Universal Cardy-Like Behavior of 3D Partition Functions from Supersymmetric Localization

Abstract

We investigate 3d $\mathcal{N}=2$ supersymmetric gauge theories on $S^1\times S^2$ and the corresponding 2d effective field theories arising in the limit of small ratio of radii, $\beta =R_{S^1}/R_{S^2}\to 0$. We evaluate the exact partition function of these theories in the framework of supersymmetric localization on curved backgrounds. As a result, we establish a finite-$N$ map between a particular, superconformal-index-inspired partition function and the topologically twisted index. Taking the large-$N$ limit of the partition functions, we reproduce the entropy functions of either spherically symmetric, magnetically charged, or rotating, electrically charged asymptotically AdS${}_4$ black holes. We then recast the problem of evaluating the 3d partition functions directly in the framework of rigid supersymmetry. By carefully tracking the background fields, we find that in the small-$\beta$ limit, the partition functions of these 3d large-$N$ superconformal field theories have a universal behavior related to the coefficients of the R-symmetry or flavor symmetry 2-point current correlation functions, thus obtaining a universal Cardy-like formula for 3d $\mathcal{N}=2$ superconformal field theories.

37 pages, 2 figures, 3 tables; V2: expositions improved in Sec. 4.3, typos corrected in Sec. 4.4; V3: published version, more discussions added, main results unchanged

References in corpus (23)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Black hole microstates in AdS${}_4$ from supersymmetric localization
• The complete superconformal index for N=6 Chern-Simons theory
• Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS${}_5$ black holes
• Contact Terms, Unitarity, and F-Maximization in Three-Dimensional Superconformal Theories
• Cardy Formulae for SUSY Theories in d=4 and d=6
• The string origin of dyonic $\mathcal{N}=8$ supergravity and its simple Chern-Simons duals
• A String Model of Black Hole Microstates
• R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization
• Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula
• EFT and the SUSY Index on the 2nd Sheet
• Black hole entropy in massive Type IIA
• Microscopic Entropy of Rotating Electrically Charged AdS${}_4$ Black Holes from Field Theory Localization
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
• Universal 3d Cardy Block and Black Hole Entropy
• High-Temperature Expansion of Supersymmetric Partition Functions
• Current Correlators and AdS/CFT Geometry
• D2-brane Chern-Simons theories: F-maximization = a-maximization
• Large $N$ Superconformal Indices for 3d Holographic SCFTs
• Universal Logarithmic Behavior in Microstate Counting and the Dual One-loop Entropy of AdS${}_4$ Black Holes
• Supersymmetric phases of AdS${}_4$/CFT${}_3$
• Holonomy Saddles and Supersymmetry
• A quantum mechanics for magnetic horizons