Abstract
In his famous 2007 paper on three-dimensional quantum gravity, Witten defined candidates for the partition functions \( \mathit Z_k(q) = \sum_{n=-k}^{\infty} \omega_k(n)q^n \) of potential extremal conformal field theories (CFTs) with central charges of the form c = 24k. Although such CFTs remain elusive, he proved that these modular functions are well defined. In this note, we point out several explicit representations of these functions. These involve the partition function p(n), Faber polynomials, traces of singular moduli, and Rademacher sums. Furthermore, for each prime p ≤ 11, the p series Zk(q), where k ε {1, . . . , p−1}∪{p+1}, possess a Ramanujan congruence. More precisely, for every non-zero integer n we have that .
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Ono, K., Rolen, L. (2021). On Witten’s Extremal Partition Functions. In: Alladi, K., Berndt, B.C., Paule, P., Sellers, J.A., Yee, A.J. (eds) George E. Andrews 80 Years of Combinatory Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-57050-7_33
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DOI: https://doi.org/10.1007/978-3-030-57050-7_33
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-57049-1
Online ISBN: 978-3-030-57050-7
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