Holographic Torsion and the Prelude to Kalb–Ramond Superconductivity

Abstract

We discuss the holographic implications of torsional degrees of freedom in the context of $\hbox{AdS}_4/\hbox{CFT}_3,$ emphasizing in particular the physical interpretation of the latter as carriers of the non-trivial gravitational magnetic field, i.e. the part of the magnetic field not determined by the frame field. As a concrete example we present a new exact four-dimensional gravitational background with torsion and argue that it corresponds to the holographic dual of a 3D system undergoing parity symmetry breaking. Finally, we compare our new gravitational background with known wormhole solutions—with and without cosmological constant—and argue that they can all be unified under an intriguing “Kalb–Ramond superconductivity” framework.

Appendices

Appendix

Parity Breaking in Three Dimensions

Consider the 3D Gross–Neveu model coupled to abelian gauge fields. The Euclidean action is⁶

$$ I=-\int d^3x\left[\bar{\psi}^a\left({\slash\!\!\!}\partial-{\rm i}e{\slash \!\!\!\!A}\right)\psi^a +\frac{G}{2N}\left(\bar{\psi}^a\psi^a\right)^2 +\frac{1}{4M}F_{\mu\nu}F_{\mu\nu}\right]. $$

(A.1)

$M$ is an UV mass scale. Introducing the usual Lagrange multiplier field $\sigma,$ whose equation of motion is $\sigma =\frac{-2G}{N}\bar{\psi}^a\psi^a$ we can make the action quadratic in the fermions

$$ I=-\int d^3x\left[\bar{\psi}^a\left({\slash\!\!\!}\partial+\sigma-{\rm i}e{\slash \!\!\!\!A}\right)\psi^a -\frac{N}{2G}\sigma^2-\frac{1}{4M}F^{\mu\nu}F_{\mu\nu}\right]. $$

(A.2)

The model possesses two parity breaking vacua distinguished by the value of the pseudoscalar order parameter $\langle\sigma\rangle.$ This is seen as follows: switching off the gauge fields momentarily one integrates over the fermions to produce a large-$N$ effective action as

$$ {\mathcal{Z}} = \int({\mathcal{D}}\sigma)e^{N\left[\hbox{Tr}\log\left({\slash\!\!\!}\partial +\sigma\right)-\frac{1}{2G}\int d^3x\sigma^2\right]}. $$

(A.3)

The path integral has a non-zero large-$N$ extremum $\sigma_*$ found by setting $\sigma =\sigma_* +\frac{1}{\sqrt{N}}\lambda$

$$ {\mathcal{Z}} =\int ({\mathcal{D}}\lambda) e^{N\left[\hbox{Tr}\log\left({\slash\!\!\!}\partial +\sigma_*\right)-\frac{1}{2G}\int d^3x \sigma_* +\frac{1}{\sqrt{N}}\left\{ \hbox{Tr}\frac{\lambda}{{\slash\!\!\!}\partial+\sigma_*}-\frac{\sigma_*}{G}\int d^3x\lambda\right\}+O(1/N)\right]} $$

(A.4)

The term in the curly brackets is the gap equation. To study it one considers a uniform momentum cutoff $\Uplambda$ to obtain

$$ \frac{1}{G}=\int^\Uplambda\frac{d^3p}{(2\pi)^3} \frac{2}{p^2+\sigma_*^2}=(\hbox{Tr} 1)\left[\frac{\Uplambda}{\pi^2}-\frac{|\sigma_*|} {\pi^2}\arctan\frac{\Uplambda}{|\sigma_*|}\right]. $$

(A.5)

Defining the critical coupling as

$$ \frac{1}{G_*}=\frac{\Uplambda}{\pi^2}, $$

(A.6)

(A.5) possesses a non-zero solution for $\sigma_*$ when $G>G_*$ given by

$$ |\sigma_*|=\frac{2\pi}{G}\left(\frac{G}{G_*}-1\right)\equiv m. $$

(A.7)

The two distinct parity breaking vacua then have

$$ \sigma_*=-\frac{2G}{N}\langle\bar{\psi}^a\psi^a\rangle =\pm m. $$

(A.8)

Going back to (A.2) one can tune $G>G_*$ and start in any of the two parity breaking vacua. Suppose we start from $\sigma_*=+m.$ To leading order in $N$ we have

$$ \begin{aligned} {\mathcal{Z}}&=\int ({\mathcal{D}}A_\mu)({\mathcal{D}} \bar{\psi}^a)({\mathcal{D}}\psi^a)\exp[{\mathcal{S}}]\\ {\mathcal{S}}&=\int d^3x\left[\bar{\psi}^a\left({\slash\!\!\!}\partial +m-{\rm i}e{\slash\!\!\!\!} A\right)\psi^a-\frac{N}{2G}m^2+O(1/\sqrt{N}) -\frac{1}{4M}F^{\mu\nu}F_{\mu\nu}\right] \end{aligned} $$

(A.9)

As is well known [45, 46] for $N$ fermions the path integral (A.9) yields an effective action for the gauge fields which for low momenta is dominated by the Chern–Simons term i.e.

$$ {\mathcal{Z}} \approx \int e^{S_{\rm CS}}, $$

(A.10)

with

$$ S_{\rm CS} =\hbox{i}\frac{ke^2}{4\pi}\int d^3x \epsilon^{\mu\nu\rho}A_\mu\partial_\nu A_\rho,\quad k=\frac{N}{2}. $$

(A.11)

Had we started from the $\sigma_*=-m$ vacuum, we would have found again (A.10–A.11), however with $k=-\frac{N}{2},$ i.e. the vacuum with $\sigma_*=-m$ yields an effective Chern–Simons action with $k=-\frac{N}{2}.$

Consider now $deforming$ the action (A.9) by the Chern–Simons term with a fixed coefficient as

$$ \begin{aligned} {\mathcal{Z}}_{\rm deform}&=\int ({\mathcal{D}}A_\mu)({\mathcal{D}} \bar{\psi}^i)({\mathcal{D}}\psi^i)\exp[{\mathcal{S}}_{def}]\\ {\mathcal{S}}_{\rm def}&={\mathcal{S}}-\hbox{i}q\int d^3x\epsilon^{\mu\nu\rho} A_\mu\partial_\nu A_\rho \end{aligned} $$

(A.12)

If $q$ is fixed to

$$ q=\frac{Ne^2}{4\pi}, $$

(A.13)

the effective action for the gauge fields resulting from the fermionic path integrals in (A.12) is going to be $equal$ to the one obtained had we started at the $\sigma_*=-m$ vacuum. In other words, deforming the $\sigma_*=+m$ vacuum with a Chern–Simons term with a fixed coefficient is equivalent to being in the $\sigma_*=-m$ vacuum. This is exactly analogous to the holographic interpretation of our torsion DW.