Chiral vector bundles

Abstract

This paper focuses on the study of a new category of vector bundles. The objects of this category, called chiral vector bundles, are pairs given by a complex vector bundle along with one of its automorphisms. We provide a classification for the homotopy equivalence classes of these objects based on the construction of a suitable classifying space. The computation of the cohomology of the latter allows us to introduce a proper set of characteristic cohomology classes: some of those just reproduce the ordinary Chern classes but there are also new odd-dimensional classes which take care of the extra topological information introduced by the chiral structure. Chiral vector bundles provide a geometric model for topological quantum systems in class AIII, namely for systems endowed with a (pseudo-)symmetry of chiral type. The classification of the chiral vector bundles over spheres and tori (explicitly computable up to dimension 4), recovers the commonly accepted classification for topological insulators of class AIII which is usually based on the K-group $K_1$. However, our classification turns out to be even richer since it takes care also for possible non-trivial Chern classes.

Appendices

Appendix A. Topology of unitary groups and Grassmann manifolds

Homotopy. The complete determination of the homotopy groups of the unitary groups $\mathbb {U}(m)$ is still an open problem. The first homotopy groups are showed in Table 3.

Table 3 First homotopy groups for the unitary groups

The homotopy of the Palais unitary group $\mathbb {U}(\infty )$ is described by the Bott periodicity (see Sect. 3.1).

From the fiber sequence $\mathrm{S}\mathbb {U}(m)\rightarrow \mathbb {U}(m){\mathop {\rightarrow }\limits ^\mathrm{det}}\mathbb {U}(1)$ one obtains the isomorphisms

$$\begin{aligned} \pi _k\big (\mathbb {U}(m)\big )\;\simeq \;\pi _k\big (\mathbb {S}^1\big )\;\oplus \; \pi _k\big (\mathrm{S}\mathbb {U}(m)\big )\; \qquad \quad k\in \mathbb {N}\cup \{0\}. \end{aligned}$$

(A.1)

Equation (A.1), along with the computation of the homotopy groups $\pi _{1}(\mathbb {S}^1)\simeq \mathbb {Z}$ and $\pi _{k}(\mathbb {S}^1)=0$ for all $k\ne 1$ [37, Proposition 4.1], provides the isomorphisms $\pi _{k}(\mathrm{S}\mathbb {U}(m))\simeq \pi _{k}(\mathbb {U}(m))$ for all $k\geqslant 2$ with the only difference $\pi _{1}(\mathrm{S}\mathbb {U}(m))=0$ opposed to $\pi _{1}(\mathbb {U}(m))\simeq \mathbb {Z}$. Table A.1 can be completed with $\pi _{0}(\mathbb {U}(m))= 0=\pi _{0}(\mathrm{S}\mathbb {U}(m))$ which simply means that the groups $\mathbb {U}(m)$ and $\mathrm{S}\mathbb {U}(m)$ are path-connected.

The computation of the homotopy groups of the Grassmann manifold $G_m(\mathbb {C}^\infty )$ can be reduced to the problem of the computation of the homotopy groups of $\mathbb {U}(m)$ by means of the fiber sequence associated to the universal classifying principal $\mathbb {U}(m)$-bundle

$$\begin{aligned} \mathbb {U}(m)\;\longrightarrow \;\mathscr {S}_m^\infty \;\longrightarrow \;G_m(\mathbb {C}^\infty ) \end{aligned}$$

(A.2)

where $\mathscr {S}_m^n:= \mathbb {U}(n)/ \mathbb {U}(n-m)$ is the Stiefel variety and $\mathscr {S}_m^\infty :=\bigcup _{n=1}^{\infty }\;\mathscr {S}_m^n$ is, as usual, the inductive limit. The universality of the space $\mathscr {S}_m^\infty$ implies its contractibility, i. e.  $\pi _k(\mathscr {S}_m^\infty )=0$ for all k [38, Chapter 8, Theorem 5.1]. This fact applied to the homotopy exact sequence induced by (A.2) gives

$$\begin{aligned} \pi _k\big ( G_m(\mathbb {C}^\infty )\big )\;\simeq \;\pi _{k-1}\big (\mathbb {U}(m)\big )\;\qquad \quad \ k\in \mathbb {N}. \end{aligned}$$

(A.3)

The connectedness of the Grassmann manifold also implies $\pi _0( G_m(\mathbb {C}^\infty ))=0$.

