# Regular Lagrangians are smooth Lagrangians

Tomohiro Asano<EMAIL_ADDRESS><EMAIL_ADDRESS>Supported
by JSPS KAKENHI Grant Number JP24K16920. Research Institute for Mathematical
Sciences, Kyoto University, Kitashirakawa-Oiwake-Cho, Sakyo-ku, 606-8502,
Kyoto, Japan.  Stéphane Guillermou<EMAIL_ADDRESS>Supported by ANR COSY (ANR-21-CE40-0002) and Centre Henri Lebesgue
(ANR-11-LABX-0020-01). UMR CNRS 6629 du CNRS Laboratoire de Mathématiques Jean
LERAY 2 Chemin de la Houssinière, BP 92208, F-44322 NANTES Cedex 3 France
Yuichi Ike<EMAIL_ADDRESS><EMAIL_ADDRESS>Supported by
JSPS KAKENHI Grant Numbers JP21K13801 and JP22H05107. Institute of Mathematics
for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka-shi, Fukuoka
819-0395, Japan.  Claude Viterbo<EMAIL_ADDRESS>Supported by ANR COSY (ANR-21-CE40-0002). Université Paris-Saclay, CNRS,
Laboratoire de mathématiques d’Orsay, 91405, Orsay, France.

###### Abstract

We prove that for any element in the $\gamma$-completion of the space of
smooth compact exact Lagrangian submanifolds of a cotangent bundle, if its
$\gamma$-support is a smooth Lagrangian submanifold, then the element itself
is a smooth Lagrangian. We also prove that if the $\gamma$-support of an
element in the completion is compact, then it is connected.

To Pierre Schapira for his $80^{th}$ birthday.

##  1 Introduction

Let $M$ be a $C^{\infty}$ closed connected manifold. The space
$\mathfrak{L}(T^{*}M)$ of smooth compact exact Lagrangian submanifolds of
$T^{*}M$ carries a distance $\gamma$, called the spectral distance (see
[Vit92, Oh97, MVZ12, HLS16]). The metric space $(\mathfrak{L}(T^{*}M),\gamma)$
is not complete, so we consider its completion. Its study was initiated in
[Hum08], pursued further in [Vit22a], and has applications to Hamilton–Jacobi
equations [Hum08], Symplectic Homogenization theory [Vit08], and to
conformally symplectic dynamics [AHV24].

The elements of the completion $\widehat{}\mathfrak{L}(T^{*}M)$ are by
definition certain equivalence classes of Cauchy sequences with respect to the
spectral norm $\gamma$. Despite their very abstract nature, they admit a
geometric incarnation called the _$\gamma$ -support_, which was introduced by
Viterbo in [Vit22a]. For a smooth Lagrangian $L\in\mathfrak{L}(T^{*}M)$ we
have $\operatorname{\gamma-supp}(L)=L$.

In this note, we give positive answers to Conjecture 8.2 of [Vit22a] and to a
question in [AHV24]: for $L_{\infty}\in\widehat{}\mathfrak{L}(T^{*}M)$,

  * (i)

if $\operatorname{\gamma-supp}(L_{\infty})=L$ for some
$L\in\mathfrak{L}(T^{*}M)$, then $L_{\infty}=L$,

  * (ii)

if $\operatorname{\gamma-supp}(L_{\infty})$ is compact, then it is connected.

### Acknowledgments

The authors are very grateful to Vincent Humilière for many helpful
discussions. This paper is the continuation of [AGHIV23].

##  2 Notations

Throughout this paper, we fix a field $\mathbf{k}$.

Let $\mathcal{L}(T^{*}M)$ denote the set of compact exact Lagrangian branes,
i.e., triples $(L,f_{L},\widetilde{G})$, where $L$ is a compact exact
Lagrangian submanifold of $T^{*}M$, $f_{L}\colon L\to\mathbb{R}$ is a function
satisfying $df_{L}=\lambda_{\mid L}$, and $\widetilde{G}$ is a grading of $L$
(see [Sei00, Vit22a]). The action of $\mathbb{R}$ on $\mathcal{L}(T^{*}M)$
given by $(L,f_{L},\widetilde{G})\mapsto(L,f_{L}-c,\widetilde{G})$ is denoted
by $T_{c}$. Let $\mathfrak{L}(T^{*}M)$ be the set of compact exact
Lagrangians, where we do not record primitives or gradings. For $L_{1},L_{2}$
in $\mathcal{L}(T^{*}M)$ we define as in [Vit22a] the spectral invariants
$c_{+}(L_{1},L_{2})$ and $c_{-}(L_{1},L_{2})$ and

$\mathrm{c}(L_{1},L_{2})=\max\\{c_{+}(L_{1},L_{2}),0\\}-\min\\{c_{-}(L_{1},L_{2}),0\\}.$

This defines a distance111Note that the definition given in [AGHIV23] is not
correct, and has to be replaced by the one above. This has been corrected in
the published version of [Vit22a]. For $L_{1},L_{2}$ in
$\mathfrak{L}(T^{*}M)$, we define

$\gamma(L_{1},L_{2})=\inf_{c\in\mathbb{R}}\mathrm{c}(L_{1},T_{c}L_{2})=c_{+}(L_{1},L_{2})-c_{-}(L_{1},L_{2}).$

Denote by $\widehat{\mathfrak{L}}(T^{*}M)$ (resp.
$\widehat{\mathcal{L}}(T^{*}M)$) the completion of $\mathfrak{L}(T^{*}M)$
(resp. $\mathcal{L}(T^{*}M)$) with respect to $\gamma$ (resp. $\mathrm{c}$).

We recall several notions and ideas from [Tam18]. We denote by $(t;\tau)$ the
canonical coordinates on $T^{*}\mathbb{R}$. The Tamarkin category
$\mathcal{T}(T^{*}M)$ is defined as the quotient category
$\mathsf{D}(\mathbf{k}_{M\times\mathbb{R}})/\mathsf{D}_{\\{\tau\leq
0\\}}(\mathbf{k}_{M\times\mathbb{R}})$, where
$\mathsf{D}_{A}(\mathbf{k}_{M\times\mathbb{R}})\coloneqq\\{F\in\mathsf{D}(\mathbf{k}_{M\times\mathbb{R}})\mid\operatorname{{\operatorname{SS}}}(F)\subset
A\\}$ is a triangulated full subcategory of
$\mathsf{D}(\mathbf{k}_{M\times\mathbb{R}})$. The Tamarkin category has a
monoidal structure. For
$F,F^{\prime}\in\mathsf{D}(\mathbf{k}_{M\times\mathbb{R}})$ we set $F\ast
F^{\prime}\coloneqq\mathrm{R}m_{!}(q_{1}^{-1}F\otimes q_{2}^{-1}F^{\prime})$,
where $q_{1},q_{2}\colon M\times\mathbb{R}^{2}\to M\times\mathbb{R}$ are the
projections and $m$ is the addition map $m(x,s,t)=(x,s+t)$. The operation
$\ast$ preserves the left orthogonal ${}^{\perp}\mathsf{D}_{\\{\tau\leq
0\\}}(\mathbf{k}_{M\times\mathbb{R}})$ and moreover $F\mapsto
F\ast\mathbf{k}_{M\times[0,+\infty[}$ is a projector onto it. This projector
induces an equivalence between $\mathcal{T}(T^{*}M)$ and
${}^{\perp}\mathsf{D}_{\\{\tau\leq 0\\}}(\mathbf{k}_{M\times\mathbb{R}})$,
with which we identify them in what follows. We also set
$\mathop{{\mathcal{H}}om}\nolimits^{*}(F,F^{\prime})\coloneqq\mathrm{R}q_{1*}\mathop{\mathrm{R}\mathcal{H}om}\nolimits(q_{2}^{-1}F,m^{!}F^{\prime})$
and denote the projection of this $\mathop{{\mathcal{H}}om}\nolimits^{*}$ onto
$\mathcal{T}(T^{*}M)$ by the same symbol. This defines an internal hom
$\mathop{{\mathcal{H}}om}\nolimits^{*}\colon\mathcal{T}(T^{*}M)^{\mathrm{op}}\times\mathcal{T}(T^{*}M)\to\mathcal{T}(T^{*}M)$.
For $c\in\mathbb{R}$, let $T_{c}\colon M\times\mathbb{R}\to M\times\mathbb{R}$
be the translation $T_{c}(x,t)=(x,t+c)$. The category $\mathcal{T}(T^{*}M)$
comes with a family of morphisms of functors
$\tau_{c}\colon\operatorname{id}\to T_{c*}$ for each $c\geq 0$ introduced by
Tamarkin. They give rise to an interleaving distance on $\mathcal{T}(T^{*}M)$
denoted $d_{\mathcal{T}(T^{*}M)}$ (see [KS18] and [AI20]).

We recall the composition of sheaves. For $F\in\mathsf{D}(\mathbf{k}_{M\times
N})$ and $G\in\mathsf{D}(\mathbf{k}_{N\times P})$, set $F\circ
G\coloneqq\mathrm{R}q_{13!}(q_{12}^{-1}F\otimes q_{23}^{-1}G)$, where $q_{ij}$
are the projections from $M\times N\times P$ to the $(i\times j)$ factors. We
also consider a mixture of $\circ$ and $\ast$: for
$F\in\mathcal{T}(T^{*}M\times T^{*}N)$, $G\in\mathcal{T}(T^{*}N\times
T^{*}P)$, we set $F\varoast
G=\mathrm{R}m_{!}\mathrm{R}q_{13!}(q_{12}^{-1}F\otimes q_{23}^{-1}G)$ where
$q_{ij}$ are projections from $M\times N\times P\times\mathbb{R}^{2}$ to
$M\times N\times\mathbb{R}$, $N\times P\times\mathbb{R}$, $M\times
P\times\mathbb{R}^{2}$ and $m$ the addition map. We set for short
$\mathcal{K}^{\varoast}(F)\coloneqq\mathcal{K}\varoast F$ for
$\mathcal{K}\in\mathcal{T}(T^{*}M^{2})$ and $F\in\mathcal{T}(T^{*}M)$.

