[Paper Review] Renyi entropy and C_T for p-forms on even spheres

J. S. Dowker|arXiv (Cornell University)|Jun 14, 2017

Black Holes and Theoretical Physics2 references3 citations

TL;DR

This paper computes the Rényi entropy and the central charge $ C_T $ for p-form gauge fields on even-dimensional spheres, using boundary conditions (absolute or relative) to define coexact forms. It derives a generating function for degeneracies and establishes duality between absolute and relative boundary conditions, yielding exact results for the entanglement entropy and $ C_T $ in even spheres.

ABSTRACT

Renyi entropy and central charge, $C_T$, are calculated for a coexact p--form on an even sphere with particular reference to the conformally invariant case. It is shown, for example, that the entanglement entropy is minus the standard conformal anomaly with no `shift' being required. The shift necessary for a conformal p--form, when using a hyperbolic technique, is predicted, on a numerical basis, to be (minus) the entanglement entropy of a conformal (p-1)-form. The central charges agree numerically with a general formula of Buchel {\it et al}.

Motivation & Objective

• To compute Rényi entropy and the central charge $ C_T $ for p-form gauge fields on even-dimensional spheres.
• To analyze degeneracies of p-forms under absolute and relative boundary conditions using a generating function formalism.
• To establish duality between absolute and relative boundary conditions for coexact forms.
• To derive exact expressions for Rényi entropy and $ C_T $ in even-dimensional spheres via spectral analysis.

Proposed method

• Uses a generating function $ d_b(p,\sigma) = \sum_{m=0}^\infty d_b(p,m) \sigma^m $ to encode degeneracies of p-forms under boundary conditions.
• Applies boundary conditions $ b = a $ (absolute) or $ b = r $ (relative) to define coexact p-forms on the fundamental domain of the group action $ \Gamma $.
• Introduces a finite, fermionic Poincaré series over form orders $ p $, though not used in the main derivation.
• Relies on spectral analysis of p-forms on spheres to compute Rényi entropy and $ C_T $, with duality between absolute and relative conditions.
• Treats $ q $ as the orbifold order, distinct from the generating variable $ \sigma $, to maintain clarity in the generating function.
• Derives results via the interplay between cohomological structure and boundary conditions, focusing on even spheres.

Experimental results

Research questions

• RQ1 How does the Rényi entropy of p-forms on even spheres depend on the form order $ p $ and boundary conditions?
• RQ2 What is the exact value of the central charge $ C_T $ for p-forms on even-dimensional spheres?
• RQ3 How do degeneracies of p-forms transform under absolute and relative boundary conditions?
• RQ4 What is the role of the generating function $ d_b(p,\sigma) $ in organizing spectral data for p-forms?
• RQ5 How does duality between absolute and relative boundary conditions manifest in Rényi entropy and $ C_T $?

Key findings

• The Rényi entropy for p-forms on even spheres is computed exactly using spectral data and boundary condition duality.
• The central charge $ C_T $ for p-forms on even spheres is derived via the generating function of degeneracies and boundary conditions.
• Degeneracies $ d_b(p,m) $ are encoded in a generating function $ d_b(p,\sigma) $, which organizes the spectrum of p-forms under $ \Gamma $-action.
• Duality between absolute ($ b = a $) and relative ($ b = r $) boundary conditions holds for coexact p-forms, enabling symmetry-based simplifications.
• The method yields exact results for Rényi entropy and $ C_T $ without relying on approximations or large-N limits.
• The orbifold order $ q $ is kept distinct from the generating variable $ \sigma $, ensuring clarity in spectral summing.