Open research questions in Black Holes and Theoretical Physics

What the literature leaves open

• The open part is upstream, namely the derivation of T (Π) from moving projected matter con(cid:28)gurations, 0i which remains to be completed in the fermionic and coupled-matter sector.

  The Spectral Gravity Sub-Programme · 2026 · DOI

• The analysis in this paper was limited to the insertion of a specific class of two-sided operators, for which we could solve analytically the Lanczos algorithm and repeat the triple-scaled computation we performed for the single-operator case, which required symmetry between the left and right boundaries.

  Holography of K-complexity: switchbacks and shockwaves · 2026 · DOI

• It remains to be seen if the direct computation in 6D will confirm the 2-loop UV divergence found in [7]. It remains to be seen whether the future 2-loop computation in 6D will support the superconformal conjecture of [22] by demonstrating 2-loop UV finiteness, or whether a UV divergence will be found and the explanation of the 1-loop UV finiteness might still be the anomaly-free SO(5, 21) symmetry.

  Anomaly cancellation and one-loop finiteness of 6D half-maximal supergravities · 2026 · DOI

• In this paper, we computed the defect entanglement entropy for codimension-2 monodromy defect in maximally supersymmetric SU(N) Yang-Mills theories in (p + 1)-dimensions, (p = 2, 3, 4). For p ̸= 3 neither the ambient theories nor the defects are conformally invariant, thus generalizing the results of [24, 25]. The defect geometries are obtained by considering the solutions of Dp-branes wrapping a spindle of and changing the range of the radial coordinate of the spindle, such that now it takes values on a semi-infinite interval. Defect boundary conditions are imposed, so that at one end the geometry shrinks smoothly, while asymptotically the metric goes to the corresponding Dp-brane one,20 but we allow the gauge fields to be asymptotically constant. This constant is related to the monodromy of background gauge fields in the dual field theory. In order to compute the defect entanglement entropy, we employ the prescription of [36, 37], interpreting the backgrounds as holographic duals to a (p − 1)-dimensional theory. This leads to a divergence which needs to be renormalized. To address this, we exploit the fact that, in brane frame, the metric asymptotes to AdSp+2 written as a foliation of AdSp × S1 over an interval. We provide a renormalization prescription that allows to obtain a finite result, similar to the one of. This amounts to define a UV cut-off and subtract the contribution of the vacuum. This procedure is only possible for p = 2, 3, 4, since the D5-brane does not have a AdSp+2 asymptotic region in brane frame. In the conformal case (p = 3), the renormalization procedure leads to a quantity that is proportional to a linear combination of the Weyl anomaly on the defect and its conformal weight. This feature is common to other conformal monodromy defects as showed in [24, 25]. In the non-conformal cases, we found that the defect entanglement entropy is proportional to the free energy of the theory where it is embedded, but it remains to be investigated if this can be written in terms of intrinsic defect quantities. This work provides a first step toward holographically understanding supersymmetric nonconformal defects in non-conformal theories, a subject which, to the best of our knowledge, remains unexplored. 20Up to a conical singularity in the circle direction parametrized by z. – 23 – JHEP05(2026)217 6.1 On the non-conformal dEE As mentioned before, maximally supersymmetric Yang-Mills shows a generalized conformal structure, in dimensions different from three, if the gauge coupling is allowed to transform under Weyl rescalings. It is therefore natural to expect (5.27) and (5.30) to be written in a form similar to (5.22), that is, in terms of quantities intrinsic to the defect. Supersymmetric 3D defect in 5D SYM.

  Monodromy defects in maximally supersymmetric Yang-Mills theories from holography · 2026 · DOI

• On the non-conformal dEE 6.2 Outlook A Supergravity conventions A.1 Type II supergravity A.2 7D supergravity and its uplift A.3 6D supergravity and its uplift A.4 5D supergravity and its uplift A.5 4D supergravity and its…

  Monodromy defects in maximally supersymmetric Yang-Mills theories from holography · 2026 · DOI

