# 3-d topological quantum memory with a power-law energy barrier

Kamil P. Michnicki  Department of Physics, University of Washignton, Seattle,
WA USA.

###### Abstract

We discuss energy barriers and their relationship to self-correcting quantum
memories. We introduce the solid code, a 3-d version of Kitaev’s surface code,
and then combine several solid codes using a technique called welding. The
resulting code is a $[[O(L^{3}),1,O(L^{\frac{4}{3}})]]$ stabilizer code with
an energy barrier of $O(L^{\frac{2}{3}})$, which is an exponential improvement
over the previous highest energy barrier in 3-d. No-go results are avoided by
breaking microscopic translation invariance.

##  I Introduction

An important problem in the quantum computing community is whether it is
possible to make a quantum version of the ferromagnetic hard disc drive. Such
a medium could be used to protect a quantum state from decoherence without the
need to actively detect and correct errors. The 4-d toric code Hamiltonian,
Dennis et al. (2002); Alicki et al. (2008), a spin system, is a theoretical
example of such a self-correcting quantum memory. It uses a macroscopic energy
barrier to prevent noise from accumulating and corrupting stored quantum
information. It is a major open question whether such a system can exist in
less than four dimensions. The problem is intimately related to the problem in
condensed matter physics of whether topological order can exist at non-zero
temperatures Nussinov and Ortiz (2008). Most results for self-correcting
quantum memories in 2-d and 3-d to date have been negative Bravyi and Terhal
(2009); Yoshida (2011); Landon-Cardinal and Poulin (2013) or use operators of
unbounded strength Hamma et al. (2009). One exception has been the cubic code
Haah (2011), but that too has an energy barrier of only $\log(L)$. In this
letter, we improve the best known energy barrier for spin Hamiltonians with
topological order from $O(\log L)$ to $O(L^{2/3})$. Both the Haah code and the
code proposed in this letter can be shown Bravyi and Haah (2011) to give a
theoretical increase in storage time when the system size is increased up to a
temperature dependent maximum. It is open whether there truly is a maximum
system size for these codes past which the storage time decreases but we
conjecture that there is. The goal is to have a storage time scaling
exponentially with the size, i.e. number of spins, of the system. The result
presented in this letter can be viewed as a stepping stone towards this goal.

To gain some intuition, consider the ferromagnetic hard disc drive. It uses
the net magnetization of a ferromagnet to store bits of information. At room
temperature the net magnetization is stable against a global change in
polarization because of a large energy barrier separating states of opposite
polarization. If the magnetization of a small domain flips, due to noise,
there will be an energy penalty proportional to the perimeter of the domain.
At sufficiently low temperatures, this tension tends to shrink it; a
phenomenon which leads to a stable classical memory that is self correcting.

The requirements for quantum memories are more stringent than those for
classical memories. Classical memories need only have a stable order parameter
while quantum memories must simultaneously hide the order parameter. Luckily,
there are many local error correcting codes that achieve this such as the 2-d
and 3-d toric code and color codes Kitaev et al. (1997); Castelnovo and Chamon
(2007); Bombin and Martin-Delgado (2006). The ground state observables of
these depend on the topology of the system, not on any local observables. In
this way they hide and therefore protect the stored superposition. Although
these systems are resilient against a certain rate of local noise, the noise
can still build-up past the point where it can affect observables on the
stored quantum state. The 4-d toric code Hamiltonian fights this build up of
errors by ensuring that any sequence of local operations that can change such
observables must have a macroscopically large energy at some point in the
sequence. Thermalization to lower energies prevents such build up of errors.
The hope is that if we can find a large energy barrier in 3-d for a local spin
system with topological order then the system might be a good quantum memory.

The toric codes, color codes, cubic code as well as the code presented in this
letter are all examples of stabilizer codes. We review stabilizer codes now.
The Pauli group is defined by

$G=\\{(i)^{k}P_{1}\otimes...\otimes
P_{n}:k\in\\{0,1,2,3\\},P_{i}\in\\{I,X,Y,Z\\}\\}.$ (1)

where $X$, $Y$ and $Z$ are single qubit Pauli operators. A stabilizer group
$S$ is a subgroup of the Pauli group such that $-I\notin S$. This implies that
there exists a subspace $\mathcal{H}_{c}$, such that for all
$\left|\psi\right>\in\mathcal{H}_{c}$ and for all $h\in S$,
$h\left|\psi\right>=\left|\psi\right>$. This subspace is what we call the code
space. The protocol to do error correction with a stabilizer code is to
measure operators from the stabilizer group and to perform error correcting
operations based on the value of the measurements. For an ideal self-
correcting quantum memory, the environment does the error correction by
thermalizing to lower energy states.

Logical operators are operators that map one codeword to another and keep
correctable states correctable. For simplicity we will choose logical
operators that are in the Pauli group and which commute with the stabilizer
group. The code presented in this letter is of a special type Calderbank and
Shor (1996). It has a generating set where each generator is either a tensor
product of exclusively $X$ operators or exclusively $Z$ operators. We will be
considering logical operators of the same type.

The energy barrier of a code is defined with respect to a Hamiltonian. The
Hamiltonian is defined in terms of a generating set $R$ such that the group
generated by multiplication of elements of $R$ generates $S$. That is,
$\langle R\rangle=S$. The Hamiltonian is $H=-\sum_{h\in R}h$. The ground state
subspace is exactly the code space of the stabilizer group $S$. The energy
barrier is defined with respect to a local sequence of errors that maps one
ground state to another, i.e. that enacts a logical operator. The minimum peak
energy over all such sequences is what we call the energy barrier.

In this letter, we will be discussing the energy barrier exclusively. But the
motivation to study the energy barrier comes from considering the lifetime of
a memory subject to thermal noise. Thermal noise is typically modeled Bravyi
and Haah (2011); Chesi et al. (2010); Alicki et al. (2008); Davies (1976) by a
set of local jump operations, i.e. the errors, which occur with a rate
proportional to $\exp(-\beta\Delta)$. $\Delta$ is the change in energy upon
the application of the error. Thus the larger the energy barrier the more
thermal noise is suppressed in this model.

Main result–There exists a local stabilizer Hamiltonian with an energy barrier
of $O(L^{\frac{2}{3}})$ where the Hamiltonian is composed of $O(L^{3})$ qubits
and the qubits are of finite density. By finite density we mean that a finite
number of qubits fit into a finite volume reference box. By local we mean that
the terms in the Hamiltonian act on a set of qubits contained in another
finite reference box.

Haah Haah (2013) proved that for local translation-invariant stabilizer codes,
the highest energy barrier is $O(\log L)$. This improves a no-go theorem by
Yoshida Yoshida (2011). So how exactly can the result in this letter hold? The
code in this letter is constructed from macroscopic blocks. Each such block
satisfies the result by Haah. We describe these blocks in section IIA. These
macroscopic blocks are joined together, welded, into a macroscopic lattice.
This welding process is described in section IIB. The code in this letter is
translation invariant over a length that grows with the system size. This
avoids the no-go results by Yoshida Yoshida (2011) and Haah Haah (2013).

##  II Solid Codes and Welding

###  II.1 Solid Codes

In this section we introduce the solid code which is a local stabilizer code
on a three dimensional lattice of qubits. We use the word qubit instead of
spin to emphasize that the subsystems of the Hamiltonian could be any two
level system. The solid code is a stabilizer code. It is the 3-d analog of a
surface code Dennis et al. (2002), i.e. a 3-d toric code Castelnovo and Chamon
(2007) with rough and smooth boundaries. We discuss its logical operators and
show that one of them has a constant energy barrier which means it cannot
store quantum information though it is an important building block of the
welded codes.

We define the generators of the solid code with respect to a graph, shown in
figure 1 (qubits are labeled by edges). The graph is a cubic lattice, i.e. a
cube composed of $d\times d\times d$ cubic primitive cells but with the
horizontal edges removed for primitive cells at the top and bottom boundaries.
These qubits are not included in our code. Terms in the Hamiltonian are
labeled by vertices and faces of this graph, where by faces we mean faces of
the primitive cells. These faces will be referred to as plaquettes. The
vertices are:

$V=\\{v=(v_{1},v_{2},v_{3}):v_{i}\in\\{1,...,N\\}\\}.$ (2)

Using the unit vectors $n_{1}=(1,0,0)$, $n_{2}=(0,1,0)$ and $n_{3}=(0,0,1)$,
the edges are:

$\begin{split}E=\\{\\{v,v+n_{3}\\}:v\in V,v_{3}\neq N\\}\\\
\cup\\{\\{v,v+n_{2}\\}:v\in V,v_{3}\neq 1,v_{3}\neq N\\}\\\
\cup\\{\\{v,v+n_{1}\\}:v\in V,v_{3}\neq 1,v_{3}\neq N\\}.\end{split}$ (3)

Let $\Gamma(v)$ be the set of edges that neighbor a vertex $v$. For each $v$
with $|\Gamma(v)|>1$ define the term $h_{v}^{X}=\prod_{e\in\Gamma(v)}X_{e}$.
Let $\partial f$ be the set of edges on the boundary of a plaquette $f$. For
each plaquette $f$ define a term $h_{f}^{Z}=\prod_{e\in\partial f}Z_{e}$. The
plaquettes along the top and bottom rough boundaries are missing one edge.
Those qubits aren’t included in our code. $|\partial f|=3$ there. Finally the
Hamiltonian is given by a sum over the elements of the set of vertices $V$ and
the set of faces $F$:

$H=\left(\sum_{v\in V:|\Gamma(v)|>1}-h_{v}^{X}\right)+\left(\sum_{f\in
F}-h_{f}^{Z}\right).$ (4)

The $X$ and $Z$-type terms overlap on an even number of qubits and so commute.
Hence the terms generate a stabilizer group and the ground state subspace of
the Hamiltonian is exactly the code space of this stabilizer group.

The set of logical operators completely determines the ground state
degeneracy. We will show that there are two distinct non-trivial logical
operator. We will denote them by $\bar{X}$ and $\bar{Z}$. Because they anti-
commute, we can only diagonalize one of them at a time. Hence the ground state
degeneracy is 2.

