[Paper Review] Elusive Worldsheet Instantons in Heterotic String Compactifications

Paul S. Aspinwall, M. Ronen Plesser|arXiv (Cornell University)|Jun 15, 2011

Black Holes and Theoretical Physics23 references9 citations

TL;DR

This paper investigates worldsheet instanton corrections to the spacetime superpotential in heterotic string compactifications on Calabi–Yau threefolds, using Landau–Ginzburg, orbifold, and nonlinear σ-model methods. It finds that while instanton corrections can lift massless gauge singlet chiral multiplets, the massless spectrum in the orbifold limit consistently matches nonlinear σ-model predictions—even in cases like the Z-manifold where instanton effects do not vanish, indicating a robust heterotic McKay correspondence.

ABSTRACT

We compute the spectrum of massless gauge singlets in some heterotic string compactifications using Landau-Ginzburg, orbifold and non-linear sigma-model methods. This probes the worldsheet instanton corrections to the quadratic terms in the spacetime superpotential. Previous results predict that some of these states remain massless when instanton effects are included. We find vanishing masses in many cases not covered by these predictions. However, we discover that in the case of the Z-manifold the corrections do not vanish. Despite this, in all the examples studied, we find that the massless spectrum in the orbifold limit agrees with the nonlinear sigma-model computation.

Motivation & Objective

• To understand the role of worldsheet instantons in destabilizing or preserving massless gauge singlet chiral multiplets in heterotic string compactifications.
• To test whether instanton corrections to the quadratic terms in the spacetime superpotential vanish or persist in specific compactifications.
• To investigate the validity of the 'cancellation miracle'—where instanton contributions cancel globally—beyond the Beasley–Witten framework.
• To verify whether the heterotic McKay correspondence holds even when instanton corrections are non-zero, using the Z-manifold as a critical example.

Proposed method

• Computes the massless spectrum of gauge singlets using nonlinear σ-models to obtain the classical (perturbative) count of h¹(End(T)).
• Applies exact conformal field theory techniques via Landau–Ginzburg models to compute the exact spectrum, including instanton effects.
• Compares the perturbative nonlinear σ-model result with the exact CFT result to isolate instanton corrections to the superpotential.
• Uses mirror symmetry to compute the exact spectrum in the Landau–Ginzburg realization of the mirror manifold.
• Analyzes the Z-manifold, a crepant resolution of T⁶/ℤ₃, to test for non-vanishing instanton corrections.
• Applies the Beasley–Witten cancellation mechanism to determine when instanton contributions vanish or cancel.

Experimental results

Research questions

• RQ1 Do worldsheet instantons generate non-zero corrections to the quadratic terms in the spacetime superpotential in heterotic compactifications?
• RQ2 In which cases does the Beasley–Witten cancellation mechanism fail to suppress instanton contributions?
• RQ3 Does the heterotic McKay correspondence remain valid when instanton corrections are non-zero, as in the Z-manifold?
• RQ4 Can the exact massless spectrum in the orbifold limit be reliably computed from the nonlinear σ-model even when instanton effects are present?
• RQ5 Is there a consistent match between the nonlinear σ-model and CFT results for the massless spectrum when instanton corrections are non-zero?

Key findings

• The Z-manifold exhibits non-vanishing worldsheet instanton corrections to the superpotential, contrary to previous expectations of cancellation.
• Despite non-zero instanton corrections, the massless spectrum in the orbifold limit agrees exactly with the nonlinear σ-model computation.
• The number of massless gauge singlet chiral multiplets is reduced from the classical h¹(End(T)) = 208 in the Z-manifold due to instanton effects.
• The cancellation mechanism observed in many Calabi–Yau compactifications does not hold universally, as demonstrated by the Z-manifold example.
• The heterotic McKay correspondence remains valid even when instanton corrections are non-zero, as the spectrum computed via orbifold methods matches the exact CFT result.
• In all studied cases, including the Z-manifold, the agreement between orbifold and nonlinear σ-model spectra confirms the robustness of the duality framework.