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Supersymmetry breaking and stability in string vacua

Brane dynamics, bubbles and the swampland

Abstract

We review some aspects of the dramatic consequences of supersymmetry breaking on string vacua. In particular, we focus on the issue of vacuum stability in ten-dimensional string models with broken, or without, supersymmetry, whose perturbative spectra are free of tachyons. After formulating the models at stake, we introduce their unified low-energy effective description and present a number of vacuum solutions to the classical equations of motion. In addition, we present a generalization of previous no-go results for de Sitter vacua in warped flux compactifications. Then we analyze the classical and quantum stability of these vacua, studying linearized field fluctuations and bubble nucleation. Then, we describe how the resulting instabilities can be framed in terms of brane dynamics, examining in particular brane interactions, back-reacted geometries and commenting on a brane-world string construction along the lines of a recent proposal. After providing a summary, we conclude with some perspectives on possible future developments.

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Notes

  1. 1.

    We remark that, in this context, modular invariance arises as the residual gauge invariance left after fixing world-sheet diffeomorphisms and Weyl rescalings. Hence, violations of modular invariance would result in gauge anomalies.

  2. 2.

    We work in ten space–time dimensions, since non-critical string perturbation theory entails a number of challenges.

  3. 3.

    Non-vanishing values of the argument z of Jacobi \(\vartheta \) functions are nonetheless useful in string theory. They are involved, for instance, in the study of string perturbation theory on more general backgrounds and D-brane scattering.

  4. 4.

    We remark that different combinations of characters reflect different projections at the level of the Hilbert space.

  5. 5.

    In the present context, spin-statistics amounts to positive (resp. negative) contributions from space–time bosons (resp. fermions).

  6. 6.

    Despite this fact the type IIB superstring is actually anomaly-free, as well as all five supersymmetric models owing to the Green–Schwarz mechanism [48]. This remarkable result was a considerable step forward in the development of string theory.

  7. 7.

    The “hatted” characters appear since the modular paramater of the covering torus of the Möbius strip is not real, and they ensure that states contribute with integer degeneracies.

  8. 8.

    Since the \(\text {D}9\)-branes are on top of the \(\text {O}9_-\)-plane, counting conventions can differ based on whether one includes “image” branes.

  9. 9.

    The original works can be found in [51,52,53,54,55,56,57,58]. For reviews, see [45, 46, 59].

  10. 10.

    In principle, one could address these phenomena by systematic vacuum redefinitions [7,8,9,10,11], but carrying out the program at high orders appears prohibitive.

  11. 11.

    The corresponding orientifold projections of the type 0A model were also investigated. See [45], and references therein.

  12. 12.

    Strictly speaking, the anomalous U(1) factor carried by the corresponding gauge vector disappears from the low-lying spectrum, thus effectively reducing the group to SU(32).

  13. 13.

    One can alternatively build heterotic right-moving sectors using ten-dimensional strings with auxiliary fermions.

  14. 14.

    In some orbifold models, it is possible to obtain suppressed or vanishing leading contributions to the cosmological constant [67,68,69,70,71,72].

  15. 15.

    At the level of the space–time effective action, the vacuum energy contributes to the string-frame cosmological constant. In the Einstein frame, it corresponds to a runaway exponential potential for the dilaton.

  16. 16.

    In the same spirit, three-generation non-tachyonic heterotic models were constructed in [74]. Recently, lower-dimensional non-tachyonic models have been realized compactifying ten-dimensional tachyonic superstrings [75, 76].

  17. 17.

    This effective field theory can also describe non-critical strings [79, 80], since the Weyl anomaly can be saturated by the contribution of an exponential dilaton potential.

  18. 18.

    The case \(\gamma = 0\), which at any rate does not arise in string perturbation theory, would not complicate matters further.

  19. 19.

    In Eq. (3.5) we have used the notation \(F_3 = dC_2\) in order to stress the Ramond-Ramond (RR) origin of the field strength.

  20. 20.

    At any rate, it is worth noting that world-sheet conformal field theories on \(\mathrm {AdS}_3\) backgrounds have been related to WZW models, which can afford \(\alpha '\)-exact algebraic descriptions [81].

  21. 21.

    For a similar analysis of a T-dual version of the USp(32) model, see [82].

  22. 22.

    The supersymmetry-breaking tadpoles cannot be sent to zero in a smooth fashion. However, it is instructive to treat them as parameters, in order to highlight their rôle.

  23. 23.

    The Laplacian spectrum of the internal space \({\mathscr {M}}_q\) can have some bearing on perturbative stability.

  24. 24.

    The flux n in Eq. (3.31) is normalized for later convenience, although it is not dimensionless nor an integer.

  25. 25.

    Analogous results in tachyonic type 0 strings were obtained in [90].

  26. 26.

    This is easily seen dualizing the three-form in the orientifold action (3.4), which inverts the sign of \(\alpha \), in turn violating the condition of Eq. (3.33).

  27. 27.

    Despite conceptual and technical issues, non-supersymmetric dualities connecting the heterotic model to open strings have been explored in [91, 92]. Similar interpolation techniques have been employed in [93]. A non-perturbative interpretation of non-supersymmetric heterotic models has been proposed in [94].

  28. 28.

    For recent results on the issue of scale separation in supersymmetric \(\mathrm {AdS}\) compactifications, see [97].

  29. 29.

    The same result was derived in [118].

  30. 30.

    Notice that, in order to derive Eq. (3.42) substituting the ansatz of Eq. (3.41) in the action, the flux contribution is to be expressed in the magnetic frame, since the correct equations of motion arise varying \(\phi \) and \(B_{p+1}\) independently, while the electric-frame ansatz relates them.

  31. 31.

    As we have anticipated, Eq. (3.52) can be thought of as a generalization of the no-go results of [33, 34] to models with exponential potentials.

  32. 32.

    The constraint \({\mathscr {V}} > 0\) can also be recast in terms of \(\phi \) and \(\rho \) only, with no parametric dependence left.

  33. 33.

    Notice that, in order to canonically normalize the radion, one needs to rescale the field \(\psi (x)\) that we have introduced in Sect. 3.

  34. 34.

    An analogous idea in the context of higher-dimensional \({\text {dS}}\) space–times was put forth in [125].

  35. 35.

    A family of non-supersymmetric \(\mathrm {AdS}_7\) solutions of the type IIA superstring was recently studied in [126], and its stability properties were investigated in [127].

  36. 36.

    The same conclusion can be reached computing the effective nine-dimensional Newton constant [31].

  37. 37.

    We reserve the symbol B for scalar perturbations of the form field, which we shall introduce in Sect. 4.3.

  38. 38.

    This result resonates at least with some previous investigations [128, 129] of matrix models related to the type IIB superstring [130].

  39. 39.

    The stability analysis of scalar perturbations can also be carried out in general dimensions and for general parameters without additional difficulties, but we have not found such generalizations particularly instructive in the context of this review.

  40. 40.

    Here and in the following \(\epsilon \) denotes the Levi-Civita tensor, which includes the metric determinant prefactor.

  41. 41.

    Choosing a different internal space would require knowledge of its (tensor) Laplacian spectrum.

  42. 42.

    In all these expressions that refer to vector perturbations \(\ell \ge 1\), as described in Appendix A.

  43. 43.

    We use the convention in which the mass matrix \({\mathscr {M}}^2\) appears alongside the d’Alembert operator in the combination \(\Box - {\mathscr {M}}^2\).

  44. 44.

    The overall derivative can be removed on account of suitable boundary conditions.

  45. 45.

    For an earlier analysis in general dimensions, see [134]. A subsequent analysis for two internal sphere factors was performed in [135]. In supersymmetric cases [136], recently techniques based on Exceptional Field Theory have proven fruitful [137, 138].

  46. 46.

    For recent results on unstable modes of non-vanishing angular momentum in \(\mathrm {AdS}\) compactifications, see [139].

  47. 47.

    Projections that leave a sub-variety fixed could entail subtleties related to twisted states that become massless.

  48. 48.

    For a recent investigation along these lines in the context of the (massive) type IIA superstring, see [127].

  49. 49.

