Q-ball-like solitons on the M2-brane with worldvolume fluxes

Solitons are stable classical solutions with finite energy to nonlinear equations that are able to mimick the behaviour of particles as well as excitations. They are classified into topological and nontopological ones. Topological solitons like monopoles, vortices, instantons and kinks have been extensively studied in the literature. Non-topological solitons have also received a lot of interest for its potential to explain dark matter among others. One simple case corresponds to Q-balls. Q-balls [1] are the generic name given to solitons, originally with spherical topology, which are invariant under a $U(1)$ global symmetry. This symmetry implies the existence of a Noether charge which provides stability against perturbations. Thus global symmetry has been associated to quantum invariants as lepton or barion number. There exists also gauged Q-balls [2, 3] in the case in which the complex scalar fields that parametrized the Q-ball are also charged under a gauge potential $U(1)$. There are also other works in which the authors have considered more general global or gauged groups [4, 5]. Dynamics of Q-balls have been studied in [6, 7] and for the gauged case [2, 8]. Their stability has also been discussed in several papers, see for example [9]. Recently, it has been discovered that it is possible to obtain solitonic solutions that simultaneously share the two types of properties. These type of solitons have been denoted as Q-monopole-ball [10].

Solitons with other topologies different from the sphere have been described to less extent [11]. For example, Q-torus, which corresponds to Q-balls with toroidal topology have been also studied in [12]. Q-balls have also been used to model dark matter [13], in the context of string theory [14]. This paper describes Q-ball solitons from the open and closed string moduli as well as other types of solitons. However, there are very few results of solitons have been reported at the M-theory level.

Classical solutions of membrane theory can be useful to better understand the M-theory properties. First examples of classical solutions to the membrane theory firstly appeared in [15]. In the context of matrix models, it was discussed in [16]. The existence of membrane solitonic solutions has been discussed in several works. For example in terms of stringy solitary waves in [17], in the context of Q-ball matrix model [18], as instantonic solutions [19], or in terms of membranes formulated on hyperkähler backgrounds [20, 21]. There are bounds for the experimental detection of Q-balls [9], though they are model dependent.

This paper presents analytical and numerical methods to obtain solitonic solutions of the M2-brane with worldvolume fluxes, specifically Q-ball-like solutions. By Q-ball-like (QBL) solutions we refer to solitonic excitations of the membrane that satisfy the characteristic QBL ansatz. We consider a supermembrane on a $M_{9}\times T^{2}$ target space with a $C_{\pm}$ flux background. This configuration possesses a magnetic $U(1)$ monopole in the compact sector [22, 23] associated with the flux content that induces the presence of a magnetic monopole over the worldvolume [24]. We search solutions to the nonlinear equations of the M2-brane Hamiltonian on this particular background focusing on the existence of QBL solitons. These solutions are described by complex scalars that model the dynamics of the excitations on the membrane with fluxes, in the non-compact transverse space to the lightcone coordinates. The Hamiltonian that describes the dynamics of the M2-brane contains on top of nontrivial topological terms, non-linear cuartic scalar contributions and a symplectic field strength whose dynamics under certain assumptions becomes coupled. We analyze two different cases: Isotropic analytical solutions that generate string-like configurations with a non-vanishing cost in energy and numerical non-isotropical solutions for the membrane full-fledged configurations.
The string-like configurations, unlike with the string-like spikes associated with the case of a membrane without central charge, carry minimal energy associated with the M2-brane with worldvolume fluxes and the mass terms and thus have dynamics. These analytical solutions of the isotropic case are found by solving the equations of motion (EOM) and the constraints of the theory: the area preserving first class constraint and the flux constraint. The flux condition induces a topological central charge restriction. The different families of solutions found satisfy the Q-ball ansatz and include the case of localized solutions. With the aid of numerical simulation, we extend those results to the non-isotropic case in the absence of a symplectic gauge field. We then obtain, QBLnuev full-fledged membrane solutions.

Having established the existence of these solutions, we characterise some of their properties. We show that the analytic families of solutions considered exhibit dispersion generically, but they also admit dispersion-free solutions.

The dynamics of the Q-ball solitons are characterised by studying their Lorentz transformation and their nonrelativistic limit, the Galilean one. As we are in the Light Cone Gauge (LCG), the transformed solutions are not automatically valid and some extra restrictions must be imposed. We obtain it for some of the analytic families of solutions considered.

