# The emergence of flagpole and flag-dipole fermions in fluid/gravity correspondence

## Abstract

The emergence of flagpole and flag-dipole singular spinor fields is explored, in the context of fermionic sectors of fluid/gravity correspondence, arising from the duality between the gravitino, in supergravity, and the phonino, in supersymmetric hydrodynamics. Generalized black branes, whose particular case consists of the AdS–Schwarzschild black brane, are regarded. The correspondence between hydrodynamic transport coefficients, and the universal absorption cross sections of the generalized black branes, is extended to fermionic sectors, including supersound diffusion constants. A free parameter, in the generalized black brane solution, is shown to control the flipping between regular and singular fermionic solutions of the equations of motion for the gravitino.

## 1 Introduction

Classical spinor fields were classified studying all the possibilities to evaluate their respective bilinear covariants that either satisfy the Fierz identities or their generalizations. This feature has introduced the Lounesto’s spinor field classification into six classes of spinor fields, assuming the U(1) gauge symmetry of quantum electrodynamics [1]. A second quantized version of such a classification was introduced in Ref. [2], where quantum spinors and their correlators provided a setup for a second quantized classification. Going further, encompassing \(\mathrm{SU}(2)\times \mathrm{U}(1)\) gauge symmetry, a new classification, embracing spinor field multiplets that represent non-Abelian gauge fields, was lately introduced in Ref. [3]. Recently, new classes of fermionic fields on 7-manifolds were derived [4, 5], regarding, in particular, the AdS\(_5\times S^5\) and AdS\(_4\times S^7\) compactifications [6], also including new fermionic solutions of M-theory compactifications with one supersymmetry [7]. These new classes emulate singular spinor fields on higher dimensions and more general signatures. Hence, it is natural to further explore the role of the spinor fields classifications in the fluid/gravity correspondence setup.

The low-energy/low-momentum limit of the AdS/CFT correspondence is also known as fluid/gravity correspondence. In this regime, the field theory side is taken to be an effective theory, hence, hydrodynamics [8]. On the other hand, the compactification of higher dimensions leads the gravitational theory to conventional General Relativity (GR), although this gives some freedom to play with extensions of GR and investigation of its dual theories. This fact has lead to successful predictions of transport coefficients in strongly coupled field theories, being the quark-gluon plasma [9] the most famous example, but not the only one, also appearing in other setups, like the graphene [10], superconductors [11], and Fermi liquids [12]. One intriguing feature of this duality is the so called KSS result [13, 14], named after Kovtun, Son and Starinets, which states that the shear viscosity to entropy density ratio is universal, in the sense that its numerical value is the same for almost all known physical systems. One exception involve a highly complex framework [15].

To lead the fluid/gravity correspondence – essentially based on bosonic fields – further, one aims to include fermionic modes into the description. To accomplish so, one refers to supersymmetry in the bulk and analyzes its effect in the boundary, describing supersymmetric hydrodynamics [16]. This setup indeed leads to predictions [17, 18, 19, 20, 21, 22] and the quest which concerns us in this work is related to the problem of whether a quantity similar to the shear viscosity to entropy ratio, associated to fermionic sectors, exists. In Ref. [23] the sound diffusion constant was first calculated in a supersymmetric holographic background and indicated that this quantity is the obvious candidate for the task, which was investigated and asserted later by [24].

The sound diffusion constant is related to the super current, which turns out to be the super partner of the energy-momentum tensor. When one considers a field theory with \(T>0\), supersymmetry is spontaneously broken and the emergence of a collective fermionic excitation called phonino arises in very general circumstances [25]. The sound diffusion is associated with the damping of this mode, i. e., imaginary part of the dispersion relation. It was computed analytically for \(\mathcal {N}=4\) supersymmetry. In the holographic setting the EOM for the gravitino in AdS\(_5\) background were solved to first order in the frequency and momentum using the retarded Green’s function of the dual supersymmetric current, from where the dispersion relation was read off.

