Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM

Abstract

We compute the grand partition function of $\mathcal{N}=4$ SYM at one-loop in the $SU(2)$ sector with general chemical potentials, extending the results of Pólya's theorem. We make use of finite group theory, applicable to all orders of $1/N_c$ expansion. We show that only the planar terms contribute to the grand partition function, which is therefore equal to the grand partition function of an ensemble of XXX${}_{\frac{1}{2}}$ spin chains. We discuss how Hagedorn temperature changes on the complex plane of chemical potentials.

34 pages, v2: typo corrected, v3: added a subsection about finite Nc, to appear in JHEP

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Cited by in corpus (7)

• The Hagedorn temperature of AdS5/CFT4 via integrability
• The Hagedorn temperature of AdS5/CFT4 at finite coupling via the Quantum Spectral Curve
• Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: Chemical potentials and deformations
• Algebraic geometry and Bethe ansatz (I) the quotient ring for BAE
• Q-systems with boundary parameters
• From Hagedorn to Lee-Yang: Partition functions of $\mathcal{N}=4$ SYM theory at finite $N$
• The $A_m^{(1)}$ Q-system