Cohomology. The cohomology ring of the Grassmann manifold

$$\begin{aligned} H^\bullet \big (G_m(\mathbb {C}^\infty ),\mathbb {Z}\big )\;\simeq \;\mathbb {Z}[\mathfrak {c}_1,\ldots ,\mathfrak {c}_m], \qquad \quad \mathfrak {c}_k\in H^{2k}\big (G_m(\mathbb {C}^\infty ),\mathbb {Z}\big ) \end{aligned}$$

(A.4)

is the ring of polynomials with integer coefficients and m generators $\mathfrak {c}_k$ of even degree [57, Theorem 14.5]. These generators $\mathfrak {c}_k$ are called universal Chern classes and there are no polynomial relationships between them. Since isomorphism classes of rank m complex vector bundles over X are classified by maps $[\varphi ]\in [X,G_m(\mathbb {C}^\infty )]$, one defines the (topological) Chern classes of a given vector bundle $\mathscr {E}\rightarrow X$ by

$$\begin{aligned} {c}_k(\mathscr {E})\;:=\;\varphi ^*(\mathfrak {c}_k)\;\in \;H^{2k}(X,\mathbb {Z})\qquad \quad k\in \mathbb {N}\; \end{aligned}$$

where $\varphi ^*:H^k(G_m(\mathbb {C}^\infty ),\mathbb {Z})\rightarrow H^k(X,\mathbb {Z})$ is the pullback induced by the classifying map $\varphi$. Isomorphic vector bundles possess the same family of Chern classes.

Remark A.1

(Postnikov sections of the Grassmann manifold) The problem of the construction of the Postnikov tower for the spaces $G_m(\mathbb {C}^\infty )$ has been firstly studied in [61]. With reference to the content of Sect. 5.4 let $\alpha _j:G_m(\mathbb {C}^\infty )\rightarrow \mathscr {G}_{j}^m$ be the j-th Postnikov section of the Grassmann manifold $G_m(\mathbb {C}^\infty )$. This section is obtained from the previous one $\mathscr {G}_{j-1}^m$ according to the (principal) fibration sequences

$$\begin{aligned} K(\pi _{j},j)\;\longrightarrow \; \mathscr {G}_{j}^m\;{\mathop {\longrightarrow }\limits ^{p_{j}}}\; \mathscr {G}_{j-1}^m\;{\mathop {\longrightarrow }\limits ^{\kappa ^{j+1}}}\; K(\pi _{j},j+1) \end{aligned}$$

(A.5)

where $\pi _{j}:=\pi _{j}(\mathscr {G}_{j}^m)$ and $\kappa ^{j+1}$ define the related Postnikov invariant. Since $\pi _{j}(\mathscr {G}_{j}^m)=0$ for j odd and $j\leqslant 2m$ one has that $\mathscr {G}_{2j-1}^m=\mathscr {G}_{2j-2}^m$ for $j\leqslant m$. Thus, one has to compute $\kappa ^{2j+1}\in H^{2j+1}(\mathscr {G}_{2j-1}^m,\pi _{2j})$ for $j\leqslant m$. Now [61, Lemma 4.4] states that for $j\leqslant m$ the invariant $\kappa ^{2j+1}$ has order $(j-1)!$. With this information we can immediately conclude that $\mathscr {G}_{1}^m\simeq \{*\}$ which implies $\kappa ^3=0$ and so $\mathscr {G}_{3}^m\simeq \mathscr {G}_{2}^m\simeq K(\mathbb {Z},2)\simeq \mathbb {C}P^\infty$ for all $m\geqslant 1$. For the determination of the next section one needs $\kappa ^5\in H^5(\mathbb {C}P^\infty , \pi _4)\simeq 0$ for $m\geqslant 2$. In conclusion one has

$$\begin{aligned} \mathscr {G}_{4}^m\;\simeq \;K(\mathbb {Z},2)\;\times \; K(\mathbb {Z},4), \qquad \quad \forall \ m\geqslant 2. \end{aligned}$$

(A.6)

Lemma 4.1 in [61] assures that $H^k(\mathscr {G}_{4}^m,\mathbb {Z})\simeq H^k(G_m(\mathbb {C}^\infty ),\mathbb {Z})$ for all $k\leqslant 4$ and with the help of the Künneth formula for cohomology and the explicit knowledge of the cohomology groups of $K(\mathbb {Z},2)$ and $K(\mathbb {Z},4)$ one obtains that for all $m\geqslant 2$

$$\begin{aligned} \begin{aligned} H^0\big (G_m(\mathbb {C}^\infty ),\mathbb {Z}\big )\;&\simeq \;H^{0}\big ( K(\mathbb {Z},2) , \mathbb {Z}\big )\;\otimes _\mathbb {Z}\; H^{0}\big ( K(\mathbb {Z},4) , \mathbb {Z}\big )\; \simeq \;\mathbb {Z}\\ H^1\big (G_m(\mathbb {C}^\infty ),\mathbb {Z}\big )\;&\simeq \;0\\ H^2\big (G_m(\mathbb {C}^\infty ),\mathbb {Z}\big )\;&\simeq \;H^{2}\big ( K(\mathbb {Z},2) , \mathbb {Z}\big )\;\simeq \; \mathbb {Z}\\ H^3\big (G_m(\mathbb {C}^\infty ),\mathbb {Z}\big )\;&\simeq \;0\\ H^4\big (G_m(\mathbb {C}^\infty ),\mathbb {Z}\big )\;&\simeq \;H^{4}\big ( K(\mathbb {Z},2) , \mathbb {Z}\big )\;\oplus \; H^{4}\big ( K(\mathbb {Z},4) , \mathbb {Z}\big )\; \simeq \;\mathbb {Z}^2. \end{aligned} \end{aligned}$$