Flagpoles and flag-dipoles are types of the so called singular spinor fields in Lounesto’s U(1) gauge classification. Flagpoles encompass neutral Majorana, and Elko, spinor fields, as well as charged spinor fields satisfying specific Dirac equations [26]. Flag-dipoles are very rare in the literature, being their first appearance in Ref. [27]. The emergence of flagpole and flag-dipole singular spinor fields in the context of fermionic sectors of fluid/gravity correspondence is here scrutinized, exploring the duality between the graviton, in a supergravity bulk setup, and the phonino, in the boundary supersymmetric hydrodynamics. These spinor fields emerge when generalized black branes are considered, whose particular case in the AdS–Schwarzschild black brane for a very particular choice of parameter. This parameter appearing in the generalized black branes shall be shown to drive the flipping that takes regular into singular spinors fields, as solutions of the equations of motion for the gravitino.

This paper is organized as follows: Sect. 2 is devoted to a brief review of the U(1) spinor field classification from bilinear covariants. The Fierz identities, and their generalizations, are discussed as well as the role of singular and regular spinor fields. In Sect. 3, the relation between hydrodynamic transport coefficients and the universal absorption cross sections in the corresponding gravity dual is provided, and then extended to the fermionic sectors. The Kubo formulæ for the gravitino transport coefficients is used. Section 4 is dedicated to explore the aforementioned results for generalized black branes and derive the supersound diffusion constants. The bulk action for the gravitino is employed in the generalized black brane background, showing how flagpole and flag-dipole fermionic fields emerge as solutions of the derived equations of motion. The free parameter, in the generalized black brane, is then analyzed, also providing the flipping between regular and singular fermionic fields.

## 2 General bilinear covariants and spinor field classes

*x*on a 4D spacetime, with cotangent basis \(\{e^\mu \}\) read

When both the scalar and the pseudoscalar vanish, a spinor field is called singular, otherwise its said to be regular. The objects in Eqs. (1b) and (1d), being 1-form fields, are named poles. Since spinor fields in the class 4, (2d), have non vanishing **K** and **J**, spinor fields in this class are called flag-dipoles, because \(\mathbf{S}\ne 0\) is a 2-form field, identified by a flag, according to Penrose. Besides, spinor fields in class 5, (2e), have a vanishing pole, \(\mathbf{K} = 0\), a non null pole, \(\mathbf{J}\ne 0\), and a non null flag, \(\mathbf{S}\ne 0\), being flagpoles. Spinor fields in class 6, (2f), present two poles, \(\mathbf{J}\ne 0\) and \(\mathbf{K}\ne 0\), and a null flag, \(\mathbf{S}=0\), corresponding therefore to a flag-dipole. Flag-dipole spinor fields were shown to be a legitimate solution of the Dirac field equation in a torsional setup [27, 29, 30], whereas Elko [31] and Majorana uncharged spinor fields represent type-5 spinors [26], although a recent example of a charged flagpole spinor has been shown to be a solution of the Dirac equation. Additional spinor fields classes, upwards of the Lounesto’s classification, complete all the formal possibilities, including ghost fields [28]. The standard, textbook, Dirac spinor field is an element of the set of regular spinors in the class 1, (2a). Besides, chiral spinor fields were shown to correspond to be elements of the class 6, (2f), of (dipole) spinor fields. Chiral spinor fields governed by the Weyl equation are Weyl spinor fields. However, the class 6 of dipole spinor fields further allocates mass dimension one spinors, whose dynamics, of course, is not ruled by the Weyl equation, as well as flagpole spinor fields in the class 5, that are not neutral and satisfy the Dirac equation. The spinor field class 5, still, is also composed by mass dimension one spinor fields [26, 31, 32]. The Lounesto’s spinor field classification was also explored in the lattice approach to quantum gravity [33]. Flipping between regular and singular spinor fields was scrutinized in Ref. [34], for very special cases.

