 3.16 for more details. The first main result of this paper is: Theorem 1.2 Let . †; / be an oriented two-dimensional FKMM-manifold in the sense of Definition 1.1. Let .Ᏹ; ‚/ be a Q-bundle of rank 2m over . †; / and 2 Map. †; SU.2// Z 2 any map which represents .Ᏹ; ‚/ in the sense of the isomorphism { 1 in (1-2). Then the map The proof of Theorem 1.2 is postponed to Section 3.6. Theorem 1.2 clearly applies to the classification of Q-bundles over the involutive torus .T 2 ; TR /, reproducing in this way results already existing in the literature. In this regard the result [21, (2.9)], previously announced in [22,II.25,page 19], deserves a special mention. The latter is in agreement with Theorem 1.2 above in view of the equality e i2 WZ † .w/ D e i2 WZ † . / (justified by the Polyakov-Wiegmann formula, see Lemma 3.17) where the map w employed in [22] is related to the map of Theorem 1.2 by the relation w D Q, with Q the constant matrix in . However, it is worth pointing out that the validity of Theorem 1.2 goes far beyond the standard case .T 2 ; TR /. For instance, Theorem 1.2 extends the classification of Q-bundles over Riemann surfaces of genus g endowed with an orientation-preserving involution with a finite set of fixed points [8,Appendix A] and this application seems to be new in the literature.
In order to describe the three-dimensional case it is worth mentioning that any Q-bundle .Ᏹ; ‚/ over the involutive space .X; / can be equivalently described by a principal Q-bundle .ᏼ; y ‚/ over the same base space (see Section 3.1) and that for principal Q-bundles there exists a notion of equivariant Q-connection (see Section 3.2). Given a Q-connection ! 2 1 Q .ᏼ; u.2m// one can define the associated Chern-Simons 3-form as specified in Definitions 3.9 and 3.14. Remarkably, under the hypotheses stipulated in Proposition 3.12, the quantity in the right-hand side of (1-4) turns out to be independent of the choice of the invariant connection ! or of the global section s W X ! ᏼ, and therefore defines an invariant for the underlying principal Q-bundle .ᏼ; y ‚/, or equivalently for the associated Q-bundle .Ᏹ; ‚/.
Let us recall that when .X; / is a three-dimensional FKMM-manifold in the sense of Definition 2.8, Proposition 2.10 applies and we have an isomorphism Vec 2m Q .X; / Ä ' Map.X ; f˙1g/=OEX; U .1/ Z 2 for all m 2 N: In the formula above, Map.X ; f˙1g/ ' Z 2 jX j denotes the set of maps from X to f˙1g (recall that X is a set of finitely many points). The group action of OEX; U .1/ Z 2 on Map.X ; f˙1g/ is given by multiplication and restriction. The map Ä which implements the isomorphism is the FKMM-invariant; see Section 2.3. Given a Q-bundle .Ᏹ; ‚/ over .X; /, its FKMM-invariant Ä.Ᏹ; ‚/ can be represented in terms of a map 2 Map.X ; f˙1g/ and one can use the product sign map to define the so-called strong Fu-Kane-Mele index It turns out that the definition above is well-posed in the sense that Ä s .Ᏹ; ‚/ only depends on the equivalence class of in Map.X ; f˙1g/=OEX; U .1/ Z 2 ; hence it defines a topological invariant for .Ᏹ; ‚/. This fact is a consequence of the second main result of this paper: Let .X; / be a three-dimensional FKMM-manifold in the sense of Definition 2.8 such that X ¤ ∅. Assume in addition that: (e) X is oriented and reverses the orientation.
The proof of Theorem 1.3 is postponed to Section 3.7. Along with Corollary 3.32, it expresses the fact that the strong index

"Quaternionic" vector bundles from a topological perspective
In this section base spaces will be considered only from a topological point of view. Henceforth, we will assume that: X is a topological space which admits the structure of a Z 2 -CW-complex. The dimension d of X is, by definition, the maximal dimension of its cells, and X is called low-dimensional if 0 6 d 6 3.
For the sake of completeness, let us recall that an involutive space .X; / has the structure of a Z 2 -CW-complex if it admits a skeleton decomposition given by gluing cells of different dimension in ascending order, and the involution permutes the cells. For a precise definition of the notion of Z 2 -CW-complex the reader can refer to [7,Section 4.5] or [1; 38]. Assumption 2.1 allows the space X to have several disconnected components. However, in the case of multiple components, we will tacitly assume that vector bundles built over X possess fibers of constant rank on the whole base space. Let us recall that a space with a CW-complex structure is automatically Hausdorff and paracompact, and it is compact exactly when it is constructed out of a finite number of cells [26]. Almost all the examples considered in this paper will concern spaces with a finite CW-complex structure.

Basic facts about "quaternionic" vector bundles
In this section we recall some basic facts about the topological category of "quaternionic" vector bundles. Furthermore, the necessary notation for the description of the various results will be fixed. We refer to [8; 10; 11; 12] for a more systematic presentation of the subject. Definition 2.2 ("quaternionic" vector bundles) A "quaternionic" vector bundle, or Q-bundle, over .X; / is a complex vector bundle W Ᏹ ! X endowed with a homeomorphism ‚W Ᏹ ! Ᏹ such that (Q 1 ) the projection is equivariant in the sense that ı ‚ D ı ; (Q 2 ) ‚ is antilinear on each fiber, ie ‚. p/ D N ‚.p/ for all 2 C and p 2 Ᏹ, where N is the complex conjugate of ; (Q 3 ) ‚ 2 acts fiberwise as multiplication by 1, namely Let us recall that it is always possible to endow Ᏹ with an (essentially unique) equivariant Hermitian metric m with respect to which ‚ is an antiunitary map between conjugate fibers [8, Proposition 2.5]. The equivariance is expressed by where Ᏹ Ᏹ WD f.p 1 ; p 2 / 2 Ᏹ Ᏹ j .p 1 / D .p 2 /g.
A vector bundle morphism between two vector bundles W Ᏹ ! X and 0 W Ᏹ 0 ! X over the same base space is a continuous map f W Ᏹ ! Ᏹ 0 which is fiber preserving in the sense that D 0 ıf and that restricts to a linear map on each fiber f j x W Ᏹ x ! Ᏹ 0 x . Complex vector bundles over X together with vector bundle morphisms define a category. The symbol Vec m C .X/ is used to denote the set of equivalence classes of isomorphic vector bundles of rank m. From these data, it is possible to define a category of Q-bundles and Q-morphisms. A Q-morphism between two Q-bundles .Ᏹ; ‚/ and .Ᏹ 0 ; ‚ 0 / over the same involutive space .X; / is a vector bundle morphism f commuting with the involutions, ie f ı‚ D ‚ 0 ıf . The set of equivalence classes of isomorphic Q-bundles of rank m over .X; / will be denoted by Vec m Q .X; /. Remark 2.3 ("real" vector bundles) By changing condition .Q 3 / in Definition 2.2 to (R) ‚ 2 acts fiberwise as the multiplication by 1, namely one ends in the category of "real" vector bundles, or R-bundles. The set of isomorphism classes of rank m R-bundles over the involutive space .X; / is denoted by Vec m R .X; /. For more details we refer to [3; 7].
In the case of a trivial involutive space .X; Id X /, one has bijections  Vec 2m Q .X; Id X / ' Vec m H .X /; Vec m R .X; Id X / ' Vec m R .X /; m 2 N; where Vec m F .X/ is the set of equivalence classes of vector bundles over X with typical fiber F m and H denotes the skew field of quaternions. The first isomorphism in  is proved in [12] -see also [8,Proposition 2.2] -while the proof of the second is provided in [3] -see also [7,Proposition 4.5]. These two results justify the names "quaternionic" and "real" for the related categories. Let x 2 X and Ᏹ x ' C m be the related fiber. In this case the restriction ‚j Ᏹ x Á J defines an antilinear map J W Ᏹ x ! Ᏹ x such that J 2 D 1 Ᏹ x . Said differently, the fibers Ᏹ x over fixed points x 2 X are endowed with a quaternionic structure; see [8,Remark 2.1]. This fact has an important consequence [8, Proposition 2.1]: Proposition 2.4 If X ¤ ∅, then every Q-bundle over .X; / has even rank.
The set Vec 2m Q .X; / is nonempty since it contains at least the trivial element in the "quaternionic" category. The rank 2m product Q-bundle over the involutive space .X; / is the complex vector bundle where the matrix Q is given by 0 1 1 0 : : : A "quaternionic" vector bundle is called Q-trivial if it is isomorphic to the product Q-bundle.
A section of a complex vector bundle W Ᏹ ! X is a continuous map s W X ! Ᏹ such that ıs D Id X . The set .Ᏹ/ of sections of Ᏹ has the structure of a left C.X /-module with multiplication given by the pointwise product .f s/.

