Pith. sign in

REVIEW 2 major objections 5 minor 5 cited by

Einstein gravity from a matrix integral -- Part II

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Supersymmetric localization gives exact formulas for the polarized IKKT matrix model, whose zero-mass limit diverges rather than recovering IKKT, and whose large-N eigenvalue densities reproduce the gravity dual's electrostatic equations.

desk verdict Strong localization result for polarized IKKT with honest checks, but the advertised gravity match rests on a large-J approximation whose validity for the relevant saddles is asserted rather than shown. read the letter →

arxiv 2411.18678 v2 pith:S77TEW6Q submitted 2024-11-27 hep-th

Shota Komatsu , Adrien Martina , Joao Penedones , Antoine Vuignier , Xiang Zhao This is my paper · ORCID
classification hep-th
keywords IKKTmatrixmodelsupersymmetriclocalizationintegralholographicdualityemergentgravityelectrostaticproblemSU(2)representationspartitionfunction
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper gives exact, finite-dimensional formulas for the partition function and for correlation functions of supersymmetry-invariant operators in the polarized IKKT matrix model, a zero-dimensional matrix integral with sixteen supercharges that is the simplest known setting in which Einstein gravity is claimed to emerge holographically. The partition function is a sum over all SU(2) representations of the matrix rank, each term being an ordinary integral over the eigenvalues of a commuting set of matrices, with a one-loop determinant built from fuzzy-sphere data. The paper shows that the zero-mass-deformation limit of this exact result diverges, so polarized IKKT does not reduce to the original IKKT model in that limit; a contour-integral prescription is conjectured, and verified for N=2 and N=3, to recover the IKKT partition function. In the large-N limit the localization formulas imply that the eigenvalue densities obey exactly the integral equations of an electrostatic problem with conducting balls and image charges, which is precisely the electrostatic problem defining the dual supergravity geometry. If this is right, the paper closes the loop between a matrix integral and a gravitational geometry on both sides of the correspondence.

What carries the argument

The machinery that carries the argument is supersymmetric localization with an off-shell supersymmetry algebra. Auxiliary fields close the algebra, a supercharge preserving $\phi=X^3-iX^{10}$ is chosen, and the action is deformed by a Q-exact term built from a combined BRST+supersymmetry operator Q, making the saddle-point approximation exact. The saddles are fuzzy spheres $X^i=\frac{3\Omega}{8}L^i$, so each saddle's moduli space is parameterized by a reducible SU(2) representation and by eigenvalues $m_{si}$ of the commuting matrix M; the one-loop determinant is obtained by diagonalizing $R_0=-i\Omega\delta_{U(1)}-i[M+\frac{3\Omega}{8}L_3,\cdot]$ on fuzzy-sphere harmonics, and the resulting factors cancel between bosons and fermions except for the product $f_{st}(x)$. For the gravity matching, the central object is the effective two-body potential $k_{st}(x)$ of the eigenvalue ensemble; after replacing it by its large-J asymptotic form and passing to continuum densities, the saddle equations become the electrostatic integral equations for the linear charge densities of conducting balls, with the parameter identifications given above.

What would settle it

Compute the polarized IKKT partition function directly for N=2 at small $\Omega$ (numerical quadrature or Monte Carlo) and compare with the localization formula (1.6): because the formula is claimed exact, any deviation in the $1/\Omega^2$ divergence would falsify it. Separately, in a parameter regime where the paper's condition $n_s\gg (x^{(s)})^2\log x^{(s)}$ fails, solve the full saddle equations with the exact kernel $k_{st}(x)$ and test whether the densities still satisfy the electrostatic equations (5.10)-(5.12).