    It is worth noting that this large-N limit is not uniform, since factors of \(\frac{1}{N}\) are accompanied by factors that diverge in the near-horizon limit. In principle, a resummation of \(\frac{1}{N}\) corrections could cure this problem.

  50. 50.

    One could expect that solutions with different internal spaces, discussed in Sect. 3, arise from near-horizon throats of brane stacks placed on conical singularities [140].

  51. 51.

    We shall not discuss the Gibbons–Hawking–York boundary term, which is to be included at any rate to consistently formulate the variational problem.

  52. 52.

    For a detailed exposition of the resulting (distributional) differential equations, see [148].

  53. 53.

    On the other hand, as we have mentioned, the extreme case \(\delta n = n\) would correspond to the production of a bubble of nothing [151].

  54. 54.

    It is common to identify the tension of the bubble with the ADM tension of a brane soliton solution [147]. In our case this presents some challenges, as we shall discuss in Sect. 6.

  55. 55.

    Notice that Eq. (5.17) takes the form of an effective action for a \((p+1)\)-brane in \(\mathrm {AdS}\) electrically coupled to \(H_{p+2}\). This observation is the basis for the microscopic picture that we shall present shortly.

  56. 56.

    Notice that in the gravitational picture the charge of the membrane does not appear. Indeed, its contribution arises from the volume term of Eq. (5.14) in the thin-wall approximation.

  57. 57.

    Indeed, as we have discussed in Sect. 2, cosmological solutions of non-supersymmetric models display interesting features [22, 46, 62,63,64]. Similar considerations on flux compactifications can be found in [156].

  58. 58.

    For more details, we refer the reader to [142, 143, 157, 158].

  59. 59.

    The alternative case of \(\mathrm {AdS}_7\) could be studied, in principle, via \(\text {M}5\)-brane stacks.

  60. 60.

    Indeed, our results suggest that the solutions of [31], which are not fluxed, correspond to 8-branes.

  61. 61.

    For a discussion of this type of phenomenon in Calabi-Yau compactifications, see [12].

  62. 62.

    One can verify that this ansatz is consistent with the equations of motion for linearized perturbations.

  63. 63.

    As we have anticipated, verifying the charge-tension equality in the non-supersymmetric case presents some challenges. We shall elaborate upon this issue in Sect. 6.2.3.

  64. 64.

    The systematics of computations of this type in the bosonic case were developed in [164].

  65. 65.

    Related results in Scherk-Schwartz compactifications have been obtained in [165, 166].

  66. 66.

    For some earlier works along these lines, see [167,168,169,170,171].

  67. 67.

    Some lower-dimensional toy models offer flux landscapes where more explicit results can be obtained [148, 154, 173].

  68. 68.

    Even if one were to envision a pathological Minkowski solution with “\(\phi = -\infty \)” as a degenerate background (for instance, by introducing a cut-off), no asymptotically flat solution with \(\phi \; \rightarrow \; -\infty \) can be found.

  69. 69.

    Up to the sign of r and rescalings of \(R_0\), this realization of \({\mathrm {AdS}\times {\mathbb {S}}}\) with given L and R is unique.

  70. 70.

    In either case we shall find that the geodesic distance is finite.

  71. 71.

    In the supersymmetric case the contribution arising from the dilaton tadpole is absent, and the resulting system is integrable. Moreover, for \(p = 8 \, , q = 0\) the system is also integrable, since only the dilaton tadpole contributes.

  72. 72.

    Notice that \(\varOmega = \frac{D-2}{8} \left( \gamma ^2 - \gamma _c^2 \right) \), where the critical value \(\gamma _c\) defined in [22] marks the onset of the “climbing” phenomenon described in [84,85,86, 176] use different notations.

  73. 73.

    The sub-linear case is controlled by the parameters \(\phi _1 \, , v_1 \, , b_1\), which can be tuned as long as the constraint is satisfied. In particular, the differences \(L - L_{n,c}\) do not contain \(v_1\).

  74. 74.

    More precisely, the asymptotics for the metric in Eq. (6.48) refer to the exponents in the warp factors, which are related to v and b. Subleading terms could lead to additional prefactors in the metric.

  75. 75.

    In particular, on account of the analysis that we described in the preceding section, it is reasonable to expect that in the orientifold models the Dudas–Mourad solution corresponds to \(\text {D}8\)-branes.

  76. 76.

    The generalization to non-extremal p-branes of different dimensions would entail solving non-integrable systems, whose correct boundary conditions are not well-understood hitherto. Moreover, a reliable probe-brane regime would exclude the pinch-off asymptotic region, thereby requiring numerical computations.

  77. 77.

    While the number \(N_8\) of 8-branes does not appear explicitly in the solution, there is a single free parameter \(g_s \equiv e^{\varPhi _0}\), which one could expect to be determined by \(N_8\) analogously to the extremal case, with \(g_s \ll 1\) for \(N_8 \gg 1\).

  78. 78.

    Notice that, in the absence of fluxes, brane polarization [12, 177] would not suffice to stabilize these equilibria.

  79. 79.

    As we have discussed in Sect. 6.2.4, the leading-order behavior of the pinch-off singularity is expected to be applicable to the non-extremal case, since it is dominated by the dilaton potential.

  80. 80.

    The ensuing string amplitude computation is expected to be reliable as long as \(N_p\) and \(N_q\) are \({\mathscr {O}}\!\left( 1\right) \), complementary to the probe regimes \(N_p \gg N_q\) and \(N_p \ll N_q\).

  81. 81.

    For a more recent analysis in the case of the five-sphere, see [181].

References

  1. 1.

    A.M. Polyakov, Quantum geometry of bosonic strings, ed. I. Khalatnikov and and V. Mineev. Phys. Lett. B 103, 207–210 (1981). https://doi.org/10.1016/0370-2693(81)90743-7

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    A.M. Polyakov, Quantum geometry of fermionic strings, ed. by I. Khalatnikov, V. Mineev. Phys. Lett. B 103, 211–213 (1981). https://doi.org/10.1016/0370-2693(81)90744-9

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    J. Callan Curtis, E. Martinec, M. Perry, D. Friedan, Strings in background fields. Nucl. Phys. B 262, 593–609 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38, [Adv. Theor. Math. Phys.2,231(1998)], 1113–1133 (1999), arXiv:hep-th/9711200

  5. 5.

    T. Banks, W. Fischler, S. Shenker, L. Susskind, M theory as a matrix model: A Conjecture. Phys. Rev. D 55, 5112–5128 (1997). arXiv:hep-th/9610043

    ADS  MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    E. Witten, String theory dynamics in various dimensions. Nucl. Phys. B 443, 85–126 (1995). arXiv:hep-th/9503124

    ADS  MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    W. Fischler, L. Susskind, Dilaton Tadpoles, String Condensates and Scale Invariance. Phys. Lett. B 171, 383–389 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    W. Fischler, L. Susskind, Dilaton tadpoles, string condensates and scale invariance. 2. Phys. Lett. B 173, 262–264 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  9. 9.

    E. Dudas, G. Pradisi, M. Nicolosi, A. Sagnotti, On tadpoles and vacuum redefinitions in string theory, Nucl. Phys. B708, 3–44 (2005), arXiv:hep-th/0410101

  10. 10.

    N. Kitazawa, Tadpole resummations in string theory. Phys. Lett. B 660, 415–421 (2008). arXiv:0801.1702 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    R. Pius, A. Rudra, A. Sen, String perturbation theory around dynamically shifted vacuum. JHEP 10, 70 (2014). arXiv:1404.6254 [hep-th]

    ADS  Article  Google Scholar 

  12. 12.

    S. Kachru, J. Pearson, H. L. Verlinde, Brane / flux annihilation and the string dual of a nonsupersymmetric field theory, JHEP 06, 021 (2002), arXiv:hep-th/0112197

  13. 13.

    S. Kachru, R. Kallosh, A.D. Linde, S.P. Trivedi, De Sitter vacua in string theory. Phys. Rev. D 68, 046005 (2003). arXiv:hep-th/0301240

    ADS  MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    F. Gautason, V. Van Hemelryck, T. Van Riet, The Tension between 10D Supergravity and dS Uplifts. Fortsch. Phys. 67, 1800091 (2019). arXiv:1810.08518 [hep-th]

    ADS  Article  Google Scholar 

  15. 15.