The above computations show that the first two universal Chern classes $\mathfrak {c}_1$ and $\mathfrak {c}_2$ can be identified with the basic class of $H^{2}( K(\mathbb {Z},2) , \mathbb {Z})$ and $H^{4}( K(\mathbb {Z},4) , \mathbb {Z})$, respectively (for a definition of basic class see e. g. [2, Definition 5.3.1]).$\blacktriangleleft$.

Also the cohomology ring of the unitary group $\mathbb {U}(m)$ is well-known. It is a classical result that

$$\begin{aligned} H^\bullet \big (\mathbb {U}(m),\mathbb {Z}\big )\;\simeq \;{\bigwedge }_\mathbb {Z}\;[\mathfrak {w}_1,\ldots ,\mathfrak {w}_{m}], \qquad \quad \mathfrak {w}_k\in H^{2k-1}\big (\mathbb {U}(m),\mathbb {Z}\big ) \end{aligned}$$

(A.7)

is the exterior algebra generated by m odd-degree classes $\mathfrak {w}_k$ [10, Théorème 19.1] or [11, Section 10]. By adhering to a modern terminology (see e. g. [62, 63]) we will refer to the $\mathfrak {w}_k$’s as the universal odd Chern classes. The description (A.7) can be generalized to the infinite unitary group $\mathbb {U}(\infty )$.

Lemma A.2

The cohomology ring of the infinite unitary group $\mathbb {U}(\infty )$

$$\begin{aligned} H^\bullet \big (\mathbb {U}(\infty ),\mathbb {Z}\big )\;\simeq \;{\bigwedge }_\mathbb {Z}\;[\mathfrak {w}_1,\mathfrak {w}_{2},\ldots ], \qquad \quad \mathfrak {w}_k\in H^{2k-1}\big (\mathbb {U}(\infty ),\mathbb {Z}\big ), \qquad k\in \mathbb {N}\end{aligned}$$

(A.8)

is the exterior algebra generated by a countable family of odd-degree classes $\mathfrak {w}_k$.

Proof

The group $\mathbb {U}(\infty )$ is by definition the limit of the direct system of inclusions

$$\begin{aligned} \mathbb {U}(1)\;{\mathop {\hookrightarrow }\limits ^{\imath _1}}\;\mathbb {U}(2)\; {\mathop {\hookrightarrow }\limits ^{\imath _2}}\;\mathbb {U}(3)\;{\mathop {\hookrightarrow }\limits ^{\imath _3}}\;\cdots \end{aligned}$$

which induces an inverse system in cohomology

$$\begin{aligned} H^\bullet \big (\mathbb {U}(1),\mathbb {Z}\big )\;{\mathop {\longleftarrow }\limits ^{\imath _1^*}}\;H^\bullet \big (\mathbb {U}(2),\mathbb {Z}\big )\;{\mathop {\longleftarrow }\limits ^{\imath _2^*}}\;H^\bullet \big (\mathbb {U}(3),\mathbb {Z}\big )\;{\mathop {\longleftarrow }\limits ^{\imath _3^*}}\;\cdots . \end{aligned}$$

(A.9)

The relation between the inverse system (A.9) and the cohomology $H^\bullet \big (\mathbb {U}(\infty ),\mathbb {Z}\big )$ is specified by the Milnor exact sequence. Let us define a cochain complex $(\mathcal {C}^\bullet ,\delta )$ by

$$\begin{aligned} \mathcal {C}^k\;:=\; \left\{ \begin{aligned}&{\prod }_j\; H^\bullet \big (\mathbb {U}(j),\mathbb {Z}\big )&\qquad \quad&j=0,1\\&0&\qquad \quad&j\geqslant 2\\ \end{aligned} \right. \end{aligned}$$

where $\prod$ is the direct product of abelian groups and the (only non-trivial) differential $\delta :\mathcal {C}^0\rightarrow \mathcal {C}^1$ is defined by

$$\begin{aligned} \delta \big (\mathfrak {a}_1,\mathfrak {a}_2,\mathfrak {a}_3,\ldots \big )\;:=\;\big (\mathfrak {a}_1-\imath _1^*(\mathfrak {a}_2),\mathfrak {a}_2-\imath _2^*(\mathfrak {a}_3),\mathfrak {a}_3-\imath _3^*(\mathfrak {a}_4),\ldots \big ), \quad \mathfrak {a}_j\in H^\bullet \big (\mathbb {U}(j),\mathbb {Z}\big ). \end{aligned}$$

By definition the inverse limit (A.9) arises as the 0-th cohomology group of the cochain complex $(\mathcal {C}^\bullet ,\delta )$, i. e. 