Stable range in low dimension
The stable rank condition for vector bundles expresses the pretty general fact that the nontrivial topology can be concentrated in a subvector bundle of minimal rank. This minimal value depends on the dimensionality of the base space and on the category of vector bundles under consideration. For complex (as well as real or quaternionic) vector bundles the stable rank condition is a well-known result; see eg [29, Chapter 9, Theorem 1.2]. The proof of the latter is based on an "obstruction-type argument" which provides the construction of a certain maximal number of global sections [29, Chapter 2, Theorem 7.1].
The latter argument can be generalized to vector bundles over spaces with involution by means of the notion of Z 2 -CW-complex [1; 38] -see also [7,Section 4.5]. A Z 2 -CW-complex is a CW-complex with a Z 2 -action that permutes the cells. The action of Z 2 on each cell is either trivial or free. Since this construction is modeled after the usual definition of CW-complex, just by replacing "points" with "Z 2 -points", (almost) all topological and homological properties valid for CW-complexes have their natural counterpart in the equivariant setting. The use of this technique is essential for the determination of the stable rank condition in the case of R-bundles [7,Theorem 4.25] and Q-bundles [10,Theorems 4.2 and 4.5].
In this section we recall the results about the stable range for R-bundles and (even rank) Q-bundles over low-dimensional base spaces. Indeed, these are the only cases of interest in the present work. Vec 2m Q .X; / ' Vec 2 Q .X; / if 2 6 d 6 5: In particular, under the hypotheses of validity of Theorem 2.5, the dimensions d D 0; 1 are trivial since in these cases only the trivial Rand Q-bundles (up to isomorphism) exist. In the cases d D 2; 3, which are the really interesting cases for this work, it is enough to study the sets Vec 1 R .X; / and Vec 2 Q .X; /.

The FKMM-invariant
Q-bundles can be classified, at least partially, by means of a characteristic class called FKMM-invariant. This topological object was first introduced in [19] and then studied and generalized in [8; 10; 11]. In this section we review the main properties of the FKMM-invariant.
Let .X; / be an involutive space and X Â X its fixed point subset. In order to introduce the FKMM-invariant one needs the equivariant Borel cohomology group of .X; / with coefficients in the local system Z.1/; ie where Â 1 is the antipodal map on the infinite sphere S 1 . The local system Z.1/ over .X; / can be identified with the product space Z.1/ ' X Z made equivariant by the Z 2 -action .x; l/ 7 ! . .x/; l/. The fixed point subset X is closed in X and -invariant. The inclusion { W X ,! X extends to an inclusion { W X ,! X of the respective homotopy quotients. The relative equivariant cohomology can be defined as usual by the identification For a more detailed description of equivariant Borel cohomology we refer to Section 3.1 of [8].
The FKMM-invariant is a map which associates the isomorphism class OE.Ᏹ; ‚/ of the Q-bundle .Ᏹ; ‚/ to a cohomology class Ä.Ᏹ; ‚/ in the relative equivariant cohomology group H 2 Z 2 .X jX ; Z.1//. The construction of the map Ä was first described in [8,Section 3.3] and then generalized in [10,Section 2.5]. In this section we will skip the details of the construction of the FKMM-invariant and we will focus only on the relevant properties of the map (2-4): (a) Isomorphic Q-bundles define the same FKMM-invariant.
The FKMM-invariant is additive with respect to the Whitney sum and the abelian structure of H 2 for each pair of Q-bundles .Ᏹ 1 ; ‚ 1 / and .Ᏹ 2 ; ‚ 2 / over the same involutive space .X; /.
For the justification of these properties we refer to [10, Section 2.6].

