$S$-duality for holographic $p$-wave superconductors

We consider the generalization of the $S$-duality transformation previously investigated in the context of the fractional quantum Hall effect (FQHE) and $s$-wave superconductivity to $p$-wave superconductivity in $2+1$ dimensions in the framework of the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. The vector Cooper condensate transforms under the $S$-duality action to the pseudovector condensate at the dual side. The $3+1$-dimensional Einstein-Yang-Mills theory, the holographic dual to $p$-wave superconductivity, is used to investigate the $S$-duality action via the $\mathrm{AdS}/\mathrm{CFT}$ correspondence. It is shown that, in order to implement the duality transformation, chemical potentials on both the electric and magnetic sides of the duality have to be introduced. A relation for the product of the non-Abelian conductivities in the dual models is derived. We also conjecture a flavor $S$-duality transformation in the holographic dual to $3+1$-dimensional QCD low-energy QCD with non-Abelian flavor gauge groups. The conjectured $S$-duality interchanges isospin and baryonic chemical potentials.

Figure 1

Two solutions $w$ of the Riccati equation in the ${\mathrm{AdS}}_4$-Schwarzschild metric: a standard solution for $\mu =3.66$, $w_1=0.21$ (left) and a new solution for $\mu =1.05$, ${\tilde{w}}_1=-2$ (right).

Figure 2

Two solutions for the gauge field $A$ of the equation of motion in the ${\mathrm{AdS}}_4$-Schwarzschild metric: a standard solution for $\mu =3.656$, $w_1=0.21$ (left) and a new solution for $\tilde{\mu }=1.05$, ${\tilde{w}}_1=-2$ (right).

Figure 3

Combination $R(z)=-w(z)\tilde{w}(z)(1+z+z^2-z^3{q}^2{)}^2$ for the two solutions of the Riccati equation in the Schwarzschild metric $q=0$. Combination $R$ being constant means that the duality relation (130) is satisfied in the AdS-RN space.

Figure 4

The duality relation for the RN-AdS BH metric in the IR and the UV asymptotics with the charge increased from $q=0$ to $q=1.5$. The duality in the IR is $-w^{\prime }(z=1)$ vs $\mu \tilde{\mu }/2(3-q^2{)}^2$ (left). The duality in the UV is $-{\tilde{w}}^{\prime }(z=0)$ vs $\mu \tilde{\mu }$ (right). Both are straight lines, which verifies the duality conditions (136) and (142).

Figure 5

The duality relation for the RN-AdS BH metric in the UV asymptotics with the charge increased from $q=0$ to $q=1.5$: $A^{\prime }(z=0)/\tilde{A}(z=0)$ vs $\tilde{\mu }$ (left) and $-{\tilde{A}}^{\prime }(z=0)/A(z=0)$ vs $\mu$ (right). Linear behavior confirms the duality conditions for the gauge fields (120).

Figure 6

The mapping between the chemical potentials, $\tilde{\mu }$ vs $\mu$, as the charge is increased from $q=0$ to $q=1.5$.

Figure 7

Two solutions $w$ of the Riccati equation in the ${\mathrm{AdS}}_5$: a standard threshold solution for $\mu =4$, $w_1=0$ (left) and a new solution for $\mu =0.001$, $w_1=-0.009$ (right). The upper curve in the right plot is $w=2/z$. The straight line in the left plot is $w=-0.0004z$.

Figure 8

Two solutions for the gauge field $A$ of the equation of motion in the ${\mathrm{AdS}}_5$: a standard threshold solution for $\mu =4$ $w_1=0$ (left) and a new solution for $\mu =0.001$, $w_1=-0.009$ (right). Two coinciding curves for numerical and analytical solutions $4z^2/(1+z^2{)}^2$ are shown in the left panel.