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The Branes: Three Viewpoints

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Gravity, a Geometrical Course

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Abstract

This chapter deals with the brane/bulk dualism. The first section contains a conceptual outline where the three sided view of branes as 1) classical solitonic solutions of the bulk theory, 2) world volume gauge-theories described by suitable world-volume actions endowed with κ-supersymmetry and 3) boundary states in the superconformal field theory description of superstring vacua is spelled out. Next a New First Order Formalism, invented by the author of this book at the beginning of the XXI century and allowing for an elegant and compact construction of κ-supersymmetric Born-Infeld type world-volume actions on arbitrary supergravity backgrounds is described. It is subsequently applied to the case of the D3-brane, both as an illustration and for the its intrinsic relevance in the gauge/gravity correspondence. The last sections of the chapter are devoted to the presentations of branes as classical solitonic solutions of the bulk theory. General features of the solutions in terms of harmonic functions are presented including also a short review of domain walls and some sketchy description of the Randall-Sundrun mechanism.

Tu se’ certo il cantor del trino regno,

Tu lo spirto magnanimo e sovrano

Cui, quasi cervo a puro fonte, io vegno.

Giovanni Marchetti

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Notes

  1. 1.

    By first string revolution it is meant the discovery by Green and Schwarz of the mechanism of anomaly cancellation which singled out five perturbatively consistent superstring models, namely:

    1. 1.

      Type II A

    2. 2.

      Type II B

    3. 3.

      Type I with SO(32) gauge group

    4. 4.

      Heterotic E8×E8

    5. 5.

      Heterotic SO(32).

    By second string revolution it is meant the series of discoveries around 1995–1996 that demonstrated that all the perturbatively consistent string models are related to each other by non-perturbative dualities pointing out to the fact that there is just one non-perturbative superstring theory.

  2. 2.

    As already mentioned in the main text, by Wess-Zumino terms it is generally understood terms of the form \( {{\int }_{{{\mathcal{W}}_{p+1}}}}{{\mathbb{A}}^{\left[ p+1 \right]}} \) where \( {{\mathcal{W}}_{p+1}} \) denotes the world volume spanned by a p-brane and \(\mathbb{A}^{[p+1]}\) denotes a suitable (p+1)-form present in the considered background supergravity.

  3. 3.

    In this section we use the notations and conventions described in Appendix B.1.

  4. 4.

    The need of a cosmological term for p-brane actions with p≠1 was first noted by Tucker and Howe in [20]. We also would like to attract the attention of the reader on the Sect. 5.3 of Volume 1 where the auxiliary fields needed to realize a systematic first order formalism in geometrical gravity were first discussed in anticipation of their essential role in supergravity.

  5. 5.

    A partial first order formalism was already introduced in the literature for Dp-branes [31, 32] in the context of the superembedding approach initiated by the Kharkov group and extensively developed also in collaborations with the Padua group and other groups [21–23]. In particular in [33, 34] an action with a partial first order formalism was introduced in the sense that there is an auxiliary F ij field for the gauge degrees of freedom but the action is “second order” in the brane coordinates x and θ, which enter through the pullback of the target space supervielbein E a.

  6. 6.

    For a general discussion of the Gaillard-Zumino formula see Chap. 8, Sects. 8.3.1–8.3.2.

  7. 7.

    In the paper quoted above the κ-supersymmetry projector presented here was originally introduced within a 2nd order formulation of the theory. It is particularly significant and rewarding that the same projector is valid also in first order formulation. As shown in the appendix the mechanism by means of which it works are very subtle and take advantage of the explicit solutions for the auxiliary fields in terms of the physical ones. In this way one finds an overall non-trivial check of all the algebraic machinery of our new first order formalism.

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  1. Dipartimento di Fisica Teorica, University of Torino, Torino, Italy

    Pietro Giuseppe Frè

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  1. Pietro Giuseppe Frè
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Frè, P.G. (2013). The Branes: Three Viewpoints. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5443-0_7

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