There are two types of charges for the solutions found: a non-topological charge related to the Noether charge of the Q-ball and a topological charge related to the existence of a monopole charge resulting from the pull-back of the target $C_{\pm}$ fluxes. The monopole charge defined on the world volume is induced by the nontrivial embedding in the compact space, and the Q-ball solitons are related to the non-compact target space. There is a non-trivial interaction between them induced by the covariant derivative terms at the level of the mass operator. This interaction leads to an enhanced stability.

Finally, we describe their superposition law and demonstrate that, as expected, they cannot be linearly superposed. However, they can be linearly composed for very restricted values of the Fourier modes.

The paper is structured as follows: Section 2 reviews the formulation of the supermembrane toroidally wrapped on $M_{9}\times T^{2}$ subject to a $C_{\pm}$ flux condition and the equations of motion to solve them. Sections 3 to 8 present our results. In Section 3, we obtain analitically a family of exact QBL string-like solutions for the isotropic case. The dispersion of solutions in the presence of a symplectic gauge field is characterized. Additionally, localized solutions for the QBL ansatz are demonstrated. Section 4 characterizes a membrane QBL soliton numerically. Firstly, we compare the numerical analysis of the isotropic case with the previously obtained analytical results by imposing the ansatz of the so-called ’family’. Then, we extend the analysis to the non-isotropic case, taking into account the complete membrane excitations, when the symplectic gauge vanishes. In the non-isotropic case, the nonlinear character of the complex scalar field becomes explicitly manifest. In section 5 the dynamics of the solution by Lorentz and Galilean transformations is discussed. Section 6 demonstrates a relation between topological and non-topological charges of the M2-brane, improving stability. In Sec. 7 we present a superposition law for the QBL family of solutions that was previously found for the isotropic case. Linear superposition of the solutions is only possible when the frequencies are strongly constrained, consistent with the nonlinear nature of the equations. The results obtained are discussed and concluded in section 8.

In this work we analyze the existence of solutions to the dynamics of an M2-brane mass operator, focusing in those of Q-ball-type. The bosonic sector that we analyze, corresponds to the supermembrane with $C$ fluxes introduced in [22, 23] that induce a topological condition denoted central charges which is associated with the presence of a magnetic monopole over the worldvolume of the membrane [24]. The M2-brane is formulated on a target space $M_{9}\times T^{2}$ in the Light Cone Gauge (LCG) [25, 26]. We define the embedding maps on the non-compact space $X^{m}(\tau,\sigma,\rho):\Sigma\to M_{9}$ with $m=2,...,8$. The maps $X^{r}(\tau,\sigma,\rho):\Sigma\to T^{2}$, where $r=9,10$, describe the embedding on the compactified 2-torus. These maps act as scalars on the M2-brane worldvolume and as vectors on the target space. The coordinates of the supermembrane worldvolume are labeled by $(\tau,\sigma,\rho)$ parametrizing $\Sigma\times\mathbb{R}$, with $\Sigma$ representing a Riemann surface of genus one and $\mathbb{R}$ parametrizing time. The maps $X^{r}$ satisfy the winding condition

where $m_{r}^{s}$ are the winding numbers and $R^{r}$ are the torus radii.

The M2-brane is restricted by a worldvolume flux condition. It corresponds to a topological condition denoted by the authors as Central Charge condition(CC) [24], satisfying that $n=det\mathbb{W}$ where $\mathbb{W}$ denotes the winding matrix of an irreducible wrapping of the M2-brane around the compactified 2-torus. The central charge condition is associated to the existence of a monopole over the worldvolume which is given by the curvature $\widehat{F}=\epsilon_{rs}dX^{r}_{h}\wedge dX^{s}_{h}$ associated to a nontrivial $U(1)$ gauge field $\widehat{F}=d\widehat{A}$ defined on the membrane worldvolume.

where $A_{T^{2}}$ denotes the 2-torus target space area.

The worldvolume flux can be also induced by the pullback of a target-space 2-form flux condition generated by the quantization of the supergravity 3-form in the LCG once that the $X^{-}$ has been eliminated from the formulation, see, for example, in [22, 23].

The fluxes of the Hamiltonian can be induced by the presence of a constant quantized 3-form $C_{\pm}$ background. The Hamiltonian of a supermembrane with central charges corresponds to the Hamiltonian of the M2-brane with worldvolume fluxes $C_{-}$. Its formulation is equivalent to the Hamiltonian of the M2-brane with central charge, plus a constant shift given by the pullback of the $C_{+}$ contribution [22, 23]

Both cases are related, being duals for the $C_{-}$ case and the central charge one. They are defined in terms of an integer $n$, so in the rest of the paper we will use the central charge formulation for simplicity in th3e resolution.