The logical operator $\bar{X}$ of the solid code resembles an open membrane.
If $\left|\psi\right>$ is in the ground subspace of $H$ then each vertex
operator satisfies $h^{X}_{v}\left|\psi\right>=\left|\psi\right>$. Similarly a
product of vertex operators has a $+1$ eigenvalue. Multiplying vertex
operators generates closed membranes of qubits and horizontal pairs of open
membranes. These open membranes can be made to be far apart so that each
membrane overlaps with a disjoint set of plaquettes. Hence each membrane
commutes with each term in the Hamiltonian, yet is not generated by them. We
conclude that the logical operator $\bar{X}$ is a tensor product of single
qubit $X$ operators on a single horizontal membrane.

The logical operator $\bar{Z}$ resembles an open string. If
$\left|\psi\right>$ is in the ground subspace then a plaquette operator
satisfies $h^{Z}_{f}\left|\psi\right>=\left|\psi\right>$. Similarly a product
of plaquette operators has a $+1$ eigenvalue. Multiplying plaquette operators
generates closed strings, strings starting and ending on the same rough
boundary and pairs of strings extending between opposite rough boundaries.
These pairs of open strings can be made to be far apart so that each string
overlaps with a disjoint set of vertex operators. Hence each open string
commutes with each term of the Hamiltonian, yet is not generated by them. We
conclude that the logical operator $\bar{Z}$ is a tensor product of single
qubit $Z$ operators on a single string extending between opposite rough
boundaries. See figure 1.

The logical operator $\bar{Z}$ for the solid code has a constant energy
barrier. Understanding why is key to doing better. Applying a $Z$ operator on
a single qubit violates either one or two terms. We call these violated terms
defects. By flipping an adjacent qubit, we satisfy that term while violating
at most one other term. In this way we can move a defect in the bulk of the
solid or annihilate it at either of the rough boundaries. By flipping a qubit
on a rough boundary, we create a single defect which we can then move to and
annihilate on the opposite boundary. This sequence has no more than a constant
energy penalty.

The sequence of errors resulting in the logical operator $\bar{Z}$ resembles a
growing string. One way to create a large energy barrier is to force this
string to split many times. To do this we need qubits such that errors on them
create three or more defects. We will be combining blocks of solid code in the
next section to achieve this.

###  II.2 Welded Solid Codes

In this section we achieve a power-law energy barrier by combining several
solid codes into a 3-d lattice. It is interesting to note that the final
lattice is not a regular lattice. This is because each block is bent and
stretched to match-up and connect with each other into a macroscopic lattice.
The procedure for combining blocks of code is called welding. We will weld
three solid codes together, analyze the shape of the logical operators and
show that the energy barrier has increased.

To gain some intuition, consider a 1-d Ising model of a finite length. Here
the Hamiltonian is $\sum_{i=0}^{n-1}-Z_{i}Z_{i+1}$. The ground state has two
degenerate eigenstates $\left|00...0\right>$ and $\left|11...1\right>$.
Suppose we want to flip all of the qubits in a sequence that minimizes the
number of defects. The best we can do is to flip the first, second, third etc…
qubits in a line until all of the qubits have been flipped. This sequence
creates a single defect and moves it from one end of the string to the other.
Welding is like combining three such strings on the last qubit, i.e. so each
string shares the $n$th qubit with each other. When we try to move a defect
past this shared qubit, it will split into two. Thus the energy increases. For
the welded solid code there is 2-d boundary between 3-d blocks of qubits.
Defects split when moving past these boundaries. But now there is a choice
about which part of the boundary we move the defect through since it is an
area not a point.

We now describe this boundary where defects split by combining three solid
codes along their rough boundaries. For each solid we identify qubits on the
bottom rough boundaries with each other. More precisely, for all
$i,j\in\\{1,...,N\\}$ the qubit labeled by $(\\{i,j,0\\},\\{i,j,1\\})$ in the
first solid code is the same as the corresponding qubit in the second and
third solid codes. Because the $X$ and $Z$-type terms no longer commute, we
update all local $Z$-type stabilizers to commute with the $X$-type
stabilizers. The result is that whenever $Z$-type stabilizers agree on the
shared qubits of the three solids, they are combined into a single operator.
Combining operators is what we call welding. We can formalize welding as
follows. For a $Z$-type operator $h$, define $Q(h)$ to be the qubits that $h$
acts on nontrivially. A set of $Z$-type operators $\\{h_{1},...,h_{n}\\}$ is
said to be welded  together into a $Z$-type operator $h$ when
$Q(h)=\cup_{i}Q(h_{i})$. If we treat the identity operator $I$ as a $Z$-type
operator, then every $Z$-type stabilizer of the new code is a welded version
of $Z$-type stabilizers from the three solid codes. Thus we call the resulting
code a welded code. A more thorough account of the theory of welding can be
found in the supplementary material and Michnicki (2012).

Next, we show that the new code, the three welded solids, encodes only a
single qubit by showing that all nontrivial $Z$-type logical operators are
equivalent. The $Z$-type logical operators from each solid get welded together
so that the new logical operator $\bar{Z}$ resembles three strings emanating
from a single qubit on the shared boundary. See figure 2. After welding, the
$Z$-type stabilizers have the following shapes: half-loops on rough
boundaries, loops in the bulk of each solid, three welded half-loops on the
shared rough boundary and pairs of logical Z operators. There can be no other
$Z$-type logical operator. This is because if a $Z$-type operator acts with an
even number of $Z$ operators on the shared qubits, then it is in the
stabilizer group and if it acts with an odd number of $Z$ operators on the
boundary, then it is equivalent the operator composed of three welded strings.
Since there is only one nontrivial $Z$-type logical operator, it follows that
the ground state degeneracy is 2, corresponding to a single logical qubit.

A defect caused by $Z$ errors would split into two moving through this shared
boundary, an increase in energy. This is because a single-qubit $Z$ operator
applied to this shared boundary creates three violated terms, one for each
solid block. Thus the energy barrier for the logical operator $\bar{Z}$ has
increased from 1 to 2.

Figure 1: A solid code with qubits represented by edges. The following
operators shaded darker: a plaquette operator, a star operator and a logical Z
operator. Figure 2: Three solid codes welded together with qubits represented
by edges. The bifurcating $\bar{Z}$ operator is shaded darker.

In order to increase the energy barrier to a power law, we generalize welding
three solids on a single boundary to welding many solids into a lattice. We
label the solids by their rough boundaries, where a rough boundary is denoted
by $W_{i}$. In the previous example we had the set of solids
$E=\\{\\{W_{1},W_{4}\\},\\{W_{2},W_{4}\\},\\{W_{3},W_{4}\\}\\}$ and they all
share the rough boundary $W_{4}$, where defects split. In this notation, the
welded rough boundaries act as“fat” vertices and the bulk of the solid codes
act as “fat” edges of a graph $G=(V=\\{W_{i}\\},E)$. We weld the solid codes
into the graph of a 3-d cubic lattice. As we will see, this gives a
particularly high energy barrier.

We deduce the energy barrier of the logical operator $\bar{Z}$ of the welded
cubic lattice by looking at the rough boundaries of each block. If there is an
odd number of $Z$ errors on the rough boundaries of a particular solid code,
then there must be at least one defect in the bulk of that solid. Hence, if we
weld the solid codes into a graph $G$, then the energy barrier of the welded
string operator will be at least as big as the energy barrier for an Ising
model Hamiltonian $H=\sum_{\\{i,j\\}\in E}-Z_{i}Z_{j}$ with precisely the same
graph $G$. If $G$ is a 3-d cubic lattice of width $R$, then the energy barrier
for the logical operator $\bar{Z}$ will be $O(R^{2})$. In fact, this bound can
be saturated provided we never create more than one defect within the bulk of
any solid.

We can deduce the energy barrier of the logical operator $\bar{X}$ in a
similar way as for the logical operator $\bar{Z}$. The logical operator
$\bar{X}$ is an $X$-type operator and hence does not get welded. It remains a
membrane. We show a lower bound for the energy barrier by considering the
contribution to the energy barrier from the vertical plaquettes only, i.e.
plaquettes in the $y$-$z$ and $z$-$x$ planes of each solid, leaving out the
plaquettes in the $x$-$y$ plane. Notice that for each vertical surface of
plaquettes, defects can move up and down without creating new vertical defects
but moving between these vertical regions creates a vertical defect in each
neighboring region, synonymously to the case of the solid regions. These
“flat” regions are connected to each other in a 2-d square lattice of width
$O(d)$, provided each solid is $O(d)$ qubits wide. Again the energy barrier of
this horizontal membrane will be given by the energy barrier of the Ising
model on a 2-d square lattice. So the energy barrier is lower bounded by
$O(d)$. This bound can be saturated, even if we include horizontal plaquettes,
provided that the membrane is grown completely horizontally and in a single
domain.

Finally, the energy barrier of solid codes welded in a cubic lattice is the
minimum of the two energy barriers: $O(d)$ and $O(R^{2})$. The total number of
qubits will be $O(d^{3})$ qubits per solid with $O(R^{3})$ solids, which leads
to the number of qubits $N\sim O(d^{3}R^{3})$. The maximum energy barrier for
a fixed number of qubits $N$ is the minimum of the $X$ and $Z$ energy
barriers. Thus the maximum energy barrier happens when $O(d)\sim O(R^{2})$,
leading to an energy barrier of $\delta E\sim O(N^{2/9})$. The qubits can be
placed in a box of side lengths of $O(L)$ so that the energy barrier is
$O(L^{2/3})$. This demonstrates our main result.

##  III Discussion

We have constructed a code that has an exponentially higher energy barrier
than the the logarithmic bound presented by Haah Haah (2013). We achieved this
by tuning the length over which the code is periodic to a macroscopic
distance.

