Comment on CFT in AdS and boundary RG flows: O(1/N) Result

TL;DR

This paper clarifies the exact equivalence between a recent series expansion for boundary anomalous dimensions in AdS/CFT and a previous analytic expression, correcting the final formula and extending the analysis to all transition types.

Contribution

It demonstrates the precise equivalence of the series expansion to the original formula and provides corrected, explicit expressions involving hypergeometric functions for all boundary transition cases.

Findings

Confirmed the series expansion matches the original expression.

Corrected the final formula involving hypergeometric functions.

Extended the analysis to all boundary transition types.

Abstract

In a recent paper [JHEP 11 (2020) 118], S. Giombi and H. Khanchandani studied the 1/N expansion of the O(N) model in semi-infinite space within the framework of conformal field theory in anti-de Sitter space. They presented a series expansion for the O(1/N) correction to the boundary anomalous dimension in the case of the ordinary transition. Although they were unable to sum the series or simplify its form analytically, they demonstrated numerically that their result matches our earlier, simple analytic expression given in Prog. Theor. Phys. 70 (1983) 1226. In this paper, we show that their series expansion is in fact exactly equivalent to our original expression. However, since the final formula in eq. (4.57) of their paper, which is expressed in terms of two different 3F2 functions, cannot produce the correct values, we derive the correct formulae involving two 3F2 functions in the Appendices. We comment on the similarity and difference between their analysis and ours in the cases of the ordinary and special transitions, and present full details of deriving the anomalous dimension for all the cases including the extraordinary transition, which were not written in our earlier paper.