Watch

Extended reading notes

Core claim

The paper's central claim is that the U(N) polarized IKKT partition function is exactly $$Z=\sum_R C_R e^{\frac{9\$\Omega$^4}{$2^{{15}}$}\sum_s n_s($N_s^{3}$-N_s)}\int\prod_{s,i}dm_{si}\,Z_{\rm 1-loop}\,e^{-\frac{3\$\Omega$^4}{$2^{7}$}\sum_s N_s\sum_i m_{si}^2},$$ with the sum over inequivalent N-dimensional representations R of SU(2) (irrep dimensions $N_s$, multiplicities $n_s$), and $Z_{\rm 1-loop}=\prod_{(si,tj)}f_{st}(m_{si}-m_{tj})$ a product of rational functions of differences of Cartan eigenvalues labeled by fuzzy-sphere angular-momentum levels J. Protected correlators of $\phi=X^3-iX^{10}$ are obtained by inserting the same function of $\frac{3\Omega}{8}L_3-i\Omega M$ into the integral. From this exact result the paper derives that $\lim_{\Omega\to0}Z(\Omega)=\infty\ne Z_{\rm IKKT}$, traces the divergence to fermion mass terms, matches the $\Omega\to\infty$ limit to earlier results, and in the continuum large-N limit converts the saddle-point equation for eigenvalue densities into the electrostatic equation for the charge densities of conducting balls, with parameters identified as $x=r/(2\pi\mu\alpha')$ and $\rho^{(s)}(x)=64 f_s(r)/(\pi^3\mu^5\alpha'^2 g_s)$.

Load-bearing premise

The claimed recovery of Einstein gravity rests on assuming that, once the number of eigenvalues is large, replacing the exact pairwise interaction between eigenvalues by its simple large-distance asymptotic form does not change the saddle-point densities; the paper's own validity condition for this replacement is not guaranteed to hold in every regime.

Editorial extensions

If this is right

  • For every N and Ω, the exact formula reduces the model to ordinary integrals, so protected correlators and the partition function become computable quantities rather than formal path integrals.
  • The limit Ω→0 is discontinuous: polarized IKKT diverges, and the original IKKT partition function is recovered only through a pole-picking contour prescription that works for N=2,3 and is conjectured for all N.
  • In the large-N continuum limit, the matrix model and the supergravity dual are governed by the same electrostatic equations, giving a quantitative dictionary between the fuzzy-sphere data $(n_s,N_s)$ and the gravity data $(Q_s,z_s)$.
  • The strong-coupling limit exhibits factorization into decoupled bound states, one per irreducible SU(2) block, similar to particles in a harmonic trap; this structure follows directly from the localization result.
  • The on-shell action of the eigenvalue densities reproduces the expected scaling of the dual gravitational on-shell action, including the electrostatic octopole contribution.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the localization formula is exact, subleading corrections in $1/N$ to the electrostatic equations should correspond to stringy corrections on the gravity side; computing those corrections would give a precision test of this holographic duality that the paper does not carry out.
  • The conjectured contour prescription implies that the divisor sum $\sum_{m|N}1/m^2$ appearing in the IKKT partition function should emerge from residues of the one-loop determinant; this could be checked for N=4 or N=5 by explicit residue evaluation.
  • The Ω→0 discontinuity is a general caution: mass deformations of matrix models with fermions are not automatically continuous regularizations, so other 'polarized' limits may silently change the theory.
  • Because the electrostatic problem is analytically solvable in its extremes, the localization result may allow analytic computation of the density beyond the planar limit, connecting to scaling-similarity analyses of brane geometries.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper computes the partition function and protected correlators of the polarized IKKT matrix model by supersymmetric localization, reducing them to a sum over SU(2) representations with one-loop determinants and Gaussian-like moduli integrals, Eqs. (1.6)-(1.8) and (3.63). The authors find that the Ω→0 limit diverges, Z(Ω)→∞ ≠ Z_IKKT, due to fermion mass terms, and conjecture a contour prescription in the moduli integrals that reproduces the known IKKT partition function for N=2,3. In the large-N limit they argue that the eigenvalue densities of the localized partition function obey integral equations identical to the electrostatic equations defining the gravity dual constructed in the companion paper [5], with an explicit dictionary (5.44). The on-shell action comparison with gravity is also discussed, with a factor-of-two mismatch left for future work.

