1. Introduction
Recently, the black objects in high energy theories have been largely investigated from supergravity theories in arbitrary dimensional (D) space-times [1–5]. The corresponding physical properties of such objects have been dealt with including the thermodynamic and the optical ones. Concretely, black hole behaviors in type IIB superstring and M-theory have been studied using different methods [6–13]. In type IIB superstring, for instance, the thermodynamic and the optical properties of the black holes in the AdS5 × S5 background have been investigated using brane physics. Exploiting analytical and numerical methods, various thermodynamical behaviors have been approached and examined in arbitrary dimensions by varying the brane number. In particular, the Hawking-Page transition has been treated with and without dark energy sectors [10, 12]. These investigations have been extended to M-theory black holes with AdSp+2 × S11−p−2 near horizon geometries. In this way, the black holes can be obtained from Mp-branes, where p = 2 and p = 5 producing solutions in 4D and 7D, respectively. Supported by even horizon telescope (EHT) observational findings, the optics of such M-theory black holes have been studied by dealing with either the shadow or the deflection angle of light rays. Precisely, the black hole shadows in the spherical M-theory compactifications with M2 and M5 branes have been examined using the M-brane number variation [10]. Using one-dimensional real curves, the 4D black hole shadows have been studied where various geometric configurations have been obtained. In particular, the D-shapes and the cardioid shapes have been found for rotating black holes [12]. Inspired by such activities, models supported by M-theory gravities have been also approached [14, 15]. Precisely, the shadows and the deflection angle of the light rays near to the black holes in the Starobinsky Bel-Robinson gravity have been discussed by introducing a new stringy parameter β [16]. It has been shown that such a parameter modifies the thermodynamics and optics of the ordinary black holes [16–18].
More recently, supersymmetric compactifications have been exploited to approach the BPS and non BPS black objects from type II superstrings and M-theory [19]. These studies have been encouraged by the fact that supersymmetry is broken in lower dimensional realistic space-times. In 5D, the BPS and non BPS black objects obtained from M-theory on various Calabi–Yau (CY) three-folds, with Kahler moduli parameters, have been investigated by calculating the relevant quantities such as the entropy and the string tension using the 4D N = 2 supersymmetry formalism combined with the attractor mechanism [20–23]. To build certain BPS and non BPS states, M2 and M5 branes wrapping two and four-cycles in CY geometries using projective hypersurfaces with two Kahler parameters have been exploited using techniques developed in [19, 24]. In this way, the relevant quantities of the black holes and the black strings have been computed and analyzed to inspect the corresponding stability behaviors. Later soon, these works have been generalized to CY models with three Kahler parameters using numerical computations via a general treatment [25].
In this work, we would like to contribute to these activities by reconsidering the study of certain behaviors of 5D black objects from the compactification of M-theory on a CY three-fold (CY3) via the N = 2 supergravity formalism. First, we construct such a CY3 as the intersection of two homogeneous polynomials of degrees (ω + 2, 1) and (2, 1) in a product of two weighted projective spaces given by ${{\mathbb{WP}}}^{4}(\omega ,1,1,1,1)\times {{\mathbb{P}}}^{1}$. By determining the allowed electric charge regions of the BPS and non BPS black hole solutions by wrapping an M2-brane on appropriate two cycles in such a CY3, we calculate their entropies as a function of ω taking a maximal value associated with the ordinary projective space ${{\mathbb{P}}}^{4}$. For generic values of ω, we find that the non BPS states are unstable matching perfectly with the results of [19]. Then, we provide a similar study of 5D black string solutions. For the allowed magnetic charge regions of the BPS and non BPS black strings, we determine the tension involving a minimal value for ${{\mathbb{P}}}^{4}$. By computing the recombination factor, we show that the non-BPS black string states are stable in the allowed regions of the magnetic charge space.
The organization of this work is as follows. In section 2 , we review briefly 5D black objects from M-theory on CY three-folds. In section 3 , we study the stability behaviors of 5D BPS and non-BPS black holes from M2-branes. In section 4 , we investigate the 5D BPS and non-BPS black strings from M5-branes. In the last section, we present our concluding remarks.
