Theorem: ind(D_spin) = 1 - g = 0 at g = 1 (Dirac on torus T^2)

Theorem (Atiyah-Singer-genus-one canonical): for a Riemann surface of genus g, the Dolbeault operator D_partial-bar (Cauchy-Riemann operator on holomorphic functions) has analytic index ind(D_partial-bar) = dim H^0 - dim H^1 = 1 - g…