1.1 A Hamiltonian $\coprod _n BO(n)$-action and Bott periodicity

In $T^*\mathbf {R}^N$, for any quadratic form $A$ on $\mathbf {R}^N$, we consider the quadratic Hamiltonian function $F_A(\mathbf {q},\mathbf {p})=A(\mathbf {p})$. Here we identify $(\mathbf {R}^N)^*$ with $\mathbf {R}^N$ using the standard inner product on $\mathbf {R}^N$. The time-1 flow $\varphi _A^1$ of the Hamiltonian vector field $X_{F_A}$ of $F_A$, defined as $\omega (X_{F_A},-)=dF_A$, sends $(\mathbf {q},\mathbf {p})$ to $(\mathbf {q}+\partial _{\mathbf {p}}A(\mathbf {p}), \mathbf {p})$. In the following, we assume that $A$ as a symmetric matrix is idempotent, i.e. there exists a subspace $E_A\subset \mathbf {R}^N$ such that $A(\mathbf {p})=|\mathrm {proj}_{E_A}\mathbf {p}|^2$. If we start with the linear Lagrangian $L_0$ that is the graph of the differential of $-\frac {1}{2}|\mathbf {q}|^2$, and do $\varphi _A^1$ to it, then the resulting Lagrangian $L_A$ is the graph of the differential of $-\frac {1}{2}|\mathbf {q}|^2+A(\mathbf {q})$. Moreover, the Maslov index of the loop starting from $L_0$, going towards $L_A$ via the Hamiltonian flow of $F_A$ and finally going back to $L_0$ through the path by decreasing the eigenvalue of $A$ from $1$ to $0$ is exactly the rank of $A$ (up to a negative sign). We can conjugate this loop by the path connecting the zero-section to $L_0$ through the graphs of differentials of $-\frac {1}{2}t|\mathbf {q}|^2, t\in [0,1]$ so then the base point is always at the zero-section.

If we take the stabilization $\varinjlim _NT^*\mathbf {R}^N$, then the above picture tells us a way to assign a based loop in the stable Lagrangian Grassmannian $U/O$ from an element in $\coprod _{n}Gr(n,\mathbf {R}^\infty )=\coprod _{n}BO(n)$, which is exactly the content of (a part of) Bott periodicity that the latter after taking group completion is the based loop space of the former. Now it is natural to assemble the Hamiltonian map for different choices of $A$ as a Hamiltonian $\coprod _{n}BO(n)$-action on $\varinjlim _NT^*\mathbf {R}^N$. Here we take the standard commutative topological monoid structure on $\coprod _{n}BO(n)$ as in [Reference HarrisHar80], where the addition operation is only defined for two perpendicular finite dimensional subspaces in $\varinjlim _N\mathbf {R}^N$.

1.2 Quantization of the Hamiltonian $\coprod _n BO(n)$-action and a natural “module”

We use microlocal sheaf theory to study the Hamiltonian $\coprod _n BO(n)$-action described above. The standard way to do this is to find a sheaf over prescribed coefficient with singular support in a conic Lagrangian lifting of the moment Lagrangian of the Hamiltonian action, called sheaf quantization (cf. [Reference Guillermou, Kashiwara and SchapiraGKS12, Reference TamarkinTam15]). Here we take the coefficient to be the universal one, the sphere spectrum $\mathbf {S}$, so sheaves are taking values in the stable $\infty$-category of spectra $\mathrm {Sp}$ (see [Reference Jin and TreumannJT24] for basic properties of microlocal sheaves of spectra). The commutative monoid structure on $\coprod _n BO(n)$ endows the associated sheaf category with a symmetric monoidal convolution structure (we will make this precise by using the category of correspondence developed in [Reference Gaitsgory and RozenblyumGR17]), and it is natural to ask the sought-for sheaf to be a commutative algebra object in it. Moreover, to establish a connection of the Hamiltonian action with Maslov index and Bott periodicity, etc., we need to quantize a ‘module’ of it generated by (the stabilization of) $L_0$ as well. We will make this more precise below.

For any smooth manifold $X$, let $T^{*,<0}(X\times \mathbf {R}_t)$ be the negative half of $T^*(X\times \mathbf {R}_t)$, where every covector has strictly negative component in ${\rm d}t$. For any exact Lagrangian submanifold $L$ in $T^*X$, a conic lifting $\mathbf {L}$ of $L$ in $T^{*,<0}(X\times \mathbf {R}_t)$ (with the Liouville 1-form $-\tau \, {\rm d}t+\alpha$) is any conic Lagrangian, i.e. invariant under the $\mathbf {R}_+$-scaling on the fibers, determined by the properties that:

• – the projection to $T^*X$ sends $\mathbf {L}\cap \{\tau =-1\}$ isomorphicallyFootnote ² to $L$.

By the conic property, the 1-form $-\tau \, {\rm d}t+\alpha$ vanishes on $\mathbf {L}$, which is equivalent to saying that $\mathbf {L}\cap \{\tau =-1\}$ is a Legendrian submanifold in the contact hypersurface $(\{\tau =-1\}\cong T^*X\times \mathbf {R}_t, {\rm d}t+\alpha )$ that projects to $L$ isomorphically. The latter is usually referred to as a Legendrian lifting of $L$. In other words, by writing $-\alpha |_{L}=df$, its Legendrian lifting (for the choice of $f$) is just the graph of $t=f(\mathbf {q}, \mathbf {p})$ with $(\mathbf {q}, \mathbf {p})\in L$, and $\mathbf {L}$ is its conification. For more details about Legendrian liftings and their conification, we refer the reader to [Reference Jin and TreumannJT24, § 3.1–3.5] (note that in [Reference Jin and TreumannJT24] there is some sign difference: we used $\alpha =\mathbf {p} d\mathbf {q}$ instead of $-\mathbf {p} d\mathbf {q}$). For any exact Lagrangian in general position, $\mathbf {L}$ is completely determined by the front projection of $\mathbf {L}$ in $X\times \mathbf {R}_t$, i.e. its image under the projection $T^{*,<0}(X\times \mathbf {R}_t)\rightarrow X\times \mathbf {R}_t$ to the base. Moreover, if $L$ is closed and the front projection of $\mathbf {L}$ is a smooth submanifold, then $\mathbf {L}$ is just half of the conormal bundle of the submanifold with negative $dt$-component, and we say it is the negative conormal bundle of the smooth submanifold.

