Exact results for five-dimensional superconformal field theories with gravity duals (1207.4359v2)

Abstract: We apply the technique of supersymmetric localization to exactly compute the $S^5$ partition function of several large $N$ superconformal field theories in five dimensions that have $AdS_6$ duals in massive type IIA supergravity. The localization computations are performed in the non-renormalizable effective field theories obtained through relevant deformations of the UV superconformal field theories. We compare the $S^5$ free energy to a holographic computation of entanglement entropy in the $AdS_6$ duals and find perfect agreement. In particular, we reproduce the $N^{5/2}$ scaling of the $S^5$ free energy that was expected from supergravity.

Exact Results for Five-Dimensional Superconformal Field Theories with Gravity Duals

This paper by Daniel L. Jafferis and Silviu S. Pufu addresses the computational intricacies of the partition functions in five-dimensional superconformal field theories (SCFTs), particularly those with gravity duals through the AdS/CFT correspondence. Utilizing supersymmetric localization, the authors derive precise results for the partition functions of these theories on $S^5$, revealing a striking agreement with computations of entanglement entropy in the corresponding AdS backgrounds.

Key Insights and Methodology

The technique of supersymmetric localization enables exact computations in SCFTs, even at strong coupling, by exploiting the properties of certain supercharges and Q-exact operators. By considering relevant deformations of the ultraviolet (UV) SCFTs, the authors manage to compare the field theory free energy computed via localization with holographic approaches. In particular, the paper reproduces the $N^{5/2}$ scaling behavior of the free energy on $S^5$ that is anticipated from supergravity predictions.

In detailing their computational framework, the paper describes how the infinite-dimensional path integral is reduced to a finite-dimensional one, focusing on the Coulomb branch configurations where the fields are constant. The evaluation of the partition function takes into account various saddle points and relies heavily on the large N limit where N is the number of coincident branes or color factors.

Numerical Results and Theoretical Implications

The authors find that the free energy F on $S^5$ scales as $$-\frac{9\sqrt{2}\pi\cdot n^{3/2} \cdot N^{5/2}}{5\sqrt{8 - N_f}}$$, providing a compelling consistency check for several theoretical conjectures:

1. The AdS/CFT duality between the field theories considered and their gravity backgrounds.
2. The equivalence between the free energy on $S^5$ and the entanglement entropy across a 3-sphere.
3. The holographic computation method for entanglement entropy.

Furthermore, the paper speculates on the broader applicability of these results via a conjecture that posits $-\Delta F$ being positive for all five-dimensional conformal field theories, bearing analogy to the F-theorem in three dimensions.

Challenges and Future Directions

The five-dimensional setting introduces unique challenges, such as gauge field localization on instanton backgrounds, whose full contributions remain incompletely understood but are argued to be suppressed at large N. Also, the presence of curvature singularities in the supergravity background imposes difficulties on straightforward calculations of the on-shell action, prompting the use of alternative approaches like entanglement entropy calculations.

Looking forward, further exploration into the landscape of 5d SCFTs with gravity duals may uncover new facets of gauge/gravity duality and spur advances in understanding the nature of strongly-coupled field dynamics in higher dimensions. Moreover, refining techniques for suppressing instanton contributions at large N could enhance computational efficiency and expand the scope of localization-based exact results in superconformal settings.

In conclusion, this paper articulates a technically refined symbiosis of localization techniques and holographic principles, providing valuable insights into the underlying structure and scaling behaviors of five-dimensional SCFTs while paving the way for deeper investigations into quantum field theories in higher dimensions.