The time that it takes for a memory to be corrupted depends not only on the
energy barrier but also on the number of error sequences that lead to a
logical operator. Bravyi and Haah (2011) derived a lower bound on the storage
time $t$ for any stabilizer code Hamiltonian with energy barrier $\delta E$ of
$t\sim\frac{e^{\beta\delta E}}{N}2^{-k(L)}$ when $N=O(L^{3})\lesssim
e^{\beta}$ using the error model of a Hamiltonian in the weak coupling limit.
This leads to a lower bound of $t\sim e^{\beta e^{\frac{2}{9}\beta}}$ for
$N\lesssim e^{\beta}$ for the welded solid code. An upper bound is not known.

Further progress might be made by considering non-periodic codes or local
dynamic models such as cellular automata decoders.

Acknowledgements. Thank you Aram Harrow and Steve Flammia for stimulating
discussion and helpful suggestions throughout the writing process, and to
Jeongwan Haah for feedback on the final manuscript. This work was supported by
NSF grant 0829937, DARPA QuEST contract FA9550-09-1-0044 and IARPA via DoI NBC
contract D11PC20167. Part of this work was carried out while visiting at MIT.

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##  IV Appendix:the Theory of Welding

Welding is a technique for combining two stabilizer groups to produce a third.
It can be used to combine the shape of their logical operators while keeping
the generating set local.

The motivation for the technique of welding comes from the problem of
combining two CSS stabilizer groups $S_{1}$ and $S_{2}$. Since they are of the
CSS type, we have the identities $S_{1}=\left<S_{1}^{X}\cup S_{1}^{Z}\right>$
and $S_{2}=\left<S_{2}^{X}\cup S_{2}^{Z}\right>$ where $S_{i}^{X}$ (resp.
$S_{i}^{Z}$) refers to the subgroup of $X$-type (resp. $Z$-type) operators of
the stabilizer group $S_{i}$. The stabilizer group $S_{1}$ is defined on
qubits $Q_{1}$ and the stabilizer group $S_{2}$ is defined on qubits $Q_{2}$.
When the condition $Q_{1}\cap Q_{2}\neq\emptyset$ is satisfied then the group
$\left<S_{1}\cup S_{2}\right>$ is not necessarily Abelian and thus not
necessarily a stabilizer group. This is because there may be anticommuting
pairs between the sets $S_{1}^{X}\cup S_{1}^{Z}$ and $S_{2}^{X}\cup
S_{2}^{Z}$. One way to get around this problem is to choose the $X$-type
subgroup to be $\left<S_{1}^{X}\cup S_{2}^{X}\right>$ and then update the
subgroups $S_{1}^{Z}$ and $S_{2}^{Z}$ to commute with these operators. This
new code is what we call a welded code.

When a $Z$-type operator $h$ commutes with each element of the set
$S_{1}^{X}\cup S_{2}^{X}$, it takes on a special form. When restricted to
qubits $Q_{1}$(or $Q_{2}$), $h$ is a $Z$-type stabilizer or nontrivial logical
operator of the stabilizer group $S_{1}$(or $S_{2}$). That is, $h$ contains
either $Z$-type stabilizers or logical operators of the stabilizer groups
$S_{1}$ and $S_{2}$ as substrings.

Generating sets can be kept local after welding provided that they satisfy
certain conditions. These conditions are called well matched and independent
on the weld. Furthermore, the nontrivial logical operators of the stabilizer
group $S_{3}$ always look like nontrivial logical operators of the stabilizer
groups $S_{1}$ and $S_{2}$ when restricted to qubits $Q_{1}$ or $Q_{2}$.

Next we make a series of definitions and observations that make these
introductory remarks more clear. Throughout these definitions
$i\in\\{1,2,3\\}$ where we are welding code 1 and code 2 into code 3.

######  Definition 1.

The group $S_{i}$ denotes a stabilizer group of the CSS type, i.e.
$S_{i}=\left<S_{i}^{X}\cup S_{i}^{Z}\right>$ for $S_{i}^{X}$ and $S_{i}^{Z}$
defined below.

######  Definition 2.

The group $S_{i}^{X}$ contains all $X$-type stabilizers of the group $S_{i}$,
i.e. $S_{i}^{X}=S_{i}\cap\left<X_{1},X_{2},...,X_{n}\right>$.

######  Definition 3.

The group $S_{i}^{Z}$ contains all $Z$-type stabilizers of the group $S_{i}$,
i.e. $S_{i}^{Z}=S_{i}\cap\left<Z_{1},Z_{2},...,Z_{n}\right>$.

Here, $X_{k}$ (or $Z_{k}$) denote a bit flip (or phase flip) on qubit $k$.
Without loss of generality, we will assume that $X$ and $Z$-type stabilizers
and logical operators have $+1$ coefficients. Generally, an $X$(or $Z$)-type
stabilizer can have a $+1$ or $-1$ coefficient. As an example,
$-X_{1}X_{2}X_{3}$ is a valid stabilizer with a $-1$ coefficient. However, the
logical operators do not depend on the choice of the eigenvalues of the
stabilizers.

######  Definition 4.

The set $H_{i}^{X}$ denotes a generating set of the group $S_{i}^{X}$, i.e.
$\left<H_{i}^{X}\right>=S_{i}^{X}$.

######  Definition 5.

The set $H_{i}^{Z}$ denotes a generating set of the group $S_{i}^{Z}$, i.e.
$\left<H_{i}^{Z}\right>=S_{i}^{Z}$.

######  Definition 6.

The group $N_{i}$ denotes the normalizer of the group $S_{i}$, i.e. $h\in
N_{i}$ if and only if $h\in\left<X_{1},...,X_{n},Z_{1},...,Z_{n}\right>$ and
for all $g\in S_{i}$, $[h,g]=0$.

######  Definition 7.

The group $N_{i}^{Z}$ contains all $Z$-type operators that commute with every
element of $S_{i}$, i.e. $N_{i}^{Z}=N_{i}\cap\left<Z_{1},...,Z_{N}\right>$.

######  Definition 8.

The set $L_{i}^{Z}$ is a minimal generating set for all $Z$-type logical
operators, i.e. it includes all $Z$-type logical operators means that
$\left<L_{i}^{Z}\cup S_{i}^{Z}\right>=N_{i}^{Z}$ and it is minimal means that
$\left<L_{i}^{Z}\right>\cap S_{i}^{Z}=\\{I\\}$.

The sets $L_{i}^{Z}$, $H_{i}^{X}$ and $H_{i}^{Z}$ are assumed to be uniquely
defined in the observations and theorems of this appendix. But any choice
satisfying the conditions of the observations and theorems will do.

######  Definition 9.

The set $Q(S)$ denotes qubits that a stabilizer code $S$ acts on non-
trivially, i.e. qubit $q\in Q(S)$ iff there exists $h\in S$ such that
$tr_{q}h=0$. To simplify notation we denote $Q(S_{i})$ as $Q_{i}$.

######  Definition 10.

The operator $\theta_{Q}(O)$ denotes the restriction of a product operator $O$
to qubits $Q$. More specifically, if $O=\otimes_{i}P_{i}$ then
$\theta_{Q}(O)=\otimes_{i\in Q}P_{i}\otimes_{j\notin Q}I_{j}$. When applied to
a set of product operators $S$, $Q(S)=\\{\theta_{Q}(h):h\in S\\}$. We simplify
the notation by defining $\theta_{Q_{1}}\equiv\theta_{1}$,
$\theta_{Q_{2}}\equiv\theta_{2}$ and $\theta_{Q_{1}\cap
Q_{2}}\equiv\theta_{12}$.

With these definitions we formalize our observation that $Z$-type operators
that commute with $S_{1}^{X}\cup S_{2}^{X}$ contain the former stabilizers or
logical operators as substrings.

######  Observation 11.

If $h\in N_{3}^{Z}$ then $\theta_{1}(h)\in N_{1}^{Z}$ and $\theta_{2}(h)\in
N_{2}^{Z}$.

###### Proof.

By definition $h$ commutes with each element of $H_{1}^{X}$. The set
$H_{1}^{X}$ has support only on qubits $Q_{1}$. Hence $\theta_{1}(h)\in
N_{1}^{Z}$. A similar argument shows that $\theta_{2}(h)\in N_{2}^{Z}$. ∎

What this observation says is that if a $Z$-type operator commutes with all of
the $X$-type operators, then it looks like a $Z$-type stabilizer or logical
operator of each code when restricted to qubits of that code. With this
motivation we define welding between two stabilizer codes of the CSS type.

######  Definition 12.

Define

$H_{3}^{X}\equiv H_{1}^{X}\cup H_{2}^{X}$ (5)
$S_{3}^{Z}\equiv\\{h\in\left<Z_{1},...,Z_{n}\right>:\theta_{1}(h)\in
S_{1}^{Z},\theta_{2}(h)\in S_{2}^{Z}\\}.$ (6)

$S_{3}=\left<H_{3}^{X}\cup S_{3}^{Z}\right>$ is said to be a welded stabilizer
group of the groups $S_{1}$ and $S_{2}$. They are welded on the qubits
$Q_{1}\cap Q_{2}$.

Not only do $Z$-type stabilizers get welded together, but also $Z$-type
logical operators get welded together.

######  Observation 13.

If $l$ is a non-trivial $Z$-type logical operator, i.e. $l\in
N_{3}^{Z}\backslash S_{3}^{Z}$, then either $\theta_{1}(l)\in
N_{1}^{Z}\backslash S_{1}^{Z}$ or $\theta_{2}(l)\in N_{2}^{Z}\backslash
S_{2}^{Z}$ or both.

###### Proof.

If $l\in N_{3}^{Z}\backslash S_{3}^{Z}$ then $l$ commutes with each element of
the groups $S_{1}^{X}$ and $S_{2}^{X}$. Hence $\theta_{1}(l)\in N_{1}^{Z}$ and
$\theta_{2}(l)\in N_{2}^{Z}$. If both $\theta_{1}(l)\in S_{1}^{Z}$ and
$\theta_{2}(l)\in S_{2}^{Z}$ then $l\in S_{3}^{Z}$ which contradicts our
assumption that it is not. Hence $\theta_{1}^{Z}(l)\in N_{1}^{Z}\backslash
S_{1}^{Z}$ or $\theta_{2}(l)\in N_{2}^{Z}\backslash S_{2}^{Z}$ or both. ∎

Next, we define a special set of conditions under which nontrivial logical
operators of the welded code exist and where the welded stabilizer group
$S_{3}$ is local provided that the stabilizers groups $S_{1}$ and $S_{2}$ are
local. Generally the welded code need not encode any qubits. That is, all
logical operators might be proportional to the identity $I$. Also it is not
guaranteed that two local codes necessarily weld into another local code. We
now define conditions under which these problems go away.