A (stabilized) conic Lagrangian lifting of the graph of the Hamiltonian action has front projection in $\coprod _n BO(n)\times \varinjlim _M\mathbf {R}^M\times \varinjlim _M\mathbf {R}^M\times \mathbf {R}_t$ consisting of points

(1.2.1)\begin{equation} (A, \mathbf{q}, \mathbf{q}+\partial_{\mathbf{p}}A(\mathbf{p}), t=A(\mathbf{p})), \quad \mathbf{q},\mathbf{p}\in \varinjlim_M\mathbf{R}^M, \end{equation}

which is a smooth subvariety. Here and after, by some abuse of notation, we will use $A$ to denote its eigenspace $E_A$ of $1$ as well, so then $A$ also represents an element in $\coprod _n BO(n)$.

Note that we can equally represent a point in (1.2.1) by

\[ (A,\mathbf{q}, \mathbf{q}+2\mathbf{p}, t=|\mathbf{p}|^2), \quad \mathbf{q}\in \varinjlim_M\mathbf{R}^M,\ \mathbf{p}\in A. \]

Then modulo the roles of $\mathbf {q}$ and $t$, which contribute trivially to the topology, the front projection is just the tautological vector bundle on $\coprod _n BO(n)$.

Let

(1.2.2) \begin{gather} G_N=\coprod_n Gr(n,\mathbf{R}^N),\quad VG_N=\text{the tautological vector bundle on } G_N,\nonumber\\ G=\varinjlim_{N}G_N=\coprod_n BO(n),\quad VG=\varinjlim_{N}VG_N. \end{gather}

A quantization of the Hamiltonian $G$-action is then given by a commutative algebra object in

(1.2.3)\begin{equation} \mathrm{Loc}(VG;\mathrm{Sp}):=\varprojlim_{N}\mathrm{Loc}(VG_{N};\mathrm{Sp}) \end{equation}

equipped with the convolution symmetric monoidal structure, whose restriction to the unit ${(A=0, \mathbf {p}=0)}$ of $VG$ is the constant sheaf $\mathbf {S}$ on a point. Here and after $\mathrm {Loc}(X;\mathrm {Sp})$ consists of locally constant cosheaves, and the limit of above is taken for the $!$-pullback along the natural embeddings $VG_{N}\hookrightarrow VG_{N'}$ for $N\leq N'$ (cf. Remark 1.7 below). A canonical candidate of such a quantization is the dualizing sheaf $\varpi _{VG}$.

Now what do we mean by to quantize a ‘module’ of the Hamiltonian action generated by the stabilization of $L_0$? First, we can quantize $L_0\subset T^*\mathbf {R}^M$ by the $*$-extension of a local system on

\[ Q^0_M:=\{s<-\tfrac{1}{2}|\mathbf{q}|^2\}\subset \mathbf{R}^M_\mathbf{q}\times\mathbf{R}_s \]

to $\mathbf {R}^M_\mathbf {q}\times \mathbf {R}_s$, whose boundary has negative conormal bundle equal to a conic lifting of $L_0$. Second, we can assemble the images of (the stabilization of) $L_0$ under the Hamiltonian $G$-action, i.e. $L_A=\varphi _A^1(L_0)$, into a single Lagrangian $\widehat {L}$ in the stable sense, and we can present a conic lifting of it as the stabilization of the negative conormal bundle of the boundary of the following domain

(1.2.4)\begin{align} \widehat{Q}_{N,M}:=\{s<-\tfrac{1}{2}|\mathbf{q}|^2+A(\mathbf{q}),\ A\in G_N\}\subset G_N\times \mathbf{R}^M_{\mathbf{q}}\times \mathbf{R}_s \end{align}

for $M\geq N$.

Now the point is that given any quantization of $L_0$, represented by an object in $\mathrm {Loc}(Q^0;\mathrm {Sp}):= \varprojlim _M \mathrm {Loc}(Q_M^0;\mathrm {Sp})$, and the canonical quantization of the Hamiltonian $G$-action $\varpi _{VG}$, we should automatically get a quantization of $\widehat {L}$, which is represented by an object in

(1.2.5)\begin{align} \mathrm{Loc}(\widehat{Q};\mathrm{Sp}):=\varprojlim_{N,M} \mathrm{Loc}(\widehat{Q}_{N,M};\mathrm{Sp}). \end{align}

This can be made precise using correspondences for sheaves. If we view $\widehat {L}$ as a ‘module’ of the Hamiltonian action generated by the stabilization of $L_0$, then the resulting quantizations of $\widehat {L}$ as above are those that we are interested in. A main question that we will answer is a characterization of these quantizations of $\widehat {L}$, for example, what are the monodromies of the corresponding local systems on $\widehat {Q}$.