2. 5D black objects in M-theory on Calabi–Yau manifolds
In this work, we reconsider the study of behaviors of 5D black objects derived from M-theory CY compactifications. Before going ahead, we present the relevant concepts which will be exploited in the present discussion of M-theory black object physical properties. To start, its is recalled that a CY three-fold is a compact Kahler complex geometry involving a Ricci-flat metric with a SU(3) holonomy group. The latter has been shown to be crucial in string theory revolutions and duality scenarios [26, 27]. In particular, the CY geometry has been explored to form a bridge between string theory in 10D and semi-realistic models in 4D with minimal supersymmetric charges being $\tfrac{1}{4}$ of the initial ones. It has been remarked that a CY three-fold has a Hodge diagram which is essential in the determination of the 4D and 5D models with N = 2 supersymmetry from Type II superstrings and M-theory, respectively. In connection with black hole building models, this manifold has been extensively used in the attractor mechanism [20–22].
At low energy limits, M-theory is modeled by a N = 1 supergravity in 11D [28–32]. Omitting the fermionic sector, it contains a metric gMN and an antisymmetric three-form field AMNP, where M, N, and P take values from zero to 10. Besides that, it involves two branes known as M2-branes and M5-branes. M-theory on CY three-folds lead to 5D N = 2 supergravity models with eight supercharges involving vector multiplets and hypermultiplets. The numbers of such multiplets are fixed by the geometric Hodge numbers h1,1 and h2,1 counting the size and the shape parameters, respectively. It has been remarked that the shape parameters are not relevant in the study of the black branes in M-theory CY compactifications by freezing hypermultiplet fields. In this way, the BPS and non BPS black states are obtained from M2 and M5-branes wrapping on two and four-cycles with h1,1 electric and magnetic charges, respectively. These solutions can be dealt with by means of a 5D N = 2 Maxwell-Einstein supergravity formalism via the following action
$\begin{eqnarray}\begin{array}{c}S=\frac{1}{2{\kappa }_{5}^{2}}\int {{\rm{d}}}^{5}x\left(R\star {\mathbb{1}}-{G}_{{IJ}}{\rm{d}}{t}^{I}\wedge \star {\rm{d}}{t}^{J}\right.\\ \,-\,\left.{G}_{{IJ}}{F}^{I}\wedge \star {F}^{J}-\frac{1}{6}{C}_{{IJK}}{F}^{I}\wedge {F}^{J}\wedge {A}^{K}\right),\end{array}\end{eqnarray}$
where I, J, and K take values from 1 to h1,1. R is the Ricci scalar and tI denote the size moduli associated with the Kahler forms ${{ \mathcal J }}_{I}$. The fields FI = dAI are Maxwell tensors corresponding to abelian vector multiplets which can be obtained from the AMNP reduction on the CY three-folds [33]. The symmetric tensor CIJK, which gives the triple intersection numbers, is a crucial geometric quantity providing the normalized volume of the CY three-fold via the relation $\begin{eqnarray}{ \mathcal V }=\displaystyle \frac{1}{3!}{C}_{{IJK}}{t}^{I}{t}^{J}{t}^{K}.\end{eqnarray}$
This quantity is linked to the CY moduli space metric via the relation $\begin{eqnarray}{G}^{{IJ}}=2\left(-{ \mathcal V }{A}^{{IJ}}+\displaystyle \frac{{t}^{I}{t}^{J}}{2}\right),\end{eqnarray}$
where AIJ denotes the matrix inverse of AIJ = CIJKtK. Roughly, the M-theory compactification on CY three-folds, in the presence of M2 and M5 branes wrapping on two and four-cycles, generates 5D electric and magnetic charged black objects, respectively. For 5D black holes, the effective potential is expressed as $\begin{eqnarray}{V}_{\mathrm{eff}}^{e}={G}^{{IJ}}{q}_{I}{q}_{J},\end{eqnarray}$
