From Hagedorn to Lee-Yang: partition functions of N $$ \mathcal{N} $$ = 4 SYM theory at finite N
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Springer
Received: May 28, Revised: August 5, Accepted: August 9, Published: October 1,
2020 2020 2020 2020
Alexander T. Kristensson and Matthias Wilhelm Niels Bohr International Academy, Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, Copenhagen Ø 2100, Denmark
E-mail: [email protected], [email protected] Abstract: We study the thermodynamics of the maximally supersymmetric Yang-Mills theory with gauge group U(N ) on R×S 3 , dual to type IIB superstring theory on AdS5 ×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 Poincar´e series of trace algebras of generic matrices. Via this Molien-Weyl formula, we also generate exact results for larger sectors. Keywords: AdS-CFT Correspondence, Confinement, Supersymmetric Gauge Theory ArXiv ePrint: 2005.06480
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)006
JHEP10(2020)006
From Hagedorn to Lee-Yang: partition functions of N = 4 SYM theory at finite N
Contents 1
2 Partition functions at infinite and finite N 2.1 The character formula 2.2 The Molien-Weyl formula
5 5 6
3 The su(2) sector and Lee-Yang behavior
8
4 Generalization to larger sectors 4.1 su(2|3) sector 4.2 sl(2) sector 4.3 psu(1, 1|2) sector 4.4 Full theory
13 13 15 18 19
5 Conclusion and outlook
19
A Derivation of the character formula
21
B Derivation of the Molien-Weyl formula
23
C Calculation of the partition function in the psu(1, 1|2) sector at N = 2
25
1
Introduction
Gauge theories exhibit a rich thermodynamic structure, much of which is still to be understood. This is even the case for what might be the simplest gauge theory, namely the maximally (N = 4) supersymmetric Yang-Mills (SYM) theory with gauge group U(N ), on R × S 3 . Via the AdS/CFT correspondence [1], a dual description of N = 4 SYM theory is given by type IIB superstring theory on AdS5 × S 5 , which has been used to study the thermodynamic properties of both theories from early on [2]. Gauß’ law dictates that the states on a compact space such as S 3 are color singlets, leading to a phase transition in N = 4 SYM theory that bears resemblance to the confinementdeconfinement transition in QCD. The compact space S 3 allows also for a direct comparison between the conformal N = 4 SYM theory
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