We now discuss how the approach of Section 5.1 complies with a more general
theory on semi-torsors. As shown in [28], one can define the notion of a frame
of a semi-torsor. Let us briefly review this here for completeness. If $G$ is
a Lie group and $F$ a $G$-semi-torsor consisting of $n$ orbits, then $G\times
I_{n}\cong F$ as (left) $G$-spaces. Such an isomorphism is always of the form
$(g,i)\mapsto gf_{i}$ for a unique tuple $\tilde{f}=(f_{1},\dots,f_{n})$ of
elements in $F$. This tuple $\tilde{f}$ we call a basis of $F$, similar to
linear algebra. Observe that $n$ elements of $F$ form a basis if and only if
each orbit in $F$ is represented exactly once. The set of all frames of $F$ we
denote by $\operatorname{Fr}(F)$. Clearly, one can translate or permute the
elements of a basis. This is captured by a natural action of the wreath
product $G\wr I_{n}$, i.e.,

$\displaystyle G\wr I_{n}\times\operatorname{Fr}(F)$
$\displaystyle\to\operatorname{Fr}(F),$
$\displaystyle((g_{1},\dots,g_{n}),\sigma)\cdot(f_{1},\dots,f_{n})$
$\displaystyle=\big{(}g_{1}f_{\sigma^{-1}(1)},\dots,g_{n}f_{\sigma^{-1}(n)}\big{)}.$

In fact, this action is free and transitive, so that, in other words,
$\operatorname{Fr}(F)$ is a $G\wr I_{n}$-torsor.

This is exactly what we did in the previous section. For the eigenvalues, we
see that $\operatorname{Spec}(A)$ is a set containing $n$ points, hence
trivially a semi-torsor with $G$ the trivial group. Each eigenvalue is thus an
orbit by itself, and a frame is thus any tuple
$\tilde{\lambda}=(\lambda_{1},\dots,\lambda_{n})$ listing all elements of
$\operatorname{Spec}(A)$ once. These are exactly the tuples we considered
before, i.e.,

$\operatorname{Fr}(\operatorname{Spec}(A))=\operatorname{Spec}(A)!.$

This space is naturally endowed with the $S_{n}$-action on $n$-tuples, making
it an $S_{n}$-torsor. For the eigenstates, we consider the space
$\operatorname{Eig}(A)$, which we already found to be a
$\mathbb{C}^{\times}$-semi-torsor. The orbits in this case are the eigenrays,
and hence a frame of $\operatorname{Eig}(A)$ is a tuple $\tilde{f}$ of
eigenvectors such that each eigenray is represented exactly once. Indeed, we
again find that $\tilde{f}$ should be an eigenframe of $A$, i.e.,

$\operatorname{Fr}(\operatorname{Eig}(A))=\mathrm{EigFr}(A).$

And this space is clearly a $\mathbb{C}^{\times}\wr~{}I_{n}$-torsor.

The next step is to lift this procedure of taking frames to the level of semi-
principal bundles. We remark that this theory closely resembles that of the
frame bundle of a vector bundle. As shown in [28], the operation of taking the
frame space induces a functor from semi-principal bundles to principal
bundles. Given a semi-principal $G$-bundle $\pi\colon B\to M$, we obtain its
frame bundle $\pi!\colon\operatorname{Fr}(B)\to M$ by applying the above
fiber-wise;

$\operatorname{Fr}(B)=\bigsqcup_{m\in M}\operatorname{Fr}(B_{m}).$

It holds that $\pi!\colon\operatorname{Fr}(B)\to M$ is a principal $G\wr
I_{n}$-bundle. Moreover, any $G$-connection on $B$ becomes a $G\wr
I_{n}$-connection on $\operatorname{Fr}(B)$ by stating that a path in
$\operatorname{Fr}(B)$ is horizontal if and only if all frame elements
traverse horizontal paths. The map $B\mapsto\operatorname{Fr}(B)$ induces a
retracting functor from the semi-principal bundles to the principal bundles.
The holonomy interpretation of the explicit permutations and holonomy matrices
we found above can now be obtained by applying this frame bundle functor to
$\operatorname{Spec}(V)$ resp. $\operatorname{Eig}(V)$.

####  5.2.1 Frame bundle of the spectrum bundle

Let us start with $\operatorname{Spec}(V)$. We found in Theorem 3.4 that
$\pi_{v}\colon\operatorname{Spec}(V)\to N(V)$ is an $I_{n}$-bundle, hence it
is a semi-principal bundle with trivial structure group. The frame bundle
functor then yields the following.

######  Proposition 5.3.

The frame bundle of $\operatorname{Spec}(V)$ is the principal $S_{n}$-bundle

$\mathrm{E}(V):=\operatorname{Fr}(\operatorname{Spec}(V))=\big{\\{}\big{(}A,\tilde{\lambda}\big{)}\in
N(V)\times\mathbb{C}^{n}\mid\tilde{\lambda}\in\operatorname{Fr}(\operatorname{Spec}(A))\big{\\}},$

with action

$\sigma\cdot(\lambda_{1},\dots,\lambda_{n})=\big{(}\lambda_{\sigma^{-1}(1)},\dots,\lambda_{\sigma^{-1}(n)}\big{)}$

and projection

$\displaystyle\pi_{\lambda}!\colon\ \mathrm{E}(V)$ $\displaystyle\to N(V),$
$\displaystyle\big{(}A,\tilde{\lambda}\big{)}$ $\displaystyle\mapsto A.$

