A Note on the Equivalence of Vafa's and Douglas's Picture of Discrete Torsion

TL;DR

This note proves the equivalence of Vafa's discrete torsion, defined via $H^2(B\Gamma,\mathrm{U}(1))$, with Douglas's D-brane approach based on projective representations, for general nonabelian $\Gamma$ actions and arbitrary genus worldsheet $\Sigma$. It builds the bridge by expressing the torus and higher-genus amplitudes through bar-resolution representatives of $H_n(\Gamma)$ and by identifying the genus-$g$ phase with the product of cocycle factors from a central extension, i.e., $s(a)s(b)=\alpha(a,b)s(ab)$ and $\xi_g=\prod_{i=1}^g s(a_i)s(b_i)s(a_i)^{-1}s(b_i)^{-1}$. The two pictures are shown to yield the same triangulation-dependent torsion factors, reducing to Vafa's $\xi_1=\alpha(a,b)/\alpha(b,a)$ on genus one and to the corresponding higher-genus expressions. This unifies the two formulations and highlights the central role of $H^2(\Gamma,\mathrm{U}(1))$ and classifying spaces in discrete torsion for orbifolds.

Abstract

For a general nonabelian group action and an arbitrary genus worldsheet we show that Vafa's old definition of discrete torsion coincides with Douglas's D-brane definition of discrete torsion associated to projective representations.

Figures (4)

• Figure 1: A simplicial decomposition of a torus.
• Figure 2: A simplicial decomposition of a genus 2 surface.
• Figure 3: A genus 2 surface with a hole cut out.
• Figure 4: Part of the simplicial decomposition of the arbitrary genus case.