TBA-like equations and Casimir effect in (non-)perturbative AdS/CFT

TL;DR

This paper computes minimal anomalous dimensions of high spin, long twist operators in strong coupling ${ m extbf{N}=4}$ SYM, extending the O(6) sigma model description to include finite size effects and Casimir corrections, with implications for string theory.

Contribution

It extends the O(6) sigma model framework to incorporate finite size effects and Casimir corrections in the analysis of high spin operators at strong coupling.

Findings

Confirmed the O(6) sigma model description at leading order.

Derived the exact effect of finite size on the spectrum.

Predicted higher-loop corrections and wrapping effects.

Abstract

We consider high spin, $s$, long twist, $L$, planar operators (asymptotic Bethe Ansatz) of strong ${\cal N}=4$ SYM. Precisely, we compute the minimal anomalous dimensions for large 't Hooft coupling $\lambda$ to the lowest order of the (string) scaling variable $\ell \sim L/ (\ln \mathcal{S} \sqrt{\lambda})$ with GKP string size $\sim\ln \mathcal{S}\equiv 2 \ln (s/\sqrt{\lambda})$. At the leading order $(\ln \mathcal{S}) \cdot \ell ^2 $, we can confirm the O(6) non-linear sigma model description for this bulk term, without boundary term $(\ln \mathcal{S})^0$. Going further, we derive, extending the O(6) regime, the exact effect of the size finiteness. In particular, we compute, at all loops, the first Casimir correction $\ell ^0/\ln \mathcal{S}$ (in terms of the infinite size O(6) NLSM), which reveals only one massless mode (out of five), as predictable once the O(6) description has been extended. Consequently, upon comparing with string theory expansion, at one loop our findings agree for large twist, while reveal for negligible twist, already at this order, the appearance of wrapping. At two loops, as well as for next loops and orders, we can produce predictions, which may guide future string computations.