**10. References**


14 Will-be-set-by-IN-TECH

the Bianchi identities (62) and (63) could lead to constraints on the admissible torsion *Tα*, as in 4D and higher dimensions. However, in 3D the situation is different: Using Appendix B, the

<sup>2</sup> *ηβ* <sup>∧</sup> *<sup>η</sup>αβ* = (−1)*<sup>s</sup> <sup>ρ</sup>*

*<sup>ν</sup>*] *ϑγ* <sup>∧</sup> *<sup>ϑ</sup><sup>μ</sup>* <sup>∧</sup> *<sup>ϑ</sup><sup>ν</sup>* <sup>=</sup> 0.

*<sup>T</sup><sup>β</sup>* <sup>∧</sup> *ηαβ* <sup>=</sup> <sup>4</sup>*κ*<sup>2</sup>

Thus the first Bianchi identity does not give any further information. The second Bianchi

which is identically zero by a similar argument, or by employing Eq. (73). Consequently, the Bianchi identities impose *no* restrictions on the axial torsion given by (10) in 3D, a fact which

[1] Anandan, J. (1994). Topological and geometrical phases due to gravitational field with curvature and torsion, *Physics Letters* A195, 284–292; (1996). Gravitational phase

[2] Baekler, P., Mielke, E. W. & Hehl, F. W. (1992). Dynamical symmetries in topological 3D

[3] Bakke, K., Furtado, C. & Nascimento, J. R. (2009). Gravitational geometric phase in the

[4] Cacciatori, S. L., Caldarelli, M. M., Giacomini, A., Klemm, D. & Mansi, D. S. (2006). Chern-Simons formulation of three-dimensional gravity with torsion and nonmetricity,

[5] Carlip, S. (1995). Lectures on (2+1)–dimensional gravity, *Journal of Korean Physical Society*

[6] Carlip, S. (1998). *Quantum gravity in* (2 + 1)*–dimensions*, Cambridge University Press. [7] Carlip, S. (1995). The (2 + 1)–dimensional black hole, *Classical Quantum Gravitation* 12, 2853–2880; (2005). Conformal field theory, (2+1)-dimensional gravity, and the BTZ black

[8] Chern, S. S. & Simons, J. (1971). Some cohomology classes in principal fiber bundles and their application to Riemannian geometry, *Proceedings of Natural Academy of Science* 68,

[9] de Juan, F. , Cortijo, A. & Vozmediano, M. A. H. (2010). Dislocations and torsion in

[10] Dereli, T. & Tucker, R. W. (1988). Gravitational interactions in 2+1 dimensions, *Classical*

<sup>2</sup> *<sup>D</sup>ηα* <sup>=</sup> <sup>2</sup>*κρ*

2<sup>2</sup> 

*ηβμνηαβγ*

*ϑγ* <sup>∧</sup> *<sup>ϑ</sup><sup>μ</sup>* <sup>∧</sup> *<sup>ϑ</sup><sup>ν</sup>* (83)

<sup>2</sup> *ηαβ* <sup>∧</sup> *<sup>η</sup><sup>β</sup>* <sup>≡</sup> 0. (84)

<sup>3</sup> *ηαβ* <sup>∧</sup> *<sup>η</sup><sup>β</sup>* <sup>≡</sup> <sup>0</sup> (85)

first Bianchi identity yields

(−1)*<sup>s</sup>*

identity (63) yields

**10. References**

28, S447–S467.

791–794.

*<sup>η</sup>αβ* <sup>∧</sup> *<sup>R</sup>*<sup>∗</sup>

*<sup>β</sup>* = (−1)*<sup>s</sup> <sup>ρ</sup>*

<sup>=</sup> <sup>−</sup>(−1)*<sup>s</sup>*

*DR <sup>α</sup>* <sup>=</sup> *<sup>ρ</sup>*

*DT<sup>α</sup>* <sup>=</sup> <sup>2</sup>*<sup>κ</sup>*

2*δ<sup>α</sup>* [*μδ γ*

*<sup>D</sup>ηα* <sup>=</sup> <sup>2</sup>*<sup>κ</sup>*

has allowed us to construct something non-trivial from the MB model.

operator and cosmic strings, *Physical Review* D53, 779–786.

presence of torsion, *The European Physical Journal* C60, 501–507.

graphene and related systems, *Nuclear Physics* B828, 625–637 .

gravity with torsion, *Il Nuovo Cimento* B107, 91–110.

*Journal Geometrical Physics* 56, 2523–2543.

*Quantum Gravitation* 5, 951–959.

hole, *Classical Quantum Gravitation* 22, R85–R124.

Furthermore, the exterior covariant derivative of Eq. (10) provides the identity


**1. Introduction**

The search for the theory of quantum gravity (QG) in 4-dimensions (4D) is one of the most significant challenges of temporary physics. The great effort and insights of many theoreticians and experimentalists resulted in the emergence of one of the greatest achievements of 20th century science, i.e. standard model of particles and fields (SM). SM (with its minimal extensions by massive neutrinos and after renormalization) describes and predicts, with enormous accuracy, almost all perturbative aspects and behaviour of interacting quantum fields and particles which place themselves in the realm of electromagnetic, strong and weak nuclear interactions, within the range of energies up to few TeV. However, gravity at quantum level is not covered by this pattern. The oldest, semiclassical, approach to QG relies on the quantization of metric field which is understood as the perturbation of the ground spacetime metric. This is exactly in the spirit of quantum field theory (QFT) as in SM. There should follow various correlation functions of physical processes where gravity at quantum level is present. There should, but actually they do not since the expressions are divergent and the theory is not renormalizable. Even the presence of supersymmetry does not change this substantially. On the other hand, we have a wonderful theory of general relativity (GR)

**Quantum Gravity Insights from Smooth** 

**4-Geometries on Trivial <sup>4</sup>**

*Institute of Physics, University of Silesia, Katowice* 

**4**

Jerzy Król

*Poland* 

which, however, is a theory of classical gravity and it prevents its quantization in 4D.

mathematics which is still not fully comprehended.

Among existing approaches to QG, superstring theory is probably the most advanced and conservative one. It attempts to follow GR and quantum mechanics as much as possible. However, superstring theory has to be formulated in 10 spacetime dimensions and on fixed, not dynamical, background. Many proposals how to reach the observed physics from 10D superstrings were worked out within the years. These are among others, compactiffication, flux stabilization, brane configuration model-buildings, brane worlds, holography or anti-de-Sitter/conformal field theory duality, i.e. AdS/CFT. There exists much ambiguity, however, with determining 4D results by these methods. Some authors estimate that there exist something about 10500 different backgrounds of superstring theory which all could be "good" candidates expressing 4D physics. This means that similar variety of possible models for true physics is predicted by superstring theory. To manage with such huge amount of "good" solutions, there was proposed to use the methods of statistical analysis to such *landscape* of possible backgrounds. Anyway, one could expect better prediction power from the fundamental theory which would unify gravity with other interactions at quantum level. On the other hand, superstring theory presents beautiful, fascinating and extremly rich