Topological classification over low-dimensional FKMM-spaces
The FKMM-invariant is an extremely efficient tool for the classification of Q-bundles in low dimensions. The first observation is that, in great generality, the FKMM-invariant is injective in low dimensions, ie when the base space has dimension 0 6 d 6 3. More precisely, as a consequence of [10, Theorems 4.7 and 4.9] one has that:  to obtain a complete classification of Q-bundles in dimension d D 0; 1; 2.
In the case d D 3, the surjectivity of the FKMM-invariant can be recovered by requiring some extra properties for the base space .X; /. In the next part of this work we will mainly focus on spaces of the following type: X is a compact Hausdorff manifold without boundary; (b) the involution preserves the manifold structure; (c) the fixed point set X consists at most of a finite collection of points; Let us observe that an involutive space .X; / which fulfills conditions (a) and (b) in Definition 2.8 is a closed manifold which automatically admits the structure of a The manifold structure and the map are tacitly assumed to be of some given regularity (eg C r or smooth). The next result provides the topological classification of Q-bundles over low-dimensional FKMM-manifolds.
; and the isomorphism (in the nontrivial cases) is given by the FKMM-invariant Ä.
The cases d D 0; 1 are a consequence of the stable condition described in Theorem 2.5. The case d D 2 follows from Theorem 2.7. Finally the new case d D 3 is proved in [11,Proposition 4.13].
Let us observe that Theorem 2.9 also holds trivially in the case of a free involution, that is, when X D ∅. In this case, as a consequence of condition (d) in Definition 2.8 one has that H 2 Therefore, as a consequence of Theorem 2.9, one concludes that an FKMM-manifold with free involution only supports the trivial Q-bundle. In order to focus on the nontrivial situations we will assume henceforth that d D 2; 3 and X ¤ ∅.
When .X; / is an FKMM-manifold, the cohomology group H 2 Z 2 .X jX ; Z.1// has an explicit representation in terms of equivalence classes of maps. As proved in [8, Lemma 3.1] one has the isomorphism where Map.X ; f˙1g/ ' Z 2 jX j is the set of maps from X to f˙1g (recall that X is a set of finitely many points) and OEX; U .1/ Z 2 denotes the set of classes of Z 2homotopy equivalent equivariant maps between the involutive space .X; / and the group U .1/ endowed with the involution given by complex conjugation. The group action of OEX; U .1/ Z 2 on Map.X ; f˙1g/ is given by restriction and multiplication. More precisely, let OEu 2 OEX; U .1/ Z 2 and s 2 Map.X ; f˙1g/. Then the action of OEu on s is given by OEu.s/ WD uj X s. By combining Theorem 2.9 with the isomorphism (2-5) one gets the following result: Proposition 2.10 Let .X; / be an FKMM-manifold of dimension d D 2; 3 and assume that X ¤ ∅. Then, the FKMM-invariant Ä induces the isomorphism In summary, the content of Theorem 2.9 and Proposition 2.10 is the following: Every Q-bundle .Ᏹ; ‚/ over an FKMM-space .X; / of dimension d D 2; 3 such that X ¤ ∅ is classified by its FKMM-invariant Ä.Ᏹ; ‚/. The latter can be represented as a map modulo the (right) multiplication by the restriction over X of an equivariant function uW X ! U .1/. The map s .Ᏹ;‚/ is called the canonical section associated to .Ᏹ; ‚/ and its construction is described in [8,Section 3.2] or [10, Section 2.2].

The Fu-Kane-Mele index
Let us focus on the nontrivial case of an FKMM-manifold .X; / of dimension d D 2; 3 such that X ¤ ∅. At the end of last section we observed that every Q-bundle .Ᏹ; ‚/ over .X; / is classified by the canonical section s .Ᏹ;‚/ 2 Map.X ; f˙1g/ modulo the action (multiplication and restriction) of an equivariant map uW X ! U .1/. Clearly .Ᏹ; ‚/ is equivalently classified by any other map 2 Map.X ; f˙1g/ in the same equivalence class of s .Ᏹ;‚/ , namely by any representative of Consider now the product sign map where Z 2 is identified with the multiplicative group f˙1g. Moreover, any Q-bundle .Ᏹ; ‚/ over .X; / is classified by the FKMM-invariant Ä.Ᏹ; ‚/ 2 f˙1g which can be computed by Ä.Ᏹ; ‚/ D …. /, where … is the product sign map (2-6) and is any representative of the class OEs .Ᏹ;‚/ 2 Map.X ; f˙1g/=OEX; U .1/ Z 2 of the canonical section.
As a byproduct of Theorem 2.11 one has that the Fu-Kane-Mele index is unambiguously defined on the whole equivalence class OEs .Ᏹ;‚/ , and the Q-bundle .Ᏹ; ‚/ is classified, up to isomorphism, by the sign …. / 2 f˙1g where 2 Map.X ; f˙1g/ is any map which differs from s .Ᏹ;‚/ by the multiplication with the restriction of an equivariant map uW X ! U.1/.
Although with some differences, the next result pairs Theorem 2.11 in dimension d D 3. It can be considered one of the main achievements of this work. Theorem 2.12 is a direct consequence of Theorem 1.3, which will be proved in Section 3.7. It is worth noting that even though Theorems 2.11 and 2.12 seem to be of topological nature, they need the manifold structure of X . In particular, Theorem 1.3 relies on differential geometric techniques.
In general the quantity Ä s .Ᏹ; ‚/ in Theorem 2.12 does not completely specify the FKMM-invariant of .Ᏹ; ‚/, but only a part of it. We refer to Ä s .Ᏹ; ‚/ as the strong component of the FKMM-invariant.
The set of Z 2 -homotopy classes of Z 2 -equivariant maps will be denoted by OEX; SU .2/ Z 2 : Let us consider also the groups where N and N are the complex conjugates of and , respectively, and the group structures are given by pointwise multiplication. The related sets of equivalence classes under Z 2 -homotopy are denoted by OEX; U .2/ 0 Z 2 and OEX; U .1/ Z 2 , respectively. By construction one has an inclusion Map.
it follows that det.G . // D det. / D 1. Moreover, the equality G . / D G . / 1 follows from a direct calculation along with the equality Q D N Q valid for maps with values in SU.2/.
The main aim of this section is to prove the following result: Then there is a natural bijection where the action of OEX; U .1/ Z 2 on OEX; SU.2/ Z 2 is defined as follows: given OE in We start with a couple of preliminary results which are valid in dimension 0 6 d 6 3.
Proof Let W Ᏹ ! X be a rank 2 Q-bundle. The low-dimensionality of the base space implies that the underlying complex vector bundle Ᏹ is isomorphic to the product bundle Two Q-structures ‚ and ‚ 0 on X C 2 , induced respectively by the maps and 0 in Map.
We are now in position to complete the proof of Theorem 2.13. For this purpose the restriction to dimensions d 6 2 will be crucial.

