# A Uniqueness Result for the Calderón Problem for $U(N)$-connections coupled
to spinors

Carlos Valero

###### Abstract

In this paper we define a Dirichlet-to-Neumann map for a twisted Dirac
Laplacian acting on bundle-valued spinors over a spin manifold. We show that
this map is a pseudodifferential operator of order $1$ whose symbol determines
the Taylor series of the metric and connection at the boundary. We go on to
show that if two real-analytic connections couple to a spinor via the
Yang–Mills–Dirac equations with appropriate boundary conditions, and have
equal Dirichlet-to-Neumann maps, then the two connections are locally gauge
equivalent. In the abelian case, the connections are globally gauge
equivalent.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Spinors and the Dirac Operator
    2. 2.2 $U(N)$-connections and bundle-valued spinors
    3. 2.3 Pseudodifferential Calculus
  3. 3 Boundary determination
  4. 4 Recovering real-analytic Yang–Mills–Dirac connections
  5. Acknowledgements

##  1 Introduction

In this paper, we consider a Calderón inverse problem for unitary connections
on Hermitian vector bundles over spin manifolds that couple to spinor fields
via the Yang–Mills–Dirac system. In particular, we consider the Dirichlet-to-
Neumann map for the twisted Dirac Laplacian acting on vector-valued spinors,
and investigate how the introduction of the spin structure affects the
recovery of the metric and connection from boundary data.

The Calderón problem has its origin in the physical question of whether one
can determine the conductivity of a medium by making measurements on the
boundary of potential functions and the induced currents. Geometrically, this
corresponds to the question of whether one can determine the metric on a
manifold with boundary, up to isometry, from knowledge of its Dirichlet-to-
Neumann map, which sends a function on the boundary to the normal derivative
of its harmonic extension. Much work has since been done on the Calderón
problem for the scalar Laplacian; we refer the reader to [18], or §1 of the
more recent [5], for a survey of uniqueness results in the literature.

Many natural extensions of this problem arise when one considers other
important second-order elliptic operators induced by some geometric structure.
For example, one may fix a vector bundle $E$ over a Riemannian manifold
$(M,g)$ with boundary, and consider a connection $\nabla$ on this vector
bundle. One may then ask to what extent the Dirichlet-to-Neumann map for the
connection Laplacian $\nabla^{*}\nabla$ determines the connection, up to gauge
equivalence. This problem has been studied in a few recent works, in which a
number of uniqueness results are proved. In [11], a Riemannian manifold,
Hermitian vector bundle, and connection are reconstructed from the hyperbolic
Dirichlet-to-Neumann map associated to the wave equation of the connection
Laplacian. In [3], the elliptic Dirichlet-to-Neumann map is considered, and it
is shown using methods of complex geometrical optics that the Dirichlet-to-
Neumann map for a connection Laplacian determines the connection up to gauge
for a class of vector bundles over special Riemannian manifolds, namely
conformally transversally anisotropic manifolds with injective ray transform.
Using new methods of geometric analysis and Runge approximation, Cekić shows
in [4] that a Hermitian vector bundle and Yang–Mills connection can be
recovered, up to gauge transformations, from the Dirichlet-to-Neumann map of
its connection Laplacian. Finally, in the recent preprint [7], the authors
reconstruct a Euclidean vector bundle and connection from the connection
Laplacian Dirichlet-to-Neumann map when all of the data is real-analytic and
the dimension of $M$ is at least $3$.

On the other hand, inverse boundary problems for first-order Dirac operators
have also been studied in the literature. One important uniqueness result is
due to Kurylev and Lassas [10] who showed that a Riemannian manifold and
super-vector bundle can be recovered from the spectrum and eigenfunctions of
the corresponding Dirac operator on the boundary. There are also many works
that consider the problem of recovering magnetic potentials from boundary data
corresponding to first-order Dirac operators. Much in the spirit of the
present paper, Salo and Tzou [14, 15] have shown, using the method of limiting
Carleman weights, that a potential and magnetic field can be recovered from
the Cauchy data of an associated Dirac equation over a compact domain in
$\mathbb{R}^{n}$. In the terminology used in the present paper, the magnetic
field corresponds to the curvature of an abelian connection.

In this paper, we consider the question of determining a $U(N)$-connection $A$
up to gauge equivalence, from the Dirichlet-to-Neumann map of the twisted
Dirac Laplacian $\operatorname{\not{\mathrm{D}}}_{A}^{2}$. More precisely, we
consider a compact spin manifold $M$ with boundary $\partial M$, and a
Hermitian vector bundle $E$ over $M$. Then the Dirac operator
$\operatorname{\not{\mathrm{D}}}$ of $M$ can be defined, acting on its bundle
of complex spinors $S$. Now given any connection $A$ on $E$, we may endow the
bundle $S\otimes E$ with the connection $\omega^{\mathrm{s}}\otimes A$, where
$\omega^{\mathrm{s}}$ is the spin connection on $S$, induced by the Levi-
Civita connection. With this connection, we may define a twisted, or
covariant, Dirac operator $\operatorname{\not{\mathrm{D}}}_{A}$, which like
$\operatorname{\not{\mathrm{D}}}$ is a first-order, elliptic, and self-adjoint
operator acting on sections of $S\otimes E$.

In order for the Dirichlet problem to be well-defined, we consider the square
of this operator, $\operatorname{\not{\mathrm{D}}}_{A}^{2}$. The Dirichlet-to-
Neumann map associated to $\operatorname{\not{\mathrm{D}}}_{A}^{2}$ can be
thus defined by sending any section $\chi$ of $S\otimes E|_{\partial M}$ to
the covariant normal derivative of its harmonic extension with respect to
$\operatorname{\not{\mathrm{D}}}_{A}^{2}$. It is straightforward to generalize
this to include non-zero mass terms and zeroth-order potentials, provided that
the Dirichlet-problem is well-defined. We summarize with the following
definition:

######  Definition 1.1.

Let $M$ be an $n$-dimensional compact spin manifold with boundary $\partial
M$, and let $g$ be a Riemannian metric on $M$. Let $S$ be the spinor bundle
associated to some fixed spin structure on $M$, and let $E$ be a Hermitian
bundle of rank $N$ on $M$. Let $A$ denote a $U(N)$-connection on $E$, and let
$Z$ be an endomorphism of $S\otimes E$. For any $m\in\mathbb{R}$ such that
$m^{2}\notin\operatorname{Spec}{\left(\operatorname{\not{\mathrm{D}}}_{A}^{2}+Z\right)}$,
we define the Dirichlet-to-Neumann map

$\Lambda_{g,A,Z,m}:C^{\infty}\left(S\otimes E|_{\partial M}\right)\to
C^{\infty}\left(S\otimes E|_{\partial M}\right)$ (1)

as follows. For $\chi\in C^{\infty}\left(S\otimes E|_{\partial M}\right)$, we
may solve the Dirichlet problem

$\begin{cases}\operatorname{\not{\mathrm{D}}}_{A}^{2}\varphi+Z\varphi-m^{2}\varphi=0,\\\
\varphi|_{\partial M}=\chi,\end{cases}$ (2)

to obtain a unique solution $\varphi\in C^{\infty}(S\otimes E)$. We then
define $\Lambda_{g,A,Z,m}(\chi):=\nabla^{A}_{\nu}\varphi\big{|}_{\partial M}$
where $\nu$ is the inward unit normal to $\partial M$.

We will often suppress subscripts on the Dirichlet-to-Neumann map that are
understood to be fixed, and indicate only the relevant ones. For example, when
we recover the connection with the background metric, mass parameter, and
endomorphism fixed in Section 4, the Dirichlet-to-Neumann map is simply
denoted $\Lambda_{A}$.

######  Remark 1.2.

The Dirichlet-to-Neumann map can be extended to a map

$\Lambda_{g,A,Z,m}:H^{\frac{1}{2}}\left(S\otimes E|_{\partial M}\right)\to
H^{-\frac{1}{2}}\left(S\otimes E|_{\partial M}\right),$ (3)

where for any vector bundle $\mathcal{E}$ and $s\in\mathbb{R}$,
$H^{s}(\mathcal{E})$ denotes the Hilbert space of $s$-Sobolev sections of
$\mathcal{E}$. That is, $H^{s}(\mathcal{E})$ denotes the space of
distributional sections of $\mathcal{E}$ that are represented by a tuple of
$H^{s}$ functions in any smooth local trivialization. Indeed, there is a
natural weak formulation of Definition 1.1 if we introduce a modified
Dirichlet-to-Neumann map $\hat{\Lambda}_{g,A,Z,m}$ as follows: for any
$\chi\in H^{\frac{1}{2}}\left(S\otimes E|_{\partial M}\right)$, we can again
solve (2) to obtain a unique $\varphi\in H^{1}(S\otimes E)$. We then define
$\hat{\Lambda}_{g,A,Z,m}(\chi)\in H^{-\frac{1}{2}}(S\otimes E|_{\partial M})$
by the property that

$\left\langle\hat{\Lambda}_{g,A,Z,m}(\chi),\zeta\right\rangle=\int_{M}\left\langle\operatorname{\not{\mathrm{D}}}_{A}\varphi,\operatorname{\not{\mathrm{D}}}_{A}\psi\right\rangle-\int_{M}\left\langle\left(Z-m^{2}\right)\varphi,\psi\right\rangle$
(4)

holds for all $\zeta\in H^{\frac{1}{2}}(S\otimes E|_{\partial M})$, where
$\psi\in H^{1}(S\otimes E)$ is any extension of $\zeta$, which exists by the
standard trace theorems for Sobolev spaces [2, §11]. In the smooth case, the
Green’s formula for the Dirac operator [12, Equation 5.7]

$\int_{M}\left\langle\operatorname{\not{\mathrm{D}}}_{A}\sigma_{1},\sigma_{2}\right\rangle=\int_{M}\left\langle\sigma_{1},\operatorname{\not{\mathrm{D}}}_{A}\sigma_{2}\right\rangle-\int_{\partial
M}\left\langle\nu\cdot\sigma_{1},\sigma_{2}\right\rangle$ (5)

implies that
$\hat{\Lambda}_{g,A,Z,m}(\chi)=-\gamma(\nu)\operatorname{\not{\mathrm{D}}}_{A}\varphi|_{\partial
M}$, where $\gamma(\nu)$ denotes Clifford multiplication by $\nu$, and so
differs from $\Lambda_{g,A,Z,m}(\chi)$ as given in Definition 1.1 by
tangential derivatives. In particular, we may say that
$\hat{\Lambda}_{g,A,Z,m}$ and $\Lambda_{g,A,Z,m}$ contain the same information
about the geometric data, such as the metric and connection. In this paper, we
thus are free to restrict ourselves to considering $\Lambda_{g,A,Z,m}$.

