[Paper Review] Higher dimensional formulation of counterterms

Marika Taylor-Robinson|ArXiv.org|Oct 16, 2001

Black Holes and Theoretical Physics3 citations

TL;DR

This paper investigates holographic renormalization in higher-dimensional asymptotically AdS₅×S⁵ and AdS₃×S³ spacetimes, demonstrating that while local boundary counterterms can cancel UV divergences, the counterterm action lacks covariance due to degenerate conformal boundaries. The key result is that counterterms depend on specific coordinate and gauge choices, yet a systematic renormalization procedure remains viable for both five- and ten-dimensional formulations of gauged supergravity.

ABSTRACT

It is by now well established that divergences of the on-shell action for asymptotically AdS solutions can be cancelled by adding covariant local boundary counterterms to the action. Here we show that although one can still renormalise the action for asymptotically $AdS_p imes S^q$ solutions using local boundary counterterms the counterterm action is not covariant since the conformal boundary is degenerate. Any given counterterm action is defined with respect to specific coordinate frame and gauge choices.

Motivation & Objective

• To extend holographic renormalization from lower-dimensional gauged supergravity to the full higher-dimensional action in AdS₅×S⁵ and AdS₃×S³ backgrounds.
• To investigate whether boundary counterterms can be formulated covariantly when the conformal boundary is degenerate due to the collapse of the compact manifold.
• To demonstrate a systematic procedure for renormalizing the on-shell action in ten-dimensional supergravity using five-dimensional counterterms and Kaluza-Klein uplift.
• To clarify the role of coordinate and gauge choices in defining counterterm actions for degenerate boundaries.

Proposed method

• Analyzes the divergences in the on-shell action for asymptotically AdS₃×S³ and AdS₅×S⁵ spacetimes using a higher-dimensional approach.
• Applies the method of holographic renormalization by introducing local boundary counterterms to cancel UV divergences.
• Uses coordinate transformations to bring the metric into a form suitable for boundary analysis near infinity.
• Matches ten-dimensional fields to five-dimensional effective fields via Kaluza-Klein ansatz and asymptotic expansions.
• Evaluates boundary contributions from brane distributions, showing that inner boundary terms vanish due to zero 5-volume of singular surfaces.
• Demonstrates that Gibbons-Hawking and brane-induced boundary terms vanish for generic D3-brane distributions, leading to a vanishing renormalized action.

Experimental results

Research questions

• RQ1 Can holographic renormalization be consistently applied to higher-dimensional supergravity actions for AdS₅×S⁵ and AdS₃×S³ backgrounds?
• RQ2 Why do boundary counterterms fail to be covariant in the presence of degenerate conformal boundaries?
• RQ3 How do coordinate and gauge choices affect the construction of counterterm actions in such backgrounds?
• RQ4 What is the role of Kaluza-Klein reduction in matching higher-dimensional solutions to lower-dimensional counterterms?
• RQ5 Why do boundary contributions from D3-brane distributions vanish in the renormalized action?

Key findings

• The renormalized action for BPS D3-brane distributions in AdS₅×S⁵ spacetime is zero, as all boundary contributions vanish due to zero 5-volume of singular surfaces.
• Counterterm actions for degenerate boundaries such as AdSₚ×S^q cannot be expressed in terms of covariant intrinsic boundary quantities, requiring explicit coordinate and gauge choices.
• The procedure for renormalizing the ten-dimensional action can be systematically extended by matching asymptotic expansions and using five-dimensional counterterms.
• Inner boundary terms from brane locations vanish because the boundary 5-volume is zero, even when the fields are singular at those points.
• The absence of finite contributions from both inner and outer boundaries confirms that the renormalized action is finite and well-defined despite non-covariant counterterms.
• The method successfully removes UV divergences in the on-shell action, even though the counterterms are not manifestly covariant due to the degeneracy of the conformal boundary.