1.3 Stratified Morse theory

Let $X$ be a smooth manifold and $\mathfrak {S}=\{S_\alpha \}$ be a Whitney stratification of $X$. Let

\[ \Lambda_\mathfrak{S}=\bigcup_{S_\alpha\in\mathfrak{S}}T^*_{S_\alpha}X \]

be the union of the conormal bundles of the strata, which is a closed conic Lagrangian in $T^*X$. Now given any $\mathfrak {S}$-constructible sheaf $\mathcal {F}$ and a covector $(x,p)$ in the smooth locus of $\Lambda _\mathfrak {S}$, denoted by $\Lambda _\mathfrak {S}^{\rm sm}$, one can define the local Morse group or microlocal stalk of $\mathcal {F}$ at $(x,p)$. The definition depends on a choice of a local function $f$ near $x$ whose differential at $x$ is $p$, and which is a Morse function restricted to the stratum containing $x$. Such a function is called a local stratified Morse function. Roughly speaking, the microlocal stalk measures how the local sections of $\mathcal {F}$ change when we move across $x$ in the way directed by $f$. One uses the microlocal stalks to define the singular support of $\mathcal {F}$, denoted by $\mathit {SS}(\mathcal {F})$, which is a closed conic Lagrangian contained in $\Lambda _\mathfrak {S}$. The microlocal stalks and singular support play an essential role in microlocal sheaf theory and the theory of perverse sheaves. For example, a central question in microlocal sheaf theory is to calculate the microlocal sheaf category associated to a conic Lagrangian (or equivalently a Legendrian), which is roughly speaking a localization of the category of sheaves whose singular support are contained in the conic Lagrangian.

The main theorem in stratified Morse theory [Reference Goresky and MacPhersonGM88] states (in part) that the microlocal stalk defined above is independent of the choice of the local stratified Morse functions $f$, in the sense that the microlocal stalks at $(x,p)$ for two different $f_1, f_2$ are isomorphic up to a shift of degree by the difference of their Morse indices along the stratum containing $x$.

We will enhance the main theorem in stratified Morse theory by exhibiting the (stabilized) monodromies of the microlocal stalks along the space of all choices of local stratified Morse functions $f$ (cf. Theorem 1.4 below). This is an application of the quantization of the Hamiltonian $G$-action and its ‘module’ described above.

1.4 The $J$-homomorphism

The $J$-homomorphism is an $E_\infty$-map

\begin{align*} J: \coprod_{n}BO(n)\rightarrow \mathrm{Pic}(\mathbf{S})\simeq \mathbf{Z}\times BGL_1(\mathbf{S}), \end{align*}

where $\mathrm {Pic}(\mathbf {S})$ is the classifying space of stable sphere bundles. If one takes the group completion of $\coprod _{n}BO(n)$, $\mathbf {Z}\times BO$, then one can uniquely extend $J$ (up to a contractible space of choices) to an $\Omega ^\infty$-map from $\mathbf {Z}\times BO$ to $\mathrm {Pic}(\mathbf {S})$. By the Thom construction, every vector bundle gives rise to a stable sphere bundle, and $J$ is the map between the classifying space of stable vector bundles to the classifying space of stable sphere bundles.

There are several equivalent models of $J$ to express its $E_\infty$-structure. A model of $J$ as an infinite loop space map from $G$ to $\mathrm {Pic}(\mathbf {S})$ is given as follows (cf. [Reference LurieLur15] for the complex $J$-homomorphism). Let $\mathrm {Vect}_{\mathbf {R}}^{\simeq }$ be the topologically enriched category of real finite-dimensional vector spaces, with morphisms being linear isomorphisms. The direct sum of vector spaces makes $N(\mathrm {Vect}_{\mathbf {R}}^{\simeq })$ into a symmetric monoidal $\infty$-groupoid. For any $V\in \mathrm {Vect}_{\mathbf {R}}^{\simeq }$, let $V^c$ be the one-point compactification of $V$ (with the marked point at $\infty$). The functor $V\mapsto \Sigma ^\infty V^c$ of forming $\mathbf {S}$-lines determines the symmetric monoidal functor

(1.4.1)\begin{equation} J: N(\mathrm{Vect}_{\mathbf{R}}^{\simeq})^\otimes\rightarrow \mathrm{Pic}(\mathbf{S})^\otimes. \end{equation}

Here we give an equivalent model of $J$ using correspondences for sheaves. Let

\[ \pi_{VG,N}: VG_N\rightarrow G_N \]

be the vector bundle projection, where $G_N, VG_N$ are defined in (1.2.2). Since $G$ and $VG$ are commutative topological monoids, their local system categories $\mathrm {Loc}(G;\mathrm {Sp})$ and $\mathrm {Loc}(VG;\mathrm {Sp})$ are equipped with the convolution symmetric monoidal structures from the symmetric monoidal functor

\begin{align*} \mathrm{Fun}(-, \mathrm{Sp}): \mathcal{S}\mathrm{pc}^{\rm op}&\to \mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}}\\ X&\mapsto \mathrm{Fun}(X,\mathrm{Sp})\simeq \mathrm{Loc}(X,\mathrm{Sp}). \end{align*}

This will agree with the symmetric monoidal structure defined in § 1.8 using correspondences in the 1-category of locally compact Hausdorff spaces and then taking inverse limits (1.2.3). Then $J$ can be viewed as a commutative algebra object in $\mathrm {Loc}(G;\mathrm {Sp})$ whose costalks are isomorphic to suspensions of the sphere spectrum. We have the following result proved in § A.4.

Proposition 1.2 (See Proposition 4.13)

The functor

\[ (\pi_{VG})_*=\varprojlim_{N}(\pi_{VG,N})_*: \mathrm{Loc}(VG;\mathrm{Sp})\rightarrow \mathrm{Loc}(G;\mathrm{Sp}) \]

is symmetric monoidal, and the local system $(\pi _{VG})_*\varpi _{VG}$ is the commutative algebra object in $\mathrm {Loc}(G;\mathrm {Sp})$ classified by $J$.

