[Paper Review] Computing 1/N^2 corrections in AdS/CFT

James T. Liu, Ruben Minasian|arXiv (Cornell University)|Oct 28, 2010

Black Holes and Theoretical PhysicsPhysics and Astronomy60 references16 citations

TL;DR

This paper computes 1/N² corrections from closed string loops in AdS/CFT compactifications of IIB string theory on Sasaki-Einstein manifolds, finding that for T¹,¹ reductions, the loop correction yields c−a=1/24, matching the contribution of a free N=2 hypermultiplet. The result is derived via dimensional reduction of the IIA one-loop CP-odd term B∧X₈ and supersymmetry, revealing a universal 1/N² effect in the dual N=1 SCFT.

ABSTRACT

Stringy corrections in AdS/CFT generally fall into the category of either α' effects or string loop effects, corresponding to 1/λand 1/N corrections, respectively, in the dual field theory. While α'^3R^4 corrections have been well studied, at least in the context of N=4 super-Yang-Mills, less is known about the 1/N^2 corrections arising from closed string loops. In this paper, we consider AdS_5 x SE_5 compactifications of the IIB string, and compute the closed string loop correction to the anomaly coefficients a and c in the dual field theory. For T^{1,1} reductions, we find the string loop correction to yield c-a=1/24, which is the contribution to c-a of a free N=2 hypermultiplet. We also comment on reductions to lower dimensional AdS theories as well as the nature of T-duality with higher derivatives.

Motivation & Objective

• To compute 1/N² corrections in AdS/CFT arising from closed string loops in IIB string theory compactifications on Sasaki-Einstein 5-folds.
• To determine the impact of these loop corrections on the anomaly coefficients a and c in the dual 4D N=1 superconformal field theory.
• To clarify the role of higher-derivative terms, particularly the CP-odd B∧X₈ term, in the effective action and their reduction to lower dimensions.
• To establish a connection between string loop effects and the spectrum of hypermultiplets in the dual field theory.

Proposed method

• Reduction of the IIA one-loop CP-odd term B∧X₈ in 10D to 5D via compactification on Calabi-Yau threefolds.
• Use of supersymmetry to lift the 5D effective action to a complete N=2 supergravity theory with vector and hypermultiplets.
• Calculation of the anomaly coefficients a and c from the 5D action, focusing on the contribution from the Chern-Simons term involving the second Chern class c₂I.
• Application of the generalized geometry framework with Courant brackets and O(1,1) T-duality to analyze the T-duality invariance of the CP-odd corrections.
• Use of the 11D supergravity action with the C₃∧X₈ term as a lift of the IIA B∧X₈ term to reconstruct the 5D action via circle reduction.
• Derivation of the 1/N² correction by identifying the loop-induced contribution to the anomaly trace anomaly c−a as arising from the hypermultiplet sector.

Experimental results

Research questions

• RQ1 What is the contribution of closed string loops to the anomaly coefficients a and c in the 4D N=1 SCFT dual to AdS₅×T¹,¹ compactifications of IIB string theory?
• RQ2 How do the 1/N² corrections from string loops compare to the known α′³R⁴ corrections in the AdS/CFT correspondence?
• RQ3 Can the CP-odd one-loop term B∧X₈ in IIA string theory be consistently reduced to 5D and related to the anomaly structure in the dual field theory?
• RQ4 What is the role of T-duality in preserving or modifying the structure of higher-derivative corrections in the effective action?
• RQ5 Does the 1/N² correction from string loops reproduce the anomaly contribution of a free N=2 hypermultiplet in the dual field theory?

Key findings

• The 1/N² correction from closed string loops in the T¹,¹ compactification yields c−a=1/24, which exactly matches the contribution of a single free N=2 hypermultiplet to the anomaly trace.
• The correction arises from the Chern-Simons term involving the second Chern class c₂I in the 5D N=2 supergravity action, induced by the compactification of the IIA one-loop CP-odd term B∧X₈.
• The result is consistent with the expectation that 1/N² effects in the field theory correspond to string loop effects in the bulk, providing a concrete realization of this correspondence.
• The T-duality invariance of the CP-odd term B∧X₈ is preserved under circle reduction, and the resulting 5D action correctly captures the anomaly structure.
• The computation confirms that the 1/N² correction is universal and independent of the specific Calabi-Yau geometry, depending only on the topological data of the Sasaki-Einstein manifold.
• The framework allows for generalization to other AdS₅×SE₅ compactifications and provides a method to compute 1/N² corrections in general AdS/CFT setups.