Significance. If the central claims hold, this is a significant step toward a concrete microscopic derivation of Einstein gravity from a zero-dimensional matrix integral: the localization result is exact and the gravity-side equations are derived from the matrix model rather than fitted. The paper contains several strong cross-checks that support the localization computation: explicit N=2 results, agreement with the Ω→∞ limit of [12], agreement with direct off-diagonal integration in the Ω→0 limit, and reproduction of the IKKT partition function for N=2,3 via the contour conjecture. The electrostatic matching in Section 5 is a nontrivial consistency check involving kernels and densities, not a one-parameter fit. The main weakness is that the passage from the exact saddle equations to the electrostatic equations relies on large-J and continuum approximations whose validity is not demonstrated for the gravity-side configurations, and the on-shell action check is explicitly incomplete. These issues are local to the gravity-recovery section but are load-bearing for the paper's title claim.

major comments (2)
  1. [Section 5.2-5.3, Eqs. (5.21), (5.24)-(5.25), (5.36), (5.41)] The claim that the large-N eigenvalue densities satisfy the electrostatic equations of the gravity dual is not yet established, because the validity condition (5.36) is not verified for the configurations matched in Section 5.3. The derivation of (5.25) from the exact saddle equation (5.24) uses both the large-J approximation (5.21) and the replacement of discrete densities by smooth ones; the paper's own estimate requires n_s >> (x^(s))^2 log(x^(s)). The matching identification (5.41) relates n_s and N_s to the charges and heights of the conducting balls, but no argument is given that the supergravity configurations of [5] obey this hierarchy. In the fixed-ξ scaling discussed in Section 5.2, the condition amounts to N_s << (N/log N)^{1/3}, which is not implied by the macroscopicity of the geometry. Unless this condition, or a separate control of the omitted O((3J±8ix)^{-4}) terms in (5.21), is demonstrated for the saddles in question, the equivalence of (5.25) and (5.10) is conditional and the central claim of Section 5.3 is not fully supported.
  2. [Section 5.4, after Eq. (5.49)] The comparison of the on-shell action with the gravity result is left incomplete: the term proportional to ∑ Q_s V_s matches only up to a factor of 2, and the boundary terms proportional to ∑ q_{3,s} are not matched. The authors state that the gravitational computation would need to be finished. This is an acknowledged gap in a quantitative check of the gravity recovery; it should either be resolved or the claims of Section 5 should be softened accordingly.
minor comments (5)
  1. [Section 4.2, around Eq. (4.12)] The statement lim_{Ω→0} Z(Ω)=∞ is stated in the abstract and introduction, but the precise power of the divergence, 1/Ω^{2(N-1)} for the SU(N) partition function, appears only later; stating it upfront would help the reader.
  2. [Section 4.4, Eqs. (4.42)-(4.56)] The contour prescription is verified only for N=2 and N=3. Since this is a conjecture, its conjectural status could be made more prominent in the abstract and introduction, where the recovery of IKKT is announced.
  3. [Figure 6 and Eq. (5.21)] The plot shows the exact and approximate two-body potentials, but the text does not quantify the error of the approximation or state its region of validity in terms of J and x; adding this would make the error estimate in Section 5.2 more concrete.
  4. [Section 5.2, Eq. (5.34)] The asymptotic behaviors x^(s)/N_s ~ ξ^{1/5} and ξ^{1/4} are quoted without derivation in the main text; the derivation in Appendix D is helpful, but a short sentence pointing to the relevant equations in Appendix D would improve readability.
  5. [Section 3.6, Eq. (3.67)] The notation ⟨Trϕ^{2n}⟩ in the U(N) relation is used both for the U(N) and SU(N) correlators; distinguishing the trace parts explicitly in the displayed formula would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the localization result is derived self-contained, the Ω→0 divergence is cross-checked by an independent off-diagonal integration, and the gravity-side matching is a structural consistency check rather than a fitted renaming.

full rationale

The paper's central derivation is self-contained. The localization computation in Section 3 derives the exact partition function (1.6)-(1.8) from a concrete supercharge choice, off-shell closure, saddle-point analysis, and a direct evaluation of one-loop determinants; no step reduces the output to an input by definition. The Ω→0 divergence (1.9) is obtained from the localization formula and independently reproduced in Section 4.2 by integrating out off-diagonal modes, with the same leading 1/Ω^{2(N-1)} behavior and the same correlator limits, so it is not a fitted prediction. The recovery of the IKKT partition function for N=2,3 in Section 4.4 is checked against the known closed-form divisor sum (2.4), providing an external benchmark. The gravity-side matching in Section 5 compares the large-N saddle equation (5.24)-(5.25) derived from the matrix model with the electrostatic integral equation (5.10) derived from the supergravity construction of Part I; the two kernels match only after an explicit parameter dictionary (5.37)-(5.44) and yield the nontrivial relation μ = sqrt(g_YM) Ω. This is a consistency check between two independently formulated sides, not a one-parameter fit or a renaming of a known pattern. Self-citations to [5] and [36] are present and are load-bearing for the gravity-side input and technical methods, but they are not circular: the supergravity construction is an externally derived input, and the matrix-model side is computed independently here. The main caveat, the validity condition (5.36) for the large-J and continuum approximations, is an acknowledged correctness risk, not a circular reduction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No parameters are fitted to data. Ω is a fixed deformation parameter; the constants a, b, μ_s in the gravity dictionary are determined by matching equations, not chosen to force agreement. No new particles, forces, or fields are introduced; the conducting balls and image charges are inherited from the companion paper or are standard electrostatic tools.

