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

We study the thermodynamics of the maximally supersymmetric Yang-Mills theory with gauge group U(N ) on ℝ × S3, dual to type IIB superstring theory on AdS5× S5. 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 $$ \mathfrak{su} $$ su (2) sector containing two complex scalars, as well as at N ≤ 5 for the $$ \mathfrak{su} $$ 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 Poincaré series of trace algebras of generic matrices. Via this Molien-Weyl formula, we also generate exact results for larger sectors.

Spherically symmetric double layers in Weyl+Einstein gravity

We study the matching conditions on singular hypersurfaces in Weyl[Formula: see text]Einstein gravity. Unlike General Relativity, the so-called quadratic gravity allows the existence of a double layer, i.e. the derivative of [Formula: see text]-function. This double layer is a purely geometrical phenomenon and it may be treated as the purely gravitational shock wave. The mathematical formalism was elaborated by Senovilla for generic quadratic gravity. We derived the matching conditions for the spherically symmetric singular hypersurface in the Weyl[Formula: see text]Einstein gravity. It was found that in the presence of the double layer, the matching conditions contain an arbitrary function. One of the consequences of such freedom is that a trace of the extrinsic curvature tensor of a singular hypersurface is necessarily equal to zero. We suggested that the [Formula: see text] and [Formula: see text] components of the surface matter energy–momentum tensor of the shell describe energy flow [Formula: see text] and momentum transfer [Formula: see text] of particles produced by the double layer itself. Moreover, the requirement of the zero trace of the extrinsic curvature tensor (mentioned above) implies that [Formula: see text], and this fact also supports our suggestion, because it means that for the observer sitting on the shell, particles will be seen created by pairs, and the sum of their momentum transfers must be zero. We found also that the spherically symmetric null double layer in the Weyl[Formula: see text]Einstein gravity does not exist at all.

Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace–Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. The book shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. It begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. The book avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. It also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, the book demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.