# The Calderón problem for the Fractional Dirac Operator

Hadrian Quan, Gunther Uhlmann

###### Abstract.

We show that knowledge of the source-to-solution map for the fractional Dirac
operator acting over sections of a Hermitian vector bundle over a smooth
closed connencted Riemannian manifold of dimension $m\geq 2$ determines
uniquely the smooth structure, Riemannian metric, Hermitian bundle and
connection, and its Clifford modulo up to a isometry. We also mention several
potential applications in physics and other fields.

##  1\. Introduction

The Calderón problem asks whether one can determine the electrical
conductivity of a medium by making voltage and current measurements at the
boundary. In the anisotropic case, that is when the conductivity depends on
direction. It is modelled by a positive-definite symmetric matrix. It was
shown in [24] that this problem is equivalent to determining a Riemannian
metric from the associated Dirichlet to Neumann (DN) map associated with
harmonic functions. Therefore this problem can be considered on a compact
Riemannian manifold with boundary and arises also in the AdS/CFT
correspondence [5]. See [33], [34] for more details and other results.

The study of the fractional Calderón problem was initiated in [15] where the
unknown potential in the fractional Schrödinger equation on a bounded domain
in the Euclidean space was determined from exterior measurements. An important
generalization was in the work [13] where the Euclidean Laplacian is replaced
by the Laplace-Beltrami operator of a Riemannian metric. Following these
works, inverse problems of recovering lower order terms for fractional
elliptic equations have been studied extensively, see for example [14], [13],
[30], [29], [2], [4], [7], [6], [8], [9], [10], [25], [26], [28] for some of
the contributions. In all of those papers, it is assumed that the leading
order coefficients are known.

In the article [12], it was solved the fractional anisotropic Calderón problem
on closed Riemannian manifolds of dimensions two and higher that the knowledge
of the local source-to-solution map for the fractional Laplacian, given an
arbitrary small open nonempty a priori known subsef a smooth closed connected
Riemannian manifold, determines the Riemannian manifold up to an isometry.
This can be viewed as a nonlocal analog of the anisotropic Calderón problem in
the setting of closed Riemannian manifolds, which is open in dimensions three
and higher.

We consider in this paper the anisotropic Calderon problem for the fractional
Dirac operator acting on sections of a Hermitian vector bundle over a smooth
closed connected Riemannian manifold $M$ of dimension $m\geq 2$. Just from
local measurements confined to an arbitrary non-empty open set, we show that
the fractional Dirac operator determines the smooth structure, Riemannian
metric, Hermitian bundle and connection, and its Clifford module, up to a
isometry which fixes the set in question. We will first briefly motivate the
concept of a generalized Dirac operator, and its fractional powers, before
explaining the types of measurements we allow, and stating our results.

In his attempts to quantize electromagnetism, Dirac defined the Dirac operator
while trying to find a 1st order differential operator such that its square
was the “Laplacian” on Minkowski space (in fact this was a factorization of
the d’Alembertian). The data of the Clifford algebra (see §2) entered into his
definition after he realized this was not possible for a scalar operator. On a
vector bundle $E$ over a Riemannian manifold $M$, the definition of a
generalized Dirac operator $\not{D}$ is motivated by the desire to capture
this property that $\not{D}\circ\not{D}=\Delta_{E}$ is a generalized Laplace-
type operator on $E$, i.e. the Bochner Laplacian $\nabla^{*}\nabla$ of some
connection up to lower order terms. This factorization of $\Delta_{E}$ implies
$\not{D}$ is elliptic and hence admits a spectral resolution
$\\{\varphi_{k},\lambda_{k}\\}$ of $L^{2}(M;E)$-orthonormal eigensections,
with discrete eigenvalues accumulating only at $\pm\infty$. From the spectral
theorem we can define for $\alpha\in(0,1)$, the fractional Dirac operator by
$\not{D}^{\alpha}\varphi_{k}=\lambda_{k}^{\alpha}\varphi_{k}$. From unique
solvability of the “Poisson equation”,

$\not{D}^{\alpha}u=f,$

for data $f$ orthogonal to $\operatorname{Ker}(\not{D})$, we can consider the
operator which maps a source function
$f\in{\mathcal{C}}^{\infty}_{0}(\mathcal{O};E|_{\mathcal{O}})\cap\operatorname{Ker}(\not{D})^{\perp}$
to the solution,

$L_{\not{D},\mathcal{O}}:f\mapsto u:=(\not{D}^{-\alpha}f)|_{\mathcal{O}},$

for $f\perp\operatorname{Ker}(\not{D})$. We refer to this operator as the
_source-to-solution map_. We prove,

######  Theorem 1.1.

Let $\mathscr{D}_{j}:=(M_{j},E_{j},g_{j},h_{E_{j}},\nabla^{E_{j}})$ for
$j=1,2$ be two Dirac bundles (see §2) over closed Riemannian manifolds of
dimension $n\geq 2$, and let $\mathcal{O}_{j}\subset M_{j}$ be non-empty open
sets. Assuming there exists a diffeomorphism
$\psi:\mathcal{O}_{1}\to\mathcal{O}_{2}$ satisfying

$L_{\not{D}_{1},\mathcal{O}_{1}}(\psi^{*}f)=\psi^{*}L_{\not{D}_{2},\mathcal{O}_{2}}(f),$

for all
$f\in{\mathcal{C}}^{\infty}_{0}(\mathcal{O}_{1};E_{1}|_{\mathcal{O}_{1}})$
orthogonal to $\operatorname{Ker}(\not{D}_{1})$. Then there is an isomorphism
of Hermitian vector bundles $\Psi:E_{1}\to E_{2}$, covering an isometry
$\psi:M_{1}\to M_{2}$ which restricts to
$\Psi|_{E_{1}|_{\mathcal{O}_{1}}}=\psi$.

######  Corollary 1.2.

Let $\mathscr{D}:=(M,E,g,h_{E},\nabla^{E})$ be a Dirac bundle over a smooth
closed manifold of dimension $m\geq 2$, with $\mathcal{O}\subset M$ a non-
empty open subset. Let $\\{\varphi_{k}\\}_{k=1}^{\infty}\subset L^{2}(M;E)$ be
the collection of positive eigensections with corresponding eigenvalues
$\\{\lambda_{k}\\}_{k=1}^{\infty}\subset\mathbb{R}^{+}$. Then the partial
spectral data plus clifford multiplication
$\mathpzc{cl}:\mathbb{C}l(TM,g)\to\operatorname{End}(E)$,

$(\mathcal{O},\varphi_{k}|_{\mathcal{O}},\lambda_{k},\mathpzc{cl}|_{T^{*}\mathcal{O}})$

determines the metric, Hermitian vector bundle, and connection, up to an
isometry.

###  1.1. Applications

Already there has been some interest in studying generalizations of classical
equations of physics with respect to a fractional time derivative, (see for
example [23], [19]).

In this work we consider instead fractional differential operators,
corresponding to fractional spatial derivatives. One place where there has
been interest in applying such nonlocal operators is in the study of particle
physics beyond the standard model, i.e. fields governed by fractional wave
equations. The work of [38] considered $n$th powers for $n\geq 2$ of the
d’Alembert operator and demonstrated that for $n>2$ the covariant wave
equations generated by $\Box_{g}^{1/n}$ generate a representation of $SU(n)$.
Along similar lines [18] developed a form of local gauge invariance for such
fractional fields and used this to deduce the Baryon mass spectrum via a
fractional extension of the classical Zeeman effect [39]. For a more recent
development of such fractional field theories see the work of [16] who suggest
that anomalous power laws for the “strange metal” properties of Cuprate can be
explained if the metal interacts with light via a gauge theory of fractional
dimension; also the work of [22] who introduce a theory of fractional
electromagnetism, which is motivated in part by a generalization of the
Caffarelli-Silvestre extension to the case of the Hodge Laplacian.

The most surprising applications relate such nonlocal operators to questions
coming from quantum information theory. A crucial first step in studying
entanglement properties of algebraic quantum field theories is the Reeh-
Schlieder theorem [37], which states all local fields in the field algebra of
spacetime are entangled with fields localized to all other regions (c.f. [31]
for a more mathematical exposition of this theorem in the language of von
Neumann algebras). The work of [36] first gave a proof of the Reeh-Schlieder
theorem for static spacetimes using only the strong anti-locality property of
$\Delta_{g}^{1/2}$. In the work of [15] they demonstrate the comparable anti-
locality property for $\Delta_{g}^{\alpha}$ for $\alpha\in(0,1)$, using
standard techniques in inverse problems (Carleman estimates, etc.). That this
entanglement property of the field algebra is equivalent to a strong unique
continuation principle for certain nonlocal operators suggests other
interesting connections between questions in inverse problems and quantum
information theory.