where qI are the corresponding electric charges. To inspect the stability behaviors, one needs to determine the potential critical points by solving the constraint $\begin{eqnarray}{D}_{I}{V}_{\mathrm{eff}}^{e}=0\qquad {D}_{I}={\partial }_{I}-\frac{2}{3{ \mathcal V }}{\tau }_{I},\end{eqnarray}$
where τI is given by $\begin{eqnarray}{\tau }_{I}=\displaystyle \frac{1}{2}{C}_{{IJK}}{t}^{J}{t}^{K},\end{eqnarray}$
describing the size of the Ith divisor ${{ \mathcal D }}_{I}$ in the CY three-fold. These divisors could be wrapped by the M5-brane to provide the 5D black strings.The critical points have been exploited to express certain thermodynamic quantities including the entropy by help of the attractor mechanism [20–22]. For the BPS states, the mass of black holes is given in terms of the attractor central charge value
$\begin{eqnarray}M={Z}_{e}{| }_{t={t}_{c}},\end{eqnarray}$
where tc indicates the critical value of t. For the non-BPS states, however, the mass is $\begin{eqnarray}M=\sqrt{\frac{3}{2}{V}_{\mathrm{eff}}{| }_{t={t}_{c}}},\end{eqnarray}$
leading to $\begin{eqnarray}{V}_{\mathrm{eff}}=\frac{2}{3}{M}^{2}.\end{eqnarray}$
Using the central charge Ze, one gets $\begin{eqnarray}S=2\pi {\left(\displaystyle \frac{{Z}_{e}{| }_{t={t}_{c}}}{3}\right)}^{3/2}.\end{eqnarray}$
The effective potential can be used to express the non BPS entropy given as follows $\begin{eqnarray}S=2\pi {\left(\frac{1}{6}{V}_{\mathrm{eff}}{| }_{t={t}_{c}}\right)}^{3/4}.\end{eqnarray}$
More details can be found in [19]. Similar techniques have been developed for the black string potential which reads as $\begin{eqnarray}{V}_{\mathrm{eff}}^{m}=4{G}_{{IJ}}{p}^{I}{p}^{J},\end{eqnarray}$
where pI are now the magnetic charges identified with the wrapping numbers of the M5-brane on ${{ \mathcal D }}_{I}$. In this way, the string tension T is given in terms of the square root of the magnetic effective potential calculated at the critical points obtained by solving the constraint ${D}_{I}{V}_{\mathrm{eff}}^{m}=0$. As suggested in [19], the stability of such 5D black objects has been discussed in terms of a ratio called the recombination factor denoted by R. The solutions are unstable for R > 1. In this situation, the black objects would prefer to decay into the BPS/non-BPS pairs. For R < 1, however, the black objects are stable enjoying the recombination of the brane/anti-brane behaviors.3. 5D black holes in M-theory CY compactifications
In this section, we study 5D black holes from a special CY3 with two Kahler parameters. The manifold, that we construct here, is considered as the intersection of two hypersurfaces in a product of two weighted projective spaces called the ambient geometry given by
$\begin{eqnarray}{ \mathcal A }={{\mathbb{WP}}}^{4}({\omega }_{1},{\omega }_{2},{\omega }_{3},{\omega }_{4},{\omega }_{5})\times {{\mathbb{WP}}}^{1}(\omega {{\prime} }_{1},\omega {{\prime} }_{2}),\end{eqnarray}$
where ωi and $\omega {{\prime} }_{i}$ are natural numbers. It is recalled that a n-dimensional weighted projective space ${{\mathbb{WP}}}^{n}$ can be used to extend the notion of the ordinary projective space ${{\mathbb{P}}}^{n}$. Indeed, ${{\mathbb{WP}}}^{n}$ is defined by considering the following identification $\begin{eqnarray}{z}_{i}\sim {\lambda }^{{\omega }_{i}}{z}_{i},\qquad i=1,\ldots ,n+1,\end{eqnarray}$
