Numerical Matrix Evidence, Classical-Spacetime Recovery, and Limits

Numerical matrix calculations can test quantitative consequences of proposed gravity duals and measure collective geometry beyond perturbation theory. They establish only what survives a complete extrapolation from regulator, finite volume or temperature, finite $N$, and sign or contour choices. A good match to one black-hole coefficient or one expanding eigenvalue profile is evidence for that shared regime, not a proof of the proposal’s nonperturbative completeness.

Required background. Approximate bulk locality and spectral data supplies recovery criteria. BFSS matrix quantum mechanics supplies the D0 thermal system. The IKKT proposal supplies the contour-sensitive emergent-geometry problem.

Helpful background. Precision, convergence, and error budgets supplies extrapolation standards. Euclidean inverse problems supplies real-time reconstruction cautions.

A regulator-to-gravity protocol

For thermal BFSS, fix the dimensionless temperature $t=T/\lambda^{1/3}$ and discretize or spectrally truncate the Euclidean time circle. At each $(t,N)$, increase the temporal cutoff until Ward identities, energy, and scalar extent converge. Control excursions along flat directions, report the fermion Pfaffian phase or phase-quenching test, and then extrapolate $N\to\infty$. Only after these steps compare with the low-temperature supergravity window, whose curvature and dilaton bounds follow from the D-brane regime analysis Itzhaki et al. 1998.

For Lorentzian IKKT, the contour and infrared constraints define the ensemble. One must vary matrix size, regulator parameters, block size, and complex-weight method before extrapolating an extent tensor. A Euclidean ensemble cannot be interpreted as Lorentzian merely by relabeling one eigenvalue as time.

Correlated uncertainties should be propagated through the entire sequence. Separate statistical error from cutoff ansatz, finite-$N$ ansatz, thermal window, phase treatment, and autocorrelation. Agreement should be tested against observables not used to tune the extrapolation.

First application: black-zero-brane internal energy

Use

$$e(t,N)=\frac{E}{N^2\lambda^{1/3}}, \qquad t=\frac{T}{\lambda^{1/3}}.$$

Classical type-IIA supergravity predicts at large $N$ and low $t$ within its validity window

$$e(t,\infty)=7.41\,t^{14/5}+c_{\alpha'}t^{23/5}+\cdots.$$

The leading coefficient and exponent are fixed by the near-extremal D0 solution; the next exponent tests the first stringy correction. A defensible analysis first obtains continuum values at each $N$ and $t$, then fits several temperature windows with both the leading-only and correction-inclusive forms, and finally checks finite-$N$ terms separately. Simulations have reproduced the leading behavior and resolved the expected higher-derivative scaling in controlled windows Berkowitz et al. 2016, Hanada et al. 2016.

This test links a nonperturbative gauge calculation to a classical black-object prediction. It does not validate temperatures where the IIA dilaton or curvature is large, and it does not test IKKT spacetime emergence.

Adversarial control: fit-window and form instability

Fit the same data over a narrow range to $At^p$, $At^{14/5}+Bt^q$, and a flexible polynomial. If $A=7.41$ appears only when $p$ is fixed, disappears when the coarsest lattice is removed, or correlates strongly with an unconstrained $q$, the claimed confirmation is not stable. Likewise, an apparent four-dimensional IKKT extent that changes with block size or contour regulator is not a continuum geometric observable.

The evidence ceiling is quantitative validation of specified observables in overlapping $N$, coupling, temperature, curvature, and regulator regimes. Numerical convergence can strongly corroborate a duality and discriminate correction terms; it cannot establish unmeasured sectors, unique bulk reconstruction, or a nonperturbative completion beyond the simulated limiting prescription.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Berkowitz, E., Rinaldi, E., Hanada, M., Ishiki, G., Shimasaki, S., and Vranas, P. (2016), “Precision Lattice Test of the Gauge/Gravity Duality at Large-$N$,” Physical Review D 94, 094501. arXiv:1606.04948.
• Hanada, M., Hyakutake, Y., Ishiki, G., and Nishimura, J. (2016), “Numerical Tests of the Gauge/Gravity Duality Conjecture for D0-Branes at Finite Temperature and Finite $N$,” Physical Review D 94, 086010. arXiv:1603.00538.
• Itzhaki, N., Maldacena, J. M., Sonnenschein, J., and Yankielowicz, S. (1998), “Supergravity and the Large $N$ Limit of Theories with Sixteen Supercharges,” Physical Review D 58, 046004. arXiv:hep-th/9802042.