##  2\. Background on Generalized Dirac operators

Let $(M^{m},E^{n},h_{E},g)$ be a Hermitian vector bundle of rank $n$ over
smooth compact Riemannian manifold without boundary. We further call this
Hermitian bundle a _Clifford module_ if $E\to M$ is bundle over $M$ equipped
with bundle morphism, known as _Clifford multiplication_ ,

$\mathpzc{cl}:\mathbb{C}l(T^{*}M,g)\to\operatorname{End}(E)$

from the unique Clifford module on $T^{*}M$ induced by $g$ to
$\operatorname{End}(E)$. As a vector bundle, $\mathbb{C}l(T^{*}M,g)$ is
isomorphic to $\Lambda^{\bullet}T^{*}M$, and as a module has an algebra
operation determined by the relation

$\mathpzc{cl}(\alpha)\mathpzc{cl}(\beta)+\mathpzc{cl}(\beta)\mathpzc{cl}(\alpha)=-2g(\alpha,\beta)\operatorname{Id}\quad\forall\alpha,\beta\in\Lambda^{\bullet}T^{*}M.$

We say the Hermitian metric is compatible with Clifford multiplication if

$h_{E}(\mathpzc{cl}(\theta)v,w)+h_{E}(v,\mathpzc{cl}(\theta)w)=0,\quad\forall
v,w\in{\mathcal{C}}^{\infty}(M;E)$

and a choice of Hermitian connection $\nabla^{E}$ on $E$ is is compatible with
Clifford multiplication if

$[\nabla^{E},\mathpzc{cl}(\theta)]=\mathpzc{cl}(\nabla^{g}\theta)$

where $\nabla^{g}$ is the Levi-Civita connection of $g$.

Given a Hermitian bundle with connection both compatible with Clifford
multiplication we can construct a _generalized Dirac operator_ $\not{D}$,
defined by

$\not{D}:{\mathcal{C}}^{\infty}(M;E)\xrightarrow{\nabla^{E}}{\mathcal{C}}^{\infty}(M;T^{*}M\otimes
E)\xrightarrow{i\mathpzc{cl}(-)}{\mathcal{C}}^{\infty}(M;E),$

and call the collected data $(M,E,g,h_{E},\nabla^{E})$ a _Dirac bundle_. The
generalized Dirac operator is a Dirac operator in the usual sense by arising
as the ‘square root’ of a generalized Laplace operator on $E$; compatibility
of the Hermitian metric and connection implies

$[\not{D},f]=[i\mathpzc{cl}\circ\nabla^{E},f]=i\mathpzc{cl}(f),$

thus $\not{D}$ has principal symbol

$\sigma_{1}(\not{D})(x,\xi)=i\mathpzc{cl}(\xi),\implies\sigma_{2}(\not{D}^{2})(x,\xi)=|\xi|_{g}^{2}\operatorname{Id}_{E},$

hence $\not{D}^{2}:=\Delta_{E}$ is a principally scalar multiple of the metric
$g$. Further, we see from this computation that both $\not{D}$ and
$\not{D}^{2}$ are principally scalar elliptic differential operators (of
orders 1 and 2 respectively).

##  3\. Fractional Dirac operator and determination of the Heat kernel

In this section we give two equivalent definitions of the fractional Dirac
operator and use one to define the source-to-solution operator associated to
an a priori known open set $\mathcal{O}\subset M$. Then, following [12], we
show that knowledge of this source-to-solution operator determines the Heat
kernel on $\mathcal{O}$.

From the symbol calculation above we see that $\not{D}$ is a symmetric
operator. Because $M$ is a closed manifold,
$\not{D}:{\mathcal{C}}^{\infty}(M;E)\to{\mathcal{C}}^{\infty}(M;E)$ is
essentially self-adjoint on its core domain of smooth sections with a self-
adjoint extension to $H^{1}(M;E)$. Unlike $\not{D}^{2}$, $\not{D}$ fails to be
a non-negative operator. On the other hand, its discrete spectrum (excluding
the zero eigenvalue) is in correspondence with the spectrum of $\not{D}^{2}$;
the discrete eigenvalues of $\not{D}$ come in positive and negative pairs
$\\{\pm\lambda_{k}\\}$ (corresponding to an eigenvalue $\lambda_{k}^{2}$ of
$\not{D}^{2}$) which we index by their absolute values

$0=\lambda_{0}<|\lambda_{1}|\leq|\lambda_{2}|\leq\cdots\nearrow+\infty$

for the distinct eigenvalues of $\not{D}$, and denote $d_{k}$ the multiplicity
of $\lambda_{k}$. Let $\\{\varphi_{k_{j}}\\}_{j=1}^{d_{k}}$ be an
$L^{2}(M;E)$-orthonormal basis for the eigenspace
$\operatorname{Ker}(\not{D}-\lambda_{k})$ corresponding to $\lambda_{k}$, and
denote $\pi_{k}:L^{2}(M;E)\to\operatorname{Ker}(\not{D}-\lambda_{k})$ for the
orthogonal projection onto the corresponding eigenspace, written as

$\pi_{k}(f)=\sum_{j=1}^{d_{k}}\langle
f,\varphi_{k_{j}}\rangle_{L^{2}(M;E)}\varphi_{k_{j}},$

for all $k=0,1,\ldots$. Here $\langle\cdot,\cdot\rangle_{L^{2}(M;E)}$ is the
$L^{2}$-inner product on sections of $E$ induced by our choice of $h_{E}$.

Fix $\alpha\in(0,1)$. Given this spectral resolution of $\not{D}$ we can
define the Fractional Dirac operator
$\not{D}^{\alpha}:{\mathcal{C}}^{\infty}(M;E)\to{\mathcal{C}}^{\infty}(M;E)$

$\not{D}^{\alpha}f=\sum_{k\in\mathbb{Z}}\lambda_{k}^{\alpha}\pi_{k}(f),$

which extends to an unbounded self-adjoint operator on $L^{2}(M;E)$ with
domain $\mathcal{D}(\not{D}^{\alpha})=H^{\alpha}(M;E)$.

Unlike the scalar case, the nullspace of $\not{D}$ may include more than just
the constant functions. Say that dim $\operatorname{Ker}(\not{D})=d_{0}$ has
orthonormal basis $\\{\varphi_{j}^{0}\\}_{j=1}^{d_{0}}$, then we can solve the
equation

$\not{D}^{\alpha}u=f,$ (3.1)

for $f\in{\mathcal{C}}^{\infty}_{0}(\mathcal{O};E)$, with $\mathcal{O}\subset
M$ open, whenever we impose

$\langle f,\varphi_{1}^{0}\rangle_{L^{2}}=\cdots=\langle
f,\varphi_{d_{0}}^{0}\rangle_{L^{2}}=0$

for a unique solution $u=u^{f}\in{\mathcal{C}}^{\infty}(M;E)$ defined by the
condition that $u^{f}\perp_{h_{E}}\operatorname{Ker}(\not{D})$. Associated to
the equation (3.1) we can define the local source-to-source solution map
$L_{M,g,\mathcal{O},\ldots}$ by

$L_{\not{D},\mathcal{O}}(f):=u^{f}|_{\mathcal{O}}=(\not{D}^{-\alpha}f)|_{\mathcal{O}}$

We can give an equivalent (spectral theoretic) definition for the Fractional
Dirac operator via the heat kernel of the square of the Dirac operator. First
note that we can define from the integral formula for the Gamma function,

$\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}dt,\implies\mu^{-\alpha}=\frac{1}{\Gamma(\alpha)}\int_{0}^{\infty}e^{-t\mu}t^{\alpha-1}dt,$

thus we write

$(\Delta_{E})^{-\alpha}=\frac{1}{\Gamma(\alpha)}\int_{0}^{\infty}e^{-t\Delta_{E}}t^{\alpha-1}dt.$

Using this formula, we have
$\not{D}^{\alpha}=\not{D}\circ\Delta_{E}^{\frac{\alpha-1}{2}}=\Delta_{E}^{\frac{\alpha-1}{2}}\not{D}$,
i.e.

$\not{D}^{\alpha}u=\frac{1}{\Gamma(\frac{1-\alpha}{2})}\int_{0}^{\infty}t^{\frac{1-\alpha}{2}}e^{-t\Delta_{E}}(\not{D}u)\frac{dt}{t}$

using the fact that $\not{D}e^{-t\Delta_{E}}=e^{-t\Delta_{E}}\not{D}$ (from
uniqueness of the solution to the heat equation).

Using this definition of $\not{D}^{-\alpha}$ we can easily extend the proof of
[12, Thm 1.5] to our setting:

######  Theorem 3.1.

Let $\mathscr{D}_{j}:=(M_{j},E_{j},g_{j},h_{E_{j}},\nabla^{E_{j}})$ for
$j=1,2$ be two Dirac bundles over closed Riemannian manifolds of dimension
$n\geq 2$, then we denote by $\not{D}_{j}$ the generalized Dirac operator
associated to $\mathscr{D}_{j}$. Let $\mathcal{O}_{j}\subset M_{j}$ be non-
empty open sets and assume that

$(\mathcal{O}_{j},g_{j}|_{\mathcal{O}_{j}}):=(\mathcal{O},g)$ (3.2)

and that there exists hermitian bundle isomorphism
$\phi:E_{1}|_{\mathcal{O}_{1}}\to E_{2}|_{\mathcal{O}_{2}}$. Assume
furthermore that

$L_{\not{D}_{1},\mathcal{O}_{1}}(f)=L_{\not{D}_{2},\mathcal{O}_{2}}(f)$ (3.3)

for all $f\in{\mathcal{C}}^{\infty}_{0}(\mathcal{O}_{1},E_{1}|_{\mathcal{O}})$
such that
$f\perp_{L^{2}(\mathcal{O};E_{1}|_{\mathcal{O}})}\operatorname{Ker}(\not{D}_{1})$,
for any smooth extension of $f$ to $M_{2}$. Then

$e^{-t\Delta_{E_{1}}}(x,y)=e^{-t\Delta_{E_{2}}}(x,y),\quad
x,y\in\mathcal{O},\;t>0$

Remark: The requirement for well-posed of the source-to-source solution maps
that $f$ be orthogonal to the space Harmonic sections is subtle: the local
bundle isomorphism $\phi$ is already sufficient to ensure that
$\text{dim}\;\operatorname{Ker}(\not{D}_{1})=\text{dim}\operatorname{Ker}(\not{D}_{2})$.
If $f\in\operatorname{Ker}(\not{D}_{2})$ then its restriction to
$\mathcal{O}_{2}$ pullbacks to a section of $E_{1}|_{\mathcal{O}_{1}}$, and in
particular is also an element of $\operatorname{Ker}(\not{D}_{1})$ as the
Dirac operator commutes with pullback. Similarly pullback by the inverse
proves the reverse inclusion. The possibility that a non-trivial Harmonic
section vanishes on the open sets in consideration is ruled out by the unique
continuation principle for the Dirac operator: if $\not{D}f=0$ and
$f|_{\mathcal{O}}=0$ then $f=0$ (see e.g. [3, Ch. 8]).