Remarkably, we observe that $\pi_{\lambda}!$ is also the pull-back of
$q\colon\mathbb{C}^{\underline{n}}\to\binom{\mathbb{C}}{n}$ along the map
$\operatorname{Spec}\colon N(V)\to\binom{\mathbb{C}}{n}$; the fiber
$\operatorname{Fr}(\operatorname{Spec}(A))$ of $\mathrm{E}(V)$ equals
$q^{-1}(\operatorname{Spec}(A))$. In addition, when a Hamiltonian operator
family $H\colon M\to\operatorname{End}(V)$ is given, it is again natural to
consider the pull-back along $H$. This yields the bundle
$\pi_{\lambda}^{H}!\colon\mathrm{E}(H)\to N(H)$, $(x,\tilde{E})\mapsto x$,
where $\tilde{E}$ is then a tuple listing the energies of $H(x)$, and is the
frame bundle of $\pi_{\lambda}^{H}$. Observe that both pull-backs together
result in the following commutative diagram, where all vertical maps are
principal $S_{n}$-bundles:

${\mathrm{E}(H)}$${\mathrm{E}(V)}$${\mathbb{C}^{\underline{n}}}$${(x,\tilde{E})}$${(H(x),\tilde{E})}$${\tilde{E}}$${N(H)}$${N(V)}$${\binom{\mathbb{C}}{n},}$${x}$${H(x)}$${\operatorname{Spec}(H(x)).}$$\scriptstyle{\pi_{\lambda}^{H}!}$$\scriptstyle{\pi_{\lambda}!}$$\scriptstyle{q}$$\scriptstyle{H}$$\scriptstyle{\operatorname{Spec}}$

We observe that $\mathrm{E}(H)$ and $\mathrm{E}(V)$ are also the natural
spaces behind the merging path method for EP detection, which we reviewed in
Section 3.2.1. Namely, the above diagram can alternatively be obtained by
completing diagram (3.2) using pull-backs.

The holonomy interpretation behind the permutations can be explained as
follows. Fix a loop $\gamma$ in $N(H)$ based at $x_{0}$, and let $\tilde{E}$
be the ordering of $\operatorname{Spec}(H(x_{0}))$ w.r.t. which we find the
permutation $\sigma$ expressing $p_{\gamma}$. Observe that the picking of the
ordering $\tilde{E}=(E_{1},\dots,E_{n})$ of $\operatorname{Spec}(H(x_{0}))$ is
equivalent to picking the point $\big{(}x_{0},\tilde{E}\big{)}$ in the fiber
of $\mathrm{E}(H)$ above $x_{0}$. This point $\big{(}x_{0},\tilde{E}\big{)}$
fixes a unique lift of $\gamma$ to $\mathrm{E}(H)$. Physically, this lift
traces all of the energies simultaneously and in a particular order. The
permutation $\sigma$ now naturally appears via the holonomy theory of
principal bundles; the end point of the lift is
$\sigma\big{(}x_{0},\tilde{E}\big{)}$, which fixes $\sigma$ uniquely. One can
alternatively argue by means of holonomy groups. We already found that
$p_{\gamma}$ lies in the holonomy group
$\operatorname{Hol}^{\operatorname{Spec}(H)}_{N(H)}(x_{0})$. Clearly,
$p_{\gamma}$ extends to a map $\tilde{p}_{\gamma}$ on frames by applying
$p_{\gamma}$ on each frame element separately, and so $\tilde{p}_{\gamma}$ is
an element of $\operatorname{Hol}^{\mathrm{E}(H)}_{N(H)}(x_{0})$. Both of
these groups are based on a point in the base space, and so consist of
“abstract” fiber automorphisms. An explicit group element is obtained by
considering the holonomy group at a point in the total space. In this case, we
move to the holonomy group at $\big{(}x_{0},\tilde{E}\big{)}$ defined as

$\operatorname{Hol}^{\mathrm{E}(H)}\big{(}x_{0},\tilde{E}\big{)}=\big{\\{}\sigma\in
S_{n}\mid\big{(}x_{0},\tilde{E}\big{)}\sim\sigma\big{(}x_{0},\tilde{E}\big{)}\big{\\}},$

where the equivalence relation $\sim$ holds if and only if the two points can
be connected by a path. The groups are related by the isomorphism
$\operatorname{Hol}^{\mathrm{E}(H)}_{N(H)}(x_{0})\to\operatorname{Hol}^{\mathrm{E}(H)}\big{(}x_{0},\tilde{E}\big{)}$,
$\tilde{p}_{\gamma}\mapsto\big{[}\tilde{p}_{\gamma}\big{(}\tilde{E}\big{)}/\tilde{E}\big{]}$,
where $[-/-]$ denotes the unique group element translating the right entry to
the left one. We summarize these findings in the following.

######  Lemma 5.4.

Given a loop $\gamma$ in $N(H)$ based at $x_{0}$, the permutation $\sigma$
expressing $p_{\gamma}$ w.r.t. the ordering $\tilde{E}$ of
$\operatorname{Spec}(H(x_{0}))$ is the holonomy element at the point
$\big{(}x_{0},\tilde{E}\big{)}$ in the frame bundle $\mathrm{E}(H)$.