where (z1, …, zn+1) are the homogeneous coordinates. ωi are called the weights and λ is a non-zero complex number. Since the weights are not relevant for the one-dimensional complex projective space ${{\mathbb{P}}}^{1}$, one should consider the following ambient space $\begin{eqnarray}{ \mathcal A }={{\mathbb{WP}}}^{4}({\omega }_{1},{\omega }_{2},{\omega }_{3},{\omega }_{4},{\omega }_{5})\times {{\mathbb{P}}}^{1}.\end{eqnarray}$
In this way, CY3 is associated with a matrix configuration which takes the following form $\begin{eqnarray}\left[\begin{array}{c}{{\mathbb{WP}}}^{4}({\omega }_{1},{\omega }_{2},{\omega }_{3},{\omega }_{4},{\omega }_{5})\\ {{\mathbb{P}}}^{1}\end{array}\right|\left|\begin{array}{cc}{d}_{1}^{1} & {d}_{1}^{2}\\ {d}_{2}^{1} & {d}_{2}^{2}\end{array}\right],\end{eqnarray}$
such that $\begin{eqnarray}{d}_{1}^{1}+{d}_{1}^{2}={\omega }_{1}+{\omega }_{2}+{\omega }_{3}+{\omega }_{4}+{\omega }_{5}\end{eqnarray}$
$\begin{eqnarray}{d}_{2}^{1}+{d}_{2}^{2}=2,\end{eqnarray}$
needed to satisfy the CY condition. For such a CY matrix configuration, CY3 can be built via the intersection of two homogeneous polynomials of degrees $({d}_{1}^{1},{d}_{2}^{1})$ and $({d}_{1}^{2},{d}_{2}^{2})$ in ${{\mathbb{WP}}}^{4}({\omega }_{1},{\omega }_{2},{\omega }_{3},{\omega }_{4},{\omega }_{5})\times {{\mathbb{P}}}^{1}$. Instead of elaborating general and complex scenarios, we restrict ourselves to the effect of only one weight on the 5D black hole behaviors. The present simplicity could help one to stay focused on the weight effect. This can be exploited to identify the associated behavior variation. However, it can be hoped that the generalization could provide relevant findings. This is beyond the scope of the present work. Roughly, the weight effect should depend on the above matrix configuration form. Up to the CY condition, various choices could be dealt with. After an examination, certain choices generate a trivial weight dependance. However, other ones lead to a relevant dependance, where the weight could appear in several physical quantities. To see that, we consider the following CY3 matrix configuration $\begin{eqnarray}\left[\begin{array}{c}{{\mathbb{WP}}}^{4}(\omega )\\ {{\mathbb{P}}}^{1}\end{array}\right|\left|\begin{array}{cc}\omega +2 & 2\\ 1 & 1\end{array}\right],\end{eqnarray}$
where we have used $\begin{eqnarray}{\omega }_{1}=\omega ,\quad {\omega }_{2}={\omega }_{3}={\omega }_{4}={\omega }_{5}=1,\end{eqnarray}$
$\begin{eqnarray}{d}_{2}^{1}={d}_{2}^{2}=1,\quad {d}_{1}^{1}=\omega +2,\quad {d}_{1}^{2}=2.\end{eqnarray}$
In this situation, CY3 is the intersection of two homogeneous polynomials of degrees (ω+2,1) and (2,1) in ${{\mathbb{WP}}}^{4}(\omega ,1,1,1,1)\times {{\mathbb{P}}}^{1}$. In this CY3 configuration, the 5D black holes can obtained from an M2-brane wrapping a non trivial 2-cycle. The wrapping numbers provide two electric charges q1 and q2 which will be important in the present section. These two charges generate a relevant ratio $q=\displaystyle \frac{{q}_{1}}{{q}_{2}}$ which will be exploited to discuss the 5D black hole stability from the M-theory CY compactification. Before that, we should compute primordial geometric quantities. In particular, we calculate the normalized volume of CY3. This can be obtained by determining the intersection numbers CIJK. Using the method of [34, 35], we find $\begin{eqnarray}{C}_{111}=\omega +4,\qquad {C}_{112}=2(\omega +2).\end{eqnarray}$