######  Definition 14.

Two generating sets $H_{1}$ and $H_{2}$ are well matched when for all
$h_{1}\in H_{1}$, there exists $h_{2}\in H_{2}$ such that
$\theta_{12}(h_{1})=\theta_{12}(h_{2})$ and for all $h_{2}\in H_{2}$ there
exists $h_{1}\in H_{1}$ such that $\theta_{12}(h_{2})=\theta_{12}(h_{1})$.

######  Definition 15.

Consider a set $W=\\{h\in H_{i}^{Z}:\theta_{12}(h)\neq I\\}$. The generating
set $H_{i}^{Z}$ is said to be independent on the weld  if for all subsets
$Y\subseteq W$ such that $Y\neq\phi$, the product $\prod_{h\in Y}h\neq I$.

######  Theorem 16.

If the sets $H_{1}^{Z}\cup L_{1}^{Z}$ and $H_{2}^{Z}\cup L_{2}^{Z}$ are well
matched and independent on the weld and if the sets $L_{1}^{Z}$ and
$L_{2}^{Z}$ are well matched and independent on the weld then:

  1. 1.

A generating set for the $Z$-type stabilizers of the stabilizer code $S_{3}$
is given by $H_{3}^{Z}\equiv\\{h_{1}h_{2}\theta_{12}(h_{1}):h_{1}\in
H_{1}^{Z},h_{2}\in H_{2}^{Z},\theta_{12}(h_{1})=\theta_{12}(h_{2})\\}$.

  2. 2.

A generating set for the $Z$-type logical operators of the stabilizer code
$S_{3}$ is given by $L_{3}^{Z}\equiv\\{l_{1}l_{2}\theta_{12}(l_{1}):l_{1}\in
L_{1}^{Z},l_{2}\in L_{2}^{Z},\theta_{12}(l_{1})=\theta_{12}(l_{2})\\}$.

Theorem 16 tells us what the stabilizers group is and what the logical
operators are. Hence we can find properties of the code such as the number of
encoded qubits. For instance, suppose stabilizer groups $S_{1}$ and $S_{2}$
have $n_{1}$ and $n_{2}$ independent $Z$-type logical operators that have no
support on qubits $Q_{1}\cap Q_{2}$ and both codes have $m$ $Z$-type logical
operators that have support on $Q_{1}\cap Q_{2}$. The welded code has
$n_{1}+n_{2}+m$ encoded qubits.

To prove theorem 16, we first prove a lemma.

######  Lemma 17.

The set $H_{3}^{Z}\cup L_{3}^{Z}$ generates every element in the groups
$\\{h\in N_{2}^{Z}:\theta_{1}(h)=I\\}$ and $\\{h\in
N_{1}^{Z}:\theta_{2}(h)=I\\}$.

###### Proof.

We will show that if $g\in H_{2}^{Z}\cup L_{2}^{Z}$ such that
$\theta_{1}(g)=I$, then $g\in H_{3}^{Z}\cup L_{3}^{Z}$ and that $H_{2}^{Z}\cup
L_{2}^{Z}$ generates all of the group $\\{h\in N_{2}^{Z}:\theta_{1}(h)=I\\}$.

If $g\in H_{2}^{Z}$ such that $\theta_{1}(g)=I$ then we can find $h_{1}\in
H_{1}^{Z}$ and $h_{2}\in H_{2}^{Z}$ such that
$\theta_{12}(h_{1})=\theta_{12}(h_{2})$ and $h_{1}h_{2}\theta_{12}(h_{1})=g$.
Namely when $h_{1}=g$ and $h_{2}=I$. Hence $g\in H_{3}^{Z}$. Similarly if
$g\in L_{2}^{Z}$ such that $\theta_{1}(g)=I$ then we can find $l_{1}\in
L_{1}^{Z}$ and $l_{2}\in L_{2}^{Z}$ such that
$\theta_{12}(l_{1})=\theta_{12}(l_{2})$ and $l_{1}l_{2}\theta_{12}(l_{1})=g$.
Namely when $l_{1}=g$ and $l_{2}=I$. Hence $g\in L_{3}^{Z}$.

The set $\\{H_{2}^{Z}\cup L_{2}^{Z}\\}$ necessarily generates every element of
the set $\\{g\in N_{2}^{Z}:\theta_{1}(g)=I\\}$. This is because no element of
$\\{H_{2}^{Z}\cup L_{2}^{Z}\\}$ with support on qubits $Q_{1}\cap Q_{2}$ can
be included in the product by independence on the weld.

Proving that the set $H_{3}^{Z}\cup L_{3}^{Z}$ generates every elements of the
group $\\{h\in N_{1}^{Z}:\theta_{2}(h)=I\\}$ is symmetric to proving that it
generates every element of the group $\\{h\in N_{2}^{Z}:\theta_{1}(h)=I\\}$. ∎

We now prove theorem 16.

###### Proof.

We first prove item 1 that $H_{3}^{Z}$ generates $S_{3}^{Z}$. We have the
identity $\theta_{1}(H_{3}^{Z})=H_{1}^{Z}$ because

$\displaystyle\theta_{1}(H_{3}^{Z})$
$\displaystyle=\\{\theta_{1}(h_{1}h_{2}\theta_{12}(h_{1})):h_{1}\in
H_{1}^{Z},h_{2}\in H_{2}^{Z},\theta_{12}(h_{1})=\theta_{12}(h_{2})\\}$
$\displaystyle=H_{1}^{Z}.$

Hence if $g\in S_{3}^{Z}$ then we can find $h\in\left<H_{3}^{Z}\right>$ such
that $\theta_{1}(hg)=I$. The operator $hg$ is a stabilizer of $S_{3}$. By
lemma 17, $hg$ must be generated by the set $H_{3}^{Z}$. Hence the operator
$g$ is generated by the set $H_{3}^{Z}$.

Next we prove item 2 that the set $L_{3}^{Z}$ generates all $Z$-type logical
operators of the stabilizer group $S_{3}$. We need to prove that
$\left<L_{3}^{Z}\cup H_{3}^{Z}\right>=N_{3}^{Z}$. We know that
$\theta_{1}(\left<L_{3}^{Z}\cup H_{3}^{Z}\right>)=\theta_{1}(N_{1}^{Z})$.
Hence if $l\in N_{3}^{Z}$ then we can find $h\in\left<L_{3}^{Z}\cup
H_{3}^{Z}\right>$ such that $\theta_{1}(hg)=I$. By lemma 17,
$hg\in\left<L_{3}^{Z}\cup H_{3}^{Z}\right>$ and hence the operator $g$ is
generated by $L_{3}^{Z}\cup H_{3}^{Z}$. ∎

Next we’ll discuss locality. We’ll show that local well-matched independent-
on-the-weld CSS stabilizer groups weld into a local stabilizer group. First
we’ll need a notion of the width of a stabilizer.

######  Definition 18.

Define the distance between qubits $q_{1},q_{2}\in Q$ to be $d(q_{1},q_{2})$.
Let the maximum width of an element in a set of Pauli operators $M$ be defined
as $R(M)\equiv max\ d(q_{1},q_{2})$ such that $q_{1},q_{2}\in Q(\\{h\\})$ for
$h\in M$.

######  Corollary 19.

If $H_{1}^{Z}$ and $H_{2}^{Z}$ are well matched and linearly independent on
the weld then

$R(H_{3}^{Z}\cup H_{3}^{X})\leq R(H_{1}^{Z}\cup H_{1}^{X})+R(H_{2}^{Z}\cup
H_{2}^{X}).$ (7)

###### Proof.

Let $h$ be an operator in $H_{3}^{X}\cup H_{3}^{Z}$ such that
$R(\\{h\\})=R(H_{3}^{X}\cup H_{3}^{Z})$. If $h\in H_{1}^{X}$ then
$R(H_{3}^{Z}\cup H_{3}^{X})=R(\\{h\\})\leq R(H_{1}^{Z}\cup
H_{1}^{X})+R(H_{2}^{Z}\cup H_{2}^{X})=R(\\{h\\})+R(H_{2}^{Z}\cup H_{2}^{X})$.
If $h\in H_{2}^{X}$ then the proof is the same as for $h\in H_{1}^{X}$ so
without lack of generality we only prove one of those cases. If $h\in
H_{3}^{Z}$ then $h=h_{1}h_{2}\theta_{12}(h_{1})$ such that $h_{1}\in
H_{1}^{Z}$, $h_{2}\in H_{2}^{Z}$ and $\theta_{12}(h_{1})=\theta_{12}(h_{2})$.
By the triangle inequality $R(\\{h_{1}h_{2}\theta_{12}(h_{1})\\})\leq
R(\\{h_{1}\\})+R(\\{h_{2}\\})\leq R(H_{1}^{Z}\cup H_{1}^{X})+R(H_{2}^{Z}\cup
H_{2}^{X})$. ∎

Corollary 19, along with theorem 16, allow us to combine the shapes of the
logical operators from two codes while keeping the generating set for the
resulting code local. One can even design local codes with large energy
barriers using this technique as demonstrated with the welded solid code.

To summarize, combining two CSS stabilizer codes so that the stabilizers
commute lead to the idea that the $Z$-type stabilizer should be updated so as
to achieve this. When the $Z$-type stabilizers are updated, they become welded
versions of the $Z$-type stabilizers from the former codes. If the codes are
well matched and independent on the weld, then we can weld generating sets of
stabilizers and logical operators. If the former generating sets are local
then so will be the welded generating set.


