[Paper Review] Tension Between a Vanishing Cosmological Constant and Non-Supersymmetric Heterotic Orbifolds

Stefan Groot Nibbelink, Orestis Loukas+3 more|arXiv (Cornell University)|Oct 25, 2017

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates whether non-supersymmetric heterotic string theories on toroidal orbifolds can achieve a perturbatively vanishing cosmological constant at one-loop level. By requiring the right-moving fermionic partition function to vanish identically in every orbifold sector—implying at least one preserved Killing spinor per sector—it demonstrates through group representation theory that no such non-supersymmetric orbifold exists in six dimensions, establishing a no-go result rooted in finite group structure.

ABSTRACT

We investigate under which conditions the cosmological constant vanishes perturbatively at the one-loop level for heterotic strings on non-supersymmetric toroidal orbifolds. To obtain model-independent results, which do not rely on the gauge embedding details, we require that the right-moving fermionic partition function vanishes identically in every orbifold sector. This means that each sector preserves at least one, but not always the same Killing spinor. The existence of such Killing spinors is related to the representation theory of finite groups, i.e. of the point group that underlies the orbifold. However, by going through all inequivalent (Abelian and non-Abelian) point groups of six-dimensional toroidal orbifolds we show that this is never possible: For any non-supersymmetric orbifold there is always (at least) one sector, that does not admit any Killing spinor. The underlying mathematical reason for this no-go result is formulated in a conjecture, which we have tested by going through an even larger number of finite groups. This conjecture could be applied to situations beyond symmetric toroidal orbifolds, like asymmetric orbifolds.

Motivation & Objective

• To determine whether non-supersymmetric heterotic string models on toroidal orbifolds can achieve a vanishing cosmological constant at one-loop order.
• To identify model-independent conditions under which the one-loop cosmological constant vanishes, independent of gauge embedding details.
• To investigate whether the existence of at least one preserved Killing spinor in every orbifold sector—necessary for vacuum energy cancellation—can be realized in non-supersymmetric settings.
• To test the mathematical feasibility of such Killing spinor conditions across all finite point groups of six-dimensional toroidal orbifolds.
• To formulate and test a conjecture on finite group representation theory that underlies the impossibility of such cancellations in non-supersymmetric models.

Proposed method

• Analyzes the one-loop string partition function in terms of twisted sectors labeled by space group elements and orbifold point groups.
• Imposes the condition that the right-moving fermionic partition function vanishes identically in every sector, which requires at least one preserved Killing spinor per sector.
• Applies representation theory of finite groups (point groups) to determine whether any group element admits a trivial representation (i.e., a preserved spinor).
• Systematically examines all inequivalent Abelian and non-Abelian point groups of six-dimensional toroidal orbifolds for the existence of such Killing spinors.
• Tests a conjecture on finite group structure that links the absence of trivial representations in all cyclic subgroups to the impossibility of vacuum energy cancellation.
• Extends the analysis to asymmetric orbifolds and considers implications beyond symmetric compactifications using character theory and branching rules.

Experimental results

Research questions

• RQ1 Can a non-supersymmetric heterotic string compactification on a toroidal orbifold yield a vanishing cosmological constant at one-loop order?
• RQ2 Is it possible for every orbifold sector to preserve at least one Killing spinor, even if not the same one, in a non-supersymmetric model?
• RQ3 What group-theoretic conditions must be satisfied for the right-moving fermionic partition function to vanish identically across all sectors?
• RQ4 Does the absence of a trivial representation in all cyclic subgroups of the point group prevent the cosmological constant from vanishing perturbatively?
• RQ5 Can the no-go result be generalized beyond symmetric orbifolds to include asymmetric constructions?

Key findings

• For all non-supersymmetric heterotic orbifolds in six dimensions, there exists at least one orbifold sector that does not admit any preserved Killing spinor.
• The requirement that the right-moving fermionic partition function vanishes identically in every sector cannot be satisfied in any non-supersymmetric toroidal orbifold due to group representation constraints.
• A conjecture is formulated stating that no finite group with non-trivial point group action can have trivial representations in all cyclic subgroups unless the group is trivial or supersymmetric.
• The conjecture was tested across a large number of finite groups, including non-Abelian ones, and no counterexamples were found, supporting the robustness of the no-go result.
• The result implies that perturbative cancellation of the cosmological constant in non-supersymmetric heterotic string models on symmetric toroidal orbifolds is impossible.
• The mathematical obstruction arises from the structure of finite group representations, particularly the absence of trivial representations in cyclic subgroups of the point group.