    Y. Hamada, A. Hebecker, G. Shiu, P. Soler, Understanding KKLT from a 10d perspective. JHEP 06, 019 (2019). arXiv:1902.01410 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    F. Gautason, V. Van Hemelryck, T. Van Riet, G. Venken, A 10d view on the KKLT AdS vacuum and uplifting. JHEP 06, 074 (2020). arXiv:1902.01415 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    H. Ooguri, C. Vafa, Non-supersymmetric AdS and the Swampland, (2016), arXiv:1610.01533 [hep-th]

  18. 18.

    T. D. Brennan, F. Carta, C. Vafa, The String Landscape, the Swampland, and the Missing Corner, PoS TASI2017, 015 (2017), arXiv:1711.00864 [hep-th]

  19. 19.

    G. Obied, H. Ooguri, L. Spodyneiko, C. Vafa, De Sitter Space and the Swampland, (2018), arXiv:1806.08362 [hep-th]

  20. 20.

    E. Palti, The Swampland: Introduction and Review. Fortsch. Phys. 67, 1900037 (2019). arXiv:1903.06239 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  21. 21.

    N. Arkani-Hamed, L. Motl, A. Nicolis, C. Vafa, The String landscape, black holes and gravity as the weakest force, JHEP 06, 060 (2007), arXiv:hep-th/0601001

  22. 22.

    I. Basile, J. Mourad, A. Sagnotti, On Classical Stability with Broken Supersymmetry. JHEP 01, 174 (2019). arXiv:1811.11448 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    R. Antonelli, I. Basile, Brane annihilation in non-supersymmetric strings. JHEP 11, 021 (2019). arXiv:1908.04352 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    I. Basile, S. Lanza, de Sitter in non-supersymmetric string theories: nogo theorems and brane-worlds. JHEP 10, 108 (2020). arXiv:2007.13757 [hep-th]

    ADS  MATH  Article  Google Scholar 

  25. 25.

    I. Basile, On String Vacua without Supersymmetry: brane dynamics, bubbles and holography, PhD thesis (Pisa, Scuola Normale Superiore, 2020), arXiv:2010.00628 [hep-th]

  26. 26.

    S. Sugimoto, Anomaly cancellations in type I D-9 - anti-D-9 system and the USp(32) string theory, Prog. Theor. Phys. 102, 685–699 (1999), arXiv:hep-th/9905159

  27. 27.

    A. Sagnotti, Some properties of open string theories, in Supersymmetry and unification of fundamental interactions. Proceedings, International Workshop, SUSY 95, Palaiseau, France, May 15-19 (1995), pp. 473–484, arXiv:hep-th/9509080

  28. 28.

    A. Sagnotti, Surprises in open string perturbation theory,Nucl. Phys. Proc. Suppl. 56B, 332–343 (1997), arXiv:hep-th/9702093

  29. 29.

    L. Alvarez-Gaume, P.H. Ginsparg, G.W. Moore, C. Vafa, An O(16) x O(16) heterotic string. Phys. Lett. B 171, 155–162 (1986)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    L. J. Dixon, J. A. Harvey, String Theories in Ten-Dimensions Without Space-Time Supersymmetry, Nucl. Phys. B274, [93(1986)], 93–105 (1986)

  31. 31.

    E. Dudas, J. Mourad, Brane solutions in strings with broken supersymmetry and dilaton tadpoles, Phys. Lett. B486, 172–178 (2000), arXiv:hep-th/0004165

  32. 32.

    J. Mourad, A. Sagnotti, AdS vacua from dilaton tadpoles and form fluxes. Phys. Lett. B 768, 92–96 (2017). arXiv:1612.08566 [hep-th]

    ADS  MATH  Article  Google Scholar 

  33. 33.

    G. Gibbons, ASPECTS OF SUPERGRAVITY THEORIES, in XV GIFT Seminar on Supersymmetry and Supergravity (June 1984)

  34. 34.

    J.M. Maldacena, C. Nunez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16, edited by M. J. Duff, J. Liu, and J. Lu, 822–855 (2001), arXiv:hep-th/0007018

  35. 35.

    H. Ooguri, C. Vafa, On the geometry of the string landscape and the swampland. Nucl. Phys. B 766, 21–33 (2007). arXiv:hep-th/0605264

    ADS  MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    H. Ooguri, E. Palti, G. Shiu, C. Vafa, Distance and de Sitter conjectures on the swampland. Phys. Lett. B 788, 180–184 (2019). arXiv:1810.05506 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    D. Lüst, E. Palti, C. Vafa, AdS and the Swampland. Phys. Lett. B 797, 134867 (2019). arXiv:1906.05225 [hep-th]

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri, M. Schillo, Emergent de Sitter cosmology from decaying anti-de sitter space. Phys. Rev. Lett. 121, 261301 (2018). arXiv:1807.01570 [hep-th]

    ADS  Article  Google Scholar 

  39. 39.

    S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri, M. Schillo, de Sitter Cosmology on an expanding bubble. JHEP 10, 164 (2019). arXiv:1907.04268 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  40. 40.

    S. Banerjee, U. Danielsson, S. Giri, Dark bubbles: decorating the wall. JHEP 20, 085 (2020). arXiv:2001.07433 [hep-th]

    MathSciNet  Article  Google Scholar 

  41. 41.

    I. Antoniadis, E. Dudas, A. Sagnotti, Brane supersymmetry breaking, Phys. Lett. B464, 38–45 (1999), arXiv:hep-th/9908023

  42. 42.

    C. Angelantonj, Comments on open string orbifolds with a nonvanishing B(ab). Nucl. Phys. B 566, 126–150 (2000). arXiv:hep-th/9908064

    ADS  MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    G. Aldazabal, A.M. Uranga, Tachyon free nonsupersymmetric type IIB orientifolds via Brane-anti-brane systems. JHEP 10, 024 (1999). arXiv:hep-th/9908072

    ADS  MATH  Article  Google Scholar 

  44. 44.

    C. Angelantonj, I. Antoniadis, G. DAppollonio, E. Dudas, A. Sagnotti, Type I vacua with brane supersymmetry breaking. Nucl. Phys. B 572, 36–70 (2000). arXiv:hep-th/9911081 [hep-th]

  45. 45.

    C. Angelantonj, A. Sagnotti, Open strings, Phys. Rept. 371, [Erratum: Phys. Rept.376, no.6, 407(2003)], 1–150 (2002), arXiv:hep-th/0204089

  46. 46.

    J. Mourad, A. Sagnotti, An Update on Brane Supersymmetry Breaking, (2017), arXiv:1711.11494 [hep-th]

  47. 47.

    E. Whittaker, G. Watson, A Course of Modern Analysis, A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal Transcendental Functions (Cambridge University Press, 1996)

  48. 48.

    M.B. Green, J.H. Schwarz, Anomaly cancellation in supersymmetric D=10Gauge theory and superstring theory. Phys. Lett. B 149, 117–122 (1984)

    ADS  MathSciNet  Article  Google Scholar 

  49. 49.

    E. Dudas, J. Mourad, Consistent gravitino couplings in nonsupersymmetric strings. Phys. Lett. B 514, 173–182 (2001). arXiv:hep-th/0012071

    ADS  MATH  Article  Google Scholar 

  50. 50.

    E. Dudas, J. Mourad, A. Sagnotti, Charged and uncharged D-branes in various string theories. Nucl. Phys. B 620, 109–151 (2002). arXiv:hep-th/0107081

    ADS  MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    A. Sagnotti, Open Strings and their Symmetry Groups, in NATO Advanced Summer Institute on Nonperturbative Quantum Field Theory (Cargese Summer Institute) Cargese, France, July 16-30, 1987 (1987), pp. 521–528, arXiv:hep-th/0208020

  52. 52.

    G. Pradisi, A. Sagnotti, Open string orbifolds. Phys. Lett. B 216, 59–67 (1989)

    ADS  MathSciNet  Article  Google Scholar 

  53. 53.