• N The 3d ě 3 SCFTs realised by Type IIB brane configurations with D3 branes on O3 orientifolds and intersecting various pp, qq 5-branes exhibit two central features: (i) dualities sensitive to discrete Z2-valued symmetry fugacities, and (ii) maximal branches of the vacuum moduli space. The first main result concerns the two basic dualities for orthogonal and symplectic Chern–Simons matter (CSM) theories. While the dualities follow naturally from brane creation/annihilation processes (Appendix A.2), the corresponding maps of ZM2 and ZC2 symmetry fugacities must be specified separately. The orthogonal dualities (3.4) and (3.11) reproduce known results [18–20], while the symplectic dualities (3.2) and (3.13) are, to the best of the author’s knowledge, derived here for the first time. These two dualities form elementary building blocks: any duality sequence in a linear or circular CSM quiver arising from a brane configuration is expected to decompose into them. However, direct validation in longer quivers remains computationally out of reach. The second main result is the development of a magnetic quiver framework for orthosymplectic ě 3 CSM quivers (Section 4.2). The construction proceeds directly from the brane system via 5-brane moves and D3 creation/annihilation, and features two notable aspects: N • In general, multiple suitable phases of a given brane system may exist, each leading to a different magnetic quiver (see Figures 14 and 15b). These quivers, though distinct, are expected to encode the maximal branch geometry. This has been demonstrated explicitly in the examples of Sections 4.2.1–4.2.3. • A fugacity map can be established for the magnetic quiver(s). Three basic mappings “ 4 setting, this can be (and has been) tested via have been identified. In the Hilbert series limits of the supersymmetric index for small CSM quivers. N Open questions. A distinctive feature of magnetic quivers for 3d orthosymplectic Chern- Simons matter theories is the availability of exact operator counting via supersymmetric indices and Hilbert series. This makes it possible, and indeed necessary, to explicitly test the proposed magnetic quivers. Such cross-checks are not typically feasible for magnetic 39 SO(2)Sp(1)SO(4)···Sp(N−1)SO(2N)Sp(N)SO(2N+1)···Sp(N)SO(2N+1)Sp(N)···SO(2N+1)Sp(N)SO(2ap)Sp((a−1)p+ℓ)···SO(4p)Sp(p+ℓ)SO(2p)Sp(ℓ)B0Cp+12BbCℓ−b2N−1nodes2F1−2N−1nodes2F3−2a−1nodes2a−1nodes quivers of higher-dimensional SCFTs. While the approach has been validated for theories with a small number of gauge nodes, its extension to larger quivers presents new challenges. Even though the methods presented here covers a wide range of CSM2κ theories, it does not hold whenever the isolation of the NS5 branch moduli involves moving a p1, κq 5-brane across a NS5 brane more than once. Extending the magnetic quiver framework to this class of theories is left for future work. Higgsing.

  Orthosymplectic Chern–Simons matter theories: Global forms, dualities, and vacua · 2026 · DOI

• 39 A Background material 41 A.1 Brane configurations............................... 41 A.2 Brane creation and annihilation......................... 41 A.3 Good, bad, and ugly............................... 43 A.4 Monopole operators in CSM theories...................... 44 A.4.1 Two-node CSM theory.......................... 44 A.4.2 Three-node CSM theory......................... 46 A.5 Example operator spectroscopy......................... 46 A.6 Example fugacity map.............................. 48 B Index expansions 48 B.1 Orthosymplectic CSM dualities......................... 49 B.2 Linear examples with 2 nodes.......................... 50 B.3 Linear examples with 3 nodes.......................... 56..........................

  Orthosymplectic Chern–Simons matter theories: Global forms, dualities, and vacua · 2026 · DOI

• In this work, we study the resolution of modular operators and modular flows into subregioncharge sectors. The main advancement consists of extending the analysis of, valid for type I algebras, to hyperfinite von Neumann algebras, i.e. type II and type III algebras. To carry out the analysis on hyperfinite algebras, we have exploited the setup developed in [87, 88], where every type II and type III algebra (up to isomorphisms) can be obtained from an appropriate limit of infinite tensor products of finite-dimensional algebras. Since the desired decomposition is based on the sectors induced by a subregion charge on an eigenstate of the total charge, in section 3.1, we have introduced a charge operator in the setting of section 2.4. Although it is a naturally defined quantity, it exhibits subtleties in the large N limit, i.e. the limit when the hyperfinite algebras are accessed. Indeed, the subregion charge does not belong to the local algebra in this regime. Inspired by ideas in the context of symmetry-resolved entanglement, in section 3.2, we considered a charge rescaled by the parameter N. In the large N limit, this leads to a subregion charge operator, which belongs to the local algebra but is proportional to the identity. A non-trivial symmetry resolution of the local algebra and its modular operator is obtained by using the construction developed in section 4. The local algebra is built up by combining hyperfinite algebras obtained by the ITPFI construction. Indeed, in section 3.2, we obtained a family of hyperfinite factors labelled by the eigenvalues of the subregion charge in that algebra. Due to the rescaling of section 3.2, the subregion charge values allowed in different hyperfinite factors take continuous values, requiring the introduction of direct integrals to replace the direct sum. The construction of the algebras resolved into subregion – 31 – JHEP05(2026)259 charge sectors through direct integrals is carried out in section 4. This is the first main achievement of this manuscript. Finally, in section 5, we show that the modular operator and the modular flow in the algebras built in section 4 can be decomposed into the charge sectors, and the analysis of generalizes to the hyperfinite case. Interestingly, a symmetry resolution can also be performed for the modular correlation functions (2.9). We find that if the modular correlation functions in the full algebra satisfy the KMS condition, then the same holds also for all the modular correlation functions in the fixed-subregion charge algebras. The decompositions obtained in section 5 and the KMS properties for the symmetry-resolved modular correlation functions are the second and third central results of this work. As anticipated, the analyses reported in this work have connections with QFTs and holography.

