Holographic studies of dense QCD matter based on the D3/D7 brane system with a phenomenologically adjusted infrared dilaton profile have produced hybrid-neutron-star equations of state (EoS) stiff enough to support stable deconfined-quark cores in two-solar-mass stars. However, the baryonic (nuclear) phase in this framework is available only in the smeared-D5-brane approximation, which yields an unphysically stiff pressure and an unphysical fourth-order onset transition, forcing the use of phenomenological nuclear EoS at low density; and only a small region of the model's parameter space has been scanned. The open problem is threefold and linked: (i) construct and numerically solve localized (both single-localized and multi-localized) D5-brane baryon-vertex embeddings coupled to the D7 flavor branes at finite baryon density, and determine whether the resulting baryonic EoS matches empirical nuclear-matter constraints near saturation density, including beta equilibrium and the symmetry energy; (ii) delineate which regions of the model parameter space (A,λ~,κ,RAdS,λt)(A,\tilde\lambda,\kappa,R_{\mathrm{AdS}},\lambda_t) admit hybrid EoS simultaneously satisfying the two-solar-mass maximum-mass constraint and the GW170817 tidal-deformability constraint; and (iii) compute how rotation and neutrino transport modify the stability and observables of stars with holographic quark cores. These deliverables jointly decide, within one bottom-up holographic framework, whether quark cores in massive neutron stars are compatible with all current nuclear and multimessenger constraints.

Background

Gauge/gravity duality (holography) models strongly coupled gauge theories by weakly coupled gravity in a higher-dimensional spacetime; in bottom-up holographic QCD, the D3/D7 system places NfN_f D7-branes (the "flavor" branes carrying the quarks) into the geometry sourced by NcN_c D3-branes, and adjusting the dilaton (a scalar field controlling the effective string coupling) in the infrared stiffens the resulting dense-matter equation of state (EoS) beyond the conformal P∝ϵP\propto\epsilon behavior. In this framework a baryon containing NcN_c quarks is represented by a "baryon vertex": a D5-brane wrapping the internal five-sphere, whose worldvolume flux must be neutralized by fundamental strings stretching to the D7-branes. Evans, Kim, Magou, Seo and Sin (JHEP 2012, arXiv:1204.5640) showed that a suitable dilaton profile stabilizes such wrapped D5-branes, giving a holographic baryonic phase. Because solving the shape of a D5-brane localized at a point in physical space is a hard nonlinear problem, all existing calculations smear the D5 charge homogeneously over space — an approximation expected to be reliable only at high density, where baryons overlap. Bitaghsir Fadafan, Cruz Rojas and Evans (arXiv:1911.12705) had earlier shown that dilaton-adjusted D3/D7 systems can produce much stiffer quark-matter EoS than the conformal-background analysis of Hoyos, Jokela, Rodríguez Fernández and Vuorinen (arXiv:1603.02943), bringing hybrid stars with quark cores within reach. Bitaghsir Fadafan, Cruz Rojas and Mager (arXiv:2512.02484) carry this program through to compact-star phenomenology: with a hybrid EoS that pairs a phenomenological nuclear phase (the Hebeler–Lattimer–Pethick–Schwenk chiral effective field theory (EFT) bands, which constrain the nuclear EoS near the saturation density of nuclear matter from low-energy nuclear physics) with a holographic quark phase, and a weakened transition parameter κ\kappa, the model produces stable hybrid stars with quark cores and maximum masses near 2M⊙2M_\odot; it also computes tidal deformabilities (the dimensionless Love number Λ\Lambda measuring how a companion's tidal field deforms the star, constrained by the gravitational-wave event GW170817 to Λ(1.4M⊙)=190−120+390\Lambda(1.4M_\odot)=190^{+390}_{-120}). Two defects of the framework are explicit in the paper. First, the smeared D5 baryon phase is "way too stiff and disagrees with phenomenology", exhibits an onset behavior Ω∼(μ−μmin⁡)4\Omega\sim(\mu-\mu_{\min})^4 (an undesirable fourth-order phase transition at the baryonic onset chemical potential μmin⁡\mu_{\min}), and simple modifications of the action or dilaton profile do not fix this; the isospin-asymmetry (symmetry) energy has, in the paper's words, "never been computed with localized holographic matter". Second, only a restricted region of the five-dimensional parameter space (A,λ~,κ,RAdS,λt)(A,\tilde\lambda,\kappa,R_{\mathrm{AdS}},\lambda_t) — the dilaton amplitude, transition dilaton scale, transition-strength parameter, AdS radius, and effective 't Hooft coupling — was scanned. In the adjacent Witten–Sakai–Sugimoto (WSS) holographic model (a different brane construction in which baryons are instanton solitons), recent work (Bartolini, Gudnason, Mager and Rebhan, arXiv:2607.19125) shows that replacing the homogeneous ansatz by localized soliton crystals dramatically improves agreement with nuclear phenomenology around saturation — evidence that localization, not parameter tinkering, is the missing ingredient, but a construction that exists only in the WSS model and not for the wrapped-D5 baryons of the D3/D7 setup.