This provides the normalized volume expression of the proposed CY3 given by $\begin{eqnarray}{ \mathcal V }=\displaystyle \frac{1}{6}{t}_{1}^{2}((\omega +4){t}_{1}+6(\omega +2){t}_{2}).\end{eqnarray}$
Exploiting the constraint ${ \mathcal V }=1$, we obtain the effective potential of the 5D black holes $\begin{eqnarray}\begin{array}{c}{V}_{\mathrm{eff}}^{e}={t}_{1}^{2}{q}_{1}^{2}+\frac{(4+\omega ){t}_{1}^{2}}{6(2+\omega )}{q}_{1}{q}_{2}\\ \ \ +\frac{(4+\omega ){t}_{1}^{2}+8(8+6\omega +{\omega }^{2}){t}_{1}{t}_{2}+24{\left(2+\omega \right)}^{2}{t}_{2}^{2}}{12{\left(2+\omega \right)}^{2}}{q}_{2}^{2},\end{array}\end{eqnarray}$
which is a quadratic polynomial in the electric charge space (q1, q2). To examine the 5D black hole behaviors derived from the proposed CY3, we need to solve the constraint ${D}_{I}{V}_{\mathrm{eff}}^{e}=0$. A calculation shows that we have the following algebraic equation $\begin{eqnarray}\begin{array}{l}\left(4(2+\omega ){q}_{2}-(2(2+\omega ){q}_{1}-(4+\omega ){q}_{2})x\right)\\ \ \ \ \times \left(12(2+\omega ){q}_{2}+(6(2+\omega ){q}_{1}+(4+\omega ){q}_{2})x\right)=0,\end{array}\end{eqnarray}$
where one has used a local variable $x=\tfrac{{t}_{1}}{{t}_{2}}.$ It has been remarked that there are two solutions which could be given in terms of the weight parameter ω. The latter can be considered as a relevant parameter in the present investigation. In what follows, the BPS and non-BPS solutions will be dealt with in terms of such a parameter. For the BPS solutions, we find $\begin{eqnarray}x=\displaystyle \frac{4(2+\omega ){q}_{2}}{2(2+\omega ){q}_{1}+(4+\omega ){q}_{2}}.\end{eqnarray}$
The positive values of the local variable x generate two allowed regions in the electric charge space which are $\begin{eqnarray}\begin{array}{c}\left\{{q}_{2}\lt 0\,\,{\rm{and}}\,\,{q}_{1}\lt \displaystyle \frac{\left(4+\omega ){q}_{2}\right.}{\left.2(2+\omega \right)}\right\},\\ \,\left\{{q}_{2}\gt 0\,\,{\rm{and}},\,{q}_{1}\gt \displaystyle \frac{\left(4+\omega ){q}_{2}\right.}{\left.2(2+\omega \right)}\right\}.\end{array}\end{eqnarray}$
However, the non-BPS black hole states correspond to the second solution $\begin{eqnarray}x=-\displaystyle \frac{12(2+\omega ){q}_{2}}{6(2+\omega ){q}_{1}+(4+\omega ){q}_{2}}.\end{eqnarray}$
This generates two allowed possible charge regions $\begin{eqnarray}\begin{array}{c}\left\{{q}_{2}\lt 0\,\,{\rm{and}}\,\,{q}_{1}\lt -\displaystyle \frac{\left(4+\omega ){q}_{2}\right.}{6(2+\omega ){q}_{1}+(4+\omega ){q}_{2}}\right\},\\ \,\left\{{q}_{2}\gt 0\,\,{\rm{and}}\,\,{q}_{1}\gt -\displaystyle \frac{\left(4+\omega ){q}_{2}\right.}{6(2+\omega ){q}_{1}+(4+\omega ){q}_{2}}\right\}.\end{array}\end{eqnarray}$
The electric charge regions of the BPS and non-BPS black holes by varying ω are illustrated in figure 1. The colored regions describe the existence of the large BPS and non-BPS black holes with non zero electric charges. It has been remarked that the size of such regions depends on ω. In the non colored regions, the black hole solutions are not allowed. The size of the regions which do not correspond to large black holes decreases with the weight ω.
Figure 1. Electric charge regions for black hole sates. |
The entropy of 5D black holes can be determined by considering the constraint ${ \mathcal V }=1$. The calculation gives
$\begin{eqnarray}{S}_{\mathrm{BPS}}=\frac{\pi }{6}\sqrt{\frac{{q}_{2}}{{\left(2+\omega \right)}^{3}}}\,(6{q}_{1}(2+\omega )-{q}_{2}(4+\omega )).\end{eqnarray}$
This computation is presented in figure 2.
