Lagrangian Cobordism and Shadow Distance
in Tamarkin Category

Let $M$ be a closed manifold. Let $L_{0},\dots,$ $L_{m-1}$ and $L^{\prime}_{0},\dots,L^{\prime}_{n-1}\subset T^{*}M$ be exact Lagrangian submanifolds with conical ends. An exact Lagrangian cobordism with conical ends $V\subset T^{*}(M\times\mathbb{R})$ between $(L_{0},\dots,L_{m-1})$ and $(L^{\prime}_{0},\dots,L^{\prime}_{n-1})$ is an exact Lagrangian with conical end $V_{\infty}$ such that $V_{\infty}\cap T^{*,\infty}(M\times[-1,1])$ is compact and

In this study, we use the Tamarkin category $\mathcal{D}(M)$, which is a localization of the sheaf category $\operatorname{\mathrm{Sh}}(M\times\mathbb{R})$. We can equip $\mathcal{D}(M)$ with an interleaving distance $d^{\prime}_{\mathcal{D}(M)}$ that is stable under Hamiltonian isotopies [AI20]. See Subsection 2.2 for details. For any exact Lagrangian $L\subset T^{*}M$, we can construct a simple sheaf $F_{L}\in\mathcal{D}(M)$ with reduced microsupport on $L$ [Gu12, JT17, Vi19, Gu23]. They are called sheaf quantizations of $L$.

Our first result is the following existence theorem of a sheaf quantization of a Lagrangian cobordism, namely, that there exists a simple sheaf with reduced microsupport in the given exact Lagrangian cobordism.

Our second result is the following iterated cone decomposition theorem of a sheaf quantization of a Lagrangian cobordism. We write

Our third result is the following inequality for the interleaving distance by the shadow distance of a Lagrangian cobordism. For a Lagrangian cobordism $V\subset T^{*}(M\times\mathbb{R})$, the shadow $\mathcal{S}(V)$ of $V$ is defined as the area of $T^{*}\mathbb{R}\setminus\mathcal{U}(V)$, where $\mathcal{U}(V)$ is the union of unbounded regions of $T^{*}\mathbb{R}\setminus p(V)$ with $p\colon T^{*}(M\times\mathbb{R})\to T^{*}\mathbb{R}$ being the projection map. It is known that in the exact setting, the Lagrangian shadow induces a metric on all Lagrangian submanifolds [CS19]. Our result in particular implies an isomorphism of the sheaves in the Tamarkin category modulo torsion objects $\mathcal{T}(M)=\mathcal{D}(M)/\mathrm{Tor}$ where sheaves related by Hamiltonian isotopies become isomorphic.

In fact, we prove Theorems 1.3 and 1.4 in more general setting, without assuming the cobordism is a Lagrangian submanifold. See the main body of the paper.

We apply the theorems above to Lagrangian intersection, recovering [BCS18, Thm. C] in the cotangent bundle case, which shows that there is an energy cost to separate or connect Lagrangians through cobordisms.

Comparing to previous Lagrangian intersection results using sheaf theoretic methods [Ike19, AISQ], we need to estimate the energy of higher homotopies in the iterated cone decomposition, which, in Lagrangian Floer theory, come from pseudo-holomorphic polygons instead of just pseudo-holomorphic strips. See the main body of the paper.

The relation between Lagrangian cobordisms and Fukaya categories and their filtrations has been studied in many previous works. Biran–Cornea [BC13, BC14] showed that (monotone) Lagrangian cobordisms induce iterated cone decompositions in the Fukaya categories. Cornea–Shelukhin [CS19] introduced the shadow distance of Lagrangian cobordisms, and later Biran–Cornea–Shelukhin [BCS18] used the shadow distance to prove results on Lagrangian intersection using the theory of weakly filtered $A_{\infty}$-categories. More recently, motivated by the geometry, Biran–Cornea–Zhang [BCZ23] developed the theory of persistence triangulated categories and showed that the Tamarkin category and the Fukaya category are persistence triangulated categories.

It is well known that the Hofer distance of $C^{1}$-close Lagrangians is related to the shadow of Lagrangian movie under Hamiltonian isotopies [Milinkovic]. Since its introduction, the Lagrangian shadow has become a standard tool in the study of quantitative aspects of Lagrangian submanifolds. See [Hicks21, HicksMak22] for interesting geometric constructions that involve the Lagrangian shadows.