# Strichartz estimates for the Dirac equation on asymptotically flat manifolds

Federico Cacciafesta  Federico Cacciafesta: Dipartimento di Matematica,
Universitá degli studi di Padova, Via Trieste, 63, 35131 Padova PD, Italy
<EMAIL_ADDRESS>,  Anne-Sophie de Suzzoni  Anne-Sophie de Suzzoni:
CMLS, École Polytechnique, CNRS, Université Paris-Saclay, 91128 PALAISEAU
Cedex, France<EMAIL_ADDRESS>and  Long Meng  Long
Meng: CERMICS, École des ponts ParisTech, 6 and 8 av. Pascal, 77455 Marne-la-
Vallée, France<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove Strichartz estimates for the Dirac equation on
asymptotically flat manifolds. The proof combines the weak dispersive
estimates proved in [4] with the Strichartz and smoothing estimates for the
wave and Klein-Gordon flows exploiting the results in [11] and [7]-[14] in the
same geometrical setting.

###### Key words and phrases:

Dirac equation, Strichartz estimates, asymptotically flat manifolds

###### 2010 Mathematics Subject Classification:

35Q41, 42B37

##  1\. Introduction

In [4], the authors started the study of dispersive dynamics of the Dirac
equation in a non-flat setting, proving local smoothing estimates in the cases
of asymptotically flat manifolds and warped products. Later on, in [5] and
[1], the authors proved respectively local and global in time weighted
Strichartz estimates for spherically symmetric manifolds: in this case, it is
indeed possible to take advantage of the so called partial wave decomposition,
which is analog of the spherical harmonics decomposition for the Dirac
operator, in order to recast the equation into a sum of “radial Dirac
equations” with potentials, for which several results are available. The
problem of proving Strichartz estimates without the assumption of spherical
symmetry on the manifold could not be solved directly, as indeed it was not
possible to apply the standard Duhamel trick to deduce them from the ones on
$\mathbb{R}^{3}$, since the equation in this framework is a first order
perturbation of the flat one. The purpose of this short paper is to fill this
gap, proving in fact Strichartz estimates for the Dirac equation on
asymptotically flat manifolds of dimension 3.

We thus consider the following Cauchy problem

(1) $\displaystyle\left\\{\begin{aligned}
&i\partial_{t}u-\mathcal{D}_{m}u=0\\\ &u(0,x)=u_{0}(x),\quad
x\in\mathbb{R}^{3}.\end{aligned}\right.$

where $u:\mathcal{M}\rightarrow\mathbb{C}^{4}$, $(\mathcal{M},g)$ is a
manifold with a Lorentzian metrics $g$ that decouples space and time. In other
words, we assume that it writes

(2) $g_{jk}=\left\\{{\begin{array}[]{ll}1&\textrm{ if }j=k=0\\\ 0&\textrm{ if
}jk=0\textrm{ and }j\neq k\\\ -h_{jk}(x)&\textrm{ otherwise.
}\end{array}}\right.$

endowed with a spin structure. We refer to Section 2 in [4] (see also [9],
section 5.6) for all the details on the construction and properties of the
Dirac operator on a manifold $\mathcal{M}$: here, we very briefly limit
ourselves to recall that the Dirac operator with mass $m\geq 0$ can be written
as

(3)
$\displaystyle\mathcal{D}_{m}=-i\gamma^{0}\gamma^{a}e^{i}_{\;a}D_{i}-\gamma^{0}m.$

where $\gamma^{j}$ denote the standard Dirac matrices, that is
$\gamma^{0}=\beta$ and $\gamma^{j}=\gamma^{0}\alpha_{j}$ for $j=1,2,3$ with

(4) $\alpha_{j}=\left(\begin{array}[]{cc}0&\sigma_{j}\\\
\sigma_{j}&0\end{array}\right),\quad\beta=\left(\begin{array}[]{cc}I_{2}&0\\\
0&-I_{2}\end{array}\right)$

and $\sigma_{j}$ are the Pauli matrices

(5) $\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\quad\sigma_{2}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right),\quad\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),$

$e^{i}_{\;a}$ is a matrix bundle satisfying

(6) $h^{ij}=e^{i}_{\;a}\delta^{ab}e^{j}_{\;b}$

where $\delta$ is the Kronecker symbol. We call it a dreibein since it is
restriction to space of a vierbein. A vierbein is a matrix bundle
$\tilde{e}^{\mu}_{\;a}$ satisfying

$\tilde{e}^{\mu}_{\;a}\eta^{ab}\tilde{e}^{\nu}_{\;b}=g^{\mu\nu};$

setting $e^{j}_{\;a}=\tilde{e}^{j}_{\;a}$ for $j,a=1,2,3$, we get indeed (6).
Note that the existence of such a dreibein is induced by the asymptotic
flatness of the manifold. The covariant derivative $D_{i}$ is given by

(7) $\displaystyle D_{0}=\partial_{0},\quad D_{j}=\partial_{j}+B_{j},\quad
j=1,2,3$

where $B_{j}$ writes

$B_{j}=\frac{1}{8}[\gamma_{a},\gamma_{b}]\Omega_{j}^{\;ab}.$

It has a purely geometric part $\omega_{j}^{\;ab}$, called the spin connection

(8)
$\omega_{j}^{\;ab}=e^{i}_{\;a}\partial_{j}e^{ib}+e_{i}^{\;a}\Gamma_{\;jk}^{i}e^{kb}$

with the Christoffel symbol (or affine connection) $\Gamma^{i}_{\;jk}$ given
by

(9)
$\Gamma_{\;jk}^{i}:=\frac{1}{2}h^{il}(\partial_{j}h_{lk}+\partial_{k}h_{jl}-\partial_{l}h_{jk}),$

and a purely algebraic part $\frac{1}{8}[\gamma_{a},\gamma_{b}]$, which is due
to the nature of the particle we consider (here a pair electron-positron) and
more specifically to its spin (here $\frac{1}{2}\oplus\frac{1}{2}$).

Finally we recall that the scalar curvature $\mathcal{R}_{h}$ writes

(10)
$\mathcal{R}_{h}:=h^{jk}\left(\partial_{i}\Gamma_{\;jk}^{i}-\partial_{k}\Gamma_{\;ji}^{i}+\Gamma_{\;jk}^{l}\Gamma^{i}_{\;il}-\Gamma^{l}_{\;ji}\Gamma_{\;kl}^{i}\right).$

For what concerns the manifold $\mathcal{M}$, we assume the following

Assumptions (A). Let $(\mathcal{M},g)$ be a 4-dimensional Lorentzian manifold
with a metrics $g$ having the structure given by (2), and $h\in
C^{\infty}(\mathbb{R}^{3})$. We assume that there exists a constant $C_{h}$
and $\sigma\in(0,1)$ such that for all $\alpha\in\mathbb{N}^{3}$ such that
$|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}\leq 3$ and all $x$,

(11) $|\partial^{\alpha}(h_{ij}(x)-\delta_{ij})|\leq C_{h}\left\langle
x\right\rangle^{-|\alpha|-1-\sigma}$

where
$\partial^{\alpha}=\partial_{1}^{\alpha_{1}}\partial_{2}^{\alpha_{2}}\partial_{3}^{\alpha_{3}}$.

######  Remark 1.1.

The assumptions above are fairly standard, and manifolds satisfying these are
usually referred to as asymptotically flat manifolds. We stress the fact that
requiring the constant $C_{h}$ to be small enough is a sufficient condition to
ensure that the manifold is non trapping (see e.g. [12], [3]): thus, we will
not have to assume this condition, that is crucial in order to have
dispersion. Besides, it is easy to see that condition (11) holds for the
inverse matrix of $h$ as well, provided the constant $C_{h}$ is sufficiently
small. Finally, let us mention the fact that the decay condition (11) might
not be optimal; in particular, the power $-|\alpha|-1-\sigma$ could be
weakened to $-|\alpha|-\sigma$ in the massless case, but we here prefer to
provide a unified and much simpler presentation of the results.

######  Remark 1.2.

According to (6), one can bound, in the sense of matrices, the square of $e$
with $h$. As $h$ is “close” to the identity, estimate (11) holds true for the
matrices $e$. Thus, under assumptions (A), it is possible to prove that there
exist constants $C_{B},\,C_{B}^{\prime},\,C_{\Gamma},\,C_{R}>0$ such that

$|B|\leq C_{B}C_{h}\left\langle x\right\rangle^{-2-\sigma},\quad|\partial
B|\leq C_{B}^{\prime}C_{h}\left\langle x\right\rangle^{-2-\sigma}$
$|\mathcal{R}_{h}|\leq C_{R}C_{h}\left\langle
x\right\rangle^{-3-\sigma},\quad|\Gamma|\leq C_{\Gamma}C_{h}\left\langle
x\right\rangle^{-2-\sigma}.$

These bounds will be proved in forthcoming Proposition 2.1.

Before stating our Theorem, let us recall the definition of admissible
Strichartz triple:

######  Definition 1.1.

In dimension $3$, the triple $(s,q,r)$ is called wave admissible if

$\frac{1}{q}=\frac{1}{2}-\frac{1}{r},\quad 2\leq q,r\leq\infty,\quad
r\neq\infty\quad s=\frac{1}{2}-\frac{1}{r}+\frac{1}{q}.$

The triple $(s,q,r)$ is called Klein-Gordon (or Schrödinger) admissible if

$\frac{2}{q}=\frac{3}{2}-\frac{3}{r},\quad 2\leq q\leq\infty,\quad 2\leq r\leq
6\quad s=\frac{1}{2}-\frac{1}{r}+\frac{1}{q}.$

We will use the standard notation $L^{p}$ for Lebesgue spaces; more precisely,
the norms in time will be denoted by $L^{p}_{t}$ and will be intended to be on
$\mathbb{R}^{+}$, and we define

$\|f\|_{L^{p}(\mathcal{M}_{h})}^{p}=\int_{\mathbb{R}^{3}}|f(x)|^{p}\sqrt{\det
h(x)}dx.$

Then we define the $H^{s}_{p}$ and $\dot{H}^{s}_{p}$ norms for
$s\in\mathbb{N}$ as

$\|f\|_{\dot{H}^{s}_{p}(\mathcal{M}_{h})}:=\|(-\widetilde{\Delta}_{h})^{s/2}f\|_{L^{p}(\mathcal{M}_{h})},\quad\|f\|_{H^{s}_{p}(\mathcal{M}_{h})}:=\|(1-\widetilde{\Delta}_{h})^{s/2}f\|_{L^{p}(\mathcal{M}_{h})}$

where $\widetilde{\Delta}_{h}$ is the standard Laplace-Beltrami operator. For
negative $s$, we define these spaces by duality and for fractional $s$, we
define these spaces by interpolation.