Problem Statement

Within the D3/D7 holographic model with a phenomenologically adjusted infrared dilaton profile defined in Bitaghsir Fadafan, Cruz Rojas and Mager (arXiv:2512.02484) — which supports hybrid neutron stars with stable deconfined-quark cores when the baryonic phase is described by phenomenological nuclear EoS — solve the following linked problems, whose common objective is a fully holographic, self-consistent hybrid-star EoS capable of deciding whether quark cores in massive neutron stars are compatible with all current constraints:

(i) Replace the smeared-D5 approximation of the baryonic sector: construct and numerically solve localized D5-brane baryon-vertex embeddings (both single-localized and multi-localized) coupled to the NfN_f D7 flavor branes at finite baryon density, including the charge-neutralizing string sources and worldvolume flux balance, and extract the resulting baryonic EoS P(μ)P(\mu) and grand potential Ω(μ,d)\Omega(\mu,d) (with dd the D5-brane density) near onset. Determine in particular whether localization cures the unphysically stiff pressure and the fourth-order onset behavior Ω∼(μ−μmin⁡)4\Omega\sim(\mu-\mu_{\min})^4 of the smeared phase, and whether a localized holographic baryonic EoS can match empirical nuclear-matter constraints near saturation density — including beta equilibrium and the isospin symmetry energy, which has never been computed with localized holographic matter.

(ii) Using the holographic baryonic EoS from (i), systematically explore the model parameter space (A,λ~,κ,RAdS,λt)(A,\tilde\lambda,\kappa,R_{\mathrm{AdS}},\lambda_t) beyond the restricted region scanned in the source paper, and delineate which regions yield hybrid EoS simultaneously compatible with the maximum-mass constraint Mmax⁡≳2M⊙M_{\max}\gtrsim 2M_\odot and the GW170817 tidal-deformability constraint Λ(1.4M⊙)=190−120+390\Lambda(1.4M_\odot)=190^{+390}_{-120}, mapping mass–radius curves and tidal deformabilities across the viable region.

(iii) Quantify how rotation and neutrino transport modify the stability and observable properties of stars with holographic quark cores in this model.

Problems (i)–(iii) are linked: (ii) and (iii) take the baryonic EoS obtained in (i) as input. Problem (i) is the central missing ingredient: without it, no fully holographic neutron-star model exists in this framework.

References

  1. Kazem Bitaghsir Fadafan, Jesús Cruz Rojas, Jonas Mager, "Properties of Stable Massive Quark Stars in Holography" (2025), arXiv:2512.02484, DOI: 10.48550/arXiv.2512.02484, https://arxiv.org/abs/2512.02484
  2. Nick Evans, Keun-Young Kim, Maria Magou, Yunseok Seo, Sang-Jin Sin, "The Baryonic Phase in Holographic Descriptions of the QCD Phase Diagram", JHEP 0909:045 (2012), arXiv:1204.5640, DOI: 10.1007/JHEP09(2012)045, https://arxiv.org/abs/1204.5640
  3. Lorenzo Bartolini, Sven Bjarke Gudnason, Jonas Mager, Anton Rebhan, "Holographic Soliton Crystals for Dense Nuclear Matter and Neutron Stars" (2026), arXiv:2607.19125, DOI: 10.48550/arXiv.2607.19125, https://arxiv.org/abs/2607.19125
  4. K. Hebeler, J. M. Lattimer, C. J. Pethick, A. Schwenk, "Equation of state and neutron star properties constrained by nuclear physics and observation", Astrophys. J. 773:11 (2013), arXiv:1303.4662, DOI: 10.1088/0004-637X/773/1/11, https://arxiv.org/abs/1303.4662
  5. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), "GW170817: Measurements of Neutron Star Radii and Equation of State", Phys. Rev. Lett. 121:161101 (2018), arXiv:1805.11581, DOI: 10.1103/PhysRevLett.121.161101, https://arxiv.org/abs/1805.11581
  6. Kazem Bitaghsir Fadafan, Jesús Cruz Rojas, Nick Evans, "Deconfined, Massive Quark Phase at High Density and Compact Stars: A Holographic Study", Phys. Rev. D 101:126005 (2020), arXiv:1911.12705, DOI: 10.1103/PhysRevD.101.126005, https://arxiv.org/abs/1911.12705
  7. Lorenzo Bartolini, Sven Bjarke Gudnason, "Symmetry energy in holographic QCD", SciPost Phys. 16:156 (2024), arXiv:2209.14309, DOI: 10.21468/SciPostPhys.16.6.156, https://arxiv.org/abs/2209.14309