\section{Introduction}

Let $\mathbf{K}^*(X) = \mathbf{K}^0(X) \oplus \mathbf{K}^1(X)$ denote the complex $K$-theory of a space $X$.  I am not sure who first proposed that when $X$ and $\hat{X}$ are a mirror pair of compact Calabi-Yau $3$-folds one should have isomorphisms
\begin{equation}
\label{eq:111}
\mathbf{K}^0(X) \cong \mathbf{K}^1(\hat{X}) \text{ and } \mathbf{K}^1(X) \cong \mathbf{K}^0(\hat{X})
\end{equation}
--- it is an instance of the string-theoretical idea \cite{MM,Moore,Witten} that ``$D$-branes have charges in $K$-theory.''  Rationally, \eqref{eq:111} is a consequence of the usual Hodge-diamond flip, but the question of whether it holds becomes interesting if $\mathbf{K}^*(X)$ or $\mathbf{K}^*(\hat{X})$ has torsion, or if one and not the other group is known to be torsion-free.  It might be interesting more generally if one searches for very natural isomorphisms, more on that in \S\ref{sec:three}.
\medskip

I believe that \eqref{eq:111} is an open problem.  Batyrev and Kreuzer in \cite{BK} gave a case-by case verification for the half-billion mirror pairs associated with 4d reflexive polytopes, actually obtaining isomorphisms in integral cohomology
\begin{equation}
\label{eq:112}
\mathrm{tors}(H^2(X,\mathbf{Z})) \cong \mathrm{tors}(H^3(\hat{X},\mathbf{Z})) \qquad \mathrm{tors}(H^4(X,\mathbf{Z})) \cong \mathrm{tors}(H^5(\hat{X},\mathbf{Z}))
\end{equation}
and deducing \eqref{eq:111} from the Atiyah-Hirzebruch spectral sequence.  But Addington \cite{Addington} has given examples of derived equivalent $3$-folds $\hat{X}$ and $\hat{X}'$ where $H^3(\hat{X},\mathbf{Z})$ and $H^3(\hat{X}',\mathbf{Z})$ have different torsion subgroups, suggesting that \eqref{eq:112} should not hold in general.

\medskip

In \S\ref{sec:two}, we will give an explicit example, by verifying \eqref{eq:111} in one new case: a $T$-dual pair of flat $3$-folds (for which homological mirror symmetry is essentially known after \cite{Abouzaid})
\[
X:= X_{1,5} \qquad \hat{X} := X_{2,12}
\]
with $\mathbf{K}^0(X) \cong \mathbf{K}^1(\hat{X})$ but $\mathrm{tors}(H^2(X,\mathbf{Z})) = (\mathbf{Z}/4)^3$ and $\mathrm{tors}(H^3(\hat{X},\mathbf{Z})) = \mathbf{Z}/4$.
\medskip

In \S\ref{sec:three} we will discuss conjectures --- some of mine and one of Ganatra's --- about the $K$-theory of Fukaya categories.

\section{$3$-folds}
\label{sec:two}

\subsection{The flat $3$-manifold $B$.}

Let $B$ denote the quotient of $\mathbf{R}^3/\mathbf{Z}^3$ by the action of $\mathbf{Z}/2 \times \mathbf{Z}/2$ whose three nontrivial operators are
\begin{equation}
\label{eq:Fed-Sch}
\begin{array}{ccc}
\alpha(x_1,x_2,x_3)& := & (x_1 + \frac{1}{2},-x_2 + \frac{1}{2},-x_3) \\
\beta(x_1,x_2,x_3)& :=& (-x_1+\frac{1}{2},-x_2,x_3+\frac{1}{2}) \\
\gamma(x_1,x_2,x_3)& :=& (-x_1,x_2+\frac{1}{2},-x_3 + \frac{1}{2})
\end{array}
\end{equation}
It is the $3$-manifold studied in \cite{HW}.  We regard it as having a basepoint at image of $0 \in \mathbf{R}^3$, and as having a flat metric given by the usual dot product on $\mathbf{R}^3$.  The fundamental group of $B$ is one of the Fedorov-Schoenflies crystallographic groups, with presentation \cite[Th. 3.5.5]{Wolf}
\begin{equation}
\label{eq:Wolf}
\begin{array}{ccc}
\alpha^2 = t_1 & \alpha t_2 = t_2^{-1} \alpha &  \alpha t_3 = t_3^{-1} \alpha \\
\beta t_1 = t_1^{-1} \beta & \beta t_2 = t_2^{-1} \beta &  \beta^2 = t_3 \\
\gamma t_1 = t_1^{-1} \gamma & \gamma^2 = t_2 & \gamma t_3 = t_3^{-1} \gamma
\end{array}
\end{equation}
and
\[
[t_1,t_2] = [t_2,t_3] = [t_3,t_1] =  \gamma \beta \alpha = 1
\]
The $t_1,t_2,t_3$ are translation operators on $\mathbf{R}^3$.  Being flat, the holonomy group of $B$ is a representation
\begin{equation}
\label{eq:holonomy}
\pi_1(B) \to \mathrm{SO}(3)
\end{equation}
Its image is isomorphic to $\mathbf{Z}/2 \times \mathbf{Z}/2$ (the group of diagonal matrices in $\mathrm{SO}(3)$).  Abelianizing \eqref{eq:Wolf} gives $H_1(B) = \mathbf{Z}/4 \oplus \mathbf{Z}/4$, and since $\alpha,\beta,\gamma$ are orientation-preserving we have by Poincar\'e duality
\begin{equation}
\label{eq:HiB}
H_0(B) = \mathbf{Z} \qquad H_1(B) = \mathbf{Z}/4 \oplus \mathbf{Z}/4 \qquad H_2(B) = 0 \qquad H_3(B) = \mathbf{Z}
\end{equation}

\subsection{Tri-elliptic $3$-fold $X_{0,4}$}
\label{subsec:X04}
Let $\tau_1,\tau_2,\tau_3$ be complex numbers with positive imaginary part, and put
\begin{equation}
E_i := \mathbf{C}/(\mathbf{Z} + \tau_i \mathbf{Z})
\end{equation}

Let $X_{0,4}$ be the quotient of $E_1 \times E_2 \times E_3$ by the complexification of the operators \eqref{eq:Fed-Sch}, i.e.
\begin{equation}
\label{eq:Fed-Sch-complex}
\begin{array}{ccc}
\alpha(z_1,z_2,z_3)& := & (z_1 + \frac{1}{2},-z_2 + \frac{1}{2},-z_3) \\
\beta(z_1,z_2,z_3)& :=& (-z_1+\frac{1}{2},-z_2,z_3+\frac{1}{2}) \\
\gamma(z_1,z_2,z_3)& :=& (-z_1,z_2+\frac{1}{2},-z_3 + \frac{1}{2})
\end{array}
\end{equation}
(We follow \cite{DW} for the name).  The projections $\mathbf{C} \to \mathbf{R}:x_i + \tau_i y_i \mapsto x_i$ descend to a map
\begin{equation}
\label{eq:X04B}
X_{0,4} \to B
\end{equation}
which is split by the subset cut out by $y_1 = y_2 = y_3 = 0$.  The translation action of
\begin{equation}
\label{eq:V}
V:= \mathbf{R}\tau_1 \times \mathbf{R}\tau_2 \times \mathbf{R}\tau_3
\end{equation}
on $\mathbf{C} \times \mathbf{C} \times \mathbf{C}$ descends to an action on $E_1 \times E_2 \times E_3$ and on $X_{0,4}$.  The action preserves the fibers  of \eqref{eq:X04B}, and determines an identification of the fiber over $b$ with the quotient of $V$ by a lattice $V_{\mathbf{Z},b} \subset V$.  We will denote the lattice over the basepoint by $M_{0,4}$, i.e.
\begin{equation}
\label{eq:M04}
M_{0,4} := V_{\mathbf{Z},0} = \mathbf{Z}\tau_1 \times \mathbf{Z}\tau_2 \times \mathbf{Z} \tau_3
\end{equation}

The action of $\pi_1(B)$ on $V$ and on $M_{0,4}$ is through the holonomy $\mathbf{Z}/2 \times \mathbf{Z}/2$ \eqref{eq:holonomy}.

\subsection{More tri-elliptic $3$-folds}
On each $E_i$ we may define a biholomorphic action of $\mathbf{Z}/2 \times \mathbf{Z}/2 \times \mathbf{Z}/2$: the three generators act by
\[
z \mapsto z+1/2 \qquad z \mapsto z + \tau_i/2 \qquad z \mapsto -z
\]
Altogether this defines an action of $(\mathbf{Z}/2)^{\times 9}$ on $E_1 \times E_2 \times E_3$.  In \cite{DW}, Donagi and Wendland classified the subgroups that act freely.  The quotient $X = (E_1 \times E_2 \times E_3) /G$ must factor as a product of a surface and an elliptic curve, or else be isomorphic to one of the foursome
\begin{equation}
\label{eq:DW-names}
X_{0,4} \qquad X_{1,5} \qquad X_{1,11} \qquad X_{2,12}
\end{equation}
where $X_{0,4}$ is as in \S\ref{subsec:X04} and the other three are defined below.  These $3$-folds are part of a more general classification problem considered in \cite{DW}, which is reflected in the weird names.  They also appear in \cite{Lange}, where they are called ``hyperelliptic $3$-folds of type (2,2).''  Some older appearances are given in \cite{DonagiSharpe}.

Each of the $3$-folds \eqref{eq:DW-names} is aspherical, and fits into a fiber sequence
\begin{equation}
\label{eq:T-here}
V/M_{I,J} \to X_{I,J} \to B
\end{equation}
where $V$ is as in \eqref{eq:V} and $M_{I,J}$ is a lattice in $V$.

\subsection{Definition}
Let $X_{1,5}$ denote the quotient of $X_{0,4}$ by the involution
\begin{equation}
\label{eq:X15quot}
\qquad (z_1,z_2,z_3) \mapsto \left(z_1 + \frac{\tau_1}{2},z_2 + \frac{\tau_2}{2},z_3 + \frac{\tau_3}{2}\right)
\end{equation}
Then
\[
M_{1,5} = M_{0,4} + \mathbf{Z}\left(\tau_1/2, \tau_2/2, \tau_3/2\right)
\]

\subsection{Definition}
Let $X_{1,11}$ denote the quotient of $X_{0,4}$ by the involution
\begin{equation}
\label{eq:X111quot}
(z_1,z_2,z_3) \mapsto \left(z_1 + \frac{\tau_1}{2},z_2 + \frac{\tau_2}{2},z_3\right)
\end{equation}
Then
\[
M_{1,11} = M_{0,4} + \mathbf{Z}\left(\tau_1/2,\tau_2/2,0\right)
\]

\subsection{Definition}
Let $X_{2,12}$ denote the quotient of $X_{0,4}$ by the $\mathbf{Z}/2 \times \mathbf{Z}/2$ group generated by the pair of involutions
\begin{equation}
\label{eq:X212quot}
(z_1,z_2,z_3) \mapsto \left(z_1 + \frac{\tau_1}{2},z_2 + \frac{\tau_2}{2},z_3\right) \text{ and }(z_1,z_2,z_3) \mapsto \left(z_1 ,z_2 + \frac{\tau_2}{2},z_3 + \frac{\tau_3}{2}\right)
\end{equation}
Then
\[
M_{2,12} = M_{0,4} + \mathbf{Z}\left\{(\tau_1/2,\tau_2/2,0),(0,\tau_2/2,\tau_3/2)\right\}
\]

\subsection{$T$-duality}
The $T$-dual fibration to $X_{I,J} \to B$, of Strominger-Yau-Zaslow, is the space of pairs $(b,L)$ where $b \in B$ and $L \in H^1(V/V_{\mathbf{Z},b},\mathrm{U}(1))$ is the isomorphism class of a rank one unitary local system on the fiber above $b$.  Let us denote it by $\hat{X}_{I,J}$.  It is another split torus fibration
\begin{equation}
\label{eq:T-hat-here}
V^*/\hat{M}_{I,J} \to \hat{X}_{I,J} \to B
\end{equation}
where $V^* := \mathrm{Hom}(V,\mathfrak{u}(1))$ and $\hat{M} \subset V^*$ is the dual lattice to $M$.  As such $\hat{X}_{I,J}$ is determined up to homotopy equivalence by the dual $\pi_1(B)$-module (equivalently, the dual $\mathbf{Z}/2 \times \mathbf{Z}/2$-module) to $M_{I,J}$.  $M_{0,4}$ and $M_{1,11}$ are self-dual, while $M_{1,5}$ and $M_{2,12}$ are dual to each other, and therefore we have homotopy equivalences
\begin{equation}
\label{eq:T-dual-IJ}
\hat{X}_{0,4} \simeq X_{0,4} \qquad \hat{X}_{1,5} \simeq X_{2,12}, \qquad \hat{X}_{1,11} \simeq X_{1,11}
\end{equation}
The homotopy equivalences \eqref{eq:T-dual-IJ} can be taken to be natural diffeomorphisms, if $X_{I,J}$ has parameters $\tau_1,\tau_2,\tau_3$ and we take the corresponding parameters for $\hat{X}_{I,J}$ to be the purely imaginary numbers $(i|\tau_1|^{-1},i|\tau_2|^{-1}, i|\tau_3|^{-1})$.

\subsection{$K$-theory}
Let $X = X_{I,J}$ and $\hat{X} = X_{I',J'}$ be a dual pair of the $3$-folds.  We wish to prove \eqref{eq:111}, that $\mathbf{K}^0(X) \cong \mathbf{K}^1(\hat{X})$ and that $\mathbf{K}^1(X) \cong \mathbf{K}^0(\hat{X})$ --- we will do so without actually computing $\mathbf{K}^*(X)$ and $\mathbf{K}^*(\hat{X})$, indeed I do not quite know what the $K$-theory of these manifolds is \S\ref{subsec:HXIJ}--\ref{subsec:AH-fil}.

Let $\mathbf{K}$ denote the complex $K$-theory spectrum.  It is an $E_{\infty}$-ring spectrum.  We write $\mathrm{Mod}(\mathbf{K})$ for the symmetric monoidal $\infty$-category of module spectra over $\mathbf{K}$, and we will study sheaves of $\mathbf{K}$-module spectra on $X$, $\hat{X}$ and related spaces.  These are stable $\infty$-categories --- for an $\infty$-category we will write $\mathrm{Maps}(c,d)$ for the space of maps and $[c,d]$ for the set of homotopy classes of maps between two objects.  We write $\Sigma$ for the suspension functor in a stable $\infty$-categories.

