Partition functions of higher derivative conformal fields on conformally related spaces

The character integral representation of one loop partition functions is useful to establish the relation between partition functions of conformal fields on Weyl equivalent spaces. The Euclidean space Sa × AdSb can be mapped to Sa+b provided Sa and AdSb are of the same radius. As an example, to begin with, we show that the partition function in the character integral representation of conformally coupled free scalars and fermions are identical on Sa × AdSb and Sa+b. We then demonstrate that the partition function of higher derivative conformal scalars and fermions are also the same on hyperbolic cylinders and branched spheres. The partition function of the four-derivative conformal vector gauge field on the branched sphere in d = 6 dimension can be expressed as an integral over ‘naive’ bulk and ‘naive’ edge characters. However, the partition function of the conformal vector gauge field on $$ {S}_q^1 $$ S q 1 × AdS5 contains only the ‘naive’ bulk part of the partition function. This follows the same pattern which was observed for the partition of conformal p-form fields on hyperbolic cylinders. We use the partition function of higher derivative conformal fields on hyperbolic cylinders to obtain a linear relationship between the Hofman-Maldacena variables which enables us to show that these theories are non-unitary.

Partition functions of p-forms from Harish-Chandra characters

We show that the determinant of the co-exact p-form on spheres and anti-de Sitter spaces can be written as an integral transform of bulk and edge Harish-Chandra characters. The edge character of a co-exact p-form contains characters of anti-symmetric tensors of rank lower to p all the way to the zero-form. Using this result we evaluate the partition function of p-forms and demonstrate that they obey known properties under Hodge duality. We show that the partition function of conformal forms in even d + 1 dimensions, on hyperbolic cylinders can be written as integral transforms involving only the bulk characters. This supports earlier observations that entanglement entropy evaluated using partition functions on hyperbolic cylinders do not contain contributions from the edge modes. For conformal coupled scalars we demonstrate that the character integral representation of the free energy on hyperbolic cylinders and branched spheres coincide. Finally we propose a character integral representation for the partition function of p-forms on branched spheres.

Hyperbolic cylinders and entanglement entropy: gravitons, higher spins, p-forms

We show that the entanglement entropy of D = 4 linearized gravitons across a sphere recently computed by Benedetti and Casini coincides with that obtained using the Kaluza-Klein tower of traceless transverse massive spin-2 fields on S1× AdS3. The mass of the constant mode on S1 saturates the Brietenholer-Freedman bound in AdS3. This condition also ensures that the entanglement entropy of higher spins determined from partition functions on the hyperbolic cylinder coincides with their recent conjecture. Starting from the action of the 2-form on S1× AdS5 and fixing gauge, we evaluate the entanglement entropy across a sphere as well as the dimensions of the corresponding twist operator. We demonstrate that the conformal dimensions of the corresponding twist operator agrees with that obtained using the expectation value of the stress tensor on the replica cone. For conformal p-forms in even dimensions it obeys the expected relations with the coefficients determining the 3-point function of the stress tensor of these fields.

Normally hyperbolic cylinders at double resonance

This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). In the non-simple homology case, the chapter demonstrates that for positive energy, there exist periodic orbits. The strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block.

New characterizations of spacelike hyperplanes in the steady state space

In this article, we deal with complete spacelike hypersurfaces immersed in an open region of the de Sitter space $\mathbb {S}^{n+1}_{1}$ which is known as the steady state space $\mathcal {H}^{n+1}$. Under suitable constraints on the behavior of the higher order mean curvatures of these hypersurfaces, we are able to prove that they must be spacelike hyperplanes of $\mathcal {H}^{n+1}$. Furthermore, through the analysis of the hyperbolic cylinders of $\mathcal {H}^{n+1}$, we discuss the importance of the main hypothesis in our results. Our approach is based on a generalized maximum principle at infinity for complete Riemannian manifolds.