###### Proof.

Choosing $\omega_{1}\Subset\mathcal{O}$ non-empty and open, with
$\omega_{2}\subset\mathcal{O}$ also non-empty open such that
$\overline{\omega}_{1}\cap\overline{\omega}_{2}=\emptyset$. For
$f\in{\mathcal{C}}^{\infty}_{0}(\mathcal{O};E|_{\mathcal{O}})$, due to (3.2),
we have for all $m=1,2,\ldots$,

$\not{D}_{1}^{2m-1}f=\not{D}_{2}^{2m-1}f=\not{D}^{2m-1}f\quad\text{on
$\omega_{1}$},$

and $\not{D}^{2m-1}f$ is orthogonal to $\operatorname{Ker}(\not{D}_{j})$ for
$j=1,2$. Further from (3.3), for all $m=1,2,3,\ldots$ that

$(\not{D}_{1}^{-\alpha}\not{D}^{2m-1}f)|_{\mathcal{O}}=(D_{2}^{-\alpha}\not{D}^{2m-1}f)|_{\mathcal{O}},$

hence

$\displaystyle\int_{0}^{\infty}t^{\frac{\alpha-1}{2}}((e^{-t\Delta_{E_{1}}}\not{D}_{1}-$
$\displaystyle
e^{-t\Delta_{E_{2}}}\not{D}_{2})\not{D}^{2m-1}f)(x)\frac{dt}{t}=\int_{0}^{\infty}t^{\frac{\alpha-1}{2}}((e^{-t\Delta_{E_{1}}}-e^{-t\Delta_{E_{2}}})\Delta^{m}f)(x)\frac{dt}{t}=0$
$\displaystyle=\int_{0}^{\infty}t^{\frac{\alpha-1}{2}}\partial_{t}^{m}((e^{-t\Delta_{E_{1}}}-e^{-t\Delta_{E_{2}}})f)(x)\frac{dt}{t}$

for $x\in\mathcal{O},\;m=1,2,\ldots$, where we have again used (3.2) and that
$\not{D}^{2}=\Delta_{E}$. We next aim to integrate by parts $m$-times, and
observe that the boundary terms vanish due to a combination of classical Heat
kernel estimates (both short and long time). The relevant Heat kernel
estimates which hold for the Heat kernel of the Dirac Laplacian
$\Delta_{E_{j}}$, are

$|e^{-t\Delta_{E_{j}}}(x,y)|\leq
Ct^{-(m+1)}e^{-\frac{cdg_{j}(x,y)^{2}}{t}},\quad 0<t<1,\;x,y\in M_{j}$

(has a slightly worse exponent in $t$ than what is known for the scalar case,
but still suffices for our purposes), and

$||e^{-t\Delta_{E_{j}}}||_{L^{1}\to L^{\infty}}\leq Ct^{-m/2},\quad t>0$

which is precisely the classical estimate of [35], equivalent to the scalar
case. The first estimate can be found in [27, Thm 3.5]. Combining these two
estimates, we can bound the integrand above,
$|\partial_{t}((e^{-t\Delta_{E_{1}}}-e^{-t\Delta_{E_{2}}})f)|(x)$, by
functions vanishing at the endpoints of the integral. Proceeding to integrate
by parts $m$-times we have that

$\int_{0}^{\infty}t^{\frac{\alpha-1}{2}-(m+1)}((e^{-t\Delta_{E_{1}}}-e^{-t\Delta_{E_{2}}})f)(x)dt=0$

and from here we can conclude that $\mathcal{F}(\chi_{[0,\infty)}\varphi)(s)$
is holomorphic with all derivatives vanishing at $s=0$, for

$\varphi(s)=\frac{((e^{s\Delta_{E_{1}}}-e^{s\Delta_{E_{2}}})f)(x)}{s^{\alpha}},$

thus $((e^{-t\Delta_{E_{1}}}-e^{-t\Delta_{E_{2}}})f)(x)=0$ for all
$x\in\omega_{2}$ and all $t>0$. By unique continuation of the Heat equation
this implies the same equality holds on all of $\mathcal{O}$. Using that
$f\in{\mathcal{C}}^{\infty}_{0}(\omega_{1})$ was arbitrary we have

$e^{-t\Delta_{E_{1}}}(x,y)=e^{-t\Delta_{E_{2}}}(x,y),\quad
x,y\in\mathcal{O},t>0$

as claimed. ∎

##  4\. Reconstruction via the Wave equation

Have shown that the source-to-source solution operators determine the Heat
kernel on the open neighborhood $\mathcal{O}$. Using the Kannai transmutation
formula

$e^{-t\Delta_{E_{j}}}(x,y)=\frac{1}{(4\pi
t)^{1/2}t}\int_{0}^{\infty}e^{-\frac{\tau}{4t}}\frac{\sin(\mathchoice{{\hbox{$\displaystyle\sqrt{\tau\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{\tau\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{\tau\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\tau\,}$}\lower
0.4pt\hbox{\vrule
height=2.15277pt,depth=-1.72223pt}}}\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.41666pt,depth=-2.73334pt}}})}{\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}}d\tau\quad t>0$

the local determination of $e^{-t\Delta_{E_{j}}}(x,y)$ and this equality imply

$\left(\frac{\sin(t\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.41666pt,depth=-2.73334pt}}})}{\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{1}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.41666pt,depth=-2.73334pt}}}}f\right)(x)=\left(\frac{\sin(t\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.41666pt,depth=-2.73334pt}}})}{\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{2}}\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}}f\right)(x),$ (4.1)

for all $t>0$ and $x\in\mathcal{O}$.

The benefit of this last equality is that the operator in (4.1) lets us
represent the solution of the initial value problem,

$\begin{cases}(\partial_{t}^{2}-\Delta_{E_{j}})u_{j}(t,x)=f(t,x),&(t,x)\in(0,\infty)\times
M_{j}\\\ u_{j}(0,x)=0,\partial_{t}u_{j}(0,x)=0&x\in M_{j}\\\ \end{cases}$
(4.2)

for data $f\in{\mathcal{C}}^{\infty}_{0}(\mathcal{O}\times M_{j};E_{j})$ via
the formula

$u_{j}(t,x)=\int_{0}^{t}\frac{\sin(t-s)\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=3.41666pt,depth=-2.73334pt}}}}{\mathchoice{{\hbox{$\displaystyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\textstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=6.83331pt,depth=-5.46667pt}}}{{\hbox{$\scriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule
height=4.78333pt,depth=-3.82668pt}}}{{\hbox{$\scriptscriptstyle\sqrt{\Delta_{E_{j}}\,}$}\lower
0.4pt\hbox{\vrule height=3.41666pt,depth=-2.73334pt}}}}fds.$

Associated to (4.2), we define the source-to-source solution map for the wave
operator on $\mathcal{O}$ by

$L_{\mathscr{D}_{j},\mathcal{O}_{j}}^{\text{wave}}:{\mathcal{C}}^{\infty}_{0}((0,\infty)\times\mathcal{O})\to{\mathcal{C}}^{\infty}_{0}([0,\infty)\times\mathcal{O}),\quad
L_{\mathscr{D}_{j},\mathcal{O}_{j}}^{\text{wave}}(f)=u_{j}|_{\mathcal{O}_{j}}$

where $u_{j}(t,x)$ is the unique solution to (4.2). And by the equality in
(4.1) we have equality of the source-to-solution maps for the wave operators
for $j=1,2$,

$L_{\mathscr{D}_{1},\mathcal{O}_{1}}^{\text{wave}}(f)=L_{\mathscr{D}_{2},\mathcal{O}_{2}}^{\text{wave}}(f).$

Now we can move to the statement of our main theorem.

######  Theorem 4.1.

Let $\mathscr{D}_{j}:=(M_{j},E_{j},g_{j},h_{E_{j}},\nabla^{E_{j}})$ for
$j=1,2$ be two Dirac bundles over closed Riemannian manifolds of dimension
$n\geq 2$, and let $\mathcal{O}_{j}\subset M_{j}$ be non-empty open sets.
Assuming there exists a diffeomorphism
$\psi:\mathcal{O}_{1}\to\mathcal{O}_{2}$ satisfying

$L_{\mathscr{D}_{1},\mathcal{O}_{1}}^{\text{wave}}(\psi^{*}f)=\psi^{*}(L_{\mathscr{D}_{2},\mathcal{O}_{2}}^{\text{wave}}f),\quad\forall
f\in{\mathcal{C}}^{\infty}_{0}((0,\infty)\times\mathcal{O}_{2};E).$

Then there is an isomorphism of Hermitian vector bundles $\Psi:E_{1}\to
E_{2}$, covering an isometry $\psi:M_{1}\to M_{2}$ which coincides with
$\Psi|_{E_{1}|_{\mathcal{O}_{1}}}=\psi$.