We put an analytic structure on $M$ and define
$\mathcal{T}_{\mathrm{lc}}(T^{*}M)$ as the subcategory of
$\mathcal{T}(T^{*}M)$ made by objects that are limits (for the interleaving
distance) of constructible sheaves. We remark that for a submanifold $N$ of
$M$, the pull-back to $N\times\mathbb{R}$ commutes with $T_{c*}$ and
$\tau_{c}$. It follows that the pull-back is a contraction and hence sends
$\mathcal{T}_{\mathrm{lc}}(T^{*}M)$ to $\mathcal{T}_{\mathrm{lc}}(T^{*}N)$.

For an object $F\in\mathcal{T}(T^{*}M)$, we define its reduced microsupport
$\operatorname{RS}(F)\subset T^{*}M$ by

$\operatorname{RS}(F)\coloneqq\overline{\rho_{t}(\operatorname{{\operatorname{SS}}}(F)\cap\\{\tau>0\\})},$

where $\operatorname{{\operatorname{SS}}}(F)\subset
T^{*}(M\times\mathbb{R}_{t})$ is the microsupport of $F$ [KS90],
$\\{\tau>0\\}\subset T^{*}(M\times\mathbb{R}_{t})$, and
$\rho_{t}\colon\\{\tau>0\\}\to T^{*}M,(x,t;\xi,\tau)\mapsto(x;\xi/\tau)$. For
a closed subset $A\subset T^{*}M$ we let $\mathcal{T}_{A}(T^{*}M)$ be the full
subcategory of $\mathcal{T}(T^{*}M)$ consisting of the $F$ with
$\operatorname{RS}(F)\subset A$. We also set
$\mathcal{T}_{\mathrm{lc},A}(T^{*}M)=\mathcal{T}_{A}(T^{*}M)\cap\mathcal{T}_{\mathrm{lc}}(T^{*}M)$.

##  3 Preliminaries

We recall that we have a quantization map
$Q\colon\mathcal{L}(T^{*}M)\to\mathcal{T}(T^{*}M)$ introduced in [Gui23,
Vit19]. This is extended to the completion
$\widehat{Q}\colon\widehat{\mathcal{L}}(T^{*}M)\to\mathcal{T}_{\mathrm{lc}}(T^{*}M)$
in [GV22]. Note that
$\widehat{Q}(T_{c}(\widetilde{L}_{\infty}))\simeq{T_{c}}_{*}\widehat{Q}(\widetilde{L}_{\infty})$.
However, an approximation argument is missing in [GV22], which we shall now
provide.

######  Proposition 3.1.

For any $\widetilde{L}_{\infty}\in\widehat{\mathcal{L}}(T^{*}M)$, one has
$\widehat{Q}(\widetilde{L}_{\infty})\in\mathcal{T}_{\mathrm{lc}}(T^{*}M)$.

###### Proof.

According to [CE12], Corollary 6.25, an element
$\widetilde{L}\in{\mathcal{L}}(T^{*}M)$, is $C^{k}$-approximated for any
$k\geq 1$ by analytic Lagrangians $\widetilde{L}_{i}$. We thus find that
$\widetilde{L}=C^{k}-\lim\widetilde{L}_{i}$ hence
$\widetilde{L}_{i}=\varphi_{i}(\widetilde{L})$ where $\varphi_{i}$ is
generated by a $C^{k}$-small Hamiltonian. According to Lemma 3.2 the distance
between $\widetilde{L}$ and $\widetilde{L}_{i}$ can then be chosen arbitrarily
small. As a result $\widetilde{L}$ is a $\gamma$-limit of analytic
Lagrangians. According to [KS90] Theorem 8.4.2, the $Q(\widetilde{L}_{i})$ are
constructible hence their limit $Q(\widetilde{L})$ is in
$\mathcal{T}_{\mathrm{lc}}(T^{*}M)$. Since $\widetilde{L}_{\infty}$ can be
written as a Cauchy sequence of elements of $\mathcal{L}(T^{*}M)$, the claim
follows. ∎

######  Lemma 3.2.

Let $h\colon T^{*}(M\times\mathbb{R})\times I\to\mathbb{R}$ be a homogeneous
Hamiltonian function and $\phi$ be the associated homogeneous Hamiltonian
isotopy. Let $K\in\mathsf{D}((M\times\mathbb{R})^{2}\times I)$ be the sheaf
associated with $\phi$ constructed in [GKS12]. Then, for any
$F\in\mathcal{T}(T^{*}M)$

$d_{\mathcal{T}(T^{*}M)}(F,K_{1}\circ F)\leq
4\int_{0}^{1}\max|h_{s}(x,t;\xi,1)|\,ds.$

###### Proof.

This lemma can be proved in a similar way to Theorem 1.2 (or 4.16) [AI20]. ∎

Let $L\in\mathfrak{L}(T^{*}M)$ and
$L_{\infty}\in\widehat{}\mathfrak{L}(T^{*}M)$. We assume that
$\operatorname{\gamma-supp}(L_{\infty})=L$ and we want to prove that
$L_{\infty}\in\mathfrak{L}(T^{*}M)$ and $L_{\infty}=L$. Let
$\widetilde{L}=(L,f_{L},\widetilde{G})\in\mathcal{L}(T^{*}M)$ be a lift of $L$
and let $\widetilde{L}_{\infty}\in\widehat{}\mathcal{L}(T^{*}M)$ be a lift of
$L_{\infty}$. In view of [AGHIV23] our assumption means that the sheaf
$F_{\widetilde{L}_{\infty}}=\widehat{Q}(\widetilde{L}_{\infty})\in\mathcal{T}(T^{*}M)$
satisfies $\operatorname{RS}(F_{\widetilde{L}_{\infty}})=L$. To prove that
$L_{\infty}=L$ it is enough to see that $F_{L_{\infty}}$ is isomorphic to
$F_{L}=Q(\widetilde{L})$, up to translation (in $t$) and shift (in grading).
To this end, we shall characterize the objects $F$ of
$\mathcal{T}_{\mathrm{lc},L}(T^{*}M)$ with
$\operatorname{{\operatorname{SS}}}(F)=T_{c}(\Lambda)$ for some
$c\in\mathbb{R}$, where $\Lambda\subset T^{*}(M\times\mathbb{R})$ is the cone
over a Legendrian lift of $L$ and $T_{c}$ also denotes the translation on
$T^{*}(M\times\mathbb{R})$ by $c$ (see Definition 3.3, Lemma 3.4 and Corollary
3.13). Explicitly

$\Lambda=\\{(x,\tau p,-f_{L}(x,p),\tau)\mid\tau>0,(x,p)\in L\\}.$

Hence $\Lambda$ is a conic Lagrangian submanifold of
$T^{*}(M\times\mathbb{R})$ contained in $\\{\tau>0\\}$. Note that the
coisotropic submanifold $\rho_{t}^{-1}(L)$ is foliated by the translates of
$\Lambda$: $\rho_{t}^{-1}(L)=\bigsqcup_{c\in\mathbb{R}}T_{c}(\Lambda)$. It is
not too difficult to see that any closed conic coisotropic subset of
$\rho_{t}^{-1}(L)$ is a union of translates of $\Lambda$. Hence for any non
zero $F\in\mathcal{T}_{L}(T^{*}M)$, $\operatorname{{\operatorname{SS}}}(F)$
contains at least $T_{c}(\Lambda)$ for some $c\in\mathbb{R}$. However we shall
not use these facts.

We denote by $q\colon M\times\mathbb{R}\to M$ the projection.

######  Definition 3.3.

Let $F\in\mathcal{T}(T^{*}M)$. We say that $F$ is _cohomologically chordless_
if

$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes q^{-1}G,T_{c*}F)\simeq 0$

for all $c<0$ and all locally constant $G\in\mathsf{D}(\mathbf{k}_{M})$ (we
say that an object of $\mathsf{D}(\mathbf{k}_{M})$ is locally constant if its
cohomology sheaves are locally constant).

We first need :

######  Lemma 3.4.

Let $F\in\mathcal{T}_{L}(T^{*}M)$ with
$\operatorname{{\operatorname{SS}}}(F)=T_{c_{0}}(\Lambda)$ for some
$c_{0}\in\mathbb{R}$ and $F|_{M\times\\{t\\}}\simeq 0$ for $t\ll 0$. Then $F$
is cohomologically chordless.

