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Gravity in 2+1 Dimensions as a Chern–Simons Theory

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Deformations of Spacetime Symmetries

Part of the book series: Lecture Notes in Physics ((LNP,volume 986))

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Abstract

In the previous chapter, we argued that the effective theory of particles coupled to gravity in 2+1 dimensions can be understood as a deformation of the standard relativistic particles theory. Now it is time to present the rigorous derivation of this result. We start with the formulation of the pure 2+1 gravity [1, 2] in terms of the Chern–Simons theory (an extensive discussion of gravity in 2+1 dimensions can be found in the book of Carlip [3]). Then we discuss the coupling of particles, following the ideas of [4, 5], to be followed by a detailed derivation of the deformed particle action, with curved momentum space, describing the relativistic particle moving in its own gravitational field. We conclude this chapter with a discussion on how deformed spacetime symmetries arise directly from quantum gravity in 2+1 dimensions.

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Notes

  1. 1.

    In higher dimensions, for Lorentz group, one conventionally uses the generators \(J_{ab}\), antisymmetric in a, b. In 2+1 dimensions, there is a duality between vectors and antisymmetric tensors, and it is convenient to make use of it defining \(J_a = \frac{1}{2}\, \epsilon _a{}^{bc}\, J_{bc}\).

  2. 2.

    In the next chapter, we will see that in 3+1 dimensions, the theory gravity which uses the analogue of (2.18) is perfectly well defined for all values of the cosmological constant.

  3. 3.

    Here and below momentum means relativistic momentum, whose components are energy and linear momentum.

  4. 4.

    We will stay within this limit for the major remaining part of this chapter.

  5. 5.

    We refer the reader to Chap. 5 for a pedagogic introduction to semi-direct product groups and the Poincaré group.

  6. 6.

    We will sometime use \((J_a)^b{}_c=-\left( \epsilon _a\right) ^b{}_c\) to denote the \(3\times 3\) matrix representation of the generators of the Lie algebra \(\mathfrak {so}(2,1)\). It will be clear from the context, which representation we have in mind.

  7. 7.

    Similar construction has been presented in the papers [11, 12], in which one uses the path integral (spin foam) formalism to integrate out the gravitational degrees of freedom in the 2+1-dimensional gravity—scalar field system. As a result, one obtains the deformed action for the scalar field on a non-commutative spacetime.

  8. 8.

    Some other useful parametrizations can be found in [15].

  9. 9.

    Note that this is not the centre of mass position.

  10. 10.

    To see this it is sufficient to recall that the conserved Noether current associated with a global symmetry with parameters \(n^\alpha \) can be most easily found by taking \(n^\alpha \) time dependent and looking for the term in variation of the Lagrangian that is proportional to the time derivative on \(n^\alpha \) (the terms proportional to \(n^\alpha \)itself vanish because this parameter generates global symmetry.).

  11. 11.

    Although the spinless part of the Lagrangian (2.107) looks exactly the same the Lagrangian of the \(\kappa \)-deformed particle that we are going to discuss in detail in Chap. 4, (4.142) there is a crucial difference between the two. In the case of \(\kappa \)-deformation, we do not impose the constraint (2.108), which makes it possible for \(\kappa \)-deformed particle to be not frozen.

  12. 12.

    In the case when spacetime has no boundaries. When boundaries are present the situation is by far more complicated and interesting. See [25] for details.

  13. 13.

    In three dimensions there is only one rotation generator with \(\rho ^{ab}=\rho ^{12}\) and its bracket with itself obviously vanishes. Equation (2.127) holds in any dimension.

  14. 14.

    We use here Euclidean not Lorentzian signature gravity for technical reasons, which are explained in [23]. It is believed that the final result can be continued back to Lorentzian case (like it is in the case of Wick rotation).

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Authors and Affiliations

  1. Department of Physics, University of Naples Federico II, Naples, Italy

    Michele Arzano

  2. Institute for Theoretical Physics, University of Wrocław, Wroclaw, Poland

    Jerzy Kowalski-Glikman

  3. National Centre for Nuclear Research, Swierk, Poland

    Jerzy Kowalski-Glikman

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Arzano, M., Kowalski-Glikman, J. (2021). Gravity in 2+1 Dimensions as a Chern–Simons Theory. In: Deformations of Spacetime Symmetries. Lecture Notes in Physics, vol 986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-63097-6_2

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