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Moonshine and String Theory

Periodic Reporting for period 4 - MST (Moonshine and String Theory)

Reporting period: 2020-03-01 to 2021-02-28

- What is the nature of the mysterious relation between modular objects and finite groups? Why are these two fundamentally different mathematical structures related to each other? What is the relation to theoretical physics, in particular string theory?
- This is curiosity driven research. Just as any research in pure mathematics, as a civilisation we explore new possibilities by continuously pushing the boundary between known and unknown.
- We would like to understand the nature, in particular in relation to string theory that stems from physics, of the novel type of moonshine recently discovered by the PI. We would also like to work towards having an overview of the general landscape of connections between modular objects and finite groups.
Here I report on the hightlights of the ERC research performed in the duration of the ERC starting grant. In the papers "Weight one Jacobi forms and umbral moonshine" and in particular in "Optimal mock Jacobi theta functions", we have clarified the nature of the modular-like objects appearing in umbral moonshine. In "K3 string theory, lattices and moonshine", we made sharp conjectures on the relation between umbral moonshine and symmetries of string theory on K3, and provided non-trivial evidence for them. In "K3 Elliptic Genus and an Umbral Moonshine Module", we exploited the connection to string theory to construct the moonshine module for an important instance of umbral moonshine. In "Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules", we constructed the moonshine modules for another four instances of umbral moonshine by employing the expressions for the umbral moonshine functions in terms of meromorphic Jacobi forms. In "Vertex operator superalgebra/sigma model correspondences: The four-torus case", we extend and shed further light on the relation between K3 sigma models, umbral moonshine, and a particular vertex operator super-algebra by establishing an analogous relation in the case of T^4 sigma models. I believe that these papers and the talks our team has given on the topics have significantly furthered the progress of the research on this topic.
As described in detail above, the research performed by the ERC team greatly expanded our understanding of umbral moonshine, in terms of the structure of the underlying finite group modules, the relation to string theory, and the nature of the modular-like objects involved. We expect these results to lead to more progress on the topic in the near future.
an illustration of umbral moonshine