It follows that we should consider $\sigma$ to lie in the group
$\operatorname{Hol}^{\mathrm{E}(H)}\big{(}x_{0},\tilde{E}\big{)}$. This
automatically accounts for the change in $\sigma$ if we choose another
ordering $\tilde{E}^{\prime}$. Geometrically, we see this amounts to picking a
different reference point in $\mathrm{E}(H)$, namely
$(x_{0},\tilde{E}^{\prime})$ instead of $\big{(}x_{0},\tilde{E}\big{)}$.
Hence, the permutation $\sigma^{\prime}$ obtained w.r.t. $\tilde{E}^{\prime}$
lies in
$\operatorname{Hol}^{\mathrm{E}(H)}\big{(}x_{0},\tilde{E}^{\prime}\big{)}$
rather than $\operatorname{Hol}^{\mathrm{E}(H)}\big{(}x_{0},\tilde{E}\big{)}$.
We observe that the relation between $\sigma$ and $\sigma^{\prime}$ is given
by the usual conjugation relation between holonomy groups. Namely, there is a
unique $\tau\in S_{n}$ such that $\tilde{E}^{\prime}=\tau\tilde{E}$, and
consequently $\sigma^{\prime}=\tau\sigma\tau^{-1}$. Note that the frame
dependence of $\sigma$ can also be regarded as a gauge dependence. The gauge
freedom is then in the choice of frame of $\operatorname{Spec}(A)$, and
different gauges are related by an $S_{n}$-symmetry. Of course, the gauge
invariant behind the permutations $\sigma$ and $\sigma^{\prime}$ is the map of
eigenvalues $p_{\gamma}$, or equivalently its frame version
$\tilde{p}_{\gamma}$.

Using the framework of holonomy groups, we can be more explicit on how
permutations arising from different loops $\gamma$ are related, as the authors
also studied in [27]. Like $\operatorname{Spec}(H)$, also $\mathrm{E}(H)$ is a
covering space of $N(H)$, and so the permutations are prescribed by the
monodromy action. With respect to the ordering $\tilde{E}$ of
$\operatorname{Spec}(H(x_{0}))$, this is captured by the homomorphism

$\displaystyle\pi_{1}(N(H),x_{0})$
$\displaystyle\to\operatorname{Hol}^{\mathrm{E}(H)}\big{(}x_{0},\tilde{E}\big{)},$
$\displaystyle[\gamma]$
$\displaystyle\mapsto\big{[}\tilde{p}_{\gamma}\big{(}\tilde{E}\big{)}/\tilde{E}\big{]}.$

Clearly this homomorphism is surjective, and its kernel consists of all
classes whose paths induce a cyclic change of eigenvectors. In addition, it
shows how the holonomy matrix of a complicated loop can be decomposed in
holonomy matrices of more elementary loops. This present argument hence
improves on a previous proof of this fact given in [27].

Let us briefly compare the holonomy groups of $\mathrm{E}(V)$ and
$\mathrm{E}(H)$. For $\mathrm{E}(V)$, every holonomy group exhausts all of
$S_{n}$ as all possible paths are present. Indeed, the bundle
$\pi_{\lambda}!\colon\mathrm{E}(V)\to N(V)$ does not admit a reduction of the
structure group. However, $\mathrm{E}(H)$ can have smaller holonomy groups,
and admit a reduction of the structure group. In an extreme case, e.g., if the
Hamiltonian family is always Hermitian w.r.t. a given inner product on $V$,
then $\mathrm{E}(H)$ is trivial as $\operatorname{Spec}(H)$ is,444An
alternative argument: if the energies of $H$ are real, then $\mathrm{E}(H)$ is
a pull-back of $\mathbb{R}^{\underline{n}}\to\binom{\mathbb{R}}{n}$, hence
trivial by Proposition A.3. and the structure group can be reduced to the
trivial group 1. Intermediate cases are also possible, and express in what way
the energy bands are connected. For example, if $k$ energy bands are
disconnected from the remaining $n-k$, clearly the structure group $S_{n}$ can
be reduced to $S_{k}\times S_{n-k}$. Practically, this can be achieved by
restricting to tuples $\tilde{E}$ in which these $k$ energy bands appear only
in the first $k$ entries.

######  Example 5.5.

Let us continue the study of the standard EP2 system as we described in
Example 3.5. We found $N(H)=\mathbb{C}\setminus\\{\pm{\rm i}\\}$, which means
$\pi_{1}(N(H),0)$ has two generators. Define generators $a_{\pm}$ to wind once
around $\pm{\rm i}$ in positive direction. We found in Example 3.6 that both
generators exchange the eigenvalues. Let us label the spectrum
$\operatorname{Spec}(H(0))=\\{\pm 1\\}$ as $(+1,-1)$. Then the permutations
arising from the EPs are expressed by the homomorphism

$\displaystyle\pi_{1}(N(H),0)$
$\displaystyle\to\operatorname{Hol}^{\mathrm{E}(H)}(0,(+1,-1)),$
$\displaystyle a_{\pm}$ $\displaystyle\mapsto(12).$

Clearly, this confirms that a permutation occurs if and only if a loop
decomposes into an odd number of $a_{\pm}$. As seen in Example 5.1, if an
extra level is present, the exact form of the permutation does indeed depend
on the labelling. In addition, we observe that for the extended system
$H^{\prime}$ presented there, the principal $S_{3}$-bundle
$\mathrm{E}(H^{\prime})$ reduces to a $S_{2}$-bundle, which is isomorphic to
$\mathrm{E}(H)$.