The algebraic results above can also be enhanced to a categorical level. Biran–Cornea [BC14] defined a category that with a functor into the cone resolution category of the Fukaya category, while Nadler–Tanaka [NadTanaka] defined a stable $\infty$-category of Lagrangian cobordisms with an exact functor into the wrapped Fukaya category [Tanaka16, Tanaka16Exact]. We expect that the sheaf quantization result here provides a way to define a functor from that category to the sheaf categories.

Finally, it is expected that the relation between Lagrangian cobordisms and Fukaya categories passes through the Waldhausen $s$-construction in algebraic $K$-theory. In particular, Biran–Cornea [BC14] conjectured an isomorphism between the Lagrangian cobordism group and the Grothendieck group of the Fukaya category. See [BCZ23-2] for the relation between the Grothendieck group and persistence of the Fukaya category. It would be interesting to study the question for the sheaf categories.

TA thanks Kaoru Ono for the helpful discussions. TA was partially supported by Innovative Areas Discrete Geometric Analysis for Materials Design (Grant No. 17H06461). YI thanks Alexandru Oancea for the fruitful discussions. He also thanks Paul Biran, Octav Cornea, and Jun Zhang for helpful discussions and clarifications. YI was partially supported by JSPS KAKENHI Grant Numbers 21K13801 and 22H05107. WL thanks Xin Jin and David Treumann for helpful discussions and clarifications and Bingyu Zhang for helpful discussions.

For a manifold $X$, we write $\mathbf{k}_{X}$ for the constant sheaf with stalk $\mathbf{k}$ and let $\operatorname{\mathrm{Sh}}(X)=\operatorname{\mathrm{Sh}}(\mathbf{k}_{X})$ denote the dg-derived category of sheaves of $\mathbf{k}$-vector spaces on $X$. One can define Grothendieck’s six operations between dg-derived categories of sheaves $\mathop{{\mathcal{H}}om}\nolimits,\allowbreak\otimes,\allowbreak f_{*},\allowbreak f^{-1},\allowbreak f_{!},\allowbreak f^{!}$ for a morphism of manifolds $f\colon X\to Y$ [Spaltenstein, Schn18]. For a locally closed subset $Z$ of $X$, we denote by $\mathbf{k}_{Z}\in\operatorname{\mathrm{Sh}}(X)$ the constant sheaf with stalk $\mathbf{k}$ on $Z$, extended by $0$ on $X\setminus Z$. Moreover, for a locally closed subset $Z$ of $X$ and $F\in\operatorname{\mathrm{Sh}}(X)$, we define

Let $X_{1},X_{2},X_{3}$ be manifolds and $\pi_{ij}\colon X_{1}\times X_{2}\times X_{3}\to X_{i}\times X_{j}$ for $i\neq j\in\{1,2,3\}$. For $K_{12}\in\operatorname{\mathrm{Sh}}(X_{1}\times X_{2})$ and $K_{23}\in\operatorname{\mathrm{Sh}}(X_{2}\times X_{3})$, their composition is defined by

Following [KS90, robalo2018lemma], for $F\in\operatorname{\mathrm{Sh}}(X)$, we let $\operatorname{{\operatorname{SS}}}(F)$ denote the microsupport of $F$, which is a conical (i.e., invariant under the action of $\mathbb{R}_{>0}$) closed subset of $T^{*}X$. The set $\operatorname{{\operatorname{SS}}}(F)$ describes the singular codirections of $F$, in which $F$ does not propagate. We also use the notation $\mathring{\operatorname{{\operatorname{SS}}}}(F)\coloneqq\operatorname{{\operatorname{SS}}}(F)\cap\mathring{T}^{*}X=\operatorname{{\operatorname{SS}}}(F)\setminus 0_{X}$.

By using microsupports, we can microlocalize the category $\operatorname{\mathrm{Sh}}(X)$. Let $A\subset T^{*}X$ be a subset and set $\Omega=T^{*}X\setminus A$. We denote by $\operatorname{\mathrm{Sh}}_{A}(X)$ the full dg-subcategory of $\operatorname{\mathrm{Sh}}(X)$ consisting of sheaves whose microsupports are contained in $A$. By the triangle inequality, the subcategory $\operatorname{\mathrm{Sh}}_{A}(X)$ is a pretriangulated subcategory. We set

the dg-quotient of $\operatorname{\mathrm{Sh}}(X)$ by $\operatorname{\mathrm{Sh}}_{A}(X)$. A morphism $u\colon F\to G$ in $\operatorname{\mathrm{Sh}}(X)$ becomes a quasi-isomorphism in $\operatorname{\mathrm{Sh}}(X;\Omega)$ if $u$ is embedded in an exact triangle $F\overset{u}{\to}G\to H\overset{+1}{\to}$ with $\operatorname{{\operatorname{SS}}}(H)\cap\Omega=\varnothing$. We also denote by $\operatorname{\mathrm{Sh}}_{(A)}(X)$ the full dg-subcategory of $\operatorname{\mathrm{Sh}}(X)$ formed by $F$ for which there exists a neighborhood $U$ of $A$ satisfying $\operatorname{{\operatorname{SS}}}(F)\cap U\subset A$.

