Picard–Lefschetz/Alien correspondence for the heat kernel

Assume that $a,b$ are sufficiently close that the heat kernel has a Borel-summable asymptotic expansion $\widetilde K_0$ associated with a minimal real geodesic $\gamma_0$. Let $\gamma_1$ be another holomorphic geodesic from $a$ to $b$, with complexified length $d_1$, and write

$$\omega=\tau\mathcal E(\gamma_1)-\tau\mathcal E(\gamma_0)=\frac{d_1^2}{4}-\frac{d_0^2}{4}.$$

Let $\widetilde K_1$ be the formal heat-kernel solution associated with $\gamma_1$. At the Stokes phase $\theta_*$, the relevant downward gradient Morse trajectories satisfy

$$\operatorname{Im}(e^{-i\theta_*}\mathcal E_{\mathbb C}(\gamma_1))=\operatorname{Im}(e^{-i\theta_*}\mathcal E_{\mathbb C}(\gamma_0)),$$ $$\operatorname{Re}(e^{-i\theta_*}\mathcal E_{\mathbb C}(\gamma_1))>\operatorname{Re}(e^{-i\theta_*}\mathcal E_{\mathbb C}(\gamma_0)).$$

They have fixed endpoints $\gamma(0,s)=a$, $\gamma(\tau,s)=b$, and converge in $H_t^1$ to $\gamma_0$ as $s\to-\infty$ and to $\gamma_1$ as $s\to+\infty$.

Picard–Lefschetz/Alien correspondence. The pointed Alien derivative of $\widetilde K_0$ at $\omega$ is

$$\dot\Delta_{\omega}^{+}\widetilde K_0=N\widetilde K_1,$$

where $N$ is the appropriate signed count of these connecting solutions.

This conjecture identifies alien derivatives of heat-kernel asymptotic expansions with Picard–Lefschetz wall-crossing data. It is motivated by the finite-dimensional correspondence, while the infinite-dimensional heat-kernel setting remains conjectural.