If $U$ is a manifold we write $\mathbf{K}_U$ for the constant sheaf of $\mathbf{K}$-module spectra on $U$, and $\omega_U$ for the orientation sheaf.

\subsection{Lemma}
\label{lem:spinc-structures}
Each of the spaces $B, X, \hat{X}, X \times_B \hat{X}$ are $\mathbf{K}$-orientable --- that is, there are isomorphisms of sheaves
\begin{equation}
\label{eq:spin-on-these}
\Sigma^{-3} \mathbf{K}_B \cong \omega_B \qquad \Sigma^{-6} \mathbf{K}_X \cong \omega_X \qquad \Sigma^{-6} \mathbf{K}_{\hat{X}} \cong \omega_{\hat{X}} \qquad \Sigma^{-9} \mathbf{K}_{X \times_B \hat{X}} \cong \omega_{X \times_B \hat{X}}
\end{equation}

\begin{proof}
Any $\mathrm{Spin}^c$-structure on a manifold induces a $\mathbf{K}$-orientation, and one way to endow an oriented flat manifold with a $\mathrm{Spin}^c$ structure is to lift its holonomy representation
\begin{equation}
\label{eq:spin-n}
\pi_1 \to \mathrm{SO}(n)
\end{equation}
along the natural homomorphism $\mathrm{Spin}^c(n) \to \mathrm{SO}(n)$.  Each of $B$, $X$, $\hat{X}$ and $X \times_B \hat{X}$ fibers over $B$, and the holonomy around any loop in those fibers is trivial, so \eqref{eq:spin-n} factors through $\pi_1(B) \to \mathrm{SO}(3)$ \eqref{eq:holonomy}.  The equations \eqref{eq:Wolf} can be solved in $\mathrm{Spin}^c(3)$,  for instance we may solve them in $\mathrm{Spin}(3)$ by taking $\alpha,\beta,\gamma$ to be the usual unit quaternions.  Then the lift of \eqref{eq:spin-n} can be taken to be the composite of $\pi_1 \to \pi_1(B) \to \mathrm{Spin}(3)$ with any lift of $\mathrm{Spin}(3) \to \mathrm{SO}(3) \to \mathrm{SO}(n)$ to $\mathrm{Spin}^c(n)$.
\end{proof}

\subsection{Local-on-$B$ identifications of $K$-theory}

Write $\mathbf{K}[U]$ for the $K$-homology spectrum and $\mathbf{K}^U$ for the $K$-cohomology spectrum of a space $U$ --- that is, $\mathbf{K}[U]$ is the smash product of $\mathbf{K}$ with the suspension spectrum of $U$ and $\mathbf{K}^U$ is the internal mapping object from $\mathbf{K}[U]$ to $\mathbf{K}$.  They are related to the $K$-homology and $K$-cohomology groups of $U$ by
\[
[\Sigma^i \mathbf{K},\mathbf{K}[U]] \cong \mathbf{K}_i(U)
\]
and
\[
\mathbf{K}^i(U) \cong [\mathbf{K}[U],\Sigma^i \mathbf{K}] \cong [\Sigma^{-i} \mathbf{K}, \mathbf{K}^U]
\]
In terms of sheaf operations, we have
\[
\Gamma_c(\omega_U) = \mathbf{K}[U] \qquad \Gamma(\mathbf{K}_U) = \mathbf{K}^U
\]

We consider the fiber square
\begin{equation}
\label{eq:this-square}
\xymatrix{
X \times_B \hat{X} \ar[r]^-g \ar[d]_-h & X \ar[d]^-{p} \\
\hat{X} \ar[r]_-{q} & B
}
\end{equation}
Factoring the maps $X \to \mathit{pt}$ and $\hat{X} \to \mathit{pt}$ through $B$ gives canonical isomorphisms
\begin{equation}
\label{eq:from-local-on-B}
\mathbf{K}[X] \cong \Gamma_c(B,p_! \omega_X) = \Gamma(B,p_!\omega_X) \qquad \mathbf{K}^{\hat{X}} \cong \Gamma(B,q_* \mathbf{K}_{\hat{X}})
\end{equation}
where we replace $\Gamma_c$ with $\Gamma$ using the compactness of $B$.  The $\mathbf{K}$-orientability of $X$ gives an identification of $\mathbf{K}[X] \cong \Sigma^{-6} \mathbf{K}^{\hat{X}}$.  So to prove \eqref{eq:111} it suffices to produce an isomorphism between $\Sigma^{-3} p_! \omega_X$ and $q_* \mathbf{K}_{\hat{X}}$.  To that end, let us study the sheaf of spaces on $B$ whose sections over $U \subset B$ are given by
\begin{equation}
\label{eq:sheaf-of-maps}
\mathrm{Maps}\left(\left(\Sigma^{-3} p_! \omega_X\right)\vert_U, \left(q_* \mathbf{K}_{\hat{X}}\right)\vert_U\right)
\end{equation}
where $\mathrm{Maps}$ is taken in the $\infty$-category of sheaves of $\mathbf{K}$-modules over $U$.

\subsection{Lemma}
If $\pi = q\circ h = p \circ g$ denotes the projection $X \times_B \hat{X} \to B$, and one fixes $\mathbf{K}$-orientations of $X$, $\hat{X}$, and $X\times_B \hat{X}$, there are natural isomorphisms
\begin{equation}
\label{eq:sheafhom2}
\mathrm{Maps}\left(\left(\Sigma^{-3} p_! \omega_X\right)\vert_U, \left(q_* \mathbf{K}_{\hat{X}}\right)\vert_U\right)
 \cong \mathrm{Maps}(\mathbf{K}[\pi^{-1}(U)],\mathbf{K})
\end{equation}
where the left-hand side is \eqref{eq:sheaf-of-maps} and on the right-hand side $\mathrm{Maps}$ is taken in the $\infty$-category of $\mathbf{K}$-modules.

\begin{proof}

\begin{eqnarray}
\quad \quad \mathrm{Maps}\left(\left(\Sigma^{-3} p_! \omega_X\right)\vert_U, \left(q_* \mathbf{K}_{\hat{X}}\right)\vert_U\right) & \cong &  \mathrm{Maps}\left(\left(\Sigma^{-3} q^* p_! \omega_X\right)\vert_{q^{-1}(U)}, \mathbf{K}_{{q^{-1}(U)}}\right) \label{eq:364} \\
& \cong & \mathrm{Maps}\left(\left(\Sigma^{-3} h_! g^* \omega_X\right)\vert_{q^{-1}(U)}, \mathbf{K}_{{q^{-1}(U)}}\right)
\label{eq:365}
\\
\label{eq:366}
& \cong & \mathrm{Maps}\left(\Sigma^{-3} \left(h_! g^* \Sigma^{-6} \mathbf{K}_{X}\right)\vert_{q^{-1}(U)}, \mathbf{K}_{{q^{-1}(U)}}\right) \\
\label{eq:367}
& \cong & \mathrm{Maps}\left(\Sigma^{-9} \left(h_! \mathbf{K}_{X \times_B \hat{X}}\right)\vert_{q^{-1}(U)}, \mathbf{K}_{{q^{-1}(U)}}\right) \\
\label{eq:368}
& \cong & \mathrm{Maps}\left(\left(h_! \omega_{X \times_B \hat{X}}\right)\vert_{q^{-1}(U)}, \mathbf{K}_{{q^{-1}(U)}}\right) \\
\label{eq:369}
& \cong & \mathrm{Maps}\left(\left(h_! \omega_{X \times_B \hat{X}}\right)\vert_{q^{-1}(U)}, \Sigma^{6}\omega_{{q^{-1}(U)}}\right) \\
\label{eq:3610}
& \cong & \mathrm{Maps}\left(\Gamma_c\left(\left(h_! \omega_{X \times_B \hat{X}}\right)\vert_{q^{-1}(U)}\right), \Sigma^{6}\mathbf{K}\right) \\
\label{eq:3611}
& \cong & \mathrm{Maps}(\mathbf{K}[(q \circ h)^{-1}(U)],\Sigma^6\mathbf{K})
\end{eqnarray}
where \eqref{eq:364} is the $(q^*,q_*)$-adjunction, \eqref{eq:365} is proper base-change, \eqref{eq:366} uses the $\mathbf{K}$-orientation of $X$,\eqref{eq:368} uses the $\mathbf{K}$-orientation of $X \times_B \hat{X}$,  \eqref{eq:369} uses the $\mathbf{K}$-orientation of $q^{-1}(U) \subset \hat{X}$, \eqref{eq:3610} uses the $(q^{-1}(U) \to \mathit{pt})_!,(q^{-1}(U) \to \mathit{pt})^!$ adjunction.  Finally one applies the Bott isomorphism $\mathbf{K} \cong \Sigma^6 \mathbf{K}$ to obtain the right-hand-side of \eqref{eq:sheafhom2}.
\end{proof}

\subsection{Poincar\'e bundle}
\label{subsec:Pe-bundle}
When $T$ and $\hat{T}$ are dual tori,  (for instance, if $T = V/M$ \eqref{eq:T-here} and $\hat{T} = V^*/\hat{M}$ \eqref{eq:T-hat-here} are fibers above the basepoint of $X \to B$ and $\hat{X} \to B$), there is a canonical pairing $H_1(T) \otimes H_1(\hat{T}) \to \mathbf{Z}$, which determines a canonical element
\begin{equation}
\label{eq:coev}
\mathrm{coev} \in H^1(T;\mathbf{Z}) \otimes H^1(\hat{T};\mathbf{Z}) \subset H^2(T \times \hat{T};\mathbf{Z})
\end{equation}
Let us say that a line bundle on $X \times_B \hat{X}$ is a ``Poincar\'e bundle'' if its restriction to a fiber is this canonical element.

The connected components of the right-hand side of \eqref{eq:sheaf-of-maps} are virtual vector bundles on $\pi^{-1}(U)$.  In particular, a line bundle on $X \times_B \hat{X}$ determines a homotopy class of maps
\begin{equation}
\label{eq:line-bundle-map}
P_{L}:\Sigma^{-3} p_! \omega_X \to q_* \mathbf{K}_{\hat{X}}
\end{equation}

\begin{lemma}
\label{lem:poincare}
If $L$ is a Poincar\'e bundle, $P_L$ is an isomorphism.
\end{lemma}

\begin{proof}
We prove that $P_L$ is an isomorphism on stalks.  More generally we prove that if $T$ and $\hat{T}$ are dual tori, a line bundle whose Chern class is \eqref{eq:coev} exhibits $\mathbf{K}^T$ and $\mathbf{K}^{\hat{T}}$ as dual objects in the monoidal category $\mathrm{Mod}(\mathbf{K})$.  Such a line bundle determines a homotopy class of maps
\begin{equation}
\label{eq:241}
\mathbf{K} \to \mathbf{K}^{T\times \hat{T}} \cong \mathbf{K}^T \otimes_{\mathbf{K}} \mathbf{K}^{\hat{T}}
\end{equation}
in $\mathrm{Mod}(\mathbf{K})$, and we will show that for all $i$ the composite
\begin{equation}
\label{eq:251}
[\Sigma^i \mathbf{K}^T,\mathbf{K}] \xrightarrow{\otimes \mathbf{K}^{\hat{T}}} [\Sigma^i \mathbf{K}^{T} \otimes_{\mathbf{K}} \mathbf{K}^{\hat{T}}, \mathbf{K}^{\hat{T}}] \xrightarrow{\eqref{eq:241}} [\Sigma^i \mathbf{K},\mathbf{K}^{\hat{T}}]
\end{equation}
is an isomorphism.

In case $T = \hat{T} = \mathrm{U}(1)$, we have canonically $\mathbf{K}^T \cong \mathbf{K} \oplus \Sigma \mathbf{K}$, $\mathbf{K}^{\hat{T}} \cong \mathbf{K} \oplus \Sigma \mathbf{K}$, and
\begin{equation}
\label{eq:TTU1}
\mathbf{K}^{T \times \hat{T}} \cong \mathbf{K} \oplus \Sigma \mathbf{K} \oplus \Sigma \mathbf{K} \oplus \Sigma^2 \mathbf{K}.
\end{equation}
Then \eqref{eq:241} is the Bott isomorphism $\mathbf{K} \cong \Sigma^2 \mathbf{K}$ onto the last summand of \eqref{eq:TTU1}, and one can check \eqref{eq:251} directly.

In the general case, the domain of \eqref{eq:251} is $\mathbf{K}_{i}(T)$ and the codomain is $\mathbf{K}^{-i}(\hat{T})$, and the square
\[
\xymatrix{
\mathbf{K}_1(\mathrm{U}(1)) \otimes_{\mathbf{Z}} \mathrm{Hom}(\mathrm{U}(1),T) \ar[r] \ar[d] & \mathbf{K}_1(T) \ar[d]^{\eqref{eq:251}}\\
\mathbf{K}^{-1}(\mathrm{U}(1)) \otimes_{\mathbf{Z}} \mathrm{Hom}(\hat{T},\mathrm{U}(1)) \ar[r] & \mathbf{K}^{-1}(\hat{T})
}
\]
commutes, where the left vertical arrow is \eqref{eq:251} for $T = \hat{T} = \mathrm{U}(1)$, tensored with the identification of cocharacters of $T$ with characters of $\hat{T}$.  The horizontal arrows induce graded ring isomorphisms
\begin{equation}
\label{eq:ring-iso}
\Lambda (\mathrm{Hom}(\mathrm{U}(1),T)) \otimes \mathbf{K}^* \to \mathbf{K}_*(T) \qquad \Lambda (\mathrm{Hom}(\hat{T},\mathrm{U}(1))) \otimes \mathbf{K}^* \to \mathbf{K}^*(T)
\end{equation}
where the multiplication on $\mathbf{K}_*(T)$ is defined using the group structure on $T$ (the Pontrjagin product), and the ring structure on $\mathbf{K}^*(\hat{T})$ is tensor product of vector bundles.  Thus we may complete the proof that \eqref{eq:251} is an isomorphism by noting that it intertwines the Pontrjagin product on $\mathbf{K}_*(T)$ with the tensor product on $\mathbf{K}^*(\hat{T})$.  A strong form of this is true but to make use of \eqref{eq:ring-iso} we only need to note that (letting $m:T \times T \to T$ denote the multiplication and $\Delta:\hat{T} \to \hat{T} \times \hat{T}$ the diagonal) the following two elements of $\mathbf{K}^0(T \times T \times \hat{T})$ are equal:
\begin{itemize}
\item The pullback of \eqref{eq:coev} along $m \times 1:T \times T \to \hat{T} \to T \times \hat{T}$
\item The pullback of \eqref{eq:coev} $\boxtimes$ \eqref{eq:coev} along the map $T \times T \times \hat{T} \to T \times \hat{T} \times T \times \hat{T}$ that carries $(t_1,t_2,\hat{t})$ to $(t_1,\hat{t},t_2,\hat{t})$
\end{itemize}
In fact these are equal in $H^2(T \times T \times \hat{T};\mathbf{Z})$.  It follows that two maps from the upper left to the lower right corner of the evident square
\[
\xymatrix{
\mathbf{K}[T] \otimes_{\mathbf{K}} \mathbf{K}[T] \ar[d] \ar[r] & \mathbf{K}^{\hat{T}} \otimes_{\mathbf{K}} \mathbf{K}^{\hat{T}} \ar[d] \\
\mathbf{K}[T] \ar[r] & \mathbf{K}^{\hat{T}}
}
\]
are homotopic, and therefore that \eqref{eq:251} is a ring homomorphism.
\end{proof}