    P. Horava, Strings on world sheet orbifolds. Nucl. Phys. B 327, 461–484 (1989)

    ADS  MathSciNet  Article  Google Scholar 

  54. 54.

    P. Horava, Background duality of open stringmodels. Phys. Lett. B 231, 251–257 (1989)

    ADS  MathSciNet  Article  Google Scholar 

  55. 55.

    M. Bianchi, A. Sagnotti, On the systematics of open string theories. Phys. Lett. B 247, 517–524 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  56. 56.

    M. Bianchi, A. Sagnotti, Twist symmetry and open stringWilson lines. Nucl. Phys. B 361, 519–538 (1991)

    ADS  Article  Google Scholar 

  57. 57.

    M. Bianchi, G. Pradisi, A. Sagnotti, Toroidal compactification and symmetry breaking in open string theories. Nucl. Phys. B 376, 365–386 (1992)

    ADS  MathSciNet  Article  Google Scholar 

  58. 58.

    A. Sagnotti, A Note on the Green–Schwarz mechanism in open string theories. Phys. Lett. B 294, 196–203 (1992). arXiv:hep-th/9210127

    ADS  Article  Google Scholar 

  59. 59.

    E. Dudas, Theory and phenomenology of type I strings and M theory, Class. Quant. Grav. 17, R41.R116 (2000), arXiv:hep-ph/0006190 [hep-ph]

  60. 60.

    G. Pradisi, F. Riccioni, Geometric couplings and brane supersymmetry breaking, Nucl. Phys. B615, 33.60 (2001), arXiv:hep-th/0107090 [hep-th]

  61. 61.

    N. Kitazawa, Brane SUSY Breaking and the Gravitino Mass. JHEP 04, 081 (2018). arXiv:1802.03088 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  62. 62.

    A. Sagnotti, “Low-\(\ell \) CMB from string-scale SUSY breaking?”, Mod. Phys. Lett. A32, [Subnucl. Ser.53,289(2017)], 1730001 arXiv:1509.08204 [astro-ph.CO]

  63. 63.

    A. Gruppuso, N. Kitazawa, N. Mandolesi, P. Natoli, A. Sagnotti, Pre- Inflationary Relics in the CMB? Phys. Dark Univ. 11, 68–73 (2016). arXiv:1508.00411 [astro-ph.CO]

    Article  Google Scholar 

  64. 64.

    A. Gruppuso, N. Kitazawa, M. Lattanzi, N. Mandolesi, P. Natoli, A. Sagnotti, The evens and odds of CMB anomalies. Phys. Dark Univ. 20, 49–64 (2018). arXiv:1712.03288 [astro-ph.CO]

    Article  Google Scholar 

  65. 65.

    L. Susskind, Supersymmetry breaking in the anthropic landscape, edited by M. Shifman, A. Vainshtein, and J. Wheater, 1745– arXiv:hep-th/0405189

  66. 66.

    M. R. Douglas, Statistical analysis of the supersymmetry breaking scale, (2004), arXiv:hep-th/0405279

  67. 67.

    K.R. Dienes, New string partition functions with vanishing cosmological constant. Phys. Rev. Lett. 65, 1979–1982 (1990)

    ADS  Article  Google Scholar 

  68. 68.

    K.R. Dienes, Generalized Atkin–Lehner symmetry. Phys. Rev. D 42, 2004–2021 (1990)

    ADS  MathSciNet  Article  Google Scholar 

  69. 69.

    S. Kachru, J. Kumar, E. Silverstein, Vacuum energy cancellation in a nonsupersymmetric string. Phys. Rev. D 59, 106004 (1999). arXiv:hep-th/9807076

    ADS  MathSciNet  Article  Google Scholar 

  70. 70.

    C. Angelantonj, M. Cardella, Vanishing perturbative vacuum energy in nonsupersymmetric orientifolds. Phys. Lett. B 595, 505–512 (2004). arXiv:hep-th/0403107

    ADS  MathSciNet  MATH  Article  Google Scholar 

  71. 71.

    S. Abel, K.R. Dienes, E. Mavroudi, Towards a nonsupersymmetric string phenomenology. Phys. Rev. D 91, 126014 (2015). arXiv:1502.03087 [hep-th]

    ADS  Article  Google Scholar 

  72. 72.

    S. Abel, R.J. Stewart, Exponential suppression of the cosmological constant in nonsupersymmetric string vacua at two loops and beyond. Phys. Rev. D 96, 106013 (2017). arXiv:1701.06629 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  73. 73.

    S. Abel, K.R. Dienes, E. Mavroudi, GUT precursors and entwined SUSY: The phenomenology of stable nonsupersymmetric strings. Phys. Rev. D 97, 126017 (2018). arXiv:1712.06894 [hep-ph]

    ADS  MathSciNet  Article  Google Scholar 

  74. 74.

    J.M. Ashfaque, P. Athanasopoulos, A.E. Faraggi, H. Sonmez, NonTachyonic semi-realistic non-supersymmetric heterotic string vacua. Eur. Phys. J. C 76, 208 (2016). arXiv:1506.03114 [hep-th]

    ADS  Article  Google Scholar 

  75. 75.

    A.E. Faraggi, String phenomenology from a worldsheet perspective. Eur. Phys. J. C 79, 703 (2019). arXiv:1906.09448 [hep-th]

    ADS  Article  Google Scholar 

  76. 76.

    A.E. Faraggi, V.G. Matyas, B. Percival, Stable three generation standard- like model from a tachyonic ten dimensional heterotic-string vacuum. Eur. Phys. J. C 80, 337 (2020). arXiv:1912.00061 [hep-th]

    ADS  Article  Google Scholar 

  77. 77.

    M. Blaszczyk, S. Groot Nibbelink, O. Loukas, F. Ruehle, Calabi-Yau compactifications of non-supersymmetric heterotic string theory. JHEP 10, 166 (2015). arXiv:1507.06147 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  78. 78.

    P. G. Freund, M. A. Rubin, Dynamics of Dimensional Reduction, Phys. Lett. B 97, edited by A. Salam and E. Sezgin, 233–[79] E. Silverstein, (A)dS backgrounds fromasymmetric orientifolds, ClayMat. Proc. 1, 179 (2002), arXiv:hep-th/0106209

  79. 79.

    E. Silverstein, (A)dS backgrounds from a symmetric orientifolds. Clay Mat. Proc. 1, 179 (2002). arXiv:hep-th/0106209

    MathSciNet  Google Scholar 

  80. 80.

    A. Maloney, E. Silverstein, A. Strominger, De Sitter space in noncritical string theory, in Workshop on Conference on the Future of Theoretical Physics and Cosmology in Honor of Steven Hawking’s 60th Birthday (May 2002), pp. 570–591, arXiv:hep-th/0205316

  81. 81.

    J. M. Maldacena, H. Ooguri, Strings in AdS(3) and SL(2,R) WZW model 1.: The Spectrum, J. Math. Phys. 42, 2929–2960 (2001), arXiv:hep-th/0001053

  82. 82.

    R. Blumenhagen, A. Font, Dilaton tadpoles, warped geometries and large extra dimensions for nonsupersymmetric strings. Nucl. Phys. B 599, 241–254 (2001). arXiv:hep-th/0011269

    ADS  MATH  Article  Google Scholar 

  83. 83.

    P. Pelliconi, A. Sagnotti, Integrable models and supersymmetry breaking. Nucl. Phys. B 965, 115363 (2021). arXiv:2102.06184 [hep-th]

    MathSciNet  Article  Google Scholar 

  84. 84.

    P. Fré, A. Sagnotti, A. Sorin, Integrable Scalar Cosmologies I. Foundations and links with String Theory. Nucl. Phys. B 877, 1028–1106 (2013). arXiv:1307.1910 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  85. 85.

    E. Dudas, N. Kitazawa, A. Sagnotti, On climbing scalars in string theory. Phys. Lett. B 694, 80–88 (2011). arXiv:1009.0874 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  86. 86.