We note that there are some natural gauge invariances that arise from
Definition 1.1, which shall be explored in greater detail in Section 2. The
first is the gauge-invariance of the connection $A$. Recall that two
connections $A$ and $A^{\prime}$ are called gauge equivalent if there exists a
unitary automorphism $G$ of $E$, otherwise called a gauge transformation, such
that their covariant derivatives are related by

$\nabla^{A^{\prime}}=G^{-1}\circ\nabla^{A}\circ G.$ (6)

We say that $A$ and $A^{\prime}$ are locally gauge equivalent about a point
$x\in M$ if there is an open neighbourhood $U$ of $x$ such that the
restrictions of $A$ and $A^{\prime}$ to $U$ are gauge equivalent.

It is easy to see that if there exists a $G$ as in (6) with $G|_{\partial
M}=\operatorname{id}$, then $\Lambda_{A^{\prime}}=\Lambda_{A}$. Indeed, if
$\varphi$ is the solution to (2) for $A$, then
$\left(\operatorname{id}_{S}\otimes G\right)^{-1}\varphi$ is the solution to
(2) with $A$ replaced by $A^{\prime}$. Therefore, suppressing
$\operatorname{id}_{S}$, we have

$\Lambda_{A^{\prime}}(\chi)=\nabla_{\nu}^{A^{\prime}}\left(G^{-1}\varphi\right)\big{|}_{\partial
M}=G^{-1}\nabla_{\nu}^{A}\varphi\big{|}_{\partial M}=\Lambda_{A}(\chi).$ (7)

Thus, given $\Lambda_{A}=\Lambda_{A^{\prime}}$, we can only every recover the
connection up to a gauge transformation that is equal to the identity on the
boundary.

Going further, we would like to say that the definition 1.1, inasmuch as it
depends on the geometry of the metric $g$, depends only on the isometry class
of $g$. However, since spin structures, and hence spinor bundles, are defined
with respect to a fixed metric, we must take care in relating the Dirichlet-
to-Neumann maps of two different metrics. In Section 2, we explain how a
diffeomorphism $\Phi:M\to M$ induces an isomorphism of associated spinor
bundles $\tilde{\Phi}:S_{\Phi^{*}g}\to S_{g}$. Then it is easy to prove that
the Dirichlet-to-Neumann map is diffeomorphism-invariant:

######  Lemma 1.3.

Let $\Phi:M\to M$ be a diffeomorphism such that $\Phi|_{\partial
M}=\operatorname{id}$. Then

$\Lambda_{\Phi^{*}g}=\tilde{\Phi}^{-1}|_{\partial
M}\circ\Lambda_{g}\circ\tilde{\Phi}|_{\partial M}.$ (8)

In this paper, we want to consider a Calderón Problem for a $U(N)$-connection
$A$, which couples to an $E$-valued spinor $\phi$ through some natural
equations arising from physics. To this end, recall that given a connection
$A$, one may define its curvature $F_{A}$. If $P$ denotes the principal
$U(N)$-bundle of unitary frames of $E$, then $F_{A}$ is an
$\operatorname{ad}{P}$-valued $2$-form. In physics, the curvature of a
connection corresponds to a force field; if the connection is abelian then its
curvature is the electromagnetic field. Now let $m\in\mathbb{R}$ be such that
$m^{2}$ is not in the Dirichlet spectrum of
$\operatorname{\not{\mathrm{D}}}_{A}^{2}$. We then assume that there exists an
$E$-valued spinor $\phi$ such that $(A,\phi)$ satisfies the following second-
order Yang–Mills–Dirac system:

$\begin{cases}\operatorname{\not{\mathrm{D}}}_{A}^{2}\phi=m^{2}\phi,\\\
d_{A}^{*}F_{A}=J(\phi),\end{cases}$ (9)

where the current $J(\phi)$ is an $\operatorname{ad}{P}$-valued $1$-form
depending quadratically o $\phi$. The system (9) arises when a gauge field
interacts with fermionic matter, as represented by the spinor field $\phi$. In
this paper we prove the following result:

######  Theorem 1.4.

Let $(M,g)$ be a compact $n$-dimensional real-analytic spin manifold with
boundary, and let $E$ be a real-analytic Hermitian vector bundle over $M$. Let
$A$ and $B$ be $U(N)$-connections on $E$, real-analytic in the interior of
$M$, and let $\phi$ and $\psi$ be $E$-valued spinors on $M$ such that
$(A,\varphi)$ and $(B,\psi)$ both satisfy the second-order Yang–Mills–Dirac
system (9), and such that $\varphi|_{\partial M}=\psi|_{\partial M}$. If
$\Lambda_{A,m}=\Lambda_{B,m}$, then $A$ and $B$ are locally gauge equivalent
about every point in $M$. If $N=1$, then $A$ and $B$ are globally gauge
equivalent.

In other words, if a connection $A$ couples to an auxiliary $E$-valued spinor
via the second-order Yang–Mills–Dirac system, then by making boundary
measurements $\Lambda_{A}(\chi)$ of other spinor fields, one can determine the
curvature of the connection at any point up to conjugation. In the abelian
case, one can determine the connection up to gauge.

##  2 Preliminaries

In this section, we recall some of the basic tools and constructions that we
need to state the central problem and prove the main result. We review the
construction of spinor bundles and the Dirac operator in Section 2.1, and
extend this to bundle-valued spinors in Section 2.2. Lastly, we review a few
key results from the theory of pseudodifferential operators in 2.3.

###  2.1 Spinors and the Dirac Operator

In this subsection, we briefly recall some basic notions of spin geometry,
such as Clifford algebras, Spin manifolds, the construction of the spinor
bundle on a Spin manifold, and the definition and fundamental properties of
Dirac operators. The important results for our purposes are the local formulae
(13) for the spin connection, and (14) for the Dirac operator. For details on
the results presented in this section, we refer the reader to [12].

######  Definition 2.1.

The Clifford algebra $\operatorname{C\ell}_{n}$ is the real associative
algebra generated by vectors in $\mathbb{R}^{n}$, with a product $\cdot$
satisfying the relation

$v\cdot w+w\cdot v=-2\left\langle v,w\right\rangle$ (10)

for all $v,w\in\mathbb{R}^{n}$, where $\left\langle\cdot,\cdot\right\rangle$
denotes the standard inner product on $\mathbb{R}^{n}$.

Note that we have a natural inclusion
$\mathbb{R}^{n}\subseteq\operatorname{C\ell}_{n}$, which can be extended to a
vector space isomorphism
$\Lambda^{*}\mathbb{R}^{n}\to\operatorname{C\ell}_{n}$ by fixing an
orthonormal basis $\\{e_{i}\\}_{i=1}^{n}$ of $\mathbb{R}^{n}$ and sending

$e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}\mapsto e_{1}\cdots e_{k}.$ (11)

In particular, we have $\dim{\operatorname{C\ell}_{n}}=2^{n}$.

Clifford algebras play an important role throughout geometry and physics. One
reason for this is that $\operatorname{C\ell}_{n}$ naturally contains the spin
group $\operatorname{Spin}(n)$, which can be defined as a double cover of the
rotation group $SO(n)$. That is, there exists a surjective Lie group
homomorphism $\lambda:\operatorname{Spin}(n)\to SO(n)$ such that
$\ker{\lambda}=\\{1,-1\\}$. Thus, $\lambda$ is a 2-sheeted covering map.

######  Remark 2.2.

It follows from the preceding that $\operatorname{Spin}(n)$ is in fact the
universal cover of $SO(n)$ for $n\geq 3$, since
$\pi_{1}(SO(n))=\mathbb{Z}_{2}$ for $n\geq 3$.

######  Definition 2.3.

Let $(M,g)$ be an oriented $n$-dimensional Riemannian manifold without
boundary. A spin structure $P_{\operatorname{Spin}}$ on $M$ is a principal
$\operatorname{Spin}(n)$-bundle, which covers the $SO(n)$-bundle of
orthonormal frames of $M$, in such a way that the covering map is compatible
with the $2$-fold covering map $\lambda:\operatorname{Spin}(n)\to SO(n)$. When
such a bundle $P_{\operatorname{Spin}}$ exists, we say $M$ is spin.

######  Remark 2.4.

Not every oriented Riemannian manifold is spin. In fact, the obstruction to
being spin is entirely contained in the second Stiefel-Whitney class of $M$.
Even when $M$ is spin, the spin structure need not be unique. See [12] for
details.

