Stability of non-extremal conifold backgrounds with sources

We present finite temperature solutions describing $N_c$ $D5$ branes wrapped on the $S^2$ of the resolved conifold in the presence of $N_f$ flavor branes sources and their backreaction i.e., $N_f/N_c\sim 1$. In these solutions the dilaton does not blow up at infinity but stabilizes to a finite value. Thus, we can use them to generate new ones with $D5$ and $D3$ charge. The resulting backgrounds are nonextremal versions of the “flavored” resolved deformed conifold. It is tempting to interpret these solutions as gravity duals of finite temperature field theories exhibiting nontrivial phenomena as Seiberg dualities, Higgsing and confinement. However, a first necessary step in this direction is to investigate their stability. We study the specific heat of these new flavored backgrounds and find that they are thermodynamically unstable. Our results on the stability also apply to some of the nonextremal backgrounds with Klebanov-Strassler asymptotics found in the literature.

Figure 1

Horizon-shot and UV-shot solutions for metric functions at $s=1$, $c_{+}=3$, $C_2=800000$. $e^k$—orange, $e^g$—blue, $e^h$—purple, $e^{8x}$—pink.

Figure 2

Horizon-shot and UV-shot solutions for metric functions at $s=1$, $c_{+}=3$, $C_2=800000$. Zoomed into region near horizon at ${\rho }_h\sim 5.097$.

Figure 3

Horizon-shot and UV-shot solutions for metric functions at $s=1$, $c_{+}=50$, $C_2=5000$.

Figure 4

Horizon-shot and UV-shot solutions for metric functions at $s=1$, $c_{+}=50$, $C_2=5000$. Zoomed into region near horizon at ${\rho }_h\sim 3.19$.

Figure 5

Temperature at the horizon versus $N_f/N_c$. $c_{+}=50$, $C_2=5000$.

Figure 6

Temperature at the horizon versus $N_f/N_c$. $c_{+}=3,C_2=800000$.

Figure 7

Temperature at the horizon versus $C_2$. $c_{+}=50$, $s=1$.

Figure 8

Temperature at the horizon versus $C_2$. $c_{+}=3$, $s=1$.

Figure 9

Temperature at the horizon versus for very large values of $C_2$. $c_{+}=3$, $s=1$.

Figure 10

Temperature at the horizon versus $\mathcal{A}/V_3$. $c_{+}=3$, $s=1$.

Figure 11

$g_{\rho \rho }$ metric element for solutions after rotation, $c_{+}=50$, $C_2=5000$. From bottom to top: $s=0$, 1, $19/10$, $51/10$, $91/10$.

Figure 12

$g_{\rho \rho }$ metric element for solutions after rotation, $c_{+}=3$, $C_2=800000$. From bottom to top: $s=0$, 1, $19/10$, $51/10$, $91/10$.

Figure 13

$g_{xx}$ (red) and $g_{tt}$ (blue) metric elements for the unflavored solution after rotation: $c_{+}=50$, $C_2=5000$, $s=0$.

Figure 14

$g_{xx}$ (red) and $g_{tt}$ (blue) metric elements for the unflavored solution after rotation: $c_{+}=3$, $C_2=800000$, $s=0$.

Figure 15

$g_{xx}$ (red) and $g_{tt}$ (blue) metric elements for flavored solutions after rotation: $c_{+}=50$, $C_2=5000$. From top to bottom: $s=0$, 1, $19/10$, $51/10$, $91/10$.

Figure 16

$g_{xx}$ (red) and $g_{tt}$ (blue) metric elements for flavored solutions after rotation: $c_{+}=3$, $C_2=800000$. From top to bottom: $s=0$, 1, $19/10$, $51/10$, $91/10$.

Figure 17

$\frac{3}{2}{g}_{\psi \psi }$ (blue), $g_{\theta \theta }$ (red), and $g_{\tilde{\theta }\tilde{\theta }}$ (yellow) metric elements for flavored solution after rotation. The different groups correspond to $s=0$ (solid), $s=2/5$ (dotted) and $s=61/10$ (dashed). $c_{+}=3$, $C_2=800000$.

Figure 18

$\frac{3}{2}{g}_{\psi \psi }$ (blue), $g_{\theta \theta }$ (red), and $g_{\tilde{\theta }\tilde{\theta }}$ (yellow) metric elements for flavored solution after rotation. The different groups correspond to $s=0$ (solid), $s=2/5$ (dotted) and $s=61/10$ (dashed). $c_{+}=50$, $C_2=50000$.

Figure 19

$d{e}_{\mathrm{bef}}/d{\mathfrak{s}}_{\mathrm{bef}}$ (red line) and $T_h$ (blue dots) plotted versus ${\mathfrak{s}}_{\mathrm{bef}}$, for the flavored case with $c_{+}=50$, $s=1$.

Figure 20

ADM energy density versus horizon temperature ($s=1$, $c_{+}=50$).

Figure 21

ADM energy density versus horizon temperature ($s=1$, $c_{+}=3$).

Figure 22

ADM energy density versus entropy density ($s=1$, $c_{+}=50$).

Figure 23

ADM energy density versus entropy density ($s=1$, $c_{+}=3$).

Figure 24

Constraint evaluated on a horizon shot solution with $c_{+}=50$, $C_2=5000$, $s=1$.

Figure 25

Constraint evaluated on same solution, near horizon at ${\rho }_h\simeq 3.194024$.

Figure 26

Horizon-shot and UV-shot $a$ and $b$ after tuning $b_0$. brown—UV-shot $a$, black—UV-shot $b$, blue—horizon shot $a$, red—horizon-shot $b$. $s=1$, $c=3$, $C_2=800000$.