$$\begin{aligned} \lim _{\leftarrow j}\ H^\bullet \big (\mathbb {U}(j),\mathbb {Z}\big )\;:=\; H^0\big (\mathcal {C}^\bullet ,\delta \big )\;=\;\mathrm{Ker}(\delta ). \end{aligned}$$

The cokernel of $\delta$ is usually called “limit 1”:

$$\begin{aligned} {\lim _{\leftarrow j}}^1\ H^\bullet \big (\mathbb {U}(j),\mathbb {Z}\big )\;:=\; H^1\big (\mathcal {C}^\bullet ,\delta \big )\;=\;\mathcal {C}^1/\mathrm{Im}(\delta ). \end{aligned}$$

The Milnor exact sequence [56, Lemma 2] states that

$$\begin{aligned} 0\;\longrightarrow \;{\lim _{\leftarrow j}}^1\ H^\bullet \big (\mathbb {U}(j),\mathbb {Z}\big )\;\longrightarrow \;H^\bullet \big (\mathbb {U}(\infty ),\mathbb {Z}\big )\;\longrightarrow \;{\lim _{\leftarrow j}}\ H^\bullet \big (\mathbb {U}(j),\mathbb {Z}\big )\;\longrightarrow \;0. \end{aligned}$$

(A.10)

Notice that each $\imath ^*_j: H^\bullet (\mathbb {U}(j+1),\mathbb {Z})\rightarrow H^\bullet (\mathbb {U}(j),\mathbb {Z})$ is surjective since $\imath ^*_j(\mathfrak {w}_k)=\mathfrak {w}_k$ for all $k=1,\ldots ,j$ and $\imath ^*_j(\mathfrak {w}_{j+1})=0$. Then, a simple argument shows that also the differential $\delta :\mathcal {C}^0\rightarrow \mathcal {C}^1$ is surjective and so the “limit 1” is trivial. This implies that $H^\bullet \big (\mathbb {U}(\infty ),\mathbb {Z}\big )\simeq \mathrm{Ker}(\delta )$ and the kernel $\mathrm{Ker}(\delta )$ is generated by elements of the form $\big (\mathfrak {w}_k,\mathfrak {w}_k,\mathfrak {w}_k,\ldots \big )$ which can be identified with the generators $\mathfrak {w}_k$ for all $k\in \mathbb {N}$. This concludes the proof. $\square$

Remark A.3

(Postnikov sections of the Palais unitary group) The Postnikov resolution of $\mathbb {U}(\infty )$ can be used to provide a different description of the universal odd Chern classes $\mathfrak {w}_k$, at least in low dimension. With reference to the technological apparatus described in Sect. 5.4 we want to compute the Postnikov sections $\alpha _j:\mathbb {U}(\infty )\rightarrow \mathscr {U}_{j}^\infty$. From $\pi _1(\mathbb {U}(\infty ))=\mathbb {Z}$ we immediately get $\mathscr {U}_{1}^\infty =K(\mathbb {Z},1)\simeq \mathbb {S}^1$. Since we know that $\pi _{2j}(\mathbb {U}(\infty ))=0$ and by using the fact that $K(0,j)=\{*\}$ (along with [37, Corollary 4.63]) we obtain from (5.7) that $\mathscr {U}_{2j}^\infty \simeq \mathscr {U}_{2j-1}^\infty$. Hence, we need to compute only the odd sections. For the computation of $\mathscr {U}_{3}^\infty$ we need the knowledge of the Postnikov invariant $\kappa ^4$ in the fiber sequence

$$\begin{aligned} K(\mathbb {Z},3)\;\longrightarrow \; \mathscr {U}_{3}^\infty \;{\mathop {\longrightarrow }\limits ^{p_{3}}}\; \mathscr {U}_{2}^\infty \simeq \mathscr {U}_{1}^\infty \;{\mathop {\longrightarrow }\limits ^{\kappa ^{4}}}\; K(\mathbb {Z},4). \end{aligned}$$

(A.11)

From its very definition we know that $\kappa ^{4}\in H^{4}(\mathscr {U}^\infty _{2}, \mathbb {Z})\simeq H^{4}(\mathbb {S}^1, \mathbb {Z})=0$. The vanishing of $\kappa ^{4}$ immediately yields

$$\begin{aligned} \mathscr {U}_{4}^\infty \;\simeq \; \mathscr {U}_{3}^\infty \;\simeq \;K(\mathbb {Z},1)\;\times \; K(\mathbb {Z},3). \end{aligned}$$

(A.12)

The next step requires the computation of the Postnikov invariant $\kappa ^5$ in the fiber sequence

$$\begin{aligned} K(\mathbb {Z},5)\;\longrightarrow \; \mathscr {U}_{5}^\infty \;{\mathop {\longrightarrow }\limits ^{p_{5}}}\; \mathscr {U}_{4}^\infty \simeq \mathscr {U}_{3}^\infty \;{\mathop {\longrightarrow }\limits ^{\kappa ^{6}}}\; K(\mathbb {Z},6). \end{aligned}$$

