N=2 Superconformal Algebra and the Entropy of Calabi-Yau Manifolds

TL;DR

This paper investigates the entropy of Calabi-Yau manifolds using N=2 superconformal algebra, revealing dimension-dependent behaviors and deriving formulas consistent with Cardy's at large dimensions.

Contribution

It introduces a novel approach to compute Calabi-Yau entropy via superconformal algebra representation theory, highlighting differences between even and odd dimensions.

Findings

CY entropy matches hyperKahler entropy for odd D.

Explicit formulas for CY entropy in even and odd D.

Entropy growth aligns with Cardy's formula at large D.

Abstract

We use the representation theory of N=2 superconformal algebra to study the elliptic genera of Calabi-Yau (CY) D-folds. We compute the entropy of CY manifolds from the growth rate of multiplicities of the massive (non-BPS) representations in the decomposition of their elliptic genera. We find that the entropy of CY manifolds of complex dimension D behaves differently depending on whether D is even or odd. When D is odd, CY entropy coincides with the entropy of the corresponding hyperKahler (D-3)-folds due to a structural theorem on Jacobi forms. In particular, we find that the Calabi-Yau 3-fold has a vanishing entropy. At D>3, using our previous results on hyperKahler manifolds, we find $S_{CY_D} \sim 2\pi \sqrt{{(D-3)^2\over 2(D-1)}n}$. When D is even, we find the behavior of CY entropy behaving as $S_{CY_D}\sim 2 \pi\sqrt{{D-1\over 2}n}$. These agree with Cardy's formula at large D.