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Superstrata

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Abstract

We give a survey of the present status of the microstate geometries called superstrata. Superstrata are smooth, horizonless solutions of six-dimensional supergravity that represent some of the microstates of the D1–D5–P black hole in string theory. They are the most general microstate geometries of that sort whose CFT dual states are identified. After reviewing relevant features of the dual CFT, we discuss the construction of superstratum solutions in supergravity, based on the linear structure of the BPS equations. We also review some of recent work on generalizations of superstrata and physical properties of superstrata. Although the number of superstrata constructed so far is not enough to account for the black-hole entropy, they give us valuable insights into the microscopic physics of black holes.

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Notes

  1. For \(\mathcal{M}=\mathrm{K3}\), \(N_1\) includes the D1-brane charge induced on the worldvolume of the D5-branes by a curvature coupling [1]. Namely \(N_1=N_1^{\mathrm{explicit}}+N_1^{\mathrm{induced}}\), \(N_1^\mathrm{induced}=-N_5\).

  2. \(J=J^3_0\in {\mathbb {Z}}/2\) where \(J^3_0\) is a generator of \(SU(2)_L\in SO(4)\) coming from the rotational symmetry in the directions transverse to the D-branes.

  3. The CFT dual states of superstrata based on the five-dimensional multi-center solutions with two centers are known, while duals of multi-center solutions with more than two centers or those of their superstratum generalizations are not known.

  4. The string-theory configurations resulting from such multistage supertube transition should be called (general) superstrata, which in general contain dipole charges that do not allow a description in terms of smooth geometry. Superstrata that do allow a geometric description, like the ones constructed in [29], should properly be called geometric superstrata, although they are normally simply called superstrata.

  5. For reviews of the D1-D5 CFT, see e.g. [49, 50].

  6. For \(\mathcal{M}=T^4\), the low-energy dynamics of the D-brane bound state can be described by a supersymmetric sigma model with target space \({\mathbb {R}}^4\times T^4\times \mathrm{Sym}^{N_1 N_5}(T^4)\), where the \({\mathbb {R}}^4\) part describes the center-of-mass motion of the D-branes in the noncompact \({\mathbb {R}}^4\), the \(T^4\) part describes worldvolume Wilson lines along the internal \(T^4\), and the \(\mathrm{Sym}^{N_1 N_5}(T^4)\) part describes the moduli space of D1-branes as instantons inside the D5 worldvolume [51, 53]. Here we are focusing on the last part. For \(\mathcal{M}=\mathrm{K3}\), the target space is \({\mathbb {R}}^4\times \mathrm{Sym}^{N_1 N_5+1}(\mathrm{K3})\) [2, 51] where the \({\mathbb {R}}^4\) part describes the center-of-mass motion in the noncompact \({\mathbb {R}}^4\) and the \(\mathrm{Sym}^{N_1 N_5+1}(\mathrm{K3})\) part describes the instanton moduli space which we are focusing on.

  7. Except for the case with \(\alpha =-\) (\(\dot{\alpha }=-\)) and \(k=1\) for which 8 left-moving (right-moving) supercharges are preserved.

  8. For \(\mathcal{M}=\mathrm{K3}\), supersymmetry implies that the number of chiral primary states do not change [60]. For \(\mathcal{M}=T^4\), such supersymmetry argument is not enough for showing that the number stays constant, although we expect that it does, on physical grounds (single-particle supergravitons and their gas must exist everywhere in the moduli space).

  9. These states are not normalized.

  10. These fields have symmetry indices as \(X^{A{\dot{A}}}(z,{\overline{z}}),\Psi ^{\alpha {\dot{A}}}(z),{\tilde{\Psi }}^{\dot{\alpha }{\dot{A}}}({\overline{z}})\) and each has four components on each strand [49, 50].