The closed one-forms $dX_{r}$ can be decomposed by a Hodge decomposition in terms of two harmonic one-forms $d\widehat{X}_{r}$ and two exact one-forms $dA_{r}$, such that

The $dX_{rh}$ is a closed one-form defined in terms of the basis of harmonic forms as $d{X}_{rh}=R^{r}m_{r}^{s}d\widehat{X}_{s}(\sigma,\rho)$. They are not physical degrees of freedom in the theory but encode topological information of the compactified manifold. The central charge condition guarantees the existence of the two independent harmonic constant one-forms, $d\widehat{X}_{s}(\sigma,\rho)$. The central charge condition also implies the existence a new dynamical degree of freedom $A_{r}$ whose associated one-form $\mathbb{A}_{r}=dA_{r}=\partial_{a}A_{r}d\sigma^{a}$, with $\partial_{a}=(\partial_{\sigma},\partial_{\rho})$ describes a symplectic gauge field over the M2-brane worldvolume. It transforms as a symplectic connection under symplectomorphisms. $dA_{r}$ are exact one-forms.

The LCG Hamiltonian of the M2-brane theory with central charge corresponds to [25, 26]

where $T$ stands for the membrane tension and $W$ is the determinant of the induced spatial part component of the foliated metric on the membrane; $\left\{A,B\right\}=\frac{1}{\sqrt{W}}\epsilon^{ab}\partial_{a}A\partial_{b}B;\$ is the symplectic bracket defined in terms of the harmonic one-forms $d\widehat{X}$ of the Riemann surface with $a,b=1,2$ and $\partial_{a}=(\partial_{\sigma},\partial_{\rho})$ and $\sqrt{W}=\frac{1}{2}\epsilon^{ab}\epsilon_{rs}\partial_{a}\widehat{X}^{r}\partial_{b}\widehat{X}^{s}$. The scalar fields $X^{m},A_{r}$ have respectively, canonical momenta $P_{m}$ and $P_{r}$. The symplectic covariant derivative is defined as $\mathcal{D}_{r}\bullet=D_{r}\bullet+\{A_{r},\bullet\}$ with $D_{r}\bullet=2\pi m_{r}^{u}\theta_{uv}R_{r}\frac{\epsilon^{{a}{b}}}{\sqrt{W}}\partial_{{a}}\widehat{X}^{v}\partial_{{b}}\bullet.$   The derivative $D_{r}$ is defined in terms of the moduli of the 2-torus, the harmonic one-forms $d\widehat{X}_{r}$, and a matrix $\theta_{uv}$ with $u,v=9,10$, related to the monodromy associated to its global description in terms of a torus bundle [27]. Therefore, the Hamiltonian contains a symplectic curvature $\mathcal{F}$ defined by

The constraints of the theory associated to the local Area Preserving Diffeomorphisms (APD) are:

and to the two APD global constraints,

In the dynamics of the M2-brane we will only be concerned with the local area preserving diffeomorphisms constraint and the flux constraint.

The dynamics of the M2-brane with worldvolume fluxes is determined by the equations of motion (EOM) and their constraints. The Lagrangian density of the theory can be obtained by performing a usual Legendre transformation of the Hamiltonian density defined as

where $\Lambda$ represents a Lagrange multiplier and $\phi=-\phi_{APD}$ represents the APD constraint given by equation (8). It contains the Lagrangian of MIM2, previously obtained in [28], plus a constant term associated to the $C_{+}$ flux contribution, denoted by $K$.

The tension has been fixed to $T=1$. Since the $K$ contribution gives a non dynamical constant term, added to the action, its equations of motion are equal to those of the M2-brane with central charges. The symplectic field strength, and the symplectic covariant derivative now run over the indices $i,j=0,r$, with $r=9,10$. $A_{0}=\Lambda$ the lagrange multiplier and $D_{0}=\partial_{\tau}$. As explained in [29, Allen-Andersson-Restuccia-2010] the gauge freedom of the system allows to fix $\Lambda=0$ and then the Lagrangian density function $\mathcal{L}=\mathcal{L}_{T}-K$ reduces to

The nonlinear system of equations that one has to solve is the following: From (12) we derive the equations of motion for the dynamical fields $X^{m}(\tau,\sigma,\rho)$ and $A_{r}(\tau,\sigma,\rho)$

subject to satisfy the APD constraint of the theory

and the worldvolume flux condition (2), that restricts the winding numbers allowed for the M2-brane.

