[Paper Review] A One-Loop Test of Quantum Supergravity

Sayantani Bhattacharyya, Alba Grassi+2 more|arXiv (Cornell University)|Oct 22, 2012

Black Holes and Theoretical Physics29 references4 citations

TL;DR

This paper tests the AdS/CFT correspondence at next-to-leading order in the 1/N expansion by computing the one-loop logarithmic correction to the partition function in eleven-dimensional supergravity on AdS₄×X₇, finding perfect agreement with the universal logarithmic term observed in ABJM-type CFTs. The correction arises solely from zero modes of the two-form ghost field, explaining its universality across different compactifications.

ABSTRACT

The partition function on the three-sphere of ABJM theory and its generalizations has, at large N, a universal, subleading logarithmic term. Inspired by the success of one-loop quantum gravity for computing the logarithmic corrections to black hole entropy, we try to reproduce this universal term by a one-loop calculation in Euclidean eleven-dimensional supergravity on AdS_4 imes X_7. We find perfect agreement between the results of ABJM theory and the eleven dimensional supergravity.

Motivation & Objective

• To test the AdS/CFT correspondence at next-to-leading order in the 1/N expansion using quantum gravity corrections.
• To determine whether the universal subleading logarithmic correction in the three-sphere partition function of ABJM-like CFTs can be reproduced via a one-loop calculation in eleven-dimensional supergravity.
• To identify the origin of the universal logarithmic term in terms of zero modes in the supergravity background AdS₄×X₇.
• To establish a direct correspondence between microscopic CFT results and macroscopic quantum gravity computations in the context of M-theory compactifications.

Proposed method

• Perform a one-loop quantum gravity calculation in Euclidean eleven-dimensional supergravity on the AdS₄×X₇ background dual to ABJM-type CFTs.
• Identify the contribution to the free energy from zero modes, focusing on the two-form anticommuting ghost field arising from the three-form gauge field quantization.
• Compute the logarithmic correction using the heat kernel method, where the absence of constant terms in the odd-dimensional heat kernel expansion implies that only zero modes contribute to the logarithmic term.
• Use the fact that the integration range for the ghost zero modes is independent of the AdS radius L, leading to a logarithmic dependence on L in the partition function.
• Apply the general framework of one-loop quantum gravity corrections to black hole entropy to the present context, adapting it to M-theory backgrounds.
• Derive the logarithmic correction as −3/2 log L by analyzing the beta function coefficient β₂ = 3 for the ghost two-form, matching the field theory result.

Experimental results

Research questions

• RQ1 Can the universal subleading logarithmic correction in the three-sphere partition function of ABJM-like CFTs be reproduced via a one-loop calculation in eleven-dimensional supergravity on AdS₄×X₇?
• RQ2 What is the origin of the universality of the logarithmic correction in the CFT partition function, and does it persist in the gravity dual?
• RQ3 Why do only zero modes contribute to the logarithmic correction in eleven-dimensional supergravity, and how does this differ from metric or other field zero modes?
• RQ4 How does the L-independence of the integration range for ghost zero modes lead to a universal log L correction independent of the X₇ manifold?
• RQ5 Can the one-loop quantum gravity framework used for black hole entropy corrections be generalized to M-theory compactifications and higher-dimensional AdS backgrounds?

Key findings

• The one-loop calculation in eleven-dimensional supergravity on AdS₄×X₇ yields a logarithmic correction of −3/2 log L to the free energy, matching exactly the universal term found in the large-N partition function of ABJM-type CFTs.
• The logarithmic correction arises exclusively from the zero modes of the two-form anticommuting ghost field, which is a consequence of the odd-dimensional nature of the spacetime and the absence of constant terms in the heat kernel expansion.
• The integration range for the ghost zero modes is independent of the AdS radius L, leading to a universal L-dependent logarithmic term that does not depend on the specific geometry of X₇.
• The result confirms the universality of the logarithmic correction: it is the same across all ABJM-like theories and all X₇ manifolds, both in the CFT and in the gravity dual.
• The computation provides a non-trivial test of the AdS/CFT correspondence at next-to-leading order in the 1/N expansion, extending the one-loop quantum gravity program from black holes to M-theory compactifications.
• The framework used here can be generalized to study logarithmic corrections in type IIA and IIB compactifications, such as AdS₆×X₄, where both zero and non-zero modes may contribute depending on the compactification manifold.