###### Proof.

This is already done in [Gui23, Lemma 12.4.4], but we sketch the proof for the
convenience of the reader. First the microsupports of $F\otimes q^{-1}G$ and
$T_{c*}F$ do not meet when $c$ runs over $\mathopen{]}-\infty,0[$, hence
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes q^{-1}G,T_{c*}F)$ is
independent of $c<0$ by a variation on the Morse theorem for sheaves [KS90,
Corollary 5.4.19] (see [Nad16] or [Gui23, Corollary 1.2.17]). We choose $a$
such that $\Lambda\subset T^{*}(M\times\mathopen{]}-a,a[)$. For $c<-2a$ we
obtain that $T_{c*}F$ is locally constant on
$\operatorname{\operatorname{supp}}(F\otimes q^{-1}G)$, say $T_{c*}F\simeq
q^{-1}G^{\prime}\simeq q^{!}G^{\prime}[-1]$ there. Then
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes q^{-1}G,T_{c*}F)$ is
isomorphic to $\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes
q^{-1}G,q^{!}G^{\prime}[-1])$. Using the adjunction $(\mathrm{R}q_{!},q^{!})$
and the projection formula $\mathrm{R}q_{!}(F\otimes
q^{-1}G)\simeq\mathrm{R}q_{!}F\otimes G$, it is then enough to check that
$\mathrm{R}q_{!}F\simeq 0$. This can be proved stalkwise:
$(\mathrm{R}q_{!}F)_{x}\simeq\mathrm{R}\Gamma_{c}(\\{x\\}\times\mathbb{R};F|_{\\{x\\}\times\mathbb{R}})$
and the vanishing follows again from the Morse theorem for sheaves since
$\operatorname{{\operatorname{SS}}}(F|_{\\{x\\}\times\mathbb{R}})\subset\\{\tau\geq
0\\}$ and $F|_{\\{x\\}\times\mathbb{R}}$ vanishes near $-\infty$. ∎

######  Lemma 3.5.

Let $(F_{i})_{i\in\mathbb{N}}$, be a convergent sequence in
$\mathcal{T}(T^{*}M)$ and set $F=\lim_{i}F_{i}$ (the limit being for the
distance $d_{\mathcal{T}(T^{*}M)}$). We assume that $F_{i}$ is cohomologically
chordless for each $i\in\mathbb{N}$. Then $F$ is cohomologically chordless.

###### Proof.

By [GV22, Proposition 6.24] (or [AI24, Theorem 4.3]), up to taking a
subsequence, there exist a sequence of positive numbers
$(\varepsilon_{i})_{i\in\mathbb{N}}$ converging to $0$ and morphisms

$f_{i}\colon T_{-\varepsilon_{i}*}F_{i}\to
T_{-\varepsilon_{i+1}*}F_{i+1},\quad u_{i}\colon T_{-\varepsilon_{i}*}F_{i}\to
F$ (3.1)

such that $u_{i+1}\circ f_{i}=u_{i}$, for all $n$, and the morphism
$\operatorname*{\operatorname{hocolim}}T_{-\varepsilon_{i}*}F_{i}\to F$
induced by the $u_{i}$’s is an isomorphism, where
$\operatorname*{\operatorname{hocolim}}$ is the sequential homotopy colimit
described in [BN93] (see also [KS06, Notation 10.5.10]). The same proposition
holds with homotopy limits instead of homotopy colimits and we can write in
the same way (taking a subsequence again)
$F\xrightarrow{\sim}\operatorname{holim}T_{\eta_{j}*}F_{j}$ for some other
sequence $(\eta_{i})_{i\in\mathbb{N}}$.

Since the tensor product commutes with direct sums, it also commutes with
homotopy colimits and we have, for any $G\in\mathsf{D}(\mathbf{k}_{M})$,
$F\otimes
q^{-1}G\simeq\operatorname*{\operatorname{hocolim}}(T_{-\varepsilon_{i}*}F_{i}\otimes
q^{-1}G)$. Recall that the category of sheaves on a topological space $X$ is a
Grothendieck category, so we may apply Lemma 3.6 and infer that
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes q^{-1}G,T_{c*}F)$ is a
homotopy limit of
$E_{i}=\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(T_{-\varepsilon_{i}*}F_{i}\otimes
q^{-1}G,T_{(\eta_{i}+c)*}F)$. For a given $c<0$ and for $i$ big enough we have
$\varepsilon_{i}+\eta_{i}+c<0$ and then $E_{i}\simeq 0$. It follows that
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes q^{-1}G,T_{c*}F)$ vanishes.
∎

######  Lemma 3.6.

Let $\mathcal{C}$ be a Grothendieck category. Let $(A_{i},f_{i})$,
$i\in\mathbb{N}$, be an inductive system in $\mathsf{D}(\mathcal{C})$, with
homotopy colimit $A$, and let $(B_{j},g_{j})$, $j\in\mathbb{N}$, be a
projective system, with homotopy limit $B$. Then
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A,B)$ is a homotopy limit of the
system $(\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A_{i},B_{i}),h_{i})$ where
$h_{i}$ is the morphism induced by composition with $f_{i}$, $g_{i}$.

###### Proof.

According to [Hov01], Theorem 2.2, the category $\mathrm{Ch}(\mathcal{C})$ of
chain complexes on $\mathcal{C}$ is a model category having homotopical
category $\mathsf{D}(\mathcal{C})$. We denote by $\mathbb{V}_{\mathbf{k}}$ the
category of $\mathbf{k}$-vector spaces.

We apply results of [CS02] where homotopy (co)limits are defined for
categories with weak equivalences. If $\mathcal{A}$ is such a category and $I$
is a small category, we have a functor
$\operatorname{holim}^{\prime}_{I}\colon\mathrm{Fun}(I,\mathcal{A})\to\mathrm{Ho}(\mathcal{A})$
(in particular
$\operatorname{holim}^{\prime}_{I}\colon\mathrm{Fun}(I,\mathrm{Ch}(\mathcal{C}))\to\mathrm{Ho}(\mathrm{Ch}(\mathcal{C}))=\mathsf{D}(\mathcal{C})$.
In the proof of Lemma 3.5 the notation $\operatorname{holim}_{I}F$ applies to
$F\in\mathrm{Fun}(I,\mathsf{D}(\mathcal{C}))$ — this is not a functor:
$\operatorname{holim}_{I}F$ is well-defined up to a non-unique isomorphism. We
use the notation $\operatorname{holim}^{\prime}_{I}F$ to avoid confusion (but
this is denoted by $\operatorname{holim}_{I}$ in [CS02]). We have
$\operatorname{holim}^{\prime}_{I}F\simeq\operatorname{holim}_{I}Q\circ F$
where $Q\colon\mathcal{A}\to\mathrm{Ho}(\mathcal{A})$ is the quotient.

We will apply Section 31.5 from [CS02] which states that if $F\colon I\times
J\to\mathcal{C}$ is a functor to a model category, then
$\operatorname{holim}^{\prime}_{I\times
J}F\simeq\operatorname{holim}_{I}\operatorname{holim}^{\prime}_{J}F\simeq\operatorname{holim}_{J}\operatorname{holim}^{\prime}_{I}F$.
In our case $I=J=\mathbb{N}^{\mathrm{op}}$. We first lift the diagram
$i\mapsto A_{i}$ to a similar diagram in the set of chain complexes on
$\mathcal{C}$. We shall use the same notation for the lift. We do the same for
$j\mapsto B_{j}$ and we may further impose that each $B_{j}$ is a complex of
injectives. We then define a functor
$F\colon(\mathbb{N}^{\mathrm{op}})^{2}\to\mathrm{Ch}(\mathbb{V}_{\mathbf{k}})$
by $F(i,j)=\operatorname{Hom}(A_{i},B_{j})$. Since the $B_{j}$’s are
injective, we have
$\operatorname{holim}^{\prime}_{i}F(i,j)\simeq\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(\operatorname*{\operatorname{hocolim}}^{\prime}_{i}A_{i},B_{j})\simeq\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A,B_{j})$
for each $j$. From the definition of $\operatorname{holim}$ we also have
$\operatorname{holim}_{j}\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A,B_{j})\simeq\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A,\operatorname{holim}^{\prime}_{j}B_{j})$.
Hence

$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A,B)\simeq\operatorname{holim}^{\prime}_{(i,j)\in(\mathbb{N}^{\mathrm{op}})^{2}}\operatorname{Hom}(A_{i},B_{j}).$

According to 31.6 (loc. cit.) for $F\colon I\to\mathcal{C}$ a functor in a
model category and $f\colon J\to I$ an initial functor, the map

$\operatorname{holim}^{\prime}_{I}F\to\operatorname{holim}^{\prime}_{J}f^{*}F$

is a weak equivalence. Using the fact that the inclusion of the diagonal
$\mathbb{N}^{\mathrm{op}}$ in $(\mathbb{N}^{\mathrm{op}})^{2}$ is initial we
get

$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A,B)\simeq\operatorname{holim}^{\prime}_{i\in\mathbb{N}^{\mathrm{op}}}\operatorname{Hom}(A_{i},B_{i})\simeq\operatorname{holim}_{i\in\mathbb{N}^{\mathrm{op}}}\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(A_{i},B_{i}).$

This concludes the proof. ∎

We now prove that, if $F\in\mathcal{T}_{\mathrm{lc},0_{M}}(T^{*}M)$ is
cohomologically chordless, then
$\operatorname{{\operatorname{SS}}}(F)=0_{M}\times(\\{c_{0}\\}\times]0,\infty[)$
for some $c_{0}\in\mathbb{R}$ (Proposition 3.8 below).

We first recall a microlocal characterization of the inverse image of sheaves
by a projection with contractible fibers.

######  Lemma 3.7.

Let $N$ be a manifold and let $I$ be an open interval (or more generally a
contractible manifold). Let $p\colon N\times I\to N$ be the projection and let
$i_{a}\colon N\times\\{a\\}\to N\times I$ be the inclusion, for $a\in I$. Then
$p^{-1}\colon\mathsf{D}(\mathbf{k}_{N})\to\mathsf{D}_{T^{*}N\times
0_{I}}(\mathbf{k}_{N\times I})$ is an equivalence of categories, with inverses
$\mathrm{R}p_{*}$ and $i_{a}^{-1}$, $a\in I$. Moreover, in the case
$N=\mathbb{R}$, these functors induce equivalences
$\mathcal{T}(\mathrm{pt})\simeq\mathcal{T}_{0_{I}}(T^{*}I)$ and
$\mathcal{T}_{\mathrm{lc}}(\mathrm{pt})\simeq\mathcal{T}_{\mathrm{lc},0_{I}}(T^{*}I)$.

