Abstract
We consider Einstein gravity with positive cosmological constant coupled with higher spin interactions and calculate Euclidean path integral perturbatively. We confine ourselves to the static patch of the 3 dimensional de Sitter space. This geometry, when Euclideanlized is equivalent to 3-sphere. However, infinite number of topological quotients of this space by discrete subgroups of the isometry group are valid Euclidean saddles as well. The case of pure Einstein gravity is known to give a diverging answer, when all saddles are included as contribution to the thermal partition functions (also interpreted as the Hartle Hawking state in the cosmological scenario). We show how higher spins, described by metric-Fronsdal fields help making the partition function finite. We find a curious fact that this convergence is not achieved by mere inclusion of spin-3, but requires spin-4 interactions.
Highlights
It was the seminal work by Hartle and Hawking [1] where a plausible candidate for the same was proposed by defining the vacuum state as a Euclidean functional integral over geometries with fixed data over some compact co-dimension-1 hyper-surface
While investigating the HartleHawking vacuum state for quantum gravity in the functional integral approach, it is natural to include all classical saddles or geometries, around which fluctuations would be evaluated in a perturbative manner
This is in the sense that if one minimally couples higher spin fields with pure gravity, the linearized higher spin action for a given spin would result into a 1-loop partition function structurally similar to that of the spin-2 case
Summary
We just discussed Lens spaces as quotients of S3 and how the geometry of S3 (or thermal de Sitter) induces the geometry on them. We keep in mind that the induced geometry (metric) is a local structure, while global properties are captured in the topological properties In this view, let us stress here that a Lens space can be constructed by gluing two solid tori at their boundaries (which are 2-tori themselves). One uses an element of the mapping class group P SL(2, Z) ≡ SL(2, Z)/Z2 ( in this case the group of modular transformations) of the toric boundaries. Another structure of relevance here would be the fundamental group π1(T 2) = Z ⊕ Z of the boundary tori. Rather the left right coset Z\P SL(2, Z)/Z is the subgroup which has injective maps to the set of Lens spaces
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