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Cosmologies with (Hidden) N=2 Supersymmetry

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Quantum Cosmology - The Supersymmetric Perspective - Vol. 1

Part of the book series: Lecture Notes in Physics ((LNP,volume 803))

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Abstract

In this chapter we discuss a rather different methodology for retrieving SQC models: not directly from N = 1 SUGRA (and therefore involving a dimensional reduction towards an N = 4 SUSY minisuperspace), but instead in an opposite direction, starting from purely bosonic configurations (e.g., general relativity and the bosonic sectors of string theories) and then building (in a consistent manner!) a minisuperspace with N = 2 SUSY, which is the simplest case [1–8].

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Notes

  1. 1.

    Whose formal expressions bear interesting similarities with those in Chap. 5.

  2. 2.

    For example, for the expression in (6.2), we need (see footnote 15 in Sect. 2.3.1)

    $$\beta_{1} \equiv \beta^{0} - 2\beta^{+}\,,$$
    ((6.3))
    $$\beta_{2} \equiv \beta^{0} + \beta^{+} + \sqrt{3}\beta^{-}\,,$$
    ((6.4))
    $$\beta_{3} \equiv \beta^{0} + \beta^{+} - \sqrt{3}\beta^{-}\,,$$
    ((6.5))

    and of course,

    $$\beta^{0} \equiv \pm \Omega = \pm \ln a = \pm \alpha\,,$$
    ((6.6))

    for the Misner–Ryan time coordinate choice(s) [15].

  3. 3.

    Note that this feature seems to influence the form of the quantum supersymmetry constraints, as will be made clear in subsequent sections.

  4. 4.

    Ω is employed for time coordinate in the ADM parametrization, whereas \(\tilde{\Omega}\) is the conformal factor function.

  5. 5.

    At this stage, the reader might find it useful to glance through Sect. 3.3 of Vol. II. In fact, most of what follows here concerns a supersymmetric extension of particle motion in a potential well, i.e., supersymmetric quantum mechanics (SQM).

  6. 6.

    Perhaps a quick glance at (6.42) may help to illustrate this point.

  7. 7.

    It is sufficient to examine the case \(\widetilde{\Omega}(q)=0\), where the metric \({\mathcal{G}}_{XY}\left(={\mathcal{G}}_{XY}^{(0)} \right)\) is flat, and U(q) is given by \(U^{(0)}(q)\) (see Exercise 6.1).

  8. 8.

    They do indeed constitute (N = 2) SUSY generators or (local) gauge constraints.

  9. 9.

    X,Y correspond to minisuperspace variables and we employ x,y for the corresponding variables in the corresponding tangent space.

  10. 10.

    When Λ = 0, we recover the analysis in Exercise 6.2. In fact, for \(\alpha\rightarrow-\infty\), near the initial singularity, the cosmological constant term is negligible and we get \({\mathsf{B}}_1\rightarrow \textrm{e}^{-{\mathsf{W}}}\), \({\mathsf{W}}=\textrm{e}^{2\alpha}/2\) with \({\mathsf{B}}_0 \rightarrow 0\). The possibility of B 0 approaching eW can be dealt with in a Bianchi IX setting (of which the FRW is a special case). If we require Ψ not to diverge for \(\beta_\pm \rightarrow\infty\) for fixed α, then it must be excluded. Note that the remaining solution is not then the Hartle–Hawking state, but agrees with a Vilenkin type solution.

  11. 11.

    Also representing the \(\check{d}(\check{d}+1)/2\) modulus (degrees of freedom) fields.

  12. 12.

    We now take D as generic, not necessarily \(D=3\), as mentioned in Sect. 6.2.1.

  13. 13.

    \(\textrm{SL}(2,\mathbb{R})\) is also possible as a subgroup of the T-duality group.

  14. 14.

    \(\Lambda^2 <0\) is required for the Euclidean action to be real (see Sects. 6.1 and 6.1.1).

  15. 15.

    Due to the quantum Hamiltonian having an additional spin term, to avoid imaginary solutions to (6.301), we restrict the analysis to the region of parameter space where \(\Upsilon>0\). This corresponds to \(\omega < -D / (D-1)\) when \(\Lambda >0\) and \(\omega >-D/(D-1)\) if \(\Lambda <0\).

  16. 16.

    The geometry of the spatial sections of the Kantowski–Sachs model is \(S^1 \times S^2\). The symmetry group of these surfaces is of the Bianchi type IX, but only acts transitively on 2D surfaces that foliate the three-space.

  17. 17.

    For simplicity, however, we consider the case where \(A=0\). This corresponds to the limit \(\varsigma \rightarrow -\infty\) and denotes a weak coupling regime for ς. In the strong limit (\(\varsigma \rightarrow +\infty\)), the term in A dominates and Λ can be positive. In the former situation, the ‘averaged’ scale factor volume (represented by α) and the dilaton are more important, while in the latter a large anisotropy dominates in the spatial directions. This allows Λ to be positive. In particular, one may have \(\varsigma \leftarrow +\infty\) and \(u \rightarrow -\infty\), with, e.g., the singularity \(a_2 \rightarrow 0\), \(a_1 \rightarrow \infty\).

  18. 18.

    A non-diagonal minisuperspace metric would mean that fermionic states could not be clearly separated after the wave function has been annihilated by the supercharges.

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  1. Depto. Fisica, Universidade da Beira Interior, Rua Marquês d’Avila e, Bolama, 6200-307, Covilhã, Portugal

    Paulo Vargas Moniz

  2. CENTRA-IST, Lisboa, Portugal

    Paulo Vargas Moniz

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Moniz, P.V. (2010). Cosmologies with (Hidden) N=2 Supersymmetry. In: Quantum Cosmology - The Supersymmetric Perspective - Vol. 1. Lecture Notes in Physics, vol 803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11575-2_6

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  • DOI: https://doi.org/10.1007/978-3-642-11575-2_6

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