In order to find the admissible M2-brane solutions, the system of equations (13), (14), (15) and (2) must be solved. For simplicity we will also assume that one of the scalar fields is constant and we will fix the harmonic sector as in [BRR2005nonperturbative],

For this ansatz $\sqrt{W}=1$. There are some differences with the case analysed in [BRR2005nonperturbative]. The main one is associated with the fact that for the supersymmetric M2-brane, the worldvolume flux condition guarantees the discreteness of the spectrum. Other difference due to the topological restriction imposed is that appearance of new single-valued degrees of freedom, $A_{r}$ that are components of a symplectic gauge connection $\mathbb{A}=dA_{r}$ in the theory. The EOM for the constant $X_{2}$ can be expressed in terms of complex variables $Z_{a}$ with $a=1,2,3$ where $Z_{a}(\tau,\sigma,\rho)=X_{2a+1}+iX_{2a+2}$. They are,

subject to the APD constraint of the theory:

In the following, we assume the Q-ball ansatz for the complex scalars $Z_{a}$. We will denote this last ansatz as ’Q-ball like’ (QBL) irrespective of its toroidal topology associated to the membrane worldvolume

The EOM for $Z_{a}$ become

The EOM for $A_{r}$ are:

where the mixed partial second order derivatives are assumed to satisfy $\partial^{2}_{\sigma\rho}=\partial^{2}_{\rho\sigma}$.

The M2-brane is subject to the area preserving diffeomorphism (APD) constraint

and subject to the flux condition that restricts the allowed winding numbers appearing in $X_{rh}$,

In this section we obtain analytic solutions to the system of equations of the M2-brane with $C_{\pm}$ fluxes (21)-(25) of the complex scalar field $Z_{a}$. In this section, for simplicity, we assume the system to be isotropic $Z_{a}=Z_{b}$ for $a\neq b$, having fixed $X_{2}=0$. This assumption trivialises the symplectic bracket for the complex scalar embedding field and correspomd to strong -like configurations but with cost in energy. The solutions still contain a non-vanishing symplectic gauge field, $\mathbb{A}=dA$ with a non-trivial curvature $\mathcal{F}$. To solve the equations of motion we assume that $A_{r}$ and $Z$ have the same spatial dependence associated with a single-valued function $f(\sigma,\rho)$. This assumption transforms the equations of motion into a system of three nonlinear coupled differential equations which must also satisfy the APD constraint and the flux constraint, i.e. a non-vanishing central charge.The configurations obtained depend non-trivially on a combination of the spatial variables. In the first two families we illustrate the simplest solutions, in which the dependence on the symplectic gauge cuvature becomes generalized and in the last family we show the existence of localized excitations with the characteristic breathing modes.

Interestingly, due to the central charge condition mass terms are generated, these dominate in the quartic potential along the flat directions and resemble a modified harmonic oscillator. They correspond to the minimum energy of the theory. These contributions are not present in compactifications without induced 2-form worldvolume fluxes. Furthermore, for the solutions obtained, the central charge condition is responsible for generating terms in the equations of motion that allow the presence of breathing modes different from zero.

In order to find analytic solutions to the M2-brane equations, we have inspected the symmetries of the theory. First of all, it is easy to see that there is a symmetry in the choice of the spatial coordinates. Furthermore, the equations are non-linear with triple and quadratic products in the partial derivatives of first and second order. By direct observation of the differential equations one can observe that a family of solutions for a given $f$ appears when first derivatives are proportionals among them and consequently the same type of relation holds for the second derivatives. Then, a natural choice is to convert the PDE of the system into an ODE-type system by means of imposing that $\partial_{\sigma}f\propto\partial_{\rho}f$-. We keep the partial derivative notation, to recall that the system truly depends on two spatial variables. We can then extend these relations into second order derivatives, then we will look for a family of ansätze that satisfy the following relations.

The membrane is closed and therefore all single-valued functions must be periodic on the coordinates ($\sigma,\rho$) of the torus. For simplicity we are assuming a rectangular 2-torus with the moduli described by the two radii ($R_{9},R_{10}$). For the $A_{r}(\sigma,\rho,\tau)$ scalar field, we impose the same type of dependence $\partial_{\sigma}A_{r}\propto\partial_{\rho}A_{r}$ at the level of the first derivatives. We impose the following relation of proportionality

The proportionality in the first derivatives imposes the following second derivatives relation

We can observe that the EOM is greatly simplified under these assumptions. For $Z_{c}$,

For $A_{r}$,

where $r\neq s$. The APD constraint (24) with the above ansatz (27) becomes:

and the topological constraint (25) remains invariant. In the following we propose two different forms to obtain a family of solutions.