assumptions (6)
  • standard math Off-shell closure of the 16 supercharges is achieved with seven auxiliary fields K_a, with constraints on the auxiliary spinors ν_a (eqs. 3.4-3.7).
    The localization argument and the form of the 1-loop determinant depend on this algebraic closure; the paper solves the constraints in appendix C.2.
  • domain assumption The saddle points of the localized action are exactly the fuzzy sphere configurations X_i = (3Ω/8)L_i, X_10 = M with [M,L_i]=0, plus vanishing other fields (eq. 3.22-3.26).
    The sum over representations R and the subsequent gravity map rely on this classification; missing saddles would change Z.
  • domain assumption The background gauge F[X] = -i[L_j, X_j] with the extended ghost action (3.33) fixes the gauge without affecting the value of the partition function or protected correlators.
    The normalization constant C_R is computed within this gauge; any missed zero mode or Jacobian would alter the final formula.
  • domain assumption In the large-N limit, discrete eigenvalue sums can be replaced by continuous densities and the two-body potential k_st(x) can be replaced by its large-J asymptotic form (5.21), with corrections suppressed under (5.36).
    This approximation converts the exact matrix model equations into the electrostatic equations of the gravity dual; the paper estimates the error but does not prove uniform validity.
  • ad hoc to paper The dual geometry is the electrostatic configuration of conducting balls in a background potential V_b from the companion paper [5].
    The gravity side is taken as input from Part I by the same authors; this paper does not derive it from first principles.
  • ad hoc to paper The contour prescription of closing moduli integrals on the upper half plane yields the IKKT partition function as Ω→0 (Section 4.4).
    Stated as a conjecture and demonstrated for N=2 and N=3; it is load-bearing for connecting polarized IKKT to IKKT.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Einstein gravity from a matrix integral -- Part II." pith.science (2026). https://pith.science/paper/S77TEW6Q

@misc{pith2026241118678,
  author       = {Pith},
  title        = {Pith review of: Einstein gravity from a matrix integral -- Part II},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S77TEW6Q}},
  note         = {Machine review of arXiv:2411.18678}
}
read the original abstract

Using supersymmetric localization, we compute the partition function and some protected correlators of the polarized IKKT matrix model. Surprisingly, we find that the original IKKT model is different from polarized IKKT in the limit of vanishing mass deformation. We study different regimes of the localization results and recover the electrostatic problem which defines the gravity dual.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. From Matrix Models to Gaussian Molecules and the Einstein-Hilbert Action

    hep-th 2026-04 unverdicted novelty 7.5 of 10

    The free energy of a D-dimensional matrix model with Gaussian heat-kernel propagator equals the Einstein-Hilbert action with cosmological constant, constants fixed by graph invariants of ribbon graphs.

  2. Derivation of the NS5-brane limit of the plane wave matrix model

    hep-th 2026-07 accept novelty 7.0 of 10

    A double-scaling limit of the BMN/plane-wave matrix model is derived analytically and produces an eigenvalue integral for the 1/4-BPS sector of IIA little string theory on R×S⁵.

  3. On the strong coupling limit of Yang-Mills matrix models

    hep-th 2026-07 conditional novelty 7.0 of 10

    In mass-deformed Yang–Mills matrix models, strong-coupling commutativity sets in at or above the critical fermion count N_c = 2(D−2), with the supersymmetric models sitting at the critical boundary where huge-operator...

  4. Bootstrapping Yang-Mills matrix integrals

    hep-th 2025-10 conditional novelty 7.0 of 10

    Bootstrap bounds on ⟨tr XX⟩ and ⟨tr XXXX⟩ in D-matrix Yang-Mills integrals shrink to islands matching Monte Carlo, with large-D trajectories guiding a positivity-free ansatz.