Here by the dualizing sheaf on a locally compact Hausdorff space $X$, we mean the constant cosheaf with cofiber $\mathbf {S}$, through Lurie's Verdier duality for sheaves of spectra (cf. [Reference LurieLur17, Theorem 5.5.5.1])

\[ \mathbb{D}: \mathrm{Shv}(X;\mathrm{Sp})\overset{\sim}{\to} \mathrm{Shv}(X;\mathrm{Sp}^{\rm op})^{\rm op}. \]

Then $\varpi _{VG}$ is the object corresponding to $\varpi _{VG_N}, N\geq 0$, under the inverse limit. We emphasize that the functor $(\pi _{VG})_*$ is not well defined if we regard $\pi _{VG}$ as a morphism in the $\infty$-category of spaces $\mathcal {S}\mathrm {pc}$, for it is not invariant under homotopies (note that $(\pi _{VG})_!$ is well defined but gives trivial information). This is a place where we need to work in the ordinary 1-category of locally compact Hausdorff spaces, and the validity of $(\pi _{VG})_*$ (which follows from base change) and its symmetric monoidal structure leads us to employ the category of correspondences.

1.5 Statement of the main results

Now we are ready to state our main results. The space $\widehat {Q}=\varinjlim _{N,M}\widehat {Q}_{N,M}$ (see (1.2.4) for the definition of $\widehat {Q}_{N,M}$) is naturally homotopy equivalent to $G$, so we can view $\widehat {Q}$ as a $G$-torsor in $\mathcal {S}\mathrm {pc}$ (i.e. a free $G$-module generated by a point), and $\mathrm {Loc}(\widehat {Q};\mathrm {Sp})\simeq \mathrm {Loc}(G;\mathrm {Sp})$ is naturally a module of $\mathrm {Loc}(G;\mathrm {Sp})$. Let $\mathrm {Loc}(\widehat {Q};\mathrm {Sp})^J$ be the stable $\infty$-category of $J$-equivariant local systems on $\widehat {Q}$, which is a full subcategory of $J$-modules in $\mathrm {Loc}(\widehat {Q};\mathrm {Sp})$. Here we use $J$ to represent the local system on $G\simeq \widehat {Q}$ that it classifies. Since $\widehat {Q}$ is a $G$-torsor, $\mathrm {Loc}(\widehat {Q};\mathrm {Sp})^J$ consists of objects $J\otimes _{\mathbf {S}}\mathcal {R}$, $\mathcal {R}\in \mathrm {Sp}$, and the mapping spectrum from $J\otimes _{\mathbf {S}}\mathcal {R}_1$ to $J\otimes _{\mathbf {S}}\mathcal {R}_2$ is isomorphic to the mapping spectrum from $\mathcal {R}_1$ to $\mathcal {R}_2$ (see Lemma 4.15). There is a general definition of $\mathrm {Loc}(Y;\mathrm {Sp})^\chi$ for any commutative topological monoid $H$ acting on a space $Y$ and $\chi : H\rightarrow \mathrm {Pic}(\mathbf {S})$ an $E_\infty$-map called a character (see Definition 4.14).

Consider the map, which is a $VG_N$-module map (cf. § 4.1.1)

\begin{align*} \psi_{N,M} & : VG_N\times Q^0_M\rightarrow \widehat{Q}_{N,M},\\ &\,(A,\mathbf{p}; \mathbf{q},s)\mapsto (A,\mathbf{q}+2\mathbf{p},s+A(\mathbf{p})),\ \mathbf{p}\in A, \end{align*}

whose stabilization represents the Hamiltonian $G$-action that sends $L_0$ to $L_A, A\in G$.

Theorem 1.3 The family of correspondences

induces a canonical equivalence

\[ \psi_*\pi^!_{Q^0}:=\varprojlim_{N,M}(\psi_{N,M})_*\pi^!_{Q_M^0}:\mathrm{Loc}(Q^0;\mathrm{Sp})\overset{\sim}{\longrightarrow} \mathrm{Loc}(\widehat{Q};\mathrm{Sp})^{J} \]

Intuitively, the theorem states that the $\varpi _{VG}$-modules in $\mathrm {Loc}(\widehat {Q};\mathrm {Sp})$ (as a module of $\mathrm {Loc}(VG;\mathrm {Sp})$) that are freely generated by local systems on $Q^0$ are characterized precisely as the $J$-equivariant local systems on $\widehat {Q}$ (cf. Corollary 4.16 for more details).

Now we state an application of the above theorem to stratified Morse theory. Although stratified Morse theory is aimed at constructible sheaves over ordinary rings, there is a direct generalization of it over ring spectra (cf. [Reference Jin and TreumannJT24]). Let $\mathrm {Shv}_\mathfrak {S}(X;\mathrm {Sp})$ be the full subcategory of $\mathrm {Shv}(X;\mathrm {Sp})$ consisting of sheaves whose restriction to each stratum $S_\alpha$ is locally constant. The notion of microlocal stalk can be directly generalized to take values in spectra. Following the notation in § 1.3, we can stabilize $X$ and $\mathfrak {S}$ as $X\times \mathbf {R}_{y_1}\times \mathbf {R}_{y_2}\times \cdots$, $\{S_\alpha \times \mathbf {R}_{y_1}\times \cdots \}$, and the space of local stratified Morse functions associated to $(x,p)$, where $x\in S_\beta$ for some $\beta$, can be stabilized by adding $-\frac {1}{2}(y_1^2+y_2^2+\cdots )$ in the newly added variables $y_1,y_2,\ldots$. Then the space of stabilized local stratified Morse functions for $(x,p)$ is canonically homotopy equivalent to $G$, through the map taking $f$ to the sum of the positive eigenspaces of the Hessian of $f|_{S_\beta \times \mathbf {R}\times \cdots }$ at $x$ in $T_xS_\beta \times \mathbf {R}\times \cdots$. We can view this space as a $G$-torsor in $\mathcal {S}\mathrm {pc}$.