The FKMM-invariant for oriented two-dimensional FKMM-manifolds
Throughout this section we will assume that the pair . †; / is an oriented twodimensional FKMM-manifold in the sense of Definition 1.1. The use of the letter † instead of X is motivated to easier connect the results discussed here with the theory developed in Section 3.4 and 3.6 When . †; / is an oriented two-dimensional FKMM-manifold, two presentations for Vec 2 Q . †; / are available. The first description, Proof We will start by proving the injectivity ofˆÄ. Suppose ; 0 2 Map. †; SU.2// Z 2 are such thatˆÄ. / DˆÄ. 0 /. We want to show the existence of a Z 2 -equivariant homotopy Q W † OE0; 1 ! SU.2/ such that Q j † f0g D and Q j † f1g D 0 . This can be done by a standard argument in homotopy theory. Let † j be the j -skeleton of † with respect to a Z 2 -CW decomposition. The 0-skeleton † 0 consists of the 0-cells of the form e 0 (a fixed cell) or Z 2 e 0 (a free cell), where e 0 D is a standard 0-cell. Accordingly, we can express † 0 as the disjoint union † 0 D † fix 0 t † free 0 . By assumption, we have † fix 0 D X . Notice that the mapˆÄ factors through where the involution # on SU .2/ is #. / D 1 , so that the fixed point set SU .2/ # D f˙1 C 2 g consists of two points. The first map is induced from the restriction 7 ! j † . The second map is the bijection induced from the obvious identification SU.2/ # ' f˙1g. It follows that j † fix 0 D 0 j † fix 0 . On the other hand, for each free 0-cell Z 2 e 0 we can find a homotopy connecting j f1g e 0 and 0 j f1g e 0 because SU .2/ is path connected. This homotopy extends to a Z 2 -equivariant homotopy connecting j Z 2 e 0 and 0 j Z 2 e 0 since the action of Z 2 on Z 2 e 0 is free. In this way, we get a Z 2 -equivariant By assumption again, the 1-skeleton † 1 is given by attaching only free 1-cells of the form Z 2 e 1 to † 0 . We already have a homotopy Q 0 j f1g @e 1 OE0;1 . This homotopy, together with j f1g e 1 and 0 j f1g e 1 , gives a map from @.f1g e 1 OE0; 1/ D .f1g @e 1 OE0; 1/ [ .f1g e 1 @OE0; 1/ which can be extended to a map from f1g e 1 OE0; 1 in view of 1 .SU .2// D 0. Extending this map equivariantly, and gathering together the maps constructed in this way for each free 1-cell, one gets a Z 2 -equivariant homotopy Q 1 W † 1 OE0; 1 ! SU.2/ which extends Q 0 and connects j † 1 with 0 j † 1 . Finally, the 2-skeleton † 2 D † is given by attaching only free 2-cells of the form Z 2 e 2 to † 1 . We already have a homotopy Q 1 j f1g @e 2 OE0;1 . This homotopy, together with j f1g e 2 and 0 j f1g e 2 , provides a map from @.f1g e 2 OE0; 1/. This extends to a map from f1g e 2 OE0; 1, The same group OE †; U .1/ Z 2 acts on both sides andˆÄ is equivariant. An inspection of the group actions shows thatˆÄ descends to a bijective homomorphism between the quotients.
In view of Theorem 2.13, one can think of a map 2 Map. †; SU.2// Z 2 as a rank 2 Q-bundle on †. Then it makes sense to talk about the "FKMM-invariant of the map ". Proposition 2.18 shows that such an invariant is indeed built through the isomorphismˆÄ. More precisely, by combining Proposition 2.18 with Theorem 2.11 one obtains where Ä. / represents the FKMM-invariant of the Q-bundle defined by the map .

Differential geometric classification of "quaternionic" vector bundles
In this section we provide differential geometric realizations of the FKMM-invariant. However, this require more structure on the involutive space .X; /. More properly, we need to pass from the topological category to the smooth category. In this section the quite general Assumption 2.1 will be replaced by the more restrictive: Assumption 3.1 (smooth category) X is a compact, path-connected, Hausdorff smooth d -dimensional manifold without boundary and with a smooth involution .
In particular, a space X which fulfills Assumption 3.1 is a closed manifold and the pair .X; / automatically admits the structure of a Z 2 -CW-complex; see eg [39,Theorem 3.6]. Observe that the notion of FKMM-manifold given in Definition 2.8 is compatible with Assumption 3.1. It is worth pointing out that the smooth condition can be relaxed to a less demanding regularity condition; for instance it is sufficient to assume that the manifold structure is C r -regular for some r 2 N. Anyway, this is only a technical detail and for a simpler presentation it is enough to focus only on the smooth case.
Let us point out that in Section 2.1 we introduced the notion of Q-bundle in the topological category meaning that all the maps involved in the various definitions are continuous functions between topological spaces. However, when the involutive space .X; / has an additional smooth manifold structure one can equivalently define Qbundles in the smooth category by requiring that all spaces involved in the definitions carry a smooth manifold structure and maps are smooth functions. However, for what concerns the problem of the classification, the two categories are equivalent [9, Theorem 2.1], namely top Vec m Q .X; / ' smooth Vec m Q .X; /: Clearly, the same holds true also in the "real" category. For more details on this point we refer to [9, Section 2].

Principal "quaternionic" bundles and related FKMM-invariant
The next definition was introduced in [9, Section 2.1].
Definition 3.2 (principal Rand Q-bundle) Let .X; / be an involutive space which satisfies Assumption 3.1 and W ᏼ ! X a (smooth) principal U .m/-bundle. We say that ᏼ has a "real" structure if there is a homeomorphism y ‚W ᏼ ! ᏼ such that: (Eq.) The bundle projection is equivariant in the sense that ı y ‚ D ı .
(Inv.) y ‚ is an involution, ie y ‚ 2 .p/ D p for all p 2 ᏼ. where R u .p/ D p u denotes the right U .m/-action and N u is the complex conjugate of u.
We say that ᏼ has a "quaternionic" structure if the structure group U .2m/ has even rank and condition ( y R) is replaced by: ( y Q) The right U .2m/-action on the fibers and the homeomorphism y ‚ fulfill the condition y ‚.R u .p// D R .u/ . y ‚.p//; for all p 2 ᏼ and u 2 U .2m/; where W U .2m/ ! U .2m/ is the involution given by and Q is the matrix (2-2).
We will often refer to principal "real" and "quaternionic" bundles with the abbreviations principal R-bundles and principal Q-bundles, respectively.

Remark 3.3
Let us notice that both the "real" and the "quaternionic" case require that y ‚ has to be an involution. This means that both principal Rand Q-bundles are examples of Z 2 -equivariant principal bundles (indeed, properties (Eq.) and (Inv.) define these objects). This is indeed a difference with respect to the vector bundle case; cf Definition 2.2.
Morphisms (and isomorphisms) between principal Rand Q-bundles are defined in a natural way: if .ᏼ; y ‚/ and .ᏼ 0 ; y ‚ 0 / are two such principal bundles over the same involutive space .X; / then an Ror Q-morphism is a principal bundle morphism f W ᏼ ! ᏼ 0 such that f ı y ‚ 0 D y ‚ıf . We will use the symbols Prin U .m/ R .X; / and Prin U .2m/ Q .X; / for the sets of equivalence classes of principal "real" and "quaternionic" bundles over .X; /, respectively. A principal R-bundle over .X; / is called trivial if it is isomorphic to the product bundle X U.m/ with trivial R-structure y ‚ 0 W .x; u/ 7 ! . .x/; N u/. In much the same way, a trivial principal Q-bundle is isomorphic to the product bundle X U .2m/ endowed with the trivial Q-structure y ‚ 0 W .x; u/ 7 ! . .x/; .u//.
A standard result says that there is an equivalence of categories between principal U.m/bundles and complex vector bundles. This equivalence is realized by the associated bundle construction along its inverse, called orthonormal frame bundle construction; see [9, Appendix B] for more details. A similar result extends to the "real" and the "quaternionic" categories [9, Proposition 2.4] leading to

