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Symmetries of Gravity in AdS\(_3\)

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BMS Particles in Three Dimensions

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Abstract

In this chapter we explore a physical model where the Virasoro group plays a key role, namely three-dimensional gravity on Anti-de Sitter (AdS) backgrounds and its putative dual two-dimensional conformal field theory (CFT). These considerations will be a basis and a guide for our study of asymptotically flat space-times in part III.

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Notes

  1. 1.

    In two dimensions one has in addition \(R_{\mu \nu }=\frac{1}{2}R g_{\mu \nu }\).

  2. 2.

    This is not a trivial requirement; for instance the function \(\sin (r^{42})/r\) is of order \({\mathcal O}(1/r)\) as \(r\rightarrow +\infty \) but its derivative is not of order \({\mathcal O}(1/r^2)\).

  3. 3.

    The term “superpotential” here has nothing to do with supersymmetry.

  4. 4.

    In practice, for constant \(\epsilon \) this gives \(\delta A_{\mu }=0\), but for fields with non-zero electric charge the transformation given by constant \(\epsilon \)’s is non-trivial.

  5. 5.

    Strictly speaking we have displayed this isomorphism here for the complexification of \(\mathfrak {so}(2,2)\), but it also holds for real Lie algebras.

  6. 6.

    The terminology is somewhat backwards, since the highest weight h is actually the lowest eigenvalue of \(L_0\) in the space of the representation; this terminological clash is standard.

  7. 7.

    The actual value of the quantum weight h may differ from the classical parameter defined in (8.66) by corrections of order \({\mathcal O}(1/c)\), so from now on it is understood that h refers to the quantum value. This subtlety will have very little effect on our discussion.

  8. 8.

    As before, the quantum values of \((h,\bar{h})\) may differ from their classical counterparts by 1 / c corrections.

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  1. Institute for Theoretical Physics, ETH Zürich, Zürich, Switzerland

    Blagoje Oblak

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Oblak, B. (2017). Symmetries of Gravity in AdS\(_3\) . In: BMS Particles in Three Dimensions. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-61878-4_8

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