The process of taking microlocal stalks has a natural translation to correspondences via the notion of contact transformations defined in [Reference Kashiwara and SchapiraKS94, A.2]. A contact transformation is just a local conic symplectomorphism that transforms a local piece of conic Lagrangian to another piece of conic Lagrangian, and the local stratified Morse functions correspond to a generic class among them (modulo some obvious equivalences that act trivially on the microlocal stalks), which we call Morse transformations.Footnote ³ These relations are established in §§ 5.1, 5.2, and 5.3. This enables us to reveal the interesting structures underlying the ‘local system of microlocal stalks’ on the space of stabilized local stratified Morse functions, by applying the category of correspondences and using Theorem 1.3. In particular, Theorem 1.4 directly follows from Theorem 5.10. Moreover, there is a generalization of Theorem 1.4 from $(x,p)\in \Lambda _{\mathfrak {S}}^{\rm sm}$ to the case that $(x,p)\in \Lambda ^{\rm sm}$ for any conic Lagrangian $\Lambda$ (see Remark 5.7).

1.6 Expected strategy of the proof of Claim 1.1

The idea of the proof is easier to express if $G=\coprod _nBO(n)$ were a group. First let us assume that this were the case. Let $L$ be an (immersed) exact Lagrangian in $T^*\mathbf {R}^N$ and let $\mathbf {L}$ be a conic lifting of $L$ in $T^{*,<0}(\mathbf {R}^N\times \mathbf {R}_t)$. The argument holds for $L$ being non-exact and can be generalized to the case when $\mathbf {R}^N$ is replaced by a smooth manifold via a standard treatment as in [Reference Abouzaid and KraghAK16]. Now we cover $L$ by small open sets $\{\Omega _i\}_{i\in I}$ whose finite intersections are all contractible if not empty (this is called a good cover), which gives a sufficiently fine covering of the front projection of $\mathbf {L}$. In [Reference Jin and TreumannJT24], we show that for each finite intersection $\bigcap _{s=1}^k\Omega _{i_s}$, the associated microlocal sheaf category with coefficient $\mathbf {S}$ is (non-canonically) equivalent to the $\infty$-category of spectra, which defines a local system of stable $\infty$-categories on $L$ with fiber equivalent to $\mathrm {Sp}$ denoted by $\mathrm {Brane}_L$. Such an equivalence of categories follows from the correspondence given by a choice of Morse transformation, and Theorem 1.4 implies that if we consider the Morse transformations altogether (again modulo some obvious equivalences), then the microlocal sheaf category associated to $\bigcap _{s=1}^k\Omega _i$ is canonically equivalent to the category of $J$-equivariant local systems of spectra on the space of Morse transformations associated to $\bigcap _{s=1}^k\Omega _i$ which is a $G$-torsor.

Let $\check {C}_\bullet L$ be the $\check {\text {C}}$ech nerve of the covering $\{\Omega _i\}_{i\in I}$, and let $G\text {-}\mathsf {torsors}$ denote the $E_\infty$-groupoid of $G$-torsors in $\mathcal {S}\mathrm {pc}$, which is equivalent to $BG$ (assuming $G$ were a group). The above implies that the classifying map for $\mathrm {Brane}_L$ factors as

\[ \check{C}_\bullet L\overset{\check{\gamma}}{\longrightarrow} G\text{-}\mathsf{torsors}\overset{\mathrm{Loc}(-;\mathrm{Sp})^J}{\longrightarrow} B\mathrm{Pic}(\mathbf{S}) \]

where the first map $\check {\gamma }$ is taking the space of Morse transformations and the latter is taking a $G$-torsor $M$ to $\mathrm {Loc}(M;\mathrm {Sp})^J$. Here we use a compatibility result of Morse transformations under the restriction from a small open set to a smaller one in the $\check {\text {C}}$ech cover. One can show that $\check {\gamma }$ is homotopic to the stable Gauss map after passing to the geometric realization $|\check {C}_\bullet L|\simeq L$, and $\mathrm {Loc}(-;\mathrm {Sp})^J$ exactly models the delooping of the $J$-homomorphism (up to the canonical involution on $BG$). This proves Claim 1.1 under the assumption that $G$ were a group.

For the actual situation in which $G=\coprod _nBO(n)$ is not a group, some more work needs to be done, which involves a more detailed study of the space $U/O$ and the compatibility of Morse transformations under restrictions (see § 5.6). For example, one technical step in [Reference JinJin20] is to replace the universal principal $\mathbf {Z}\times BO$-bundle $E(\mathbf {Z}\times BO)\to U/O$ by an $\infty$-category of quadruples $(\mathcal {U}, Q_\flat,\,{}^{\backprime }{G}, M)$, denoted by $\mathsf {QHam}(U/O)$, over $\mathrm {Open}(U/O)^{\rm op}$, where $\mathcal {U}$ is an open subset of $U/O$, $Q_\flat$ is a (stabilized) quadratic form that plays a similar role as $A_S$ in (5.4.2) with $T_{(0,0)}L$ replaced by $\ell \in \mathcal {U}$, $^{\backprime }{G}$ is a commutative algebra object in $\mathrm {Corr}(\mathrm {S}_{\mathrm {LCH}})$ (see § 1.7 below) that plays a similar role as $G$ (or $G^{A^\perp }$ (5.6.1) depending on the choice of $A_S$), and $M$ is a free module of $^{\backprime }{G}$ generated by $Q_\flat$ (parametrizing a space of Morse transformations that work for all $\ell \in \mathcal {U}$). A morphism between two objects $(\mathcal {U}_i, Q^{(i)}_\flat$, $^{\backprime }{G}_{(i)}, M_{(i)})$ is roughly given by $\mathcal {U}_2\hookrightarrow \mathcal {U}_1$, an inclusion $^{\backprime }{G}_{(1)}\hookrightarrow \,{}^{\backprime }{G}_{(2)}$ that is a commutative algebra homomorphism, and a $^{\backprime }{G}_{(1)}$-module map $M_{(1)}\hookrightarrow M_{(2)}$ that sends $Q_\flat ^{(1)}$ to some appropriate element in $M_{(2)}$. The key point of introducing $\mathsf {QHam}(U/O)$ is that the quantization results contained in this paper will be assembled into a natural functor $F: \mathsf {QHam}(U/O)\to \mathrm {Fun}(\Delta ^1, \mathcal {P}\mathrm {r}_{\mathrm {st}}^{\mathrm {L}})$ that sends $(\mathcal {U}, Q_\flat, \,{}^{\backprime }{G}, M)$ to an equivalence of categories analogous to that in Theorem 1.3 (more precisely, see [Reference JinJin20, Proposition 4.13]), and it further induces a functor $\mathrm {Open}(U/O)^{\rm op}\to \mathrm {Fun}(\Delta ^1, \mathcal {P}\mathrm {r}_{\mathrm {st}}^{\mathrm {L}})$ by right Kan extension. To give an appropriate definition of $\mathsf {QHam}(U/O)$ that works for our purposes, we must allow $Q_\flat$ and $^{\backprime }{G}$ to be more flexible than $A_S$ and $G^{A^\perp }$ that are considered in § 5.6. For example, since our $\ell$ is moving in $\mathcal {U}$, we cannot expect (5.4.2) to hold for all $\ell \in \mathcal {U}$. There are other more subtle reasons that impose both flexibility and technical conditions on $Q_\flat$, $^{\backprime }{G}$ and the morphisms between the objects in $\mathsf {QHam}(U/O)$ (cf. [Reference JinJin20, § 4.1]).