Our main result is then the following:

######  Theorem 1.2.

Let $(\mathcal{M},g)$ be as given by assumptions (A). Then the massless Dirac
flow satisfies the Strichartz estimate:

(12)
$\displaystyle\|e^{it\mathcal{D}}u_{0}\|_{L^{q}_{t}\dot{H}^{1-s}_{r}(\mathcal{M}_{h})}\lesssim\|u_{0}\|_{\dot{H}^{1}(\mathcal{M}_{h})}$

for all wave admissible triple $(s,q,r)$, while in the massive case we have

(13)
$\displaystyle\|e^{it\mathcal{D}_{m}}u_{0}\|_{L^{q}_{t}H^{1/2-s}_{r}(\mathcal{M}_{h})}\lesssim\|u_{0}\|_{H^{1}(\mathcal{M}_{h})}\quad(m\neq
0)$

for all Klein-Gordon admissible triple $(s,q,r)$ such that $q>2$.

Let us briefly comment on the strategy of our proof, which is short, but
relies on several different recent results. The idea consists in squaring
equation (1) in order to obtain a system of wave or Klein-Gordon equations on
the manifold $(\mathcal{M},g)$ (depending on whether $m=0$ or $m>0$), and then
combining the estimates for such flows on asymptotically flat manifolds with
the standard argument based on Duhamel formula and local smoothing estimates
to control the “perturbative terms”. This trick is in fact widely used for the
study of several properties of the Dirac equation, as the Dirac operator is
indeed built as a suitable “square root” of the Laplacian. Anyway, we should
stress the fact that this strategy comes with two main difficulties: one is in
that the Laplace-Beltrami operator obtained after the squaring procedure is
not the standard (or “scalar”) one, but it is a “spinorial” Laplace operator
(the covariant derivative is not the same as the covariant derivatives for
scalar or vector fields). Therefore, it will not be possible to apply directly
the existing results, and we will have somehow to estimate the difference of
the solutions to the “scalar” and the “spinorial” wave/Klein-Gordon equations.
This difference contains a first order term, and this represents the second
difficulty, as indeed we will have to rely on a local smoothing at the “first
order” level.

Acknowledgments. F.C. and L.M. acknowledge support from the University of
Padova STARS project “Linear and Nonlinear Problems for the Dirac Equation”
(LANPDE), and AS. dS. is supported by the ANR project ESSED ANR-18-CE40-0028.

##  2\. Proof of Theorem 1.2

We begin with proving the statements in Remark 1.2.

######  Proposition 2.1.

Assume that $C_{h}\ll 1$. The dreibein $e$ exists and can be chosen such that
there exist constants $C_{B}$ and $C_{B}^{\prime}$ such that for all $x$, we
have

$|B(x)|\leq C_{B}C_{h}\left\langle x\right\rangle^{-\sigma-2},\quad|\partial
B(x)|\leq C_{B}^{\prime}C_{h}\left\langle x\right\rangle^{-\sigma-3}.$

What is more, there exist constants $C_{R}$ and $C_{\Gamma}$ such that for all
$x$, we have

$|\mathcal{R}_{h}(x)|\leq C_{R}C_{h}\left\langle
x\right\rangle^{-\sigma-3},\quad|\Gamma|\leq C_{\Gamma}C_{h}\left\langle
x\right\rangle^{-\sigma-2}.$

###### Proof.

The scalar curvature does not depend on $e$ and writes

$\mathcal{R}_{h}=h^{jk}(\partial_{i}\Gamma^{i}_{\;jk}-\partial_{k}\Gamma^{i}_{\;ji}+\Gamma^{l}_{\;jk}\Gamma^{i}_{\;li}-\Gamma^{l}_{\;ij}\Gamma^{i}_{kl})$

with the affine connection given by (9). Therefore for all $x$, we have

$|\mathcal{R}_{h}(x)|\leq
C_{R}^{\prime}(|\partial\Gamma(x)|+|\Gamma(x)|^{2}).$

Indeed, we can choose $C_{h}$ small enough such that for all $x$,
$|h^{-1}(x)|\leq 2$. Therefore,

$|\Gamma(x)|\leq C_{\Gamma}|h^{\prime}(x)|\leq C_{\Gamma}C_{h}\left\langle
x\right\rangle^{-2-\sigma}$

and since $(h^{-1})^{\prime}(x)=-h^{-1}(x)h^{\prime}(x)h^{-1}(x)$, choosing
$C_{h}\ll 1$, we get

$|\partial\Gamma(x)|\leq
C_{\Gamma}^{\prime}(|h^{\prime}(x)|^{2}+|h^{\prime\prime}(x)|)\leq
C_{\Gamma}^{\prime}C_{h}\left\langle x\right\rangle^{-3-\sigma}.$

We deduce

$|\mathcal{R}_{h}(x)|\leq C_{R}C_{h}\left\langle x\right\rangle^{-3-\sigma}.$

We look for $e$ a matrix bundle such that

$h^{ij}(x)=e^{i}_{\;a}(x)\delta^{ab}e^{j}_{\;b}(x)$

for all $x$. This can be rewritten as

$h^{ij}(x)=e^{ia}(x)\delta_{ab}e^{jb}(x).$

If we restrict $(e^{ia})_{1\leq i,a\leq 3}$ to be symmetric, this rewrites as

$h=e^{2}.$

As it is well-known, $e\mapsto e^{2}$ seen as a map from the the symmetric
matrices to the symmetric matrices is $\mathcal{C}^{\infty}$ and its
differential at the identity is twice the identity of the symmetric matrices.
Therefore, it can be reversed into a $\mathcal{C}^{\infty}$ map $F$ (a square
root) around the identity since $Id^{2}=Id$. We choose $C_{h}$ small enough
such that for all $x$, $h(x)$ lies in $K$ a compact subset of the definition
set of $F$ and we choose $e(x)=F(h(x))$ for all $x$. By the increment theorem,
we have

$|e(x)-Id|\leq\sup_{K}|DF||h(x)-Id|.$

We also have that for all $x$,

$|e^{\prime}(x)|\leq\sup_{K}|DF|\,|h^{\prime}(x)|,\quad|e^{\prime\prime}(x)|\leq\sup_{K}|D^{2}F|\,|h^{\prime}(x)|^{2}+\sup_{K}|DF|\,|h^{\prime\prime}(x)|,\\\
|e^{\prime\prime\prime}(x)|\leq\sup_{K}|D^{3}F|\,|h^{\prime}(x)|^{3}+3\sup_{K}|D^{2}F|\,|h^{\prime}(x)|\,|h^{\prime\prime}(x)|+\sup_{K}|DF|\,|h^{\prime\prime\prime}(x)|.$

Therefore, we deduce that for all $x$,

$|e(x)-Id|\leq C_{e}C_{h}\left\langle
x\right\rangle^{-\sigma-1},\quad|e^{\prime}(x)|\leq C_{e}C_{h}\left\langle
x\right\rangle^{-2-\sigma},\\\ |e^{\prime\prime}(x)|\leq
C_{e}C_{h}\left\langle
x\right\rangle^{-3-\sigma},\quad|e^{\prime\prime\prime}(x)|\leq
C_{e}C_{h}\left\langle x\right\rangle^{-4-\sigma}.$

We deduce

$|\omega(x)|\leq C_{\omega}C_{h}\left\langle
x\right\rangle^{-2-\sigma},\quad|B(x)|\leq C_{B}C_{h}\left\langle
x\right\rangle^{-2-\sigma},\quad\textrm{and}\quad|\partial B(x)|\leq
C_{B}^{\prime}C_{h}\left\langle x\right\rangle^{-3-\sigma}.$

∎

We now recall the connection between the Dirac and the wave/Klein-Gordon
equations on manifolds, and recall the local smoothing estimate for it. This
is the first main ingredient in our proof.

######  Theorem 2.2.

If $u$ is a solution to (1), then it also solves the following

(14) $\displaystyle\left\\{\begin{aligned}
&\partial_{t}^{2}u+m^{2}u-\Delta_{h}u+\frac{1}{4}\mathcal{R}_{h}u=0,\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3},\\\
&\partial_{t}u(0,x)=i\mathcal{D}_{m}u_{0}(x),\quad u(0,x)=u_{0}(x),\quad
x\in\mathbb{R}^{3}\end{aligned}\right.$

where $\Delta_{h}=D^{i}D_{i}$ with $D_{i}$ given by (7). Besides,

(15)
$\displaystyle\Delta_{h}v=\widetilde{\Delta}_{h}v+B^{i}\partial_{i}v+\widetilde{D}^{i}B_{i}v+B^{i}B_{i}v$

where $\widetilde{\Delta}_{h}$ is the Laplace-Beltrami operator for scalars
tensor the 4-dimensional identity matrix,
$\widetilde{D}^{i}\Psi_{k}=\partial^{i}\Psi_{k}-\Gamma^{l\;i}_{\;k}\Psi_{l}$
and $B^{i}=h^{ij}B_{j}$.