###### Proof.

Proposition 2.7.8 of [KS90] says that $p^{-1}$ and $\mathrm{R}p_{*}$ give
equivalences between $\mathsf{D}(\mathbf{k}_{N})$ and
$\mathsf{D}_{p}(\mathbf{k}_{N\times I})$, where the latter category is the
subcategory of $\mathsf{D}(\mathbf{k}_{N\times I})$ whose objects restrict to
constant sheaves on the fibers. Now Proposition 5.4.5 of [KS90] says that
$\mathsf{D}_{p}(\mathbf{k}_{N\times I})$ coincides with
$\mathsf{D}_{T^{*}N\times 0_{I}}(\mathbf{k}_{N\times I})$. Since
$i_{a}^{-1}\circ p^{-1}\simeq\operatorname{id}_{\mathsf{D}(\mathbf{k}_{N})}$,
we deduce that $i_{a}^{-1}$ is also an inverse to $p^{-1}$.

The functors $p^{-1}$ and $i_{a}^{-1}$ commute with
$-\ast\mathbf{k}_{[0,+\infty[}$ and we deduce
$\mathcal{T}(\mathrm{pt})\simeq\mathcal{T}_{0_{I}}(T^{*}I)$. Moreover they
send constructible sheaves to constructible sheaves and are $1$-Lipschitz with
respect to the interleaving distance. Hence they also induce the last
equivalence of the lemma. ∎

######  Proposition 3.8.

Let $F\in\mathcal{T}_{\mathrm{lc},0_{M}}(T^{*}M)$ such that $F$ is
cohomologically chordless. Then there exists $c_{0}$ and a locally constant
sheaf $G_{0}$ on $M$ such that $F\simeq
G_{0}\boxtimes\mathbf{k}_{[c_{0},+\infty[}$.

###### Proof.

(i) For $c\in\mathbb{R}$ we set
$G_{c}=\mathrm{R}q_{*}\mathop{\mathrm{R}\mathcal{H}om}\nolimits(F,T_{c*}F)$.
By Lemma 3.7, for any open ball $B\subset M$, we have
$F|_{B\times\mathbb{R}}\simeq p^{-1}F^{\prime}$ for some
$F^{\prime}\in\mathcal{T}_{\mathrm{lc}}(\mathrm{pt})$, where $p\colon
B\times\mathbb{R}\to\mathbb{R}$ is the projection. It follows that
$\mathop{\mathrm{R}\mathcal{H}om}\nolimits(F,T_{c*}F)|_{B\times\mathbb{R}}\simeq
p^{-1}\mathop{\mathrm{R}\mathcal{H}om}\nolimits(F^{\prime},T_{c*}F^{\prime})$.
By base change we deduce that $G_{c}|_{B}$ is constant. Hence $G_{c}$ is
locally constant. We also have the adjunction isomorphism

$\displaystyle\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes
q^{-1}G_{c},T_{c*}F)$
$\displaystyle\simeq\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(q^{-1}G_{c},\mathop{\mathrm{R}\mathcal{H}om}\nolimits(F,T_{c*}F))$
(3.2)
$\displaystyle\simeq\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(G_{c},\mathrm{R}q_{*}\mathop{\mathrm{R}\mathcal{H}om}\nolimits(F,T_{c*}F))=\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(G_{c},G_{c}).$

Since $F$ is cohomologically chordless, it follows that $G_{c}\simeq 0$ for
any $c<0$.

(ii) Let $x\in M$ be given and let $B$ be a small ball around $x$. With the
same notations as in (i) we have
$\mathop{\mathrm{R}\mathcal{H}om}\nolimits(F,T_{c*}F)|_{B\times\mathbb{R}}\simeq
p^{-1}\mathop{\mathrm{R}\mathcal{H}om}\nolimits(F^{\prime},T_{c*}F^{\prime})$
and the base change formula gives
$\mathrm{R}\Gamma(B;G_{c})\simeq\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F^{\prime},T_{c*}F^{\prime})$.
For $c<0$ we thus obtain
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F^{\prime},T_{c*}F^{\prime})\simeq
0$. Let us check that this implies $F^{\prime}\simeq
E\otimes\mathbf{k}_{[c_{0},+\infty[}$ for some constant sheaf $E$ on
$\mathbb{R}$ and some $c_{0}\in\mathbb{R}$.

By [GV22, Corollary B.12] we have a decomposition
$F^{\prime}\simeq\bigoplus_{j\in\mathcal{I}}\mathbf{k}_{[a_{j},b_{j}[}[d_{j}]$,
where $\mathcal{I}$ is a countable set and $a_{j}\in\mathbb{R}$,
$b_{j}\in\mathbb{R}\cup\\{+\infty\\}$, $d_{j}\in\mathbb{Z}$. If $F^{\prime}$
is not of the form $E\otimes\mathbf{k}_{[c_{0},+\infty[}$, then there exists
$n$ with $b_{n}\not=+\infty$ or there exist $n,m$ with $a_{n}\not=a_{m}$ (say
$a_{n}<a_{m}$). In the first case we write
$F^{\prime}\simeq\mathbf{k}_{[a_{n},b_{n}[}[d_{n}]\oplus F^{\prime\prime}$ and
see that
$H(c):=\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(\mathbf{k}_{[a_{n},b_{n}[},\mathbf{k}_{[c+a_{n},c+b_{n}[})$
is a direct summand of
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F^{\prime},T_{c*}F^{\prime})$. By
Lemma 3.9 below $H(c)\simeq\mathbf{k}[-1]$ for $a_{n}-b_{n}<c<0$. The second
case is similar, with the use of the fact that
$\operatorname{Hom}(\mathbf{k}_{[a_{n},+\infty[},\mathbf{k}_{[c+a_{m},+\infty[})\simeq\mathbf{k}$
for $a_{n}-a_{m}<c$. In both cases we have
$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F^{\prime},T_{c*}F^{\prime})\not=0$
and get a contradiction. Hence $F^{\prime}\simeq
E\otimes\mathbf{k}_{[c_{0},+\infty[}$ for some constant sheaf $E$ and
$c_{0}\in\mathbb{R}$, as claimed.

(iii) Summing up, we have for any $x\in M$ and ball $B$ around $x$ an
isomorphism $F|_{B\times\mathbb{R}}\simeq p^{-1}(E\otimes
k_{[c_{0},+\infty[})$ where $p\colon B\times\mathbb{R}\to\mathbb{R}$ is the
projection, $E\in\mathsf{D}(\mathbf{k})$ and $c_{0}\in\mathbb{R}$. Since $M$
is connected, $c_{0}$ does not depend on $x$. It follows that $F$ is supported
on $M\times[c_{0},+\infty[$, hence
$\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{[c_{0},+\infty[},F)\xrightarrow{\sim}F$.

Let us set $G_{0}=\mathrm{R}q_{*}F$. The image of $\operatorname{id}_{G_{0}}$
by the adjunction isomorphisms

$\operatorname{Hom}(G_{0},\mathrm{R}q_{*}F)\simeq\operatorname{Hom}(q^{-1}G_{0},F)\\\
\simeq\operatorname{Hom}(q^{-1}G_{0},\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{[c_{0},+\infty[},F))\simeq\operatorname{Hom}(q^{-1}G_{0}\otimes\mathbf{k}_{[c_{0},+\infty[},F)$

gives a morphism $u\colon q^{-1}G_{0}\otimes\mathbf{k}_{[c_{0},+\infty[}\to
F$. By (ii) it is locally an isomorphism, hence it is an isomorphism. ∎

######  Lemma 3.9.

Let $a,c\in\mathbb{R}$ and $b,d\in\mathbb{R}\cup\\{+\infty\\}$ with $a<b$,
$c<d$. We have

$\displaystyle\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{[a,b[},\mathbf{k}_{[c,d[})$
$\displaystyle\simeq\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{[a,b[\,\cap\,]c,d]},\mathbf{k}_{\mathbb{R}})$
$\displaystyle\simeq\begin{cases}\mathbf{k}_{[c,b]}&\text{if }a\leq c<b\leq
d,\\\ \mathbf{k}_{]a,d[}&\text{if }c<a<d<b,\\\
\mathbf{k}_{\\{a\\}}[-1]&\text{if }a=d,\\\ \mathbf{k}_{I}&\text{else, where
$I$ is half closed or empty}\end{cases}$

and in particular

$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(\mathbf{k}_{[a,b[},\mathbf{k}_{[c,d[})\simeq\begin{cases}\mathbf{k}&\text{if
}a\leq c<b\leq d,\\\ \mathbf{k}[-1]&\text{if }c<a\leq d<b,\\\
0&\text{else.}\end{cases}$

###### Proof.

For an interval $I$ with non empty interior let us write
$I^{*}=(\overline{I}\setminus I)\cup\operatorname{\operatorname{Int}}(I)$ (in
words, we turn closed ends into open ones and conversely). Then
$\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{I},\mathbf{k}_{\mathbb{R}})\simeq\mathbf{k}_{I^{*}}$.
In particular
$\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{[a,b[},\mathbf{k}_{[c,d[})\simeq\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{[a,b[},\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{]c,d]},\mathbf{k}_{\mathbb{R}}))\simeq\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{[a,b[}\otimes\mathbf{k}_{]c,d]},\mathbf{k}_{\mathbb{R}})$,
which gives the first isomorphism. The second one follows by a case by case
check, together with the additional isomorphism
$\mathop{\mathrm{R}\mathcal{H}om}\nolimits(\mathbf{k}_{\\{a\\}},\mathbf{k}_{\mathbb{R}})\simeq\mathbf{k}_{\\{a\\}}[-1]$.
The last assertion is obtained by taking global sections. ∎

We recall that we have quantization maps
$Q\colon\operatorname{\operatorname{DHam}}(T^{*}M)\to\mathcal{T}(T^{*}M^{2})$
introduced in [GKS12]. For
$\varphi\in\operatorname{\operatorname{DHam}}(T^{*}M)$ and
$\mathcal{K}_{\varphi}=Q(\varphi)$, the action of $\mathcal{K}_{\varphi}$ on
$\mathcal{T}(T^{*}M)$,
$F\mapsto\mathcal{K}_{\varphi}^{\varoast}(F)=\mathcal{K}_{\varphi}\varoast F$,
is an auto-equivalence of category and we have
$\operatorname{RS}(\mathcal{K}_{\varphi}^{\varoast}(F))=\varphi(\operatorname{RS}(F))$.