We follow the proof of [17], (which itself was an application to closed
manifolds of the boundary control method of Belishev [1] combined with the
crucial unique continuation method of Tataru [32]) extended to the bundle
case. The general structure of the proof is as follows

  1. (1)

The source-to-source solution operator
$(\mathcal{O},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$ determines the
distance function $d_{g}$ on $\mathcal{O}\times\mathcal{O}$

  2. (2)

The _source-to-source solution data_
$(\mathcal{O},g|_{\mathcal{O}},h_{E}|_{\mathcal{O}},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$
determines the _distance data_ $(\mathcal{O},g|_{\mathcal{O}},R(M))$

  3. (3)

The _distance data_ $(\mathcal{O},g|_{\mathcal{O}},R(M))$ determines the
topology, smooth structure and Riemannian structure of $(M,g)$

  4. (4)

New: The distance data determines the isomorphism class of $E\to M$ and
Hermitian metric $h_{E}$

  5. (5)

New: The distance data determines the Hermitian connection $\nabla^{E}$, and
thus the homomorphism of clifford multiplication from the identity
$[\nabla^{E},\mathpzc{cl}(\theta)]=\mathpzc{cl}(\nabla^{g}\theta)$

This uses several facts about the solution to the linear wave equation on
sections of $E$: the first is its finite speed of propagation.

######  Theorem 4.2.

Let $T>0$, and $p\in M$ be open and define the open cone

$C_{T,p}:=\\{(t,x)\in(0,T)\times M:d_{g}(x,p)<T-t\\}.$

Let $f\in L^{2}(\mathbb{R}\times M;E)$ and suppose $u$ solves

$\begin{cases}(\partial_{t}^{2}-\Delta_{E})u=f&(0,\infty)\times M\\\
f|_{C_{T,p}}=0\\\ u|_{\\{t=0\\}\times
B_{T}(p)}=\partial_{t}u|_{\\{t=0\\}\times B_{T}(p)}=0\end{cases}$

then $u|_{C_{T,p}}=0$.

To give the statement of the relevant unique continuation principle we also
define

$M(T,\mathcal{O})=\\{x\in M:d_{g}(x,\mathcal{O})\leq T\\}$

for the domain of dependence of the wave equation. As a result of the Carleman
estimates established in [11], the proof of the unique continuation theorem is
as in the scalar case since the wave equation being considered is principally
scalar (i.e. $\partial_{t}^{2}-\Delta_{E}$) see section 2.5 of [20],

######  Theorem 4.3.

Let $T>0$, and $\mathcal{O}\subset M$ be open and bounded. Let
$u\in{\mathcal{C}}^{\infty}_{0}(\mathbb{R}\times M;E|_{\mathcal{O}})$. Let

$\begin{cases}(\partial_{t}^{2}-\Delta_{E})u=0&(0,2T)\times
M(T,\mathcal{O})\\\ u|_{(0,2T)\times\mathcal{O}}\equiv 0\end{cases}$

Then $u|_{C(T,\mathcal{O})}\equiv 0$, for

$C(T,\mathcal{O})=\\{(t,x)\in(0,2T)\times
N:d_{g}(x,\mathcal{O})\leq\min\\{t,2T-t\\}\\}$

Given the proper form of unique continuation we should be able to prove
density of solutions with sources from the set,

$\mathcal{F}(T,\mathcal{O})=\\{f\in{\mathcal{C}}^{\infty}_{0}(\mathbb{R}\times
M;E)|\operatorname{supp}(f)\subset(0,T)\times\mathcal{O}\\}$

######  Theorem 4.4 (Approximate controllability).

Let $\mathcal{O}\subset M$ be open and bounded. For $T>0$ the set

$\mathcal{W}_{T}=\\{Wf(T,\cdot):f\in\mathcal{F}(T,\mathcal{O})\\}$

is dense in $L^{2}(M(T,\mathcal{O});E)$. Further, by considering time
reparametrization, we obtain that
$\\{Wf(T,\cdot):f\in{\mathcal{C}}^{\infty}_{0}((T-r,T)\times\mathcal{O};E)$ is
dense in $L^{2}(M(r,\mathcal{O});E)$ for all $r>0$.

###### Proof.

From the finite speed of propagation we have that $\mathcal{W}_{T}\subset
L^{2}(M(T,\mathcal{O});E)$. Thus it suffices to show that the orthogonal
complement of $\mathcal{W}_{T}$ contains only the origin. Let $\phi\in
L^{2}(M(T,\mathcal{O});E)$ satisfy $\langle
Wf(T,\cdot),\phi\rangle_{L^{2}(M;E)}=0$ for all
$f\in\mathcal{F}(T,\mathcal{O})$. Let
$u\in{\mathcal{C}}^{\infty}(\mathbb{R}\times M)$ solve

$\begin{cases}(\partial_{t}^{2}-\Delta_{E})u=0,&(0,T)\times M\\\
u|_{t=T}=0\;\partial_{t}u|_{t=T}=\phi\end{cases}$

From Green’s identities we have

$\langle f,u\rangle_{L^{2}((0,T)\times
M;E)}=\langle(\partial_{t}^{2}-\Delta_{E})Wf,u\rangle_{L^{2}((0,T)\times
M;E)}-\langle Wf,(\partial_{t}^{2}-\Delta_{E})u\rangle_{L^{2}((0,T)\times
M;E)}=0$

thus $u\equiv 0$ on $(0,T]\times\mathcal{O}$ by density of
$\mathcal{F}(T,\mathcal{O})\subset L^{2}((0,T)\times\mathcal{O};E)$. Extending
$u$ across $t=T$ by the odd reflection $u(t,x)=-u(2T-t,x)$, and denoting the
extension by $U$ we have that it satisfies

$\begin{cases}(\partial_{t}^{2}-\Delta_{E})U=0&(0,2T)\times M\\\
U|_{t=T}=0,\;\partial_{t}U|_{t=T}=\partial_{t}u|_{t=T}=\phi\end{cases}$

by our odd reflection, thus $U|_{(0,2T)\times\mathcal{O}}\equiv 0$. Now by
theorem 4.3 we conclude that $U|_{C(T,\mathcal{O})}\equiv 0$, in particular
since $\\{T\\}\times M(T,\mathcal{O})\subset C(T,\mathcal{O})$ we have
$\phi|_{M(T,\mathcal{O})}=\partial_{t}U|_{\\{T\\}\times
M(T,\mathcal{O})}\equiv 0$ as claimed. ∎

Having proven this fact, the proofs of 1) 2) and 3) from [17] generalize
immediately to the bundle-valued case. One lemma they depend on is the
Blagovestchenskii identity:

######  Lemma 4.5 (Blagovestchenskii Identity).

Let $(M,g)$ be complete. Let $T>0$ and $\mathcal{O}\subset M$ open and
bounded. Let $f,h\in\mathcal{F}(2T,\mathcal{O})$, then

$\langle Wf(T,\cdot),Wg(T,\cdot)\rangle_{L^{2}(M;E)}=\langle
f,(JL_{\mathscr{D},\mathcal{O}}^{\text{wave}}-(L_{\mathscr{D},\mathcal{O}}^{\text{wave}})^{*}J)g\rangle_{L^{2}((0,T)\times
M;E)}$

where $J:L^{2}((0,2T);E)\to L^{2}((0,T);E)$ is the time averaging operator
$J\phi(t)=\tfrac{1}{2}\int_{t}^{2T-t}\phi(s)ds$.

Another such lemma is that the source-to-source solution operator determines
the distance function:

######  Lemma 4.6.

Let $(M,g)$ and $\mathcal{O}\subset M$ be open and bounded. Then
$(\mathcal{O},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$ determines the
distance function $d_{g}$ on $\mathcal{O}\times\mathcal{O}$.

###### Proof.

Set $x,y\in\mathcal{O}$, and choose an auxiliary metric $d_{0}$ on
$\mathcal{O}$ which induces the same metric space topology on $\mathcal{O}$ as
$g$. Let $\varepsilon>0$ and consider
$\mathcal{B}_{x,\varepsilon}:=(0,\infty)\times B_{\varepsilon}^{d_{0}}(x)$ and

$t_{\varepsilon}:=\inf\\{t>0:\exists
f\in{\mathcal{C}}^{\infty}_{0}(\mathcal{B}_{x,\varepsilon};E),\;\operatorname{supp}{(L_{\mathscr{D},\mathcal{O}}^{\text{wave}}f)}(t,\cdot)\cap
B_{\varepsilon}^{d_{0}}(y)\neq\emptyset\\}.$

From finite propagation speed and lemma (4.4) we have

$t_{\varepsilon}=\text{dist}_{g}(B_{\varepsilon}^{d_{0}}(x),B_{\varepsilon}^{d_{0}}(y))$

and thus $\lim_{\varepsilon\to 0}t_{\varepsilon}=d_{g}(x,y)$. ∎

Let $\gamma_{y,\xi}(t)$ be the time-$t$ flow of the unique unit speed geodesic
starting at $y$ with initial velocity vector $\xi$. Several facts about
geometric determination in our setting will rely on the notion of cut time,

$\tau(y,\xi)=\sup\\{t>0:d_{g}(y,\gamma_{y,\xi}(t))=t\\}.$

Note that we have

$M=\\{\gamma_{y,\xi}(t)\in M:y\in\mathcal{O},\;\xi\in
S_{y}M,\;t<\tau(y,\xi)\\}$ (4.3)

for $\mathcal{O}\subset M$ open and non-empty. In other words any point in $M$
can be reached by geodesics originating in $\mathcal{O}$ which do not meet the
cut locus.