\subsection{Theorem}
\label{th:111}
Let $X$ and $\hat{X}$ be as in \eqref{eq:T-dual-IJ}.  Then \eqref{eq:111} holds, i.e.
\[
\mathbf{K}^0(X) \cong \mathbf{K}^1(\hat{X}) \text{ and } \mathbf{K}^1(X) \cong \mathbf{K}^0(\hat{X})
\]

\begin{proof}
After \eqref{eq:from-local-on-B} and Lemma \ref{lem:poincare}, it suffices to construct a Poincar\'e bundle on $X \times_B \hat{X}$.  The fundamental group $\pi_1(B)$ acts on $H^2(T \times \hat{T};\mathbf{Z}) = H^2(V/M \times V^*/\hat{M};\mathbf{Z})$, and the canonical class \eqref{eq:coev} is fixed by this action.  We will prove the existence of a Poincar\'e bundle by showing that the map
\begin{equation}
\label{eq:surj}
H^2(X \times_B \hat{X};\mathbf{Z}) \to H^2(T \times \hat{T};\mathbf{Z})^{\pi_1(B)}
\end{equation}
is a surjection.  As $X \times_B \hat{X}$ is a $K(\pi,1)$-space, the domain of \eqref{eq:surj} is isomorphic to the cohomology of the fundamental group $\pi_1(X \times_B \hat{X})$.  To prove that it is a surjection is equivalent to showing that the differentials
\[
d_2:H^2(T \times \hat{T};\mathbf{Z})^{\pi_1(B)} \to H^2(\pi_1(B); H^1(T \times \hat{T};\mathbf{Z})
\]
and
\[
d_3:\ker(d_2) \to H^3(B;\mathbf{Z})
\]
vanish, in the Serre spectral sequence of the fibration $X \times_B \hat{X} \to B$.  Let us denote this spectral sequence by ${}^{X\hat{X}}E_r^{st}$.  We similarly denote the Serre spectral sequence of $X \to B$ by ${}^X E_r^{st}$ and of $\hat{X} \to B$ by ${}^{\hat{X}} E_r^{st}$.

Since the fibration has a section, all of $H^3(B;\mathbf{Z})$ must survive to the $E_{\infty}$-page, so $d_3$ vanishes.  The codomain of $d_2$ is
\begin{equation}
\label{eq:dirsumdec}
H^2(\pi_1(B),H^1(T;\mathbf{Z})) \oplus H^2(\pi_1(B),H^1(\hat{T};\mathbf{Z}))
\end{equation}
The sections of $X \to B$ and $\hat{X} \to B$ induce maps $X \to X \times_B \hat{X}$ and $\hat{X} \to X \times_B \hat{X}$ that commute with the projections to $B$, which in turn induce maps of spectral sequences
\begin{equation}
\label{eq:mapsss}
{}^X E_r^{st} \to {}^{X \hat{X}} E_r^{st} \qquad {}^{\hat{X}} E_r^{st} \to {}^{X \hat{X}} E_r^{st}
\end{equation}
The direct sum decomposition \eqref{eq:dirsumdec} is induced by \eqref{eq:mapsss} on $E_2^{21}$, thus we can complete the proof by showing that ${}^X E_2^{21}$ and ${}^{\hat{X}} E_2^{21}$ survive to the $E_{\infty}$-pages, i.e. that
\begin{equation}
{}^X E_2^{02} \to {}^X E_2^{21} \qquad {}^{\hat{X}} E_2^{02} \to {}^{\hat{X}} E_2^{21}
\end{equation}
are both zero.  Now ${}^X E_2^{02} = H^0(\pi_1;H^2(T;\mathbf{Z})) = 0$ and $H^0(\pi_1;H^2(\hat{T};\mathbf{Z})) = 0$:  $H^2(T;\mathbf{Z}) \cong M \subset V$ and $H^2(\hat{T};\mathbf{Z}) \cong \hat{M} \subset V^*$ as $\pi_1$-modules, and $\pi_1$ acts on $V$ and $V^*$ without invariants ($V$ and $V^*$ split as the sum of the three nontrivial characters $\pi_1 \to \mathrm{GL}_1(\mathbf{R})$).
\end{proof}

\subsection{Cohomology of $X_{I,J}$}
\label{subsec:HXIJ}
The two-vertex regular cell complex structure on $S^1$, with vertices at $0$ and $\pi$, is preserved by the action of $\mathbf{Z}/2 \times \mathbf{Z}/2$ generated by
\[
\theta \mapsto \theta+ \pi \qquad \theta \mapsto -\theta
\]
Each of the $3$-folds \eqref{eq:DW-names} can be written as a quotient of a torus $T^6 = S^1 \times S^1 \times S^1 \times S^1 \times S^1 \times S^1$ by the free action of an elementary abelian $2$-group that preserves the product cell structure.  The cellular cochain complex of $T^6$ is a complex of free $\mathbf{Z}[G]$-modules
\begin{equation}
\label{eq:T6complex}
\mathbf{Z}^{64}\to\mathbf{Z}^{384} \to \mathbf{Z}^{960} \to \mathbf{Z}^{1280} \to \mathbf{Z}^{960} \to \mathbf{Z}^{384} \to \mathbf{Z}^{64}
\end{equation}
of $\mathbf{Z}[G]$-rank $2^6 \binom{6}{i}/|G|$ in degree $i$.  Passing to invariants gives a cochain complex for the cohomology of $T^6/G$, small enough to handle by computer --- I used sage.  Besides $H^0 = H^6 = \mathbf{Z}$ and $H^1 = 0$, we have
\[
\begin{array}{|c|c|c|c|c|c|}
\hline
& H^2 & H^3 & H^4 & H^5  \\
\hline
X_{0,4} & \mathbf{Z}^3 \oplus (\mathbf{Z}/4)^2 \oplus (\mathbf{Z}/2)^3 & \mathbf{Z}^8 \oplus (\mathbf{Z}/2)^3 & \mathbf{Z}^3 \oplus (\mathbf{Z}/2)^3 & (\mathbf{Z}/4)^2 \oplus (\mathbf{Z}/2)^3 \\
\hline
X_{1,5} & \mathbf{Z}^3 \oplus (\mathbf{Z}/4)^3 & \mathbf{Z}^8 \oplus (\mathbf{Z}/2)^2 & \mathbf{Z}^3 \oplus (\mathbf{Z}/2)^2 & (\mathbf{Z}/4)^3 \\
\hline
X_{1,11}  & \mathbf{Z}^3 \oplus (\mathbf{Z}/4)^2 \oplus (\mathbf{Z}/2)^2 & \mathbf{Z}^8 \oplus (\mathbf{Z}/2)^2 & \mathbf{Z}^3 \oplus (\mathbf{Z}/2)^2 & (\mathbf{Z}/4)^2 \oplus (\mathbf{Z}/2)^2  \\
\hline
X_{2,12}  & \mathbf{Z}^3 \oplus (\mathbf{Z}/4)^2 \oplus (\mathbf{Z}/2)^2 & \mathbf{Z}^8 \oplus \mathbf{Z}/4 & \mathbf{Z}^3 \oplus \mathbf{Z}/4 & (\mathbf{Z}/4)^2 \oplus (\mathbf{Z}/2)^2 \\
\hline
\end{array}
\]
The top row was previously computed in \cite{BCDP}, and the $H^5$ (equivalently, $H^2$) columns in \cite{DW}.

\subsection{Atiyah-Hirzebruch filtration}
\label{subsec:AH-fil}
Let $X$ be a connected closed manifold of real dimension $6$.  $\mathbf{K}^*(X)$ carries the Atiyah-Hirzebruch filtration
\begin{equation}
\label{eq:AH}
\begin{array}{ccccccccc}
\mathbf{K}^0(X) & = & F^0 \mathbf{K}^0(X) & \supset & F^2 \mathbf{K}^0(X) & \supset & F^4 \mathbf{K}^0(X) & \supset & F^6 \mathbf{K}^0(X)  \\
\mathbf{K}^1(X) & = & F^1 \mathbf{K}^1(X) & \supset & F^3\mathbf{K}^1(X) & \supset & F^5 \mathbf{K}^1(X)
\end{array}
\end{equation}
where $F^k(\mathbf{K}^*(X))$ consists of those classes that vanish when restricted to any $(k-1)$-dimensional submanifold.  The associated graded pieces of this filtration are the groups at the last page of the Atiyah-Hirzebruch spectral sequence:
\[
E_2^{st} = H^s(X,\mathbf{K}^t(\mathit{pt})) \implies E_{\infty}^{st} = F^{s+t} \mathbf{K}^s(X)/F^{s+t + 1} \mathbf{K}^s(X)
\]
If $X$ is oriented, then the spectral sequence degenerates immediately: $E_2^{st} = E_{\infty}^{st}$.   The argument is given in \cite{BrDi}  --- let us briefly repeat the argument here.  Since $\mathbf{K}^t(\mathit{pt}) = 0$ for $t$ odd, all the even differentials $d_{2p}$ vanish.  In general, $d_{2p-1}$ vanishes on $H^i(X,\mathbf{Z})$ for $i \leq 2p-2$ \cite[\S 7]{Atiy}, so on a 6-dimensional complex the only possible nonvanishing differential is $d_3:H^3(X,\mathbf{Z}) \to H^6(X,\mathbf{Z})$.  Even this differential must vanish if $H^6(X,\mathbf{Z})$ has no torsion \cite[\S 2.4]{AtHi}, i.e. if $X$ is orientable.

Plausibly, whenever $X$ is a Calabi-Yau $3$-fold, or even just admits a Spin structure, the Atiyah-Hirzebruch filtration might split: that is, there might be a $\mathbf{Z}/2$-graded isomorphism between
\[
\mathbf{K}^0(X)\oplus \mathbf{K}^1(X) \text{ and } \bigoplus H^i(X;\mathbf{Z})
\]
This is claimed in \cite{Doran}, but I believe the proof there has a gap (discussed in \S\ref{subsec:chern-classes}).  I do not know whether the filtration on $\mathbf{K}^*(X_{I,J})$ splits: if it does, one could conclude Theorem \ref{th:111} directly from the computations in \S\ref{subsec:HXIJ}.  On an oriented 6-manifold one necessary and sufficient condition for the filtration of $\mathbf{K}^0(X)$ to split is the existence of a function $\varphi:H^2(X;\mathbf{Z}) \to H^4(X;\mathbf{Z})$ that obeys
\[
\varphi(c+c') - \varphi(c) - \varphi(c') = c \cup c'
\]
(For instance, we could take $\varphi(c) = c^2/2$ if we could divide by $2$).  The problem of computing the cup product on $H^*(X_{I,J};\mathbf{Z})$ also arose in \cite{BCDP}.  Determining this by computer is more difficult --- the problem is that, although the cup product on $H^*(T^6)$ is induced by a (noncommutative) ring structure on the cochains \eqref{eq:T6complex}, the groups $G$ do not act by ring automorphisms.  One can solve this by passing to the barycentric subdivision of $S^1$ (which induces a subdivision of $(S^1)^{\times 6}$), but the resulting chain complexes are too big to treat in a simple-minded way.

\subsection{Chern classes}
\label{subsec:chern-classes}
A virtual vector bundle has a well-defined Chern class, giving us maps
\begin{equation}
\label{eq:chern-classes}
c_i:\mathbf{K}^0(X) \to H^{2i}(X,\mathbf{Z}) \qquad c_i^\Sigma:\mathbf{K}^1(X) \to H^{2i -1}(X,\mathbf{Z})
\end{equation}
The second map $c_i^{\Sigma}$ is the composite of
\[
\mathbf{K}^1(X) \cong \mathbf{K}^2(\Sigma X) \cong \mathbf{K}^0(\Sigma X) \xrightarrow{c_i} H^{2i}(\Sigma X,\mathbf{Z}) = H^{2i-1}(X,\mathbf{Z})
\]
Except for $c_0$, the functions $c_i$ of \eqref{eq:chern-classes} are not group homomorphisms, they instead obey the Cartan formula $c_n(V+W) = c_n(V) c_0(W) + c_{n-1}(V) c_1(W) + \cdots + c_0(V) c_n(W)$.  The $i$th Chern class becomes a group homomorphism on $F^{2i} \mathbf{K}^0(X)$, since $c_j(E) = 0$ for any $j < i$ and $E \in F^{2i} \mathbf{K}^0(X)$.  As all nontrivial cup products in $H^*(\Sigma X;\mathbf{Z})$ vanish, the Cartan formula shows that $c_i^{\Sigma}:\mathbf{K}^1(X) \to H^{2i - 1}(X,\mathbf{Z})$ are group homomorphisms.

Lemma 4.1 of \cite{Doran} asserts that, when $X$ is a closed oriented $6$-manifold, the map
\begin{equation}
\label{eq:DM41}
(c_2,c_3):F^4 \mathbf{K}^0(X) \to H^4(X,\mathbf{Z}) \oplus H^6(X,\mathbf{Z})
\end{equation}
is an isomorphism onto
\begin{equation}
\label{eq:DM41-im}
\{(c_2,c_3) \mid \mathrm{Sq}^2(c_2) = c_3\}
\end{equation}
where $\mathrm{Sq}^2:H^4(X,\mathbf{Z}/2) \to H^6(X,\mathbf{Z}/2)$ is a Steenrod operation.  Lemma 4.2 of \cite{Doran} asserts that the map
\begin{equation}
\label{eq:DM42}
(c_1,c_2,c_3):F^2\mathbf{K}^0(X) \to H^2(X,\mathbf{Z}) \oplus H^4(X,\mathbf{Z}) \oplus H^6(X,\mathbf{Z})
\end{equation}
is an isomorphism onto
\begin{equation}
\label{eq:DM42-im}
\{(c_1,c_2,c_3) \mid \mathrm{Sq}^2(c_2) = c_3 + c_1 c_2 + c_1^3 \}
\end{equation}
I believe that \eqref{eq:DM41-im} is correct, but \eqref{eq:DM42-im} is not.  For example, if $X$ is the quintic $3$-fold, the virtual vector bundle $\mathcal{O}(1) - \mathcal{O}$ belongs to $F^2 \mathbf{K}^0(X)$ and has $(c_1,c_2,c_3) = (h,0,0)$, where $h$ is the hyperplane section of $X \subset \mathbf{P}^4$.  But $h^3 = 5 \in H^6(X,\mathbf{Z})$, which is nonzero in $H^6(X,\mathbf{Z}/2)$.