######  Lemma 3.10.

Let $\varphi\in\operatorname{\operatorname{DHam}}(T^{*}M)$ and
$\mathcal{K}_{\varphi}=Q(\varphi)$. Let $F\in\mathcal{T}(T^{*}M)$ be
cohomologically chordless. Then $\mathcal{K}_{\varphi}^{\varoast}(F)$ is
cohomologically chordless.

###### Proof.

Since $\mathcal{K}_{\varphi}^{\varoast}$ is an equivalence, we have

$\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(\mathcal{K}_{\varphi}^{\varoast}(F\otimes
q^{-1}G),\mathcal{K}^{\varoast}(T_{c*}F))\simeq\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(F\otimes
q^{-1}G,T_{c*}F).$

Hence it is enough to check that $\mathcal{K}_{\varphi}^{\varoast}$ commutes
with $T_{c*}$, which is clear by the definition of $\varoast$, and that

$\mathcal{K}_{\varphi}^{\varoast}(F\otimes
q^{-1}G)\simeq\mathcal{K}_{\varphi}^{\varoast}(F)\otimes q^{-1}G.$

Since $\varphi$ is the time $1$ of some isotopy, both sides of this
isomorphism are restrictions at time $1$ of sheaves in
$\mathcal{T}_{A}(T^{*}(M\times\mathbb{R}))$, where $A\subset
T^{*}(M\times\mathbb{R})$ is given by
$A=\\{(x,\xi,s,\sigma)\mid\sigma=h(x,\xi,s)\\}$, with $h$ the Hamiltonian
function of $\varphi$. Both sheaves coincide at time $0$ and the result
follows from a uniqueness property in this situation (see for example
Corollary 2.1.5 in [Gui23]). ∎

We equip $\operatorname{\operatorname{DHam}}(T^{*}M)$ with the sheaf-theoretic
spectral metric $\gamma^{s}$ defined as

$\gamma^{s}(\varphi,\varphi^{\prime})=d_{\mathcal{T}(T^{*}M^{2})}(\mathcal{K}_{\varphi},\mathcal{K}_{\varphi^{\prime}}).$

Denote by $\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$ the
completion of $\operatorname{\operatorname{DHam}}(T^{*}M)$ with respect to
$\gamma^{s}$. By the completeness of $\mathcal{T}(T^{*}M^{2})$ with respect to
$d_{\mathcal{T}(T^{*}M^{2})}$ [AI24, GV22], we can extend the map $Q$ as

$\widehat{Q}\colon\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)\to\mathcal{T}(T^{*}M^{2}).$
(3.3)

We still denote by
$\mathcal{K}_{\varphi_{\infty}}\coloneqq\widehat{Q}(\varphi_{\infty})$ for
$\varphi_{\infty}\in\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$.

Let us make some remarks on the quantization
$\widehat{Q}\colon\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)\to\mathcal{T}_{\mathrm{lc}}(T^{*}M^{2})$
recalled above. As a completion of a group,
$\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$ is a group. The
category $\mathcal{T}_{\mathrm{lc}}(T^{*}M^{2})$ is not a group but it comes
with the operation $\varoast$ which is associative and has
$\mathbf{k}_{\Delta_{M}\times[0,+\infty[}$ as a neutral element.

######  Lemma 3.11.

Let $\varphi_{\infty}\in\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$
and $\mathcal{K}_{\varphi_{\infty}}=\widehat{Q}(\varphi_{\infty})$. Then
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}\colon\mathcal{T}(T^{*}M)\to\mathcal{T}(T^{*}M)$,
$F\mapsto\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F)\coloneqq\mathcal{K}_{\varphi_{\infty}}\varoast
F$ is an equivalence of categories. Moreover, if $F\in\mathcal{T}(T^{*}M)$ is
cohomologically chordless, so is
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F)$.

###### Proof.

We find that $\mathcal{K}_{\varphi_{\infty}}$ has an inverse with respect to
$\varoast$ given by
$\mathcal{K}_{\varphi_{\infty}}^{-1}=\widehat{Q}(\varphi_{\infty}^{-1})$. The
first assertion is then clear. Writing $\varphi_{\infty}$ as a limit of a
Cauchy sequence $(\varphi_{n})_{n}$ of
$\operatorname{\operatorname{DHam}}(T^{*}M)$, the sequence
$\mathcal{K}_{\varphi_{n}}^{\varoast}(F)$ converges to
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F)$. Hence the second assertion
follows from Lemmas 3.10 and 3.5. ∎

Now we extend Proposition 3.8 to the case of a general exact Lagrangian $L$.
To reduce the problem to Proposition 3.8 we shall use a result Arnaud,
Humilière, and Viterbo [AHV24].

######  Theorem 3.12 (cf. [AHV24]).

Let $L\in\mathfrak{L}(T^{*}M)$ be a compact exact Lagrangian submanifold of
$T^{*}M$. Then, there exists
$\varphi_{\infty}\in\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$ such
that $\varphi_{\infty}(L)=0_{M}$, where both sides should be understood as
elements in $\widehat{\mathfrak{L}}(T^{*}M)$. Moreover, the functor
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}\colon\mathcal{T}(T^{*}M^{2})\to\mathcal{T}(T^{*}M^{2})$
sends $\mathcal{T}_{L}(T^{*}M)$ to $\mathcal{T}_{0_{M}}(T^{*}M)$.

The theorem is proved in [AHV24] for the completion of
$\operatorname{\operatorname{DHam}}(T^{*}M)$ with respect to the usual
spectral metric $\gamma$. In Appendix A, we give a proof for the sheaf-
theoretic spectral metric $\gamma^{s}$. Note that if $\mathbf{k}$ is of
characteristic $2$ or $M$ is spin, then $\gamma^{s}$ coincides with the usual
spectral metric $\gamma$ (see [GV22]).

######  Corollary 3.13.

Let $\widetilde{L}\in\mathcal{L}(T^{*}M)$ be a lift of
$L\in\mathfrak{L}(T^{*}M)$ and set $F_{\widetilde{L}}=Q(\widetilde{L})$. Let
$\varphi_{\infty}\in\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$ be
given by Theorem 3.12 and set
$\mathcal{K}_{\varphi_{\infty}}=\widehat{Q}(\varphi_{\infty})\in\mathcal{T}_{\mathrm{lc}}(T^{*}M^{2})$.

  1. $\mathrm{(i)}$

There exist a locally constant sheaf $G_{0}$ on $M$, of rank $1$, and
$c_{0}\in\mathbb{R}$ such that
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F_{\widetilde{L}})\simeq
G_{0}\boxtimes\mathbf{k}_{[c_{0},+\infty[}$.

  2. $\mathrm{(ii)}$

Let $F\in\mathcal{T}_{\mathrm{lc},L}(T^{*}M)$ such that $F$ is cohomologically
chordless. Then there exists $c_{1}$ and a locally constant sheaf $G_{1}$ on
$M$ such that $F\simeq T_{c_{1}*}(F_{\widetilde{L}}\otimes q^{-1}G_{1})$.

###### Proof.

Let $F$ be as in (ii). By Theorem 3.12
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F)\in\mathcal{T}_{\mathrm{lc},0_{M}}(T^{*}M)$,
and by Lemma 3.11 it is cohomologically chordless. By Proposition 3.8, we
deduce $\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F)\simeq
G_{F}\boxtimes\mathbf{k}_{[c_{F},+\infty[}$ for some $c_{F}\in\mathbb{R}$ and
some locally constant sheaf $G_{F}$ on $M$.

By Lemma 3.4 the sheaf $F_{\widetilde{L}}$ satisfies the property in (ii). In
particular, we have (i) with $c_{0}=c_{F_{\widetilde{L}}}$ and
$G_{0}=G_{F_{\widetilde{L}}}$, but we have to check that $G_{0}$ is of rank
$1$. We set
$F_{1}=(\mathcal{K}_{\varphi_{\infty}}^{-1})^{\varoast}(\mathbf{k}_{[c_{0},+\infty[})$.
Then $\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F_{1}\otimes
q^{-1}G_{0})\simeq\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F_{\widetilde{L}})$.
Hence $F_{1}\otimes q^{-1}G_{0}\simeq F_{\widetilde{L}}$. Restricting to
$M\times\\{t\\}$ for $t\gg 0$ we obtain $(F_{1}|_{M\times\\{t\\}})\otimes
G_{0}\simeq F_{\widetilde{L}}|_{M\times\\{t\\}}\simeq\mathbf{k}_{M}$, which
implies that $G_{0}$ is of rank $1$. This proves (i).