In the following, we give a brief overview of the category of correspondences, the results we have obtained and how we apply these results to the quantization process.

1.7 The category of correspondences and the functor of taking sheaves of spectra

For any ordinary 1-category $\mathcal {C}$ that admits fiber products, one can define an ordinary 2-category $\mathbf {Corr}(\mathcal {C})$ (cf. [Reference Gaitsgory and RozenblyumGR17]) with the same objects as $\mathcal {C}$ and whose set of morphisms from an object $X$ to $Y$ consists of correspondences

(1.7.1)

A 2-morphism between 1-morphisms is presented by the left one of the following commutative diagrams (

), and a composition of two 1-morphisms is presented by the outer correspondence on the right.

(1.7.2)

There is a higher categorical analogue of this where $\mathcal {C}$ is an $\infty$-category and $\mathbf {Corr}(\mathcal {C})$ is an $(\infty,2)$-category. One can also forget the non-invertible 2-morphisms, and get an $\infty$-category $\mathrm {Corr}(\mathcal {C})$. If $\mathcal {C}$ is (symmetric) monoidal, then $\mathbf {Corr}(\mathcal {C})$ is (symmetric) monoidal, and so is $\mathrm {Corr}(\mathcal {C})$.

In [Reference Gaitsgory and RozenblyumGR17], the authors introduce the $(\infty,2)$-category of correspondences and use it to give a systematic treatment of the six-functor formalism for dg-categories of quasi/ind-coherent sheaves. A similar treatment without using $(\infty,2)$-categories has been taken in [Reference Liu and ZhengLZ12]. The approach of [Reference Gaitsgory and RozenblyumGR17] can be adapted to the case of sheaves of spectra, given the foundations on $\infty$-topoi [Reference LurieLur09] and stable $\infty$-categories [Reference LurieLur17]. Let $\mathrm {S}_{\mathrm {LCH}}$ be the ordinary 1-category of locally compact Hausdorff spaces, equipped with the Cartesian symmetric monoidal structure, i.e. the tensor product is just the Cartesian product. The starting point is the symmetric monoidal functor (cf. [Reference LurieLur09])

\begin{align*} \mathrm{Shv}\mathcal{S}:\ &(\mathrm{S}_{\mathrm{LCH}})^{\rm op}\rightarrow \mathcal{P}\mathrm{r}^{\mathrm{L}}\\ & X\mapsto \mathrm{Shv}(X;\mathcal{S})\\ &(f: X\rightarrow Y)\mapsto (f^*: \mathrm{Shv}(Y;\mathcal{S})\rightarrow\mathrm{Shv}(X;\mathcal{S})), \end{align*}

where $\mathrm {Shv}(X;\mathcal {S})$ denotes the (presentable) $\infty$-category of sheaves of spaces on $X$ and $\mathcal {P}\mathrm {r}^{\mathrm {L}}$ denotes the $\infty$-category of presentable $\infty$-categories with continuous functors. Taking stabilizations of $\mathrm {Shv}(X;\mathcal {S})$ to $\mathrm {Shv}(X;\mathrm {Sp})$ and running the approach in [Reference Gaitsgory and RozenblyumGR17], one gets all six functors, and one has the following statement that is crucial for our purposes mentioned above.

Theorem 1.5 There is a canonical symmetric monoidal functor

\[ \mathrm{ShvSp}: \mathrm{Corr}(\mathrm{S}_{\mathrm{LCH}})\rightarrow \mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}} \]

sending $X$ to $\mathrm {Shv}(X;\mathrm {Sp})$ and any correspondence (1.7.1) to the functor $g_!f^*: \mathrm {Shv}(X;\mathrm {Sp})\rightarrow \mathrm {Shv}(Y;\mathrm {Sp})$.

Here $\mathcal {P}\mathrm {r}_{\mathrm {st}}^{\mathrm {L}}$ is the $\infty$-category of presentable stable $\infty$-categories. The theorem will be proved in § 3. Now to define a symmetric monoidal structure on $\mathrm {Shv}(X;\mathrm {Sp})$ for some $X\in \mathrm {S}_{\mathrm {LCH}}$, it suffices to endow $X$ with the structure of a commutative algebra object in $\mathrm {Corr}(\mathrm {S}_{\mathrm {LCH}})$. This is exactly the motivation for us to develop general and concrete constructions of (commutative) algebra objects in § 2, together with their modules, (right-lax) homomorphisms between algebras, etc. in $\mathbf {Corr}(\mathcal {C})$, for a Cartesian symmetric monoidal $\infty$-category $\mathcal {C}$.