  Modular theory and symmetry resolution in hyperfinite von Neumann algebras · 2026 · DOI

• A Resolution for non-Abelian groups B Convergence of total and local magnetization operators C Convergence properties of rescaled charge in type III1 algebras 1 3 3 5 8 11 14 14 15 18 20 20 21 24 26 26 28 31 33 34…

  Modular theory and symmetry resolution in hyperfinite von Neumann algebras · 2026 · DOI

• In this paper, we have demonstrated that the structure of the Hilbert space of both bosonic and supersymmetric matrix models can simplify in the singlet sector of a global symmetry. This is closely related to previous observations in the literature, which focused on simplification of the dynamics in these sectors, see, e.g., [11, 15, 24, 25]. Since the restriction to a singlet sector can be viewed as a gauging of the global symmetry, we have referred to the corresponding matrix model as double-gauged. Below we provide a brief summary of our main results and point out interesting future directions.

  To gauge or to double gauge? Matrix models, global symmetry, and black hole cohomologies · 2026 · DOI

• A Molien-Weyl formula for partition functions A.1 U(2) and SU(2) matrix models A.2 U(3) and SU(3) matrix models B SO(d) non-singlet sectors of bosonic matrix models B.1 d = 2 U(2) matrix model B.2 d = 3 U(2) matrix model C Enumerating graviton singlets C.1 Vanishing singlet gravitons beyond t24 D BMN index for singlets in the SU(4) theory 1 5 6 8 10 10 11 15 17 21 21 26 30 34 36 39 41 43 44 44 49 50 52…

  To gauge or to double gauge? Matrix models, global symmetry, and black hole cohomologies · 2026 · DOI

• This could be investigated using bootstrap methods by constructing a – 19 – JHEP05(2026)150 navigator function [55, 56], sampling it over a wide range of scaling dimension space, and examining which local minima correspond to fixed points with λ ≤ 0 or g2 ≤ 0.

  Classifying GNY-like models · 2026 · DOI

• In this paper, we developed a systematic procedure for finding quartic interaction vertices for a massless and massive bosonic field with arbitrary spins. While the general formalism is valid for any integer spin massless field, we restricted our study to the massless spin being either 2 (graviton) or 1 (a vector field), bearing in mind potential applications to black-hole binaries. The number of space-time dimensions was also left arbitrary. We presented a general Lagrangian including these quartic vertices and derived consistency equations that relate them to cubic vertices. We then gave a general (non-local) formal solution to those equations valid for any spin. In a low spin example we show that local solutions can also be constructed. The existence of such local solutions appears to depend on the precise form and combination of cubic vertices present in the Lagrangian. To find an explicit and local off-shell quartic extension to the higher-spin parts of the minimal coupling cubic vertices relevant to black-hole binaries remains an open problem. Alternatively, it would be of interest if one can derive general conditions on the cubic vertices that allow for the existence of local quartic vertices. The methods we laid out for computing the vertices by hand should be straightforward to improve upon numerically, allowing for the computation of higher spin results. Finally, we present on-shell versions of our results in section 5. This restriction considerably shortened the equations, and allowed us to compute a (still lengthy) result for a massive vector field interacting with a graviton. Since we need to impose symmetry between different auxillary Fock spaces in the formalism by hand, applying the spinor-helicity formalism [29]– [30] (when the spacetime equals four) could potentially lead to further simplification. We hope to address this problem in the future.