  11. For recent progress, see [68, 69].

  12. For supersymmetric solutions that do not preserve this symmetry, see [45].

  13. There are also supersymmetric solutions with a timelike Killing vector, but they are not relevant for the microstates of the D1-D5-P black hole whose Killing spinor squares to a null Killing vector [70].

  14. The six-dimensional exterior derivative acting on \(u,v,x^m\), although nothing depends on u.

  15. For example, when one relates 6D and 5D solutions, other choices are more convenient; see [43].

  16. The \(A\ge 6\) components [5, 7] break the symmetry of \(\mathcal{M}\).

  17. The absolute value square of the Fourier coefficient is proportional to \(N^\psi _k\) with a non-trivial coefficient. For the precise map, see, e.g., [37].

  18. Generally, the large N scaling is \(b_I^{\mathsf {k}}\sim A^\psi _{k,m,n,f}N^{-1/2}\sim (N^{\psi }_{k,m,n,f}/N)^{1/2}\) [7, 61, 62].

  19. If one wants to consider some other background state \(\Psi _{\mathrm{bg}}\), then one needs to study the spectrum of linearized supergravity around the background geometry dual to \(\Psi _{\mathrm{bg}}\), in order to carry out the procedure of this section.

  20. Here we using the NS language, appropriate for the \(\mathrm{AdS}_3\times S^3\) background.

  21. This coiffuring for low-frequency source is more non-trivial than the high-frequency one. For low-frequency coiffuring, the term in \(Z_1\) to be turned on is proportional to \(\Delta _{k_1-k_2,m_1-m_2,n_1-n_2}\), whereas one naively expects terms proportional to \(\Delta _{k_1,m_1,n_1}\Delta _{k_2,m_2,n_2}= \Delta _{k_1+k_2,m_1+m_2,n_1+n_2}\), the second-layer source being quadratic in \(Z_I,\Theta _I\).

  22. The expression (4.24) can be regarded as a sort of triple hypergeometric function.

  23. Their solutions include generalization to excitations around the orbifold \((\mathrm{AdS}_3\times S^3)/{\mathbb {Z}}_p\) with \(p\ge 1\), but here we are setting \(p=1\).

  24. Having a single mode turned on in the bulk means that, on the boundary, infinitely many modes are turned on. Namely, the corresponding CFT state has \((J^+_{-1})^k{|{++}\rangle }_{k+1}\), \((J^+_{-1})^k{|{--}\rangle }_{k-1}\) and \((J^+_{-1})^k{|{00}\rangle }_k\) turned on not just for one value of k but for all integer multiples of k.

  25. They also present orbifolded superstrata of “Style 1” mentioned in section 4.3.8.

  26. It was argued that these D0-branes puff out into M2-branes whose Landau level degeneracy accounts for the entropy of the MSW black hole [93, 95]. These M2-branes are supposed to wrap a non-trivial \(S^2\) in the geometry and are sometimes dubbed supereggs. However, it was shown that such M2-branes will violate charge conservation [96] and/or break the supersymmetry [97] preserved by the MSW black hole. Therefore, these superegg M2-branes and their Landau levels cannot be the precise description of the microstates.

  27. This correlation function, being really a four-point function, is not protected.

  28. The spectrum in the (1, 0, n) geometry was studied in [83] before.

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Acknowledgements

I thank Iosif Bena, Nejc Čeplak, Stefano Giusto, Emil Martinec, Rodolfo Russo, David Turton and Nick Warner for fruitful collaborations and for sharing illuminating insights. I also thank Pierre Heidmann, Daniel Mayerson and Alexander Tyukov for valuable discussions. The work of MS was supported in part by JSPS KAKENHI Grant Numbers 16H03979, and MEXT KAKENHI Grant Numbers 17H06357 and 17H06359.

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  1. Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8602, Japan

    Masaki Shigemori

  2. Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-Ku, Kyoto, 606-8502, Japan

    Masaki Shigemori

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Shigemori, M. Superstrata. Gen Relativ Gravit 52, 51 (2020). https://doi.org/10.1007/s10714-020-02698-8

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