Genus two partition functions of chiral conformal field theories

TL;DR

This work analyzes genus-2 vacuum amplitudes in chiral self-dual CFTs at c=24k, proving that a modular-invariant genus-2 partition function yields infinitely many relations among structure constants, all consequences of OPE associativity and genus-1 modular covariance. It develops two finite, coordinate-friendly expansions (Schottky and sewn-tori) to express genus-2 data in terms of a finite basis, and shows these invariants are constrained by a completeness theorem. The extremal ansatz is shown to admit a consistent genus-2 amplitude for all k, but such a result relies only on genus-1-consistency data, not on deeper higher-genus constraints. The paper also highlights nontrivial higher-genus conditions that contracted Jacobi identities may or may not satisfy, and argues that fully reconstructing the Jacobi structure from higher-genus data imposes strong, nontrivial requirements that become decisive only at large genus.

Abstract

A systematic analysis of the genus two vacuum amplitudes of chiral self-dual conformal field theories is performed. It is explained that the existence of a modular invariant genus two partition function implies infinitely many relations among the structure constants of the theory. All of these relations are shown to be a consequence of the associativity of the OPE, as well as the modular covariance properties of the torus one-point functions. Using these techniques we prove that for the proposed extremal conformal field theories at c=24k a consistent genus two vacuum amplitude exists for all k, but that this does not actually check the consistency of these theories beyond what is already testable at genus one.

Figures (4)

• Figure 1: To the left, the geometric interpretation of the Schottky coordinates $p_1$ and $p_2$; the third coordinate $x$ is given by the cross section of the insertion points. To the right, the geometric interpretation of the sewn tori coordinates $q_1$, $q_2$, $\epsilon$.
• Figure 2: A graphical representation of the set ${\cal P}^{(*)}_k$ for $k=13$. Each (either white or black) circle in the diagram denotes a pair $(h_1,l)$ for which we can find a $h_2\leq h_1$ with $0\leq l \leq h_1+h_2$. Black circles denote pairs $(h_1,l)$ for which at least one such choice of $h_2$ corresponds to an element in ${\cal P}^{(*)}_k$.
• Figure 3: The two possible pants decompositions of a genus $2$ surface. They correspond to the expansions in the coordinates $q_1,q_2,\epsilon$ (left) and $p_1,p_2,x$ (right), and the associated invariants are $\mathcal{D}_{h_1,h_2;l}$ (left) and $\mathcal{C}^{(0)}_{h_1,h_2;l}$ (right).
• Figure 4: A fundamental domain for a Schottky group. Each generator $\gamma_i$, $i=1,\ldots,g$, with fixed points $a_i,r_i$, maps the circle $C_{-i}$ to the circle $C_{i}$.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Lemma 1