\section{Conjectures}
\label{sec:three}

It should be possible to choose the isomorphisms \eqref{eq:111} to intertwine additional structures on $X$ and $\hat{X}$.

\subsection{$K$-homology}
In fact \eqref{eq:111} is expected for any mirror pair of Calabi-Yau manifolds of odd complex dimension.  If $X$ and $\hat{X}$ have even complex dimension, then we expect $\mathbf{K}^i(X) \cong \mathbf{K}^i(\hat{X})$ for $i = 0,1$.  I think the right way to organize these expectations is as an equivalence of $\mathbf{K}$-module spectra:
\begin{equation}
\label{eq:spectrum}
\Sigma^{-n} \mathbf{K}[X] \cong \mathbf{K}^{\hat{X}}
\end{equation}
where $n$ is the complex dimension of $X$, $\Sigma$ denotes suspension, $\mathbf{K}[?]$ denotes the $K$-homology spectrum and $\mathbf{K}^?$ denotes the $K$-cohomology spectrum.  The $K$-homology and $K$-cohomology of a compact almost complex manifold are naturally identified, and $\mathbf{K}$-theory is $2$-periodic, so \eqref{eq:spectrum} implies \eqref{eq:111} by taking homotopy groups.  Two $\mathbf{K}$-module spectra are isomorphic if and only if their homotopy groups are isomorphic, so the converse is true as well.  But using $K$-homology in place of $K$-cohomology seems to go with the grain of homological mirror symmetry, in a way that we will explain.

\subsection{The large volume and large complex structure limits}
\label{subsec:lvllcsl}
For the rest of the paper we will be treating the symplectic geometry of $X$ and the complex geometry of $\hat{X}$. And we will assume that the symplectic form on $X$ has integral cohomology class $[\omega] \in H^2(X;\mathbf{Z})$.  The isomorphism class of line bundles whose Chern class is $[\omega]$ gives a unit in $\mathbf{K}^0(X) := \pi_0(\mathbf{K}^X)$, and (using the $\mathbf{K}^X$-module structure on $\mathbf{K}[X]$) a homotopy class of automorphisms of $\mathbf{K}[X]$.  The corresponding homotopy class of automorphisms of $\mathbf{K}^{\hat{X}}$ is a monodromy operator one obtains by putting $\hat{X}$ in a family $\hat{X}_t$, where $t$ runs through a punctured disk.

The Seidel strategy \cite{Seidel} for proving HMS is to prove it first in a limit --- one takes a hyperplane section $D$ of the line bundle on $X$, and the special fiber $\hat{X}_0$ at the center of the family $\hat{X}_t$, so that there is a mirror relationship between $X -D$ and $\hat{X}_0$.  $X-D$ is called the ``large volume limit'' and $\hat{X}_0$ is called the ``large complex structure limit'' of the mirror pair.  In such a case I conjecture (I am not sure how originally) that
\begin{equation}
\label{eq:at-limit}
\Sigma^{-n} \mathbf{K}[X-D] \cong \mathbf{K}^{\hat{X}_0}
\end{equation}
as $\mathbf{K}$-modules.  For the noncompact $X-D$ or the singular $\hat{X}_0$, it is now necessary to pay attention to the difference between $\mathbf{K}$-homology and $\mathbf{K}$-cohomology.
\medskip

\noindent
{\bf Example.}  The case when $\hat{X} \subset \mathbf{C} P^{n+1}$ is a degree $n+2$ hypersurface furnishes a standard example.  A mirror $X$ to $\hat{X}$ is obtained by resolving the singularities of an anticanonical hypersurface in a weighted projective $(n+1)$-space.  The limits $X -D$ and $\hat{X}_0$ can be described directly: $X -D \subset (\mathbf{C}^*)^{n+1}$ is any sufficiently generic hypersurface whose Newton polytope is the standard reflexive lattice simplex, e.g.
\begin{equation}
\label{eq:dual-ntic}
X-D := W^{-1}(0), \quad W:(x_0,\ldots,x_n) \mapsto x_0 + \cdots + x_n + \frac{1}{x_0\cdots x_n} - 1
\end{equation}
and $\hat{X}_0$ is the union of the coordinate hyperplanes
\begin{equation}
\label{eq:ntic}
\hat{X}_0 := \{[x_0,\ldots,x_n] \in \mathbf{C}P^{n+1} \mid x_0 \cdots x_n = 0\}
\end{equation}
For these examples, \eqref{eq:at-limit} can be deduced from a similar equivalence
\begin{equation}
\label{eq:LG}
\Sigma^{-n-1} \mathbf{K}[(\mathbf{C}^*)^{n+1},W^{-1}(0)] \cong \mathbf{K}^{\mathbf{C}P^{n+1}}
\end{equation}
and from the long exact sequence of a pair.  The left-hand side of \eqref{eq:LG} denotes the $K$-homology of the pair $((\mathbf{C}^*)^{n+1},W^{-1}(0))$, which has the same homotopy type as a bouquet of spheres --- one $(n+1)$-sphere for each critical point of $W$.  Note that \eqref{eq:LG} can be seen as a third variant of \eqref{eq:spectrum}, as $((\mathbf{C}^*)^{n+1},W)$ is the Landau-Ginzburg mirror to projective space).

\subsection{$T$-duality}
\label{subsec:T-duality}
Homotopy classes of maps $\Sigma^{-n} \mathbf{K}[X] \to \mathbf{K}^{\hat{X}}$ are naturally identified with classes in the $n$th $K$-cohomology group $\mathbf{K}^n(X \times \hat{X})$.  So if one wants to prove that $\Sigma^{-n} \mathbf{K}[X]$ and $\mathbf{K}^{\hat{X}}$ are isomorphic, one should investigate classes in $\mathbf{K}^n(X \times \hat{X})$.  \S\ref{subsec:Pe-bundle} gives the example at the heart of SYZ --- a distinguished isomorphism class of line bundles on $T \times \hat{T}$ that (regarded as an element of $\mathbf{K}^0(T \times \hat{T})$ induces an isomorphism
\begin{equation}
\label{eq:KUT-duality}
\mathbf{K}[T] \cong \mathbf{K}^{\hat{T}}
\end{equation}
when $T$ and $\hat{T}$ are dual tori.

When $X$ and $\hat{X}$ are mirror Calabi-Yaus of real dimension $2n$, fibering over the same base $B$ with dual torus fibers, this suggests that $\mathbf{K}[X]$ and $\mathbf{K}^{\hat{X}}$ could be identified by a virtual vector bundle on $X \times_B \hat{X}$ whose restriction to each fiber gives \eqref{eq:KUT-duality} --- a ``Poincar\'e bundle.'' The primary obstacle to doing this is that it is not clear what this virtual ``bundle'' should look like on singular fibers.  Indeed it should not be a bundle at all, but a class in $K$-homology $\mathbf{K}_{3n}(X \times_B \hat{X})$ --- this group has a pushforward map to $\mathbf{K}_{3n}(X \times \hat{X})$, which is isomorphic to $\mathbf{K}^n(X \times \hat{X})$ using the $\mathbf{K}$-orientations of $X$ and $\hat{X}$.

Even after discarding the singular fibers, or when they are just absent, there may be a Leray obstruction to finding the Poincar\'e bundle.  In the flat cases of \S\ref{sec:two}, this was simple but not exactly tautological.  At the large volume/large complex structure limit, the singular fibers can disappear, so that every fiber is a smooth torus (though the dimensions of these tori can jump); more precisely one can in some cases \cite{RSTZ} write $X - D$ as the homotopy colimit of a diagram of commutative Lie groups and homomorphisms, and $\hat{X}_0$ as the homotopy colimit of the diagram of dual groups (perhaps orbifolds), in this generality the Leray obstruction might be interesting.

As to singular fibers, it's been known for a long time what the necessary class in $\mathbf{K}_{3n}$ looks like when $n = 2$, by hyperkahler rotating until $X \times_B \hat{X} \subset X \times \hat{X}$ is algebraic \cite{k3,BrMa}.  For higher even $n$, finding these Poincar\'e bundles is a more difficult algebraic geometry problem, even when the same hyperkahler techniques are available \cite{Arinkin, ADM}.  In general, especially for $n$ odd, the class in $\mathbf{K}_{3n}(X \times_B \hat{X})$ cannot be algebraic; it would be interesting to describe it when $X \to B$ and $\hat{X} \to B$ are a dual pair of Gross's ``well-behaved'' singular $T^3$-fibrations \cite{Gross}.

\subsection{Blanc's invariant}
In \cite{Blanc}, Blanc showed how to compute the topological $K$-theory $\mathbf{K}^Y$ of a complex algebraic variety $Y$ in a noncommutative fashion --- that is, Blanc introduced an invariant $\mathbf{K}_{\mathrm{Blanc}}(\mathcal{C}) \in \mathrm{Mod}(\mathbf{K})$ for a $\mathbf{C}$-linear dg category $\mathcal{C}$, and showed
\begin{equation}
\label{eq:blanc}
\mathbf{K}_{\mathrm{Blanc}}(\mathrm{Perf}(Y)) \cong \mathbf{K}^Y
\end{equation}
It is desirable to understand Blanc's invariant for categories arising from symplectic manifolds --- Fukaya categories and microlocal sheaf categories.  When $X$ is compact, K\"ahler with integer K\"ahler class, and Calabi-Yau, then Ganatra has conjectured that $\mathbf{K}_{\mathrm{Blanc}}(\mathrm{Fuk}(X))$ recovers the complex $K$-theory of $X$ whenever $\mathrm{Fuk}(X)$ is smooth and proper.  The last condition is motivated by results of \cite{Toen} (which state that when $Y$ is a compact complex manifold, $\mathrm{Perf}(Y)$ is smooth and proper if and only if $Y$ is algebraic) and the failure of \eqref{eq:blanc} for complex analytic manifolds that are not algebraic.

There is a basic problem with formulating Ganatra's conjecture precisely, or formulating any question about $\mathbf{K}_{\mathrm{Blanc}}(\mathrm{Fuk}(X))$ at all.  The Fukaya category of a symplectic manifold is not automatically defined over the complex numbers, but over a large Novikov field (we will call it $\mathfrak{N}$).

\subsection{Achinger-Talpo and Blanc's invariant for $\mathbf{C}((t))$-linear categories}
\label{subsec:achtal}
The $\mathbf{C}$-linear structure on a dg category $\mathcal{C}$ enters in Blanc's construction in an essential way, but for a compact symplectic manifold it is not usually possible to reduce the linear structure of $\mathrm{Fuk}(X)$ from $\mathfrak{N}$ to $\mathbf{C}$.  Recent work of Achinger-Talpo, and also of Robalo and Antieau-Heller, allow for a definition of $\mathbf{K}_{\mathrm{Blanc}}(\mathcal{C})$ when $\mathcal{C}$ is defined over $\mathbf{C}((t))$ --- this version is adapted to Seidel's relative Fukaya category and to Ganatra's conjecture.

If $\mathcal{O} \subset \mathbf{C}((t))$ is the coordinate ring of an affine curve, and $Y \to \mathrm{Spec}(\mathcal{O})$ is a dominant map of algebraic varieties, then $\mathbf{K}^{Y_a}$ has a local monodromy automorphism (call it $m$) at $t = 0$ whenever $Y_a$ is the fiber above a point $a$ close to $t = 0$.  We seek a computation of the pair $(\mathbf{K}^{Y_a},m)$ that is both noncommutative and formal, in the sense that it depends only on the $\mathbf{C}((t))$-linear category $\mathrm{Perf}(Y \times_{\mathcal{O}} \mathbf{C}((t)))$.  To define such a pair $(\mathbf{K}^{Y_a},m)$ is equivalent to defining a $\mathbf{K}$-module object of the $\infty$-category $\mathcal{S}_{/S^1}$.

For any field $F$, let $\mathrm{MV}_F$ denote the $\infty$-category underlying the Morel-Voevodsky model structure for $\mathbf{A}^1$-homotopy theory \cite[Def. 2.1]{MV}.  Let $\mathrm{MV}_F[(\mathbf{P}^1)^{-1}]$ denote the stable $\infty$-category underlying the Morel-Voevodsky model category of motivic spectra over $F$ (\cite[Def. 5.7]{Voevodsky} or \cite[Def. 2.38]{Robalo}).  If $D$ is an $F$-linear triangulated dg category, let $\mathbf{k}_{\mathrm{mot}}(D) \in \mathrm{MV}_F[(\mathbf{P}^1)^{-1}]$ denote the motivic refinement of the algebraic $K$-theory spectrum (as in \cite[Prop. 3.2]{AnHe}.  An embedding $F \to \mathbf{C}$ induces a functor (preserving direct products and all small colimits)
\[
b^*:\mathrm{MV}_F \to \mathcal{S}
\]
where $\mathcal{S}$ denotes the $\infty$-category of spaces, and a similar functor on spectra that we will also denote by $b^*$ ($b$ for ``Betti'').  When $F = \mathbf{C}$, the Blanc $K$-theory of $D$ is $\mathbf{K}_{\mathrm{Blanc}}(D) := \mathbf{K} \otimes_{\mathbf{ku}} b^*\mathbf{k}_{\mathrm{mot}}(D)$, where $\mathbf{ku}$ denotes the connective complex $K$-theory spectrum.

\begin{thm*}[Achinger-Talpo \cite{AchingerTalpo}]
There is a functor $\mathrm{MV}_{\mathbf{C}((t))} \to \mathcal{S}_{/S^1}$ making the following diagram commute:
\begin{equation}
\xymatrix{
\mathrm{MV}_{\mathbf{C}} \ar[r]^-{b^*}  \ar[d]_-{\times_{\mathbf{C}} \mathbf{C}((t))} & \mathcal{S} \ar[d]^-{\times S^1} \\
\mathrm{MV}_{\mathbf{C}((t))} \ar[r]_-{b_t^*} & \mathcal{S}_{/S^1}
}
\end{equation}
\end{thm*}
The functor $b_t^*$ carries the Morel-Voevodsky space $\mathbf{Z} \times B\mathrm{GL} \in \mathrm{MV}_{\mathbf{C}((t))}$ \cite[p. 138]{MV} representing algebraic $K$-theory to $\mathbf{Z} \times \mathrm{BU} \times S^1$.  It also carries $\mathbf{P}^1$ to $S^2 \times S^1$, and so induces a map to spectra in $\mathcal{S}_{/S^1}$.  Thus one can define the Blanc $K$-theory of a $\mathbf{C}((t))$-linear category $\mathcal{C}$ to be
\begin{equation}
\mathbf{K} \otimes_{\mathbf{ku}} b_t^*\mathbf{k}_{\mathrm{mot}}(\mathcal{C})
\end{equation}