Figure 2. BPS entropy behaviors in terms of ω. |
It follows from this figure that the entropy decreases with the weight ω. In fact, it starts from a maximal value corresponding to the ordinary projective space ${{\mathbb{P}}}^{4}$. This maximal value increases with the electric charge q1. For large values of ω, the entropy vanishes. Calculations give the expression of the entropy of the non-BPS black holes which reads as
$\begin{eqnarray}{S}_{\mathrm{non}-\mathrm{BPS}}=\frac{\pi }{6}\sqrt{\frac{|{q}_{2}|}{{(2+\omega )}^{3}}}\,|6{q}_{1}(2+\omega )-{q}_{2}(4+\omega )|\end{eqnarray}$
exhibiting similar behaviors with respect to the ω dependance. The relevant difference appears in the allowed electric charge regions of the associated black hole moduli space.The stability behaviors of the non-BPS black holes can be approached via the recombination factor R which is firstly introduced in [19]. It has been suggested that R is the ratio of the non-BPS black hole mass to the M2-brane mass wrapping the associated piecewise calibrated two-cycle. For R > 1, the non-BPS black hole is unstable. The non-BPS states would prefer to decay into the associated BPS and anti-BPS constituent states. For R < 1, however, the constituent BPS-anti-BPS pairs could recombine to provide stable non-BPS states in the black hole spectrum. Using the result of [19], we can determine the recombination factor R by considering the ratio of the non-BPS black hole mass Vc to the mass of M2-brane wrapping the associated piecewise calibrated two-cycle ${V}_{{c}^{\cup }}$. Exploiting the critical value tc, it can be expressed as follows
$\begin{eqnarray}R=\displaystyle \frac{{V}_{c}}{{V}_{{c}^{\cup }}}{\Space{0ex}{2.85ex}{0ex}| }_{t={t}_{c}},\end{eqnarray}$
where Vc represents the non-BPS black hole mass depending on the effective potential square via the relation ${V}_{c}=\sqrt{\tfrac{3}{2}{V}_{\mathrm{eff}}}$. After computations, we obtain $\begin{eqnarray}{V}_{c}=\displaystyle \frac{3{q}_{2}{t}_{2}(-6(\omega +2){q}_{1}+(\omega +4){q}_{2})}{6(\omega +2){q}_{1}+(\omega +){q}_{2}}.\end{eqnarray}$
The mass of an M2-brane wrapping the associated piecewise calibrated two-cycle ${V}_{{c}^{\cup }}$ reads as $\begin{eqnarray}{V}_{{c}^{\cup }}={t}_{1}| {q}_{1}| +t2| {q}_{2}| ={t}_{2}(| {q}_{1}| x+| {q}_{2}| ).\end{eqnarray}$
Using the critical solution of the non BPS objects and the first allowed charge region, this quantity is found to be $\begin{eqnarray}{V}_{{c}^{\cup }}={t}_{2}{q}_{2}\displaystyle \frac{-18(\omega +2){q}_{1}-(\omega +4){q}_{2}}{6{q}_{1}(\omega +2)+{q}_{2}(\omega +4)}.\end{eqnarray}$
Combining these relations, we get the recombination factor $\begin{eqnarray}R=1-\displaystyle \frac{4(\omega +4)}{18(\omega +2)q+(\omega +4)},\end{eqnarray}$
where one has used the electric charge ratio $q=\tfrac{{q}_{1}}{{q}_{2}}$ for mixed signs of q1 and q2. A close examination shows that this ratio is constrained by $-\tfrac{\omega +4}{6(\omega +2)}\lt q\lt -\tfrac{\omega +4}{18(\omega +2)}$. For this range, we illustrate the black hole recombination factor in figure 3.
Figure 3. Recombination factor of the non-BPS black hole states by varying ω. |
It follows from this figure that, for generic values of ω with a negative charge ratio q as required by the above regions, the non-BPS black hole states are unstable preferring to decay into the corresponding BPS and anti-BPS brane objects. This matches perfectly with the recent results suggesting that all the non-BPS black hole states are unstable [19].
