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

Abstract

We consider high spin, $s$, long twist, $L$, planar operators (asymptotic Bethe Ansatz) of strong $\mathcal{N}=4$ 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.

Version 2 with: new exact expression for the Casimir energy derived (beyond the first two loops of the previous version); UV theory formulated and analysed extensively in the Appendix C; origin of the O(6) NLSM scattering clarified; typos correct and references added

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• Asymptotic Bethe Ansatz on the GKP vacuum as a defect spin chain: scattering, particles and minimal area Wilson loops
• Bethe Ansaetze for GKP strings
• On the scattering over the GKP vacuum
• One-loop spectroscopy of semiclassically quantized strings: bosonic sector
• Strong Wilson polygons from the lodge of free and bound mesons
• Exploring superconformal Yang-Mills theories through matrix Bessel kernels
• A Young diagram expansion of the hexagonal Wilson loop (amplitude) in $\mathcal{N}=4$ SYM