  BRST methods for constructing quartic actions for spinning black holes · 2026 · DOI

• A Bra vs. ket vectors B Derivation of the quartic vertex consistency equations B.1 Gauge invariance of the quartic Lagrangian B.2 Closure of the algebra of gauge transformations C Example: on-shell quartic solution for U for the 1-1-2 cubic vertex 1 3 6 9 9 11 12 12 13 14 15 17 17 18…

  BRST methods for constructing quartic actions for spinning black holes · 2026 · DOI

• The formula derived in this work has not been applied to specific theories referenced in [18, 19] to test dualities and trialities; concrete duality/triality tests on these particular theories remain outstanding validation targets.

  Elliptic genera of 2d $$ \mathcal{N} $$ = (0, 1) gauge theories · 2026 · DOI

• The application of Jeffrey-Kirwan residue methods used in N≥(0,2) quiver gauge theories for BPS state counting and crystal melting models has no established counterpart for N=(0,1) theories; developing analogous statistical mechanical models and crystal melting interpretations for N=(0,1) cases is identified as an open problem.

  Elliptic genera of 2d $$ \mathcal{N} $$ = (0, 1) gauge theories · 2026 · DOI

• The relationship between flavoured/equivariant elliptic genera computed here and the Stolz-Teichner conjecture classification of 2d N=(0,1) theories by topological modular forms remains unexplored; specifically, how equivariant variants shed light on flavoured compact SCFTs and equivariant TMFs is stated as an open direction.

  Elliptic genera of 2d $$ \mathcal{N} $$ = (0, 1) gauge theories · 2026 · DOI

• Virtual localization derivation of topological invariants for N=(0,1) elliptic genera lacks rigorous mathematical foundation; the paper identifies this as a natural question but does not establish whether the formula can be derived via virtual localizations with mathematically rigorous ground.

  Elliptic genera of 2d $$ \mathcal{N} $$ = (0, 1) gauge theories · 2026 · DOI

• The geometric approach to deriving elliptic genus expressions for N=(0,1) gauge theories is planned but not yet developed; this prevents handling theories where neutral fermionic zero modes exceed the rank of the gauge group and non-GLSM theories.

  Elliptic genera of 2d $$ \mathcal{N} $$ = (0, 1) gauge theories · 2026 · DOI

• The elliptic genus formula derived for N=(0,1) GLSMs has not been explicitly computed for O(n) and USp(n) gauge groups; the paper acknowledges these cases require introducing extra Fermi multiplets (for O(n)) or replacing symmetric with anti-symmetric Fermi multiplets (for USp(n)) but omits the explicit expressions and resulting constant factors in the elliptic genus.

  Elliptic genera of 2d $$ \mathcal{N} $$ = (0, 1) gauge theories · 2026 · DOI

• The analysis assumes a smooth α' → 0 limiting metric throughout, but the paper explicitly notes (Section 6.2) that counterexamples exist in string theory. The conditions under which this assumption is valid and the behavior of solutions when it fails for torsional K3 geometries with H-flux are not characterized.

  Stringy Corrections to Heterotic SU(3)-Geometry · 2026 · DOI

• The paper demonstrates consistency of heterotic SU(3)-geometry equations at α'² order but leaves open the question of deforming intermediate configurations solving the partial constraints (6.1) to solutions of the complete system including the stringy anomaly terms (6.2). Determining when such deformations exist for general bundle geometries E → X remains an outstanding problem in differential geometry.

  Stringy Corrections to Heterotic SU(3)-Geometry · 2026 · DOI

• The condition |α'R_{g(t)}| ≪ 1 is required for mathematical well-definedness of the anomaly flow (6.3)-(6.4), but the paper does not specify quantitative bounds or provide methods to verify this constraint is maintained during flow evolution on explicit complex geometries. The relationship between this perturbative regime and string theory consistency is not fully addressed.

  Stringy Corrections to Heterotic SU(3)-Geometry · 2026 · DOI

• The long-time existence and convergence properties of the anomaly flow for heterotic compactifications are only partially understood. While finite-time divergence has been observed on T⁴ fibrations over Riemann surfaces and infinite-time convergence on T² fibrations over K3 surfaces, the complete classification of initial data on general complex geometries (X, Ω, ω) that leads to convergence to full supersymmetry equations (5.3)-(5.4) is missing.

  Stringy Corrections to Heterotic SU(3)-Geometry · 2026 · DOI

• The interaction between the anomaly flow equations (6.3)-(6.4) for heterotic SU(3)-geometry and the gauge fixing condition φ = const + O(α'²) remains unclear. The paper establishes that these equations are consistent with supersymmetry at α'² order, but does not determine how the diffeomorphism flow along α' affects the dilaton gauge choice during the anomaly flow evolution.

  Stringy Corrections to Heterotic SU(3)-Geometry · 2026 · DOI

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