On Witten’s Extremal Partition Functions

Abstract

In his famous 2007 paper on three-dimensional quantum gravity, Witten defined candidates for the partition functions

$$\begin{aligned} Z_k(q)=\sum _{n=-k}^{\infty }w_k(n)q^n \end{aligned}$$

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 $p\le 11$, 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

$$\begin{aligned} w_k(pn) \equiv 0{\left\{ \begin{array}{ll} \pmod {2^{11}}\ \ \ \ &{}{\mathrm{if}}\ p=2,\\ \pmod {3^5} \ \ \ \ &{}{\mathrm{if}}\ p=3,\\ \pmod {5^2}\ \ \ \ &{}{\mathrm{if}}\ p=5,\\ \pmod {p} \ \ \ \ &{}{\mathrm{if}}\ p=7, 11. \end{array}\right. } \end{aligned}$$