From Hagedorn to Lee-Yang: Partition functions of $\mathcal{N}=4$ SYM theory at finite $N$

Abstract

We study the thermodynamics of the maximally supersymmetric Yang-Mills theory with gauge group U(N) on R x S^3, dual to type IIB superstring theory on AdS_5 x S^5. While both theories are well-known to exhibit Hagedorn behavior at infinite N, we find evidence that this is replaced by Lee-Yang behavior at large but finite N: the zeros of the partition function condense into two arcs in the complex temperature plane that pinch the real axis at the temperature of the confinement-deconfinement transition. Concretely, we demonstrate this for the free theory via exact calculations of the (unrefined and refined) partition functions at N<=7 for the su(2) sector containing two complex scalars, as well as at N<=5 for the su(2|3) sector containing 3 complex scalars and 2 fermions. In order to obtain these explicit results, we use a Molien-Weyl formula for arbitrary field content, utilizing the equivalence of the partition function with what is known to mathematicians as the Poincare series of trace algebras of generic matrices. Via this Molien-Weyl formula, we also generate exact results for larger sectors.

arXiv comment34 pages, 3 figures, 2 ancillary files; v2: some references and clarifications added, matches published version

References in corpus (54)

• Review of AdS/CFT Integrability: An Overview
• Quantum Gravity Partition Functions in Three Dimensions
• The Hagedorn Transition, Deconfinement and N=4 SYM Theory
• Counting Gauge Invariants: the Plethystic Program
• Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics
• Exact Correlators of Giant Gravitons from dual N=4 SYM
• A first order deconfinement transition in large N Yang-Mills theory on a small 3-sphere
• Diagonal multi-matrix correlators and BPS operators in N=4 SYM
• Exact Multi-Matrix Correlators
• Black holes in 4d $\mathcal{N}=4$ Super-Yang-Mills
• SQCD: A Geometric Apercu
• Phase Diagram of N=4 Super-Yang-Mills Theory with R-Symmetry Chemical Potentials
• Counting BPS operators in N=4 SYM
• Exact Multi-Restricted Schur Polynomial Correlators
• Review of AdS/CFT Integrability, Chapter VI.1: Superconformal Symmetry
• Asymptotic growth of the 4d $\mathcal{N}=4$ index and partially deconfined phases
• A proposal of the gauge theory description of the small Schwarzschild black hole in AdS${}_5\times$S${}^5$
• Surprisingly Simple Spectra
• Spin Matrix Theory: A quantum mechanical model of the AdS/CFT correspondence
• Quantum Mechanical Sectors in Thermal N=4 Super Yang-Mills on RxS^3
• An Integrability Primer for the Gauge-Gravity Correspondence: an Introduction
• Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in N=4 Super Yang-Mills
• Anatomy of Deconfinement
• A Pendant for Polya: The One-Loop Partition Function of N=4 SYM on R x S^3
• Submatrix deconfinement and small black holes in AdS
• Free Fermions and Thermal AdS/CFT
• Lee-Yang zero distribution of high temperature QCD and Roberge-Weiss phase transition
• Partial Deconfinement
• Decoupling limits of N=4 super Yang-Mills on R x S^3
• The Hagedorn temperature of AdS5/CFT4 via integrability
• Probing QCD Phase Structure by Baryon Multiplicity Distribution
• The Hagedorn temperature of AdS5/CFT4 at finite coupling via the Quantum Spectral Curve
• Lee-Yang zeros in lattice QCD for searching phase transition points
• Nonplanar integrability at two loops
• Free particles from Brauer algebras in complex matrix models
• Thermal phase transition in Yang-Mills matrix model
• Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: Chemical potentials and deformations
• Two loop partition function for large N pure Yang-Mills theory on a small three-sphere
• More Pendants for Polya: Two loops in the SU(2) sector
• Negative specific heat from non-planar interactions and small black holes in AdS/CFT
• T-Reflection
• The 2-loop partition function of large N gauge theories with adjoint matter on S^3
• Search of QCD phase transition points in the canonical approach of the NJL model
• Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM
• String Bits at Finite Temperature and the Hagedorn Phase
• Hagedorn Temperature in Superstring Bits and SU(N) Characters
• Poincare series of some pure and mixed trace algebras of two generic matrices
• Towards extremely dense matter on the lattice
• Color Characters for White Hot String Bits
• Temperature-reflection I: field theory, ensembles, and interactions
• One-Loop Partition Functions in Deformed $\mathcal{N}=4$ SYM Theory
• Temperature-reflection II: Modular Invariance and T-reflection
• Finite N corrections to white hot string bits
• Partial deconfinement in gauge theories

Cited by in corpus (4)

• Aspects of Non-Relativistic Quantum Field Theories
• Spectral form factor for free large $N$ gauge theory and strings
• Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: Chemical potentials and deformations
• All-orders asymptotics of tensor model observables from symmetries of restricted partitions