## Download Canonical Wick rotations in 3-dimensional gravity by Riccardo Benedetti, Francesco Bonsante PDF

By Riccardo Benedetti, Francesco Bonsante

The authors improve a canonical Wick rotation-rescaling conception in third-dimensional gravity. This comprises: a simultaneous type: this exhibits how maximal globally hyperbolic area instances of arbitrary consistent curvature, which admit a whole Cauchy floor and canonical cosmological time, in addition to advanced projective buildings on arbitrary surfaces, are all various materializations of 'more basic' encoding buildings; Canonical geometric correlations: this indicates how house occasions of alternative curvature, that proportion a similar encoding constitution, are with regards to one another by way of canonical rescalings, and the way they are often reworked via canonical Wick rotations in hyperbolic 3-manifolds, that hold definitely the right asymptotic projective constitution. either Wick rotations and rescalings act alongside the canonical cosmological time and feature common rescaling services. those correlations are functorial with appreciate to isomorphisms of the respective geometric different types

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Denotes the Witten-Reshetikhin-Turaev invariant and a complete topological QFT has been developed that embodies these last partition functions (see [59]). In the papers [9] a so called quantum hyperbolic ﬁeld theory QHFT was developed by using the bordism category (including particles) that we have depicted above. In fact this is a countable family of exact QFT’s, indexed by odd integers N ≥ 1. The building blocks are the so called matrix dilogarithms, which are automorphisms of CN ⊗ CN , associated to oriented hyperbolic ideal tetrahedra encoded by their cross-ratio moduli and equipped with an additional decoration.

It is convenient to consider the following modiﬁcation of Segre embedding S : P1 × P1 ([v], [w]) → [v ⊗ (Ew)] ∈ P3 28 2. GEOMETRY MODELS II I III IV Figure 2. The product structure on ∂X−1 . With respect to such a new embedding, the action of P SL(2, R) × P SL(2, R) on ∂X−1 is simply (A, B)(x, y) = (Ax, By). In what follows, we will consider the identiﬁcation of the boundary of X−1 with P1 × P1 given by S. The product structure on ∂X−1 given by S is preserved by the isometries of X−1 . e. a causal structure) on ∂X−1 .

Since the retraction r : U(1) → Σ is a Lipschitz map, Σ is connected by spacelike Lipschitz paths. Thus, we can consider the (a priori) pseudo-distance δ of Σ deﬁned by δ(x, y) = inf{ (c)| c Lipschitz path in Σ joining x to y} where (c) denotes the length of c. In fact it is a distance, and (Σ, δ) can be regarded as the limit of the level surfaces U(a) of T as a → 0. 6. [21] The function δ is a distance. Moreover if δa denotes the distance on U(a) we have that δa → δ as a → 0 in the following sense.