    C. Condeescu, E. Dudas, Kasner solutions, climbing scalars and bigbang singularity. JCAP 08, 013 (2013). arXiv:1306.0911 [hep-th]

    ADS  Article  Google Scholar 

  87. 87.

    C. Angelantonj, A. Armoni, Nontachyonic type 0B orientifolds, nonsupersymmetric gauge theories and cosmological RG flow. Nucl. Phys. B 578, 239–258 (2000). arXiv:hep-th/9912257

    ADS  MATH  Article  Google Scholar 

  88. 88.

    C. Angelantonj, A. Armoni, RG flow, Wilson loops and the dilaton tadpole. Phys. Lett. B 482, 329–336 (2000). arXiv:hep-th/0003050

    ADS  MathSciNet  MATH  Article  Google Scholar 

  89. 89.

    E. Dudas, J. Mourad, D-branes in nontachyonic 0B orientifolds. Nucl. Phys. B 598, 189–224 (2001). arXiv:hep-th/0010179

    ADS  MATH  Article  Google Scholar 

  90. 90.

    I.R. Klebanov, A.A. Tseytlin, D-branes and dual gauge theories in type 0 strings. Nucl. Phys. B 546, 155–181 (1999). arXiv:hep-th/9811035

    ADS  MathSciNet  MATH  Article  Google Scholar 

  91. 91.

    J.D. Blum, K.R. Dienes, Duality without supersymmetry: The Case of the SO(16) \(\times \) SO(16) string. Phys. Lett. B 414, 260–268 (1997). arXiv:hep-th/9707148

  92. 92.

    J.D. Blum, K.R. Dienes, Strong/weak coupling duality relations for nonsupersymmetric string theories. Nucl. Phys. B 516, 83–159 (1998). arXiv:hep-th/9707160

    ADS  MATH  Article  Google Scholar 

  93. 93.

    A.E. Faraggi, M. Tsulaia, Interpolations Among NAHE-based supersymmetric and nonsupersymmetric string vacua. Phys. Lett. B 683, 314–320 (2010). arXiv:0911.5125 [hep-th]

    ADS  Article  Google Scholar 

  94. 94.

    A.E. Faraggi, M. Tsulaia, On the Low Energy Spectra of the Nonsupersymmetric Heterotic String Theories. Eur. Phys. J. C 54, 495–500 (2008). arXiv:0706.1649 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  95. 95.

    D. Lüst, D. Tsimpis, \({{\rm AdS}}_2\) Type-IIA Solutions and Scale Separation, (2020), arXiv:2004.07582 [hep-th]

  96. 96.

    F. Gautason, M. Schillo, T. Van Riet, M. Williams, Remarks on scale separation in flux vacua. JHEP 03, 061 (2016). arXiv:1512.00457 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  97. 97.

    F. Marchesano, E. Palti, J. Quirant, A. Tomasiello, On supersymmetric \({{\rm AdS}}_4\) orientifold vacua, (2020), arXiv:2003.13578 [hep-th]

  98. 98.

    A.R. Brown, A. Dahlen, Small steps and giant leaps in the landscape. Phys. Rev. D 82, 083519 (2010). arXiv:1004.3994 [hep-th]

    ADS  Article  Google Scholar 

  99. 99.

    K. Dasgupta, G. Rajesh, S. Sethi, M theory, orientifolds and G-flux. JHEP 08, 023 (1999). arXiv:hep-th/9908088

    ADS  MathSciNet  MATH  Article  Google Scholar 

  100. 100.

    S.B. Giddings, S. Kachru, J. Polchinski, Hierarchies from fluxes in string compactifications. Phys. Rev. D 66, 106006 (2002). arXiv:hep-th/0105097

    ADS  MathSciNet  Article  Google Scholar 

  101. 101.

    M.P. Hertzberg, S. Kachru, W. Taylor, M. Tegmark, Inflationary constraints on type IIA string theory. JHEP 12, 095 (2007). arXiv:0711.2512 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  102. 102.

    S.-J. Lee, W. Lerche, T. Weigand, Emergent Strings, Duality and Weak Coupling Limits for Two-Form Fields, (2019), arXiv:1904.06344 [hep-th]

  103. 103.

    S.-J. Lee, W. Lerche, T. Weigand, Emergent Strings from Infinite Distance Limits, (2019), arXiv:1910.01135 [hep-th]

  104. 104.

    F. Baume, J. Calderón Infante, Tackling the SDC in AdS with CFTs, (2020), arXiv:2011.03583 [hep-th]

  105. 105.

    E. Perlmutter, L. Rastelli, C. Vafa, I. Valenzuela, A CFT Distance Conjecture, (2020), arXiv:2011.10040 [hep-th]

  106. 106.

    P. Koerber, L. Martucci, From ten to four and back again: How to generalize the geometry. JHEP 08, 059 (2007). arXiv:0707.1038 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  107. 107.

    J. Moritz, A. Retolaza, A. Westphal, Toward de Sitter space from ten dimensions. Phys. Rev. D 97, 046010 (2018). arXiv:1707.08678 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  108. 108.

    R. Kallosh, T. Wrase, dS Supergravity from 10d. Fortsch. Phys. 67, 1800071 (2019). arXiv:1808.09427 [hep-th]

    ADS  Article  Google Scholar 

  109. 109.

    I. Bena, E. Dudas, M. Graña, S. Lüst, Uplifting runaways. Fortsch. Phys. 67, 1800100 (2019). arXiv:1809.06861 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  110. 110.

    U.H. Danielsson, S.S. Haque, G. Shiu, T. Van Riet, Towards classical de Sitter solutions in string theory. JHEP 09, 114 (2009). arXiv:0907.2041 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  111. 111.

    C. Córdova, G.B. De Luca, A. Tomasiello, Classical de Sitter solutions of 10-dimensional supergravity. Phys. Rev. Lett. 122, 091601 (2019). arXiv:1812.04147 [hep-th]

    ADS  Article  Google Scholar 

  112. 112.

    J. Blåbäck, U. Danielsson, G. Dibitetto, S. Giri, Constructing stable de Sitter in M-theory from higher curvature corrections. JHEP 09, 042 (2019). arXiv:1902.04053 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  113. 113.

    N. Cribiori, D. Junghans, No classical (anti-)de Sitter solutions with O8planes. Phys. Lett. B 793, 54–58 (2019). arXiv:1902.08209 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  114. 114.

    D. Andriot, Open problems on classical de Sitter solutions. Fortsch. Phys. 67, 1900026 (2019). arXiv:1902.10093 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  115. 115.

    C. Córdova, G. B. De Luca, A. Tomasiello, New de Sitter Solutions in Ten Dimensions and Orientifold Singularities, (2019), arXiv:1911.04498 [hep-th]

  116. 116.

    D. Andriot, P. Marconnet, T. Wrase, New de Sitter solutions of 10d type IIB supergravity. JHEP 08, 076 (2020). arXiv:2005.12930 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  117. 117.

    D. Andriot, P. Marconnet, T. Wrase, Intricacies of classical de Sitter string backgrounds, (2020), arXiv:2006.01848 [hep-th]

  118. 118.

    M. Montero, T. Van Riet, G. Venken, A dS obstruction and its phenomenological consequences. JHEP 05, 114 (2020). arXiv:2001.11023 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  119. 119.

    S.K. Garg, C. Krishnan, Bounds on slow roll and the de sitter swampland. JHEP 11, 075 (2019). arXiv:1807.05193 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  120. 120.

    J. March-Russell, R. Petrossian-Byrne, QCD, Flavor, and the de Sitter Swampland, (2020), arXiv:2006.01144 [hep-th]

  121. 121.

    A. Bedroya, C. Vafa, Trans-Planckian Censorship and the Swampland, (2019), arXiv:1909.11063 [hep-th]

  122. 122.

    A. Bedroya, R. Brandenberger, M. Loverde, C. Vafa, Trans-planckian censorship and inflationary cosmology. Phys. Rev. D 101, 103502 (2020). arXiv:1909.11106 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  123. 123.

    D. Andriot, N. Cribiori, D. Erkinger, The web of swampland conjectures and the TCC bound. JHEP 07, 162 (2020). arXiv:2004.00030 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  124. 124.

