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Modular Constraints on Calabi-Yau Compactifications

Christoph A. Keller, Hirosi Ooguri

TL;DR

The work addresses non-BPS constraints in two-dimensional N=(2,2) sigma-models with Calabi–Yau targets, showing that large total Hodge numbers force a dense non-BPS sector with low total weight. It combines the extended N=2 superconformal algebra, elliptic genus, and modular invariance to bound the low-lying non-BPS spectrum, and then strengthens these bounds with a differential-operator (medium-temperature) approach. A universal bound Δ_total < 0.656 emerges for sufficiently large h_total, with a linear growth in the number of such states in h_total, and asymptotic improvements toward Δ_total = 1/2 in the large-h_total limit; for moderate h_total, operator-based refinements sharpen the constraints and extend them to CY d-folds. The results connect Calabi–Yau geometry to constraints on massive string states, informing the string landscape by predicting a proliferating non-BPS sector as Hodge numbers increase.

Abstract

We derive global constraints on the non-BPS sector of supersymmetric 2d sigma-models whose target space is a Calabi-Yau manifold. When the total Hodge number of the Calabi-Yau threefold is sufficiently large, we show that there must be non-BPS primary states whose total conformal weights are less than 0.656. Moreover, the number of such primary states grows at least linearly in the total Hodge number. We discuss implications of these results for Calabi-Yau geometry.

Modular Constraints on Calabi-Yau Compactifications

TL;DR

The work addresses non-BPS constraints in two-dimensional N=(2,2) sigma-models with Calabi–Yau targets, showing that large total Hodge numbers force a dense non-BPS sector with low total weight. It combines the extended N=2 superconformal algebra, elliptic genus, and modular invariance to bound the low-lying non-BPS spectrum, and then strengthens these bounds with a differential-operator (medium-temperature) approach. A universal bound Δ_total < 0.656 emerges for sufficiently large h_total, with a linear growth in the number of such states in h_total, and asymptotic improvements toward Δ_total = 1/2 in the large-h_total limit; for moderate h_total, operator-based refinements sharpen the constraints and extend them to CY d-folds. The results connect Calabi–Yau geometry to constraints on massive string states, informing the string landscape by predicting a proliferating non-BPS sector as Hodge numbers increase.

Abstract

We derive global constraints on the non-BPS sector of supersymmetric 2d sigma-models whose target space is a Calabi-Yau manifold. When the total Hodge number of the Calabi-Yau threefold is sufficiently large, we show that there must be non-BPS primary states whose total conformal weights are less than 0.656. Moreover, the number of such primary states grows at least linearly in the total Hodge number. We discuss implications of these results for Calabi-Yau geometry.

Paper Structure

This paper contains 19 sections, 82 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The extended $N=2$ superconformal algebra
  3. Representations of the extended $N=2$ superconformal algebra
  4. Calabi-Yau partition functions
  5. Modular invariance: the direct way
  6. A first attempt
  7. Using Hodge numbers
  8. Upper bounds
  9. Generalization to Calabi-Yau $d$-folds
  10. Differential operators
  11. Differential operators and the medium temperature expansion
  12. Constraints on massless representations
  13. Constraints on massive representations
  14. Calabi-Yau threefolds
  15. Constraints from the ordinary partition function
  16. ...and 4 more sections

Figures (5)

  • Figure 1: Upper bound for the total weight $\Delta_{{\rm total}}=\Delta + \bar{\Delta}$ of the lowest lying massive field for a Calabi-Yau threefold as a function of the total Hodge number $h_{{\rm total}}=h^{1,1}+h^{2,1}$.
  • Figure 2: The Hodge diamond of a Calabi-Yau of dimension $d$. The unshaded and shaded regions give positive and negative contributions to the right-hand side of (\ref{['higherCY']}) respectively.
  • Figure 3: $\Delta_0^{Q\bar{Q}}$ for the three combinations of massive representations.
  • Figure 4: $\Delta_0^{Q\bar{Q}}$ for the various combinations of massive representations.
  • Figure 5: Upper bound for the total weight $\Delta_{\rm total}$ of the lowest lying massive field for a Calabi-Yau threefold with enhanced symmetry.