(A.13)

In this case the Postnikov invariant is an element of $\kappa ^{6}\in H^{6}(\mathscr {U}^\infty _{4}, \mathbb {Z})\simeq H^{6}(\mathbb {S}^1\times K(\mathbb {Z},3) , \mathbb {Z})$. The knowledge of the non trivial cohomology $H^{k}(\mathbb {S}^1, \mathbb {Z})\simeq \mathbb {Z}$ if $k=0,1$ and the use of the Künneth formula for cohomology provide

$$\begin{aligned} H^{6}\big (\mathscr {U}^\infty _{4}, \mathbb {Z}\big )&\simeq \;\Big (\mathbb {Z}\;\otimes _\mathbb {Z}\; H^{6}\big ( K(\mathbb {Z},3) , \mathbb {Z}\big )\Big )\;\oplus \;\Big (\mathbb {Z}\;\otimes _\mathbb {Z}\; H^{5}\big ( K(\mathbb {Z},3) , \mathbb {Z}\big )\Big )\\&\simeq \mathbb {Z}\;\otimes _\mathbb {Z}\;\mathbb {Z}_2\;\simeq \mathbb {Z}_2. \end{aligned}$$

where we used the explicit results $H^{5}( K(\mathbb {Z},3) , \mathbb {Z})=0$ and $H^{6}( K(\mathbb {Z},3) , \mathbb {Z})\simeq \mathbb {Z}_2$ computed in [14, Section 18]. This is compatible with the more general result [3, Lemma 5] which states that $\kappa ^{6}$ is a (non-trivial) element of order 2. This implies that

$$\begin{aligned} \mathscr {U}_{6}^\infty \;\simeq \; \mathscr {U}_{5}^\infty \;\simeq \;\Big (K(\mathbb {Z},1)\;\times \; K(\mathbb {Z},3)\Big )\;\times _{\kappa ^6}\; K(\mathbb {Z},5). \end{aligned}$$

(A.14)

where the last product is “twisted” by the action of the non trivial class $\kappa ^6$. Evidently the construction becomes more and more involved when the degree of the Postnikov section increases. Let us consider now the implication of (A.12) for the interpretation of the first two generators of $\mathbb {U}(\infty )$. By construction the map $\alpha _4:\mathbb {U}(\infty )\rightarrow \mathscr {U}_{4}^\infty$ is a 5-equivalence, hence the Whitehead’s second theorem⁵ assures that $H^k(\mathbb {U}(\infty ),\mathbb {Z})\simeq H^k(\mathscr {U}_{4}^\infty ,\mathbb {Z})$ for all $k\leqslant 4$. With the help of the Künneth formula for cohomology and the explicit knowledge of cohomology of $K(\mathbb {Z},1)$ and $K(\mathbb {Z},3)$ one computes

$$\begin{aligned} \begin{aligned} H^0\big (\mathbb {U}(\infty ),\mathbb {Z}\big )\;&\simeq \;H^{0}\big ( K(\mathbb {Z},1) , \mathbb {Z}\big )\;\otimes _\mathbb {Z}\; H^{0}\big ( K(\mathbb {Z},3) , \mathbb {Z}\big )\; \simeq \;\mathbb {Z}\\ H^1\big (\mathbb {U}(\infty ),\mathbb {Z}\big )\;&\simeq \;H^{1}\big ( K(\mathbb {Z},1) , \mathbb {Z}\big )\;\otimes _\mathbb {Z}\; H^{0}\big ( K(\mathbb {Z},3) , \mathbb {Z}\big )\; \simeq \;H^{1}\big ( K(\mathbb {Z},1) , \mathbb {Z}\big )\;\simeq \;\mathbb {Z}\\ H^2\big (\mathbb {U}(\infty ),\mathbb {Z}\big )\;&\simeq \;0\\ H^3\big (\mathbb {U}(\infty ),\mathbb {Z}\big )\;&\simeq \;H^{0}\big ( K(\mathbb {Z},1) , \mathbb {Z}\big )\;\otimes _\mathbb {Z}\; H^{3}\big ( K(\mathbb {Z},3) , \mathbb {Z}\big )\;\simeq \;H^{3}\big ( K(\mathbb {Z},3) , \mathbb {Z}\big )\; \simeq \;\mathbb {Z}\\ H^4\big (\mathbb {U}(\infty ),\mathbb {Z}\big )\;&\simeq \;H^{1}\big ( K(\mathbb {Z},1) , \mathbb {Z}\big )\;\otimes _\mathbb {Z}\; H^{3}\big ( K(\mathbb {Z},3) , \mathbb {Z}\big )\; \simeq \;\mathbb {Z}. \end{aligned} \end{aligned}$$

