What is the Todd class of an orbifold?

          @misc{ pirsa_PIRSA:16040074,
            doi = {10.48660/16040074},
            url = {https://pirsa.org/16040074},
            author = {Caldararu, Andrei},
            keywords = {Mathematical physics},
            language = {en},
            title = {What is the Todd class of an orbifold?},
            publisher = {Perimeter Institute},
            year = {2016},
            month = {apr},
            note = {PIRSA:16040074 see, \url{https://pirsa.org}}
          }
          

Abstract

The Todd class enters algebraic geometry in two places, in the Hirzebruch-Riemann-Roch formula and in the correction of the HKR isomorphism needed to make the Hochschild cohomology isomorphic to polyvector field cohomology (Kontsevich’s claim, proved by Calaque and van den Bergh). In the case of orbifolds the Riemann-Roch formula is known, but not the analogue of Kontsevich’s result. However, we can try to use the former as a guide towards a conjectural formulation for the latter. The problem with this approach is that in the case of an orbifold it is not obvious what the Todd class actually is. This happens because the Riemann-Roch formula mixes the Todd class with the Chern character and it is difficult to separate one from the other. In my talk I shall discuss what the study of loop groups of orbifolds predicts the correct Todd class to be, and then I shall explain how the orbifold Riemann-Roch formula can be rewritten to make this prediction consistent.