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Causal Dynamical Triangulation

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Space-Time Foliation in Quantum Gravity

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Abstract

In this chapter we argue exactly solvable 2-dimensional Causal Dynamical Triangulations (CDT) and their generalization called generalized CDT. In 2.1 we introduce new multicritical models which describe the matter-coupled version of CDT and generalized CDT; especially we focus on the third-order multicritical point where the conformal field theory with the central charge, \(c=-22/5\), couples to (the generalized) CDT. In 2.2 we extend the generalized CDT based on the string field theory of the generalized CDT in such a way that the space-time foliation is preserved; the wavefunction of the Universe in the extended model can be obtained perturbatively. We also show that there exist a matrix model description of the extended model.

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Notes

  1. 1.

    This value is considered to be the gravity-dressed edge singularity of the dimer model [2]. In the dimer model, \(\sigma = 1/6\).

  2. 2.

    While completing the article [4] we were informed by Zohren that he and Atkin [9] have obtained results which are identical to some of our results. We thank Stefan for informing us of these results prior to publication.

  3. 3.

    The authors derived the more general result with arbitrary \(\alpha \), but here we restricted our situation to that with \(\alpha =0\).

  4. 4.

    In [27], the authors derived the general saddle-point equation beyond the large-\(N\) limit. The general saddle-point equation indeed coincides with the SDE with arbitrary \(\alpha \) by the treatment, \(\alpha = 1/N^{2}\).

  5. 5.

    In fact, it is possible to include the interactions, \(\int dL\psi ^{\dagger }_{-}(L) \fancyscript{H}_{0}(L,\varLambda _{\text {cdt}})\psi _{+}(L)\) and its spin-flipped term. However, because of the \(\mathbb {Z}_{2}\)-symmetry as to the spin reflection, such terms merely cause a constant shift of the string coupling constant, so that we have not included these terms in the Hamiltonian.

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Authors and Affiliations

  1. Department of Physics, Nagoya University, Nagoya, Japan

    Yuki Sato

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  1. Yuki Sato
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Correspondence to Yuki Sato .

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Sato, Y. (2014). Causal Dynamical Triangulation. In: Space-Time Foliation in Quantum Gravity. Springer Theses. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54947-5_2

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