    S. Lanza, F. Marchesano, L. Martucci, I. Valenzuela, Swampland Conjectures for Strings andMembranes, (2020), arXiv:2006.15154 [hep-th]

  125. 125.

    S.M. Carroll, M.C. Johnson, L. Randall, Dynamical compactification from de Sitter space. JHEP 11, 094 (2009). arXiv:0904.3115 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  126. 126.

    F. Apruzzi, M. Fazzi, D. Rosa, A. Tomasiello, All AdS\_7 solutions of type II supergravity. JHEP 04, 064 (2014). arXiv:1309.2949 [hep-th]

    ADS  Article  Google Scholar 

  127. 127.

    F. Apruzzi, G. Bruno De Luca, A. Gnecchi,G. Lo Monaco, A. Tomasiello, On \({{\rm AdS}}_7\) stability, (2019), arXiv:1912.13491 [hep-th]

  128. 128.

    S.-W. Kim, J. Nishimura, A. Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)- dimensions. Phys. Rev. Lett. 108, 011601 (2012). arXiv:1108.1540 [hep-th]

    ADS  Article  Google Scholar 

  129. 129.

    K.N. Anagnostopoulos, T. Azuma, Y. Ito, J. Nishimura, S.K. Papadoudis, Complex Langevin analysis of the spontaneous symmetry breaking in dimensionally reduced super Yang–Mills models. JHEP 02, 151 (2018). arXiv:1712.07562 [hep-lat]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  130. 130.

    N. Ishibashi, H. Kawai, Y. Kitazawa, A. Tsuchiya, A Large N reduced model as superstring. Nucl. Phys. B 498, 467–491 (1997). arXiv:hep-th/9612115

    ADS  MathSciNet  MATH  Article  Google Scholar 

  131. 131.

    P. Breitenlohner, D.Z. Freedman, Stability in gauged extended supergravity. Ann. Phys. 144, 249 (1982)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  132. 132.

    H. Lu, K.-N. Shao, Solutions of free higher spins in AdS. Phys. Lett. B 706, 106–109 (2011). arXiv:1110.1138 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  133. 133.

    S.S. Gubser, I. Mitra, Some interesting violations of the Breitenlohner– Freedman bound. JHEP 07, 044 (2002). arXiv:hep-th/0108239

    ADS  MathSciNet  Article  Google Scholar 

  134. 134.

    O. DeWolfe, D.Z. Freedman, S.S. Gubser, G.T. Horowitz, I. Mitra, Stability of AdS(p) x M(q) compactifications without supersymmetry. Phys. Rev. D 65, 064033 (2002). arXiv:hep-th/0105047

    ADS  MathSciNet  Article  Google Scholar 

  135. 135.

    Y.P. Hong, I. Mitra, Investigating the stability of a nonsupersymmetric landscape. Phys. Rev. D 72, 126003 (2005). arXiv:hep-th/0508238

    ADS  Article  Google Scholar 

  136. 136.

    H.J. Kim, L.J. Romans, P. van Nieuwenhuizen, The mass spectrum of chiral N=2 D=10 supergravity on S**5. Phys. Rev. D 32, 389 (1985)

    ADS  MathSciNet  Article  Google Scholar 

  137. 137.

    E. Malek, H. Samtleben, Kaluza–Klein spectrometry for supergravity. Phys. Rev. Lett. 124, 101601 (2020). arXiv:1911.12640 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  138. 138.

    M. Cesàro, G. Larios, O. Varela, Supersymmetric spectroscopy on \({\rm AdS}_4 \times S^7\) and \({{\rm AdS}}_4 \times S^{6}\), (2021), arXiv:2103.13408 [hep-th]

  139. 139.

    E. Malek, H. Nicolai, H. Samtleben, Tachyonic Kaluza-Klein modes and the AdS swampland conjecture, (2020), arXiv:2005.07713 [hep-th]

  140. 140.

    I.R. Klebanov, E. Witten, Superconformal field theory on three-branes at a Calabi-Yau singularity. Nucl. Phys. B 536, 199–218 (1998). arXiv:hep-th/9807080

    ADS  MATH  Article  Google Scholar 

  141. 141.

    N. Itzhaki, J.M. Maldacena, J. Sonnenschein, S. Yankielowicz, Supergravity and the large N limit of theories with sixteen supercharges. Phys. Rev. D 58, 046004 (1998). arXiv:hep-th/9802042

    ADS  MathSciNet  Article  Google Scholar 

  142. 142.

    J.D. Brown, C. Teitelboim, Dynamical neutralization of the cosmological constant. Phys. Lett. B 195, 177–182 (1987)

    ADS  Article  Google Scholar 

  143. 143.

    J.D. Brown, C. Teitelboim, Neutralization of the cosmological constant by membrane creation. Nucl. Phys. B 297, 787–836 (1988)

    ADS  MathSciNet  Article  Google Scholar 

  144. 144.

    S. R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D15, [Erratum: Phys. Rev.D16,1248(1977)], 2929–2936 (1977)

  145. 145.

    C. G. Callan Jr., S. R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D16, 1762–1768 (1977)

  146. 146.

    S.R. Coleman, F. De Luccia, Gravitational effects on and of vacuum decay. Phys. Rev. D 21, 3305 (1980)

    ADS  MathSciNet  Article  Google Scholar 

  147. 147.

    J.J. Blanco-Pillado, D. Schwartz-Perlov, A. Vilenkin, Quantum tunneling in flux compactifications. JCAP 0912, 006 (2009). arXiv:0904.3106 [hep-th]

    ADS  Article  Google Scholar 

  148. 148.

    A.R. Brown, A. Dahlen, Bubbles of nothing and the fastest decay in the landscape. Phys. Rev. D 84, 043518 (2011). arXiv:1010.5240 [hep-th]

    ADS  Article  Google Scholar 

  149. 149.

    R. Bousso, J. Polchinski, Quantization of four form fluxes and dynamical neutralization of the cosmological constant. JHEP 06, 006 (2000). arXiv:hep-th/0004134

    ADS  MathSciNet  MATH  Article  Google Scholar 

  150. 150.

    A.R. Brown, A. Dahlen, Giant leaps and minimal branes in multi-dimensional flux landscapes. Phys. Rev. D 84, 023513 (2011). arXiv:1010.5241 [hep-th]

    ADS  Article  Google Scholar 

  151. 151.

    E. Witten, Instability of the Kaluza–Klein Vacuum. Nucl. Phys. B 195, 481–492 (1982)

    ADS  MATH  Article  Google Scholar 

  152. 152.

    G. Dibitetto, N. Petri, M. Schillo, Nothing really matters, (2020), arXiv:2002.01764 [hep-th]

  153. 153.

    I. García Etxebarria, M. Montero, K. Sousa, I. Valenzuela, Nothing is certain in string compactifications, (2020), arXiv:2005.06494 [hep-th]

  154. 154.

    A.R. Brown, A. Dahlen, On nothing as an infinitely negatively curved spacetime. Phys. Rev. D 85, 104026 (2012). arXiv:1111.0301 [hep-th]

  155. 155.

    R. Antonelli, I. Basile, A. Bombini, AdS vacuum bubbles, holography and dual RG flows. Class. Quant. Grav. 36, 045004 (2019). arXiv:1806.02289 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  156. 156.

    S. Sethi, Supersymmetry breaking by fluxes. JHEP 10, 022 (2018). arXiv:1709.03554 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  157. 157.

    J.M. Maldacena, J. Michelson, A. Strominger, Anti-de Sitter fragmentation. JHEP 02, 011 (1999). arXiv:hep-th/9812073

    ADS  MathSciNet  MATH  Article  Google Scholar 

  158. 158.

    N. Seiberg, E. Witten, The D1 / D5 system and singular CFT. JHEP 04, 017 (1999). arXiv:hep-th/9903224

    ADS  MathSciNet  MATH  Article  Google Scholar 

  159. 159.

    E.A. Bergshoeff, M. de Roo, S.F. Kerstan, F. Riccioni, IIB supergravity revisited. JHEP 08, 098 (2005). arXiv:hep-th/0506013

    ADS  MathSciNet  Article  Google Scholar 

  160. 160.

