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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 | |
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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 | |
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Academic field: | Physics |
Specialties: |
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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:
Reports on this Submission
Report
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)
Report
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
Recommendation
Ask for minor revision