The Stolz–Teichner Conjecture and Supermoonshine

A characteristic feature of any theory of quantum gravity is that spacetime can fluctuate. But not every change in spacetime is allowed. In the context of String Theory—our leading candidate for a theory of quantum gravity—the set of spacetime-changing processes corresponds to the set of deformations of the quantum field theory (QFT) living on the surface of the string.

According to the Stolz–Teichner conjecture, the space of such deformations is captured by a mathematical object known as topological modular forms (TMF). The conjecture assigns a group, TMF_v, to the set of QFTs with gravitational anomaly v (a quantity related to the dimension of the spacetime in String Theory). The elements of the groups correspond to deformation classes of QFTs, and the groups satisfy the remarkable periodicity TMF_v = TMF_v+576, which means that QFTs with gravitational anomalies v and v+576 admit identical classes of deformations. This periodicity suggests the existence of a special class in TMF_-576 known as the “periodicity class,” with the property that every class of TMF_v-576 is obtained from a unique class in TMF_v by taking the product with the periodicity class.

Furthermore, the theory of TMF predicts that certain coefficients of elliptic genera—torus partition functions with Ramond boundary conditions along the spatial direction—satisfy a divisibility property determined by their gravitational anomaly v. In our paper, we test this prediction for a certain chiral supersymmetric conformal field theory with v = -24, known as Duncan’s Supermoonshine module. We also checked the conjecture for tensor products of Supermoonshine with various discrete symmetries gauged. Such symmetries included S_n and A_n permutation symmetries, as well as non-anomalous cyclic subgroups of the Conway global symmetry of the theory. In all cases, we found a match with the predictions of Stolz–Teichner.

An important intermediate result of our work was the development of a closed-form formula for the elliptic genera of alternating orbifolds. This formula was given in two forms: the first similar to the formula of Dijkgraaf, Moore, Verlinde, and Verlinde for symmetric product orbifolds, and the second based on generalized Hecke operators.

Our motivation for developing these formulas was the construction of a physical realization of the periodicity class of TMF. In particular, we first considered a tensor product of 24 copies of Duncan’s Supermoonshine module, giving a theory with v = -576, as well as symmetric orbifolds thereof. This gave rise to a periodicity element, but with the undesirable feature of being decomposable. In order to obtain an indecomposable peridiocity element, we argue that it is necessary to combine alternating and Co₁ orbifolds, thereby necessitating a closed formula for the former. We hope that our proposal can be verified in the near future, when the relevant McKay–Thompson data for Supermoonshine is computed.

In conclusion, this study is a step toward a better understanding of TMF and its implications for physics.