\subsection{Doing without Blanc's invariant}
\label{subsec:dowithout}
Like any spectrum, $\mathbf{K}^Y$ fits into Sullivan's arithmetic square (\cite[Prop. 3.20]{Sullivan} or \cite[Prop. 2.9]{Bousfield})
\begin{equation}
\label{eq:sullivan}
\xymatrix{
\mathbf{K}^Y \ar[r] \ar[d] & \ar[d] \prod_p L_{\hat{p}} \mathbf{K}^Y \\
L_{\mathbf{Q}} \mathbf{K}^Y \ar[r] & L_{\mathbf{Q}} \prod_p  L_{\hat{p}} \mathbf{K}^Y
}
\end{equation}
which is homotopy Cartesian.  Here $L_{\mathbf{Q}}$ denotes the rationalization and $L_{\hat{p}}$ the $p$-completion of a spectrum.  Thomason's descent theorem shows that, when $Y$ is a complex algebraic variety, $L_{\hat{p}}\mathbf{K}^Y$ can be recovered from the algebraic $K$-theory spectrum of $\mathrm{Perf}(Y)$:
\begin{equation}
\label{eq:thomason}
L_{\hat{p}} \mathbf{K}^Y \cong L_{K(1),p} \mathbf{K}_{\mathrm{alg}}(\mathrm{Perf}(Y))
\end{equation}
From this point of view, Blanc's theorem is equivalent to a ``noncommutative'' construction of $L_{\mathbf{Q}} \mathbf{K}^Y$ and of the map $L_{\mathbf{Q}} \mathbf{K}^Y \to L_{\mathbf{Q}} \prod_p L_{K(1),p} \mathbf{K}_{\mathrm{alg}}(\mathrm{Perf}(Y))$.  If one is merely interested in the isomorphism type of $\mathbf{K}^Y$, then Thomason allows it to be recovered from $\mathbf{K}_{\mathrm{alg}}(\mathrm{Perf}(Y))$ only.

If $\mathcal{C}$ is linear over an algebraically closed extension of $\mathbf{C}$, and $p$ is any prime, then $L_{K(1),p} \mathbf{K}_{\mathrm{alg}}(\mathcal{C})$ is a $L_{\hat{p}} \mathbf{K}$-module in a natural way.  So a weaker form of Ganatra's conjecture can be formulated without invoking any form of Blanc's construction, this way: if $X$ is a compact symplectic manifold of dimension $2n$, with a smooth and proper $\mathfrak{N}$-linear Fukaya category, then for every prime $p$ the pair of $L_{\hat{p}} \mathbf{K}$-module spectra
\[
L_{K(1),p} \mathbf{K}_{\mathrm{alg}}(\mathrm{Fuk}(X)) \text{ and } \Sigma^{-n} L_{\hat{p}} \mathbf{K}[X]
\]
are isomorphic.  Maybe it's appropriate to call the desired equivalence of spectra a homological mirror analog of Thomason's \eqref{eq:thomason}.

\subsection{The Euler pairings} Let $\psi^{-1}:\mathbf{K} \to \mathbf{K}$ denote the natural $E_{\infty}$-ring map that carries a virtual vector space to its complex conjugate.  It induces an autoequivalence on $\mathrm{Mod}(\mathbf{K})$, the $\infty$-category of $\mathbf{K}$-modules.

The $2n$-manifolds $X$ and $\hat{X}$ have distinguished $\mathbf{K}$-orientations --- that is, there is a distinguished class in $\mathbf{K}_{2n}(X)$ and in $\mathbf{K}_{2n}(\hat{X})$ that maps to a generator of $\mathbf{K}_{2n}(X,X - x_0)$ and of $\mathbf{K}_{2n}(\hat{X},\hat{X} - x_0)$.  Denote these classes by $[X]$ and $[\hat{X}]$ --- one is determined by the complex structure on $\hat{X}$ and the other by any choice of compatible almost complex structure on $X$.  The action of the line bundle fixes $[X]$ and the action of the monodromy operator fixes $[\hat{X}]$.  They induce a further structure on $\Sigma^{-n}\mathbf{K}[X]$ and $\mathbf{K}^{\hat{X}}$, namely the ``Euler pairings''
\begin{equation}
(\psi^{-1} \Sigma^{-n} \mathbf{K}[X]) \otimes_{\mathbf{K}} \Sigma^{-n} \mathbf{K}[X] \to \mathbf{K} \qquad (\psi^{-1} \mathbf{K}^{\hat{X}}) \otimes_{\mathbf{K}} \mathbf{K}^{\hat{X}} \to \Sigma^{-2n} \mathbf{K}
\end{equation}
Under \eqref{eq:blanc} and the desired equivalence between $\Sigma^{-n} \mathbf{K}[X]$ and the Blanc $K$-theory of $\mathrm{Fuk}(X)$, these maps should be induced by the Hom structures on these categories, suggesting the purely topological problem of choosing \eqref{eq:spectrum} so that the pairings match.
On $\pi_0$ this problem is closely related to Iritani's $\Gamma$-conjectures, or to the rationality question of \cite[\S 2.2.7]{KKP}.

If $M_1$ and $M_2$ are $\mathbf{K}$-module spectra, write $B_n(M_1,M_2)$ for the spectrum of maps from $(\psi^{-1} M_1) \otimes M_2$ to $\Sigma^{-n} \mathbf{K}$.  This is a nondegenerate symmetric bilinear spectrum-valued functor on $\mathrm{Mod}(\mathbf{K})$, it would be interesting to know the $L$-theory of $B_n$.

\subsection{Exact manifolds}
\label{subsec:exact-manifolds}
If $X$ is a Weinstein manifold, a version of the Fukaya category generated by exact Lagrangian submanifolds is naturally defined over any coefficient ring (not just for $\mathfrak{N}$-algebras).  The same is true for the category of sheaves with a microsupport condition (my comfort zone).  In either case the coefficient ring can be taken to be $\mathbf{C}$ and one may apply Blanc's construction without worrying about the Novikov parameter.  I propose the following analogue of Ganatra's conjecture:

\begin{conj*}[Assembly]
Let $Q$ be a $d$-dimensional $\mathrm{Spin}^c$-manifold, let $\Lambda \subset T^* Q$ be a conic Lagrangian, and let $U$ be an open subset of $Q$.  Let $\mathrm{Sh}_{\Lambda}^w(U,\mathbf{C}) \subset \mathrm{Sh}(U,\mathbf{C})$ be Nadler's wrapped variant \cite{N-wrapped} of the category of sheaves with microsupport in $\Lambda$.
\begin{enumerate}
\item There is a natural map
\begin{equation}
\label{eq:conj}
\Sigma^{-d} \mathbf{K}[T^* U,T^* U - \Lambda] \to \mathbf{K}_{\mathrm{Blanc}}(\mathrm{Sh}_{\Lambda}^w(U,\mathbf{C})),
\end{equation}
that is covariantly functorial for open embeddings
\item Whenever $\mathrm{Sh}_{\Lambda}^w(U,\mathbf{C})$ is homologically smooth and proper, \eqref{eq:conj} is an isomorphism.
\end{enumerate}
\end{conj*}

I expect that one can formulate a similar conjecture for the wrapped and partially wrapped Fukaya categories of a Weinstein manifold $X$ --- a natural map
\[
\Sigma^{-d} \mathbf{K}^{\eta}[X,X -\Lambda]) \to \mathbf{K}(\mathrm{Fuk}_{\Lambda}^w(X))
\]
where $\Lambda$ is the skeleton and $\eta$ is a twisting parameter, presumably trivialized on the cotangent bundle of a $\mathrm{Spin}^c$-manifold.

\subsection{String topology}
Known results on homological mirror symmetry for toric varieties \cite{Kuwagaki}, combined with computations like \eqref{eq:LG} give an indurect route to equivalences
\begin{equation}
\label{eq:cannot-hold}
\mathbf{K}_{\mathrm{Blanc}}(\mathrm{Sh}_\Lambda^w(Q;\mathbf{C})) \cong \Sigma^{-d} \mathbf{K}[T^*Q,T^* Q - \Lambda]
\end{equation}
in some examples where $Q$ is a compact torus.  But the case where $Q$ is arbitrary and $\Lambda = Q$ is the zero section (we may call this the ``string topology case'' after \cite{Abouzaid2}) shows that \eqref{eq:cannot-hold} cannot hold in general.  Let us discuss this class of examples in more detail.

If $\Lambda \subset T^* Q$ is the zero section, then $\mathrm{Sh}^w_{\Lambda}(Q;\mathbf{C})$ is naturally equivalent to the category of left dg-modules over
\begin{equation}
\label{eq:COQ}
\mathbf{C}[\Omega Q] := C_*(\Omega Q;\mathbf{C})
\end{equation}
the $\mathbf{C}$-valued chains on the based loop space of $Q$.  This quasi-isomorphism type of this algebra knows the rational homotopy type of $Q$, but nothing more, so one cannot expect to recover from it the $K$-theory of $Q$.

Nevertheless, the \emph{algebraic} $K$-theory of $\mathbf{C}[\Omega Q]$ is a variant of Waldhausen's $A$-theory of $Q$, and is the target of an assembly map \cite[\S 3.2]{Waldhausen}.  More generally, for any ring or ring spectrum $R$ there is a natural map
\begin{equation}
\label{eq:waldhausen}
\mathbf{K}_{\mathrm{alg}}(R)[Q] \to \mathbf{K}_{\mathrm{alg}}(R[\Omega Q])
\end{equation}
Letting $R$ run through $\mathbf{C}$-algebras and taking realizations should produce a map $\mathbf{K}[Q] \to \mathbf{K}_{\mathrm{Blanc}}(\mathbf{C}[\Omega Q])$.   A $\mathrm{Spin}^c$-structure on $Q$ gives an identification of $\mathbf{K}[Q]$ with $\mathbf{K}[T^* Q, T^*Q - Q]$, the domain of \eqref{eq:conj}.

In Waldhausen's setting, the failure of the assembly map to be an isomorphism is very interesting.  When $R$ is the sphere spectrum, the cone on \eqref{eq:waldhausen} (whose codomain is called the $A$-theory of $Q$) is Hatcher's ``Whitehead spectrum''  \cite{Hatcher} that encodes the higher simple homotopy of $Q$, see \cite{Waldhausen} and Lurie's notes available at \url{math.harvard.edu/~lurie/281.html}.  When $R$ is a $\mathbf{C}$-algebra, or anything else, I don't know if there is a similar interpretation.

\subsection{Speculation about the length filtration}
\label{subsec:metric}
I wonder whether one could recover the complex $K$-theory of an exact manifold from a suitable absolute version of the Fukaya category, even if this category is not homologically smooth.  (``Absolute'' means ``not relative,'' i.e. not defined over $\mathbf{C}$ or $\mathbf{C}((t))$ but only over the full Novikov field.)  It would require a version of Blanc's construction that treats the Novikov parameter in a more interesting way than \S\ref{subsec:achtal}--\S\ref{subsec:dowithout}, and one could hope that in this more interesting treatment the assembly map would become an isomorphism.  I will explain what I mean by making an explicit string-topology-style conjecture along these lines.  I have no evidence for it, but I will make some remarks after stating the conjecture.
\medskip

Let $Q$ be a Riemannian manifold, and let $\Omega_{q_0} Q$ be the space of rectifiable loops in $Q$ that start and end at a basepoint $q_0$.  We will treat the basepoint a little more carefully than at the end of \S\ref{subsec:exact-manifolds}, in order to make a point about it later.  The metric endows the chain algebra $\mathbf{C}[\Omega_{q_0} Q]$ \eqref{eq:COQ} with an $\mathbf{R}$-indexed filtration: for each $t \in \mathbf{R}$ we let $F_{<t} \Omega_{q_0} Q \subset \Omega_{q_0} Q$ denote the space of loops of length less than $t$, and put
\[
F_{<t} \mathbf{C}[\Omega_{q_0} Q] := \mathbf{C}[F_{<t} \Omega_{q_0} Q]
\]

\begin{conj*}[Length and $K$-theory]
Let $(Q,q_0)$ and $(Q',q'_0)$ be compact, pointed Riemannian manifolds and suppose that there is a quasi-isomorphism of dg algebras
\begin{equation}
\label{eq:iso-lakt}
C_*(\Omega_{q_0} Q,\mathbf{C}) \cong C_*(\Omega_{q'_0} Q',\mathbf{C})
\end{equation}
that for all $t$ carries $F_{<t} C_*(\Omega_{q_0} Q;\mathbf{C})$ quasi-isomorphically to $F_{<t} C_*(\Omega_{q'_0} Q';\mathbf{C})$
\begin{equation}
\label{eq:iso-lakt-filt}
C_*(\Omega_{q_0} Q;\mathbf{C}) \xrightarrow{\sim} C_*(\Omega_{q'_0} Q';\mathbf{C})
\end{equation}
Then $\mathbf{K}_*(Q) \cong \mathbf{K}_*(Q')$.
\end{conj*}

A suitable Rees construction on the filtered dg algebra $F_{<\bullet}\mathbf{C}[\Omega_{q_0} Q]$ might give an $\mathfrak{N}$-algebra that generates the absolute wrapped Fukaya category of the unit disk bundle in $T^* Q$.  The real conjecture, which I do not know how to formula precisely, is that there is a procedure similar to Blanc's for extracting a $\mathbf{K}$-module from such a category, and that on the Fukaya category of the disk bundle of a Riemannian (or merely Finsler?) $Q$, it outputs the $K$-homology of $Q$.  (In particular, the notion of equivalence used in \eqref{eq:iso-lakt} is stronger than necessary: a Morita-style notion would be more appropriate.  For instance if $q_0$ and $q_1$ are different points of $Q$, there is not likely to be any quasi-isomorphism between $\mathbf{C}[\Omega_{q_0} Q]$ and $\mathbf{C}[\Omega_{q_1} Q]$ that preserves lengths, but the length-filtered space of paths from $q_0$ to $q_1$ could provide the Morita equivalence.)
\medskip

Let us give a reason to doubt the conjecture, followed by something more optimistic.  If $Q$ is simply-connected, one recovers $\mathbf{C}[\Omega_q Q]$, up to quasi-isomorphism, as the cobar construction of the coalgebra of chains on $Q$ \cite[\S 2]{JMoore}.  The cobar construction has a natural filtration which seems to ``coarsely'' recover the legnth filtration on $\mathbf{C}[\Omega_q Q]$, regardless of the metric.  Under the identification with the cobar complex of $\mathbf{C}[Q]$, the loops of metric length $m$ are sandwiched between the cobars of word length $b_1 m$ and $b_2 m$, where $b_1$ and $b_2$ are constants independent of $m$.  So any way of recovering the $K$-theory of $Q$ would require knowledge of the exact numerical values of the breaks in the $\mathbf{R}$-indexed filtration.

