Narrow Results

Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $𝔤$, and let ${{ℳ}}_G^0$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $\kappa$ on $C$ which is even in case $dim𝔤$ is odd. We show that there is a nonempty Zariski open substack ${{𝒰}}_{\kappa }$ of ${{ℳ}}_G^0$ such that $H^i(C,\mathrm{\text{ad}}(E_G)\otimes \kappa )=0$, $i=1,2$, for all $E_G\in {{𝒰}}_{\kappa }$. It is shown that any such $E_G$ has a canonical connection. It is also shown that the tangent bundle $T{U}_{\kappa }$ has a natural splitting, where $U_{\kappa }$ is the restriction of ${{𝒰}}_{\kappa }$ to the semi-stable locus. We also produce an isomorphism between two naturally occurring ${\mathrm{\Omega }}_{M_G^{rs}}^1$-torsors on the moduli space of regularly stable $M_G^{rs}$.

In the present paper, we introduce spin groupoids associated with the degree two Stiefel-Whitney classes, and we also introduce spinor torsors equipped with certain connections compatible with an action of spin groupoid. Using the connection, we propose a Dirac-like operator acting on the space of smooth sections of the spinor torsor. We can show that its Dirac-Laplacian admits a heat kernel in a formal sense.