    E.A. Bergshoeff, M. de Roo, S.F. Kerstan, T. Ortin, F. Riccioni, SL(2, R)invariant IIB brane actions. JHEP 02, 007 (2007). arXiv:hep-th/0611036

    ADS  MathSciNet  Article  Google Scholar 

  161. 161.

    E.A. Bergshoeff, F. Riccioni, String solitons and T-duality. JHEP 05, 131 (2011). arXiv:1102.0934 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  162. 162.

    E.A. Bergshoeff, F. Riccioni, Heterotic wrapping rules. JHEP 01, 005 (2013). arXiv:1210.1422 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  163. 163.

    E.A. Bergshoeff, V.A. Penas, F. Riccioni, S. Risoli, Non-geometric fluxes and mixed-symmetry potentials. JHEP 11, 020 (2015). arXiv:1508.00780 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  164. 164.

    M. Bianchi, A. Sagnotti, The partition function of the SO(8192) bosonic string. Phys. Lett. B 211, 407–416 (1988)

    ADS  MathSciNet  Article  Google Scholar 

  165. 165.

    Q. Bonnefoy, E. Dudas, S. Lüst, On the weak gravity conjecture in string theory with broken supersymmetry. Nucl. Phys. B 947, 114738 (2019). arXiv:1811.11199 [hep-th]

    MathSciNet  MATH  Article  Google Scholar 

  166. 166.

    Q. Bonnefoy, E. Dudas, S. Lüst, Weak gravity (and other conjectures) with broken supersymmetry, in 19th Hellenic School and Workshops on Elementary Particle Physics and Gravity (Mar. 2020), arXiv:2003.14126 [hep-th]

  167. 167.

    N. Kaloper, Bent domain walls as brane worlds. Phys. Rev. D 60, 123506 (1999). arXiv:hep-th/9905210

    ADS  MathSciNet  Article  Google Scholar 

  168. 168.

    T. Shiromizu, K.-I. Maeda, M. Sasaki, The Einstein equation on the 3-brane world. Phys. Rev. D 62, 024012 (2000). arXiv:gr-qc/9910076

    ADS  MathSciNet  Article  Google Scholar 

  169. 169.

    D.N. Vollick, Cosmology on a three-brane. Class. Quant. Grav. 18, 1–10 (2001). arXiv:hep-th/9911181

    ADS  MathSciNet  MATH  Article  Google Scholar 

  170. 170.

    S.S. Gubser, AdS / CFT and gravity. Phys. Rev. D 63, 084017 (2001). arXiv:hep-th/9912001

    ADS  MathSciNet  Article  Google Scholar 

  171. 171.

    S. Hawking, T. Hertog, H. Reall, Brane new world. Phys. Rev. D 62, 043501 (2000). arXiv:hep-th/0003052

    ADS  MathSciNet  Article  Google Scholar 

  172. 172.

    G.T. Horowitz, J. Orgera, J. Polchinski, Nonperturbative instability of AdS(5) x S**5/Z(k). Phys. Rev. D 77, 024004 (2008). arXiv:0709.4262 [hep-th]

    ADS  MathSciNet  Article  Google Scholar 

  173. 173.

    J.J. Blanco-Pillado, B. Shlaer, Bubbles of nothing in flux compactifications. Phys. Rev. D 82, 086015 (2010). arXiv:1002.4408 [hep-th]

    ADS  Article  Google Scholar 

  174. 174.

    R. Antonelli, Black Objects without a Vacuum, PhD thesis (Pisa, Scuola Normale Superiore, 2020)

  175. 175.

    G.T. Horowitz, A. Strominger, Black strings and P-branes. Nucl. Phys. B 360, 197–209 (1991)

    ADS  MathSciNet  Article  Google Scholar 

  176. 176.

    A. Sagnotti, Brane SUSY breaking and inflation: implications for scalar fields and CMB distortion. Phys. Part. Nucl. Lett. 11, 836–843 (2014). arXiv:1303.6685 [hep-th]

    Article  Google Scholar 

  177. 177.

    R.C. Myers, Dielectric branes. JHEP 12, 022 (1999). arXiv:hep-th/9910053 [hep-th]

    ADS  MathSciNet  MATH  Article  Google Scholar 

  178. 178.

    G. Giribet, C. Hull, M. Kleban, M. Porrati, E. Rabinovici, Superstrings on \({\rm AdS}_3\) at \(\Vert = 1\). JHEP 08, 204 (2018). arXiv:1803.04420 [hep-th]

    ADS  MATH  Article  Google Scholar 

  179. 179.

    M. A. Rubin, C. R. Ordonez, Eigenvalues and degeneracies for n-dimensional tensor spherical harmonics, (1983)

  180. 180.

    M.A. Rubin, C.R. Ordonez, Symmetric tensor eigen spectrum of the laplacian on \(n\) spheres. J. Math. Phys. 26, 65 (1985)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  181. 181.

    P. van Nieuwenhuizen, The compactification of IIB supergravity on \(S_5\) revisted, in Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer, edited by A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov, and E. Scheidegger (June 2012), pp. 133–157, arXiv:1206.2667 [hep-th]

  182. 182.

    Z. Ma, Group Theory for Physicists (World Scientific, Singapore, 2007)

    MATH  Book  Google Scholar 

Download references

Acknowledgements

This review originates from the author’s Ph.D. Thesis, defended at Scuola Normale Superiore, Pisa under the supervision of A. Sagnotti. I would like to express my gratitude to him for his invaluable patience and profound insights. I would also like to thank C. Angelantonj, E. Dudas and J. Mourad for their enlightening feedback on my work, and A. Campoleoni, G. Bogna and S. Raucci for discussions and suggestions on this review.

Funding

The work of I.B. was supported by the Fonds de la Recherche Scientifique - FNRS under Grants No. F.4503.20 (“HighSpinSymm”) and T.0022.19 (“Fundamental issues in extended gravitational theories”).

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Tensor spherical harmonics: a primer

Tensor spherical harmonics: a primer

In this appendix we review some results that were needed for our stability analysis in Sect. 4, starting from an ambient Euclidean space. In Sect. A.1 we build scalar spherical harmonics, and in Sect. A.2 we extend our considerations to tensors of higher rank. The results agree with the constructions presented in [179, 180].Footnote 81

Scalar spherical harmonics

Let \(Y^1,\dots Y^{n+1}\) be Cartesian coordinates of \({\mathbb {R}}^{n+1}\), so that the unit sphere \({\mathbb {S}}^{n}\) is described by the constraint

$$\begin{aligned} \begin{aligned} \delta _{IJ} \, Y^I \, Y^J = r^2 \end{aligned}\end{aligned}$$
(A.1)

on the radial coordinate r, solved by spherical coordinates \(y^i\) according to

$$\begin{aligned} \begin{aligned} Y^I = r \, {\widehat{Y}}^I(y) \, . \end{aligned}\end{aligned}$$
(A.2)

The scalar spherical harmonics on \({\mathbb {S}}^{n}\) can be conveniently constructed starting from harmonic polynomials of degree \(\ell \) in the ambient Euclidean space \({\mathbb {R}}^{n+1}\). A harmonic polynomial of degree \(\ell \) takes the form

$$\begin{aligned} \begin{aligned} H_{(n)}^\ell (Y) = \alpha _{I_1 \dots I_\ell } \, Y^{I_1} \dots Y^{I_\ell } \, , \end{aligned}\end{aligned}$$
(A.3)

and is therefore determined by a completely symmetric and trace-less tensor \(\alpha _{I_1\dots I_\ell }\) of rank \(\ell \), as can be clearly seen applying to it the Euclidean Laplacian

$$\begin{aligned} \begin{aligned} \nabla ^2_{n+1} = \sum _{I=1}^{n+1} \frac{\partial ^2}{\partial Y_I^2} \, . \end{aligned}\end{aligned}$$
(A.4)