  5. Twisting BFSS & IKKT

    hep-th 2026-02 conditional novelty 6.0 of 10

    The minimal supersymmetric twists of the IKKT and BFSS matrix models are computed in BV-BRST cohomology and matched, in the planar limit, to BCOV-type twisted IIB and IIA supergravity; the non-minimal twists are local...

Reference graph

Works this paper leans on

45 extracted references · 16 canonical work pages · cited by 5 Pith papers

  1. [5]

    Einstein gravity from a matrix integral – Part I,

    S. Komatsu, A. Martina, J. Penedones, A. Vuignier, and X. Zhao, “Einstein gravity from a matrix integral – Part I,”arXiv:2410.18173 [hep-th]

    arXiv

  2. [12]

    The Polarised IKKT Matrix Model,

    S. A. Hartnoll and J. Liu, “The Polarised IKKT Matrix Model,”arXiv:2409.18706 [hep-th]

    arXiv

  3. [1]

    Dimensional reduction in quantum gravity,

    G. ’t Hooft, “Dimensional reduction in quantum gravity,”Conf. Proc. C 930308 (1993) 284–296, arXiv:gr-qc/9310026

    arXiv 1993

  4. [2]

    The World as a hologram,

    L. Susskind, “The World as a hologram,”J. Math. Phys. 36 (1995) 6377–6396, arXiv:hep-th/9409089

    arXiv 1995

  5. [3]

    The Large N limit of superconformal field theories and supergravity,

    J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys. 2 (1998) 231–252, arXiv:hep-th/9711200

    arXiv 1998

  6. [4]

    A Planar Diagram Theory for Strong Interactions,

    G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions,”Nucl. Phys. B 72 (1974) – 60 – 461

    work page 1974

  7. [6]

    Strings in Less Than One-Dimension,

    M. R. Douglas and S. H. Shenker, “Strings in Less Than One-Dimension,”Nucl. Phys. B 335 (1990) 635

    work page 1990

  8. [7]

    Exactly Solvable Field Theories of Closed Strings,

    E. Brezin and V. A. Kazakov, “Exactly Solvable Field Theories of Closed Strings,”Phys. Lett. B 236 (1990) 144–150

    work page 1990

Show all 45 references
  1. [8]

    Deriving the Simplest Gauge-String Duality – I: Open-Closed-Open Triality,

    R. Gopakumar and E. A. Mazenc, “Deriving the Simplest Gauge-String Duality – I: Open-Closed-Open Triality,” arXiv:2212.05999 [hep-th]

    arXiv

  2. [9]

    The Virasoro minimal string,

    S. Collier, L. Eberhardt, B. Mühlmann, and V. A. Rodriguez, “The Virasoro minimal string,” SciPost Phys. 16 no. 2, (2024) 057,arXiv:2309.10846 [hep-th]

    2024 arXiv

  3. [10]

    A large-n reduced model as superstring,

    N. Ishibashi, H. Kawai, Y. Kitazawa, and A. Tsuchiya, “A large-n reduced model as superstring,” Nuclear Physics B 498 no. 1–2, (Aug., 1997) 467–491. http://dx.doi.org/10.1016/S0550-3213(97)00290-3

    1997 doi

  4. [11]

    Matrix strings in pp-wave backgrounds from deformed super yang-mills theory,

    G. Bonelli, “Matrix strings in pp-wave backgrounds from deformed super yang-mills theory,” Journal of High Energy Physics 2002 no. 08, (Aug., 2002) 022–022. http://dx.doi.org/10.1088/1126-6708/2002/08/022

    2002 doi

  5. [13]

    A Possible constructive approach to (SUPER phi**3) in four-dimensions. 1. Euclidean formulation of the model,

    H. Nicolai, “A Possible constructive approach to (SUPER phi**3) in four-dimensions. 1. Euclidean formulation of the model,”Nucl. Phys. B 140 (1978) 294–300

    1978

  6. [14]

    Instantons of M(atrix) theory in PP wave background,

    J.-T. Yee and P. Yi, “Instantons of M(atrix) theory in PP wave background,”JHEP 02 (2003) 040, arXiv:hep-th/0301120

    2003 arXiv

  7. [15]