We note that the space $\mathrm{E}(V)$ has appeared in previous papers. We
already remarked that [34] mentioned the relevance of the monodromy action in
the theory of EPs. The covering used there is of the form $\mathrm{E}(H)$,
using the one-to-one correspondence between eigenvalues and eigenprojectors
for non-degenerate operators. We also recognize $\mathrm{E}(V)$ as the space
$\widetilde{\mathfrak{M}}$ defined in [21]. However, where [21] views
$\mathrm{E}(V)$ primarily as an extended parameter space, we consider it
rather as the bundle over $N(V)$ which yields the permutations due to EPs by
its holonomy.

####  5.2.2 Frame bundle of the eigenvector bundle

We now proceed by considering the frame bundle of $\pi_{\lambda
v}\colon\operatorname{Eig}(V)\to N(V)$. This is a semi-principal
$\mathbb{C}^{\times}$-bundle by Theorem 4.4 and endowed with the
$\mathbb{C}^{\times}$-connection $\omega$ as in Proposition 4.7. Applying the
frame bundle functor then yields the following.

######  Proposition 5.6.

The frame bundle of $\operatorname{Eig}(V)$ is the principal
$\mathbb{C}^{\times}\wr~{}I_{n}$-bundle

$\mathrm{EigFr}(V)=\big{\\{}\big{(}A,\tilde{f}\big{)}\in
N(V)\times\operatorname{Fr}(V)\mid\tilde{f}\text{ is an eigenframe of
}A\big{\\}},$

where the element $\big{(}A,\tilde{f}\big{)}$ can also be written as
$\big{(}A,\tilde{\lambda},\tilde{f}\big{)}$ with $\tilde{\lambda}$ listing the
eigenvalues of $A$ in the order of $\tilde{f}$. The
$\mathbb{C}^{\times}\wr~{}I_{n}$-action on $\mathrm{EigFr}(V)$ reads

$(\tilde{z},\sigma)\cdot\big{(}A,\tilde{\lambda},\tilde{f}\big{)}=\big{(}A,\sigma\tilde{\lambda},\tilde{z}\big{(}\sigma\tilde{f}\big{)}\big{)}$

and the projection is

$\displaystyle\pi_{\lambda v}!\colon\ \mathrm{EigFr}(V)$ $\displaystyle\to
N(V),$ $\displaystyle\big{(}A,\tilde{\lambda},\tilde{f}\big{)}$
$\displaystyle\mapsto A.$

Furthermore, $\mathrm{EigFr}(V)$ is naturally endowed with the
$\mathbb{C}^{\times}\wr~{}I_{n}$-connection

$\omega!_{\left(A,\tilde{\lambda},\tilde{f}\right)}=\omega_{(A,\lambda_{1},f_{1})}\oplus\dots\oplus\omega_{(A,\lambda_{n},f_{n})}.$

Recall that we already found the principal
$\mathbb{C}^{\times}\wr~{}I_{n}$-bundle
$\Xi\colon\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}\to N(V)$ in
Section 2.2. We observe that $\Xi$ and $\pi_{\lambda v}!$ are reformulations
of each other, i.e., they are canonically isomorphic bundles.

######  Lemma 5.7.

The map

$\displaystyle\mathrm{EigFr}(V)$
$\displaystyle\to\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}},$
$\displaystyle\big{(}A,\tilde{\lambda},\tilde{f}\big{)}$
$\displaystyle\mapsto\big{(}\tilde{f},\tilde{\lambda}\big{)}$

is a canonical isomorphism of bundles over $N(V)$. Hence, in particular,
$\mathrm{EigFr}(V)$ is canonically isomorphic to
$\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}$ as a
$\mathbb{C}^{\times}\wr~{}I_{n}$-manifold, and consequently $\mathrm{E}(V)$ is
canonically diffeomorphic to
$\operatorname{PFr}(V)\times\mathbb{C}^{\underline{n}}$.

Again, given a Hamiltonian family $H$, it is more natural to study the pull-
back bundle. This yields the space $\mathrm{EigFr}(H)$, whose elements we
write as $\big{(}x,\tilde{E},\tilde{\psi}\big{)}$. The connection form
$\omega!$ also carries over to the connection form $\omega_{H}!$. Note that
$\mathrm{EigFr}(H)$ may admit a reduction of the structure group, e.g., to
$U(1)^{n}$ when $H$ is Hermitian. We can summarize the pull-back and relation
with $\Xi$ in diagram (5.1).

${\mathrm{EigFr}(H)}$${\mathrm{EigFr}(V)}$${\operatorname{Fr}(V)\times\mathbb{C}^{\underline{n}}}$${\big{(}x,\tilde{E},\tilde{\psi}\big{)}}$${\big{(}H(x),\tilde{E},\tilde{\psi}\big{)}}$${\big{(}\tilde{\psi},\tilde{E}\big{)}}$${N(H)}$${N(V)}$${N(V),}$${x}$${H(x)}$${H(x).}$$\scriptstyle{\pi_{\lambda
v}^{H}!}$$\scriptstyle{\sim}$$\scriptstyle{\pi_{\lambda
v}!}$$\scriptstyle{\Xi}$$\scriptstyle{H}$$\scriptstyle{\mathrm{id}}$ (5.1)