Now we come back to $F$ as in (ii). We have

$\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F)\simeq
G_{F}\boxtimes\mathbf{k}_{[c_{F},+\infty[}\simeq
T_{c_{1}*}(G_{1}\otimes(G_{0}\boxtimes\mathbf{k}_{[c_{0},+\infty[})),$

where $c_{1}=c_{F}-c_{0}$ and $G_{1}=G_{F}\otimes G_{0}^{-1}$. Hence
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}(F)\simeq\mathcal{K}_{\varphi_{\infty}}^{\varoast}(T_{c_{1}*}(F_{\widetilde{L}}\otimes
q^{-1}G_{1}))$ and (ii) follows. ∎

##  4 Regular Lagrangians are smooth Lagrangians

######  Theorem 4.1.

Let $L_{\infty}\in\widehat{}\mathfrak{L}(T^{*}M)$. We assume that
$L=\operatorname{\gamma-supp}(L_{\infty})$ is a compact exact Lagrangian
submanifold of $T^{*}M$. Then $L_{\infty}=L$ in $\mathfrak{L}(T^{*}M)$.

###### Proof.

We lift $L_{\infty}$ to
$\widetilde{L}_{\infty}\in\widehat{\mathcal{L}}(T^{*}M)$ and set
$F_{\infty}=\widehat{Q}(\widetilde{L}_{\infty})$. By definition,
$\widetilde{L}_{\infty}$ is the equivalence class of a Cauchy sequence
$(L_{n})_{n}$ in $\mathcal{L}(T^{*}M)$ and, by [GV22] or [AI24], the sequence
of associated sheaves $F_{L_{n}}$ converges in $\mathcal{T}(T^{*}M)$ to
$F_{\infty}$. By [AGHIV23] we know that $\operatorname{RS}(F_{\infty})=L$. We
lift $L$ into $\widetilde{L}\in\mathcal{L}(T^{*}M)$ and let
$F_{\widetilde{L}}$ be the associated sheaf. By Lemma 3.5 and Corollary 3.13
we have $F_{\infty}\simeq T_{c_{1}*}(F_{\widetilde{L}}\otimes q^{-1}G_{1})$
for some $c_{1}\in\mathbb{R}$ and some locally constant sheaf $G_{1}$ on $M$.
Let $x\in M$ be given. The sequence $F_{L_{n}}|_{\\{x\\}\times\mathbb{R}}$
converges to $F_{\infty}|_{\\{x\\}\times\mathbb{R}}$ and
$\mathrm{R}\Gamma(\\{x\\}\times\mathbb{R};F_{L_{n}}|_{\\{x\\}\times\mathbb{R}})\simeq\mathbf{k}$
for all $n$. Hence
$\mathrm{R}\Gamma(\\{x\\}\times\mathbb{R};F_{\infty}|_{\\{x\\}\times\mathbb{R}})\simeq\mathbf{k}$.
It follows that $(G_{1})_{x}\simeq\mathbf{k}$. The same kind of argument shows
that
$\mathrm{R}\Gamma(M;G_{1})\simeq\mathrm{R}\Gamma(M\times\mathbb{R};F_{\infty})\simeq\mathbf{k}$.
Hence $G_{1}\simeq\mathbf{k}_{M}$ and $F_{\infty}\simeq
T_{c_{1}*}F_{\widetilde{L}}$, which implies
$\widetilde{L}_{\infty}=T_{c_{1}}(\widetilde{L})$, hence $L_{\infty}=L$. ∎

######  Remark 4.2.

Note that for $L=0_{M}$ in $T^{*}M$ and for a manifold $M$ satisfying a
certain condition (denoted by $(\star)$ in [Vit22]), this theorem follows from
Theorem 8.6 of [Vit22a](version 2) as a consequence of Theorem 6.3 in [Vit22].
This was removed from the published version of [Vit22a] and included in
[Vit22].

##  5 Compact $\gamma$-supports are connected

######  Theorem 5.1.

Let $L_{\infty}\in\widehat{\mathfrak{L}}(T^{*}M)$ and assume that
$\operatorname{\gamma-supp}(L_{\infty})$ is compact. Then
$\operatorname{\gamma-supp}(L_{\infty})$ is connected.

######  Lemma 5.2.

For any $\widetilde{L}_{\infty}\in\widehat{\mathcal{L}}(T^{*}M)$, one has
$\operatorname{End}_{\mathcal{T}(T^{*}M)}(F_{\widetilde{L}_{\infty}})\simeq\mathbf{k}$.

###### Proof.

Write $\widetilde{L}_{\infty}$ as the equivalence class of a Cauchy sequence
$(\widetilde{L}_{n})_{n}$, where $\widetilde{L}_{n}\in\mathcal{L}(T^{*}M)$.
Then, we find that
$\mathrm{R}q_{*}\mathop{{\mathcal{H}}om}\nolimits^{*}(F_{\widetilde{L}_{n}},F_{\widetilde{L}_{n}})\simeq\mathrm{R}\Gamma(M;\mathbf{k}_{M})\otimes\mathbf{k}_{[0,+\infty[}$
for any $n$ by [Vit19, Proposition 9.11]. Since
$\gamma(\widetilde{L}_{n},\widetilde{L}_{\infty})\to 0$ and
$\mathop{{\mathcal{H}}om}\nolimits^{*}$ is continuous for the interleaving
distance, we find that
$\mathrm{R}q_{*}\mathop{{\mathcal{H}}om}\nolimits^{*}(F_{\widetilde{L}_{n}},F_{\widetilde{L}_{n}})$
converges to
$\mathrm{R}q_{*}\mathop{{\mathcal{H}}om}\nolimits^{*}(F_{\widetilde{L}_{\infty}},F_{\widetilde{L}_{\infty}})$.
This implies that

$d_{\mathcal{T}(\mathrm{pt})}(\mathrm{R}q_{*}\mathop{{\mathcal{H}}om}\nolimits^{*}(F_{\widetilde{L}_{\infty}},F_{\widetilde{L}_{\infty}}),\mathrm{R}\Gamma(M;\mathbf{k}_{M})\otimes\mathbf{k}_{[0,+\infty[})=0,$

from which we deduce
$\mathrm{R}q_{*}\mathop{{\mathcal{H}}om}\nolimits^{*}(F_{\widetilde{L}_{\infty}},F_{\widetilde{L}_{\infty}})\simeq\mathrm{R}\Gamma(M;\mathbf{k}_{M})\otimes\mathbf{k}_{[0,+\infty[}$
by [GV22, Proposition B.8]. Thus, we obtain

$\operatorname{Hom}(F_{\widetilde{L}_{\infty}},F_{\widetilde{L}_{\infty}})\simeq
H^{0}\mathop{\mathrm{R}\mathrm{Hom}}\nolimits(\mathbf{k}_{[0,+\infty[},\mathrm{R}q_{*}\mathop{{\mathcal{H}}om}\nolimits^{*}(F_{\widetilde{L}_{\infty}},F_{\widetilde{L}_{\infty}}))\simeq\mathbf{k},$
(5.1)

which proves the lemma. ∎

######  Lemma 5.3.

Let $U$ be an open subset of $T^{*}M$ and $F\in\mathcal{T}(T^{*}M)$. Assume
that $\operatorname{RS}(F)\cap U$ is compact and
$\operatorname{RS}(F)\cap\partial U=\varnothing$, where $\partial U$ is the
topological boundary defined as $\overline{U}\setminus U$. Then one has an
exact triangle

$P(U)\varoast F\to F\to Q(U)\varoast F\xrightarrow{+1}$ (5.2)

with $\operatorname{RS}(P(U)\varoast F)=\operatorname{RS}(F)\cap U$ and
$\operatorname{RS}(Q(U)\varoast
F)=\operatorname{RS}(F)\cap(T^{*}M\setminus\overline{U})$. Here
$P(U),Q(U)\colon\mathcal{T}(T^{*}M)\to\mathcal{T}(T^{*}M)$ are the microlocal
projectors associated with $U$ (see [Chi17, Zha21, Zha23]).

###### Proof.

Set $A_{1}\coloneqq\operatorname{RS}(F)\cap U$ and
$A_{2}\coloneqq\operatorname{RS}(F)\setminus A_{1}$. To show $P(U)\varoast
F\in\mathcal{T}_{A_{1}}(T^{*}M)$ and $Q(U)\varoast
F\in\mathcal{T}_{A_{2}}(T^{*}M)$, we use (a version of) Kuo’s description of
projectors by microlocal wrapping [Kuo23, KSZ23].