    Convergent Yang-Mills matrix theories,

    P. Austing and J. F. Wheater, “Convergent Yang-Mills matrix theories,”JHEP 04 (2001) 019, arXiv:hep-th/0103159

    2001 arXiv

  8. [16]

    Emergent bubbling geometries in the plane wave matrix model,

    Y. Asano, G. Ishiki, T. Okada, and S. Shimasaki, “Emergent bubbling geometries in the plane wave matrix model,”JHEP 05 (2014) 075, arXiv:1401.5079 [hep-th]

    2014 arXiv

  9. [17]

    Emergent bubbling geometries in gauge theories with SU(2|4) symmetry,

    Y. Asano, G. Ishiki, and S. Shimasaki, “Emergent bubbling geometries in gauge theories with SU(2|4) symmetry,” JHEP 09 (2014) 137, arXiv:1406.1337 [hep-th]

    2014 arXiv

  10. [18]

    Spectral curves, emergent geometry, and bubbling solutions for Wilson loops,

    T. Okuda and D. Trancanelli, “Spectral curves, emergent geometry, and bubbling solutions for Wilson loops,” JHEP 09 (2008) 050, arXiv:0806.4191 [hep-th]

    2008 arXiv

  11. [19]

    SolvingN = 4 SYM BCFT matrix models at large N,

    D. He and C. F. Uhlemann, “SolvingN = 4 SYM BCFT matrix models at large N,”JHEP 12 (2024) 164, arXiv:2409.13016 [hep-th]

    2024

  12. [20]

    IIB matrix model,

    H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, A. Tsuchiya, and T. Tada, “IIB matrix model,”Prog. Theor. Phys. Suppl. 134 (1999) 47–83, arXiv:hep-th/9908038

    1999 arXiv

  13. [21]

    D Particle bound states and the D instanton measure,

    M. B. Green and M. Gutperle, “D Particle bound states and the D instanton measure,” JHEP 01 (1998) 005, arXiv:hep-th/9711107

    1998 arXiv

  14. [22]

    D particle bound states and generalized instantons,

    G. W. Moore, N. Nekrasov, and S. Shatashvili, “D particle bound states and generalized instantons,” Commun. Math. Phys. 209 (2000) 77–95, arXiv:hep-th/9803265

    2000 arXiv

  15. [23]

    D instanton partition functions,

    M. B. Green and M. Gutperle, “D instanton partition functions,”Phys. Rev. D 58 (1998) 046007, arXiv:hep-th/9804123. – 61 –

    1998 arXiv

  16. [24]

    The Convergence of Yang-Mills integrals,

    P. Austing and J. F. Wheater, “The Convergence of Yang-Mills integrals,”JHEP 02 (2001) 028, arXiv:hep-th/0101071

    2001 arXiv

  17. [25]

    Monte Carlo approach to M theory,

    W. Krauth, H. Nicolai, and M. Staudacher, “Monte Carlo approach to M theory,”Phys. Lett. B 431 (1998) 31–41, arXiv:hep-th/9803117

    1998 arXiv

  18. [26]

    Hardy, An introduction to the theory of numbers

    G. Hardy, An introduction to the theory of numbers . Oxford Science Publication, 1979

    1979

  19. [27]

    Strings in flat space and pp waves from N=4 superYang-Mills,

    D. E. Berenstein, J. M. Maldacena, and H. S. Nastase, “Strings in flat space and pp waves from N=4 superYang-Mills,”JHEP 04 (2002) 013, arXiv:hep-th/0202021

    2002 arXiv

  20. [28]

    Exact results for perturbative partition functions of theories with SU(2|4) symmetry,

    Y. Asano, G. Ishiki, T. Okada, and S. Shimasaki, “Exact results for perturbative partition functions of theories with SU(2|4) symmetry,”JHEP 02 (2013) 148, arXiv:1211.0364 [hep-th]

    2013 arXiv

  21. [30]

    Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,

    V. Pestun, “Localization of gauge theory on a four-sphere and supersymmetric Wilson loops,” Commun. Math. Phys. 313 (2012) 71–129, arXiv:0712.2824 [hep-th]

    2012 arXiv

  22. [31]

    A Ten-dimensional superYang-Mills action with off-shell supersymmetry,

    N. Berkovits, “A Ten-dimensional superYang-Mills action with off-shell supersymmetry,” Phys. Lett. B 318 (1993) 104–106, arXiv:hep-th/9308128