Further, the cut distance and distance between points can be determined using
only the source-to-source solution data

$(\mathcal{O},g|_{\mathcal{O}},h_{E}|_{\mathcal{O}},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$

namely as shown by [17] in the scalar case,

######  Proposition 4.7.

For any $y\in\mathcal{O}$, $\xi\in S_{y}M$ we can find the cut time
$\tau(y,\xi)$ from the source-to-source solution data
$(\mathcal{O},g|_{\mathcal{O}},h_{E}|_{\mathcal{O}},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$.

Further, given $z,y\in\mathcal{O}$, $\xi\in T_{y}\mathcal{O}$, $||\eta||=1$
and $r<\tau(y,\eta)$. Then
$(\mathcal{O},g|_{\mathcal{O}},h_{E}|_{\mathcal{O}},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$
determines $d_{g}(p,z)$ where $\gamma_{y,\xi}(r)=z$.

###### Proof.

For $y,\xi$ fixed the geodesic segment $\gamma_{y,\xi}([0,s])$ is determined
by the given data for $s$ small. Choosing $s>0$ sufficiently small that
$\gamma_{y,\xi}([0,s])\subset\mathcal{O}$, we observe that the condition

there exists $\varepsilon>0$ such that
$B_{r+\varepsilon}(x)\subset\overline{B_{s+r}(y)}$ (4.4)

determines the cut distance by the forumla,

$\tau(y,\xi)=\inf\\{s+r>0:r,s>0,\gamma_{y,\xi}([0,s])\subset\mathcal{O},\text{
\eqref{ball-cond} holds}\\}.$

This follows from its contrapositive, arguing via geodesic continuation of the
arc $\gamma_{y,\xi}([0,s])$ and the triangle inequality. So $\tau(y,\xi)$ is
determined by relation (4.4).

Define $S_{\varepsilon}(x,r)=(T-(r-\varepsilon),T)\times B_{\varepsilon}(x)$.
Using finite propagation speed and lemma (4.4) we can show that for $p,y,z\in
M$, $\varepsilon>0$, and $\ell_{p},\ell_{y},\ell_{z}>\varepsilon$ the
condition that

$B_{\ell_{p}}(p)\subset\overline{B_{\ell_{y}}(y)\cup B_{\ell_{z}}(z)}$ (4.5)

is equivalent to the existence of
$\\{f_{j}\\}\subset{\mathcal{C}}^{\infty}_{0}(S_{\varepsilon}(y,\ell_{y})\cup
S_{\varepsilon}(z,\ell_{z});E)$ for every
$f\in{\mathcal{C}}^{\infty}_{0}(S_{\varepsilon}(p,\ell_{p});E)$ such that

$||Wf(T,\cdot)-Wf_{j}(T,\cdot)||_{L^{2}(M;E)}\xrightarrow{j\to\infty}0.$ (4.6)

Now choosing $\varepsilon>0$ sufficiently small that $B_{\varepsilon}(y)\cup
B_{\varepsilon}(z)\subset\mathcal{O}$, and setting $y=z$,
$\ell_{y}=\ell_{z}=s+r$, and $\ell_{p}=r+\varepsilon$, we see that relation
(4.4) is equivalent to (4.6). However, using the Blagovestchenskii identity we
see that
$(\mathcal{O},g|_{\mathcal{O}},h_{E}|_{\mathcal{O}},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$
determines (4.6) and thus the cut time as claimed.

The proof of the second claim follows similarly, by considering $s\in(0,r)$
such that $\gamma_{y,\xi}([0,s])\subset\mathcal{O}$, and set
$\gamma_{y,\xi}(s)=p$. Then we can show that

$d_{g}(p,z)=\inf_{R>0}\\{\eqref{3ell}\text{holds with
$\ell_{p}=r-s+\varepsilon$, $\ell_{y}=r$, $\ell_{z}=R$ for some $R$ and
$\varepsilon>0$}\\}$

by first observing that $d_{g}(p,z)$ is less than or equal to this infimum
(since $r<\tau(y,\xi)$), and then showing this infimum is attained by a
similar argument as above, using the Blagovestchenskii identity and lemma 4.4.
From this formulation of $d_{g}(p,z)$ we have that it is determined by the
source-to-solution data. ∎

Using this last proposition and the fact of $\eqref{exp-closure}$, we have
that
$(\mathcal{O},g|_{\mathcal{O}},h_{E}|_{\mathcal{O}},L_{\mathscr{D},\mathcal{O}}^{\text{wave}})$
determines the family of distance functions,

$R(M):=\\{d_{g}(x,\cdot)|_{\mathcal{O}}:x\in
M\\}\subset\mathcal{C}^{0}(\overline{\mathcal{O}}).$

In Helin Lassas Oksanen Saksala they prove that this set can be topologized as
a smooth Riemannian manifold, isometric to $(M,g)$, and serving as a
background space depending only on the behavior of solutions in $\mathcal{O}$,
they show the two Riemannian manifolds are isometric. Their proof, depending
only on the distance functions, has no dependence on whether or not the given
wave equation is scalar or bundle-valued, and thus we obtain:

######  Proposition 4.8.

Let $\mathscr{D}_{j}:=(M_{j},E_{j},g_{j},h_{E_{j}},\nabla^{E_{j}})$ for
$j=1,2$ be two Dirac bundles over closed Riemannian manifolds of dimension
$n\geq 2$, and let $\mathcal{O}_{j}\subset M_{j}$ be non-empty open sets.
Assuming there exists a diffeomorphism
$\psi:\mathcal{O}_{1}\to\mathcal{O}_{2}$ satisfying

$L_{\mathscr{D}_{1},\mathcal{O}_{1}}^{\text{wave}}(\psi^{*}f)=\psi^{*}(L_{\mathscr{D}_{2},\mathcal{O}_{2}}^{\text{wave}}f),\quad\forall
f\in{\mathcal{C}}^{\infty}_{0}((0,\infty)\times\mathcal{O}_{2};E).$

Then $(M_{1},g_{1})$ and $(M_{2},g_{2})$ are isometric Riemannian manifolds.

Having shown the two Riemannian manifolds are isometric, it remains to show
that we can also recover the Hermitian bundle structure and connection.

######  Lemma 4.9.

Let $f,h\in\mathcal{F}(2T,\mathcal{O})$, for $T>0,\;\mathcal{O}\subset M$
open. Then $L_{\mathscr{D},\mathcal{O}}^{\text{wave}}$ determines the inner
products

$\langle Wf(T,\cdot),Wg(T,\cdot)\rangle_{L^{2}(M;E)}.$

Further, for any $x\in M(T,\mathcal{O})$ there exists functions

$\\{g_{\ell}\\}_{\ell=1}^{\emph{Rk}\;E}\subset\mathcal{F}(2T,\mathcal{O})$

such that $\\{Wg_{\ell}(T,x))\\}_{\ell=1}^{\emph{Rk}\;E}$ forms an orthonormal
basis of $E_{x}$.

###### Proof.

The first claim follows immediately from the Blagovestchenskii identity above.
For the second claim we note that it suffices to prove $E_{x}$ is spanned by
the vectors $Wg(T,x)$, for $g\in\mathcal{F}(T,\mathcal{O})$. Thus we are left
to show that if $e\in E_{x}$ and

$\langle e,Wg(T,x)\rangle_{h_{E}}=0,\quad g\in\mathcal{F}(T,\mathcal{O})$

then $e=0$. By duality, applying $W^{*}:H^{-s}(M;E)\to
H^{-s^{\prime}}((0,T)\times M;E)$ to $e\delta_{x}$ gives a solution to the
initial value problem

$\begin{cases}(\partial_{t}^{2}-\Delta_{E})u=0\\\
u|_{t=T}=0,\;\partial_{t}u|_{t=T}=e\delta_{x}\end{cases},$

i.e. $W^{*}(e\delta_{x})=u$. Hence we have $t\mapsto\langle
u(t,x),g(t,x)\rangle_{h_{E}}=0$ for all $t\in(0,T)$. Combining this with the
initial condition on $u$ we see $u|_{(0,T]\times M}\equiv 0$. Extending $u$ by
the odd reflection across $t=T$ we obtain a section now satisfying
$(\partial_{t}^{2}-\Delta_{E})u=0$ on $(0,2T)\times M$, thus by 4.3 we have
$e=0$ as claimed. ∎

In particular this implies that $h_{E}$ restricted to $E|_{M(T,\mathcal{O})}$
for $T>0$ sufficiently small is determined by
$L_{\mathscr{D},\mathcal{O}}^{\text{wave}}$.

