A Pendant for Polya: The One-Loop Partition Function of N=4 SYM on R x S^3

Abstract

We study weakly coupled SU(N) N = 4 super Yang-Mills theory on R x S^3 at infinite N, which has interesting thermodynamics, including a Hagedorn transition, even at zero Yang-Mills coupling. We calculate the exact one-loop partition function below the Hagedorn temperature. Our calculation employs the representation of the one-loop dilatation operator as a spin chain Hamiltonian acting on neighboring sites and a generalization of Polya's counting of `necklaces' (gauge-invariant operators) to include necklaces with a `pendant' (an operator which acts on neighboring beads). We find that the one-loop correction to the Hagedorn temperature is delta ln T_H = + lambda/8 pi^2.Comment: 39 pages, harvmac. v2: references and some clarifications added, v3: proof of (3.28) correcte