A spin manifold is endowed with a distinguished complex vector bundle $S$
called the spinor bundle. It is defined as follows. First, fix a spin
structure $P_{\operatorname{Spin}}$ on $M$. Let
$\rho:\operatorname{C\ell}_{n}\to\operatorname{End}{\mathbb{C}^{k}}$ be an
irreducible $\operatorname{C\ell}_{n}$-module. It follows from the structure
of $\operatorname{C\ell}_{n}$ that $k=\left[\frac{n}{2}\right]$. Then $\rho$
restricts to a representation of $\operatorname{Spin}(n)$, called the spinor
representation. The spinor bundle $S$ is then the bundle associated to
$P_{\operatorname{Spin}}$ and the spinor representation,
$S:=P_{\operatorname{Spin}}\times_{\rho}\mathbb{C}^{k}$. The sections of $S$
are called (complex) spinors. One can also define real spinors but they shall
not concern us here.

######  Remark 2.5.

The spinor representations have the property that
$-1\in\operatorname{Spin}(n)$ acts non-trivially. Since $\lambda(-1)=1$, where
$\lambda:\operatorname{Spin}(n)\to SO(n)$ is the $2$-fold covering map
described above, it follows that the spinor representations do not descend to
representations of $SO(n)$. Therefore, the spinor representations are in some
sense the simplest representations that do not correspond to representations
of $SO(n)$.

Since the typical fibre of $S$ is a $\operatorname{C\ell}_{n}$-module, it
follows that $S$ is bundle of modules over $\operatorname{C\ell}(M)$, the
Clifford bundle of $M$, which is defined by
$\operatorname{C\ell}(M)_{x}:=\operatorname{C\ell}(T_{x}M)$. In particular,
since $TM\subseteq\operatorname{C\ell}(M)$, we have a map
$\gamma:TM\to\operatorname{End}{S}$, called Clifford multiplication. It is
possible to endow $S$ with a Hermitian metric such that $\gamma(e)$ is skew-
symmetric for all unit vectors $e\in TM$. We assume $S$ has such a metric
henceforward.

So far we have presented these definitions for a spin manifold without
boundary. A spin manifold with boundary $M$ is defined to be any closed domain
of a spin manifold $N$, whose spin structure is the restriction of the spin
structure on $N$. All of the above definitions then extend to this setting.

Now, the Levi-Civita connection $\omega$ on the oriented orthonormal frame
bundle of $M$ lifts to a connection $\omega^{\mathrm{s}}$ on any given spin
structure, which we call the spin connection. This connection induces a
covariant derivative $\nabla^{\mathrm{s}}$ on sections of $S$, which can be
explicitly described with respect to a local trivialization as follows.

Let $(e_{i})_{i}$ be a local orthonormal frame for $M$, and let $\omega^{i}_{\
j}$ be the Levi-Civita connection $1$-form with respect to this frame, so that

$\nabla_{X}e_{j}=\omega^{i}_{\ j}(X)e_{i}.$ (12)

This frame can be lifted to a local section of the spin structure
$P_{\operatorname{Spin}}$, which we can regard as a local frame
$(\sigma_{\alpha})_{\alpha}$ for $S$ (although there is another lifted frame,
namely $(-\sigma_{\alpha})_{\alpha}$, the choice of lift is immaterial here).
Then, with respect to this frame of spinors, the spin connection takes the
form

$\nabla^{\mathrm{s}}\sigma_{\alpha}=-\frac{1}{2}\sum_{i<j}\omega^{i}_{\
j}\otimes\gamma(e_{i})\gamma(e_{j})\sigma_{\alpha}.$ (13)

The structure of a Clifford module allows us to define the Dirac operator
$\operatorname{\not{\mathrm{D}}}$ on sections of $S$ as follows. For
$\varphi\in\Gamma(S)$, let $(e_{i})_{i}$ be any orthonormal frame in an open
set $U$. Then, in $U$,

$\operatorname{\not{\mathrm{D}}}\varphi:=\sum_{i=1}^{n}\gamma(e_{i})\nabla^{s}_{e_{i}}\varphi.$
(14)

This definition does not depend on the choice of orthonormal frame. The Dirac
operator plays an crucial role in physics, where it occurs in the equations of
motion for spinor fields, which represent fermionic matter. The square of the
Dirac operator satisfies the famous Lichnerowicz formula,

$\operatorname{\not{\mathrm{D}}}^{2}\varphi=\left(\nabla^{\mathrm{s}}\right)^{*}\nabla^{\mathrm{s}}\varphi+\frac{1}{4}R\varphi$
(15)

where $\left(\nabla^{s}\right)^{*}$ denotes the formal adjoint of $\nabla^{A}$
with respect to the $L^{2}$-inner product, and $R$ is the scalar curvature of
$g$. Thus one can transfer questions about the Dirac Laplacian
$\operatorname{\not{\mathrm{D}}}^{2}$ to questions about the spin connection
Laplacian at the expense of a curvature term.

###  2.2 $U(N)$-connections and bundle-valued spinors

We now want to introduce an auxiliary Hermitian vector bundle $(E,h)$ equipped
with a connection $A$, whose curvature $F_{A}$ corresponds to some physical
force. Moreover, we want to introduce a mechanism to couple this connection to
spinor fields, corresponding to the physical interaction between the force
field $F_{A}$ and fermions. If the connection is abelian, then its curvature
is the familiar electromagnetic field.

Let $(E,h)$ be a Hermitian vector bundle of rank $N$, which is associated to
its bundle of complex orthonormal frames $P$, which is a principal
$U(N)$-bundle. Consider the bundle $S\otimes E$, the bundle of $E$-valued
spinors, which is associated to the principal bundle
$P_{\operatorname{Spin}}\times_{M}P$. Given a $U(N)$-connection $A$ on $P$, we
can equip $S\otimes E$ with the connection $\omega^{\text{s}}\otimes A$. The
corresponding covariant derivative on sections of $S\otimes E$ is denoted by
$\nabla^{A}$. Thus, with respect to a local trivialization of $E$, an
$E$-valued spinor $\psi$ is given by a tuple of $N$ complex spinors, and

$\left(\nabla_{X}^{A}\psi\right)^{a}=\nabla_{X}^{\mathrm{s}}\psi^{a}+A^{a}_{\
b}(X)\psi^{b}\ \ \ \ a\in\\{1,\dots,N\\},$ (16)

where $A^{a}_{\ b}$ are the components of the $\mathfrak{u}(N)$-valued
$1$-form representing the connection $A$ in this local trivialization. We can
also define a twisted Dirac operator acting on $\psi\in\Gamma(S\otimes E)$ by

$\operatorname{\not{\mathrm{D}}}_{A}\psi:=\sum_{i=1}^{n}\gamma(e_{i})\nabla^{A}_{e_{i}}\psi$
(17)

where $(e_{i})_{i}$ is any local orthonormal frame on $M$. Note that Clifford
multiplication on $E$-valued spinors acts on the $S$ factor; that is, we have
identified $\gamma(e_{i})$ with $\gamma(e_{i})\otimes\operatorname{id}$. With
respect to a local trivialization of $E$, we can view $\psi\in\Gamma(S\otimes
E)$ as $N$ complex spinors, and the twisted Dirac operator takes the form

$\left(\operatorname{\not{\mathrm{D}}}_{A}\psi\right)^{a}=\sum_{i=1}^{n}\gamma(e_{i})\nabla^{s}_{e_{i}}\psi^{a}+A^{a}_{\
b}(e_{i})\gamma(e_{i})\psi^{b},\ \ \ \ a\in\\{1,\dots,N\\}.$ (18)

The twisted Dirac operator also satisfies a twisted Lichnerowicz formula,

$\operatorname{\not{\mathrm{D}}}_{A}^{2}\psi=\left(\nabla^{A}\right)^{*}\nabla^{A}\psi+\frac{1}{4}R\psi+\mathfrak{F}_{A}\cdot\psi,$
(19)

where the curvature operator $\mathfrak{F}_{A}:S\otimes E\to S\otimes E$ is
defined by

$\mathfrak{F}_{A}(\sigma\otimes\eta):=\frac{1}{2}\sum_{j,k=1}^{n}\left(\gamma(e_{j})\gamma(e_{k})\sigma\right)\otimes\left(F_{A}(e_{j},e_{k})\eta\right).$
(20)

A gauge transformation is a section of $U(E)$, the bundle of unitary
automorphisms of $E$. These sections form a group called the gauge group,
which we denote $\mathscr{G}(E)$. A gauge transformation $G\in\mathscr{G}(E)$
acts on a connection $A$ by taking it to the connection $A^{\prime}$ defined
by

$\nabla^{A^{\prime}}=G^{-1}\circ\nabla^{A}\circ G.$ (21)

Two connections $A$ and $A^{\prime}$ are considered to be gauge equivalent if
they lie in the same $\mathscr{G}(E)$-orbit. The notion of local gauge
equivalence over an open set is similarly defined using the restricted bundles
and connections. If $A^{\prime}$ is related to $A$ by a local gauge
transformation $G$, then with respect to a local trivialization, the
connection $1$-forms representing $A$ and $A^{\prime}$ are related by

$A^{\prime}=G^{-1}AG+G^{-1}dG.$ (22)

Note that $G^{-1}dG$ is indeed a $\mathfrak{u}(n)$-valued $1$-form.

###  2.3 Pseudodifferential Calculus

The proof of Theorem 1.4 requires a few key results from the theory of
pseudodifferential operators, which we recall briefly here. For details and
proofs, we refer the reader to [17], for example.