4. Black string behaviors in M-theory on CY three-folds
Here, we consider the 5D black string behaviors in the proposed CY3. These black branes are obtained by wrapping a dual M5-brane on a generic divisor ${ \mathcal D }$ which provide the BPS and non-BPS states with two magnetic charges p1 and p2. For such a CY3, we find that the effective potential can be expressed as
$\begin{eqnarray}{V}_{\mathrm{eff}}^{m}={v}_{11}{p}_{1}^{2}+{v}_{12}{p}_{1}{p}_{2}+{v}_{22}{p}_{2}^{2}\end{eqnarray}$
where one has $\begin{eqnarray}\begin{array}{rcl}{v}_{11} & = & \displaystyle \frac{1}{6}{t}_{1}^{2}\left(24{\left(2+\omega \right)}^{2}{t}_{2}^{2}+8{t}_{1}{t}_{2}(8+6\omega +{\omega }^{2})\right.\\ & & \left.+\,{t}_{1}^{2}{\left(4+\omega \right)}^{2},),,\right)\\ {v}_{12} & = & \displaystyle \frac{2}{3}{t}_{1}^{4}(2+\omega )(4+\omega ),\\ {v}_{22} & = & 2{t}_{1}^{4}{\left(2+\omega \right)}^{2}.\end{array}\end{eqnarray}$
To examine the 5D black string behaviors, the constraint ${D}_{I}{V}_{\mathrm{eff}}^{m}=0$ should be solved. This leads to $\begin{eqnarray}({p}_{1}-{{xp}}_{2})\left(3{p}_{1}(2+\omega )\right.\left.\,+\,x\left({p}_{1}(4+\omega )+3{p}_{2}(2+\omega )\right)\right)=0,\end{eqnarray}$
where one has used the local variable $x=\displaystyle \frac{{t}_{1}}{{t}_{2}}$. For the BPS black string states, the allowed magnetic charges corresponding to the positive values of such a local variable is required by $\begin{eqnarray}x=\displaystyle \frac{{p}_{1}}{{p}_{2}}.\end{eqnarray}$
For these BPS stringy solutions, we have two possible regions in the magnetic charge space $\begin{eqnarray}\left\{{p}_{1}\lt 0\,\,{\rm{and}}\,\,{p}_{2}\lt 0\right\},\,\,\left\{{p}_{1}\gt 0\,\,{\rm{and}}\,\,{p}_{2}\gt 0\right\}.\end{eqnarray}$
The non-BPS black string states are associated with the second solution of the above algebraic equation in the magnetic charge space $\begin{eqnarray}x=-\displaystyle \frac{3(2+\omega ){p}_{2}}{(4+\omega ){p}_{1}+3(2+\omega ){p}_{2}}.\end{eqnarray}$
Similarly, the positive values of this local variable of the CY moduli space provide two possible regions given by $\begin{eqnarray}\begin{array}{l}\left\{{p}_{2}\lt 0\,\,{\rm{and}}\,\,{p}_{1}\lt -\displaystyle \frac{3(2+\omega ){p}_{2}}{4+\omega }\right\},\\ \qquad \left\{{p}_{2}\gt 0\,\,{\rm{and}}\,\,{p}_{1}\gt -\displaystyle \frac{3(2+\omega ){p}_{2}}{4+\omega }\right\}.\end{array}\end{eqnarray}$
To see the corresponding behaviors, the allowed magnetic charge regions for the BPS and non-BPS black strings are depicted in figure 4.
Figure 4. Allowed charge regions of BPS and non-BPS black strings by varying ω. |
It has been observed from this figure that the allowed regions of the non-BPS black strings depends slightly on ω. Precisely, it has been remarked that the size regions increases with ω. There are also some regions which are not associated with black strings. Their sizes decrease with ω. For the BPS solutions, the tension of the string is found to be
$\begin{eqnarray}{T}_{\mathrm{BPS}}={\left(\sqrt{6}{p}_{1}^{2}((4+\omega ){p}_{1}+6(2+\omega ){p}_{2})\right)}^{\tfrac{1}{3}}.\end{eqnarray}$
The variation of this function is illustrated in figure 5.
