The first conformal Dirac eigenvalue on 2-dimensional tori

We show that τ(T2,χ)=2π if χ is “the” non-trivial spin structure on T2. In order to calculate this invariant, we study the infimum as a function on the spin-conformal moduli space and we show that the infimum converges to 2π at one end of the spin-conformal moduli space.