Non-Negativity of BMN Two-Point Functions With Three String Modes

TL;DR

This paper verifies that BMN two-point functions with three string modes are always non-negative at genus two, supporting a conjecture related to the positivity in the holographic correspondence.

Contribution

It provides detailed calculations and numerical verification of the non-negativity of BMN two-point functions with three string modes at genus two.

Findings

BMN two-point functions are non-negative at genus two.

Numerical tests confirm the conjecture for higher genus cases.

Supports the holographic dictionary proposal for string modes.

Abstract

Recently, we proposed a novel entry of the pp-wave holographic dictionary, which equated the Berenstein-Maldacena-Nastase (BMN) two-point functions in free $\mathcal{N}=4$ super-Yang-Mills theory with the norm squares of the quantum unitary transition amplitudes between the corresponding tensionless strings in the infinite curvature limit, for the cases with no more than three string modes in different transverse directions. A seemingly highly non-trivial conjectural consequence, particularly in the case of three string modes, is the non-negativity of the BMN two-point functions at any higher genus for any mode numbers. In this paper, we further perform the detailed calculations of the BMN two-point functions with three string modes at genus two, and explicitly verify that they are always non-negative through mostly extensive numerical tests.