Numerical tests of the gauge/gravity duality conjecture for D0-branes at finite temperature and finite $N$

According to the gauge/gravity duality conjecture, the thermodynamics of gauge theory describing D-branes corresponds to that of black branes in superstring theory. We test this conjecture directly in the case of D0-branes by applying Monte Carlo methods to the corresponding gauge theory, which takes the form of the Banks-Fischler-Shenker-Susskind (BFSS) matrix quantum mechanics. In particular, we take the continuum limit by extrapolating the UV cutoff to infinity. First we perform simulations at large $N$ so that string loop corrections can be neglected on the gravity side. Our results for the internal energy exhibit the temperature dependence consistent with the prediction including the ${\alpha }^{\prime }$ corrections. Next we perform simulations at small $N$ but at lower temperature so that the ${\alpha }^{\prime }$ corrections can be neglected on the gravity side. Our results are consistent with the prediction including the leading string loop correction, which suggests that the conjecture holds even at finite $N$.

Figure 1

(Left) The internal energy $E/N^2$ for $N=16$ is plotted against $1/\mathrm{\Lambda }$ for $T=0.5$, 0.6, 0.7, 0.8, 0.9, 1.0 from the bottom to the top. The solid and dashed lines represent fits to $E/N^2=\text{const}$ and $E/N^2=a+b/\mathrm{\Lambda }$, respectively. (Right) The internal energy obtained by extrapolation to $\mathrm{\Lambda }=\infty$. The solid line represents the prediction of type IIA supergravity $E/N^2=7.41T^{2.8}$. The dashed line represents a fit $E/N^2=7.41T^{2.8}+a{T}^{4.6}$ with $a=-5.58(1)$ obtained in Ref. [18]. The dash-dotted line represents a fit $E/N^2=7.41T^{2.8}+a{T}^p$ with $a=-0.90(26)\times 10$ and $p=4.74(35)$ obtained in Ref. [24].

Figure 2

The quantity $y=(7.41T^{2.8}-E/N^2)/7.41T^{2.8}$ representing the deviation of the internal energy from the leading prediction from supergravity is plotted against $x=T^{1.8}$. The solid line and the dashed line represent fits to $y=a_{1}x+a_2{x}^{5/3}$ using the data within the range $0.5\le T\le 0.7$ and $0.5\le T\le 0.9$, respectively, obtained by the linear fit.

Figure 3

(Top left) The distribution of $R^2$ for $N=4$, $T=0.10$, and $\mathrm{\Lambda }=10$ using $c=100$ and $R_{\mathrm{cut}}^2=4.2$ in (4.2). (Top right) The function $E(x)/N^2$ for the same set of parameters. (Bottom) The extent of the bound states estimated by $x_{\min }$ as described in the text is plotted as a function of $T$ for $N=4$ and $\mathrm{\Lambda }=16$. We also plot the expectation value $⟨R^2⟩$ obtained from configurations with $R^2\le x_{\min }$.

Figure 4

(Left) The internal energy of the bound states is plotted against $1/\mathrm{\Lambda }$ for $N=4$, $T=0.10$. The straight line represents a fit to the behavior $E=E_{\text{gauge}}+\mathrm{const}/\mathrm{\Lambda }$, and the value of $E_{\text{gauge}}$ obtained by the fit is plotted at $\mathrm{\Lambda }=\infty$. (Right) The internal energy $E_{\text{gauge}}/N^2$ obtained from our results by extrapolation to $\mathrm{\Lambda }=\infty$ is plotted against $T$. The curves represent $E_{\text{gauge}}/N^2=E_{\text{gravity}}/N^2+(c{T}^{-2.6}+\tilde{c}{T}^p)/N^4$, where the parameters $c$, $\tilde{c}$, and $p$ are obtained by fits as described at the end of Sec. 4b. The solid, dashed, and dash-dotted lines correspond to $N=5$, 4, and 3, respectively.

Figure 5

(Left) The difference $(E_{\text{gauge}}-E_{\text{gravity}})/N^2$ between the internal energy obtained by the gauge theory and that predicted from the gravity side is plotted against $1/N^4$, where the straight lines represent fits to a linear behavior. (Right) The internal energy $E_{\text{gauge}}/N^2$ obtained by the gauge theory is plotted against $1/N^2$. The dashed lines represent fits to the behavior $E_{\text{gauge}}/N^2=7.41T^{2.8}-5.76T^{0.4}/N^2+\mathrm{const}/N^4$, which is expected from the gravity side.

Figure 6

(Left) The parameter $c_1$ obtained by fitting our results to $E_{\text{gauge}}/N^2=7.41T^{2.8}+c_1/N^2+c_2/N^4$ is plotted against $T$. The dashed line represents the behavior $f(T)=-5.76T^{0.4}$ expected from the gravity side. The data point for $T=0.12$ does not have an error bar since we have only two data points ($N=3$ and $N=4$) for the two-parameter fit. (Right) The parameter $c_2$ obtained by fitting our results to $E_{\text{gauge}}/N^2=7.41T^{2.8}-5.76T^{0.4}/N^2+c_2/N^4$ is plotted against $T$. The dashed line represents a fit to the behavior $c_2=c{T}^{-2.6}+\tilde{c}{T}^p$ with $c=0.032(2)$, $\tilde{c}=0.51(64)\times 10^5$, and $p=3.7(6)$.