Let $C_{c}^{\infty}(U)$ be the poset of compactly supported smooth functions
on $U$ and $H_{\bullet}\colon\mathbb{N}\to C_{c}^{\infty}(U)$ be a final
functor satisfying $H_{n}\equiv n$ on a neighborhood of $A_{1}$ for each $n$.
The microlocal projector $Q(U)$ is described as (see after (3.1) for the
notation $\operatorname*{\operatorname{hocolim}}$)

$Q(U)\simeq\operatorname*{\operatorname{hocolim}}_{n\in\mathbb{N}}\mathcal{K}_{H_{n}}.$
(5.3)

Since $\operatorname*{\operatorname{hocolim}}$ commutes with $\varoast$,
$Q(U)\varoast
F\simeq\operatorname*{\operatorname{hocolim}}_{n}(\mathcal{K}_{H_{n}}\varoast
F)$. Since $dH_{n}$ vanishes on a neighborhood of $\operatorname{RS}(F)$,
$\operatorname{RS}(\mathcal{K}_{H_{n}}\varoast
F)=\operatorname{RS}(F)=A_{1}\cup A_{2}$. Hence
$\operatorname{RS}(\operatorname*{\operatorname{hocolim}}_{n}(\mathcal{K}_{H_{n}}\varoast
F))\subset A_{1}\cup A_{2}$. On the other hand,
$\operatorname{RS}(Q(U)\varoast F)\subset T^{*}M\setminus U$ by a formal
property of the projector $Q(U)$ and hence,

$\operatorname{RS}(Q(U)\varoast F)\subset(T^{*}M\setminus U)\cap(A_{1}\cup
A_{2})=A_{2}.$ (5.4)

Since the morphism $F\to Q(U)\varoast F$ is an isomorphism on
$T^{*}M\setminus\overline{U}$, $(P(U)\varoast
F)\cap(T^{*}M\setminus\overline{U})=\varnothing$. The triangle inequality for
microsupports shows $\operatorname{RS}(P(U)\varoast F)=A_{1}$. ∎

######  Lemma 5.4.

Let $F\in\mathcal{T}(T^{*}M)$ and assume that $\operatorname{RS}(F)$ is
decomposed into two compact disjoint subsets $A_{1}$ and $A_{2}$. Then there
exist $F_{1},F_{2}\in\mathcal{T}(T^{*}M)$ such that
$\operatorname{RS}(F_{i})=A_{i}$ and $F\simeq F_{1}\oplus F_{2}$.

###### Proof.

Take an open neighborhood $U$ of $A_{1}$ such that $\overline{U}\cap
A_{2}=\varnothing$. Applying Lemma 5.3, we have an exact triangle in
$\mathcal{T}(T^{*}M)$

$P(U)\varoast F\to F\to Q(U)\varoast F\xrightarrow{+1}$ (5.5)

with $P(U)\varoast F\in\mathcal{T}_{A_{1}}(T^{*}M)$ and $Q(U)\varoast
F\in\mathcal{T}_{A_{2}}(T^{*}M)$. Set $F_{1}\coloneqq P(U)\varoast F$ and
$F_{2}\coloneqq Q(U)\varoast F$. Since $A_{1}\cap A_{2}=\varnothing$ and each
$A_{i}$ is compact, by Tamarkin’s separation theorem we have
$\operatorname{Hom}_{\mathcal{T}(T^{*}M)}(F_{2},F_{1}[1])=0$. Then by the
above exact triangle, $F\simeq F_{1}\oplus F_{2}$. ∎

######  Remark 5.5.

Results similar to Lemma 5.3 also appear in [Zha24]. Lemma 5.4 is a wide
generalization of [Gui23, Proposition 3.3.2], where the result is local and
the sets $A_{1}$, $A_{2}$ are supposed “unknotted”.

###### Proof of Theorem 5.1.

Suppose that $\operatorname{\gamma-supp}(L_{\infty})$ is decomposed into two
non-empty compact disjoint subsets $A_{1}$ and $A_{2}$. Let
$\widetilde{L}_{\infty}$ be a lift of $L_{\infty}$. Set
$F_{\widetilde{L}_{\infty}}=Q(\widetilde{L}_{\infty})\in\mathcal{T}(T^{*}M)$.
By a result of [AGHIV23], we have $\operatorname{\gamma-
supp}(L_{\infty})=\operatorname{RS}(F_{\widetilde{L}_{\infty}})$. By Lemma
5.4, there exist $F_{i}\in\mathcal{T}_{A_{i}}(T^{*}M)$ such that
$F_{\widetilde{L}_{\infty}}\simeq F_{1}\oplus F_{2}$. Since
$F_{\widetilde{L}_{\infty}}$ is indecomposable by Lemma 5.2, either $F_{1}$ or
$F_{2}$ is zero. This is a contradiction. ∎

######  Remarks 5.6.

  1. $\mathrm{(i)}$

The connectedness does not hold for elements in
$\widehat{\mathfrak{L}}(T^{*}M)$ with non-compact $\gamma$-support. Indeed,
consider the situation in Figure 5.1. Since $f_{j}$ $C^{0}$-converges to $f$,
$L_{j}=\mathrm{graph}(df_{j})$ $\gamma$-converges to some $L_{\infty}$ in
$\widehat{\mathfrak{L}}(T^{*}S^{1})$. Clearly the $\gamma$-support of
$L_{\infty}$ is the union of the two connected components represented in
Figure 5.1(b).

  2. $\mathrm{(ii)}$

One can construct examples of sequences $F_{i}$ of elements in
$\mathcal{T}(T^{*}M)$ such that $\operatorname{RS}(F_{i})$ remain in a fixed
compact set, are connected, $F_{i}$ converges to $F$ for
$d_{\mathcal{T}(T^{*}M)}$, but $\operatorname{RS}(F)$ is not connected. As a
result $F$ is not in the image of $\widehat{Q}$.

$f_{j}$$f=\lim f_{j}$

(a)

$L_{j}=\mathrm{graph}(df_{j})$$L_{\infty}$

(b)

Figure 5.1: An element $L_{\infty}=\gamma\mathchar 45\relax\lim(L_{j})$ in
$\widehat{\mathfrak{L}}(T^{*}S^{1})$ with non-compact and disconnected
$\gamma$-support. Figure (b) is the differential of Figure (a), the
$f_{j},L_{j}$ correspond to the dashed curves, $f,L_{\infty}$ to the
continuous curves.

##  Appendix A The weak nearby Lagrangian conjecture for the sheaf-theoretic
spectral metric

In this appendix, we prove a sheaf-theoretic version of a result of Arnaud,
Humilière, and Viterbo [AHV24]. The following theorem is for the sheaf-
theoretic spectral metric $\gamma^{s}$.

######  Theorem A.1 (cf. [AHV24]).

Let $L\in\mathfrak{L}(T^{*}M)$ be a compact exact Lagrangian submanifold of
$T^{*}M$. Then, there exists
$\varphi_{\infty}\in\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$ such
that $\varphi_{\infty}(L)=0_{M}$, where both sides should be understood as
elements in $\widehat{\mathfrak{L}}(T^{*}M)$. Moreover, the functor
$\mathcal{K}_{\varphi_{\infty}}^{\varoast}\colon\mathcal{T}(T^{*}M^{2})\to\mathcal{T}(T^{*}M^{2})$
sends $\mathcal{T}_{L}(T^{*}M)$ to $\mathcal{T}_{0_{M}}(T^{*}M)$.

The zero-section $0_{M}$ is the fixed point set of the canonical Liouville
flow. We set $\psi_{0}^{s}$ to be the time-$s$ map of the canonical Liouville
flow and set $\psi_{0}\coloneqq\psi_{0}^{1}$. The map $\psi_{0}^{s}$ is the
multiplication by $e^{s}$ on each fiber. By [AHV24, Proposition 7.3], we see
that $L$ is also the fixed point set of the Liouville flow of another
Liouville 1-form which coincides with the canonical Liouville form outside a
compact subset. Set $\psi_{1}$ to be the time-1 map of this latter Liouville
flow.

We will find $\varphi_{\infty}$ as a fixed point of a contraction on
$\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$. To construct the
contraction, we first note the following.

######  Lemma A.2.

Let $\psi\colon T^{*}M\to T^{*}M$ be a diffeomorphism such that
$\psi^{*}\lambda=a\lambda$ for some $a>0$. Moreover, let $H\colon T^{*}M\times
I\to\mathbb{R}$ be a compactly supported function and set $\tilde{H}\coloneqq
a^{-1}H\circ\psi$. Then,
$\psi^{-1}\circ\phi^{H}_{s}\circ\psi=\phi^{\tilde{H}}_{s}$ and
$\lambda(X_{\tilde{H}_{s}})=a^{-1}\lambda(X_{H_{s}})\circ\psi$ for $s\in I$.

###### Proof.

For $s\in I$, we have

$\displaystyle\iota_{X_{\tilde{H}_{s}}}\omega$
$\displaystyle=-d(a^{-1}H_{s}\circ\psi)$ $\displaystyle=-a^{-1}\psi^{*}dH_{s}$
$\displaystyle=a^{-1}\psi^{*}(\iota_{X_{H_{s}}}\omega)$
$\displaystyle=a^{-1}\iota_{\psi^{*}X_{H_{s}}}(\psi^{*}\omega)$
$\displaystyle=\iota_{\psi^{*}X_{H_{s}}}\omega,$

where $\psi^{*}X\coloneqq(d\psi)^{-1}X\circ\psi$. This shows that
$X_{\tilde{H}_{s}}=\psi^{*}X_{H_{s}}$, which implies
$\phi^{H}_{s}\circ\psi=\psi\circ\phi^{\tilde{H}}_{s}$. Moreover, since
$\lambda=a^{-1}\psi^{*}\lambda$, we have

$\iota_{X_{\tilde{H}_{s}}}\lambda=\iota_{\psi^{*}X_{H_{s}}}\lambda=a^{-1}\iota_{\psi^{*}X_{H_{s}}}(\psi^{*}\lambda)=a^{-1}\psi^{*}\iota_{X_{H_{s}}}\lambda,$
(A.1)

which proves the second equality. ∎

######  Lemma A.3.

For a compactly supported function $H\colon T^{*}M\times I\to\mathbb{R}$, set
$K_{\phi^{H}}\in\mathsf{D}(M^{2}\times I\times\mathbb{R})$ be the sheaf
quantization of the Hamiltonian isotopy $\phi^{H}$. Then, for such a function
$H$, one has $K_{\psi_{0}^{-1}\circ\phi^{H}\circ\psi_{0}}\simeq
f_{*}K_{\phi^{H}}$, where $f$ is defined as

