Higher derivative corrections in holographic Zamolodchikov-Polchinski theorem

TL;DR

This paper investigates higher derivative corrections in the holographic realization of the Zamolodchikov–Polchinski theorem, which relates scale invariance to conformal invariance in unitary, Poincaré-invariant theories. It shows that a generalized strict null energy condition (extending the usual NEC to higher-derivative gravity) suffices to force any scale-invariant bulk with $d$-dimensional Poincaré symmetry to have AdS$_{d+1}$ isometry, thereby ensuring conformal invariance of the dual field theory. The analysis extends from pure Einstein gravity to theories with higher-derivative terms, demonstrating that non-conformal, scale-invariant configurations are forbidden by the generalized NEC, with concrete illustrations via Gauss–Bonnet–Einstein gravity and a massive vector model. The work also strengthens the conceptual link between the Zamolodchikov–Polchinski theorem and the holographic $c$-theorem, suggesting unified constraints on holographic duals and their field-theoretic implications.

Abstract

We study higher derivative corrections in holographic dual of Zamolodchikov-Polchinski theorem that states the equivalence between scale invariance and conformal invariance in unitary d-dimensional Poincare invariant field theories. From the dual holographic perspective, we find that a sufficient condition to show the holographic theorem is the generalized strict null energy condition of the matter sector in effective (d+1)-dimensional gravitational theory. The same condition has appeared in the holographic dual of the "c-theorem" and our theorem suggests a deep connection between the two, which was manifested in two-dimensional field theoretic proof of the both.