These breaks in the length filtration are a kind of homological, based version of the length spectrum of the metric.  The genuine length spectrum is known to recover the Laplace eigenvalues of $Q$, if the metric is generic \cite{DuistermaatGuillemin}.  Bergeron and Venkatesh have observed that similar spectral data can see a little bit of the homotopy type of $Q$ beyond the rational homotopy type \cite{BeVe}.  Specifically the Cheeger-Muller theorem gives a formula for the alternating product
\begin{equation}
\label{eq:tors-prod}
\prod_i \#\mathrm{tors}H^i(Q;\mathbf{Z})^{(-1)^i}
\end{equation}
in terms of the Laplace-de Rham eigenvalues and the volumes of the images of $H^i(Q;\mathbf{Z})$ in the spaces of harmonic $i$-forms.  In another ``coarse'' sense, (perhaps a related one?) these eigenvalues are given by the Weyl law --- it is their exact numerical values that are needed to recover \eqref{eq:tors-prod}.
\medskip

\noindent
{\bf Example.}  Let $Q$ be a nontrivial $\mathrm{SU}(2)$-bundle over $S^4$.  Any degree one map $Q \to S^7$ induces a quasi-isomorphism
\begin{equation}
\label{eq:S7example}
\mathbf{C}[\Omega_{q_0} Q] \cong \mathbf{C}[\Omega_{x_0} S^7].
\end{equation}
But if the Chern class of the bundle is $m \geq 2$, there is a little bit of torsion in the $K$-theory of $Q$: $\mathbf{K}_1(Q) = \mathbf{Z} \oplus \mathbf{Z}/m$ (while $\mathbf{K}_0(Q) = \mathbf{Z}$, and $\mathbf{K}_0(S^7) = \mathbf{K}_1(S^7) = \mathbf{Z}$).  The conjecture predicts that there is no metric on $Q$ for which \eqref{eq:S7example} preserves the length filtration.  The possibly spurious comparison made in the remarks above is that, since $\eqref{eq:tors-prod} = m$, the Laplace-de Rham spectra of $Q$ and of $S^7$ are never exactly the same for any choice of metrics.

\subsection*{Acknowledgements}
I thank Mohammed Abouzaid and Sheel Ganatra for sharing their ideas about the $K$-theory of Fukaya categories, Piotr Achinger and Mattias Talpo for their paper about $\mathbf{C}((t))$-schemes, and Nick Addington and Ben Antieau for some corrections and other advice.  I have also benefited from discussions with Ron Donagi, Mauricio Romo, Paul Seidel, Jake Solomon, Semon Rezchikov, and Arnav Tripathy.  I was supported by NSF-DMS-1811971.


\section{Introduction}
Materials with coupled magnetic and dielectric properties have become the topic of intensive research in recent times. Double perovskite material have shown potential as promising candidate in this research field and have exhibited novel phenomenon such as ferromagnetism, magneto-dielectric, multiferroic, magnetoelectric, ferromagnetic semiconductors etc.\cite{ndi, vilar, yang, dass, ndia, de, ghara, ilyas4} However, the magnetic and electric coupling is rather weak and appears at low temperature thus making it difficult to use for application purpose. The ferromagnetic ordering along with strong insulating behavior of some 3$d$ based double perovskite is eye caching and can exhibit exotic phenomenon.\cite{fuh, azu}

Recently, several double perovskites have been extensively studied and have exhibited exotic phenomena. For instance, partially disordered ferromagnetic semiconductor La$_2$NiMnO$_6$ reportedly exhibited colossal magnetodielectricity and multiglass properties.\cite{choudhary} This material also exhibits magnetocapacitance close to room temperature\cite{rogado} and exhibit novel behavior with doping.\cite{hoss2} Lu$_2$CoMnO$_6$ with up-up-down-down ($\uparrow\uparrow\downarrow\downarrow$) magnetic structure exhibits ferroelectricity which originates from the exchange-striction mechanism.\cite{zhang} La$_2$Mn(Ni/Fe)O$_6$ shows ferromagnetic ordering and high refrigerant capacity.\cite{gau} In these compounds the Ni/Mn are ferromagnetically ordered via super exchange interactions mediated by oxygen. Materials with Co/Mn at B$^\prime$/B$^\prime\prime$ are again an interesting class of double perovskite materials with interesting physical properties. Double provskit material like La$_2$CoMnO$_6$ shwows ferromagnetic behavior around $\sim$220 K and shows magnetocapacitance and these properties draw the interest of researches to investigate Co/Mn based perovskites.  Tb$_2$CoMnO$_6$ is an anisotropic magnetic material and shows giant rotating magnetocaloric effect \cite{moon}. Further, pyroelectric/magnetoelectric properties along with ferromagnetic transition have been observed for (Y/Er)$_2$CoMnO$_6$.\cite{bisco1, bisco2} Structural and magnetic properties have been studied in single crystal (Ho,Tm,Yb,Lu)$_2$CoMnO$_6$ materials using neutron diffraction and DC magnetization.\cite{bisco3} Mean field type ferromagnetism and occurrence of Griffith phase has also been observed in nano-crystaline Pr$_2$CoMnO$_6$.\cite{ilyas1, ilyas2} Raman study on bulk (Pr, Ho)$_2$CoMnO$_6$ reveals strong spin-phonon coupling in these similar materials.\cite{wliu, ilyas3, silva} Further, in bulk double perovskites dielectric properties have been reported and found thermally activated relaxation mechanism and non-debye's behavor.\cite{nath,jwchen, ilyas3, jgjwang} Despite so many interesting properties of Co/Mn based double perovskites these materials have not been studied in much detail especially in thin films and nano-crystalline form.

Here we have chosen Sm$_2$CoMnO$_6$ to investigate its physical properties in nano crystalline form. It is worth noting that the charge state Co$^{2+}$ and Mn$^{4+}$  give an ordered monoclinic phase structure whereas $Co^{3+}$ and $Mn^{3+}$ would result in B-site disordered phase. Further, in ordered phase of Sm$_2$CoMnO$_6$ the Co$^{2+}$-O-Mn$^{4+}$ would couple via superexchange interaction and give rise to a ferromagnetic ordering.\cite{good} Whereas for disordered phase moments favour antiferromagnetic spin arrangment. Furthermore, it is expected that with decreasing ionic radii of rare earth ions at A-site the structrual modification will take place and magnetic ordering temperature decreases. Sm$^{3+}$ ion is smaller than the La$^{3+}$  and Pr$^{3+}$ ions we expect reduced magnetic phase transition temperature.\cite{vasi} Further we investigate nano-crystalline Sm$_2$CoMnO$_6$ sample so the effect of grain size will also effect the physical properties specially dielectric behavior of sample.

In this paper, we have studied nano-crystalline sol-gel prepared Sm$_2$CoMnO$_6$. Structural, magnetic, dielectric and transport properties have been studied in detail. Our aim in to investigate the physical properties of nano-crystalline Sm$_2$CoMnO$_6$. Further, we will compare the obtained result to results obtained for bulk material in earlier literature. Magnetic ordering, spin-phonon coupling and dielectric response are much focused in this study.

\begin{figure}[th]
	\centering
		\includegraphics[width=8cm, height=12cm]{Fig1.eps}
\caption{(Color online) Comparative x-ray diffraction analysis by Rietveld refinement for nano-crystalline Sm$_2$CoMnO$_6$ (a) orthorhombic, (b) monoclinic phase. (c) SEM micrograph for nano-crystalline Sm$_2$CoMnO$_6$.}
	\label{fig:Fig1}
\end{figure}

\section{Experimental details}
The nano-crystalline sample of Sm$_2$CoMnO$_6$ was prepared by the sol-gel method as adopted in earlier work.\cite{ilyas1}  Sm$_2$O$_3$, Co(NO$_3$)$_2$.6H$_2$O and C$_4$H$_6$MnO$_4$.4H$_2$O  were taken in the stoichiometric ratio and dissolved in water. Sm$_2$O$_3$ is insoluble in the water, so we added HNO$_3$ drop by drop with continuous stirring on a magnetic stirrer with heating 70 $^o$C temperature until a clear solution is obtained. These solutions were then poured into a 400 ml beaker half-filled with the citric acid solution in water. The beaker with the obtained solution was then placed on the magnetic satire at 90 $^o$C temperature for 24 hours until the gel is formed. Then the temperature was increased to 200 $^o$C to dry up the gel. The sample was then collected from the beaker and crushed into fine powder by grinding in mortar and pestle. For the phase purity of the sample, the fine powder was then heated in a furnace at 800 $^o$C and 900 $^o$C for 12 hours.

The x-ray diffraction was taken for structural characterization and to confirm the phase purity of the sample using Rigaku made differectometer MiniFlex600 . XRD data was collected at room temperature in 2$\theta$ range 10$^o$ -90$^o$ with step size of 0.02$^o$ and scan rate of 2$^o$/min. Crystal structural analysis was done by  Rietveld refinement of XRD data using Fullprof program. The XPS measurements were performed with base pressure in the range of $10^{-10}$ mbar using a commercial electron energy analyzer (Omnicron nanotechnology) and a non-monochromatic Al$_{K\alpha}$ X-ray source (h$\nu$ = 1486.6 eV). The XPSpeakfit software was used to analyze the XPS data. The samples used for XPS study are in pallet form where an ion beam sputtering has been done on the samples to expose clean surface before measurements. Magnetic data was collected by Physical Properties Measurement System by Cryogenic Inc. Labram-HR800 micro-Raman spectrometer with diode laser having wavelength ($\lambda$) = 473 nm have been used to record the temperature dependent Raman spectra. This spectrometer use grating with 1800 groves/mm and CCD detector with a high resolution of $\sim$ 1 cm$^{-1}$. A THMS600 stage from Linkam UK have been used for temperature variation with the stability of $\pm$0.1 K for the low temperature Raman measurements. Dielectric measurements in the frequency range from 1 Hz to 5.6 MHz were performed using a computer controlled dielectric spectrometer.

\section{Result and Discussions}

\subsection{Structural study}
X-ray study was done for structural characterization of Sm$_2$CoMnO$_6$ and data was recorded.  Further detailed structural study of the sample was done by Rietveld refinement of XRD data using Fullprof program.  XRD patterns along with Rietveld refinement for orthorhombic and monoclinic phases are shown in the Fig. 1a and 1b. We have performed Rietveld refinement of XRD data for both orthorhombic and monoclinic crystal structures however the fitting for orthorhombic phase is not as good as for monoclinic. The R-factor R$_{exp}$, R$_{wp}$  and $\chi^{2}$ for orthorhombic structure comes out 15.0, 19.7 and  1.78 respectively. However, the rietveld refinement of XRD pattern with monoclinic phase gives reasonable good fitting  with R$_{wp}$, R$_{exp}$ and $\chi^{2}$  values as 18.6, 14 and 1.5 respectively. These values are acceptable and shows sample is chemically pure and single phase.\cite{bhatti1, bhatti2} Further, results confirm Sm$_2$CoMnO$_6$ sample adopt monoclinic structure with \textit{P2$_1$/n} space group. The sample has unit cell parameters are $a$ = 5.330624 $\AA$, $b$ = 5.498045 $\AA$, $c$ =  7.566482 $\AA$ and $\beta$ is  89.98(5) $^o$ with unit cell volume is 221.75 $\AA^3$. Further, to obtain the particle size in the nano-crystalline sample of Sm$_2$CoMnO$_6$ we have performed the scanning electron microscopy. Fig. 1c shows the SEM image obtained for the nano-crystalline structure. The SEM image is analyzed using ImageJ software. We observed that the average crystallite size for present compound is $\sim$84.17 nm.

\begin{figure}[t]
	\centering
		\includegraphics[width=8cm, height=12cm]{Fig2.eps}
\caption{(Color online) (a) The XPS core level spectra of Co 2$p$. (b) XPS core level spectra of Mn 2$p$. (c) XPS core level spectra of Sm 3$d$. In the figure the red solid line is the overall envolp of the XPS spectrum and the other colored solid lines are the other respective fitted peaks.}
	\label{fig:Fig2}
\end{figure}

\subsection{X-ray photo-electron spectroscopy (XPS)}
The physical properties of a compound is largely described by the oxidation state of cations present in material. XPS is a vital tool to understand the charge state of cations. We have employed the XPS to study the cationic charge state of Co, Mn and Sm in Sm$_2$CoMnO$_6$. XPS spectrum of Co 2$p$ is shown in Fig. 2a, it is evident that there are two peaks located at 780.7 eV and 796.01 eV for CO 2$p$$_{3/2}$ and Co 2$p$$_{1/2}$  respectively resulting from spin orbital splitting of 2$p$ orbital with 15.3 eV. Beside the Co 2$p$ peaks two satellite peaks have also been observed close to Co 2$p$ peaks and are in agreement resulting with in literature.\cite{wang, qiu, xia}  The peaks locations of Co 2$p$ core level indicates that the Co cations are presents in +2 oxidation states.

The measured XPS spectrum for Mn 2$p$ core levels along with peak fitting is shown in Fig. 2b. The Mn 2$p$ spectrum shows two distinct peaks located at 642 eV and 654 eV corresponding to Mn 2$p$$_{3/2}$ and 2$p$$_{1/2}$ resulted from spin orbital splitting of 2$p$ orbitals with splitting energy of 12 eV. There are two small peaks at the higher energy side is observed in fit the data and believed due to satellite correction. The resulting corroborates with the reported literature.\cite{ida, sachoo, cao} The peak position reveals that the Mn cation is present in +4 oxidation state.