We can now view the holonomy interpretation of the map on eigenstates as
follows. Again, we fix a family $H$ and a loop $\gamma$ in $N(H)$ based at
some reference $x_{0}$. We then pick an eigenframe $\tilde{\psi}$ of
$H(x_{0})$, which fixes the tuple $\tilde{E}$ of corresponding energies. That
is, we pick a point $\big{(}x_{0},\tilde{E},\tilde{\psi}\big{)}$ in the fiber
of $\mathrm{EigFr}(V)$ above $x_{0}$. This point fixes a unique lift of
$\gamma$ to $\mathrm{EigFr}(V)$, which traces all states in the tuple $\psi$
simultaneously. The end point of this lift then equals
$(\tilde{z},\sigma)\big{(}x_{0},\tilde{E},\tilde{\psi}\big{)}$, where
$(\tilde{z},\sigma)\in\mathbb{C}^{\times}\wr~{}I_{n}$ we obtained as before.
Hence it lies in the holonomy group

$\operatorname{Hol}^{\mathrm{EigFr}(H)}\big{(}x_{0},\tilde{E},\tilde{\psi}\big{)}=\big{\\{}(\tilde{z},\sigma)\in\mathbb{C}^{\times}\wr~{}I_{n}\mid\big{(}x_{0},\tilde{E},\tilde{\psi}\big{)}\sim(\tilde{z},\sigma)\big{(}x_{0},\tilde{E},\tilde{\psi}\big{)}\big{\\}},$

where the equivalence relation $\sim$ means that the points can be connected
by a horizontal path in $\mathrm{EigFr}(H)$, or equivalently, that there is a
path in $N(H)$ whose lift to $\mathrm{EigFr}(H)$ through one point ends at the
other. Again, we consider the holonomy group at a point in the total space
rather than in the base space.

Clearly, $\sigma$ is the same permutation as we would find when looking only
at the energies. This we already found in Proposition 4.18, which is based on
the map $\pi_{v}\colon\operatorname{Eig}(V)\to\operatorname{Spec}(V)$.
However, we can also phrase this on the frame bundles as follows. Namely, the
frame bundle functor turns $\pi_{v}$ into the bundle map
$\pi_{v}!\colon\mathrm{EigFr}(V)\to\mathrm{E}(V)$,
$\big{(}A,\tilde{\lambda},\tilde{f}\big{)}\mapsto\big{(}A,\tilde{\lambda}\big{)}$.
This map is equivariant w.r.t. the canonical quotient
$\mathbb{C}^{\times}\wr~{}I_{n}\to S_{n}$, and is compatible with the
connections. Fixing a point $\big{(}A,\tilde{\lambda},\tilde{f}\big{)}$, this
quotient reappears as a map on holonomy groups
$\operatorname{Hol}^{\mathrm{EigFr}(V)}\big{(}A,\tilde{\lambda},\tilde{f}\big{)}\to\operatorname{Hol}^{\mathrm{E}(V)}\big{(}A,\tilde{\lambda}\big{)}$.
This map simply extracts $\sigma$, which confirms that $\sigma$ is the
holonomy from $\mathrm{E}(V)$. We summarize the above findings as follows.

######  Lemma 5.8.

Given a loop $\gamma$ in $N(H)$ based at $x_{0}$, the element
$(\tilde{z},\sigma)$ expressing $P_{\gamma}$ w.r.t. the eigenframe
$\tilde{\psi}$ of $H(x_{0})$ is the holonomy element at the point
$(x_{0},\tilde{\psi})$ in the frame bundle $\mathrm{EigFr}(H)$. Writing
$\tilde{E}$ for the corresponding ordering of $\operatorname{Spec}(H(x_{0}))$,
$\sigma$ is the permutation representing $p_{\gamma}$ w.r.t. $\tilde{E}$.

The element $(\tilde{z},\sigma)$ thus lies in the group
$\operatorname{Hol}^{\mathrm{EigFr}(H)}(x_{0},\tilde{\psi})$. We can view the
picking of $\tilde{\psi}$ as a gauge freedom, just like the picking of
$\tilde{E}$ earlier. A change of gauge thus corresponds to switching holonomy
group, with the usual similarity transformation of the holonomy matrix. The
gauge invariant in this case is the map $P_{\gamma}$. Indeed, $P_{\gamma}$ is
defined without using any gauge, hence is trivially independent of gauge.

Let us then take a closer look at the phase factors $\tilde{z}$ appearing in
the holonomy matrix. Given the parallel transport formalism, each of these may
be called a “geometric” phase factor. However, we also see that they are gauge
dependent. Namely, the order gets permuted by changing the order of the
eigenframe $\tilde{\psi}$, and scaling of the individual states in
$\tilde{\psi}$ affects the phase factors in $\tilde{z}$ corresponding to non-
cyclic evolution. On the other hand, $\tilde{z}$ is subject to constraints
coming from the characteristic phases of $P_{\gamma}$. Parametrizing
$z_{j}=\exp({\rm i}\alpha_{j})$, if we follow a state until it returns to its
original ray, its acquired phase is $\sum_{\text{cycle}}\alpha_{i}$, where we
sum over the indices belonging to the corresponding cycle. Hence
$\sum_{\text{cycle}}\alpha_{i}$ must be equal to the characteristic phase of
the cycle, independent of the chosen gauge. Note however that this does not
yield a canonical phase for a non-cyclic state after following $\gamma$ once;
in one gauge all $\alpha_{i}$ in the cycle could be equal, whereas in another
all except one could vanish.

In terms of the holonomy matrix, we can express this as follows. By permuting
$\tilde{\psi}$ the phase factors can be moved within the matrix, which is
subject to the underlying cycle structure of the permutation. The matrix is
block-diagonal if and only if states from the same minimal union are listed
adjacently. The size of a block is $k\times k$, with $k$ the length of the
corresponding cycle. The $k$ individual phase factors within this block can be
gauged to arbitrary values, with the only condition that their product is
fixed.

