Abstract
We study the manifestly covariant and local 1-loop path integrals on Sd+1 for general massive, shift-symmetric and (partially) massless totally symmetric tensor fields of arbitrary spin s ≥ 0 in any dimensions d ≥ 2. After reviewing the cases of massless fields with spin s = 1, 2, we provide a detailed derivation for path integrals of massless fields of arbitrary integer spins s ≥ 1. Following the standard procedure of Wick-rotating the negative conformal modes, we find a higher spin analog of Polchinski’s phase for any integer spin s ≥ 2. The derivations for low-spin (s = 0, 1, 2) massive, shift-symmetric and partially massless fields are also carried out explicitly. Finally, we provide general prescriptions for general massive and shift-symmetric fields of arbitrary integer spins and partially massless fields of arbitrary integer spins and depths.
Highlights
Arising as the leading saddle point for the gravitational Euclidean path integral with a positive cosmological constant, the sphere plays a dominant role in the study of quantum gravity in de Sitter space [1,2,3,4,5,6,7,8,9,10,11,12]
After reviewing the cases of massless fields with spin s = 1, 2, we provide a detailed derivation for path integrals of massless fields of arbitrary integer spins s ≥ 1
We focus on the vector part of the 1-loop path integral
Summary
Arising as the leading saddle point for the gravitational Euclidean path integral with a positive cosmological constant, the sphere plays a dominant role in the study of quantum gravity in de Sitter space [1,2,3,4,5,6,7,8,9,10,11,12]. To generalize the formula (1.1) for wider classes of field contents, one must first obtain their correct functional determinant expressions This is the main motivation for this work. Generalize all known results to any massive, shift-symmetric and (partially) massless totally symmetric tensor fields of arbitrary spin s ≥ 0 in any dimensions d ≥ 2. Analogous expressions are derived for any massive (equation (5.19)), shift-symmetric (equation (6.22)) and partially massless (equation (7.11)) totally symmetric tensor fields of arbitrary spin s ≥ 0 in any dimensions d ≥ 2. With these precise expressions one could derive a formula analogous to (1.1) for more general representations. The higher spin invariant bilinear form is reviewed in appendix C
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