Let $\mathcal {P}\mathrm {r}_{\mathrm {st}}^{\mathrm {R}}\simeq (\mathcal {P}\mathrm {r}_{\mathrm {st}}^{\mathrm {L}})^{\rm op}$ be the $\infty$-category of presentable stable $\infty$-categories with limit preserving functors. There is a canonical identification $\mathrm {Corr}(\mathrm {S}_{\mathrm {LCH}})\simeq \mathrm {Corr}(\mathrm {S}_{\mathrm {LCH}})^{\rm op}$ by flipping the correspondences (cf. [Reference Gaitsgory and RozenblyumGR17, Chapter 9, 2.2]). Hence, by taking opposite categories on the source and target of $\mathrm {ShvSp}$, we get the following.

Corollary 1.6 There is a canonical symmetric monoidal functor

\[ \mathrm{ShvSp}^!_*: \mathrm{Corr}(\mathrm{S}_{\mathrm{LCH}})\rightarrow \mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{R}} \]

sending $X$ to $\mathrm {Shv}(X;\mathrm {Sp})$ and any correspondence (1.7.1) to the functor $g_*f^!: \mathrm {Shv}(X;\mathrm {Sp})\rightarrow \mathrm {Shv}(Y;\mathrm {Sp})$.

Now let

(1.7.3)\begin{align} (\mathrm{Loc})^!_*: \mathrm{Corr}(\mathrm{S}_{\mathrm{LCH}})_{\mathsf{fib}, \mathsf{all}}&\rightarrow \mathcal{P}\mathrm{r}_{\mathrm{st}}^{\mathrm{L}},\\ X&\mapsto \mathrm{Loc}(X;\mathrm{Sp}) \nonumber \end{align}

be the functor induced from $\mathrm {ShvSp}^!_*$, which restricts the vertical arrows in correspondences to locally trivial fibrations and takes each $X$ to the full subcategory of locally constant cosheaves. Note that here we can write the target as $\mathcal {P}\mathrm {r}_{\mathrm {st}}^{\mathrm {L}}$ instead of $\mathcal {P}\mathrm {r}_{\mathrm {st}}^{\mathrm {R}}$, since all the functors involved are continuous. In our applications, we will mainly use the functor $\mathrm {Loc}^!_*$, which is symmetric monoidal. Then (as mentioned before), the limit (1.2.3) and (1.2.5) are taken with respect to $!$-pullbacks.

1.8 Applications of the category of correspondences to quantizations

For simplicity, we only explain the application of correspondences to define the symmetric monoidal convolution structure on $\mathrm {Loc}(VG;\mathrm {Sp})$ and the module structure on $\mathrm {Loc}(\widehat {Q};\mathrm {Sp})$. Other applications follow from a similar line of ideas. In order to apply the functor $\mathrm {Loc}_*^!$ whose source is $\mathrm {Corr}(\mathrm {S}_{\mathrm {LCH}})_{\mathsf {fib},\mathsf {all}}$, we need to first work with $VG_N$ and $\widehat {Q}_{N,M}$ and then show the compatibility of the structures with taking limit over $N$ and $(N,M)$, respectively. In fact, using our approach we can show that the symmetric monoidal structure and module structure already exist at finite levels, i.e. on $\mathrm {Loc}(VG_N;\mathrm {Sp})$ and $\mathrm {Loc}(\widehat {Q}_{N,M};\mathrm {Sp})$, respectively.

The convolution structures at finite levels come from the following correspondences

(1.8.1)

where: (i) $\iota _{N}$ is the embedding of the subvariety of pairs $(A_1,\mathbf {p}_1;A_2,\mathbf {p}_2), \mathbf {p}_i\in A_i$ where $A_1\perp A_2$, and $a_{N}$ is the natural addition $(A_1\oplus A_2, \mathbf {p}_1+\mathbf {p}_2)$; (ii) $\jmath _{N,M}$ is defined similarly as $\iota _N$ and

\[ a_{N,M}: (A_1,\mathbf{p}_1; A_0,\mathbf{q}, s)\mapsto (A_1\oplus A_0, \mathbf{q}+2\mathbf{p}_1,s+A_1(\mathbf{p}_1)), \mathbf{p}_1\in A_1, A_1\perp A_0. \]

The structure functors are given by

\begin{gather*} (a_{N})_*\iota_{N}^!\circ\boxtimes:\mathrm{Loc}(VG_N;\mathrm{Sp})\otimes \mathrm{Loc}(VG_N;\mathrm{Sp})\rightarrow \mathrm{Loc}(VG_N;\mathrm{Sp}), \\ (a_{N,M})_*\jmath_{N,M}^!\circ\boxtimes: \mathrm{Loc}(VG_N;\mathrm{Sp})\otimes \mathrm{Loc}(\widehat{Q}_{N,M})\rightarrow \mathrm{Loc}(\widehat{Q}_{N,M}). \end{gather*}

However, to give a precise definition of the symmetric monoidal structure on $\mathrm {Loc}(VG;\mathrm {Sp})$ and the module structure on $\mathrm {Loc}(\widehat {Q};\mathrm {Sp})$, it is not enough to just list the functors above: there are coherent homotopies that encode the commutativity and associativity properties and their compatibility with taking limit, which is an infinite amount of data.

