On Witten's extremal partition functions

Abstract

In his famous 2007 paper on three dimensional quantum gravity, Witten defined candidates for the partition functions $$Z_k(q)=\sum _{n=-k}^{\infty }{w}_k(n)q^n$$ of potential extremal 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 $p\le 11$, the $p$ series $Z_k(q)$, where $k\in \{1,\dots ,p-1\}\cup \{p+1\},$ possess a Ramanujan congruence. More precisely, for every non-zero integer $n$ we have that $$w_k(pn)\equiv 0\{\begin{array}{llccc}(\mathrm{mod}{2}^{11})& \text{if}p=2,(\mathrm{mod}{3}^5)& \text{if}p=3,(\mathrm{mod}{5}^2)& \text{if}p=5,(\mathrm{mod}p)& \text{if}p=7,11.\end{array}$$

8 pages