Now we can define a semigroup $\varphi^{t}:Hom_{Dolb}(E_{1},E_{2})\to Hom_{Dolb}(E_{1},E_{2}),0\leq t<+\infty$. This is an analog of the semigroup $\phi^{t}$ in the non-archimedean case. First, we identify the sector $\underline{Hom}_{{\cal A}(Y)}^{alg,[\gamma]}(E_{1},E_{2})_{y}$ with $\underline{\Omega}^{\ast}(Y,Hom(\rho_{1},\rho_{2}))_{y}$ as above. Let us recall from the non-archimedean part, that to the homotopy class of a path $\gamma$ we canonically associated a closed $1$-form $\mu_{\gamma}=\int_{\gamma}\omega$, where $\omega$ is the symplectic form on $X^{\vee}$. Using the Riemannian metric $g_{Y}$ on $Y$ we assign to $\mu$ a vector field $\xi_{\gamma}$ on $Y$. In a local trivialization it is given by $grad((f_{1}-f_{2}+\langle m(\gamma),\cdot\rangle)/\varepsilon)$. Then the infinitesimal action of $\varphi^{t}$ is defined as the Lie derivative $Lie_{\xi_{\gamma}}$. Different Fourier components (sectors) move on $Y$ with with different speeds in different directions. Hence the picture is more complicated than in the case of Morse theory.

One can show that the generator $\Delta={d\over{dt}}|_{{t=0}}\varphi^{t}$ is a second order differential operator on $Hom_{Dolb}(E_{1},E_{2})$. When $g_{Y}$ is a flat metric and $f_{1}=f_{2}=0$ one can find the following explicit formula for $\Delta$:

$$\Delta=i\sum_{j}{\partial^{2}\over{\partial x_{j}\partial y_{j}}}-{1\over{\varepsilon}}\sum_{j}{\partial^{2}\over{\partial x_{j}^{2}}}.$$

It seems plausible that there is an extension of the semigroup $\varphi^{t}=e^{t\Delta}$, $t\geq 0$, from $Hom_{{\cal A}(Y)}^{alg}(E_{1},E_{2})$ to the whole space of morphisms $Hom_{{\cal A}(Y)}(E_{1},E_{2})$. Notice that $\Delta$ is not self-adjoint, and its real part is not elliptic. Nevertheless, we expect that the semigroup operator $\varphi^{t}$ converges as $t\to+\infty$ to a “projector” as in the case of Morse theory.

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Addresses:

M.K.: IHES, 35 route de Chartres, F-91440, France

maxim@ihes.fr

Y.S.: Department of Mathematics, KSU, Manhattan, KS 66506, USA

soibel@math.ksu.edu