The category of correspondences is an ideal tool to resolve the above issues. The upshot is that all the homotopy coherence data can be packaged inside the category of correspondences where things can be checked on the nose. For example, to present the desired symmetric monoidal structure on $\mathrm {Loc}(VG_N;\mathrm {Sp})$, we can construct a commutative algebra object in $\mathrm {Corr}(\mathrm {S}_{\mathrm {LCH}})_{\mathsf {fib},\mathsf {all}}$ whose underlying object is $VG_N$ and whose multiplication rule is given by (1.8.1). To show that the inclusion $\iota _{N,N'}: VG_N\hookrightarrow VG_{N'}$, $N'\geq N$ induces a symmetric monoidal functor $\iota _{N,N'}^!: \mathrm {Loc}(VG_{N'};\mathrm {Sp})\rightarrow \mathrm {Loc}(VG_N;\mathrm {Sp})$, we just need to show that $\iota _{N,N'}$ can be upgraded to an algebra homomorphism from $VG_{N'}$ to $VG_N$ as commutative algebra objects. In view of the following theorem (stated informally), these can be encoded by discrete data.

Let $\mathrm {Fin}_*$ denote the ordinary 1-category of pointed finite sets, where any morphism between two pointed sets preserves the marked point.

Indeed, we construct $\mathrm {Fin}_*$-objects $VG_N^\bullet$ and morphisms $VG_N^\bullet \hookrightarrow VG_{N'}^\bullet$ and apply the above theorem to show the symmetric monoidal structure on $\mathrm {Loc}(VG;\mathrm {Sp})$. The module structure on $\mathrm {Loc}(\widehat {Q})$ can be obtained appealing to a similar theorem as above for modules. We make a couple of remarks. First, one can also construct a simplicial object $VG_N^\bullet$ that represents an associative algebra object in $\mathrm {Corr}(\mathrm {S}_{\mathrm {LCH}})_{\mathsf {fib},\mathsf {all}}$, but it is not a Segal object (it is not even a Segal object in $\mathcal {S}\mathrm {pc}$), so the above theorem strictly generalizes the construction from Segal objects in [Reference Gaitsgory and RozenblyumGR17]. Second, the vertical map $a_{N,M}$ of the right correspondence in (1.8.1) is not proper, so one cannot regard the correspondence as in $\mathcal {S}\mathrm {pc}$.

1.9 Related results and future work

Claim 1 in [Reference Jin and TreumannJT24] was motivated by the Nadler–Zaslow theorem (see [Reference Nadler and ZaslowNZ09, Reference NadlerNad09]) and the notion of brane structures on an exact Lagrangian submanifold $L\subset T^*X$ in Floer theory (cf. [Reference SeidelSei08]), whose obstruction classes are given by the Maslov class in $H^1(L,\mathbf {Z})$ and the second (relative) Stiefel–Whitney class in $H^2(L;\mathbf {Z}/2\mathbf {Z})$. The obstruction classes are exactly the obstruction to the triviality of the composite map

\[ L\overset{\gamma}{\rightarrow}U/O\overset{BJ}{\rightarrow} B\mathrm{Pic}(\mathbf{S})\overset{\varphi_{\mathbf{Z}}}{\rightarrow} B\mathrm{Pic}(\mathbf{Z})\simeq S^1\times B^2(\mathbf{Z}/2\mathbf{Z}), \]

where $\varphi _{\mathbf {Z}}$ is induced from the canonical commutative ring spectrum homomorphism $\mathbf {S}\rightarrow \mathbf {Z}$.

For $L$ a compact exact Lagrangian submanifold in the cotangent bundle of a compact manifold, it is a prediction from Arnold's nearby Lagrangian conjecture that the classifying map $L\rightarrow B\mathrm {Pic}(\mathbf {S})$ is trivial. This is confirmed in [Reference Abouzaid and KraghAK16] whose approach is based on Floer (homotopy) theory (cf. [Reference Cohen, Jones and SegalCJS95]). Using sheaf quantization over $\mathbf {Z}$, it is proved in [Reference GuillermouGui23] that this is the case when $\mathbf {S}$ is replaced by $\mathbf {Z}$. The sheaf of brane structures in our setting is called the Kashiwara–Schapira stack there. In the subsequent work [Reference JinJin20], we prove the triviality of the classifying map using microlocal sheaf theory over $\mathbf {S}$.

The $E_\infty$-structure of $BJ$ in Claim 1.1 is used when the Gauss map $\gamma$ has an $E_\infty$-structure or more generally $E_n$-structure. An example that is crucial to the sheaf-theoretic approach to symplectic topology is the $E_\infty$-map $U\rightarrow U/O$ that comes from taking stabilization of the linear symplectic group actions on $T^*\mathbf {R}^N$. In [Reference LurieLur15], it is conjectured that the obstruction to define a Fukaya category over $\mathbf {S}$ on a symplectic $2n$-manifold $M$ is the composition

(1.9.1)\begin{equation} M\longrightarrow BU\overset{B^2J_{\mathbf{C}}}{\longrightarrow} B^2\mathrm{Pic}(\mathbf{S}), \end{equation}

where the first map is the stabilization of the classifying map for the tangent bundle as an $U(n)$-bundle, and $J_{\mathbf {C}}: \mathbf {Z}\times BU\rightarrow \mathrm {Pic}(\mathbf {S})$ is the complex $J$-homomorphism. By the compatibility of the complex $J$-homomorphism with the real $J$-homomorphism, one has $B^2J_{\mathbf {C}}$ is homotopic to the delooping of the classifying map of brane structures $U\rightarrow U/O\rightarrow B\mathrm {Pic}(\mathbf {S})$.

The analogue of the Fukaya category in sheaf theory for a Weinstein manifold $M$ with a choice of Lagrangian skeleton $\Lambda$ is the microlocal sheaf category along $\Lambda$. We will show in a future work that the obstruction to defining a microlocal sheaf category along $\Lambda$ is exactly the obstruction given by (1.9.1) and this is based on the $E_1$-structure on the classifying map $U\rightarrow B\mathrm {Pic}(\mathbf {S})$. There is a different approach to this using $h$-principle given in [Reference ShendeShe21, Reference Nadler and ShendeNS20].