The above result shows that $\mathfrak {w}_1$ and $\mathfrak {w}_2$ can be identified with the basic class of $H^{1}( K(\mathbb {Z},1) , \mathbb {Z})$ and $H^{3}( K(\mathbb {Z},3) , \mathbb {Z})$, respectively (for a definition of basic class see e. g. [2, Definition 5.3.1]). As a final comment we can observe that (A.12) also describes the 3-rd and 4-th Postnikov sections of $\mathbb {U}(m)$ for all $m\geqslant 2$ while (A.14) describes the 5-th and 6-th Postnikov sections of $\mathbb {U}(m)$ for all $m\geqslant 3$ (stable regime). $\blacktriangleleft$

Appendix B. Fiber bundle identification of the classifying space

The definition (3.2) says that the spaces $\chi _m(\mathbb {C}^n)$ are subspaces of $G_m(\mathbb {C}^n)\times \mathbb {U}(n)$. These spaces can be also identified with the total spaces of suitable fiber bundles. For this aim, let us recall a standard construction: For any Lie group $\mathbb {G}$ and any principal $\mathbb {G}$-bundle $\pi : \mathscr {P} \rightarrow X$ we use the adjoint action of $\mathbb {G}$ on $\mathbb {G}$ to get the associated fiber bundle

$$\begin{aligned} {\mathrm{Ad}}(\mathscr {P})\;:=\;\mathscr {P}\;\times _{\mathrm{Ad}}\;\mathbb {G}\;=\;(\mathscr {P}\;\times \;\mathbb {G})/\mathbb {G} \end{aligned}$$

where $g\in \mathbb {G}$ acts on $(p,h)\in \mathscr {P}\times \mathbb {G}$ by $(p,h)\mapsto (p\cdot g^{-1},g\cdot h\cdot g^{-1})$. It is a well-known fact that sections of ${\mathrm{Ad}}(\mathscr {P})\rightarrow X$ are in one to one correspondence with automorphisms of $\mathscr {P}$ [38, Chapter 7, Section 1].

Let us also recall that the Grassmann manifold and its tautological $\mathbb {U}(m)$-frame bundle (Stiefel variety) $\mathscr {S}_m^n\rightarrow G_m(\mathbb {C}^n)$ can be realized as quotient spaces:

$$\begin{aligned} G_m(\mathbb {C}^n)\simeq \mathbb {U}(n)/\big (\mathbb {U}(m)\times \mathbb {U}(n-m)\big ), \qquad \quad \mathscr {S}_m^n\simeq \mathbb {U}(n)/ \mathbb {U}(n-m), \end{aligned}$$

(B.1)

where $\mathbb {U}(m)$ and $\mathbb {U}(n-m)$ act (on the right) on $\mathbb {U}(n)$ through the inclusions of $\mathbb {C}^m$ and $\mathbb {C}^{m-n}$ into $\mathbb {C}^n=\mathbb {C}^m\oplus \mathbb {C}^{n-m}$. Let $\Sigma \in G_m(\mathbb {C}^n)$ be a subspace of $\mathbb {C}^n$ of dimension m. Given orthonormal basis $mathbb {v}:=(\mathrm{v}_1,\ldots ,\mathrm{v}_m)$ and $mathbb {w}:=(\mathrm{w}_1,\ldots ,\mathrm{w}_{n-m})$ of $\Sigma \subset \mathbb {C}^n$ and $\Sigma ^\bot \subset \mathbb {C}^n$, respectively one can form an element $(mathbb {v},mathbb {w})\in \mathbb {U}(n)$ by arraying the vectors as columns, i. e. 

$$\begin{aligned} (mathbb {v},mathbb {w})\;:=\; \left( \begin{array}{cccccc} | &{} &{} | &{} | &{} &{} | \\ \mathrm{v}_1 &{} \cdots &{} \mathrm{v}_m &{} \mathrm{w}_1 &{} \cdots &{} \mathrm{w}_{n-m} \\ | &{} &{} | &{} | &{} &{} | \end{array}\right) . \end{aligned}$$

(B.2)

According to this notation $mathbb {v}$ and $mathbb {w}$ can be considered an $n\times m$ and an $n\times (m-n)$ matrices, respectively. The pair $(mathbb {v},mathbb {w})$ provides a representative for $([mathbb {v}],[mathbb {w}])=\Sigma \in G_m(\mathbb {C}^n)$ according to the quotient description of the Grassmann manifold in (B.1). Here, the equivalence relations are naturally given by $mathbb {v}\sim mathbb {v}\cdot a$ and $mathbb {w}\sim mathbb {w}\cdot b$ for some $a\in \mathbb {U}(m)$ and $b\in \mathbb {U}(n-m)$. Similarly, we can use $(mathbb {v},[mathbb {w}])\in \mathscr {S}_m^n$ for a point in the Stiefel variety according to the quotient representation in (B.1) and an element $c\in \mathbb {U}(m)$ in the structure group can be represented by a block diagonal matrix

$$\begin{aligned} \left( \begin{array}{c|c}c &{} 0 \\ \hline 0 &{} mathbb {1}_{n-m}\end{array}\right) \;\subset \;\mathbb {U}(n). \end{aligned}$$

In this way points in ${\mathrm{Ad}}\big (\mathscr {S}_m^n\big )$ are given by equivalence classes $[(mathbb {v},mathbb {w}),c]$ with respect to the equivalence relation $((mathbb {v},mathbb {w}\cdot b),c)\sim ((mathbb {v}\cdot a^{-1},mathbb {w}),a\cdot c\cdot a^{-1})$ for some $a\in \mathbb {U}(m)$ and $b\in \mathbb {U}(n-m)$.