Let $W\subseteq\mathbb{R}^{n}$ be open. Then the symbol class $S^{m}(W)$ is
defined to be the space of all functions $p\in
C^{\infty}(W\times\mathbb{R}^{n})$ satisfying for all
$\alpha,\beta\in\mathbb{N}^{n}$,

$\left|\partial_{\xi}^{\alpha}\partial_{x}^{\beta}p(x,\xi)\right|\leq
C_{\alpha,\beta}\left\langle\xi\right\rangle^{m-|\alpha|},$ (23)

where $\left\langle\xi\right\rangle:=\sqrt{1+|\xi|^{2}}$ is the usual
regularization of $|\xi|$. The symbol class $S^{m}(W,\mathbb{C}^{k\times k})$
is then the space of all matrix-valued functions whose entries are in
$S^{m}(W)$. Each $p\in S^{m}(W,\mathbb{C}^{k\times k})$ yields a map
$P:C^{\infty}_{c}(W,\mathbb{C}^{k})\to C^{\infty}(W,\mathbb{C}^{k})$ given by

$(Pw)(x):=\int e^{ix\cdot\xi}p(x,\xi)\widehat{w}(\xi)\,d\xi,$ (24)

where $\widehat{w}$ denotes the Fourier transform of $w$. We say that
$P\in\Psi^{m}(W,\mathbb{C}^{k})$ if $P$ has the form (24) for a symbol $p\in
S^{m}(W,\mathbb{C}^{k\times k})$. A pseudodifferential operator is called
classical if its symbol is given by an asymptotic series of the form

$p(x,\xi)=\sum_{j\geq 1}p_{m_{j}}(x,\xi)$ (25)

where each term $p_{m_{j}}$ is positive-homogeneous of degree $m_{j}$ in
$\xi$, and the sequence of real numbers $m_{j}$ is decreasing to $-\infty$.

Now let $E$ be a vector bundle of rank $k$ over a smooth manifold $M$, and let
$\mathscr{D}^{\prime}(E)$ be the space of $E$-valued distributions on $M$. A
map $P:C^{\infty}_{0}(M,E)\to C^{\infty}(M,E)$ is called a pseudodifferential
operator of order $M$ if for every chart $W$ of $M$ and every local
trivialization of $E$ over $W$, the induced map is in
$\Psi^{m}(W,\mathbb{C}^{k})$. The space of pseudodifferential operators on $E$
of order $m$ is denoted $\Psi^{m}(M,E)$. If $P\in\Psi^{m}(M,E)$ for all
$m\in\mathbb{R}$, then we call $P$ a smoothing operator. The space of
smoothing operators is denoted $\Psi^{-\infty}(M,E)$. Moreover, if
$P\in\Psi^{m}(M,E)$, then it extends to a map
$\mathscr{E}^{\prime}(E)\to\mathscr{D}^{\prime}(E)$, and if $M$ is compact,
then $P$ extends to a map $H^{s}(E)\to H^{s-m}(E)$.

We often work with pseudodifferential operators modulo $\Psi^{-\infty}(M,E)$,
since then composition of pseudodifferential operators becomes well-defined.
Thus, we will treat two pseudodifferential operators as equivalent if their
difference is a smoothing operator. Each equivalence class then corresponds to
a symbol modulo $S^{-\infty}$, the intersection of all symbol classes $S^{m}$.
Note that in particular two classical pseudodifferential operators differ by a
smoothing operator if and only if their formal symbols are equal modulo
$S^{-\infty}$.

if $P_{i}\in\Psi^{m_{i}}(M,E)$ for $i=1,2$, then their composition
$Q:=P_{1}\circ P_{2}$ is well-defined modulo smoothing operators, and its
symbol $q$ modulo $S^{-\infty}$ is given by

$q(x,\xi)\sim\sum_{\alpha}\frac{1}{\alpha!}\partial_{\xi}^{\alpha}p_{1}(x,\xi)D_{x}^{\alpha}p_{2}(x,\xi).$
(26)

##  3 Boundary determination

In this section, we prove that the Dirichlet-to-Neumann map
$\Lambda_{g,A,Z,m}$ is a pseudodifferential operator of order $1$ whose symbol
with respect to a local trivialization determines the Taylor series of $g$,
$Q$, and $A$ at the boundary, when $A$ is in an appropriate gauge. We shall
see upon applying the recipe of Lee and Uhlmann that unlike the analogous
proofs for the scalar Laplacian [13] or the connection Laplacian [4], we need
not place any restrictions on the metric. In particular, we shall see that
because the connection coefficients of $\omega^{s}\otimes A$ involve the
Levi–Civita connection, and thus the derivatives of the metric, the Taylor
series of the metric can be recovered without fixing a representative in its
conformal class as is done in [4].

######  Theorem 3.1.

The Dirichlet-to-Neumann map in Definition 1.1 is an elliptic
pseudodifferential operator of order $1$. Moreover, in any local
trivialization where $A_{n}=0$, the total symbol of $\Lambda_{g,A,Z,m}$
determines the Taylor series of $g$, $A$, and $Z$ at the boundary.

It is worth stating once the precise meaning of Theorem 3.1 in the context of
spinor bundles, though we immediately drop this level of precision in the
interest of clarity.

Suppose we have a Riemannian metric $g$ on $M$ inducing a spinor bundle $S$
for a fixed choice of spin structure. Then for a connection $A$ on $E$, and an
endomorphism $Z$ of $S\otimes E$, we can construct the corresponding
Dirichlet-to-Neumann map $\Lambda_{g,A,Z,m}$ on $S|_{\partial M}$. Suppose we
have also another Riemannian metric $\tilde{g}$ on $M$, which induces another
spinor bundle $\tilde{S}$ corresponding to a choice of spin structure. Then
for any connection $\tilde{A}$ on $E$, and endomorphism $\tilde{Z}$ of
$\tilde{S}\times E$, we can construct the corresponding Dirichlet-to-Neumann
map $\Lambda_{\tilde{g},\tilde{A},\tilde{Z},m}$ on $\tilde{S}|_{\partial M}$.
To say that these two Dirichlet-to-Neumann maps are equal is of course to mean
that they are equal up to isomorphism, but we must take care to specify what
kind of isomorphism.

######  Definition 3.2.

We say that the Dirichlet-to-Neumann maps $\Lambda_{g,A,Z,m}$ and
$\Lambda_{\tilde{g},\tilde{A},\tilde{Z},m}$ corresponding to data $(g,A,Z)$
and $(\tilde{g},\tilde{A},\tilde{Z})$ respectively are isomorphic if there
exists an isomorphism $\Phi:S|_{\partial M}\to\tilde{S}|_{\partial M}$ induced
by an isomorphism of the $SO(n)$-structures of $g$ and $\tilde{g}$, restricted
to $\partial M$, such that

$\Lambda_{\tilde{g},\tilde{A},\tilde{Z},m}=\Phi\circ\Lambda_{g,A,Z,m}\circ\Phi^{-1}.$
(27)

This is equivalent to saying that the local matrix representations of
$\Lambda_{g,A,Z,m}$ and $\Lambda_{\tilde{g},\tilde{A},\tilde{Z},m}$ are equal
when we choose local trivializations of $S|_{\partial M}$ and
$\tilde{S}|_{\partial M}$ induced by orthonormal frames for $g$ and
$\tilde{g}$ respectively.

The precise meaning of Theorem 3.1 is then that if two sets of data $(g,A,Z)$
and $(\tilde{g},\tilde{A},\tilde{Z})$ lead to equivalent Dirichlet-to-Neumann
maps, then the Taylor series of $g$ and $\tilde{g}$ are equal at the boundary,
as are the Taylor series of $Z$ and $\tilde{Z}$, and moreover, after possibly
making a gauge transformation of $\tilde{A}$, the Taylor series of $A$ and
$\tilde{A}$ are equal at the boundary in any local trivialization where
$A_{n}=0$ and $\tilde{A}_{n}=0$ near the boundary.

The proof of Theorem 3.1 follows the recipe of Lee and Uhlmann, which is by
now standard in the literature. The idea is to factor the second-order
differential operator $\operatorname{\not{\mathrm{D}}}_{A}^{2}+Z-m^{2}$ into a
product of first-order pseudodifferential operators modulo smoothing:

$\operatorname{\not{\mathrm{D}}}_{A}^{2}+Z-m^{2}\sim\left(D_{n}+i(E-\theta_{n})-iB\right)\left(D_{n}-i\theta_{n}+iB\right)$
(28)

where $D_{n}=-i\partial_{n}$,
$E=-\frac{1}{2}g^{\alpha\beta}\partial_{n}g_{\alpha\beta}$,
$\theta_{n}=A_{n}+\omega^{\mathrm{s}}_{n}$, and $B$ is some pseudodifferential
operator to be determined. We can inductively solve for the total symbol of
$B$ to show that such a pseudodifferential operator indeed exists. Then, using
this factorization and the theory of generalized heat equations, one can show
that $B|_{\partial M}\sim\Lambda_{g,A,Z,m}$. Finally, we note that the
inductive procedure used to determine the symbol of $B$ in terms of the known
data $(g,A,Z)$ can be inverted to inductively solve for the normal derivatives
of this data at the boundary.

