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

Abstract

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.

Partition functions of p-forms from Harish-Chandra characters

“…In fact the integrand is precisely the bulk character of the conformal form on the sphere S d+1 . This is consistent with earlier observations found by studying entanglement entropy [14][15][16][17][18][19]. Entanglement entropy of theories with gauge symmetries evaluated using partition functions on hyperbolic cylinders contain only the bulk contributions and misses out on the edge terms.…”

“…Then only the bulk character of each of the p-form appears in (4.37), the edge modes are missing. This behaviour was observed in the study of the entanglement entropies of the Maxwell in d + 1 = 4 [14-16, 28, 29] field as well as the 2 form in d + 1 = 6 [17][18][19] and seen by Partition functions on the corresponding hyperbolic cylinders miss the edge modes. Here we see that this is true for conformal forms in all even dimensions and it can be seen at the level of the integrand in the character integral representation.…”

“…The interesting thing to note from (4.14) and (4. 19) is that we stared with the SO(2, d − 1) Harish-Chandra characters on AdS d . The sum over all the Kaluza-Klein modes resulted in SO(1, d + 1) Harish-Chandra characters.…”

“…In [19] a procedure to fix gauge on hyperbolic cylinders was introduced. Using this method the gauge invariant partition functions of the conformal 1-form on S 1 q × AdS 3 and the 2form on S 1 q × AdS 5 was obtained.…”

“…The conclusion in both the cases was the gauge invariant partition function can be thought of as the partition function of the tower of Kaluza-Klein tower of co-exact p-forms with the Hodge-de Rham Laplacian along the AdS directions. Though this observation was explicitly demonstrated only for the 1-form in d+1 = 4 and 2form in d + 1 = 6, the method developed in [19] is such that the result can be extrapolated to conformal p = d−1 2 on the hyperbolic cylinder S 1 q × AdS d . To summarise the gauge invariant partition function of conformal p forms is given by the following determinant…”