Traces of Singular Moduli and Moonshine for the Thompson group

          @misc{ pirsa_PIRSA:15040119,
            doi = {10.48660/15040119},
            url = {https://pirsa.org/15040119},
            author = {Harvey, Jeffrey},
            keywords = {Quantum Fields and Strings},
            language = {en},
            title = {Traces of Singular Moduli and Moonshine for the Thompson group},
            publisher = {Perimeter Institute},
            year = {2015},
            month = {apr},
            note = {PIRSA:15040119 see, \url{https://pirsa.org}}
          }
          

Abstract

We observe a relationship between the representation theory of the Thompson sporadic group and a weakly holomorphic modular form of weight one-half that appears in Zagier's work on traces of singular moduli and Borcherds products. We conjecture the existence of an infinite dimensional graded module for the Thompson group and use the observed relationship to propose a McKay-Thompson series for each conjugacy class of the Thompson group and then construct weakly holomorphic weight one-half forms at higher level that coincide with the proposed McKay-Thompson series. We also observe a discriminant property in this conjectured moonshine for the Thompson group that is closely related to the discriminant property conjectured to exist in Umbral Moonshine.