###### Proof of Theorem 3.1.

For this proof, we let $x$ denote the coordinates in a fixed boundary chart
for $g$, and write $x=(x^{\prime},x^{n})$, where $x^{n}$ is the normal
coordinate, and $x^{\prime}=(x^{1},\dots,x^{n-1})$ are the tangential
coordinates. Greek letters run over $\\{1,\dots,n-1\\}$ while Latin indices
run over $\\{1,\dots,n\\}$. So in particular, $(x^{\alpha},0)$ form
coordinates on the boundary. We let $D_{k}:=-i\partial_{k}$. Finally, let
$\theta$ be the matrix valued $1$-form representing the connection
$\omega^{\mathrm{s}}\otimes A$ in a local trivialization to be determined,
where $\omega^{\mathrm{s}}$ is the spin connection. By using the Lichnerowicz
formula

$\operatorname{\not{\mathrm{D}}}_{A}^{2}=\left(\nabla^{A}\right)^{*}\nabla^{A}+\frac{1}{4}R+\frac{1}{2}F_{A}$
(29)

and separating out the normal derivatives from the tangential derivatives, we
can write

$\displaystyle\operatorname{\not{\mathrm{D}}}_{A}^{2}+Z-m^{2}$
$\displaystyle=D_{n}^{2}+i(E-2\theta_{n})D_{n}+Q_{2}+Q_{1}+Q_{0}$ (30)

where $E=-\frac{1}{2}g^{\alpha\beta}\partial_{n}g_{\alpha\beta}$, and where

$\displaystyle Q_{2}$
$\displaystyle:=-g^{\alpha\beta}\partial_{a}\partial_{\beta},$ (31)
$\displaystyle Q_{1}$
$\displaystyle:=-g^{\alpha\beta}\theta_{\alpha}\partial_{\beta}+g^{\alpha\beta}\Gamma_{\alpha\beta}^{\gamma}\partial_{\gamma},$
(32) $\displaystyle Q_{0}$
$\displaystyle:=-g^{ij}\left(\partial_{i}\theta_{j}\right)-g^{ij}\theta_{i}\theta_{j}+g^{ij},\Gamma^{k}_{ij}\theta_{k}+\frac{1}{4}R+\frac{1}{2}F_{A}+Z-m^{2}.$
(33)

Here, $\Gamma^{i}_{\ jk}$ are the Christoffel symbols of $g$ corresponding to
the boundary chart. Note that $Q_{i}$ is an $i$-th order differential operator
involving only tangential derivatives. We want to show that there exists a
pseudodifferential operator $B(x,D^{\prime})$ of order $1$ such that

$\operatorname{\not{\mathrm{D}}}_{A}^{2}+Z-m^{2}=\left(D_{n}+i(E-\theta_{n})-iB\right)\left(D_{n}-i\theta_{n}+iB\right)+\mathscr{S}$
(34)

where $\mathscr{S}$ is a smoothing operator. In writing $B(x,D^{\prime})$, we
mean that $B$ is a map $[0,\epsilon)\to\Psi(\partial M)$, which for each value
$x_{n}$ yields a pseudodifferential operator $B(x_{n},x^{\prime},D^{\prime})$
on the corresponding tangential leaf. In particular we have a well-defined
notion of restricting $B$ to the boundary, $B(0,x^{\prime},D^{\prime})$. The
motivation for this factorization comes from the following Lemma, a proof of
which is given following the current proof.

######  Lemma 3.3.

If $B(x,D^{\prime})$ exists, then $B|_{\partial
M}=\Lambda_{g,A,Z,m}+\mathscr{S}$ where $\mathscr{S}$ is a smoothing operator.

An important implication of the preceding Lemma is that
$B(0,x^{\prime},D^{\prime})$ has the same total symbol as the Dirichlet-to-
Neumann map, and so in doing symbol computations, it suffices to work with the
operator $B$ appearing in the factorization.

We now proceed with the proof of Theorem 3.1. In order to show that such a
$B(x,D^{\prime})$ exists, we will use the factorization (34) to determine a
formal symbol whose corresponding pseudodifferential operator satisfies (34)
in each degree. So, by Equations (30) and (34), we have

$\left(D_{n}+i(E-\theta_{n})-iB\right)\left(D_{n}-i\theta_{n}+iB\right)+\mathscr{S}=D_{n}^{2}+i(E-2\theta_{n})D_{n}+Q_{2}+Q_{1}+Q_{0}$
(35)

where $\mathscr{S}$ is a smoothing operator. Expanding the left-hand side of
(35), we see it equals

$D_{n}^{2}+i(E-2\theta_{n})D_{n}+i[D_{n},B]-\partial_{n}\theta_{n}+(E-\theta_{n})\theta_{n}-[B,\theta_{n}]-EB+B^{2}.$
(36)

Replacing the left-hand side of Equation (35) with (36) and rearranging, we
get

$i[D_{n},B]-[B,\theta_{n}]-EB+B^{2}=Q_{2}+Q_{1}+Q_{0}^{\prime}$ (37)

where

$Q_{0}^{\prime}:=Q_{0}+\partial_{n}\theta_{n}-(E-\theta_{n})\theta_{n}.$ (38)

We want to consider the total symbols of the left and right-hand sides of
(37). Let $b(x,\xi^{\prime})$ be the symbol of $B(x,D^{\prime})$, and
similarly for $Q_{2},Q_{1}$ and $Q_{0}^{\prime}$. Then Equation (37) implies

$\partial_{n}b+[\theta_{n},b]+i\sum_{\alpha}\left(\partial_{\xi_{\alpha}}b\right)\left(\partial_{x^{\alpha}}\theta_{n}\right)-Eb+\sum_{\nu}\frac{1}{\nu!}\left(\partial_{\xi^{\prime}}^{\nu}b\right)\left(D_{x^{\prime}}^{\nu}b\right)=q_{2}+q_{1}+q_{0}^{\prime},$
(39)

where $\nu$ runs over multi-indices. Let us assume that $b(x,\xi^{\prime})$
has a formal symbol given by

$b(x,\xi^{\prime})=\sum_{k\leq 1}b_{k}(x,\xi^{\prime})$ (40)

where $b_{k}(x,\xi^{\prime})$ is homogeneous in $\xi^{\prime}$ of degree $k$
away from $0$. Plugging (40) into (39) and taking the degree $m$ part of the
equation, we get

$b_{1}^{2}=q_{2}$ (41)
$2b_{1}b_{0}+\partial_{n}b_{1}-Eb_{1}+\sum\left(\partial_{\xi^{\prime}}b_{1}\right)\left(D_{x^{\prime}}b_{1}\right)=q_{1}$
(42)
$2b_{1}b_{-1}+\partial_{n}b_{0}+[\theta_{n},b_{0}]+i\sum_{\alpha}\left(\partial_{\xi_{\alpha}}b_{1}\right)\left(\partial_{x^{\alpha}}\theta_{n}\right)-Eb_{0}+\sum_{\begin{subarray}{c}j+k=|\nu|\\\
0\leq j,k\leq
1\end{subarray}}\frac{1}{\nu!}\left(\partial_{\xi^{\prime}}^{\nu}b_{j}\right)\left(D_{x^{\prime}}^{\nu}b_{k}\right)=q_{0}^{\prime},$
(43)

and

$2b_{1}b_{m-1}=-\partial_{n}b_{m}-[\theta_{n},b_{m}]-i\sum_{\alpha}\left(\partial_{\xi_{\alpha}}b_{m+1}\right)\left(\partial_{x^{\alpha}}\theta_{n}\right)+Eb_{m}-\sum_{\begin{subarray}{c}j+k-|\nu|=m\\\
m\leq j,k\leq
1\end{subarray}}\frac{1}{\nu!}\left(\partial_{\xi^{\prime}}^{\nu}b_{j}\right)\left(D_{x^{\prime}}^{\nu}b_{k}\right)$
(44)

for $m\leq-1$. Note that since
$q_{2}(x,\xi^{\prime})=g^{\alpha\beta}\xi_{\alpha}\xi_{\beta}$, Equation (41)
yields

$b_{1}(x,\xi^{\prime}):=\pm|\xi^{\prime}|_{g}.$ (45)

In particular, we can choose the principal symbol $b_{1}$ to be a negative
scalar, as was needed in the proof of the claim above. It is clear from
Equations (42)-(44) that for $m<1$, we can inductively determine $b_{m-1}$
from the first $|m|+1$ terms in (40). Therefore, since we have a formal symbol
that satisfies (39), the corresponding pseudodifferential operator satisfies
(35). This completes the proof that the pseudodifferential operator $B$
exists, and hence that the Dirichlet-to-Neumann map is an elliptic
pseudodifferential operator of order $1$.

In the other direction, suppose we know the Dirichlet-to-Neumann map
$\Lambda_{g,A,Z,m}$ associated to data $(g,A,Z,m)$. Then we know its total
symbol in any boundary chart and local trivialization of $E$ over the
boundary. By using equations (41) – (44), we can work backwards and
inductively extract the boundary data from the homogeneous terms in the
expansion of the symbol. To do this, we fix a boundary chart for $\partial M$,
and an orthonormal frame $e_{\alpha}$ on $\partial M$, which we extend into
$M$ via parallel transport along $e_{n}$, the inward pointing normal. This
orthonormal frame $e_{i}$ now induces a local trivialization of the spinor
bundle near $\partial M$ in which the spin connection takes the form (13). We
also pick a local trivialization of $E$.

To start with, the restriction of the metric at the boundary $g|_{\partial M}$
is encoded in the function $q_{2}$ and can therefore be recovered from the
principal symbol $b_{1}$ as per equation (41). Upon simplifying and
rearranging, Equation (42) yields

$b_{0}=-\frac{i}{2}g^{\alpha\beta}\theta_{\alpha}\frac{\xi_{\beta}}{|\xi|}+\frac{1}{2}\left(Eg^{\alpha\beta}-\frac{1}{2}\partial_{n}g^{\alpha\beta}\right)\frac{\xi_{\alpha}\xi_{\beta}}{|\xi|^{2}}+F_{0}\left(g_{\alpha\beta}|_{\partial
M}\right)$ (46)

for all $\xi\in T^{*}\partial M\setminus\\{0\\}$, where $F_{1}$ is some
function of its arguments involving only tangential derivatives along
$\partial M$. In particular, $F_{1}$ is known once $g|_{\partial M}$ is known.
Thus, from $b_{0}$ we can determine the first two terms on the right-hand-side
of (46). Moreover, since these first two terms have opposite parity with
respect to $\xi$, we can determine each term separately by substituting
different values for $\xi$. Therefore, knowing $g^{\alpha\beta}|_{\partial
M}$, we can recover

$\theta_{\alpha}|_{\partial M}=\operatorname{id}_{S}\otimes
A_{\alpha}|_{\partial M}-\frac{1}{4}\omega^{i}_{\
j}(\partial_{\alpha})|_{\partial
M}\gamma(e_{i})\gamma(e_{j})\otimes\operatorname{id}_{E}$ (47)

where $\omega^{i}_{\ j}(\partial_{\alpha})$ denotes the connection $1$-form
for the Levi-Civita connection with respect to the orthonormal frame $e_{i}$,
evaluated on the coordinate vector $\partial_{\alpha}$. By the Clifford
relations, the matrices $\gamma(e_{i})\gamma(e_{j})$ occurring in the second
term of (47) are traceless. We can therefore recover $A_{\alpha}$ as a partial
trace over the spinor bundle:

$A_{\alpha}(0,x^{\prime})=\frac{1}{\operatorname{rank}{S}}\operatorname{Tr}_{S}\theta_{\alpha}|_{\partial
M}.$ (48)

Since $A_{n}=0$ in this gauge by assumption, we have recovered $A|_{\partial
M}$. Knowing $A|_{\partial M}$, from Equation (47) we can recover the matrix

$-\frac{1}{4}\omega^{i}_{\ j}(\partial_{\alpha})|_{\partial
M}\gamma(e_{i})\gamma(e_{j}).$ (49)

Using the fact that $\gamma(e_{i})\gamma(e_{j})$ provide an orthonormal set of
matrices with respect to the trace inner product, we can extract
$\omega^{i}_{\ j}(\partial_{\alpha})|_{\partial M}$. The final obstacle is to
obtain the Christoffel symbols of $g$ with respect to the boundary normal
chart. For this, let $h$ be the matrix defined by
$\partial_{\alpha}=h^{\beta}_{\ \alpha}e_{\beta}$. Then, if $\Gamma^{i}_{\
jk}$ denotes the connection coefficients in the boundary normal chart, we have

$\Gamma^{i}_{\ \alpha j}=\left(h^{-1}\right)^{i}_{\ k}\left(\omega^{k}_{\
\ell}(\partial_{\alpha})\right)h^{\ell}_{\ j}+\left(h^{-1}\right)^{i}_{\
k}\partial_{\alpha}h^{k}_{\ j}.$ (50)

Since $h$ is known on the boundary, so is $\omega^{i}_{\
j}(\partial_{\alpha})|_{\partial M}$, and we can recover $\Gamma^{i}_{\alpha
j}$. In particular, we recover the normal derivative of the metric at the
boundary, since

$\Gamma^{n}_{\alpha\beta}=-\frac{1}{2}\partial_{n}g_{\alpha\beta}.$ (51)

Thus we have recovered $A_{\alpha}|_{\partial M}$ $\partial_{n}g|_{\partial
M}$. The next equation (43) leads to an equation of the form

$\displaystyle b_{-1}$
$\displaystyle=-\frac{i}{2}g^{\alpha\beta}\partial_{n}\theta_{\alpha}\frac{\xi_{\beta}}{|\xi|^{2}}+\left(T_{-1}^{\alpha\beta}\left(g|_{\partial
M},\partial_{n}g|_{\partial M},\partial_{n}^{2}g|_{\partial
M},\partial_{n}A|_{\partial M}\right)+Z|_{\partial
M}g^{\alpha\beta}\right)\frac{\xi_{\alpha}\xi_{\beta}}{|\xi|^{3}}$ (52)
$\displaystyle\ \ \ \ \ \ +F_{-1}\left(g|_{\partial
M},\partial_{n}g|_{\partial M},A|_{\partial M}\right).$ (53)

So as before, we can recover $\partial_{n}\theta_{\alpha}|_{\partial M}$.
Moreover, since the local trivialization of $S$ is induced by an orthonormal
frame, the matrices $\gamma(e_{i})$ are constant, and thus
$\partial_{n}\theta_{\alpha}$ takes the form

$\partial_{n}\theta_{\alpha}=-\frac{1}{4}\partial_{n}\left(\omega^{i}_{\
j}(\partial_{\alpha})\right)\gamma(e_{i})\gamma(e_{j})\otimes\operatorname{id}_{E}+\operatorname{id}_{S}\otimes\partial_{n}A_{\alpha}.$
(54)

Taking a partial trace over $S$ as before, we can determine
$\partial_{n}A_{\alpha}|_{\partial M}$, and therefore also
$\partial_{n}\left(\omega^{i}_{\ j}(\partial_{\alpha})\right)$. Using the fact
that $e_{\beta}$ is parallel along $\partial_{n}$, and the fact that we have
determined $\Gamma^{i}_{\ jk}|_{\partial M}$, it follows that
$\partial_{n}h|_{\partial M}$ is known. Therefore, taking the normal
derivative of the transformation law (50) for the connection coefficients, we
get

$\partial_{n}\Gamma^{i}_{\ \alpha j}=\left(h^{-1}\right)^{i}_{\
k}\partial_{n}\left(\omega^{k}_{\ \ell}(\partial_{\alpha})\right)h^{\ell}_{\
j}+S\left(h|_{\partial M},\partial_{n}h|_{\partial M},g|_{\partial
M},\partial_{n}g|_{\partial M}\right),$ (55)

where $S$ is a function of known quantities. We can therefore recover the
normal derivatives of the Christoffel symbols of $g$ in the boundary normal
chart, and in particular, we can recover

$\partial_{n}\Gamma^{n}_{\ \alpha\beta}|_{\partial
M}=-\frac{1}{2}\partial_{n}^{2}g_{\alpha\beta}|_{\partial M}.$ (56)

Thus, having recovered the second derivatives of the metric at $\partial M$,
and the first derivatives of the connection at $\partial M$, the only unknown
remaining is the endomorphism $Z|_{\partial M}$ in the even term of (52),
which is now easily recovered.

Continuing in this fashion, at step $m$ we get an equation of the form

$\displaystyle b_{1-m}$
$\displaystyle=-\frac{i}{2}g^{\alpha\beta}\partial^{m-1}_{n}\theta_{\alpha}\frac{\xi_{\beta}}{|\xi|^{m}}+\left(T_{1-m}^{\alpha\beta}+\partial_{n}^{m-2}Z|_{\partial
M}\right)\frac{\xi_{\alpha}\xi_{\beta}}{|\xi|^{m+1}}$ $\displaystyle\ \ \
+F_{1-m}\left(g_{\alpha\beta}|_{\partial
M},\dots,\partial^{m-1}_{n}g_{\alpha\beta}|_{\partial M},A_{\alpha}|_{\partial
M},\dots,\partial_{n}^{m-2}A_{\alpha}|_{\partial M},Z|_{\partial
M},\dots,\partial_{n}^{m-3}Z|_{\partial M}\right),$ (57)

where $T_{1-m}^{\alpha\beta}$ is a known quantity depending on derivatives of
$g$ up to order $m$, derivatives of $A$ up to order $m-1$, and derivatives of
$Z$ up to order $m-3$. Therefore, as before we can recover
$\partial_{n}^{m-1}\theta_{\alpha}$, and by taking traces,
$\partial_{n}^{m-1}A_{\alpha}$ and $\partial_{n}^{m-1}\omega^{i}_{\
j}(\partial_{\alpha})$. Then, taking derivatives of the transformation law, we
can recover $\partial_{n}^{m-1}\Gamma^{n}_{\ \alpha\beta}$ as above. In this
fashion we recover the normal derivatives of $A$ and $g$ at the boundary. ∎

We now give a proof of Lemma 3.3, which is essentially identical to the proof
given in [13]; we nonetheless include it here in the interest of being self-
contained.

###### Proof of Lemma 3.3.

Let us suppose that a first-order pseudodifferential operator
$B(x,D^{\prime})$ exists that satisfies Equation (34). Let $\chi\in
H^{\frac{1}{2}}(S|_{\partial M})$, and let $\varphi\in\mathscr{D}^{\prime}(S)$
be the solution to the Dirichlet problem (2). Since
$(\operatorname{\not{\mathrm{D}}}_{A}^{2}+Z-m^{2})\varphi=0$, the
factorization (34) yields the following system of equations:

$\displaystyle\psi:$ $\displaystyle=\left(D_{n}-i\theta_{n}+iB\right)\varphi,\
\ \varphi|_{x^{n}=0}=\chi$ (58) $\displaystyle h$
$\displaystyle=\left(D_{n}+i(E-\theta_{n})-iB\right)\psi$ (59)

where $h$ is some smooth spinor field near $\partial M$ given by
$h:=-\mathscr{S}\psi$. Note that although the regularity of $\psi$ is not a
priori known, we know that $h$ is smooth since $\mathscr{S}$ is a smoothing
operator. Writing $t:=T-x^{n}$, we can write Equation (59) as

$\partial_{t}\psi-(B-E+\theta_{n})\psi=-ih.$ (60)

Equation (60) is a generalized backwards heat equation. Now, by elliptic
regularity, we know that $\varphi$ is smooth in the interior, and therefore so
is $\psi$ by equation (58). In particular, $\psi|_{x^{n}=T}$ is smooth. We’ll
see later that we can choose the principal symbol of $B$ to be a negative
scalar, which implies that the backwards heat equation (60) with initial
condition $\psi|_{x_{n}=T}$ is well-posed. Thus the solution operator is a
smoothing operator, and since $\psi|_{x_{n}=T}$ is smooth, so is $\psi$. In
particular, $\psi|_{x^{n}=0}$ is smooth. So, let us define an operator $R$ by

$R\chi:=\psi|_{\partial M}.$ (61)

Then we have that $R$ is smoothing by construction, and moreover

$\displaystyle R\chi:=\psi|_{\partial M}$
$\displaystyle=\left(\left(D_{n}-i\theta_{n}+iB(x,D^{\prime})\right)\varphi\right)\big{|}_{\partial
M}$ $\displaystyle=-i\left(\nabla_{n}^{A}\varphi\right)\big{|}_{\partial
M}+iB(0,x^{\prime},D^{\prime})\chi$
$\displaystyle=-i\Lambda_{g,A,Z,m}\chi+iB(0,x^{\prime},D^{\prime})\chi.$ (62)

So indeed, we have $B|_{\partial M}=\Lambda_{g,A,Z,m}$ modulo smoothing. ∎

######  Remark 3.4.

The proof of Theorem 3.1, unlike in the case of the scalar or connection
Laplacian, holds for $\dim{M}=2$. Moreover, it is unnecessary to normalize the
metric in order to obtain the Taylor series of the endomorphism $Z$ at the
boundary, as is done for the connection Laplacian in [4]. These observations
can be attributed to the fact that conformal covariance of the Dirac operator
does not extend to even powers thereof; see [6] for details.