The classification of spinor fields, according to the bilinear covariants, must not be restricted to the U(1) gauge symmetry of quantum electrodynamics. In fact, a more general classification, based on the \(\mathrm{SU}(2)\times \mathrm{U}(1)\) gauge symmetry, embraces multiplets and provide new fermionic possibilities in the electroweak setup [3].

In the next section, the relation between hydrodynamic transport coefficients and the universal absorption cross sections in the corresponding gravity dual is provided, and then extended to the fermionic sectors. A generalized black brane background shall be introduced, providing the first steps for the emergence of flagpole and flag-dipole singular spinor fields. The hydrodynamic transport coefficients, in the fermionic sectors, shall be briefly reviewed. The supersound diffusion constant, for the generalized black brane, shall be also studied, leading to the AdS–Schwarzschild results [23, 37], in a very particular limit.

## 3 Black hole absorption cross sections and fermionic sectors

*s*and

*a*the respective entropic and areal densities, one may employ the Klein–Gordon equation of a massless scalar, as an equation of motion for the low-energy absorption cross section associated with \(h_{12}\), yielding the horizon area density \(\sigma (0)=a\) [24].

*S*, that accounts the transverse component of the supersymmetric current [46, 47, 48], by \( \int \left( \overline{S}P^-\Psi _b + \overline{\Psi }_bP^+S\right) \,d^4x \) [24]. The chiral projector \(P^\pm =\frac{1}{2}\left( 1\pm \upgamma ^{4}\right) \) is employed, where \(\upgamma ^4\) denotes the gamma matrix corresponding to the radial AdS\(_5\) coordinate. Taking a fermion at rest in the boundary implies that

*P*are the energy density and pressure of the fluid, respectively. Reference [49] interprets \(\uprho \) as a sound-like excitation, the phonino. The quantities \(\mathrm{D}_\mathrm{s}\) and \(\mathrm{D}_\upsigma \) play the role of transport coefficients that govern the phonino, that has a speed dissipation given by \(v_\mathrm{s}=\frac{P}{\upepsilon }\) [25, 50]. Equation (16) can be rewritten, taking into account to the spin-representations of O(3),

*x*-coordinates implicitly by \(\frac{d}{dx} = k(r) \rho (r)r^3 \frac{d}{dr}\) [24], such that \(d_r \rho =2m\rho \sqrt{B}\), implies that \(\lim _{r\rightarrow \infty }\rho = 1\). Therefore, the \(n=0\) fermionic mode can be rescaled as \(\mathfrak {F}_0 = e^{-m\int \sqrt{B}\,dr}F_0\), yielding

*T*is the black brane temperature.

## 4 Supersound diffusion constant from the transverse gravitino

*k*,

## 5 Concluding remarks

The occurrence of flag-dipole fermions in physics is very rare, comprising features that approach regular spinor fields, although being singular ones, in the Lounesto’s classification (2a–2f) according the bilinear covariants. Besides the two previous examples in the literature, here flag-dipole solutions corresponding to the spinor part of the gravitino field (37) were obtained, together with flagpole spinor fields. They were derived as solutions of the bulk action for the gravitino field, in the background of black brane solutions that generalize the AdS–Schwarzschild one. Besides, the generalized black brane has a free parameter driving the singular spinor field solutions, which can flip between regular and singular spinor fields. The relation between hydrodynamic transport coefficients and the universal absorption cross sections in the corresponding gravity dual was also studied in the fermionic sectors of the fluid/gravity correspondence. The Kubo formulæ was employed to derive the transport coefficients for the gravitino and its dual, the phonino. In the limit where the generalized black brane parameter tends to the unit, these results are in full compliance with the ones in Ref. [24] for the AdS–Schwarzschild black brane. The supersound diffusion constants were also discussed through the solutions of the equations of motion for the gravitino.

## Notes

### Acknowledgements

PM thanks to CAPES and UFABC. RdR is grateful to the National Council for Scientific and Technological Development—CNPq (Grant no. 303293/2015-2), and to FAPESP (Grant no. 2017/18897-8), for partial financial support. The authors thank to A. J. Ferreira–Martins for fruitful discussions.

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