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Statistical Physics of the Polarised IKKT Matrix Model

by Sean A. Hartnoll, Jun Liu

Submission summary

Authors (as registered SciPost users): Sean Hartnoll
Submission information
Preprint Link: https://arxiv.org/abs/2504.06481v2  (pdf)
Date submitted: June 4, 2025, 12:13 a.m.
Submitted by: Hartnoll, Sean
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
Approaches: Theoretical, Computational

Abstract

The polarised IKKT matrix model is the worldpoint theory of $N$ D-instantons in a background three-form flux of magnitude $\Omega$, and promises to be a highly tractable model of holography. The matrix integral can be viewed as a statistical physics partition function with inverse temperature $\Omega^4$. At large $\Omega$ the model is dominated by a matrix configuration corresponding to a 'polarised' spherical D1-brane. We show that at a critical value of $\Omega^2 N$ the model undergoes a first order phase transition, corresponding to tunneling into a collection of well-separated D-instantons. These instantons are the remnant of a competing saddle in the high $\Omega$ phase corresponding to spherical $(p,q)$ fivebranes. We use a combination of numerical and analytical arguments to capture the different regimes of the model.

Author indications on fulfilling journal expectations

  • Provide a novel and synergetic link between different research areas.
  • Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
  • Detail a groundbreaking theoretical/experimental/computational discovery
  • Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Awaiting resubmission

Reports on this Submission

Report #2 by Anonymous (Referee 2) on 2025-7-15 (Invited Report)

Report

The authors studied the polarised IKKT matrix model in the Euclidean signature, which can be regarded as a statistical system. They numerically showed that the system exhibits a phase transition from the maximally irrep saddle to the trivial saddle around Omega^2 N ~ 1 with decreasing Omega and that the transition is very likely to be first order. The dual gravity picture for the maximally irrep saddle cannot be approximated by classical and non-stringy gravitational theory at any value of Omega, while that for the trivial saddle can within a certain window of Omega. They pointed out that the phase transition implies there is no regime of Omega where the classical supergravity description is dominant since the transition occurs at Omega lower than the window for the trivial saddle. They also discussed the regime where the description by classical probe five-branes is valid and showed the eigenvalue distribution of the moduli can be naturally interpreted as a uniform six-sphere by comparing their analysis of the (p,q) five-brane theory with the matrix-model result.

The analyses in the paper provide new insights into the gauge/gravity duality for the (polarised) IKKT model and will affect our general understanding of emergent spacetime. Therefore, I think the article is suitable for publication.

Recommendation

Publish (meets expectations and criteria for this Journal)

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Report #1 by Anonymous (Referee 1) on 2025-7-6 (Invited Report)

Report

This paper discusses the polarized IKKT model, which has been attracting attention as a new type of supersymmetric models that have gravity dual descriptions. It provides new results and discusses the impact of their results on the gravity dual descriptions in detail. I would recommend its publication after some minor revisions suggested below are implemented.

The IKKT matrix model was proposed in 1997 as a nonperturbative formulation of superstring theory, and as such, it has been investigated by many people both analytically and numerically. I suggest the authors to quote some important papers in this direction since it would help the readers understand in what context the IKKT model has been investigated so far.

The polarized IKKT model is a model that one can obtain by deforming the original IKKT matrix model in such a way that 16 supersymmetries are maintained. The recent excitement about this SUSY deformed model is that the dual geometry corresponding to each saddle point of the matrix model has been identified. Unlike the conventional duality between gauge theory and gravity theory pioneered by Maldacena, the matrices in the IKKT model do not depend on time. Hence there is neither space nor time on the gauge theory side. It is this aspect that makes the new duality more interesting. However, the duality discussed so far is purely in the Euclidean setup. It would be even more exciting if this duality is extended to the Lorentzian setup since one may then investigate the emergence of not only space but also time.
While this goes beyond the scope of the present paper, it would be better to be mentioned at least as an interesting future prospect.

Another important aspect of the polarized IKKT model is that one can obtain the partition function explicitly by using the localization technique making use of the supersymmetries that are preserved in the deformation. Thus the partition function reduces to the integration over the moduli parameters around each saddle, which can be evaluated by Monte Carlo sampling much more easily than simulating the original matrix model. Using such a method, the authors discover a phase transition at some critical Omega, where Omega is the deformation parameter, which is most likely of first order. This may be supported by the standard finite size scaling, which in the present case amounts to confirming the shift of the critical point with O(1/N^2). Since the authors have results for some values of N, they may try to see if the shift is consistent with this scaling.

At larger Omega, the maximal fuzzy sphere configuration dominates since it gives the minimal action. One should note that by rescaling the matrices as A->Omega A, Psi->Omega Psi, one can factor out the Omega dependence of the action as Omega^4, so that at large Omega, the classical solutions should dominate. This is not mentioned in this paper, however. At small Omega, the almost-trivial saddles dominate. As they emphasize in this paper,
this statement should be taken with care since it does not refer to the dominant matrix configurations in the original matrix integral at small Omega. The authors discuss this point carefully by considering the grand canonical ensemble, which is tractable at small Omega.

Finally the authors discuss the gravity dual picture in detail and find that the supergravity picture is only valid in some region of Omega for the subdominant saddle point. This is a bit unfortunate, but it is not something that diminishes the value of this paper.

I have some more suggestions for improvements. First I strongly recommend them to write the IKKT matrix model and its supersymmetric deformation in the Introduction to make the paper self-contained. This will make it easier to discuss some past work on the model.

Second I noticed some typos. I think the authors should use a spell-checker to avoid problems like "non-backreating" mentioned below.

p.3, below eq.(4)
The constant ....
This is not a full sentence.

p.4, caption of Fig.1
"The various regimes shown are discussed throughout the paper, this figure may be useful as a roadmap."
I think "and" is needed after comma.

p.17, section 7
non-backreating -> non-backreacting

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