[Paper Review] Strings vs Spins on the Null Orbifold

K. Surya Kiran, Chethan Krishnan+2 more|arXiv (Cornell University)|Aug 14, 2014

Black Holes and Theoretical Physics23 references4 citations

TL;DR

This paper investigates the null orbifold singularity in 2+1D higher spin gravity and string theory, showing that its Chern-Simons holonomy is trivial despite the geometry's singularity. It constructs a higher spin resolution using a Kundt spacetime with vanishing curvature invariants and demonstrates that UV divergences in the 4-point tachyon amplitude vanish in the tensionless limit, leaving only physical IR divergences. The work supports the idea that higher spin theories can resolve cosmological singularities and may capture aspects of flat space string theory in the tensionless regime.

ABSTRACT

We study the null orbifold singularity in 2+1 d flat space higher spin theory as well as string theory. Using the Chern-Simons formulation of 2+1 d Einstein gravity, we first observe that despite the singular nature of this geometry, the eigenvalues of its Chern-Simons holonomy are trivial. Next, we construct a resolution of the singularity in higher spin theory: a Kundt spacetime with vanishing scalar curvature invariants. We also point out that the UV divergences previously observed in the 2-to-2 tachyon tree level string amplitude on the null orbifold do not arise in the $α^\prime o \infty$ limit. We find all the divergences of the amplitude and demonstrate that the ones remaining in the tensionless limit are physical IR-type divergences. We conclude with a discussion on the meaning and limitations of higher spin (cosmological) singularity resolution and its potential connection to string theory.

Motivation & Objective

• To investigate whether the null orbifold singularity in 2+1D flat space can be resolved in higher spin theory, despite its singular metric.
• To analyze the behavior of string scattering amplitudes on the null orbifold in the tensionless ($\alpha' \to \infty$) limit, particularly focusing on UV and IR divergences.
• To determine whether the Chern-Simons holonomy of the null orbifold reflects its singular nature, and whether it can be trivial despite the geometry's pathology.
• To explore the connection between higher spin theory and the tensionless limit of string theory, especially in the context of cosmological singularity resolution.

Proposed method

• Using the Chern-Simons formulation of 2+1D Einstein gravity, the authors compute the holonomy of the null orbifold and find its eigenvalues to be trivial.
• Constructing a higher spin resolution via a Kundt spacetime supported by higher spin fields, which exhibits vanishing scalar curvature invariants.
• Analyzing the 4-point tachyon scattering amplitude in string theory on the null orbifold, with a detailed scan of divergences in momentum and $\alpha'$-dependent parameters.
• Applying OPE techniques and contour integration to evaluate vertex operator matrix elements, particularly focusing on the $U$-integral structure and its dependence on energy ordering.
• Using generalized Fresnel integrals to evaluate oscillatory integrals arising from vertex operator overlaps, showing that non-analytic behavior arises only when $U > U_1 + U_2$.
• Relating the $\alpha = 0$ divergence in the amplitude to the Jacobian of the OPE, showing it arises when $U_2 - U_3 = U_4 - U_1$, corresponding to $\alpha = 0$.

Experimental results

Research questions

• RQ1 Does the Chern-Simons holonomy of the null orbifold reflect its singular geometry, or can it be trivial despite the pathology?
• RQ2 Can a higher spin theory provide a classical resolution of the null orbifold singularity, and if so, what are the geometric and field-theoretic properties of such a resolution?
• RQ3 Do UV divergences in the 4-point tachyon amplitude on the null orbifold persist in the tensionless ($\alpha' \to \infty$) limit, or are they tamed?
• RQ4 Are the remaining divergences in the tensionless limit physical IR divergences, and do they arise from on-shell intermediate string states?
• RQ5 What is the role of higher spin symmetries in resolving cosmological singularities, and how does this relate to the tensionless limit of string theory?

Key findings

• The Chern-Simons holonomy of the null orbifold has trivial eigenvalues, demonstrating that singular geometry need not be reflected in holonomy.
• A higher spin resolution of the null orbifold is realized as a Kundt spacetime with vanishing scalar curvature invariants, supported by higher spin gauge fields.
• All UV divergences in the 4-point tachyon amplitude on the null orbifold disappear in the $\alpha' \to \infty$ limit, indicating no unphysical UV pathology in the tensionless regime.
• The surviving divergences in the low-tension limit are IR divergences, arising when $U_1 + U_2 = U_3 + U_4$, consistent with physical expectations from on-shell intermediate states.
• The structure of divergences in the null orbifold amplitude closely parallels that in the Milne universe, suggesting a universal origin in string theory on covering Minkowski space.
• The non-analytic behavior in the amplitude is shown to be an IR effect, arising only when $U > U_1 + U_2$, and vanishes otherwise due to oscillatory integral cancellation.