Statistical Physics of the Polarised IKKT Matrix Model

TL;DR

The paper studies the polarised IKKT matrix model as a canonical partition function with deformation parameter $\Omega$ acting as an inverse temperature. Using supersymmetric localisation, the partition function is rewritten as a sum over SU(2) representations with distinct on-shell actions, allowing a detailed exploration of the competition between maximally irreducible fuzzy-sphere saddles and highly reducible, almost-trivial saddles. Numerical evidence reveals a first-order large-$N$ phase transition at $\left(\frac{3 \Omega^2 N}{2^8}\right)^2 \approx 0.58$, separating a high-$\Omega$ D1-brane dominated regime from a low-$\Omega$ D-instanton gas regime. The authors develop intermediate analyses via Casimir distributions and a grand-canonical treatment, and construct a fivebrane saddle in a cavity background to connect representation content to spacetime brane charges, outlining the regimes where probe branes or supergravity descriptions are valid. Overall, the work illuminates how holographic spacetime data emerge from partitions of the D-instanton charge and clarifies the limitations of semi-classical backgrounds in this matrix-model setting.

Abstract

The polarised IKKT matrix model is the worldpoint theory of $N$ D-instantons in a background three-form flux of magnitude $Ω$, 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 $Ω^4$. At large $Ω$ the model is dominated by a matrix configuration corresponding to a 'polarised' spherical D1-brane. We show that at a critical value of $Ω^2 N$ 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 $Ω$ phase corresponding to spherical $(p,q)$ fivebranes. We use a combination of numerical and analytical arguments to capture the different regimes of the model.

Figures (8)

• Figure 1: Phase structure of the polarised IKKT model. Top row: spacetime physics of the irreducible representation, which is dominant at large $\Omega^2 N$ (shaded red). Middle row: spacetime physics of the maximally reducible or 'trivial' representation, which is dominant at small $\Omega^2 N$ (shaded yellow). Bottom row: appropriate description of the matrix degrees of freedom. The various regimes shown are discussed throughout the paper, and so this figure may be useful as a roadmap. Matrix perturbation theory is discussed in Hartnoll:2024csr.
• Figure 2: Logarithm of the partition function against the deformation parameter $\Omega$. Data points are generated using (\ref{['eq:forplot']}) and $\mathcal{N} = 100$, with the values of $N$ shown in the legend. Dashed lines show the large $N$ expressions (\ref{['eq:Zirred']}) and (\ref{['eq:Ztriv']}) for the large and small $\Omega$ asymptotics, respectively.
• Figure 3: The $\mathfrak{su}(2)$ Casimir against the deformation parameter $\Omega$. Data points are as in Fig. \ref{['fig:Zplot']}.
• Figure 4: Probability distribution of the Casimir for small values of $\Omega^2 N$, calculated with $N=20$. Each point gives the probability of the Casimir, not the probability density. Solid lines are obtained from the full partition function, while the translucent lines are obtained using the small deformation approximation (\ref{['eq:intsmall']}). The two sets of curves are almost indistinguishable.
• Figure 5: Probability distribution of the Casimir for several very small values of $\Omega^2 N$, calculated at $N=50$ using the small deformation approximation (\ref{['eq:intsmall']}).