"Quaternionic" connections and curvatures
Connections with "quaternionic" and "real" structures have been studied in Section 2.2 of [9]. We review here the basic definitions and the main properties of these objects. For a reminder about the theory of connections we refer to the classic monographs [33; 34]; see also [9,Appendix B].
We consider principal bundles in the smooth category W ᏼ ! X endowed with a "real" or "quaternionic" structure y ‚W ᏼ ! ᏼ over the involutive space .X; /. The structure group is U .m/ (m even in the "quaternionic" case) and u.m/ is the related Lie algebra. The symbol ! 2 1 .ᏼ; u.m// will be used for the connection 1-forms associated to given horizontal distributions p 7 ! H p of ᏼ. We observe that the Lie algebra u.m/ has two natural involutions: a real involution u.m/ 3 7 ! N 2 u.m/ and a quaternionic involution u.2m/ 3 7 ! . / WD Q N Q 2 u.2m/. Here 2 u.m/ is any anti-Hermitian matrix of size m and the matrix Q was defined in . Finally, given a k-form 2 k .ᏼ; Ꮽ/ with values in some structure Ꮽ (module, ring, algebra, group, etc) and a smooth map f W ᏼ ! ᏼ, we denote by f WD ı f the pullback of with respect to the map f (and f W T ᏼ ! T ᏼ is the differential, or pushforward, of vector fields). Given a u.m/-valued k-form 2 k .ᏼ; u.m//, we define the complex conjugate form N pointwise, ie N p .w 1 p ; : : : ; w k p / WD p .w 1 p ; : : : ; w k p / for every k-tuple fw 1 p ; : : : ; w k p g of tangent vectors at p 2 ᏼ. It follows that f N D f for every smooth map f W ᏼ ! ᏼ. Similarly, if 2 k .ᏼ; u.2m//, we define . / pointwise by . / p .w 1 p ; : : : ; w k p / WD Q p .w 1 p ; : : : ; w k p / Q. Hence, one has that .f / D f . /. Definition 3.6 ("real" and "quaternionic" equivariant connections) Let .X; / be an involutive space that satisfies Assumption 3.1 and W ᏼ ! X a smooth principal U .m/-bundle over X endowed with a "real" or a "quaternionic" structure y ‚W ᏼ ! ᏼ as in Definition 3.2. A connection 1-form ! 2 1 .ᏼ; u.m// is said to be equivariant if N ! D y ‚ ! in the "real" case or .!/ D y ‚ ! in the "quaternionic" case. Equivariant connections in the "real" case are called "real" connections (or R-connections). Similarly, the "quaternionic" connections (or Q-connections) are the equivariant connections in the "quaternionic" category. Let F ! be the curvature associated to the equivariant connection ! by the structural equation According to [9,Proposition 2.22] one has that F ! obeys the equivariant constraints .F ! / D y ‚ F ! ("quaternionic" case): Let fF˛2 2 .U˛; g/g be the collection of local 2-forms which provides the local description of the curvature F ! -in the sense of [9,Theorem C.2]. When ! is equivariant, it holds true that (3)(4) F˛D F˛("real" case); .F˛/ D F˛("quaternionic" case):

Chern-Simons form and "quaternionic" structure
In this section we discuss some aspects of Chern-Simons theory defined over (compact) manifolds without boundary in the presence of a Q-structure. For a comprehensive introduction to Chern-Simons theory we refer to [15; 28].
Let W ᏼ ! X be a (smooth) principal U .m/-bundle and ! 2 1 .ᏼ; u.m// a connection 1-form. The Chern-Simons 3-form CS.!/ 2 3 .ᏼ/ associated to ! is defined by where Tr is the usual trace on m m matrices. The 3-form CS.!/ is sometimes called Chern-Simons Lagrangian. A direct computation shows that the result of applying the exterior differential to CS.!/ can be expressed in terms of the curvature F ! 2 2 .ᏼ; u.m// according to (3)(4)(5)(6) dCS.!/ WD 1 4 2 Tr.F !^F! / 2 4 .ᏼ/: The following result will be used several times in the continuation of this work. Lemma 3.8 Assume that W ᏼ ! X admits a (smooth ) section s W X ! ᏼ and let g W X ! U .m/ be a (smooth ) map. Define a new section s g W X ! ᏼ using the right action of U.m/, that is, s g .x/ WD s.x/ g.x/. Then the two pullbacks s g CS.!/; s CS.!/ 2 3 .X / are related by the equation s g CS.!/ D s CS.!/ C 1 8 2 dTr.s !^dg 1 g/ C ƒ.g/; where ƒ.g/ 2 3 .X/ is given by (3)(4)(5)(6)(7)(8) ƒ.g/ WD 1 24 2 Tr..g 1 dg/^3/: Proof The proof is essentially a computation which is based on the two relations s g CS.!/ D CS.s g !/ and s g ! D g 1 .s !/g C g 1 dg. Therefore, by exploiting the cyclicity of the trace, one can check that CS.g 1 .s !/g C g 1 dg/ D CS.s !/ 1 8 2 dTr.s !^g 1 dg/ 1 24 2 Tr..g 1 dg/^3/: The identity 0 D d.g 1 g/ D dg 1 g C g 1 dg concludes the computation. The following result shows that the Chern-Simons invariant is well defined. where the integer N g defines the "degree" of the map g. With a similar argument one can show that cs.!/ D cs.! 0 / if ! and ! 0 are related by the transformation induced by an element of the gauge group.
When a principal U .2m/-bundle W ᏼ ! X is endowed with a Q-structure y ‚, it is natural to use an equivariant Q-connection ! 2 A Q .ᏼ/ to define the Chern-Simons 3-form CS.!/. The Q-structure y ‚ induces a symmetry of CS.!/. Proof The equivariance of ! means that y ‚ ! D Q N !Q 1 D Q. t !/Q 1 , where we used N ! D t ! since the form ! takes value in the Lie algebra u.2m/. The cyclicity of the trace provides The identity t ! 1^t ! 2 D . 1/ q 1 q 2 t .! 2^!1 / is valid for each pair ! 1 2 q 1 .ᏼ; u.2m// and ! 2 2 q 2 .ᏼ; u.2m// and the invariance of the trace under the operation of taking the transpose imply To conclude the proof it is enough to observe that Tr .!^!/ D 0 due to the anticommutation relation of 1-forms.
The invariance of CS.!/ expressed in Lemma 3.11 has an important implication on the Chern-Simons invariant in low dimensions, provided that certain conditions are met. (iii) if .ᏼ; y ‚/ admits a global equivariant section then cs.!/ D 0 for any ! 2 A Q .ᏼ/.
Proof (i) Let s W X ! ᏼ be a global section. Since ‚ .s/ WD y ‚ıs ı generally differs from s, there is a (unique) map g W X ! U .2m/ such that ‚ .s/ D s g. Then where N g WD R X ƒ.g/ 2 Z. This implies that 2cs.!/ D 0, ie cs.!/ 2˚0; 1 2 « .
(ii) Let ! 0 be a second equivariant connection and consider the map Clearly cs.! t / is a polynomial (hence continuous) function in t. On the other hand cs.! t / 2˚0; 1 2 « since ! t is equivariant. This implies that cs.! t 1 / D cs.! t 2 / for all t 1 ; t 2 2 OE0; 1 and in particular cs.!/ D cs.! 0 /. The following definition is justified by item (ii) of Proposition 3.12.
Definition 3.14 (intrinsic Chern-Simons invariant) Let .ᏼ; y ‚/ be a U .2m/ Qbundle over the involutive manifold .X; / such that X has dimension d D 3, reverses the orientation of X and ᏼ admits a global section. Then the quantity cs.ᏼ; y ‚/ WD cs.!/ for some ! 2 A Q .ᏼ/ does not depend on the choice of ! 2 A Q .ᏼ/ and defines an intrinsic (Chern-Simons) invariant for .ᏼ; y ‚/.