XPS spectra of Sm 3$d$ core level along with the fitting of peaks are shown in Fig. 2c. It is quite evident from the figure that the two distinct spin-orbital split peaks Sm 3$d$$_{5/2}$ and Sm 3$d$$_{3/2}$ are located at 1082.16 eV and 1109.33 eV respectively with at spin-orbital splitting energy of 27.17 eV. The detail analysis of XPS spectrum reveals that the Sm cation are present in +3 oxidation states as reported in literature.\cite{qliu, duan} Thus the XPS study reveals the oxidation state and the spin orbital splitting energy of the compounds present in the material. The observed oxidation state Co$^{2+}$, Mn$^{4+}$ and Sm$^{3+}$ suggests the crystallization of ordered phase of Sm$_2$CoMnO$_6$. Further, in the XPS data we do not see any contribution from other cationic valencies corresponding to the peak positions for Mn$^{3+}$ or Co$^{3+}$ unlike seen in the literature.\cite{qiu, xia, sachoo, cao, haung}

\begin{figure}[th]
	\centering
		\includegraphics[width=8cm]{Fig3.eps}
\caption{(Color online) (a) Temperature dependent magnetization data $M(T)$ shown for Sm$_2$CoMnO$_6$ measured at different fields. (b) $M(T)$ data plotted in terms of inverse susceptibility $\chi^{-1}$, solid line is fitting due to modified Curie Weiss Law and dotted line shows conventional Curie Weiss Law fit. Inset shows dM/dT vs $T$ plot showing $T_c$ for Sm$_2$CoMnO$_6$.}
	\label{fig:Fig3}
\end{figure}

\begin{figure}[t]
	\centering
		\includegraphics[width=8cm]{Fig4.eps}
\caption{(Color online) Isothermal magnetization $M(H)$ data collected at 2 K in applied field of $\pm$ 50 kOe for Sm$_2$CoMnO$_6$.}
	\label{fig:Fig4}
\end{figure}

\subsection{Magnetization study}
Fig. 3a shows the temperature dependent magnetization ($M(T)$) data measured in both zero field cooled ($ZFC$) and field cooled ($FC$) mode. $ZFC$ is measured in 300 K to 2 K range in an applied field of 100 Oe whereas $FC$ data is measure at three different applied magnetic fields (see Fig. 3a). It is quite evident that with decreasing temperature the magnetic moment ($M$) in $M(T)$ is steady till 160 K, however, with further decrease in temperature the magnetic moment begin to rise sharply. It is well known that the ordered Sm$_2$CoMnO$_6$ double perovskite is expected to exhibit ferromagnetic ordering. Since, from XPS study we observed that Mn and Co cations are in +4 and +2 oxidation states respectively with outer electronic configuration Mn$^{4+}$ (t$_{2g}^{3}$e$_{g}^{0}$), Co$^{2+}$ (t$_{2g}^{5}$e$_{g}^{2}$) these cations take part in superexchange interaction and are ferromagnetically ordered.  The sharp rise in moment below 160 K is marked by paramagnetic to ferromagnetic phase transition in Sm$_2$CoMnO$_6$. With further decrease in temperature, large bifurcation in $M_{ZFC}$ and $M_{FC}$ appears. $M_{ZFC}$ curve shows a peak like behavior around $T_c$ as shown in Fig. 3a, with further decreasing temperature the moment remain steady. However, $M_{FC}$ shows a typical ferromagnetic like feature. The nano-crystalline Sm$_2$CoMnO$_6$ shows a magnetic phase transition from PM-FM with T$_c$ $\sim$148 K, as obtained from point of inflection in $dM/dT$ vs $T$ plot shown in inset Fig. 3b. We have measured $M_{FC}$ at different applied field and observe that with increasing field, $T_c$ shifts to higher field which is due to forced spin arrangement in field direction.

To understand the magnetic behavior in further detailed we have plotted magnetization data in terms of temperature dependent inverse magnetic susceptibility i.e. $\chi^{^{-1}}$ vs $T$ as shown in Fig. 3b. In paramagnetic region close to above $T_c$ we observe that the inverse magnetic susceptibility ($\chi^{^{-1}}$) shows deviation from linearity unlike as expected for paramagnetic system by Curie Weiss law (see dotted line). However, such deviation in $\chi^{^{-1}}$ close to $T_c$  is observed and expected for double perovskite compounds which can be understood by using modified Curie Weiss law described as:\cite{booth}

\begin{eqnarray}
\chi = \frac{C_{TM}}{T - \theta_{TM}} + \frac{C_{RE}}{T - \theta_{RE}}
\end{eqnarray}

where $C_{TM}$, $C_{RE}$ and $\theta_{TM}$,  $\theta_{RE}$ are the Curie Constants and paramagnetic Curie temperatures respectively, here subscript TM and RE represent transition metal ions and rare earth ions. We have fitted the susceptibility data of  Sm$_2$CoMnO$_6$ at high temperature above $T_c$. The fitting parameters obtained from fitting of $\chi^{-1}$ with Eq. 1 in Fig. 3b were used to calculate Curie constant and $\theta_P$. The values obtained for $C_{TM}$ and  $\theta_{TM}$ for Sm$_2$CoMnO$_6$ are 3.081 emu K$^{-1}$ mole$^{-1}$ Oe$^{-1}$ and 146.96 K respectively. The large positive value of $\theta_P$ signifies that the ferromagnetic ordering is present in the sample. Further, effective magnetic moment is calculated using formula  $\mu_{eff}$ = $\sqrt{3 C_{TM} k_{B}/N}$ where $C_{TM}$ is obtained from modified Curie Weiss fitting in Fig. 3b. The value of $\mu_{eff}$ is 4.91474 $\mu_B$/f.u. Further, from fitting we have obtained the $C_{RE}$ value for rare earth ions as 15.953 emu K$^{-1}$ mole$^{-1}$ Oe$^{-1}$ which is close to the value of Sm$^{3+}$ free ions.

Isothermal magnetization $M(H)$ data have been collected at 2 K up to $\pm$50 kOe applied magnetic field as shown in Fig. 4. $M(H)$ curve shows hysteresis which is signature of ferromagnetic ordering. However the $M(H)$ curve is not symmetric. Further the magnetic moment does not show any signature of saturation even at highest applied magnetic field of 50 kOe. The magnetic moment at 50 kOe is about 3.1 $\mu$$_B$/f.u. where as the remanent magnetization and coercive force is 1.7 $\mu$$_B$/f.u. and 10 kOe respectively.

Reducing particle size in nano-scale does effect the magnetic properties of compounds in great deal. Many materials with particles dimensions at nano-scale shows drastic change in magnetic properties in comparison to their bulk form, especially manganites show superparamagnetism, spin-glass behavior, core-shall spin structure etc.\cite{tzhang} It is well established that the reducing particle size effect unit cell dimensions, which effect the Metal-Oxygen bond length and bond angles hence effect the magnetic properties.\cite{rer}  Finally, to understand the effect of nano-size on magnetism we made a comparison of the magnetic parameters with bulk as reported in literature.\cite{nath} It is found that in nano form Sm$_2$CoMnO$_6$ sample have large Curie temperature 146.96 K whereas for bulk it is found to be around 122 K \cite{nath}. The experimentally obtained value of effective magnetic moment found to be lower than bulk as reported.\cite{nath} The effect on magnetic properties can be described due to surface ferromagnetism on small nano particles as found in case of doped perovskite.\cite{hoss1}

\begin{figure*}[t]
	\centering
		\includegraphics[width=14cm]{Fig5.eps}
\caption{(Color online) (a) Raman spectra of Sm$_2$CoMnO$_6$ measured at different temperatures.   (b) shows the line shape and its Lorentzian fitting of A$_{1g}$ and B$_{2g}$ Raman modes at 496 and 636 cm$^{-1}$ respectively.}
	\label{fig:Fig5}
\end{figure*}

\begin{figure}[t]
	\centering
		\includegraphics[width=8cm]{Fig6.eps}
\caption{(Color online) Temperature Variation of (a) Raman shift (b) FWHM for Raman mode at 636 cm$^{-1}$ corresponding to stretching of Co/MnO$_6$ for Sm$_2$CoMnO$_6$. The solid line is fitting due to Eq. 2}
	\label{fig:Fig6}
\end{figure}

\subsection{Temperature dependent Raman study}

Fig. 5a shows the temperature dependent Raman spectra taken at selective temperatures across magnetic transition. The Raman spectrum are taken at 10 K, room temperature and across the magnetic transition at close temperature intervals. It is evident from the Fig. 5a the important feature in the Raman spectra are the prominent Raman modes at 636 and 496 cm$^{-1}$ corresponding to B$_{2g}$ stretching mode and A$_{1g}$ breathing mode respectively. These Raman modes are due to stretching, bending and rotation of (Co/Mn)O$_6$ octhadera. It is known from the theoretical lattice dynamics that the strong sharp peak at 636 cm$^{-1}$ originates from a symmetric stretching of the (Co/Mn)O$_6$ octahedra, whereas the band at around 496 cm$^{-1}$ describes a mixed type vibration of antisymmetric stretching and bending.\cite{ilive} Additionally, the modes at $\sim$1278 cm$^{-1}$  represent the second-order overtones of the breathing mode.\cite{meyer} The temperature dependent Raman spectra shows modification in intensity of peaks well as their peak positions corresponding to each modes. Further, it is well known that the behavior of ordered and disordered phases in double perovskite can be understand using Raman spectra, such detail study have been carried out on La$_2$CoMnO$_6$ where mode at 636 cm$^{-1}$ seems prominent and assigned to a monoclinic P2$_1$/n structure.\cite{ilive, meyer} The present study shows similar frequency results.

To understand the presence of spin phonon coupling in present compound, we have analyzed the Raman data of present series with the following anharmonic decay model:\cite{harish}

\begin{eqnarray}
\omega(T) = \omega{_0} - A\left[1 + \frac{2}{exp\left(\frac{\hbar\omega_0}{2k_BT}\right) - 1}\right]
\end{eqnarray}

\begin{eqnarray}
\Gamma(T) = \Gamma{_0} - B\left[1 + \frac{2}{exp\left(\frac{\hbar\omega_0}{2k_BT}\right) - 1}\right]
\end{eqnarray}

where $\omega_0$ and $\Gamma_0$ are the intrinsic frequency  and line width of the optical mode, A and B are the anharmonic coefficients. $\omega(T)$  and $\Gamma(T)$ describes expected temperature dependence of a phonon mode frequency and line width  due to anharmonic phonon-phonon scattering.

The temperature dependent Raman spectra shown in Fig. 5a have been analyzed using Lorentz functions. From the fitting of these spectra, peak position of Raman modes and line width have been obtained. Fig. 6a shows the temperature dependent peak position of Raman mode corresponding to stretching mode. The mode frequency increases with decreasing temperature, and shows a deviation around 150 K which is magnetic phase transition and further increases down to lowest measured temperature. To understand further we have fitted the phonon mode frequency with Eq.2 anharmonic decay model (solid red line in Fig 6a). We observed that around $T_c$ phonon frequency shows a deviation from anharmonic behavior and such behavior is due to spin phonon coupling present in materials. Such results have been reported in many other materials where at magnetic transition the material shows strong deviation from anharmonic behavior.\cite{sandi, grana, lave} Similarly, the line width as a function of temperature is plotted and shown in Fig. 6b. The line width is also fitted with the anharmonic model given in Eq. 2. It is found that the line width shows a deviation across $T_c$ in a similar fashion as mode frequency does. The deviation of mode frequency and line width form anharmonic behavior around magnetic ordering temperature is due to the additional scattering resulting from spin-phonon interaction.

\begin{figure*}[th]
	\centering
		\includegraphics[width=14cm]{Fig7.eps}
\caption{(Color online) Temperature dependent (a) real part of complex dielectric permittivity ($\epsilon$$^\prime$) (b) loss tangent (tan $\delta$) measure for Sm$_2$CoMnO$_6$ in the temperature range of 20 K to 300 K at various frequencies.}
	\label{fig:Fig7}
\end{figure*}

\begin{figure}[th]
	\centering
		\includegraphics[width=8cm]{Fig8.eps}
\caption{(Color online) Variation of relaxation time against normalized temperature i.e ln $\tau$ vs 1000/T obtained from tengant loss in Fig. 7(b).}
	\label{fig:Fig8}
\end{figure}

\subsection{Dielectric study}

Fig. 7a and 7b represent the temperature dependent real and imaginary part of complex dielectric permittivity $\epsilon^{\prime}$ and tan $\delta$  respectively measured in the temperature range 20 K to 300 K at different frequencies for Sm$_2$CoMnO$_6$. Further, for relaxor systems with relaxation mechanisms, each relaxation component will correspond to plateaus in $\epsilon^{\prime}$(T) and respond with peaks in tan $\delta$. For this material we have observed that with increasing temperature the $\epsilon^{\prime}$ increases,  at low temperature the $\epsilon^{\prime}$ increases slowly however with increasing temperature $\epsilon^{\prime}$ increases sharply. The high response of dielectric permeability is due to grain boundary effect.\cite{jwchen, mansuri}  Further, with increasing frequencies $\epsilon^{\prime}$ decreases sharply. The higher value of $\epsilon^{\prime}$ at low frequency is attributed to the accumulation of the charges at grain boundaries  On careful observation of tangent loss curve tan $\delta$ it is clearly seen that there is a broad hump at low temperatures which is feature of relaxor phenomenon. The observed relaxation is frequency dependent and shift to higher temperature with increasing frequency. This clearly indicates that the relaxation peaks in the dielectric loss is due to thermally activated mechanisms. The resonance condition defined as $\omega_p$$\tau_p$ = 1 where $\omega$ = 2$\pi$$f$ is defined as resonance frequency.

The relaxation mechanisms and its origin can be analyzed by fitting the peaks in tan $\delta$ with the Arrhenius law given as follow:
\begin{eqnarray}
 \tau_{tan \delta} = \tau_0 exp(\frac{-E_{\alpha}}{K_B T})
\end{eqnarray}
\begin{eqnarray}
 \tau_{tan \delta} = \frac{1}{2 \pi f_{tan \delta}})
\end{eqnarray}

 where, $T$ is the temperature where peak occurs in tangent loss curve at a particular frequency $f_{tan \delta}$, $\tau_0$ and $E_{\alpha}$ are characteristic relaxation temperature and activation energy respectively and, $k_B$ is the Boltzmann constant. Fig. 8 shows the plot of dielectric loss peaks as a function of absolute temperature. From the fitting parameters of the data using Eq. 4 we have calculated activation energy $E_{\alpha}$ = 0.14 eV.

\begin{figure*}[th]
	\centering
		\includegraphics[width=14cm]{Fig9.eps}
\caption{(Color online) Frequency dependent (a) real part of complex dielectric permittivity ($\epsilon$$^\prime$) (b) loss tangent (tan $\delta$) measure for Sm$_2$CoMnO$_6$ at various temperatures between 25 K and 300 K in the frequency range 1 Hz to 5.6 MHz.}
	\label{fig:Fig9}
\end{figure*}

To further understand the dielectric response we have measured the frequency dependent $\epsilon$$^\prime$ and tan $\delta$ over the frequency range 1 Hz to 5.5 MHz for Sm$_2$CoMnO$_6$ at different temperatures. In the Fig. 9a we observed that Sm$_2$CoMnO$_6$ exhibits high dielectric constant at low frequency and at high temperature. The dielectric spectrum shown in Fig. 9a clearly shows two plateaus well separated by dispersion. The separate plateau in Fig. 9a are attributed to static and optical dielectric constant. Further, the frequency dependent dielectric constant shows the dispersion which moves to higher frequency with increasing temperature. In Fig. 9b we have shown the tan $\delta$ as a function of frequency at different temperature in the range of 20 K to 300 K. It is observed that at low frequency the loss is high which decreases with increasing frequency and decreasing  temperature.