The core of this appendix is the proof of the following identifications:

Proposition B.1

There are natural homeomorphisms

$$\begin{aligned} \chi _m(\mathbb {C}^n)\;\simeq \; {\mathrm{Ad}}\big (\mathscr {S}_m^n\big ), \qquad \text {and}\qquad \mathbb {B}_\chi ^m\;\simeq \; {\mathrm{Ad}}\big (\mathscr {S}_m^\infty \big ), \end{aligned}$$

where, as usual, $\mathscr {S}_m^\infty$ denotes the inductive limit obtained by the inclusions $\mathscr {S}_m^n\subset \mathscr {S}_m^{n+1}\subset \cdots \$.

However, we start first with a technical preliminary result.

Lemma B.2

For each pair of integers $1\leqslant m\leqslant n$ there is a bijective continuous map

$$\begin{aligned} \vartheta \;:\;{\mathrm{Ad}}\big (\mathscr {S}_m^n\big )\;\longrightarrow \; \chi _m(\mathbb {C}^n). \end{aligned}$$

(B.3)

Proof

Given an element $(mathbb {v},mathbb {w})\in \mathbb {U}(n)$ as in (B.2) and a $c\in \mathbb {U}(m)$ there is a unique unitary matrix $u(mathbb {v};mathbb {w};c)\in \mathbb {U}(n)$ such that $u(mathbb {v};mathbb {w};c)\cdot (mathbb {v},mathbb {w})=(mathbb {v}\cdot c,mathbb {w})$. Such a matrix is explicitly given by

$$\begin{aligned} u(mathbb {v};mathbb {w};c)\;:=\;(mathbb {v},mathbb {w})\cdot \left( \begin{array}{c|c}c &{} 0 \\ \hline 0 &{} mathbb {1}_{n-m}\end{array}\right) \cdot (mathbb {v},mathbb {w})^{-1}. \end{aligned}$$

By construction $u(mathbb {v};mathbb {w};c)$ preserves the m-dimensional subspace $([mathbb {v}],[mathbb {w}])=\Sigma \subset \mathbb {C}^n$ spanned by the columns of $mathbb {v}$. Therefore, the pair $(\Sigma ,u):=(([mathbb {v}],[mathbb {w}]),u(mathbb {v};mathbb {w};c))$ provides a point in $\chi _m(\mathbb {C}^n)$. This allows to define the map

$$\begin{aligned} \begin{aligned} \vartheta \;:\;&\;\mathscr {S}_m^n\times \mathbb {U}(m)&\;\longrightarrow \;&\ \ \chi _m(\mathbb {C}^n)&\\&\;\big ((mathbb {v},[mathbb {w}]),c\big )&\;\longmapsto \;&\ \ (\Sigma ,u)&. \end{aligned} \end{aligned}$$

(B.4)

This map factors through the equivalence relation that defines ${\mathrm{Ad}}\big (\mathscr {S}_m^n\big )$ since $([mathbb {v}],[mathbb {w}])=([mathbb {v}\cdot a],[mathbb {w}])=\Sigma$ and

$$\begin{aligned} u\big (mathbb {v}\cdot a;mathbb {w}; a \cdot c\cdot a^{-1}\big )\;=\;u\big (mathbb {v};mathbb {w};c\big )\;=\;u\big (mathbb {v};mathbb {w}\cdot b;c\big ), \qquad \quad \forall \ a\in \mathbb {U}(m),\ \ b\in \mathbb {U}(n-m) \end{aligned}$$

Hence, the prescription (B.4) defines a map like (B.3). The map $\vartheta$ is also continuous. Indeed, the topology of ${\mathrm{Ad}}\big (\mathscr {S}_m^n\big )$ is induced from $\mathbb {U}(n)\times \mathbb {U}(m)$ by a quotient, and that of $\chi _m(\mathbb {C}^n)$ is induced from $G_m(\mathbb {C}^n)\times \mathbb {U}(n)$ by an inclusion. Thus, to prove that $\vartheta$ is continuous, it suffices to prove that the composition of the following maps is continuous:

(B.5)

The natural projection ${\mathrm{pr}}$ and the inclusion $\imath$ are continuous by construction. Moreover, the composition of the three maps is easily computed to be $((mathbb {v},mathbb {w}),c)\mapsto (([mathbb {v}],[mathbb {w}]),u(mathbb {v};mathbb {w};c))$ which is evidently continuous. This, in turn, implies the continuity of $\vartheta$.