Here, we recall the Tamarkin category $\mathcal{D}(X)$ and the interleaving distance.

We write $(x;\xi)$ for a local homogeneous coordinate system on $T^{*}X$ and $(t;\tau)$ for the homogeneous coordinate system on $T^{*}\mathbb{R}_{t}$. Define the maps

For $F,G\in\operatorname{\mathrm{Sh}}(X\times\mathbb{R}_{t})$, one sets

Note that the functor $\star$ is a left adjoint to $\mathop{{\mathcal{H}}om}\nolimits^{\star}$.

Tamarkin proved that the localized category $\operatorname{\mathrm{Sh}}(X\times\mathbb{R}_{t};\{\tau>0\})$ is equivalent to both the left orthogonal ${}^{\perp}\operatorname{\mathrm{Sh}}_{\{\tau\leq 0\}}(X\times\mathbb{R}_{t})$ and the right orthogonal $\operatorname{\mathrm{Sh}}_{\{\tau\leq 0\}}(X\times\mathbb{R}_{t})^{\perp}$:

We set $\Omega_{+}\coloneqq\{\tau>0\}\subset T^{*}(X\times\mathbb{R}_{t})$ and define a map $\rho\colon\Omega_{+}\to T^{*}X$ as $(x,t;\xi,\tau)\mapsto(x;\xi/\tau)$.

For $F\in\mathcal{D}(X)$, we take the canonical representative $P_{l}(F)\in{}^{\perp}\operatorname{\mathrm{Sh}}_{\{\tau\leq 0\}}(X\times\mathbb{R}_{t})$ unless otherwise specified. Note that if $F\in{}^{\perp}\operatorname{\mathrm{Sh}}_{\{\tau\leq 0\}}(X\times\mathbb{R}_{t})$, then $\mathop{{\mathcal{H}}om}\nolimits^{\star}(F,G)\in\operatorname{\mathrm{Sh}}_{\{\tau\leq 0\}}(X\times\mathbb{R}_{t})^{\perp}$. Thus $\mathop{{\mathcal{H}}om}\nolimits^{\star}$ induces an internal Hom functor $\mathop{{\mathcal{H}}om}\nolimits^{\star}\colon\mathcal{D}(X)^{\mathrm{op}}\times\mathcal{D}(X)\to\mathcal{D}(X)$. We also often regard $\mathop{{\mathcal{H}}om}\nolimits^{\star}(F,G)$ as in ${}^{\perp}\operatorname{\mathrm{Sh}}_{\{\tau\leq 0\}}(X\times\mathbb{R}_{t})$ by applying the projector $P_{l}$.

Now we introduce a pseudo-distance $d^{\prime}_{\mathcal{D}(X)}$ on the Tamarkin category $\mathcal{D}(X)$. Note that our distance $d^{\prime}_{\mathcal{D}(X)}$ here is different from the distances used in [AI20, AISQ, AI22Complete]. Here we use weakly $(a,b)$-isomorphic to obtain an isomorphism in $\mathcal{T}(X)$ (see Remark 2.4 below), while in [AI20, AISQ] the authors only use $(a,b)$-interleaving in the definition. Nevertheless, all the results in [AI20, AISQ] hold for this stronger notion [AI20, Remark 4.5]. For $c\in\mathbb{R}$, consider the translation map $T_{c}\colon X\times\mathbb{R}\to X\times\mathbb{R},\,T_{c}(x,t)=(x,t+c)$. By abuse of notation, we write $T_{c}\coloneqq{T_{c}}_{*}\colon\operatorname{\mathrm{Sh}}(X\times\mathbb{R}_{t})\to\operatorname{\mathrm{Sh}}(X\times\mathbb{R}_{t})$ and the induced functor ${T_{c}}_{*}\colon\mathcal{D}(X)\to\mathcal{D}(X)$. For $c\leq d$, there exists a natural transformation $\tau_{c,d}\colon T_{c}\to T_{d}$ between the functors on $\mathcal{D}(X)$.