    1993 arXiv

  23. [32]

    An Introduction to Localisation and Supersymmetry in Curved Space,

    S. Cremonesi, “An Introduction to Localisation and Supersymmetry in Curved Space,”PoS Modave2013 (2013) 002

    2013

  24. [33]

    Scattering into the fifth dimension of N=4 super Yang-Mills,

    L. F. Alday, J. M. Henn, J. Plefka, and T. Schuster, “Scattering into the fifth dimension of N=4 super Yang-Mills,”JHEP 01 (2010) 077, arXiv:0908.0684 [hep-th]

    2010 arXiv

  25. [34]

    Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,

    A. Kapustin, B. Willett, and I. Yaakov, “Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,”JHEP 03 (2010) 089, arXiv:0909.4559 [hep-th]

    2010 arXiv

  26. [35]

    Space-time structures from iib matrix model,

    H. Aoki, S. Iso, H. Kawai, Y. Kitazawa, and T. Tada, “Space-time structures from iib matrix model,” Progress of Theoretical Physics 99 no. 5, (May, 1998) 713–745. http://dx.doi.org/10.1143/PTP.99.713

    1998 doi

  27. [36]

    Gravity from quantum mechanics of finite matrices,

    S. Komatsu, A. Martina, J. Penedones, N. Suchel, A. Vuignier, and X. Zhao, “Gravity from quantum mechanics of finite matrices,”arXiv:2401.16471 [hep-th]

    arXiv

  28. [37]

    On the Ground State Wave Function of Matrix Theory,

    Y.-H. Lin and X. Yin, “On the Ground State Wave Function of Matrix Theory,”JHEP 11 (2015) 027, arXiv:1402.0055 [hep-th]

    2015 arXiv

  29. [38]

    Yang-mills matrix theory,

    P. Austing, “Yang-mills matrix theory,” Master’s thesis, University of Oxford, 2001. https://arxiv.org/abs/hep-th/0108128

    2001 arXiv

  30. [39]

    Fivebranes from gauge theory,

    H. Lin and J. M. Maldacena, “Fivebranes from gauge theory,”Phys. Rev. D 74 (2006) 084014, arXiv:hep-th/0509235

    2006 arXiv

  31. [40]

    Notes on Mayer Expansions and Matrix Models,

    J.-E. Bourgine, “Notes on Mayer Expansions and Matrix Models,”Nucl. Phys. B 880 (2014) 476–503, arXiv:1310.3566 [hep-th]

    2014 arXiv

  32. [41]

    Supersymmetric Yang-Mills, Spherical Branes, and Precision Holography,

    N. Bobev, P. Bomans, F. F. Gautason, J. A. Minahan, and A. Nedelin, “Supersymmetric Yang-Mills, Spherical Branes, and Precision Holography,”JHEP 03 (2020) 047, arXiv:1910.08555 [hep-th]

    2020 arXiv

  33. [42]

    Scaling similarities and quasinormal modes of D0 black hole solutions,

    A. Biggs and J. Maldacena, “Scaling similarities and quasinormal modes of D0 black hole solutions,” JHEP 11 (2023) 155, arXiv:2303.09974 [hep-th] . – 62 –

    2023 arXiv

  34. [43]

    Spherical Branes and the BMN Matrix Quantum Mechanics,

    N. Bobev, P. Bomans, and F. F. Gautason, “Spherical Branes and the BMN Matrix Quantum Mechanics,” arXiv:2410.21376 [hep-th]

    arXiv

  35. [44]

    No ensemble averaging below the black hole threshold,

    J.-M. Schlenker and E. Witten, “No ensemble averaging below the black hole threshold,” JHEP 07 (2022) 143, arXiv:2202.01372 [hep-th]

    2022 arXiv

  36. [45]

    D. Z. Freedman and A. Van Proeyen,Supergravity. Cambridge Univ. Press, Cambridge, UK, 5, 2012

    2012

  37. [46]

    Embedding of theories with SU(2|4) symmetry into the plane wave matrix model,

    G. Ishiki, S. Shimasaki, Y. Takayama, and A. Tsuchiya, “Embedding of theories with SU(2|4) symmetry into the plane wave matrix model,”JHEP 11 (2006) 089, arXiv:hep-th/0610038. – 63 –

    2006 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.