\subsection{Impedance spectroscopy}

Impedance spectroscopy is vital and informative technique to understand and distinguish the contributions to the electric and dielectric properties from  grain, grain boundaries and electrode-sample contact interface. Various involve in relaxation mechanisms can be identified by plotting impedance in a complex plan at various temperature. The complex impedance describe by the equation:\cite{fang}

\begin{eqnarray}
Z^* = Z^\prime + jZ^{\prime\prime}
\end{eqnarray}

\begin{eqnarray}
 Z^\prime = \frac{R}{1 + (\omega\tau)^2}
\end{eqnarray}

\begin{eqnarray}
Z^{\prime\prime} = \frac{\omega R \tau}{1 + (\omega\tau)^2}
\end{eqnarray}
where Z$^*$ is complex impedance, Z$^\prime$ and Z$^\prime\prime$ are real and imaginary parts of impedance respectively. R is resistance, $\omega$ is angular frequency and $\tau$ is relaxaton time.
Fig. 10a shows the real part of complex impedance ($Z^{\prime}$) plotted as a function of frequency in the frequency range 1 Hz to 5.6 MHz at various temperatures between 50 K to 300 K. For clarity both the axis are in logarithmic scales. It is quite evident from the figure that $Z^{\prime}$ decreases with increasing temperature. At low temperature $Z^{\prime}$ gradually decreases with increasing frequency, however at temperatures above 100 K, $Z^{\prime}$ initially remains independent of frequency then at higher frequency it began to decreases. Further, the frequency independent region moves to higher frequency with increasing temperature. Further, it is observed that at higher frequency and high temperature $Z^{\prime}$ is almost similar. This feature is possibly due to the release of accumulated space charges at high temperatures hence contribute to the enhancement of conduction in this material at high temperature. Imaginary part of impedance ($Z^{\prime\prime}$) is shown in Fig. 10b for wide frequency range. $Z^{\prime\prime}$ shows a peak which attain $Z^{\prime\prime}_{max}$ for all the curves measured at different temperatures, further it is evident that the peak moves towards higher frequency with increasing temperature. The peak shift to higher frequency with increasing temperature suggests that the relaxation time constant decreases with increasing temperature.

\begin{figure*}[th]
	\centering
		\includegraphics[width=14cm]{Fig10.eps}
\caption{(Color online) (a) Frequency dependent real part of impedance $Z^{\prime}$ measure at different temperatures. (b) Frequency dependent imaginary part of impedance $Z^{\prime\prime}$ measure at various temperatures. (c), (d) real $Z^{\prime}$ and imaginary part $Z^{\prime\prime}$ plotted in terms of Nyquist plot $Z^{\prime}$ vs $Z^{\prime\prime}$.}
	\label{fig:Fig10}
\end{figure*}

\begin{figure}[th]
	\centering
		\includegraphics[width=8cm]{Fig11.eps}
\caption{(Color online) Variation of relaxation time against normalized temperature i.e ln $\tau$ vs 1000/T obtained from Z$^\prime\prime$ plot.}
	\label{fig:Fig11}
\end{figure}

We know that the most probable relaxation time ($\tau$) can be determined for relaxation system by identification of the position of the loss peak in the $Z^{\prime\prime}$ vs log ($f$) plots using the relation:

\begin{eqnarray}
\tau = \frac{1}{\omega} = \frac{1}{2 \pi f}
\end{eqnarray}

where $\tau$ is relaxation time and $f$ is the relaxation frequency.
To further understand the relaxation behavior we have plotted the relaxation time $\tau$ vs inverse temperature 10$^3$/$T$ (K$^{-1}$). Fig. 11 shows the temperature variation of $\tau$, it is observed that the relaxation time follows Arrhenius behavior given as:

\begin{eqnarray}
\tau_b = \tau_0 exp \left( \frac{-E_\alpha}{k_BT} \right)
\end{eqnarray}

where $\tau_0$ is the pre-exponential factor, k$_B$ the Boltzmann constant and $T$ the absolutes temperature. The peak frequency $f_p$ obtained from the peak values of Z$^{\prime\prime}$ in Fig. 10b is plotted as relaxation time as a function of absolutes temperature shown in Fig. 11. The data is very well fitted with the Eq. 10 Arrhenius law for thermally activated relaxation mechanism. From the fitting parameters the activation energy (E$_{\alpha}$) have been calculated and is found to be 0.17 eV.

Fig. 10c and 10d shows the Z$^{\prime}$ vs Z$^{\prime\prime}$ in the form of Nyquist plots at some selective temperatures measure in the wide frequency range 1 Hz to 5.6 MHz. It is quite evident from the figures that the plot gives the semicircle in whole range of temperature. But the circle are compressed and their center are not on Z$^\prime$ axis. Which shows that the system is deviates from deal Debye's model, because for ideal Debye system the semicircles are supposed to have their Z$^\prime$ axis. The Nyqust plot give lot of information about grains, grain boundaries and electrode effect etc.  Further, we observed the semicircle in Nyquist plot shows a decrease in radius with increasing temperature which reflect that the resistivity decreases wth increasing temperature. Further the depression in semicircle are due to polarization effect. The non-Debye behavior account for grain boundaries, grain, stress and strain present in material. In this case we observed only one depress semicircle which indicate that the grain effect deviate the system from deal Debye behavior.

\begin{figure*}[th]
	\centering
		\includegraphics[width=14cm]{Fig12.eps}
\caption{(Color online) (a) Variation of real part of electrical modulus ($M^\prime$) with temperature. (b) Imaginary part of electrical modulus $M^{\prime\prime}$ as a function of temperature.}
	\label{fig:Fig12}
\end{figure*}

\begin{figure}[th]
	\centering
		\includegraphics[width=8cm]{Fig13.eps}
\caption{(Color online) Variation of relaxation time against normalized temperature i.e ln $\tau$ vs 1000/T obtained from M$^\prime\prime$ plot.}
	\label{fig:Fig13}
\end{figure}

\subsection{Electric modulus}
Information of interface polarization, relaxation time, electrical conductivity and grain boundary
conduction effects etc can be deduced from electrical modulus of materials. Figs. 12a and 12b show the temperature dependent real (M$^{\prime}$) and imaginary (M$^{\prime\prime}$) part of electrical modulus obtained at selective frequencies for Sm$_2$CoMnO$_6$ in the temperature range of 20 K to 300 K.

Fig. 12a shows variation of real part of electrical modulus as a function of temperature at different frequencies. It is observed that M$^\prime$ ncreases wth decreasing temperature at low temperatures all the curves of M$^\prime$ measured at different frequencies merge in to one curve. Similar feature is observed at high temperatures also. There is large dispersion in M$^\prime$ at intermediate temperatures around 150 K which may be activated by magnetic ordering. The M$^\prime$ increases with increasing frequency, but with temperature the behavior remain same. Fig. 12b shows variation of $M^{\prime\prime}$ with frequency at selected temperatures. Once again, $M^{\prime\prime}$ spectroscopy plot reveals relaxation phenomena in the material. The maximum value ($M^{\prime\prime}$) of the $M^{\prime\prime}$ peak shifts to higher frequency, which suggests that hopping of charge carriers is predominantly thermally activated. Asymmetric broadening of the peak indicates spread of relaxation with different time constants, which once again suggests the material is non-Debye-type. In Fig. 13 we have plotted the $\tau$ vs absolute temperature $T$  calculated from the frequency for which peak value M$^{\prime\prime}_{max}$ appears in Fig. 12b by using the relaton $\tau$ = 1/2 $\pi$ f. We observed that the relaxation time as a function of absolute temperature (1000/T) gives a stright line which is well obayed by Arhenioud behavior. From the fittng parameters we have obtained activaton energy as E$_\alpha$ = 0.11 eV.

\subsection{Electric AC conductivity}

To further understand the charge hoping and electrical properties we have investigated the AC conductivity in Sm$_2$CoMnO$_6$. The AC conductivity is calculated by using relation $\sigma_{ac} = \epsilon_0 \omega \epsilon^{\prime\prime}$.\cite{sing} Fig. 14a shows the variation of AC conductivity with frequency i.e. $\sigma_{ac}$ vs $f$ at some selective temperatures in the range 50 K to 300 K. It is evident from the figure that at low frequencies the conductivity is independent of frequency and gives a plateau region at all temperatures. In this region of frequency the conduction is mainly dominated by DC conductivity ($\sigma_{dc}$). However, at higher frequencies the conductivity increases with increasing frequency. It is further notable that the  plateau region marked by dc conductivity in the Fig. 14a extends to higher frequencies with increasing temperature. The frequency independent region also suggests that the hopping charge carriers are absent at low frequencies. The ac conductivity at high frequency in this case obey Jonscher's Universal Power Law given as follow:\cite{thakur}

\begin{eqnarray}
\sigma_{ac} = \sigma_{dc} + A\omega^n
\end{eqnarray}

when A is a temperature dependent constant, $\omega$ = 2$\pi$$f$ and $n$ is the power law exponent which generally varies between 0 and 1 depending upon temperature. The value of power law exponent $n$ represent the extent of interaction between mobile ions and lattice around. For non-interacting Debye system $n$ = 1 and with decreasing $n$ value the interaction is expected to increase between lattice surrounding and mobile ions. Further the constant A  gives the degree of polarizibility.

\begin{figure}[th]
	\centering
		\includegraphics[width=8cm]{Fig14.eps}
\caption{(Color online) (a) Frequency dependence plot of the ac conductivity ($\sigma_{ac}$) for temperatures ranging from 50 K to 300 K are shown for Sm$_2$CoMnO$_6$. Solid red lines are fitting due to Eq. 11. (b) Shows the temperature variation of the power law exponent $n$.}
	\label{fig:Fig14}
\end{figure}

\begin{figure}[t]
	\centering
		\includegraphics[width=8cm]{Fig15.eps}
\caption{(Color online) The variation of $\sigma_{ac}$ with absolute temperature (10$^3$/T) is shown for Sm$_2$CoMnO$_6$. The solid line are due to fitting with Eq. 12.}
	\label{fig:Fig15}
\end{figure}

The frequency dependent $\sigma_{ac}$  is fitted with the power law as shown in Fig. 14a. The conductivity data is fitted well  in full frequency range 1 Hz to 5.5 MHz. From the fitting parameters the exponent factor $n$ is obtained for all temperatures and is plotted in Fig. 14b. It ss evident from the figure that the $n$ value at room temperature is around 0.5 which increases with decreasing temperature. However we found that below 150 K the conductivity data does not obey the Power law behavior.

To further understand the conduction mechanism we have measured temperature dependent AC conductivity.  The variation of AC conductivity with absolute temperature i.e.  $\sigma_{ac}$ vs 10$^3$/T at some selective frequencies is shown in Fig. 15. We observed that with increasing frequency the conductivity increases.

\begin{eqnarray}
\sigma_{ac} = \sigma_{0}exp\left(\frac{-E_\alpha}{k_BT}\right)
\end{eqnarray}
where $\sigma_{0}$ is pre-exponent factor, $k_B$ is Boltzmann constant and $E_\alpha$ is the activation energy.

We have fitted the conductivity data with the Eq. 12. It is found that the conductivity data is well fitted at high temperatures for all frequency. At low temperature the AC conductivity is independent of temperature and form a steady region but does depend on frequency. From the fitting parameter we calculated the activation energy which decreases with increasing frequency  listed as for E$_{\alpha}$(1MHz) = 2.15 eV, E$_{\alpha}$(100 kHz) = 2.67 eV, E$_{\alpha}$(10 kHz) = 2.86  eV, E$_{\alpha}$(1 kHz) = 2.87 eV, E$_{\alpha}$(100 Hz) = 2.88 eV.

\section{Conclusion}
Nano-crystalline Sm$_2$CoMnO$_6$ sample is successfully prepared by the sol-gel method. Structural, magnetic and dielectric properties are studies in detail. Structural study by x-ray diffraction confirms that the sample is in single-phase and chemically pure.  Rietveld refinement shows that Sm$_2$CoMnO$_6$ crystallized in monoclinic crystal structure and adopt \textit{P2$_1$/n} space group. Magnetization study reveals that  Sm$_2$CoMnO$_6$ is a ferromagnetic material that undergoes paramagnetic to ferromagnetic phase transition around 148 K marked as transition temperature ($T_c$). The effective magnetic moment obtained from experimental data is close to the theoretical value for the spin only system in the high spin state for Co/Mn sublattice. It is further observed that the Sm$^{3+}$ spin ordered in opposite direction to the Co/Mn sublattice thus giving rise to antiferromagnetic ordering at low temperature. Temperature dependent Raman study shows deviation from anharmonic behavior of frequency and line width for A$_{g1}$ Raman mode around $T_c$ which confirms the presence of spin-phonon coupling in Sm$_2$CoMnO$_6$. The dielectric response is studied in detail we observed that the material has a high dielectric constant at low temperature. The dielectric loss shows a relaxation process due to grains, which is thermally activated in nature with an activation energy of 0.14 eV. The impedance spectroscopy and electrical modulus study further confirm that the relaxation process is thermally activated in nature where the relaxation time follows Arrhenius behavior. AC conductivity has been studied with both frequency and temperature dependence. we found that the activation energy decreases with increasing frequency. Further, the exponent factor obtained from conductivity analysis shows that the system deviates from the Debye's model.

\section{Acknowledgment}
We acknowledge MNIT Jaipur, India for XPS data and AIRF (JNU) for magnetic measurement facilities. We acknowledge UGC-DAE-Consortium Indore and Dr. V. G. Sathe for Raman data. We also acknowledge Dr. A. K. Pramanik for dielectric measurement and UPEA-II funding for LCR meter. We thank Saroj Jha
and Dr. Ruchita Pal for help in recording data. Author Ilyas Noor Bhatti acknowledge University Grants Commission, India for financial support.