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Black Brane Evaporation Through D-Brane Bubble Nucleation

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Black Brane Evaporation Through D-Brane Bubble Nucleation HIP-2021-20/TH Black brane evaporation through D-brane bubble nucleation Oscar Henriksson∗ Department of Physics and Helsinki Institute of Physics P.O. Box 64, FI-00014 University of Helsinki, Finland Abstract We study the process of black brane evaporation through the emission of D-branes. Gravi- tational solutions describing black branes in asymptotically anti-de Sitter spacetimes, which are holographically dual to field theory states at finite temperature and density, have previously been found to exhibit an instability due to brane nucleation. Working in the setting of D3-branes on the conifold, we construct static Euclidean solutions describing this nucleation to leading order | D3-branes bubbling off the horizon. Furthermore, we analyze the late-time dynamics of such a D3-brane bubble as it expands and find a steady-state solution including the wall profile and its speed. CONTENTS I. Introduction 2 II. Gravity solutions and field theory dual 3 III. The D3-brane effective action 4 IV. Bubble nucleation 6 arXiv:2106.13254v1 [hep-th] 24 Jun 2021 V. Late time expansion 8 VI. Discussion 11 Acknowledgments 12 References 12 ∗ oscar.henriksson@helsinki.fi 1 I. INTRODUCTION Black holes are unstable to evaporation due to Hawking radiation. Though this fact is on firm theoretical footing, there are still many open questions regarding the precise details of this evaporation, some of which may require a complete theory of quantum gravity to settle. String theory is a strong candidate for such a theory. Besides strings, this framework also introduces other extended objects known as branes. These dynamical objects also gravitate, and a large number of them can be described in the supergravity limit of string theory as a black brane. In analogy with Hawking radiation, one can ask if such black branes can be unstable to emission of the individual branes from which they are formed. This has indeed been found to occur [1{6], for example by computing the effective potential for a brane probe and finding a global minimum outside the horizon. In these cases the horizon and the true minimum are typically separated by a potential barrier; thus, the emission must proceed through bubble nucleation [7, 8]. It is interesting to understand such brane nucleation in more detail, both to better understand the decay of black branes in string theory, and when a holographic field theory dual exists, to better understand the dynamics of bubble nucleation at strong coupling. Here, we study a specific example of such a nucleation event; a black brane, built from a stack of D3-branes at a conifold singularity, which emits a D3-brane. We focus on two different aspects; the initial nucleation of the brane as a localized bubble stretching outward from the horizon, and the late-time steady-state expansion of this brane into the bulk. We do this by first deriving an effective action describing the embedding of the D3-brane in the black brane geometry. Then, we use this to find static configurations with O(3) symmetry, whose Euclidean action gives the leading semi-classical contribution to the nucleation rate. Next, assuming a steady-state solution, we solve for the late-time expansion of the now approximately planar brane. In particular this lets us obtain the terminal velocity and profile of the brane. Throughout, we use the probe approximation, neglecting the backreaction of the nucleating D3-brane on the black brane background. Since we work in the asymptotically anti-de Sitter (AdS) near-horizon limit of the black brane geometry, we can use the AdS/CFT correspondence [9] to learn about the dual Klebanov-Witten (KW) gauge theory [10]. In particular, the black brane solutions de- scribe KW theory at finite temperature and density, and the D3-brane emission is dual 2 to the spontaneous breaking of the gauge group through the condensation of a scalar field corresponding to the radial position of the brane. Had the minimum of the brane's effec- tive potential been at a finite radius (as is the case in some very similar setups [5, 6]) the nucleation would have resulted in a phase transition to a (metastable) finite-density Higgs phase, analogous to color superconductivity in QCD [11]. For our D3-branes the minimum is instead located at the AdS boundary; this seems to indicate a fatal instability of the finite density KW theory, with the dual black brane slowly disappearing as one D-brane after the other nucleates and travels towards infinity (at least until the combined backreaction of the emitted branes can no longer be ignored). II. GRAVITY SOLUTIONS AND FIELD THEORY DUAL 1;1 We study asymptotically AdS5 ×T black brane solutions of type IIB supergravity found by Herzog et al. [4]. Defining forms with legs on T 1;1, 1 !2 ≡ (sin θ1dθ1 ^ dφ1 − sin θ2dθ2 ^ dφ2) 2 (1) g5 ≡ d + cos θ1dφ1 + cos θ2dφ2 ; the ansatz for the 10D metric is η η −4η 2 2 − 5 χ 2 2 χ e 2 2 2 e 2 2 2 e 2 ds = L e 3 ds + L e (dθ + sin θ dφ ) + (dθ + sin θ dφ ) + g ; (2) 10 5 6 1 1 1 6 2 2 2 9 5 2 with ds5 an asymptotically AdS5 line element, and L the corresponding AdS radius. Mean- 4 while, the self-dual 5-form field strength takes the form F5 = L (F + ∗F), with 2 1 F = − !2 ^ !2 ^ g5 − p dA ^ g5 ^ !2 ; 27 9 2 2η− 4 χ (3) − 20 χ e 3 ∗F = − 4e 3 volM − p (∗5dA) ^ !2 : 3 2 Here ∗5 denotes the Hodge dual with respect to ds5, and A is a U(1) gauge field. This ansatz provides a consistent truncation of type IIB supergravity to a 5D theory containing the metric, the gauge field A and two scalar fields χ and η. The resulting equations of motion have asymptotically AdS5 charged black brane solutions; their 5D metric can be written in the form dr2 ds2 = −g(r)e−w(r)dt2 + + r2d~x2 ; (4) 5 g(r) 3 3 with g(r) going to zero at the horizon radius r = rH . These solutions were constructed numerically and studied in [4, 6], and we refer the interested reader there for more details. 1;1 String theory on asymptotically AdS5 × T is holographically dual to the KW theory [10], and the supergravity limit we study describes it in the large-N, strong coupling limit. KW theory is an N = 1 superconformal theory with gauge group SU(N) × SU(N) and ¯ ¯ matter fields Aα and Bα~ (α; α~ = 1; 2) in the (N; N) and (N;N) representation, respectively. It provides the low-energy description of N D3-branes placed on the tip of the cone with base T 1;1 (the conifold). Among the global symmetries a certain U(1) factor is often referred to as \baryonic" since the only gauge invariant operators charged under it are heavy, having conformal dimensions of order N. The corresponding conserved current is dual to the U(1) gauge field in the gravity truncation above. Thus, the black branes we study are dual to states of KW theory at finite temperature and baryonic chemical potential, computed by standard methods as w(r ) − H 0 e 2 g (r ) T = H and µ = Φ(r ! 1) : (5) 4π Since the theory is conformal, the physics depends only on the dimensionless parameter T/µ. The lightest baryonic operators in KW theory involve determinants of the bifundamental matter fields, which are dual to D3-branes wrapped on a 3-cycle on T 1;1 [12]. As the effective potential for these wrapped D3's does not show an instability [4], many of the usual ways a charged black brane can become unstable near extremality are ruled out. The only known instability is instead due to the emission of D3-branes parallel to the original stack, which is what we study in the following. III. THE D3-BRANE EFFECTIVE ACTION We imagine that one of the D3-branes that build up the black brane localizes somewhere in the bulk. Such a D3-brane acts as a domain wall, which in the dual KW theory effectively changes the rank of the two SU(N) gauge groups by one [12]. Thus, if the energy can be lowered by separating a D3-brane from the black brane, the gauge symmetry in KW theory is spontaneously broken. 4 The action of a D3-brane is the sum of a DBI and a WZ term [13], Z Z 4 p SD3 = −T3 d ξ − det P [G] + T3 P [C4] ; (6) where P [::: ] denotes the pullback of spacetime fields to the brane worldvolume. The dilaton is constant and has been absorbed into the tension of the brane, which is given by 1 27 N T3 = 3 4 = 2 4 ; (7) (2π) gsls 32π L where we used the relationship for the quantization of 5-form flux on the conifold background [4]. Letting the brane extend in the directions parallel to the horizon, we parameterize its worldvolume by the spacetime coordinates ft; ~xg. The brane will be located on a constant and arbitrary point on T 1;1, while the embedding in the radial direction is taken to be a general function R = R(t; ~x). In the following, we denote derivatives with respect to t by a dot and with respect to xi by @i, i = 1; 2; 3, while the components of the 10D metric will be denoted by Gµν. Taking the derivative of the brane embedding function X with respect to t gives the vector _ _ _ 2 _ 2 X = @t + R@r ! X = Gtt + GrrR ; (8) while the derivative with respect to the spatial directions gives 2 2 @iX = @i + (@iR)@r ! (@iX ) = Gii + Grr(@iR) : (9) We can also form the contractions _ _ X· @iX = GrrR(@iR) and @iX· @jX = Grr(@iR)(@jR) : (10) The induced line element on the brane worldvolume can then be written as 2 _ 2 2 X _ X ds4 = X dt + 2 X· @iX dt dxi + (@iX ) · (@jX )dxidxj ; (11) i i;j and the DBI term in (6) becomes v u ! Grr Grr p u 3 _ 2 X 2 − det g4 = t−GttGxx 1 + R + (@iR) Gtt Gxx i (12) s w(R) 4 3 − 1 w(R)− 10 χ(R) e 1 X = L R e 2 3 g(R) − R_ 2 + (@ R)2 : g(R) R2 i i 5 For the WZ term we need the pullback of the 4-form potential defined by F5 = dC4, the relevant component of which is 4 L a4(r) dt ^ dx1 ^ dx2 ^ dx3 ⊂ C4 ; (13) where a4(r) is found by integrating w 20χ 0 3 − 2 − 3 a4(r) = 4r e (14) with the condition that it goes to zero on the horizon.
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