Figure 5. BPS brane tension by varying ω. |
It has been remarked that TBPS augments with ω starting from a minimal value associated with ${{\mathbb{P}}}^{4}$. Fixing ω, TBPS increases with ∣p1∣. Similar calculations can provide the expression of the BPS black string tension. Indeed, it is given by
$\begin{eqnarray}{T}_{\mathrm{Non}-\mathrm{BPS}}={\left(| \sqrt{6}{p}_{1}^{2}((4+\omega ){p}_{1}+6(2+\omega ){p}_{2})| \right)}^{\tfrac{1}{3}}.\end{eqnarray}$
To examine the stability behaviors, we compute the recombination factor R for such black strings. The latter is given by the ratio of the black string tension to that of the minimal size of the associated piecewise calibrated divisor. In terms of a magnetic charge ratio $p=\tfrac{{p}_{1}}{{p}_{2}}$, we can calculate such a recombination factor R. This can be determined by the ratio of the non-BPS black string tension T to the volume ${V}_{{D}^{\cup }}$ being the minimum volume piecewise calibrated representative of the class [D] $\begin{eqnarray}D={p}_{1}{{ \mathcal J }}_{1}+{p}_{2}{{ \mathcal J }}_{2},\end{eqnarray}$
where ${{ \mathcal J }}_{1}$ and ${{ \mathcal J }}_{2}$ denote the Kahler forms of ${{\mathbb{WP}}}^{4}(\omega ,1,1,1,1)$ and ${{\mathbb{P}}}^{1}$, respectively. In this way, the recombination factor reads as $\begin{eqnarray}R=\displaystyle \frac{T}{{V}_{{D}^{\cup }}},\end{eqnarray}$
where one has used ${V}_{{D}^{\cup }}={A}_{1}| {p}_{1}| +{A}_{1}| {p}_{2}| $ and AI = CIJKtJtK = 2τI describing the size of the divisor in CY3. Concretely, we get $\begin{eqnarray}R=\sqrt{\displaystyle \frac{3}{2}}\left(\displaystyle \frac{(4+\omega ){p}^{2}+6(2+\omega )p}{6(2+\omega ){p}^{2}+(4+\omega )p+12(2+\omega )}\right),\end{eqnarray}$
where the charge ratio p is constrained by $-\tfrac{3(2+\omega )}{4+\omega }\lt p\lt 0$ as required by the allowed magnetic charge regions. For the proposed CY3, the possible range of the magnetic charges could be extended to −3 < p < 0 by considering large weight values. In this range, the recombination factor of the non-BPS black string states is illustrated in figure 6 by varying ω.
Figure 6. Recombination factor of the non-BPS black string states as a function of ∣p∣ by varying ω. |
For generic values of ω, it follows from this figure that R < 1. This shows that the non-BPS black string states are stable in the allowed magnetic charge regions. In this way, they enjoy the recombination of the brane/anti-brane behaviors.
5. Conclusions and discussions
Using N = 2 supergravity formalism, we have investigated certain physical behaviors of 5D black objects via the compactification of M-theory on a special Calabi–Yau three-fold. We have first built such a manifold using the techniques of the projective spaces. In particular, the manifold has been considered as the intersection of two homogeneous polynomials of degrees (ω + 2, 1) and (2, 1) in a product of weighted projective spaces given by ${{\mathbb{WP}}}^{4}(\omega ,1,1,1,1)\times {{\mathbb{P}}}^{1}$ encoded in a matrix CY configuration given by equation (3.7 ). The critical points obtained from the effective potential have been used to identify the allowed electric charge regions of the 5D BPS and non BPS black hole solutions. These region states have been derived by wrapping an M2-brane on appropriate two cycles in such a CY manifold by varying the weight ω. Then, we have calculated the entropy of the obtained solutions taking a maximal value corresponding to the ordinary projective space ${{\mathbb{P}}}^{4}$. For generic values of ω, we have shown that the non BPS states, associated with the allowed electric charge regions, are unstable by computing the black hole recombination factor. This matches perfectly with the previous works suggesting that all non BPS black holes are unstable [19]. Finally, we have elaborated a similar study for black stringy solutions. For the allowed magnetic charge regions of the BPS and non BPS black strings, we have determined the tension as a function of ω with a minimal value for ${{\mathbb{P}}}^{4}$. By computing the stringy recombination factor, we have revealed that the non-BPS black string states are stable in the allowed magnetic charge regions.
This work comes up with certain open questions. It could be possible to consider other geometries with non trivial holonomy groups. Other CY technologies could be exploited to unveil extra data associated with such activities by considering more than one weight. Mirror symmetry could find a place in such black hole studies. Motivated by our recent works on optical behaviors of black holes, the shadow and the deflection angle of light near to the obtained solutions could be approached using the developed techniques in arbitrary dimensions. We hope come back to such questions in future investigations.