Let $u$ be a solution of (1), then

(16) $\displaystyle\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}+\|\left\langle
x\right\rangle^{-1/2-}\widetilde{\nabla}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}u_{0}\|_{L^{2}(\mathcal{M}_{h})}$

where $\widetilde{\nabla}$ denotes the gradient for scalar fields, that is

$\|\left\langle
x\right\rangle^{-1/2-}\widetilde{\nabla}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}^{2}:=\int_{\mathcal{M}}\left\langle
x\right\rangle^{-1-}h^{ij}\left\langle\partial_{i}u,\partial_{j}u\right\rangle_{\mathbb{C}^{4}}dx.$

###### Proof.

The relation between the Dirac equation and system (14) is shown in Corollary
1 and Formula (34) in [4]. Furthermore, by Proposition 4 in [4], we have

(17)
$\displaystyle\mathcal{D}_{m}^{2}=m^{2}-\Delta_{h}+\frac{1}{4}\mathcal{R}_{h}.$

Let us now prove the local smoothing estimate (16). It is shown in [4, Theorem
1.1] that

(18) $\displaystyle\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}+\|\left\langle
x\right\rangle^{-1/2-}\nabla
u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}_{m}u_{0}\|_{L^{2}(\mathcal{M}_{h})}.$

Recall that $\nabla=(D_{1},D_{2},D_{3})=\widetilde{\nabla}+B$ where
$B=(B_{1},B_{2},B_{3})$. According to Proposition 2.1,

$\|\left\langle
x\right\rangle^{-1/2-}Bu\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\leq
C_{B}C_{h}\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}_{m}u_{0}\|_{L^{2}(\mathcal{M}_{h})}.$

Thus,

$\displaystyle\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}+\|\left\langle
x\right\rangle^{-1/2-}\widetilde{\nabla}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}$
$\displaystyle\leq\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}+\|\left\langle
x\right\rangle^{-1/2-}\nabla
u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}+\|\left\langle
x\right\rangle^{-1/2-}Bu\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}.$

Hence

$\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}+\|\left\langle
x\right\rangle^{-1/2-}\widetilde{\nabla}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}_{m}u_{0}\|_{L^{2}(\mathcal{M}_{h})}$

and this concludes the proof. ∎

As a second ingredient, we need to provide suitable Strichartz and local
smoothing estimates for the solutions to the “auxiliary” systems of wave and
Klein-Gordon equations. We state them in the next two Theorems.

######  Theorem 2.3 (Strichartz estimates for wave/Klein-Gordon).

Let $(\mathcal{M},g)$ be a 4-dimensional Lorentzian manifold satisfying
assumptions (A) and let $u$ be a solution to the following system:

(19) $\displaystyle\left\\{\begin{aligned}
&\partial_{t}^{2}u+m^{2}u-\widetilde{\Delta}_{h}u=0,\quad(t,x)\in{(t,x)\in\mathcal{M}},\\\
&\partial_{t}u(0,x)=u_{1}(x),\quad u(0,x)=u_{0}(x),\quad
x\in\mathbb{R}^{3}.\end{aligned}\right.$

If $m=0$, then u satisfies

(20)
$\displaystyle\|u\|_{L^{q}_{t}\dot{H}^{1-s}_{r}(\mathcal{M}_{h})}\lesssim\|u_{0}\|_{\dot{H}^{1}(\mathcal{M}_{h})}+\|u_{1}\|_{L^{2}(\mathcal{M}_{h})}$

for any wave admissible triple $(s,q,r)$.

If $m>0$, then $u$ satisfies

(21)
$\displaystyle\|u\|_{L^{q}_{t}H^{1/2-s}_{r}(\mathcal{M}_{h})}\lesssim\|u_{0}\|_{\dot{H}^{1/2}(\mathcal{M}_{h})}+\|u_{1}\|_{H^{-1/2}(\mathcal{M}_{h})}.$

for any Klein-Gordon admissible triple $(s,q,r)$.

###### Proof.

When $m=0$, this is just Theorem 1.4 in [11]. When $m>0$, the Strichartz
estimate follows from the global-in-time Strichartz estimate on non-trapping
conic manifold (i.e. scattering manifold) in [14, Theorem 1.1]. It is shown in
[8, Remark 1.2] that any asymptotically flat space $(\mathbb{R}^{3},h)$ with
decay estimates $|\partial^{\alpha}(h-\delta)|\leq C_{j}\left\langle
x\right\rangle^{-|\alpha|-1}$ is also asymptotically conic (see also [10,
Remark 1.5]). Hence we can deduce the result in the case $m>0$. ∎

######  Theorem 2.4 (Local smoothing estimates for wave/Klein-Gordon).

Let $(\mathcal{M},g)$ be a 4-dimensional Lorentzian manifold satisfying
assumptions (A). Then the following estimates hold

(22) $\displaystyle\|\left\langle
x\right\rangle^{-1/2-}e^{it\sqrt{-\widetilde{\Delta}_{h}}}f\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|f\|_{L^{2}(\mathcal{M}_{h})},$

for any $f\in L^{2}(\mathcal{M}_{h})$, and

(23) $\displaystyle\|\left\langle
x\right\rangle^{-1/2}e^{it\sqrt{m^{2}-\widetilde{\Delta}_{h}}}f\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|(1-\widetilde{\Delta}_{h})^{1/4}f\|_{L^{2}(\mathcal{M}_{h})}$

for any $f$ such that $(1-\widetilde{\Delta}_{h})^{1/4}f\in
L^{2}(\mathcal{M}_{h})$.

###### Proof.

Let us consider the following unitary transform

$\displaystyle\mathcal{V}:L^{2}(\mathbb{R}^{3},\sqrt{\det h(x)}dx)\to
L^{2}(\mathbb{R}^{3},dx),\quad v\mapsto(\det h(x))^{1/4}v.$

The transformation $\mathcal{V}$ sends $-\widetilde{\Delta}_{h}$ to

$P=-\mathcal{V}\widetilde{\Delta}_{h}\mathcal{V}^{-1}=-(\det
h(x))^{1/4}\widetilde{\Delta}_{h}(\det h(x))^{-1/4}.$

Let us start with the massless case. Let
$u:=e^{it\sqrt{-\widetilde{\Delta}_{h}}}f\in L^{2}(\mathcal{M},dg)$ and let
$v=\mathcal{V}u$; then $v\in
L^{2}(\mathbb{R}^{+}_{t}\times\mathbb{R}^{3},dx)$, and

$\displaystyle\partial_{t}^{2}u-\widetilde{\Delta}_{h}u=0\Leftrightarrow\partial_{t}^{2}v-Pv=0.$

According to [2, Theorem 1.3] or [11, Page 24, Section 6], we get

$\displaystyle\|\left\langle
x\right\rangle^{-1/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}=\|\left\langle
x\right\rangle^{-1/2-}v\|_{L^{2}(\mathbb{R}^{+}\times\mathbb{R}^{3})}\lesssim\|\left\langle
x\right\rangle^{-1/2-}v(0)\|_{L^{2}(\mathbb{R}^{3})}=\|f\|_{L^{2}(\mathcal{M}_{h})}.$

For the second estimate (massive case), according to [13, Formula (3.5) and
Proposition 3.1] (taking $V=0$), we know that $\left\langle
x\right\rangle^{-1}$ is $-\widetilde{\Delta}_{h}$-smooth, i.e.,

$\|\left\langle
x\right\rangle^{-1}e^{-it\widetilde{\Delta}_{h}}f\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|f\|_{L^{2}(\mathcal{M}_{h})}.$

It follows from [7, Theorem 2.2 and Theorem 2.4] that

$\|\left\langle
x\right\rangle^{-1/2}e^{it(-\sqrt{m^{2}-\widetilde{\Delta}})}f\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|(m^{2}-\widetilde{\Delta}_{h})^{1/4}f\|_{L^{2}}$

and this concludes the proof. ∎

We are now in a position to prove our main result.

###### Proof of Theorem 1.2..

According to (14) and (15), by Duhamel Formula, we can write for any $m\geq 0$

(24) $\displaystyle
u(t,x):=e^{it\mathcal{D}_{m}}u_{0}=\dot{W}_{m}(t)u_{0}+iW_{m}(t)\mathcal{D}_{m}u_{0}+\int_{0}^{t}W_{m}(t-s)(\Omega_{1}(u)(s)+\Omega_{2}u(s))ds$

where

$W_{m}(t)=\frac{\sin(t\sqrt{m^{2}-\widetilde{\Delta}_{h}})}{\sqrt{m^{2}-\widetilde{\Delta}_{h}}},\quad\dot{W}_{m}=\partial_{t}W_{m}$

and

(25)
$\Omega_{1}(u):=2B^{i}\partial_{i}u,\quad\Omega_{2}:=\partial^{i}B_{i}+B^{i}B_{i}-\Gamma_{\;i}^{j\;i}B_{j}-\frac{1}{4}\mathcal{R}_{h}.$

We deal with the massless and massive cases separately, starting with the
former. According to Christ-Kiselev Lemma [6] and Theorem 2.3,

$\left\|\int_{0}^{T}W_{m}(t-s)(\Omega_{1}(u)(s)+\Omega_{2}u(s))ds\right\|_{L^{q}_{t}\dot{H}^{1-s}_{r}(\mathcal{M}_{h})}\\\
\lesssim\left\|\frac{e^{it\sqrt{-\widetilde{\Delta}_{h}}}}{\sqrt{-\widetilde{\Delta}_{h}}}\int_{0}^{T}e^{-is\sqrt{-\widetilde{\Delta}_{h}}}(\Omega_{1}(u)(s)+\Omega_{2}u(s))ds\right\|_{L^{q}_{t}\dot{H}^{1-s}_{r}(\mathcal{M}_{h})}\\\
\lesssim\left\|\int_{0}^{T}e^{-is\sqrt{-\widetilde{\Delta}_{h}}}(\Omega_{1}(u)(s)u(s)+\Omega_{2}u(s))ds\right\|_{L^{2}(\mathcal{M}_{h})}.$

By the dual form of (22),

$\left\|\int_{0}^{T}e^{-is\sqrt{-\widetilde{\Delta}_{h}}}(\Omega_{1}(u)(s)u(s)+\Omega_{2}u(s))ds\right\|_{L^{2}(\mathcal{M}_{h})}\lesssim\|\left\langle
x\right\rangle^{1/2+}(\Omega_{1}(u)(s)u+\Omega_{2}u)\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}.$