##  4 Recovering real-analytic Yang–Mills–Dirac connections

From here on, we fix a background metric $g$ on a compact spin manifold $M$
with boundary $\partial M$, and let $S$ be the spinor bundle of $M$. Let $E$
be a Hermitian vector bundle of rank $N$ over $M$, associated to its principal
$U(N)$-bundle $P$ of unitary frames, and let $A$ be a $U(N)$-connection on
$P$, whose spinorial Dirichlet-to-Neumann map $\Lambda_{A}$ is given. We want
to introduce an auxiliary $E$-valued spinor $\phi$ to which $A$ couples in a
physically interesting way, and whose boundary value is known, and investigate
the extent to which $\Lambda_{A}$ determines the connection $A$ modulo the
action of the gauge group. To this end, we introduce the Yang–Mills–Dirac
equations:

$\operatorname{\not{\mathrm{D}}}_{A}\phi=m\phi$ (63) $d_{A}^{*}F_{A}=J(\phi),$
(64)

where $m\in\mathbb{R}$, and $J:\Gamma(S\otimes
E)\to\Lambda^{1}(M,\operatorname{ad}{P})$ associates to each $E$-valued spinor
$\varphi$ its current, and is defined as follows: for
$\varphi\in\Gamma(S\otimes E)$, there is a unique
$J(\varphi)\in\Lambda^{1}(M,\operatorname{ad}{P})$ such that

$\left\langle\varphi,\xi\cdot\varphi\right\rangle=\left\langle
J(\varphi),\xi\right\rangle$ (65)

for all $\xi\in\Lambda^{1}(M,\operatorname{ad}{P})$. Note that for any gauge
transformation $G$, we have $J(G\varphi)=GJ(\varphi)G^{-1}$. The physical
significance of Equations (63)–(64) is that they represent a gauge field $A$
interacting with some matter field $\phi$ of mass $m$, by means of its charge
current $J(\phi)$. For details on the Yang–Mills–Dirac system, we refer the
reader to [1].

Because we are concerned primarily with the Dirichlet-to-Neumann map of the
second-order operator $\operatorname{\not{\mathrm{D}}}_{A}^{2}$, we will
assume that the connection $A$ and the auxiliary spinor field $\phi$ satisfy
the more general second-order system:

$\operatorname{\not{\mathrm{D}}}_{A}^{2}\phi=m^{2}\phi$ (66)
$d_{A}^{*}F_{A}=J(\phi).$ (67)

Note that every solution of (63)–(64) is a solution of (66)–(67), but not vice
versa.

In this section, we prove the following result:

######  Theorem 4.1.

Let $A$ and $B$ be $U(N)$-connections as above, real-analytic in the interior,
and let $\phi$ and $\psi$ be $E$-valued spinors on $M$ such that $(A,\phi)$
and $(B,\psi)$ both satisfy the second-order Yang–Mills–Dirac system
(66)–(67), and such that $\phi|_{\partial M}=\psi|_{\partial M}$. If
$\Lambda_{A,m}=\Lambda_{B,m}$, then $A$ and $B$ are locally gauge equivalent
about any point in $M$.

###### Proof.

We work in a real-analytic trivialization over an open set $U$ intersecting
the boundary $\partial M$. We can find a smooth global gauge transformation
$F$ satisfying

$\begin{cases}\partial_{n}F=-A_{n}F\ \text{in\ }U\\\ F|_{\partial
M}=\operatorname{id},\end{cases}$ (68)

which is moreover real-analytic in $U$, since $A_{n}$ is real-analytic there.
By applying this gauge transformation $F$, we get a pair
$(A^{\prime},\phi^{\prime})$ that is real-analytic in $U$, such that
$A_{n}^{\prime}=0$ in the local trivialization over $U$. Moreover, since
$F|_{\partial M}=\operatorname{id}$, it follows that
$\Lambda_{A^{\prime}}=\Lambda_{A}$. Therefore, doing the same for $B$, we may
assume without loss of generality that $A$ and $B$ are smooth connections,
satisfying $A_{n}=B_{n}=0$ in the local trivialization over $U$, and real-
analytic there.

Since $\Lambda_{A}=\Lambda_{B}$, Theorem 3.1, implies that in this local
trivialization we have

$\partial_{n}^{k}(A-B)\big{|}_{\partial M}=0,\ \ k\geq 0.$ (69)

That is, $A$ and $B$ have the same Taylor series at the boundary in this local
trivialization.

We want to extend this observation to $\phi$ and $\psi$. Note that by
assumption $\phi|_{\partial M}=\psi|_{\partial M}$. Using this fact, as well
as the fact that $A_{n}=B_{n}=0$ in $U$, the equality
$\Lambda_{A}=\Lambda_{B}$ yields

$\partial_{n}(\phi-\psi)|_{\partial
M}=\left(\nabla^{A}_{n}\phi-\nabla^{B}_{n}\psi\right)|_{\partial
M}=\Lambda_{A}\left(\phi|_{\partial M}\right)-\Lambda_{B}\left(\psi|_{\partial
M}\right)=0.$ (70)

So $\partial_{n}(\phi-\psi)|_{\partial M}=0$ in this local trivialization.
Moreover, (66) yields

$\partial_{n}^{2}(\phi-\psi)|_{\partial
M}=\left(\operatorname{\not{\mathrm{D}}}_{A}^{2}\phi-\operatorname{\not{\mathrm{D}}}_{B}^{2}\psi\right)|_{\partial
M}-\left(\phi-\psi\right)|_{\partial M}=0,$ (71)

where we again have used $\partial_{n}(\phi-\psi)|_{\partial M}=0$,
$(\phi-\psi)|_{\partial M}=0$, and $\partial_{n}^{k}(A-B)|_{\partial M}=0$ for
all $k$. By taking derivatives of (66), similar arguments yield
$\partial_{n}^{k}(\phi-\psi)|_{\partial M}=0$.

Now we want to make a local gauge transformation of $A$ in $U$ so that the new
connection $A^{\prime}$ satisfies $d^{*}A^{\prime}=0$. The key ingredient in
this step is the Cauchy–Kovalevskaya theorem; this is one place where the
real-analyticity hypothesis plays a crucial role. To this end, we recall the
following well-known result; see [8, Theorem 5.4], for example:

######  Lemma 4.2.

Let $\mathfrak{g}$ be a matrix Lie algebra. Then for
$S:\mathbb{R}^{n}\to\mathfrak{g}$, we have

$d(e^{S})=e^{S}\left(\frac{1-e^{-\operatorname{ad}{S}}}{\operatorname{ad}{S}}\right)(dS)$
(72)

where $\operatorname{ad}{S}\in\operatorname{End}{\mathfrak{g}}$ is the
endomorphism $X\mapsto[S,X]$.

For $S\in\mathfrak{u}(N)$, let us denote the endomorphism in brackets acting
on $dS$ in (72) by $\Theta(S)$. Note that $\Theta(S)$ is defined by a power
series in $\operatorname{ad}{S}$ and is therefore real-analytic in $S$.
Moreover, note that $\Theta(0)=\operatorname{id}$, and therefore $\Theta(S)$
is invertible for small values of $S$.

Now, let us consider the following Cauchy problem for $S:U\to\mathfrak{u}(N)$,

$\begin{cases}d^{*}\left(e^{-S}Ae^{S}+e^{-S}d(e^{S})\right)=0,\\\
S|_{U\cap\partial M}=0,\\\ \partial_{n}S|_{U\cap\partial M}=0.\end{cases}$
(73)

Using Lemma 4.2, Equation (73) can be written in the form

$\begin{cases}-g^{ij}\nabla_{i}\nabla_{j}S=\Theta(S)^{-1}F(x,S,dS;A)\\\
S|_{U\cap\partial M}=0,\\\ \partial_{n}S|_{U\cap\partial M}=0,\end{cases}$
(74)

where

$\displaystyle F(x,S,dS;A)$
$\displaystyle=g^{ij}e^{-S}\Theta(-S)(\partial_{i}S)A_{j}e^{S}+g^{ij}e^{-S}\left(\nabla_{i}A_{j}\right)e^{S}+g^{ij}e^{-S}A_{j}e^{S}\Theta(S)(\partial_{i}S)$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+g^{ij}(D\Theta)(S)(\partial_{i}S,\partial_{j}S).$ (75)

Note that since $S\mapsto\Theta(S)$ is real-analytic near $S=0$, as are the
connection $A$ and metric $g$, the function $F(x,S,dS;A)$ is real-analytic in
a neighbourhood of $(x,0,0;A)$ for $x\in U\cap\partial M$. Therefore, since
$\Theta(S)^{-1}$ is well-defined and real-analytic in a neighbourhood of
$S=0$, the Cauchy–Kovalevskaya theorem (see [16, §16.4] for example) yields,
after possibly shrinking $U$, the existence of a real-analytic solution
$S:U\to\mathfrak{u}(N)$ to (74).

