[Paper Review] On the elliptic genera of manifolds of Spin(7) holonomy

Nathan Benjamin, Sarah M. Harrison+3 more|arXiv (Cornell University)|Dec 8, 2014

Black Holes and Theoretical Physics35 references4 citations

TL;DR

This paper computes the elliptic genus of manifolds with Spin(7) holonomy by decomposing it into characters of the ${\cal SW}(3/2,2)$ superconformal algebra at central charge $c=12$, revealing suggestive connections to sporadic groups—particularly the Mathieu group $M_{24}$—which are made precise in a companion paper. The work extends the moonshine phenomenon observed in K3 compactifications to Spin(7) manifolds via modular forms and representation theory.

ABSTRACT

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The $\mathcal{N}=1$ superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a $c=12$ realization of the symmetry group $\mathcal{SW}(3/2,2)$. In this paper, we compute the characters of this supergroup, and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper.

Motivation & Objective

• To compute the elliptic genus of manifolds with Spin(7) holonomy using modular functions under the congruence subgroup $\Gamma_\theta$.
• To decompose the elliptic genus into irreducible characters of the $c=12$ ${\cal SW}(3/2,2)$ superconformal algebra.
• To explore potential connections between the representation theory of the ${\cal SW}(3/2,2)$ algebra and sporadic finite groups, particularly $M_{24}$.
• To provide a character decomposition that generalizes the moonshine phenomenon observed in K3 compactifications to higher-dimensional Calabi-Yau-like geometries.

Proposed method

• Derive the elliptic genus of a Spin(7) manifold as a one-parameter family of modular functions under $\Gamma_\theta$, with the parameter fixed by the Euler characteristic $\chi(X)$.
• Construct unitary highest weight representations of the ${\cal SW}(3/2,2)$ algebra at $c=12$, including both massive and massless characters.
• Use Jacobi theta functions and the Dedekind eta function to express the characters and modular forms involved in the decomposition.
• Apply the commutation relations of the ${\cal SW}(3/2,2)$ algebra to determine the structure of Verma modules and identify subsingular vectors.
• Perform a character decomposition of the elliptic genus into irreducible representations of the ${\cal SW}(3/2,2)$ algebra using explicit modular forms.
• Compare the coefficients in the decomposition to dimensions of irreducible representations of sporadic groups, particularly $M_{24}$, to identify suggestive numerical patterns.

Experimental results

Research questions

• RQ1 How can the elliptic genus of a Spin(7) manifold be expressed as a modular function under $\Gamma_\theta$?
• RQ2 What is the complete set of characters for the $c=12$ ${\cal SW}(3/2,2)$ superconformal algebra, and how do they decompose the elliptic genus?
• RQ3 Are there numerical or structural connections between the coefficients in the character decomposition and representations of sporadic groups?
• RQ4 Can the moonshine-like phenomenon observed in K3 compactifications be extended to Spin(7) manifolds through the ${\cal SW}(3/2,2)$ algebra?

Key findings

• The elliptic genus of a Spin(7) manifold is a modular function under $\Gamma_\theta$, parameterized by the Euler characteristic $\chi(X)$.
• The massive characters of the ${\cal SW}(3/2,2)$ algebra are confirmed to match the conjecture in [7], providing a consistent character set for the algebra.
• The massless characters of the ${\cal SW}(3/2,2)$ algebra are explicitly constructed and shown to contribute to the decomposition of the elliptic genus.
• The decomposition of the elliptic genus into ${\cal SW}(3/2,2)$ characters reveals numerical patterns resembling sums of dimensions of irreducible representations of $M_{24}$, particularly in the coefficients of the $h = n + \frac{1}{4}$ terms.
• The paper identifies a subsingular vector in the $|h=\frac{1}{2}, x=\frac{1}{2}\rangle$ Verma module with $L_0$ eigenvalue 4 and $X_0$ eigenvalue 20, aiding in character classification.
• The results suggest a deep, yet not yet fully understood, connection between the geometry of Spin(7) manifolds, the representation theory of the ${\cal SW}(3/2,2)$ algebra, and sporadic finite groups, which is further explored in a companion paper.