Next we show how to construct sections of $E$ out of double sequences of
functions supported in $(T-r,T)\times\mathcal{O}$. The following lemma is
adapted from [21]:

######  Lemma 4.10.

Let $\mathcal{O}\subset M$ be open and let $x\in M$. Then there exists
$y\in\mathcal{O},\xi\in S_{y}M$ such that $\gamma_{y,\xi}(t)=x$ for
$t<\tau(y,\xi)$. We define $t_{k}=t+\tfrac{1}{k}$, and set

$Y_{k}=\mathcal{O}\cap B_{1/k}(y),\quad
X_{k}=M_{\theta<1/k}(Y_{k},t_{k})\setminus M(\mathcal{O},t).$

where

$M_{\theta<1/k}(Y_{k},t_{k}):=\\{z\in M(Y_{k},t_{k}):\exists\widetilde{y}\in
Y_{k},d_{g}(z,\widetilde{y})<t-\tfrac{1}{k},\;\exists\eta\in
T_{y}M,\gamma_{y,\eta}(\tfrac{1}{k})=y,\;\theta(\xi,\eta)<\tfrac{1}{k}\\}$

with $\theta(\cdot,\cdot)$ equal to the angle between two vectors in $T_{y}M$
with respect to the inner product induced by $g$. Suppose
$\Phi^{x}=\\{f_{jk}\\}_{j,k=1}^{\infty}\subset{\mathcal{C}}^{\infty}((T-t_{k},T)\times
Y_{k};E)$ satisfies

  1. (1)

For each $k=1,2,\ldots$ the sequence $\\{Wf_{jk}(T,\cdot)\\}$ converges weakly
in $L^{2}(M;E)$ to function supported in $X_{k}$.

  2. (2)

There exists $C>0$ such that

$||Wf_{jk}(T,\cdot)||_{L^{2}(M;E)}\leq\frac{C}{\emph{Vol}(X_{k})^{1/2}}$

  3. (3)

The limit $\lim_{j,k\to\infty}\langle
Wf_{jk}(T,\cdot),Wg(T,\cdot)\rangle_{L^{2}(M;E)}$ exists for every
$g\in\mathcal{F}(2T,\mathcal{O})$.

Then there is a vector $e(x;\Phi^{x})\in E_{x}$ such that

$\lim_{j,k\to\infty}\langle Wf_{jk}(T,\cdot),\phi\rangle_{L^{2}(M;E)}=\langle
e(x;\Phi^{x}),\phi(x)\rangle_{h_{E}},\quad\phi\in{\mathcal{C}}^{\infty}(M;E)$

This lemma is a consequence of lemma (4.9), and otherwise the proof is the
same as given in [21], using the fact that from its construction
$\text{diam}(X_{k})\to 0$ and $X_{k}\to x$. Given this criterion for
constructing vectors of $e\in E_{x}$ we observe that such a double sequence
$\Phi^{x}=\\{f_{jk}\\}_{j,k=1}^{\infty}$ satisfying the hypotheses of lemma
(4.10) exists by applying lemma (4.4) to construct a sequence for each
$k=1,2,\ldots$

$Wf_{jk}\xrightarrow{j\to\infty}\frac{1_{X_{k}}e}{\operatorname{Vol}{X_{k}}}$

where $1_{X_{k}}$ is the indicator function of $X_{k}$. This sequence clearly
satisfies the conclusions of lemma (4.10), as claimed. Further, if we choose
$\Phi^{x}$ for $x\in U$ such that

$x\mapsto\langle e(x;\Phi^{x}),Wg(T,\cdot)\rangle_{h_{E}},\quad
g\in\mathcal{F}(2T,\mathcal{O})$

is smooth, then the assignment $e(x)=e(x;\Phi^{x})$ will be a smooth section
of $E|_{U}$ as this implies $e$ depends smoothly in $x$ with respect to an
orthonormal frame $\\{Wg_{\ell}(T,\cdot)\\}_{\ell=1}^{\text{Rk}E}$ by lemma
(4.9).

######  Theorem 4.11.

Let $\mathcal{O}\subset M$ be open, non-empty, and bounded. Then the Dirac
bundle $\mathscr{D}$ comprised of $(M,g)$, Hermitian vector bundle with
connection $(E,h_{E},\nabla^{E})$ over $M$, and clifford multiplication
$\mathpzc{cl}:\mathbb{C}l(T^{*}M,g)\to\operatorname{End}{E}$ are all
determined by the source-to-source solution operator
$L_{\mathscr{D},\mathcal{O}}^{\text{wave}}$ and the Hermitian vector bundle
$E|_{\mathcal{O}}$.

###### Proof.

Having already shown the Riemannian manifold $(M,g)$ is determined by
$L_{\mathscr{D},\mathcal{O}}^{\text{wave}}$, so it remains to show the given
data also determines the Hermitian bundle with connection. By making
$\mathcal{O}$ smaller if needed, we may assume that $\mathcal{O}\subset M$ is
contractible and thus $E|_{\mathcal{O}}$ is trivial.

For each $x\in\mathcal{O}$ we can choose a double sequence
$\Phi^{x}=\\{f_{jk}^{x}\\}_{j,k=1}^{\infty}$ satisfying the criteria of lemma
4.10 and hence converges to a prescribed $e\in E_{x}$. If we further require
that this family $(\Phi^{x})_{x\in\mathcal{O}}$ be chosen such that

$x\mapsto\langle e(x;\Phi^{x}),Wg(T,\cdot)\rangle_{h_{E}},\quad
g\in\mathcal{F}(2T,\mathcal{O}),$

is smooth, then $e(x)=e(x;\Phi^{x})$ will be smooth in $\mathcal{O}$ as well.

Beginning with an orthonormal frame $\\{e_{\ell}(x)\\}$ for
$E|_{\mathcal{O}}$, we can similarly choose for each of the
$\ell=1,\ldots,\text{Rk}\;E$ families of double sequences $\Phi_{\ell}^{x}$
representing $e_{\ell}(x)$. Thus the coefficient functions $h_{i,j}(x)=\langle
e_{i}(x),e_{j}(x)\rangle_{h_{E}}=\delta_{ij}(x)$ are determined by
$L_{\mathscr{D},\mathcal{O}}^{\text{wave}}$, and in this trivialization
$E|_{\mathcal{O}}$ we can express the Hermitian metric as $\langle
v,w\rangle_{h_{E}}=\langle
v^{\ell}e_{\ell}(x),w^{\kappa}e_{\kappa}(x)\rangle_{h_{E}}=\sum
v^{\ell}\overline{w^{\ell}}$.

To determine the bundle and metric globally we need only determine the bundle
transition functions. From compactness $\operatorname{inj}_{g}>c$, and thus
there exists a finite cover $\\{B_{c}(p_{i})\\}_{i=1}^{N}$ of $M$. Further,
for each $x\in M$ there exists $y\in\mathcal{O}$ and $\xi\in S_{y}M$ such that
$\gamma_{y,\xi}(t)=x$ with $t<\tau(y,\xi)$. Since each
$E|_{B_{c}(p_{i})}\simeq B_{c}(p_{i})\times E_{p_{i}}$ is trivial, it remains
only to determine the transition functions
$T^{ij}\in{\mathcal{C}}^{\infty}(B_{c}(p_{i})\cap
B_{c}(p_{j});\operatorname{End}(E_{p}))$. With respect to the chosen
orthonormal frame $e_{\ell}(x)$ over $B_{c}(p_{i})\cap B_{c}(p_{j})$ we can
write this matrix valued function
$T^{ij}(x)=(T_{\ell,\kappa}^{ij}(x))_{\ell,\kappa=1}^{\text{Rk}E}$ in terms of
the frame, i.e. by using the local expression of the metric $h_{E}$ over
$B_{c}(p_{i})\cap B_{c}(p_{j})$. The frame $e_{\ell}(x)$, being generated by
applying lemma 4.10 to points $x\in B_{c}(p_{i})\cap B_{c}(p_{j})$, is thus
determined by $L_{\mathscr{D},\mathcal{O}}^{\text{wave}}$ as above. Thus we
have recovered the transition functions, hence the entire Hermitian bundle and
metric.

Having determined the transition functions, to determine the connection it
suffices to determine $\nabla^{E}$ on a trivialization $E|_{U}\simeq U\times
E_{p}$ where $\nabla^{E}=d+A$, for $A\in\Omega^{1}(U;E)$. This portion of the
argument follows that of Kurylev Oksanen Paternain and we include it for
completeness. Choose $T>0$ sufficiently large that $U\subset
M(2T,\mathcal{O})$. Since $Wf(T,\cdot)$ is determined by
$L_{\mathscr{D},\mathcal{O}}^{\text{wave}}$ for all
$f\in\mathcal{F}(2T,\mathcal{O})$, and the wave equation (4.2) is time-
translation invariant, we have also determined $Wf(t,\cdot)$ for $t\in(0,T)$.
Differentiating twice in $t$ gives $\Delta_{E}Wf(t,\cdot)$ for $t\in(0,T)$.
Hence by duality

$\langle f(T,\cdot),\phi\rangle_{L^{2}(M;E)}=\langle
Wf(T,\cdot),\Delta_{E}\phi\rangle_{L^{2}(M;E)}\quad\phi\in{\mathcal{C}}^{\infty}_{0}(U;E),$

thus by lemma (4.4) we can determine and densely approximate $L^{2}(U;E)$ by
the functions $Wf(T,\cdot)$ thus determining $\Delta_{E}\phi$. Now, because
$\Delta_{E}$ differs from $(\nabla^{E})^{*}\nabla^{E}$ by a zeroth order term,
and $\nabla^{E}=d+A$ over $U$, we compute

$(\nabla^{E})^{*}\nabla^{E}\phi=d^{*}d\phi-2(A,d\phi)+(d^{*}A)u-(A,Au),$

and conclude that the principal part of $(\nabla^{E})^{*}\nabla^{E}$ is
$d^{*}d$ and its first order part is $2(A,d\phi)=2g^{jk}A_{j}\partial_{k}\phi
dx^{k}$. Because the principal part depends only on the metric $g$, we can
determine $d^{*}d$ in $U$. For $x\in U$ we consider
$\phi(x)=\phi_{\ell}^{k}(x)$ such that $\phi(x)=0$ and
$\partial_{j}\phi(x)=\delta_{jk}e_{\ell}(x)$. Thus we can determine the
endomorphism piece $A$ of the connection from the first order term in
$(\nabla^{E})^{*}\nabla^{E}\phi(x)$, i.e. determine $A\in\Omega^{1}(U;E)$ from
the determination of

$((\nabla^{E})^{*}\nabla^{E}-d^{*}d)\phi=-2(A,d\phi)=-2g^{ik}(x)A_{i}e_{\ell}(x)$

for $x\in U$.

