Physics Letters B 776 (2018) 236–241 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Hairy AdS black holes with a toroidal horizon in 4D Einstein-nonlinear σ -model system ∗ Marco Astorino a, Fabrizio Canfora b, Alex Giacomini c, , Marcello Ortaggio d,c a Universidad Adolfo Ibanez, Viña del Mar, Chile b Centro de Estudios Científicos (CECS), Casilla 1469, Valdivia, Chile c Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Edificio Emilio Pugin, cuarto piso, Campus Isla Teja, Valdivia, Chile d Institute of Mathematics of the Czech Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic a r t i c l e i n f o a b s t r a c t Article history: An exact hairy asymptotically locally AdS black hole solution with a flat horizon in the Einstein-nonlinear Received 19 October 2017 sigma model system in (3+1) dimensions is constructed. The ansatz for the nonlinear SU(2) field is Received in revised form 16 November 2017 regular everywhere and depends explicitly on Killing coordinates, but in such a way that its energy– Accepted 21 November 2017 momentum tensor is compatible with a metric with Killing fields. The solution is characterized by a Available online 24 November 2017 discrete parameter which has neither topological nor Noether charge associated with it and therefore Editor: M. Cveticˇ represents a hair. A U (1) gauge field interacting with Einstein gravity can also be included. The thermodynamics is analyzed. Interestingly, the hairy black hole is always thermodynamically favoured with respect to the corresponding black hole with vanishing Pionic field. © 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction modynamic properties of solitons (see [22]). These are necessary ingredients to build Pionic black holes with non-trivial thermody- The nonlinear sigma model is a useful theoretical tool with namics. applications ranging from quantum field theory to statistical me- Following the strategy devised in those references, we will con- chanics systems like quantum magnetism, the quantum Hall effect, struct a class of analytic black hole solutions of the Einstein-- 3 meson interactions, super fluid He, and string theory [1]. The nonlinear sigma model system with intriguing geometrical prop- most relevant application of the SU(2) non-linear sigma model erties. This family of black holes possesses a flat horizon1 and a in particle physics is the description of the dynamics of Pions at discrete hairy parameter. The thermodynamics can be analysed ex- + low energy in 3 1 dimensions (see for instance [2]; for a detailed plicitly. The first law is satisfied and an interesting feature of these review [3]). Consequently, the analysis of the coupling of the non- black holes is that the hairy solution has always less free energy linear sigma model to General Relativity is extremely important than the corresponding vacuum solution: thus, the present analysis both from the theoretical and from the phenomenological point suggests that the coupling of Pions with the gravitational field can of view. On the other hand, due to the complexity of the field act as a sort of catalysis for the Pions themselves. This family of equations (which usually reduce to a non-linear system of coupled black holes with flat horizons can also be generalized to the case in PDEs), the Einstein-nonlinear sigma model system has been ana- which there is a U (1) gauge field coupled to Einstein gravity. The lyzed mostly relying on numerical analyses (classic references are interesting thermodynamical features of these black holes remain [4–8]). However, the recent generalization of the boson star ansatz to in this case as well. This is qualitatively similar (in a phenomeno- SU(2)-valued scalar fields (introduced in [9–19] and [20–22]) al- logical setting such as the Einstein–Pions system) to the recent lows to construct also non-trivial gravitating soliton solutions (see findings of [23] in which a family of asymptotically flat black holes in particular [10] and [16]) as well as to analyse explicitly ther- with non-Abelian hair has been presented which are thermody- namically favoured over the Reissner–Nordström solution. * Corresponding author. E-mail addresses: [email protected] (M. Astorino), [email protected] (F. Canfora), [email protected] (A. Giacomini), [email protected] 1 Black holes with planar horizons have recently