Then by (16) and Remark 1.2, we have

$\|\left\langle
x\right\rangle^{1/2+}\Omega_{2}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\left\langle
x\right\rangle^{2+}\Omega_{2}\|_{L^{\infty}}\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}u_{0}\|_{L^{2}(\mathcal{M}_{h})}.$

Then by (16) and Remark 1.2, we have

$\|\left\langle
x\right\rangle^{1/2+}\Omega_{1}(u)\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\left\langle
x\right\rangle^{1+}B\|_{L^{\infty}_{x}}\|\left\langle
x\right\rangle^{-1/2-}\widetilde{\nabla}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}u_{0}\|_{L^{2}(\mathcal{M}_{h})}.$

Thus thanks to Lemma A.1,

$\|u\|_{L^{q}_{t}\dot{H}^{1-s}_{r}(\mathcal{M}_{h})}\lesssim\|u\|_{\dot{H}^{1}(\mathcal{M}_{h})}+\|\mathcal{D}u_{0}\|_{L^{2}(\mathcal{M}_{h})}\lesssim\|u\|_{\dot{H}^{1}(\mathcal{M}_{h})}.$

Now we consider the massive case. According to Christ-Kiselev lemma [6], since
we restrict to the case $q>2$, and Theorem 2.3,

$\left\|\int_{0}^{T}W_{m}(t-s)(\Omega_{1}(u)(s)+\Omega_{2}u(s))ds\right\|_{L^{q}_{t}H^{1/2-s}_{r}(\mathcal{M}_{h})}\\\
\leq\left\|\frac{e^{it\sqrt{m^{2}-\widetilde{\Delta}_{h}}}}{\sqrt{m^{2}-\widetilde{\Delta}_{h}}}\int_{0}^{T}e^{-is\sqrt{m^{2}-\widetilde{\Delta}_{h}}}(\Omega_{1}(u)(s)+\Omega_{2}u(s))ds\right\|_{L^{q}_{t}H^{1/2-s}_{r}(\mathcal{M}_{h})}\\\
\leq\left\|\int_{0}^{T}e^{-is\sqrt{m^{2}-\widetilde{\Delta}_{h}}}(\Omega_{1}(u)(s)+\Omega_{2}u(s))ds\right\|_{H^{-1/2}(\mathcal{M}_{h})}.$

By the dual form of (23),

$\left\|\int_{0}^{T}e^{-is\sqrt{m^{2}-\widetilde{\Delta}_{h}}}(\Omega_{1}(u)(s)+\Omega_{2}u(s))ds\right\|_{H^{-1/2}(\mathcal{M}_{h})}\lesssim\|\left\langle
x\right\rangle^{1/2}(B^{i}\partial_{i}u+\Omega_{2}u)\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}.$

Then by (16) and Lemma A.1, we have

$\|\left\langle
x\right\rangle^{1/2}\Omega_{2}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\left\langle
x\right\rangle^{2+}\Omega_{2}\|_{L^{\infty}}\|\left\langle
x\right\rangle^{-3/2-}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}_{m}u_{0}\|_{L^{2}(\mathcal{M}_{h})}\lesssim\|u_{0}\|_{H^{1}(\mathcal{M}_{h})}.$

Then by (16), Remark 1.2 and Lemma A.1, we have

$\|\left\langle
x\right\rangle^{1/2}\Omega_{1}(u)\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\left\langle
x\right\rangle^{1+}B\|_{L^{\infty}_{x}}\|\left\langle
x\right\rangle^{-1/2-}\widetilde{\nabla}u\|_{L^{2}_{t}L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}u_{0}\|_{L^{2}}\lesssim\|u_{0}\|_{H^{1}(\mathcal{M}_{h})},$

so that

$\|u\|_{L^{q}_{t}\dot{H}^{1/2-s}_{r}(\mathcal{M}_{h})}\lesssim\|u_{0}\|_{H^{1/2}(\mathcal{M}_{h})}+\|\mathcal{D}_{m}u_{0}\|_{H^{-1/2}(\mathcal{M}_{h})}+\|u_{0}\|_{H^{1}(\mathcal{M}_{h})}\lesssim\|u_{0}\|_{H^{1}(\mathcal{M}_{h})}.$

and this concludes the proof. ∎

##  Appendix A Norm estimate

Here we give some results concerning the relationship between the standard
Sobolev norm and the one induced by the Dirac operator, which is used in the
proof of Theorem 1.2.

######  Lemma A.1.

Under assumptions (A), for $m\geq 0$,

$\|(m^{2}-\widetilde{\Delta}_{h})^{1/2}u\|_{L^{2}(\mathcal{M}_{h})}\lesssim\|\mathcal{D}_{m}u\|_{L^{2}(\mathcal{M}_{h})}\lesssim\|(m^{2}-\widetilde{\Delta}_{h})^{1/2}u\|_{L^{2}(\mathcal{M}_{h})}.$

###### Proof.

Unsing that $\tilde{\Delta}_{h}$ is self-adjoint on $L^{2}(\mathcal{M}_{h})$,
we have

$\|(m^{2}-\tilde{\Delta}_{h})^{1/2}u\|_{L^{2}(\mathcal{M}_{h})}^{2}=\left\langle(m^{2}-\tilde{\Delta}_{h})u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}.$

We also have

$-\left\langle\tilde{\Delta}_{h}u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}=h^{ij}\left\langle\partial_{i}u,\partial_{j}u\right\rangle_{L^{2}(\mathcal{M}_{h})},$

so that

$-\left\langle\tilde{\Delta}_{h}u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}=\delta^{ij}\left\langle\partial_{i}u,\partial_{j}u\right\rangle_{L^{2}(\mathcal{M}_{h})}+(h^{ij}-\delta^{ij})\left\langle\partial_{i}u,\partial_{j}u\right\rangle_{L^{2}(\mathcal{M}_{h})}.$

Using Cauchy-Schwarz inequality and the fact that $|h^{ij}-\delta^{ij}|\ll 1$
yields

$\delta^{ij}\left\langle\partial_{i}u,\partial_{j}u\right\rangle_{L^{2}(\mathcal{M}_{h})}\lesssim-\left\langle\tilde{\Delta}_{h}u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}\lesssim\delta^{ij}\left\langle\partial_{i}u,\partial_{j}u\right\rangle_{L^{2}(\mathcal{M}_{h})}.$

As the Dirac operator is self-adjoint on $L^{2}(\mathcal{M}_{h})$, we have

$\|\mathcal{D}_{m}u\|_{L^{2}(\mathcal{M}_{h})}^{2}=\left\langle\mathcal{D}_{m}^{2}u,u\right\rangle_{L^{2}(\mathcal{M}_{h})},$

and thanks to the identity
$\mathcal{D}_{m}^{2}=m^{2}-\tilde{\Delta}_{h}+\Omega_{1}+\Omega_{2}$ with
$\Omega_{1},\;\Omega_{2}$ given by (25) we get

$\|\mathcal{D}_{m}u\|_{L^{2}(\mathcal{M}_{h})}^{2}=\left\langle(m^{2}-\tilde{\Delta}_{h})u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}+\left\langle(\Omega_{1}(u)+\Omega_{2}u),u\right\rangle_{L^{2}(\mathcal{M}_{h})}.$

Using the skew-symmetry of $B_{i}$, we get

$\left\langle
B^{i}\partial_{i}u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}=-\left\langle\partial_{i}u,B^{i}u\right\rangle_{L^{2}(\mathcal{M}_{h})},\quad\left\langle
B_{i}\partial^{i}u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}=-\left\langle\partial^{i}u,B_{i}u\right\rangle_{L^{2}(\mathcal{M}_{h})}$

By Cauchy-Schwarz inequality again we get

$|\left\langle\Omega_{1}(u),u\right\rangle_{L^{2}(\mathcal{M}_{h})}|\leq\sum_{i}\|\partial_{i}u\|_{L^{2}(\mathcal{M}_{h})}\|B\left\langle
x\right\rangle\|_{L^{\infty}}\|\left\langle
x\right\rangle^{-1}u\|_{L^{2}(\mathcal{M}_{h})}.$

Now, as the $L^{2}(\mathcal{M}_{h})$ and the $L^{2}(\mathbb{R}^{3})$ norms are
equivalent (due to the assumptions on $h$), we use Hardy inequality to get

$|\left\langle\Omega_{1}(u),u\right\rangle_{L^{2}(\mathcal{M}_{h})}|\lesssim\|B\left\langle
x\right\rangle\|_{L^{\infty}}\|(m^{2}-\tilde{\Delta}_{h})^{1/2}u\|_{L^{2}(\mathcal{M}_{h})}^{2}.$

Similarly,

$|\left\langle\Omega_{2}u,u\right\rangle_{L^{2}(\mathcal{M}_{h})}|\lesssim\|\left\langle
x\right\rangle^{2}\Omega_{2}\|_{L^{\infty}}\|(m^{2}-\tilde{\Delta}_{h})^{1/2}u\|_{L^{2}(\mathcal{M}_{h})}^{2}.$

Now, as $\|\left\langle x\right\rangle B\|_{L^{\infty}}\ll 1$ (see Proposition
2.1) and

$\|\left\langle x\right\rangle^{2}\Omega_{2}\|_{L^{\infty}}\leq\|\left\langle
x\right\rangle^{2}(\partial^{i}B_{i}+B^{i}B_{i}-\Gamma^{j\;i}_{\;i}B_{j}-\frac{1}{4}\mathcal{R}_{h})\|_{L^{\infty}}\ll
1,$

we finally get that

$\|(m^{2}-\tilde{\Delta}_{h})^{1/2}u\|_{L^{2}(\mathcal{M}_{h})}^{2}\lesssim\|\mathcal{D}_{m}u\|_{L^{2}(\mathcal{M}_{h})}^{2}\lesssim\|(m^{2}-\tilde{\Delta}_{h})^{1/2}u\|_{L^{2}(\mathcal{M}_{h})}^{2}$

and this concludes the proof.

∎

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