In order to prove that $\vartheta$ is bijective we construct its inverse $\varrho$. Let us start with a point $(\Sigma ,u)\in \chi _m(\mathbb {C}^n)$ and choose orthonormal basis $mathbb {v}$ and $mathbb {w}$ such that $([mathbb {v}],[mathbb {w}])=\Sigma$ as above. Since $u\in \mathbb {U}(n)$ preserves $\Sigma$ there is a unique $c(mathbb {v};u)\in \mathbb {U}(m)$ such that $u\cdot mathbb {v}=mathbb {v}\cdot c(mathbb {v};u)$. Such a matrix is explicitly given by

$$\begin{aligned} c(mathbb {v};u)\;:=\;^t\overline{mathbb {v}} \cdot u\cdot mathbb {v} \end{aligned}$$

where $^t\overline{mathbb {v}}$ denotes the transpose of the matrix $\overline{mathbb {v}}$, complex conjugated of $mathbb {v}$. Now we define

$$\begin{aligned} \begin{aligned} \varrho \;:\;&\;\chi _m(\mathbb {C}^n)&\;\longrightarrow \;&\ \ \ \ \ \ \ \ \ {\mathrm{Ad}}\big (\mathscr {S}_m^n\big )&\\&\;(\Sigma ,u)&\;\longmapsto \;&\ \ \big ((mathbb {v},[mathbb {w}]),c(mathbb {v};u)\big )&, \end{aligned} \end{aligned}$$

(B.6)

where $(mathbb {v},[mathbb {w}])\in \mathbb {U}(n)/ \mathbb {U}(n-m)$ identifies an element of $\mathscr {S}_m^n$. The map $\varrho$ does not depend on the choice of a frame $mathbb {w}$ for $\Sigma ^\bot$ (as denoted by the square brackets). Moreover, if $mathbb {v}'=mathbb {v}\cdot a$ with $a\in \mathbb {U}(m)$ is a new frame for $\Sigma$, a direct computation shows $c(mathbb {v}';u)=a^{-1}\cdot c(mathbb {v};u)\cdot a$. These two facts prove that $\varrho$ really maps into ${\mathrm{Ad}}\big (\mathscr {S}_m^n\big )$. The proof that $\varrho =\vartheta ^{-1}$ follows now from a direct, as well trivial, verification. $\square$

Proof of Proposition B.1

Lemma B.2 proves the existence of a continuous bijection $\vartheta :{\mathrm{Ad}}\big (\mathscr {S}_m^n\big )\rightarrow \chi _m(\mathbb {C}^n)$. Since ${\mathrm{Ad}}\big (\mathscr {S}_m^n\big )$ is compact (it is the quotient of a compact group) and $\chi _m(\mathbb {C}^n)$ is Hausdorff (it is the subspace of a Hausdorff space), $\vartheta$ turns out to be a homeomorphism (see e. g. [40, Chapter I, Section 8]). This homeomorphism is compatible with taking the direct limits, so that it induces a homeomorphism $\vartheta :{\mathrm{Ad}}\big (\mathscr {S}_m^\infty \big )\rightarrow \mathbb {B}_\chi ^m$. $\square$

From the construction of the identifications in Proposition B.1 it follows that the mapping $\chi _m(\mathbb {C}^n)\rightarrow G_m(\mathbb {C}^n)$ in (3.4) and $\mathbb {B}_\chi ^m\rightarrow G_m(\mathbb {C}^\infty )$ in (3.7) agree with the bundle maps ${\mathrm{Ad}}\big (\mathscr {S}_m^n\big )\rightarrow G_m(\mathbb {C}^n)$ and ${\mathrm{Ad}}\big (\mathscr {S}_m^\infty \big )\rightarrow G_m(\mathbb {C}^\infty )$, respectively. Summarizing, one has the fiber sequence

$$\begin{aligned} \mathbb {U}(m)\;\longrightarrow \;{{\mathrm{Ad}}}\big (\mathscr {S}_m^\infty \big )\simeq \mathbb {B}_\chi ^m\;{\mathop {\longrightarrow }\limits ^{\pi }}\; G_m\big (\mathbb {C}^\infty \big ) \end{aligned}$$

(B.7)

with projection $\pi$ given by (3.7).

When $m=1$, just by exploiting the fact that $\mathbb {U}(1)$ is abelian, one gets the following immediate consequence of Proposition B.1.

Corollary B.3

$$\begin{aligned} \mathbb {B}_\chi ^1\;\simeq \;{\mathrm{Ad}}\big (\mathscr {S}_1^\infty \big )\;\simeq \;\mathbb {C}P^\infty \;\times \;\mathbb {U}(1). \end{aligned}$$