Remark 3.15 (a formula for the Chern-Simons invariant)
Let .X; / be a threedimensional involutive manifold satisfying the assumption H 2 Z 2 .X; Z.1// D 0. As a consequence of Lemma 2.14 and the isomorphism (3-1), any U .2m/ Q-bundle .ᏼ; y ‚/ over .X; / can be represented by a smooth map W X ! U .2m/ such that D Q N 1 Q. The average construction applied to the trivial connection on the product bundle [9,Example 2.15] gives an equivariant connection !, whose pullback under the trivial section s is s ! D

Wess-Zumino term in absence of boundaries
In the last section we described the Chern-Simons invariant in the case of threedimensional base manifolds without boundary. In the case of manifolds with boundary the Chern-Simons invariant itself depends on the choice of a section while the difference of the values of the Chern-Simons invariants depends only on the topological information on the boundary. This information is detected by the so-called Wess-Zumino term. The latter is a topological quantity initially defined in the context of certain two-dimensional conformal field theories known as Wess-Zumino-Witten models. An excellent introduction to the theory of Wess-Zumino-Witten models is provided by the lecture notes [20]. The presentation of the properties of the Wess-Zumino term given here follows mainly [15].
where ƒ. Q / WD 1 24 2 Tr. Q 1 d Q / 3 according to the notation (3-8), X † is any compact three-dimensional oriented manifold whose boundary coincides with †, ie @X † D †, and Q W X † ! SU .2/ is any extension of .
Notice that the extended manifold X † and the extended section Q in Definition 3.16 always exist. The existence of X † follows from the vanishing of the second bordism group, 1 2 D 0 [40, Section 7]. The existence of Q is due to k .SU .2// D 0 for k D 0; 1; 2 plus a standard application of the Oka's (type) principle to pass from continuous sections to smooth sections. Finally, the condition W † ! SU .2/ can be relaxed by asking that the section W † ! U.2/ possesses a determinant section det. /W † ! U .1/ which is nullhomotopic.
The well-posedness of Definition 3.16 is justified in the following result.
Proof Given † and W † ! SU.2/ as in Definition 3.16, consider two extended manifolds X † and X 0 † such that @X † D † D @X 0 † , and two extended sections Q and Q 0 such that Q j † D D Q 0 j † . By reversing the orientation of X 0 † and then gluing it with X † along † one obtains a compact oriented three-dimensional manifold X WD . X 0 † /tX † , where the minus sign indicates the reversal of the orientation. Similarly, Q and Q 0 can be glued together to define a section X WD . Q t Q 0 /W X ! SU .2/. It is well known that Z On the other hand, one has that Z where the minus sign is justified by the inversion of the orientation. Thus, since the Wess-Zumino term WZ † . / is defined modulo an integer, it can be computed equivalently through the pair X † ; Q or the pair X 0 † ; Q 0 . The Polyakov-Wiegmann formula for WZ † . 1 2 / follows from an explicit computation. By taking extensions of 1 and 2 one computes ƒ. 1 2 / ƒ. 1 / ƒ. 2 / directly. Then, integration over 1 The existence of X † can be also justified by observing that closed oriented two-dimensional manifolds are classified by the genus, and a genus g surface is always the boundary of a three-dimensional manifold. For instance the sphere S 2 is the boundary of the three-dimensional disk D 3 . Similarly the torus T 2 is the boundary of the manifold S 1 D 2 . The same occurs for higher genus surfaces. X † and an application of Stokes' theorem to obtain the integral on the boundary † provide the final result.
From formula (3)(4)(5)(6)(7) and Stokes' theorem one immediately deduces the following result: Lemma 3.18 Let X be a compact oriented manifold of dimension d D 3 with nonempty boundary † WD @X . Let W ᏼ ! X be a principal U .2/-bundle equipped with a connection ! and a global (smooth ) section s W X ! ᏼ. Let g W X ! U .2/ be any (smooth ) map such that det.g/W Tr.s !^dg 1 g/ C WZ † .gj † / mod Z:

Wess-Zumino term in presence of boundaries
In the rest of this work we will be interested in calculating the Wess-Zumino term through "cutting and pasting". To set up the machinery, we need to extend the definition of the Wess-Zumino term for two-dimensional manifolds with boundary. To do that, let us observe that associated to a compact oriented one-dimensional manifold S without boundary (a union of circles), there exists a Hermitian line bundle W ᏸ S ! Map.S; SU .2//. The specific structure of this line bundle will be not used in this work and for this reason the details of the construction of ᏸ S will be only sketched. The interested reader can refer to [15,Appendix A] or to [35, Section 1.3] for a more rigorous presentation.
Given S , consider a two-dimensional manifold D S (a disjoint union of disks) with boundary @D S D S along with the space Map.D S ; SU .2//. Given an element Q in Map.D S ; SU .2//, its restriction WD Q j S defines an element in Map.S; SU .2//. Let Q 1 ; Q 2 2 Map.D S ; SU.2// be two maps which agree on the boundary S , namely such that 1 D 2 . Such two maps can be glued together to produce a map .1;2/ WD Q 1 t Q 2 on the two-dimensional manifold without boundary † S WD . D S / t D S obtained by gluing two copies of D S (with opposite orientation) along the common boundary. As a consequence the quantity WZ † S . .1;2/ / turns out to be well defined according to (ii) The product of fibers (3)(4)(5)(6)(7)(8)(9) defined by the isometry is associative.
All the ingredients are now available for extending Definition 3.16 to manifolds with boundary. To introduce the next result it is worth mentioning that given a complex vector bundle Ᏹ ! X, its conjugate Ᏹ ! X is the complex vector bundle whose underlying total space agrees with Ᏹ as a set, but with inverted complex structure with respect to the multiplication by scalars z 2 C. If Ᏹ is endowed with a Hermitian metric, then so is Ᏹ. This allows the identification of Ᏹ with the dual vector bundle Ᏹ .
Proposition 3.20 (orientation) (i) Let S be a compact oriented one-dimensional manifold without boundary, and S the same manifold with reversed orientation. Then there exists a natural isometric isomorphism ᏸ S ' ᏸ S : (ii) Let † be a compact oriented two-dimensional manifold with boundary, and † the same manifold with reversed orientation. Then for any W † ! SU .2/, Property (i) of Proposition 3.20 is a direct consequence of the construction of the space ᏸ S . Property (ii) follows from Definition 3.19 under the isometry described in (i).
Remark 3.21 (central extension of the loop group) Definition 3.19 will be mainly applied to two-dimensional manifolds † such that @ † ' S 1 . In this case we will write ᏸ S 1 instead of ᏸ @ † . The set Map.S 1 ; SU.2// endowed with the pointwise multiplication is known as the loop group of SU.2/ [43], and will be denoted here by Loop SU .2/ . The total space S.ᏸ S 1 / of the principal U .1/-bundle (also known as circle bundle) associated to ᏸ S 1 inherits a group structure from the product of fibers (3)(4)(5)(6)(7)(8)(9). This gives rise to a central extension of Loop SU.2/ , Let 0 W † ! SU.2/ be the constant map with value the identity matrix 1 C 2 2 SU .2/. By definition of the product of fibers (3)(4)(5)(6)(7)(8)(9) one has that OE 0 ; e i2 WZ † D . 0 t 0 / acts as the unit of the group S.ᏸ S 1 /. Therefore, by invoking Definition 3.19 one obtains that e i2 WZ † . 0 / 2 ᏸ S 1 provides the unit of the central extension S.ᏸ S 1 /. For a more complete description of this central extension the reader is referred to [15; 35; 43].
The link between Definitions 3.16 and 3.19 is provided by the following result.
Although simplified, the version of the gluing property described in Proposition 3.22 is sufficient for the purposes of this work. Indeed, the gluing property will be mainly applied to the situation described below.