Finally, having determined every element of the Dirac bundle $\mathcal{D}$
except for the Clifford multiplication homomorphism
$\mathpzc{cl}:\mathbb{C}l(T^{*}M,g)\to\operatorname{End}(E)$ we observe that
having determined both $\nabla^{E}$ and $\nabla^{g}$ we have fixed
$\mathpzc{cl}$ due to the Clifford compatibility condition that

$[\nabla^{E},\mathpzc{cl}(\theta)]=\mathpzc{cl}(\nabla^{g}\theta),\quad\theta\in
T^{*}M.$

∎

###### Proof of Corollary 1.2.

We follow the proof of [17, Cor 2]. To obtain the total spectral data on
$\mathcal{O}$ we notice that not only are the remaining negative eigenvalues
simply minus of the positive eigenvalues, but the unknown negative
eigensections are simply the image of the positive eigensections under the
_Chirality operator_ ,

$\gamma=i^{\lceil\frac{m}{2}\rceil}\mathpzc{cl}(e_{1})\mathpzc{cl}(e_{2})\ldots\mathpzc{cl}(e_{m})\in\operatorname{End}(E_{x}),$

where $\\{e_{j}\\}_{j=1}^{m}$ is any choice of orthonormal basis of
$T_{x}\mathcal{O}$. Thus, because $\not{D}\varphi=\lambda\varphi$ implies
$\not{D}(\gamma\varphi)=-\lambda\;\gamma\varphi$ we now have knowledge of all
eigensections restricted to $\mathcal{O}$. Writing the
$\not{D}^{2}=\Delta_{E}$ eigensections as
$\psi_{k}(x)=\varphi_{k}(x)-\gamma\circ\varphi_{k}(x)$, we now have
$\\{\lambda_{k}^{2},\psi_{k}(x)\\}_{j=1}^{\infty}$ as a spectral resolution of
$\Delta_{E}$ on $L^{2}(M;E)$. From here, as in [17], given any
$f\in{\mathcal{C}}^{\infty}_{0}(\mathbb{R}^{+}\times\mathcal{O})$, we can
calculate the approximate ‘Fourier coefficients’ of the wave source-to-
solution operator $\mathcal{I}_{k}(t)=\langle
w^{f}(t,\cdot),\psi_{k}\rangle_{L^{2}}$, and from there recover the
approximate source-to-solution operator
$\sum\limits_{k=1}^{\infty}\widetilde{\mathcal{I}}_{k}(t)\varphi_{k}(x)=W[h\cdot
f](t,x)$ (i.e. the solution associated to a conformal change of source) with
$\widetilde{\mathcal{I}}_{k}(t)=\int_{0}^{t}\int_{\mathcal{O}}\frac{\sin(\lambda_{k}t)}{\sin(\lambda_{k})}f(s,x)\varphi_{j}(x)\;h\cdot
dV_{g}(x)ds$. From here the result follows from 1.1 (after determining the
conformal factor $h$, as in [17, Thm 6]). ∎

##  5\. acknowledgements

Travel funding in support of this work was provided to HQ by the Milliman
committee of UW math. The research of GU was partly supported by NSF, a Simons
Fellowship, a Walker Family Endowed Professorship at UW and a Si-Yuan
Professorship at IAS, HKUST.

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# The Guts of nearly fibered knots

Zhenkun Li  Department of Mathematics, Stanford University
<EMAIL_ADDRESS>and  Fan Ye  Department of Mathematics, Harvard
University<EMAIL_ADDRESS>
###### Abstract.

The guts of a knot is an invariant defined for the knot complement by Agol-
Zhang. Nearly fibered knots, which are defined as knots whose Floer homology
has dimension two in the top Alexander grading, were introduced by Baldwin-
Sivek. In this note, we provide three models for the guts of nearly fibered
knots in the 3-sphere. As a corollary, the nearly fibered condition can be
purely topologically characterized and is independent of the specific version
of Floer theory.

##  1\. Introduction

By work of Ni [11] and Kronheimer-Mrowka [7], a knot $K\subset S^{3}$ is
fibered if and only if its knot homology in any branch of Floer theory is
$1$-dimensional in the top Alexander grading. Hence it is natural to ask what
happens if the top grading summand of the knot homology is $2$-dimensional.
Recently, Baldwin-Sivek in [3] introduced the following definition.

######  Definition 1.1.

A knot $K\subset S^{3}$ is said to be nearly fibered (in the Heegaard Floer
sense) if

$\widehat{HFK}(S^{3},K,g(K);\mathbb{Q})\cong\mathbb{Q}^{2}.$

Their definition is stated with Heegaard Floer theory, but we can also define
nearly fibered knots in the instanton sense by requiring

$KHI(S^{3},K,g(K))\cong\mathbb{C}^{2},$

where $KHI$ denotes the instanton knot homology [7] of $K\subset S^{3}$.

In this note, we show that the nearly fibered condition has a purely
topological criterion, and is independent of the branches of Floer theory. To
better describe this criterion, we use the notion of guts of knots recently
introduced by Agol-Zhang [1].

Given a knot $K\subset S^{3}$, we can view its complement $S^{3}\backslash
N(K)$ as a sutured manifold with its whole boundary being the suture. We can
pick a maximal collection of pair-wise disjoint and pair-wise non-parallel
minimal-genus Seifert surfaces $S$ of $K$, and perform a sutured manifold
decomposition

(1.1) $S^{3}\backslash N(K)\stackrel{{\scriptstyle
S}}{{\leadsto}}(M^{\prime},\gamma^{\prime}).$

We can then pick a maximal collection of pair-wise disjoint and pair-wise non-
parallel non-trivial product annuli $A$ inside $(M^{\prime},\gamma^{\prime})$
and perform a second sutured manifold decomposition

(1.2) $(M^{\prime},\gamma^{\prime})\stackrel{{\scriptstyle
A}}{{\leadsto}}(M,\gamma)\sqcup(M_{1},\gamma_{1}).$

Here $(M_{1},\gamma_{1})$ is a product sutured manifold, and any component of
$(M,\gamma)$ is not a product.

######  Definition 1.2 ([1]).

The guts of a knot $K\subset S^{3}$ is defined to be the sutured manifold
$(M,\gamma)$.

######  Theorem 1.3 ([1, Theorem 1.1]).

The guts of a knot $K\subset S^{3}$ is well-defined, i.e., independent of the
choices of maximal collections of Seifert surfaces and product annuli in the
construction.

In this note, we prove that a knot $K\subset S^{3}$ is nearly fibered if and
only if its guts falls into one of the three basic models described below.
Note that in our paper we only state and prove the theorem in instanton
theory, but a similar argument applies to Heegaard Floer theory as well.

######  Theorem 1.4.

Suppose $K\subset S^{3}$ is a knot of genus $g$. Let $(M,\gamma)$ be its guts.
Then we have

$KHI(S^{3},K,g)\cong\mathbb{C}^{2},$

if and only if its guts $(M,\gamma)$ falls into one of the following three
models:

  1. (M1)

$M$ is a solid torus and $\gamma$ consists of four longitudes.

  2. (M2)

$M$ is a solid torus and $\gamma$ consists of two curves of slope $2$.

  3. (M3)

$M$ is the complement of the right handed trefoil and $\gamma$ consists of two
curves of slope $-2$.

We have the following corollary.

######  Corollary 1.5.

A knot is nearly fibered in the instanton sense if and only if it is nearly
fibered in the Heegaard Floer sense.

Acknowledgement The authors would like to thank John A. Baldwin and Steven
Sivek for helpful conversations and for telling us about their work on
classifying the genus-one nearly fibered knots.

##  2\. Proofs and comments

###### Proof of Theorem 1.4.

We first prove the necessary condition. Suppose $K\subset S^{3}$ is a
genus-$g$ nearly fibered knot. Let $(M,\gamma)$ be its guts. We first study
the sutured manifold decomposition in (1.1).

Claim. Any maximal collection of pair-wise disjoint and pair-wise non-parallel
minimal-genus Seifert surfaces in fact contains only one Seifert surface.