Let us treat in more detail how a seemingly Abelian connection $\omega_{H}!$
can still induce non-Abelian holonomy. That is, given the technique of path-
ordered exponential and the diagonal form of $\omega_{H}!$, one may wonder how
the end result can be non-diagonal. In short, the reason is that if the loop
$\gamma$ lifts to non-cyclic states, then any instantaneous eigenframe returns
with a twist, and this twist yields an additional permutation matrix making
the theory non-Abelian. That is, in order to write $\omega_{H}!$ as a diagonal
gauge potential, in practice one would need to extend the initial eigenframe
$\tilde{\psi}$ to a path of instantaneous eigenframes over $\gamma$. Upon
returning to $x_{0}$, one cannot use the initial eigenframe $\tilde{\psi}$
again due to the presence of non-cyclic states. Consequently, one needs to
correct for the difference between final and initial eigenframe. This basis
transformation can be expressed by an element in
$\mathbb{C}^{\times}\wr~{}I_{n}$, and thus permutation matrices enter the end
result. Naturally, this issue can only occur when $\mathrm{E}(H)$ is non-
trivial and $\gamma$ is a non-contractible loop, and so is completely absent
in case $\pi_{1}(N(H),x_{0})$ is trivial.

Let us also compare $\mathrm{EigFr}(V)$ to earlier work in [21], which is
likewise aimed to provide a holonomy description for geometric phases in the
presence of EPs. There, one considers cyclic states under the adiabatic
evolution due to a change of a Hamiltonian, which may be non-Hermitian. The
general geometric model consists of $n$ complex line bundles
$L^{1},\dots,L^{n}$ over $\mathrm{E}(V)$, there called
$\widetilde{\mathfrak{M}}$, where each line bundle corresponds to a certain
self-chosen part of the energy bands. A main disadvantage of this approach is
that the connection information is spread over the $L^{k}$. Hence, calculation
of a lift can in general not be done in a single $L^{k}$, i.e., multiple line
bundles need to be used. That is, formally the $L^{k}$ do not support a
holonomy interpretation for geometric phases in the presence of EPs. In
contrast, we found that $\mathrm{EigFr}(V)$ is a single bundle allowing to
lift a path directly from $N(V)$, which can moreover be used to treat non-
cyclic states.

It is also the case that the theory of $\mathrm{EigFr}(V)$ provides a
geometric model for the off-diagonal phases reported by Manini and Pistolesi
[20], and extends the generalized permutation matrices found by Tanaka, Cheon
and Kim [35]. The generalization of their work to non-Hermitian Hamiltonians
can be obtained from $\mathrm{EigFr}(H)$ in the following way.

First, let us clarify that there are 3 main classes of paths that are
considered in this theory. In the above, we considered loops $\gamma$ only,
i.e., the final point equals the initial point $x_{0}$. In contrast, [20]
starts by considering general paths, so that the final point $x_{1}$ need not
equal $x_{0}$. They then consider a more special case, which is also treated
in [35] and there called an “adiabatic loop”. This term refers to a path
$\gamma$ for which $x_{1}$ need not equal $x_{0}$, but still
$\operatorname{Eig}(H(x_{0}))=\operatorname{Eig}(H(x_{1}))$ (as subsets of
$V$, strictly speaking not as fibers of $\operatorname{Eig}(H)$). We thus
observe that an “adiabatic loop” need not be a “loop” in the sense we use, and
forms an intermediate subset:

$\left\\{\begin{gathered}\text{loops;}\\\
x_{0}=x_{1}\end{gathered}\right\\}\subset\left\\{\begin{gathered}\text{``adiabatic
loops'';}\\\
\operatorname{Eig}(H(x_{0}))=\operatorname{Eig}(H(x_{1}))\end{gathered}\right\\}\subset\left\\{\begin{gathered}\text{general
paths;}\\\ \text{no relation }x_{0},x_{1}\end{gathered}\right\\}\\!.$

Let us now demonstrate that $\mathrm{EigFr}(V)$ also describes the theory of
the general paths, so that we obtain the “adiabatic loops” as a special case.
Let $\gamma$ be any path in $N(H)$, so belonging to the most right set in the
above comparison. Lifting to $\mathrm{EigFr}(V)$, it induces the parallel
transport map

$\tilde{P}_{\gamma}\colon\
\mathrm{EigFr}(H(x_{0}))\to\mathrm{EigFr}(H(x_{1})).$

More explicitly, given an initial eigenframe $\tilde{\psi}$ of $H(x_{0})$, one
obtains a lift $\tilde{\Gamma}$ of $\gamma$ starting at $\tilde{\psi}$, and
the final frame is $\tilde{\psi}^{\prime}=\tilde{P}_{\gamma}(\tilde{\psi})$.
However, as $\tilde{\psi}^{\prime}$ and $\tilde{\psi}$ are (generally) not in
the same fiber, the two frames cannot be compared using the bundle structure
of $\mathrm{EigFr}(H)$. The solution is to consider the frames by themselves;
one neglects the fact that they are associated to different values of the
system parameters. Clearly, any two frames of $V$ are related by a unique
matrix; in this case there is a unique $U\in\mathrm{GL}(n,\mathbb{C})$ such
that

$\tilde{\psi}^{\prime}=U\tilde{\psi}.$

Mathematically, this argument is captured by employing the canonical
projection $\mathrm{EigFr}(V)\to\operatorname{Fr}(V)$, and using that
$\operatorname{Fr}(V)$ is a $\mathrm{GL}(n,\mathbb{C})$-torsor. We also remark
that if both $\tilde{\psi}$ and $\tilde{\psi}^{\prime}$ are orthonormal w.r.t.
some inner product, then $U$ is indeed unitary, and can be calculated by
taking inner products between the states, as used in [20].

