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Large N Free Energy of 3d N=4 SCFTs and AdS/CFT

Benjamin Assel, John Estes, Masahito Yamazaki

TL;DR

The paper tests the AdS$_4$/CFT$_3$ correspondence for 3d $\mathcal{N}=4$ SCFTs $T^{\rho}_{\hat{\rho}}[SU(N)]$ by computing the CFT free energy from the $S^3$ partition function and matching it to the on-shell type IIB gravity action on the dual backgrounds. The prototypical case $T[SU(N)]$ yields a leading behavior $F \sim \frac{1}{2}N^2\ln N$, and this scaling persists across a broader family, with a closed-form coefficient depending on partition data in the general $\hat{p}=1$ limit; the gravity calculation reproduces the same leading term. The work demonstrates a nontrivial GKPW check in the large-$N$ limit and identifies the geometric origin of the $\ln N$ enhancement, while highlighting subleading discrepancies likely arising from brane singularities and higher-curvature/string-loop effects. Overall, the results establish the leading AdS$_4$/CFT$_3$ matching for this class and suggest a maximal large-$N$ free energy within the family.

Abstract

We provide a non-trivial check of the AdS_4/CFT_3 correspondence recently proposed in arXiv:1106.4253 by verifying the GKPW relation in the large N limit. The CFT free energy is obtained from the previous works (arXiv:1105.2551, arXiv:1105.4390) on the S^3 partition function for 3-dimensional N=4 SCFT T[SU(N)]. This is matched with the computation of the type IIB action on the corresponding gravity background. We unexpectedly find that the leading behavior of the free energy at large N is 1/2 N^2 ln N. We also extend our results to richer theories and argue that 1/2 N^2 ln N is the maximal free energy at large N in this class of gauge theories.

Large N Free Energy of 3d N=4 SCFTs and AdS/CFT

TL;DR

The paper tests the AdS/CFT correspondence for 3d SCFTs by computing the CFT free energy from the partition function and matching it to the on-shell type IIB gravity action on the dual backgrounds. The prototypical case yields a leading behavior , and this scaling persists across a broader family, with a closed-form coefficient depending on partition data in the general limit; the gravity calculation reproduces the same leading term. The work demonstrates a nontrivial GKPW check in the large- limit and identifies the geometric origin of the enhancement, while highlighting subleading discrepancies likely arising from brane singularities and higher-curvature/string-loop effects. Overall, the results establish the leading AdS/CFT matching for this class and suggest a maximal large- free energy within the family.

Abstract

We provide a non-trivial check of the AdS_4/CFT_3 correspondence recently proposed in arXiv:1106.4253 by verifying the GKPW relation in the large N limit. The CFT free energy is obtained from the previous works (arXiv:1105.2551, arXiv:1105.4390) on the S^3 partition function for 3-dimensional N=4 SCFT T[SU(N)]. This is matched with the computation of the type IIB action on the corresponding gravity background. We unexpectedly find that the leading behavior of the free energy at large N is 1/2 N^2 ln N. We also extend our results to richer theories and argue that 1/2 N^2 ln N is the maximal free energy at large N in this class of gauge theories.

Paper Structure

This paper contains 16 sections, 62 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Summary of the Results
  3. Review of $T^{\rho}_{\hat{\rho}}[SU(N)]$ Theories
  4. Large $N$ Free Energy
  5. CFT Analysis
  6. The $S^3$ Partition Function
  7. $T[SU(N)]$
  8. $T^{\rho}_{\hat{\rho}}[SU(N)]$
  9. Gravity Analysis
  10. Summary of the Gravity Solution
  11. The Gravity Action
  12. $T[SU(N)]$
  13. $T^{\rho}_{\hat{\rho}}[SU(N)]$
  14. Subleading Terms
  15. Barnes $G$-function
  16. ...and 1 more sections

Figures (3)

  • Figure 1: We decompose the young diagram corresponding to $\rho$ into $p$ blocks, see \ref{['rhoconvention']}.
  • Figure 2: The infinite strip with logarithmic singularities corresponding to stacks of five-branes. The upper (red) singularities correspond to D5-branes, while the lower (blue) singularities correspond to NS5-branes. The geometry smoothly caps off into an $S^6$ as $x \rightarrow \pm \infty$.
  • Figure 3: Geometry of the $T[SU(N)]$ dual background represented by the strip with two 5-brane singularities at positions $\pm \delta \sim \pm \frac{1}{2} \log N$. In the large $N$ limit the stacks go to $\pm$ infinity.