Therefore, $e^{S}$ yields a local gauge transformation over $U$. Applying this
gauge transformation to $A$, we get a new connection
$A^{\prime}:=S^{-1}AS+S^{-1}dS$ over $U$, which satisfies $d^{*}A^{\prime}=0$.
Moreover, since the Yang–Mills–Dirac equations (66)–(67) are gauge-invariant,
the pair $(A^{\prime},\phi^{\prime})$, where $\phi^{\prime}:=e^{-S}\phi$,
continues to satisfy the Yang–Mills–Dirac system. In particular, the pair
$(A^{\prime},\phi^{\prime})$ satisfies the following nonlinear elliptic system
over $U$,

$\begin{cases}\operatorname{\not{\mathrm{D}}}_{A^{\prime}}^{2}\phi^{\prime}=m^{2}\phi^{\prime},\\\
d_{A^{\prime}}^{*}F_{A^{\prime}}=J(\phi^{\prime}),\\\
dd^{*}A^{\prime}=0.\end{cases}$ (76)

In the same manner, we may conclude, after possibly shrinking $U$ again, that
there exists a function $T:U\to\mathfrak{u}(N)$ such that the connection
$B^{\prime}$ defined by $B^{\prime}:=T^{-1}BT+T^{-1}dT$, and the spinor
$\psi^{\prime}:=e^{-T}\psi$, also satisfy the nonlinear elliptic system (76)
over $U$.

We now want to show that $S$ and $T$ have the same Taylor series at the
boundary. To this end, note that since $\Theta(0)=\operatorname{id}$, Equation
(4) yields $\Theta^{-1}(0)F(x,0,0;A)=d^{*}A(x)$ for any $x\in\partial M$.
Furthermore, since $S|_{\partial M}=T|_{\partial M}=0$ and
$\partial_{n}S|_{\partial M}=\partial_{n}T|_{\partial M}=0$, it follows that
$dS|_{\partial M}=dT|_{\partial M}=0$. These observations, along with (74)
give us

$\partial_{n}^{2}(S-T)|_{\partial M}=\left(\Delta S-\Delta T\right)|_{\partial
M}=\left(F(\cdot,0,0;A)-F(\cdot,0,0^{\prime}B)\right)|_{\partial
M}=\left(d^{*}A-d^{*}B\right)|_{\partial M}=0,$ (77)

the last equality holding since $A$ and $B$ have the same Taylor series at the
boundary. By taking derivatives of (74) and using
$\partial_{n}^{k}(A-B)|_{\partial M}=0$ for all $k$, we get
$\partial_{n}^{k}(T-S)=0$ for all $k$. Therefore, it follows that
$\partial_{n}^{k}(A^{\prime}-B^{\prime})|_{\partial M}=0$ and
$\partial_{n}^{k}(\phi^{\prime}-\psi^{\prime})|_{\partial M}=0$ for all $k\geq
0$. Since $(A^{\prime},\phi^{\prime})$ and $(B^{\prime},\psi^{\prime})$ both
satisfy (76) in $U$, and have equal Taylor series at the boundary, it follows
from the unique continuation principle for elliptic systems with scalar
principal part (see [9, Theorem 3.5.2] for example) that
$(A^{\prime},\phi^{\prime})=(B^{\prime},\psi^{\prime})$ in $U$. In particular,
this shows that $(A,\phi)$ is gauge equivalent to $(B,\psi)$ over $U$, by a
gauge transformation that is equal to the identity on $\partial M$.

We have so far shown that the original pairs $(A,\phi)$ and $(B,\psi)$ are
locally gauge equivalent near the boundary $\partial M$ by a real-analytic
gauge transformation. We now want to extend this to the interior of $M$. For
this final step, we use the real-analyticity of $A$ and $B$ in the interior as
follows.

First, we have the following lemma from [4], which is stated there for the
case $\mathcal{L}=d_{A}^{*}d_{A}$ on some Hermitian vector bundle $E$. The
proof easily extends to the case
$\mathcal{L}=\operatorname{\not{\mathrm{D}}}_{A}^{2}-m^{2}$ presently under
consideration, provided of course that
$m^{2}\notin\operatorname{Spec}{\operatorname{\not{\mathrm{D}}}_{A}^{2}}.$

######  Lemma 4.3.

Let $\beta\subseteq M^{\mathrm{int}}$ be an embedding of $[0,1]$ into $M$.
Then there exists smooth sections $\varphi_{1},\dots,\varphi_{m}$, harmonic
with respect to $\mathcal{L}$, and having
$\operatorname{supp}{\left(\varphi_{i}|_{\partial M}\right)}\subseteq\Gamma$
for some given non-empty open subset $\Gamma\subseteq\partial M$, that form a
frame for $E\otimes S$ at every point along $\beta$.

######  Remark 4.4.

The proof involves solving the Dirichlet problem for $\mathcal{L}$, and then
extending the resulting sections to global ones. See [4, Lemma 6.1] for
details.

Armed with Lemma 4.3, we fix a point $x\in M$, and let $\beta$ be a path from
$x$ to a point $y$ near $\partial M$ such that $A$ and $B$ are locally gauge
equivalent near $y$. We consider a tubular neighbourhood $W$ of $\beta$, over
which $S\otimes E$ is trivial. Since $A$ is real-analytic, the harmonic
sections $\varphi_{i}$ given by Lemma 4.3 are real-analytic. Therefore, they
provide a real-analytic trivialization of $S\otimes E$ over $U$.

Now let $\psi_{i}$ be the solution to

$\begin{cases}\operatorname{\not{\mathrm{D}}}_{B}^{2}\psi_{i}=m^{2}\psi_{i}\\\
\psi_{i}|_{\partial M}=\varphi_{i}|_{\partial M}.\end{cases}$ (78)

Since $B$ is real-analytic, so are the $\psi_{i}$. Let us construct the
endomorphism

$H_{0}=\sum_{j}\psi_{j}\otimes\varphi_{j}^{\flat}$ (79)

where $\varphi_{j}^{\flat}$ indicates the dual frame. Then $H_{0}$ is real-
analytic in $W$. Moreover, near $y$, we know that there exists a real-analytic
gauge transformation $G$ taking $A$ to $B$, satisfying $G|_{\partial
M}=\operatorname{id}$. Let $G_{0}:=\operatorname{id}\otimes G$. Then we have
that $G_{0}^{-1}\varphi_{i}=\psi_{i}$ near $\partial M$. Indeed, both
$G_{0}^{-1}\varphi_{i}$ and $\psi_{i}$ satisfy the same elliptic equation,

$\operatorname{\not{\mathrm{D}}}_{B}^{2}\left(G_{0}^{-1}\varphi_{i}\right)=G_{0}^{-1}\operatorname{\not{\mathrm{D}}}_{A}^{2}\varphi_{i}=m^{2}G_{0}^{-1}\varphi_{i}$
(80)

and boundary condition $G_{0}^{-1}\varphi_{i}|_{\partial
M}=\psi_{i}|_{\partial M}$. Also, by equality of the Dirichlet-to-Neumann
maps,

$\nabla^{B}_{n}\left(G_{0}^{-1}\varphi_{i}\right)|_{\partial
M}=G_{0}^{-1}\nabla^{A}_{n}\varphi_{i}|_{\partial
M}=\nabla^{B}\psi_{i}|_{\partial M}.$ (81)

So again, by unique continuation, we have that
$G_{0}^{-1}\varphi_{i}=\psi_{i}$ near $\partial M$. This is precisely the
property of $H$ defined in (79). Therefore $H=G_{0}$ near $\partial M$. In
particular, taking partial traces, we find that

$G=\frac{1}{\operatorname{rank}{S}}\operatorname{Tr}_{S}{H_{0}}.$ (82)

Since $G$ is unitary near $\partial M$, it follows from real-analyticity that
the right-hand of (82) is unitary in $W$. Denoting this right-hand-side by
$H$, it follows that $H$ is a unitary automorphism of $E$ over $W$. Moreover,
since $B=H^{-1}AH+H^{-1}dH$ near $\partial M$, it follows again from real-
analyticity that this holds over $W$. This proves that $A$ and $B$ are gauge
equivalent about any point in $M$, and are thus locally gauge equivalent. ∎

######  Corollary 4.5.

Let $A$ and $B$ be as above. If $rank{E}=1$, then $A$ and $B$ are globally
gauge equivalent via a gauge transformation $G$ satisfying $G|_{\partial
M}=1$.

###### Proof.

In the case $N=1$, global gauge transformations are elements of
$C^{\infty}(M,S^{1})$. The same arguments use in preceding proof show that
$(A,\phi)$ is locally gauge equivalent to $(B,\psi)$ everywhere in $M$. That
is, about every point in $M$ there exists a locally defined smooth
$S^{1}$-valued function satisfying $f\phi=\psi$. Since $\phi$ and $\psi$ are
solutions to elliptic equations, the set on which they do not vanish is dense.
It thus follows that one can patch these local $S^{1}$-valued functions to
obtain a global well-defined $S^{1}$-valued function, which is by definition a
gauge transformation from $A$ to $B$, and equal to $1$ on $\partial M$. ∎

######  Remark 4.6.

The ease with which we obtain the preceding corollary can be attributed to the
fact that Theorem 1.4 yields not only a gauge transformation between the
connections, but also between the spinors, on which the gauge action takes a
particularly simple form when $N=1$.

Here, as well as in the proof of Theorem 1.4, we have seen how the
introduction of a spinor field coupled to a unitary connection affects the
recovery of the connection up to gauge from the Dirichlet-to-Neumann map of
its twisted Dirac Laplacian. This motivates the following question: in what
ways does the introduction of a spin structure affect the recovery of other
geometric structures from boundary data associated to a Dirac Laplacian. For
instance, regarding the Calderón problem for the metric, it would be
worthwhile to study if the spin structure can be exploited to facilitate the
recovery of the metric up to isometry, even in dimension $2$, from the
Dirichlet-to-Neumann map of the Dirac Laplacian.

## Acknowledgements

The author wishes to acknowledge financial support from the Natural Sciences
and Engineering Research Council of Canada in the form of a Canada Graduate
Scholarship for Doctoral studies, as well as Professor Niky Kamran for his
helpful suggestions and comments.

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