attracted a lot of attention due (M. Ortaggio). to their applications in holography, see, e.g., [24–26]. https://doi.org/10.1016/j.physletb.2017.11.051 0370-2693/© 2017 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. M. Astorino et al. / Physics Letters B 776 (2018) 236–241 237 The paper is organized as follows. In section 2, the Einstein- Defining the quadratic combination Nonlinear σ -model is introduced and a convenient parametrization is described. The exact hairy black hole solutions are constructed S := i j =∇ 0∇ 0 +∇ i∇ i = ∇ i∇ j μν δijRμ Rν μY ν Y μY ν Y Gij(Y ) μY ν Y , in section 3. In section 4, the thermodynamic behaviour of the solutions is analyzed. In the last section 5, some conclusions are (11) drawn. with 2. The action Yi Y j Gij := δij + (12) 1 − Y k Y We consider the Einstein-non-linear sigma model system in k four dimensions, with a possible cosmological constant. This de- being the S3 metric of the target space,2 we obtain that the action scribes the low-energy dynamics of pions, whose degrees of free- (3) reads dom are encoded in an SU(2) group-valued scalar field U [3]. The √ action of the system is 4 1 i μ j SPions =−K d x −g Gij(∇μY )(∇ Y ) , (13) 2 S = SG + SPions, (1) while the energy–momentum tensor (6) takes the form where the gravitational action SG and the nonlinear sigma model action SPions are given by 1 Tμν = K Sμν − gμνS . (14) √ 2 1 4 SG = d x −g(R − 2), (2) 16π G The second of (9) means that Y I := (Y 0, Y i ) define a round unit √ K 4 μ 3-sphere in the internal space. A useful set of coordinates (H, A, G) SPions = d x −g Tr R Rμ . (3) 4 in the internal space is defined by Here we have defined Y 0 = cos H sin A, Y 1 = sin H cos G, = −1∇ ∈ Rμ U μU , U SU(2), (4) Y 3 = cos H cos A, Y 2 = sin H sin G, (15) while R is the Ricci scalar, G is Newton’s constant, is the cos- where H ∈[0, π/2], while A ∈ [0, 2πk ], G ∈ [0, 2πk ] are both mological constant and the parameter K is positive. In our con- 1 2 periodic Killing coordinates of S3, with k and k positive integers ventions c = h¯ = 1, the space–time signature is (−, +, +, +) and 1 2 (there is clearly some redundancy in this choice of the periodicities Greek indices run over space–time. In the case of the non-linear = = 3 sigma model on flat space–times, the coupling constant K is fixed – the standard choice k1 k2 1already covers the whole S – experimentally. On the other hand, its true meaning in the context but this will become physically meaningful later on). In [16] and of AdS physics is to introduce a new scale. Indeed, it is natural to [22] it has been shown that similar parametrizations are extremely expect this new scale will be a relevant quantity in the thermody- useful both in curved and in flat spaces. The tensor Sμν (11) takes namics of the black holes which will be analyzed in the following the form sections. Our results confirm this expectation (we thank the anony- mous referee for this comment). 2 2 Sμν = (∇μ H)(∇ν H)+cos H(∇μ A)(∇ν A)+sin H(∇μG)(∇ν G). The resulting Einstein equations are (16) Gμν + gμν = 8π GTμν, (5) If one defines the following combinations of the Killing coordi- where Gμν is the Einstein tensor and the energy–momentum ten- nates sor is = + = − K 1 α + G A, − G A, (17) Tμν =− Tr Rμ Rν − gμν R Rα . (6) 2 2 the field equations (7) can be written in the compact form3 For the nonlinear SU(2) sigma model, Tμν can be seen to satisfy 1 the dominant and strong energy conditions [27]. Finally, the matter H − sin(2H)∇ + ·∇ − = 0, (18) field equations are 2 sin(2H) + + 2∇ H · [∇ − + cos(2H)∇ +] = 0, (19) μ ∇ Rμ = 0. (7) sin(2H) − + 2∇ H · [∇ + + cos(2H)∇ −] = 0. (20) In the following, it will be useful to write Rμ as SU(2)-valued scalar fields may possess a non-trivial topolog- j ical charge which, mathematically, is a suitable homotopy class Rμ = iRμσ j, (8) or winding number W . Its explicit expression as an integral over where σ j are the Pauli matrices. Furthermore, we adopt the stan- a suitable three-dimensional hypersurface can be found, e.g., dard parametrization of the SU(2)-valued scalar U (xμ) in [1]. However, in the present paper we will consider only con- figurations with W = 0. ±1 μ 0 μ j μ 0 2 i U (x ) = Y (x )I ± iY (x )σ j , Y + Y Yi = 1 , (9) × where I is the 2 2identity matrix. From (8) one thus finds 2 For the special field configuration Y 0 = 0, eq.