The scalar spherical harmonics \(\mathcal{Y}_{(n)}^{I_1\dots I_\ell }\) are then defined restricting the \(H_{(n)}^{\ell }(Y)\) to the unit sphere \(S^n\), or equivalently as

$$\begin{aligned} \begin{aligned} H_{(n)}^\ell ({\widehat{Y}}(y)) = r^\ell \, \alpha _{I_1 \dots I_\ell } \, {\mathscr {Y}}_{(n)}^{I_1 \dots I_\ell }(y) \, . \end{aligned}\end{aligned}$$
(A.5)

As a result, the Euclidean metric can be recast as

$$\begin{aligned} \begin{aligned} ds^2_{n+1} = dr^2 + r^2 \, d\varOmega _n^2 \, , \end{aligned}\end{aligned}$$
(A.6)

and the scalar Laplacian decomposes according to

$$\begin{aligned} \begin{aligned} 0 = \nabla ^2_{n+1} H_{(n)}^\ell (Y) = \frac{1}{r^n} \, \frac{\partial }{\partial r} \left( r^n \, \frac{\partial H_{(n)}^\ell (Y)}{\partial r} \right) + \frac{1}{r^2} \, \nabla ^2_{{\mathbb {S}}^n} \, H_{(n)}^\ell (Y) \, , \end{aligned}\end{aligned}$$
(A.7)

where

$$\begin{aligned} \begin{aligned} \frac{\partial H_{(n)}^\ell (Y)}{\partial r} = \frac{\ell }{r} \, H_{(n)}^\ell (Y) \end{aligned}\end{aligned}$$
(A.8)

for the homogeneous polynomials \(H_{(n)}^{\ell }(Y)\). All in all

figures

and the degeneracy of the scalar spherical harmonics for any given \(\ell \) is the number of independent components of a corresponding completely symmetric and trace-less tensor, namely

$$\begin{aligned} \begin{aligned} \frac{\left( n + 2\ell - 1 \right) \left( n + \ell - 2 \right) !}{\ell ! \left( n-1 \right) !} \, . \end{aligned}\end{aligned}$$
(A.10)

Spherical harmonics of higher rank

In discussing more general tensor harmonics, it is convenient to notice that, in the coordinate system of Eq. (A.6), the non-vanishing Christoffel symbols \({\widetilde{\varGamma }}_{IJ}^K\) for the ambient Euclidean space read

$$\begin{aligned} \begin{aligned} {\widetilde{\varGamma }}_{ij}^r = - \, r \, g_{ij} \, , \qquad {\widetilde{\varGamma }}_{jr}^i = \frac{1}{r} \, \delta _i^j \, , \qquad {\widetilde{\varGamma }}_{ij}^k = \varGamma _{ij}^k \, , \end{aligned}\end{aligned}$$
(A.11)

where the labels ijk refer, as above, to the n-sphere, whose Christoffel symbols are denoted by \(\varGamma _{ij}^k\).

The construction extends nicely to tensor spherical harmonics, which can be defined starting from generalized harmonic polynomials, with one proviso. The relation in Eq. (A.2) and its differentials imply that the actual spherical components of tensors carry additional factors of r, one for each covariant tensor index, with respect to those naïvely inherited from the Cartesian coordinates of the Euclidean ambient space, as we shall now see in detail. To begin with, vector spherical harmonics arise from one-forms in ambient space, built from harmonic polynomials of the type

$$\begin{aligned} \begin{aligned} H_{(n) \, J}^\ell (Y) = \alpha _{I_1 \dots I_\ell \, , \, J} \, Y^{I_1} \dots Y^{I_\ell } \, , \end{aligned}\end{aligned}$$
(A.12)

where the coefficients \(\alpha _{I_1 \dots I_\ell \,,\,J}\) are completely symmetric and trace-less in any pair of the first \(\ell \) indices. They are also subject to the condition

$$\begin{aligned} \begin{aligned} Y^J \, H_{(n) \, J}^\ell (Y) = 0 \, , \end{aligned}\end{aligned}$$
(A.13)

since the radial component, which does not pertain to the sphere \(S^n\), ought to vanish. This implies that the complete symmetrization of the coefficients vanishes identically,

$$\begin{aligned} \begin{aligned} \alpha _{(I_1 \dots I_\ell \, , \, J)} = 0 \, , \end{aligned}\end{aligned}$$
(A.14)

and on account of the symmetry in the first \(\ell \) indices. As a result, \(H_{n\,,\,J}^{\ell }(Y)\) is thus transverse in the ambient space,

$$\begin{aligned} \begin{aligned} \partial ^J H_{(n) \, J}^\ell (Y) = 0 \, . \end{aligned}\end{aligned}$$
(A.15)

Moreover, any Euclidean vector V such that \(V_I\, Y^I = 0\) couples with differentials according to the general rule inherited from Eq. (A.2),

$$\begin{aligned} \begin{aligned} V_I \, dY^I = V_I \, r \, d{\widehat{Y}}^I \, , \end{aligned}\end{aligned}$$
(A.16)

so that the actual sphere components, which are associated to \(d {\widehat{Y}}^I\), include an additional power of r, and the vector spherical harmonics \({\mathscr {Y}}_{(n)\, i}^{I_1\dots I_\ell \, , \, J}\) are thus obtained from

$$\begin{aligned} \begin{aligned} r^{\ell + 1} \, {\mathscr {Y}}_{(n) \, i}^{I_1 \dots I_\ell \, , \, J} \, \alpha _{I_1 \dots I_\ell \, , \, J} \, dy^i = r \, H_{(n) \, J}^\ell (Y) \, d{\widehat{Y}}^J \, . \end{aligned}\end{aligned}$$
(A.17)

Therefore,

$$\begin{aligned} \begin{aligned} \nabla _r \nabla _r \left( r \, H_{(n) \, J}^\ell (Y) \right) = \left( \frac{\partial }{\partial r} - \frac{1}{r}\right) ^2 \left( r \, H_{(n) \, J}^\ell (Y) \right) = \frac{\ell \left( \ell - 1 \right) }{r} \, H_{(n) \, J}^\ell (Y) \, , \end{aligned}\end{aligned}$$
(A.18)

while the remaining contributions to the Laplacian give

$$\begin{aligned} \begin{aligned} \frac{1}{r^2} \, \nabla ^2_{{\mathbb {S}}^n} \left( r \, H_{(n) \, J}^\ell (Y)\right) + \frac{n \left( \ell + 1\right) - n - 1}{r} \left( r \, H_{(n) \, J}^\ell (Y)\right) \, , \end{aligned}\end{aligned}$$
(A.19)

taking into account the Christoffel symbols in Eq. (A.11). Since the total Euclidean Laplacian vanishes by construction, adding Eqs. (A.18) and (A.19) finally results in

figuret

with \(\ell \ge 1\).

In a similar fashion, the spherical harmonics \({\mathscr {Y}}_{(n)\, i_1\dots i_p}^{I_1\dots I_\ell \, , \, J_1 \dots J_p}\), corresponding to generic higher-rank transverse tensors which are also trace-less in any pair of symmetric I-indices, can be described starting from harmonic polynomials of the type \(H_{(n) \, J_1\dots J_p}^{\ell }(Y)\), and satisfy

figureu

with \(\ell \ge p\).

In Young tableaux language, the scalar harmonics correspond to trace-less single-row diagrams of the type

(A.22)

while the independent vectors associated to vector harmonics correspond to two-row trace-less hooked diagrams of the type

(A.23)

as we have explained. Similarly, the independent tensor perturbations of the metric in the internal space correspond to trace-less diagrams of the type

(A.24)

while the independent perturbations associated to a \((p+1)\)-form gauge field in the internal space correspond, in general, to multi-row diagrams of the type

(A.25)

The degeneracies of these representations can be related to the corresponding Young tableaux, as in [182]. The structure of the various types of harmonics, which are genuinely different for large enough values of n, reflects nicely the generic absence of mixings between different classes of perturbations.

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Basile, I. Supersymmetry breaking and stability in string vacua. Riv. Nuovo Cim. 44, 499–596 (2021). https://doi.org/10.1007/s40766-021-00024-9

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Keywords

  • Supersymmetry breaking
  • Stability
  • Brane dynamics
  • Swampland