Of course, the matrix $U$ is not unique; if we change $\tilde{\psi}$ to
$G\tilde{\psi}$, with $G\in\mathbb{C}^{\times}\wr~{}I_{n}$ a generalized
permutation matrix, then
$\tilde{\psi}^{\prime}=\tilde{P}_{\gamma}\big{(}\tilde{\psi}\big{)}$ becomes
$\tilde{P}_{\gamma}\big{(}G\tilde{\psi}\big{)}=G\tilde{P}_{\gamma}\big{(}\tilde{\psi}\big{)}=G\tilde{\psi}^{\prime}$,
and so $U$ becomes $GUG^{-1}$. That is, $U$ is unique up to conjugation by
$\mathbb{C}^{\times}\wr~{}I_{n}$. When looking for invariants, if we write
$G=(\tilde{g},\tau)$, we may set $\tau=\mathrm{id}$; non-trivial $\tau$ simply
rearrange the elements of $U$. It follows that $U_{ij}$ becomes
$g_{i}U_{ij}g_{j}^{-1}$. Hence, one obtains an invariant quantity for each
cycle in $S_{n}$ by taking the corresponding product, e.g., $(12)$ yields the
invariant $U_{12}U_{21}$ and $(123)$ yields the invariant
$U_{12}U_{23}U_{31}$, in addition to the diagonal elements $U_{kk}$ which are
also invariant. Whenever such a product is non-zero, its argument is a gauge-
invariant phase, as reported in [20].

The “adiabatic loops” are then the special case where $U$ is also a
generalized permutation matrix, which was studied further in [35]. The
similarity of this case with our study of the holonomy on $\mathrm{EigFr}(V)$
is straightforward to explain; in both cases
$\mathrm{EigFr}(H(x_{0}))=\mathrm{EigFr}(H(x_{1}))$, so that we are studying
an automorphism of a semi-torsor. The method above, based on the matrix $U$,
then reduces to the argument of Section 5.1, and invariant phases follow from
Proposition 4.16.

##  6 Discussion

In this paper we introduced a framework that facilitates the mathematical
description of the adiabatic quantum mechanics of finite-dimensional, non-
degenerate but otherwise arbitrary and hence not necessarily Hermitian
Hamiltonians. We started with a preparatory study of the space of non-
degenerate operators, which was summarized in diagram (2.6). The following
step was to find the geometry behind the adiabatic change of energies of a
Hamiltonian family $H$. This is captured by the covering space formed by the
energy bands, where we restrict to system parameter values where $H$ is non-
degenerate. Moreover, the specific covering needed for $H$ we found as the
pull-back of an abstract yet explicit model covering, which makes the covering
property and hence bundle structure immediate for any smooth Hamiltonian
family $H$.

We continued by introducing a bundle for the eigenstates. A remarkable
property of this bundle is that it is not a principal bundle, but has the
structure of a semi-principal bundle, which is described in [28]. That is,
each of its fibers consists of a collection of eigenrays rather than a single
eigenray. We found a natural connection on this bundle, which provides a
parallel transport description of the adiabatic geometric phase as found by
Garrison and Wright [12]. Because of the semi-principal structure, this
incorporates non-cyclic states as well. This formalism hence extends previous
holonomy formulations, which are based on principal bundles. Among these are
the first holonomy interpretation reported by Simon [32], which treats the
Hermitian case with separate energy bands, and the formalism in [21], which
considered the non-Hermitian case restricted to cyclic states. We furthermore
found that the Aharonov–Anandan formalism [1] for general Hermitian systems
can be obtained by a reduction from the present formalism using a given inner
product. Moreover, we found that the dynamical phase can be included as a non-
geometric addition, which means that the full adiabatic evolution of a state
can be calculated naturally on this bundle. We then obtained a generalized
quantum geometric tensor, which we found to be directly related to covariant
derivatives.

We highlight that the presented formalism, as it allows one to also describe
non-cyclic states by means of parallel transport, is a natural formalism for
the study of exceptional points. The description including geometric phases is
then given on the full eigenstate bundle, whereas the permutations of energies
can already be accurately described with the monodromy theory of the energy
band covering. Both of these spaces have a semi-principal bundle structure,
which is the key to incorporate non-cyclic behavior. However, by switching to
a multi-state approach, one can describe the permutations of energies and the
accompanying geometric phase factors simultaneously in a more explicit way,
namely via holonomy matrices. This is given by a straightforward application
of the frame bundle technique as described in [28]. This provides a more
explicit holonomy description of state evolution, also in the non-cyclic case
enabled by exceptional points.

##  Appendix A Bundle structure of tuples of distinct complex numbers

If one considers $n$ distinct complex numbers, one can do so with or without
an ordering of these numbers. Let us start with the ordered one. This brings
us to the space

$\mathbb{C}^{\underline{n}}=\big{\\{}(z_{1},\dots,z_{n})\in\mathbb{C}^{n}\mid
i\neq j\implies z_{i}\neq z_{j}\big{\\}},$

where we borrow the notation of falling factorials; if we would replace
$\mathbb{C}$ by a finite set $F$, then
$|F^{\underline{n}}|=|F|(|F|-1)\cdots(|F|-n)$, which in combinatorics is known
as the falling factorial $|F|^{\underline{n}}$. The space
$\mathbb{C}^{\underline{n}}$ is a submanifold of $\mathbb{C}^{n}$, and more is
true.