Classification via Wess-Zumino term in dimension two
In this section the description of rank 2 Q-bundles over an oriented two-dimensional FKMM-manifold . †; / obtained in Sections 2.6 and 2.7 will be combined with the theory of the Wess-Zumino term described in Sections 3.4 and 3.5 in order to prove that the Wess-Zumino term completely classifies Vec 2 Q . †; /.
The following three preliminary results are needed.  where … is the product sign map defined by (2-7).
Proof The proof of Lemma 2.17 contains the recipe to construct a map 2 Map. †; SU .2// Z 2 for each 2 Map. † ; f˙1g/ such thatˆÄ. / D . Let † D fx 1 ; : : : ; x n g be a labeling for the fixed point set. Let i 2 Map. † ; f˙1g/ be defined by i .x j / D 1 2ı ij . Let i WD i be the element in Map. †; SU .2// Z 2 such thatˆÄ. i / D i . Note that i takes the value 1 C 2 outside the disk D i . It follows that 1 ; : : : ; n commute pointwise, and the pointwise product of the i is in Map. †; SU .2// Z 2 . Then, by construction, each can be expressed as the pointwise product of a certain number of i . Let us assume that D i 1 i k . Since the supports of the differential forms 1 i d i are pairwise disjoint, the Polyakov-Wiegmann formula (see Lemma 3.17) provides The next task is to evaluate the generic term WZ † . i /. For that, the construction in Remark 3.23 will be applied. Given x i 2 † , consider a small disk D i † such that .D i / D D i and x i 2 D i is the only fixed point. The restriction i j D i has by construction the following property: i j D i .x i / D 1 C 2 and i j D i .x/ D C1 C 2 if x 2 @D i . By an equivariant diffeomorphism, D i can be identified with the closed unit disk D C endowed with the involution z 7 ! z and the map i j D i can be identified with the map D described in the proof of Lemma 2.17. By gluing two copies D and D 0 of the same disk along the common boundary S 1 one obtains that D t D 0 is identifiable with the equivariant sphere S 2 with involution .k 0 ; k 1 ; k 2 / 7 ! .k 0 ; k 1 ; k 2 / which fixes only the two poles .˙1; 0; 0/. Consequently, given the constant map 0 W D 0 ! 1 C 2 , one has that the map D t 0 W D t D 0 ! SU .2/ can be identified with the equivariant map W S 2 ! SU.2/ such that .˙1; 0; 0/ D˙1 C 2 . Since the conditions described in Remark 3.23 are met, one has that A possible realization for is (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) .k 0 ;  (3)(4)(5)(6)(7)(8)(9)(10)(11)(12). The computation of WZ S 2 . / with given by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) is as follows: Consider the map Q W S 3 ! SU .2/ defined by (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) Q .k 0 ; k 1 ; Let S 3 C WD fk 2 S 3 j k 3 > 0g be the upper hemisphere. Then @S 3 C ' S 2 and Q j @S 3 C D . Since S 3 C is just a half-sphere, one gets by a direct computation that As a consequence, e i2 WZ † . i / D e i2 WZ We are now in position to prove the first main result of this work.

Classification via Chern-Simons invariant in dimension three
The main aim of this section is to provide the proof of Theorem 1.3. This proof is facilitated by a particular presentation of principal Q-bundles over .X; /. Suppose that X D fx 1 ; : : : ; x n g consists of n points. Thanks to the slice theorem [27, Chapter I, Section 3], for each i D 1; : : : ; n one can find a closed -invariant disk D i centered at x i such that D i \ D j D ∅ for i ¤ j and each D i is equivariantly homotopic to the standard unit disk in R 3 with antipodal involution .x/ D x. Define X D WD G i D1;:::;n D i ; X 0 WD XnInt.X D /; so that X D X 0 [ X D . Given any map ' W X 0 \ X D ! U .2/ one can glue together the product bundles over X 0 and X D to form a principal U .2/-bundle over X, Then the principal U .2/-bundle ᏼ ' gives rise to a principal Q-bundle.

