On Witten's extremal partition functions

TL;DR

This paper explores explicit representations of Witten's extremal partition functions for potential extremal CFTs, revealing their modular properties, explicit formulas, and Ramanujan congruences for primes up to 11.

Contribution

It provides explicit formulas involving partition functions, Faber polynomials, and traces of singular moduli, and establishes Ramanujan congruences for these functions at small primes.

Findings

Explicit representations of extremal partition functions using known mathematical objects.

Identification of Ramanujan congruences for primes p ≤ 11.

Modular properties and explicit formulas for the functions Z_k(q).

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 $p\leq 11$, 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 $$ w_k(pn) \equiv 0\begin{cases} \pmod{2^{11}}\ \ \ \ &{\text {\rm if}}\ p=2, \pmod{3^5} \ \ \ \ &{\text {\rm if}}\ p=3, \pmod{5^2}\ \ \ \ &{\text {\rm if}}\ p=5, \pmod{p} \ \ \ \ &{\text {\rm if}}\ p=7, 11. \end{cases} $$