######  Lemma A.1.

The space $\mathbb{C}^{\underline{n}}$ is an $($algebraic$)$ open and dense
submanifold of $\mathbb{C}^{n}$.

###### Proof.

For $i\neq j$, define the function $m_{ij}\colon\mathbb{C}^{n}\to\mathbb{C}$,
$(z_{1},\dots,z_{n})\mapsto z_{i}-z_{j}$, and write
$D(m_{ij})=\big{\\{}(z_{1},\dots,z_{n})\in\mathbb{C}^{n}\mid
m_{ij}(z_{1},\dots,z_{n})\neq 0\big{\\}}$. Then
$\mathbb{C}^{\underline{n}}=\cap_{i\neq j}D(m_{ij})$ expresses the space as a
finite intersection of (algebraic) open subsets, hence is open in
$\mathbb{C}^{n}$ and so a submanifold. It is also dense; given a tuple with
repeated numbers, these become unequal by an arbitrarily small perturbation. ∎

We can now proceed to sets of $n$ distinct complex numbers, which is the
unordered variant of the ordered tuple case as just discussed. The idea is to
view the unordered sets as a quotient of the tuple space. This quotient is
specified by a permutation action; upon identifying rearranged tuples, clearly
the ordering vanishes. The action is the usual permutation of slots, i.e., for
$\sigma\in S_{n}$ and $(z_{1},\dots,z_{n})\in\mathbb{C}^{\underline{n}}$ we
have

$\sigma\cdot(z_{1},\dots,z_{n})=(z_{\sigma^{-1}(1)},\dots,z_{\sigma^{-1}(n)}).$

This action is smooth as it is the restriction of the smooth permutation
action on $\mathbb{C}^{n}$ to an invariant submanifold. Taking the quotient
results in the space of subsets of $\mathbb{C}$ with $n$ distinct complex
numbers, which we write as

$\binom{\mathbb{C}}{n}:=\mathbb{C}^{\underline{n}}/S_{n}=\big{\\{}S\subset\mathbb{C}\mid|S|=n\big{\\}}.$

Again, we borrow our notation from combinatorics. We write $q$ for the
quotient map $\mathbb{C}^{\underline{n}}\to\binom{\mathbb{C}}{n}$. Clearly,
the fiber of $q$ above a certain set in $\binom{\mathbb{C}}{n}$ consists of
all the possible orderings of its elements. Via $q$ one obtains a unique
manifold structure on $\binom{\mathbb{C}}{n}$, and in this way $q$ is
principal bundle.

######  Proposition A.2.

There is a unique manifold structure on $\binom{\mathbb{C}}{n}$ such that the
quotient map $q\colon\mathbb{C}^{\underline{n}}\to\binom{\mathbb{C}}{n}$ is a
surjective submersion. The permutation action on $\mathbb{C}^{\underline{n}}$
thus induces the principal bundle

${S_{n}}$${\mathbb{C}^{\underline{n}}}$${\binom{\mathbb{C}}{n}.}$$\scriptstyle{q}$

Moreover, this bundle is non-trivial for $n>1$.

###### Proof.

Clearly the $S_{n}$-action on $\mathbb{C}^{\underline{n}}$ is free and proper,
hence the quotient map defines a principal bundle. Assume it is trivial for
$n>1$ for contradiction. This would imply that $\mathbb{C}^{\underline{n}}$
and $\binom{\mathbb{C}}{n}\times S_{n}$ are homeomorphic. However, as
$\mathbb{C}^{\underline{n}}$ and $\binom{\mathbb{C}}{n}$ are path-connected
but $S_{n}$ is not, this cannot be. ∎

The non-triviality of the bundle is related to the topology of $\mathbb{C}$
not being compatible with a total ordering. To expand on this, observe that
for any manifold $M$ one can define the space of distinct tuples
$M^{\underline{n}}$. This similarly has a $S_{n}$-action defined on it, and
one obtains a principal bundle over the space $\binom{M}{n}$ of unordered
subsets of $n$ elements. If $M$ has an order topology given by a total
ordering, such as $\mathbb{R}$, this bundle is trivial. Indeed, given any
element $S\in\binom{M}{n}$, then there is a standard way to list the elements
of $S$, e.g., from lowest to highest. This induces triviality as follows.

######  Proposition A.3.

If a manifold $M$ is orderable, then the principal $S_{n}$-bundle
$M^{\underline{n}}\to\binom{M}{n}$ is trivial for all $n$.

###### Proof.

Let $\leq$ be a total ordering on $M$ compatible with the topology. Define an
ordering map $o\colon\binom{M}{n}\to M^{\underline{n}}$ by the condition that

$o(\\{m_{1},\dots,m_{n}\\})=(m_{1},\dots,m_{n}),\qquad\text{where}\quad
m_{1}<\dots<m_{n}.$

As all the $m_{i}$ are distinct this is well-defined, and by assumption on the
topology of $M$ also continuous. It follows that $o$ is a global section of
the principal bundle, which is thus trivial. ∎

### Acknowledgements

The authors thank the anonymous referees whose careful remarks contributed to
the quality of the paper.

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