###### Proof of the Claim.

Suppose $S$ is a minimal-genus Seifert surface of $K$. We can perform a
sutured manifold decomposition of $S^{3}\backslash N(K)$ along $S$:

$S^{3}\backslash N(K)\stackrel{{\scriptstyle
S}}{{\leadsto}}(S^{3}\backslash[-1,1]\times S,\\{0\\}\times\partial S).$

By the proof of [7, Proposition 7.16], we know that there is an isomorphism

(2.1) $SHI(S^{3}\backslash[-1,1]\times S,\\{0\\}\times\partial S)\cong
KHI(S^{3},K,g)\cong\mathbb{C}^{2}.$

If there is another minimal-genus Seifert surface $S^{\prime}$ that is
disjoint from $S$ and is not parallel to $S$, then $S^{\prime}$ also induces a
surface in $(S^{3}\backslash[-1,1]\times S,\\{0\\}\times\partial S)$, which
implies the sutured manifold is not horizontally prime. From the instanton
version of [7, Proposition 6.5 and Proposition 6.6], we know that one of the
two pieces obtained from $(S^{3}\backslash[-1,1]\times S,\\{0\\}\times\partial
S)$ by cutting along $S^{\prime}$ must have $1$-dimensional sutured homology,
because $2$ is a prime number. From [7, Theorem 7.18], that piece is a product
sutured manifold, which contradicts the assumption that $S^{\prime}$ is not
parallel to $S$. ∎

Now the claim above and Theorem 1.3 imply that the sutured manifold
$(M^{\prime},\gamma^{\prime})$ in (1.1) can be taken to be simply the
complement of $S$:

$(M^{\prime},\gamma^{\prime})=(S^{3}\backslash[-1,1]\times
S,\\{0\\}\times\partial S).$

Next, we study the sutured manifold decomposition (1.2). Note that by
construction $(M_{1},\gamma_{1})$ is a product sutured manifold so from [7,
Theorem 7.18] and the instanton version of [7, Proposition 6.5 and Proposition
6.7], we know that

$SHI(M,\gamma)\cong SHI(S^{3}\backslash[-1,1]\times S,\\{0\\}\times\partial
S)\cong\mathbb{C}^{2}.$

Also, the same argument as above shows that $(M,\gamma)$ is horizontally
prime. Thus we conclude that $(M,\gamma)$ is reduced in the sense of [5,
Definition 2.12]. Then [4, Corollary 1.6] applies and we conclude that

$g(\partial M)\leq 2-1=1.$

If $g(\partial M)=0$, since $M$ is irreducible, we know $M=B^{3}$. Then for
any possible $\gamma$ on $\partial M$, we cannot have
$SHI(M,\gamma)\cong\mathbb{C}^{2}$. Hence we must have $\partial M\cong
T^{2}$. It is well-known that any torus in $S^{3}$ bounds a solid torus. Hence
we have two cases.

Case 1. The manifold $M$ is a solid torus.

The instanton Floer homology of any sutured solid torus can be found in [8,
Section 4.3]. So the only two models are the ones as stated in (M1) and (M2).

Case 2. The manifold $S\backslash M$ is a solid torus, i.e., there is a knot
$J\subset S^{3}$ so that $M\cong S^{3}\backslash N(J)$.

Suppose $\gamma$ has $2n$ components. Let $\gamma_{2}$ be the union of two
components of $\gamma$. By the proof of [6, Theorem 3.1], we know that

$SHI(M,\gamma)\cong\mathbb{C}^{2^{n-1}}\otimes SHI(M,\gamma_{2}).$

Since

$SHI(M,\gamma)\cong\mathbb{C}^{2},$

either $n=2$ and $SHI(M,\gamma_{2})\cong\mathbb{C}$ or $n=1$. For the former
case, from [7, Theorem 7.18] we know $M$ must also be a solid torus which
reduces to Case 1. For the latter case, we further divide it into two sub-
cases.

Case 2.1. Each component of $\gamma$ represents a generator of $\ker
i_{*}\subset H_{1}(\partial M)\cong\mathbb{Z}^{2}$, where

$i_{*}:H_{1}(\partial M)\rightarrow H_{1}(M)$

is the map induced by the natural inclusion

$i:\partial M\hookrightarrow M.$

In this case, if $H_{2}(M,\partial M)$ is generated by a disk, then
$(M,\gamma)$ is non-taut and $SHI(M,\gamma)=0$ by the adjunction inequality
(c.f. [7, Proposition 7.5]). If any properly embedded surface representing a
generator of $H_{2}(M,\partial M)$ has genus at least $1$, then we can pick
any such generator, say $T\subset M$, of the minimal genus. The surface $T$
then induces a grading on $SHI(M,\gamma)$, and we know from [4, Lemma 6.2]
that we have two taut decompositions

$(M,\gamma)\stackrel{{\scriptstyle\pm T}}{{\leadsto}}(M_{\pm},\gamma_{\pm}).$

Since $g(T)\geq 1$, we know from the proof of [7, Proposition 7.11] that there
is an inclusion

$SHI(M_{+},\gamma_{+})\oplus SHI(M_{-},\gamma_{-})\hookrightarrow
SHI(M,\gamma).$

Furthermore, both $\gamma_{+}$ and $\gamma_{-}$ contain at least three
components that are parallel to each other. From [7, Theorem 7.12] and the
proof of [6, Theorem 3.1] that

$\dim SHI(M_{\pm},\gamma_{\pm})\geq 2.$

As a result, we have

$\dim SHI(M,\gamma)\geq 4.$

which leads to a contradiction in this case.

Case 2.2. Components of $\gamma$ do not represent generators of $\ker i_{*}$.

Let $Y$ be the Dehn filling of $M$ along a component of $\gamma$. Then [10,
Proposition 1.4] implies

$\dim I^{\sharp}(Y)\leq\dim SHI(M,\gamma)=2.$

Note that the proof of [10, Proposition 1.4] also implies that

$\dim I^{\sharp}(Y)\equiv\dim SHI(M,\gamma)~{}{\rm mod}~{}2.$

Hence by [12, Corollary 1.4], we know that

$\dim I^{\sharp}(Y)=2=|H_{1}(Y)|.$

Note that by [2, Theorem 1.10], the unknot and the right-handed trefoil are
the only two knots on which the positive Dehn surgeries induce instanton
L-spaces $Y$ with $|H_{1}(Y)|=2$. The case of unknot still reduces to Case 1.
The case of the right-handed trefoil is a new one. By [2, Theorem 1.1, Table
1], the surgery slope must be $2$. Hence $\gamma$ consists of curves of slope
$2$, which concludes the proof of the necessary condition.

Finally, the sufficient condition follows immediately from the first
isomorphism in (2.1) and the fact that gluing a product sutured manifold other
than a $3$-ball to an arbitrary sutured manifold via identification of a
suture does not change the sutured instanton Floer homology. (c.f. [7,
Proposition 6.7]) ∎

######  Remark 2.1.

We have the following comments which strengthen the description of the guts in
Theorem 1.4.

  1. (1)

We can compute the Euler characteristic in each of the three models. From [6],
for the first model, we have

$\chi(KHI(S^{3},K,g(K)))=\chi(SHI(M,\gamma))=0.$

As a result, we know that the symmetrized Alexander polynomial $\Delta_{K}(t)$
of $K$ has degree at most $g(K)-1$. On the other hand, if $(M,\gamma)$ is one
of the rest two models, we can compute as in [9] that

$\chi(SHI(M,\gamma))=\pm 2.$

As a result, we know that $\Delta_{K}(t)$ has degree $g(K)$ and the top non-
zero coefficients are $\pm 2$.

  2. (2)

The proof of [3, Lemma 3.4] implies that the Seifert surface complement
$(S^{3}\backslash[-1,1]\times S,\\{0\\}\times\partial S)$ admits no product
annuli whose boundary has a component that is parallel to the suture
$\\{0\\}\times\partial S$ on $\partial(S^{3}\backslash[-1,1]\times S)$. As a
result, we can further conclude that $\partial F$ must have one more component
than $\gamma$, and all but one components of $\partial F$ are glued to all of
$\gamma$. This actually rules out one model in the case $g(K)=1$ as in the
following example.

######  Example 2.2.

When $g(K)=1$, we know that

$S\cong(R_{+}(\gamma)\cup\\{1\\}\times F).$

Since in all three models we have $\chi(R_{+}(\gamma))=0$, we know that

$\chi(F)=-1.$

From part (2) of the Remark 2.1, we know that $\partial F$ has one more
component than $\gamma$. Then $\chi(F)$ rules out the model in which $\gamma$
has four components. As a result, we only have two models:

  * •

$M$ is the complement of the unknot and $\gamma$ consists of two curves of
slope $2$.

  * •

$M$ is the complement of the right handed trefoil and $\gamma$ consists of two
curves of slope $2$.

Furthermore, in this case, the surface $F$ must be a pair of pants. Yet gluing
such a thickened pair of pants to $(M,\gamma)$ along two of the three boundary
components is equivalent to gluing a product $1$-handle to $(M,\gamma)$.
Turning this around, we know that $(M,\gamma)$ being one of the above two
models is obtained from the complement of the Seifert surface by a disk
decomposition. This coincides with the discussion in [3, Section 1.2] right
above [3, Theorem 5.1]. Note that these two models do exist: for example, they
give rise to the knot $5_{2}$ in Rolfsen’s table and the 2-twisted Whitehead
double of the right-handed trefoil with positive clasp.

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