Loop Corrections to Supergravity on $AdS_5 \times S^5$ (1706.02388v2)

Abstract: We consider the four-point correlator of the stress-energy tensor multiplet in ${\cal N}=4$ SYM. In the planar limit and at large 't Hooft coupling such correlator is given by the corresponding holographic correlation function in IIB supergravity on $AdS_5 \times S^5$. We consider subleading corrections in the number of colours, i.e. order $1/N^4$, at large 't Hooft coupling. This corresponds to loop corrections to the supergravity result. Consistency conditions, most notably crossing symmetry, constrain the form of such corrections and lead to a complete determination of the spectrum of leading twist intermediate operators.

• The paper computes subleading 1/N⁴ corrections to the spectrum of twist operators in N=4 SYM using crossing symmetry constraints.
• The method leverages analytic bootstrap techniques to manage operator mixing among double trace operators in the planar limit.
• The findings refine AdS/CFT predictions by accurately quantifying loop effects in the supergravity approximation.

Loop Corrections to Supergravity on $AdS_5 \times S^5$

The paper "Loop Corrections to Supergravity on $AdS_5 \times S^5$" by Luis F. Alday and Agnese Bissi explores the intricate arena of loop corrections within the framework of supergravity on $AdS_5 \times S^5$. The investigation is rooted in the well-established $AdS/CFT$ correspondence, specifically concerning $\mathcal{N}=4$ Super Yang-Mills (SYM) theory at strong 't Hooft coupling. The primary focus is on the four-point correlator of the stress-energy tensor multiplet within this theory, examined through the lens of loop corrections in the planar limit.

Background and Objective

The $AdS/CFT$ correspondence posits a duality between ${\cal N}=4$ SYM in four dimensions and type IIB string theory on the $AdS_5 \times S^5$ background. However, only certain segments of non-protected quantities have been systematically analyzed over the years, particularly within the planar and strong coupling regimes. This paper endeavors to extend the analysis by considering subleading corrections proportional to $1/N^4$, corresponding to loop effects in the supergravity representation.

Methodology

To derive loop corrections, the authors harness consistency conditions such as crossing symmetry, which significantly constrain the potential forms that these corrections can take. The analysis pays special attention to the intermediate operators with leading twist, engaging sophisticated techniques from the analytic bootstrap to manage operator mixings among double trace compressions, a challenging aspect at subleading orders.

Key Results

The paper delineates the subleading corrections to the spectrum of twist operators $\Delta_{n,\ell}$ and elucidates upon the OPE coefficients. Primary numerical outcomes include expressions for $\Delta_{0,2}$ and $\Delta_{0,4}$, which are methodically obtained as:

• $$\Delta_{0,2} = 6 - \frac{4}{N^2}- \frac{45}{N^4} + \cdots$$
• $$\Delta_{0,4} = 8 - \frac{48}{25}\frac{1}{N^2}-\frac{12768}{3125}\frac{1}{N^4}+\cdots$$

These results are indicative of the profound impact of loop corrections at order $1/N^4$, presenting novel insights into operator dimensions at large 't Hooft couplings.

Implications and Future Directions

The implications of these findings are manifold. From a theoretical standpoint, they add depth to our understanding of the $AdS/CFT$ correspondence, opening pathways to scrutinize quantum measurements in gravity. On a practical level, the methods refined here could be adapted for analyzing other gauge theories in similar holographic frameworks.

The paper also forecasts potential avenues for developing theories of quantum gravity by employing analytic bootstrap techniques initiated in this investigation. This paper expects to spark further inquiry into subleading corrections and quantum computations beyond tree-level amplitudes in $AdS$ spaces.

Conclusion

Luis F. Alday and Agnese Bissi, through this meticulous exploration, contribute significantly to the narrative of loop corrections in supergravity, laying groundwork for robust research into deeply interconnected quantum field theories and holographic principles. The examination of non-trivial sectors in $\mathcal{N}=4$ SYM at subleading orders, dictated by crossing symmetries and other phenomena, marks a sophisticated chapter in the ongoing discourse of holography and quantum gravity.