$f\colon M^{2}\times\mathbb{R}\times I\to M^{2}\times\mathbb{R}\times
I;(x_{1},x_{2},t,s)\mapsto(x_{1},x_{2},e^{-1}t,s).$ (A.2)

###### Proof.

Set $\tilde{H}\coloneqq e^{-1}H\circ\psi_{0}$. We also define

$u_{H,s}(p)\coloneqq\int_{0}^{s}(H_{s^{\prime}}-\lambda(X_{H_{s^{\prime}}}))(\phi^{H}_{s^{\prime}}(p))\,ds^{\prime}$
(A.3)

for $H\colon T^{*}M\times I\to\mathbb{R}$ and $s\in I$. Then, by Lemma A.2, we
have

$\displaystyle u_{\tilde{H},s}(p)$
$\displaystyle=\int_{0}^{s}(\tilde{H}_{s^{\prime}}-\lambda(X_{\tilde{H}_{s^{\prime}}}))(\psi_{0}^{-1}\phi^{H}_{s^{\prime}}\psi_{0}(p))\,ds^{\prime}$
$\displaystyle=e^{-1}\int_{0}^{s}(H_{s^{\prime}}-\lambda(X_{H_{s^{\prime}}}))(\phi^{H}_{s^{\prime}}(\psi_{0}(p)))\,ds^{\prime}$
$\displaystyle=e^{-1}u_{H,s}(\psi_{0}(p)).$

We shall estimate the microsupports of
$K_{\psi_{0}^{-1}\circ\phi^{H}\circ\psi_{0}}$ and $f_{*}K_{\phi^{H}}$. On the
one hand, we have

$\displaystyle\operatorname{{\mathring{\operatorname{{\operatorname{SS}}}}}}(K_{\psi_{0}^{-1}\circ\phi^{H}\circ\psi_{0}})$
$\displaystyle={}$
$\displaystyle\left\\{((x^{\prime};\xi^{\prime}),(x;-\xi),(u_{\tilde{H},s}(x;\xi/\tau),\tau),(s,-\tau\tilde{H}_{s}(\phi^{\tilde{H}}_{s}(x;\xi/\tau))))\mathrel{}\middle|\mathrel{}(x^{\prime};\xi^{\prime}/\tau)=\phi^{\tilde{H}}_{s}(x;\xi/\tau)\right\\}$
$\displaystyle={}$ $\displaystyle\begin{aligned}
&\left\\{((x^{\prime};\xi^{\prime}),(x;-\xi),(e^{-1}u_{H,s}(x;e\xi/\tau),\tau),(s,-\tau
e^{-1}H_{s}(\phi^{H}_{s}(x;e\xi/\tau))))\right|\\\
&\quad\left.(x^{\prime};e\xi^{\prime}/\tau)=\phi^{H}_{s}(x;e\xi/\tau)\right\\}.\end{aligned}$

On the other hand, we have

$\displaystyle\operatorname{{\mathring{\operatorname{{\operatorname{SS}}}}}}(f_{*}K_{\phi^{H}})$
$\displaystyle={}$ $\displaystyle\begin{aligned}
&\left\\{((x^{\prime};\xi^{\prime}),(x;-\xi),(e^{-1}u_{H,s}(x;\xi/\tau),e\tau),(s,-\tau
H_{s}(\phi^{H}_{s}(x;\xi/\tau))))\right|\\\
&\quad\left.(x^{\prime};\xi^{\prime}/\tau)=\phi^{H}_{s}(x;\xi/\tau)\right\\}\end{aligned}$
$\displaystyle={}$ $\displaystyle\begin{aligned}
&\left\\{((x^{\prime};\xi^{\prime}),(x;-\xi),(e^{-1}u_{H,s}(x;e\xi/\tilde{\tau}),\tilde{\tau}),(s,-\tilde{\tau}e^{-1}H_{s}(\phi^{H}_{s}(x;e\xi/\tilde{\tau}))))\right|\\\
&\quad\left.(x^{\prime};e\xi^{\prime}/\tilde{\tau})=\phi^{H}_{s}(x;e\xi/\tilde{\tau})\right\\}.\end{aligned}$

Since $(f_{*}K_{\phi^{H}})|_{s=0}\simeq\mathbf{k}_{\Delta\times\\{0\\}}$, by
the uniqueness of the sheaf quantization ([GKS12]), we conclude. ∎

###### Proof of Theorem A.1.

By Lemma A.3, for $\varphi\in\operatorname{\operatorname{DHam}}(T^{*}M)$, we
have
$\mathcal{K}_{\psi_{0}^{-1}\circ\varphi\circ\psi_{0}}=f_{*}\mathcal{K}_{\varphi}$,
where $\mathcal{K}_{\varphi}=Q(\varphi)$ and $f$ is defined, by abuse of
notation, as

$f\colon M^{2}\times\mathbb{R}\to
M^{2}\times\mathbb{R},\qquad(x_{1},x_{2},t)\mapsto(x_{1},x_{2},e^{-1}t).$
(A.4)

This implies, setting
$h=\psi_{0}^{-1}\circ\psi_{1}\in\operatorname{\operatorname{DHam}}(T^{*}M)$,
that the map

$T\colon\operatorname{\operatorname{DHam}}(T^{*}M)\to\operatorname{\operatorname{DHam}}(T^{*}M),\qquad\varphi\mapsto\psi_{0}^{-1}\circ\varphi\circ\psi_{1}=\psi_{0}^{-1}\circ\varphi\circ\psi_{0}\circ
h$ (A.5)

is a contraction. Note that
$h=\psi_{0}^{-1}\circ\psi_{1}\in\operatorname{\operatorname{DHam}}(T^{*}M)$ is
proved in the proof of [AHV24, Theorem 7.4]. Indeed, we have

$\displaystyle
d_{\mathcal{T}(T^{*}M^{2})}(\mathcal{K}_{T\varphi},\mathcal{K}_{T\varphi^{\prime}})$
$\displaystyle=d_{\mathcal{T}(T^{*}M^{2})}(\mathcal{K}_{\psi_{0}^{-1}\circ\varphi\circ\psi_{0}\circ
h},\mathcal{K}_{\psi_{0}^{-1}\circ\varphi^{\prime}\circ\psi_{0}\circ h})$
(A.6)
$\displaystyle=d_{\mathcal{T}(T^{*}M^{2})}(\mathcal{K}_{\psi_{0}^{-1}\circ\varphi\circ\psi_{0}}\varoast\mathcal{K}_{h},\mathcal{K}_{\psi_{0}^{-1}\circ\varphi^{\prime}\circ\psi_{0}}\varoast\mathcal{K}_{h})$
$\displaystyle=d_{\mathcal{T}(T^{*}M^{2})}(\mathcal{K}_{\psi_{0}^{-1}\circ\varphi\circ\psi_{0}},\mathcal{K}_{\psi_{0}^{-1}\circ\varphi^{\prime}\circ\psi_{0}})$
$\displaystyle=d_{\mathcal{T}(T^{*}M^{2})}(f_{*}\mathcal{K}_{\varphi},f_{*}\mathcal{K}_{\varphi^{\prime}})$
$\displaystyle=e^{-1}d_{\mathcal{T}(T^{*}M^{2})}(\mathcal{K}_{\varphi},\mathcal{K}_{\varphi^{\prime}}),$

where the last equality follows from the fact that an $(a,b)$-isomorphism for
$(\mathcal{K}_{\varphi},\mathcal{K}_{\varphi^{\prime}})$ gives an
$(e^{-1}a,e^{-1}b)$-isomorphism for
$(f_{*}\mathcal{K}_{\varphi},f_{*}\mathcal{K}_{\varphi^{\prime}})$ and vice
versa.

Hence, the map $T$ extends to the completion as a contraction, which we will
also denote by
$T\colon\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)\to\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$.
Therefore, there exists a unique fixed point
$\varphi_{\infty}\in\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$ so
that $\psi_{0}^{-1}\varphi_{\infty}=\varphi_{\infty}\psi_{1}^{-1}$, where both
sides should be understood as actions on $\widehat{\mathfrak{L}}(T^{*}M)$.
Since $L\in\widehat{\mathfrak{L}}(T^{*}M)$ is the unique fixed point of the
action of $\psi_{1}^{-1}$ and $0_{M}\in\widehat{\mathfrak{L}}(T^{*}M)$ is the
unique fixed point of the action of $\psi_{0}^{-1}$, we find
$\varphi_{\infty}(L)=0_{M}$.

By its construction, the element
$\varphi_{\infty}\in\widehat{\operatorname{\operatorname{DHam}}}(T^{*}M)$ is
represented as the sequence $(\varphi_{n})_{n\in\mathbb{N}}$ with
$\varphi_{n}\coloneqq\psi_{0}^{-n}\circ\psi_{1}^{n}$. Thus, we obtain

$\liminf_{n}\varphi_{n}(L)=\liminf_{n}\psi_{0}^{-n}\circ\psi_{1}^{n}(L)=\liminf_{n}\psi_{0}^{-n}(L)=0_{M}.$
(A.7)

Since taking the microsupport is “continuous” by [GV22, Proposition 6.26], for
any $F\in\mathcal{T}_{L}(T^{*}M)$, we have

$\operatorname{RS}(K_{\varphi_{\infty}}^{\varoast}(F))\subset\liminf_{n}\varphi_{n}(L)=0_{M},$
(A.8)

which proves the result. ∎

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