Heat kernel asymptotics and analytic torsion on non-degenerate CR manifolds

Let $X$ be a compact oriented CR manifold of dimension $2n+1$, $n \ge 1$, with a nondegenerate Levi form of constant signature $(n_-, n_+)$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish the small time asymptotics of the heat kernel of Kohn Laplacian. Suppose that condition $Y(q)$ fails, we establish the small time asymptotics of the kernel of the difference of the heat operator and Szeg\H{o} projector. As an application we define the analytic torsion on a compact oriented CR manifold of dimension $2n+1$, $n \ge 1$, with a nondegenerate Levi form. Let $L^k$ be the $k$-th power of a CR complex line bundle $L$ over $X$. We also establish asymptotics of the analytic torsion with values in $L^k$ under certain natural assumptions.