Exact Slope and Interpolating Functions in ABJM Theory

TL;DR

The paper addresses the problem of obtaining an exact slope function in planar ABJM theory by leveraging the Quantum Spectral Curve (QSC) for ABJM. It derives an exact, near-BPS slope function $\gamma_L(h)$ expressed as a determinant-ratio depending on the interpolating function $h(\lambda)$ and validates it against known weak-coupling Luscher corrections and strong-coupling quasi-classical results. A key contribution is the conjectured exact relation for $h(\lambda)$ obtained by matching the slope-function structure to localization results, yielding explicit weak- and strong-coupling expansions and highlighting a possible matrix-model interpretation. Overall, the work provides a nonperturbative test of ABJM integrability, connects localization and integrability in a new way, and furnishes concrete predictions for short-operator dimensions in the near-BPS regime.

Abstract

Using the Quantum Spectral Curve approach we compute exactly an observable (called slope function) in the planar ABJM theory in terms of an unknown interpolating function h(λ) which plays the role of the coupling in any integrability based calculation in this theory. We verified our results with known weak coupling expansion in the gauge theory and with the results of semi-classical string calculations. Quite surprisingly at strong coupling the result is given by an explicit rational function of h(λ) to all orders. By comparing the structure of our result with that of an exact localization-based calculation for a similar observable in JHEP 1006 (2010) 011 we conjecture an exact expression for h(λ).

Figures (1)

• Figure 1: Antiperiodic functions $\rho_{2,3}(u)$ have infinitely many short cuts. In the variable $y=e^{2\pi u}$ only two cuts remain.