Lemma 3.27
Assume that the hypotheses of Theorem 1.3 are met. Any principal U .2/ Q-bundle .ᏼ; y ‚/ over .X; / is isomorphic to a principal U .2/ Q-bundle ᏼ ' of the type (3-14) for a map ' 2 Map.X 0 \ X D ; U .2// Z 2 which satisfies the following property: Let ' i WD 'j @D i be the restriction of ' on the boundary @D i ' S 2 of the disk D i for every i D 1; : : : ; n. Then, either ' i is equivariantly homotopic to the equivariant map ' W S 2 ! U .2/ defined by where S 2 is a Z 2 -space with the antipodal involution, or ' i is the constant map at Proof Since each connected component D i of X D is equivariantly contractible, the principal Q-bundle ᏼj X D is trivial. By construction, the involution on X 0 is free; thus ᏼj X 0 is trivial as well. This fact follows from [10, Theorem 4.7(2)] along with the assumption H 2 Z 2 .X; Z.1// D 0 which implies the triviality of even rank Q-bundles over spaces with free involutions. The passage from vector bundles to principal bundles is then justified by the isomorphism (3-1). Let s X D and s X 0 be global sections (ie trivializations) of ᏼj X D and ᏼj X 0 , respectively. From these sections one gets the map ' W X 0 \ X D ! U .2/ defined by the restriction on X 0 \ X D of the (pointwise) product s 1 X D s X 0 . The map ' is equivariant by construction and defines the principal Q-bundle ᏼ ' as given in (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). The isomorphism ᏼ ' ᏼ ' is a manifestation of the fact that ᏼ and ᏼ ' have the same system of transition functions. By the homotopy property of Q-bundles, the Q-isomorphism class of ᏼ ' only depends on the Z 2 -homotopy class of '. By [8, Corollary 4.1] one has OES 2 ; U .2/ Z 2 ' Z 2 , meaning that every equivariant map from the sphere S 2 with the antipodal involution into the space U .2/ with involution g 7 ! Q N gQ is Z 2 -homotopy equivalent to the constant map at 1 C 2 or to the map ' . Since X 0 \ X D is a disjoint union of antipodal spheres, the map ' restricted to each disconnected component can be equivariantly deformed to one of these two maps.
Remark 3.28 Lemma 3.27 deserves two comments. First of all it is worth noticing that the map ' constructed in the proof of the lemma can be always deformed to a smooth map providing in this way a smooth principal Q-bundle ᏼ ' which represents ᏼ in the smooth category. This is a manifestation of the equivalence between continuous and smooth category discussed in [9, Theorem 2.1]. The second observation refers to the content of Remark 2.16. In fact in view of the stable rank condition described in Theorem 2.5 one has that the representation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) must be valid also for principal U .2m/ Q-bundle. In the higher rank case the isomorphism reads (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15) ᏼ ' ᏼ ' WD .X 0 U .2m// t ' 0 .X D U .2m//; where the equivariant map ' 0 W X 0 \ X D ! U .2m/ factors as In view of the Lemma 3.27 one can assume that ᏼ has been of the form (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) since the beginning. With this presentation in hand, the next task is to compute the FKMMinvariant of ᏼ. As a preliminary fact, let us recall that the FKMM-invariant of a principal Q-bundle .ᏼ; y ‚/ is defined as the FKMM-invariant of the associated Qbundle .Ᏹ; ‚/; see Definition 3.4. The FKMM-invariant measures the difference of two trivializations of the sphere bundle of det.Ᏹ/j X . This is the same as measuring the difference of two trivializations of det.Ᏹ/j X . Lemma 3.29 Assume that the hypotheses of Theorem 1.3 are met. Let .ᏼ; y ‚/ be a principal U .2/ Q-bundle and ' 2 Map.X 0 \ X D ; U .2// Z 2 the equivariant map which represents the principal Q-bundle according to Lemma 3.27. Then the FKMMinvariant of .ᏼ; y ‚/ is represented by the function WD det.'/j X . More precisely, one has that Ä.ᏼ; y ‚/ D OE 2 Map.X ; f˙1g/=OEX; U .1/ Z 2 : Proof Starting from the representation (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), one has that det.ᏼ/ D .X 0 U .1// t det.'/ .X D U.1//: From this expression one infers that the canonical invariant section s .ᏼ; y ‚/ of det.ᏼ/j X is given by s .ᏼ; y ‚/ D .x; 1/ 2 X U.1/ det.ᏼ/; while a global invariant section s of det.ᏼ/ is given by where u X 0 W X 0 ! U .1/ and u D W X D ! U .1/ are two equivariant maps satisfying u X 0 D u D det.'/ on X 0 \ X D . Accordingly, it follows from Lemma 3.27 that det.' i /W X 0 \ D i ! U .1/ is a constant map at 1 or 1, where ' i WD 'j D i . Therefore, one can choose u X 0 to be the constant map at 1 and u D to be the locally constant map such that u D j D i D˙1 if det.' i / D˙1. Then, it follows that the FKMM-invariant is represented by u D j X D det.'/j X .
Proof By construction, X 0 D D ' on X 0 \ X D . Thus, the proof of the claim reduces to the problem of extending ' W @X 0 ! U .2/ to a smooth map Q ' W X 0 ! U .2/ such that Q 'j @X 0 D '. Indeed, given such a Q ', the proof can be completed by setting D D 1 C 2 and X 0 D Q '. To prove the existence of Q ', notice that the three-manifold X 0 admits a CW decomposition in which the dimension of each cell is at most 3. The homotopy groups i .U .2// are trivial for i D 0; 2. The map detW U .2/ ! U .1/ induces an isomorphism 1 .U .2// ' 1 .U .1// ' Z. Since det.'/ is nullhomotopic by construction, one concludes that ' extends to a continuous map Q ' 0 W X 0 ! U.2/. However, the isomorphism between continuous category and smooth category ensures the existence of a smooth map Q ' W X 0 ! U .2/, approximating the continuous map Q ' 0 , that satisfies Q 'j @X 0 D '.
Given an invariant connection ! on .ᏼ; y ‚/, one sets The two local expressions are related by The following result contains the key computation for the proof of Theorem 1.3. where ! X D is defined by (3-18) from any invariant connection !.
We are now in position to provide the proof of the second main result of this work. WZ @D i .' i / mod Z: It holds that WZ @D i .' i / D 1 when ' i D 1 C 2 (obvious!) and WZ @D i .' i / D 1 2 when ' i is homotopic to the map ' in Lemma 3.27. The proof of the latter equality is contained in the proof of Lemma 3.25. In fact the map ' coincides with the map (3-12) and a possible extension Q ' on the upper hemisphere of S 3 can be realized by the prescription (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13). In conclusion, one obtains that e i2 cs.ᏼ; y ‚/ D ….det.'/j X /: The proof is finally completed by the result in Lemma 3.29. Theorem 1.3 has a surprising consequence. Proof One needs to shows that the homomorphism …W Map.X ; f˙1g/ ! Z 2 given by the product sign map satisfies …. j X / D …. / for any map W X ! Z 2 and any equivariant map W X ! U .1/. Consider the principal U .2/ Q-bundle ᏼ ' generated according to (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) where the map ' is related to as follows: ' is the constant map at 1 C 2 on the disk D i if .x i / D 1 or ' agrees with ' on the boundary of D i if .x i / D 1. By construction the map provides a representative of the FKMMinvariant of ᏼ ' ; see Lemma 3.27. In a similar way the map 0 WD represents the FKMM-invariant of an associated principal U .2/ Q-bundle ᏼ ' 0 . Since ' and ' 0 belong to the same class in Map.X ; f˙1g/=OEX; U .1/ Z 2 it follows that ᏼ ' and ᏼ ' 0 have the same FKMM-invariant. However, under the hypotheses of Theorem 1.3 the FKMMinvariant is an isomorphism (Proposition 2.10); hence ᏼ ' and ᏼ ' 0 are isomorphic. By the naturality of the Chern-Simons invariant, cs.ᏼ ' ; y ‚ ' / D cs.ᏼ ' 0 ; y ‚ ' 0 /. The proof of the claim then follows in view of formula (1)(2)(3)(4)(5)(6).