Thermal Phase in Bubbling Geometries

Transcription

1 Thermal Phase in Bubbling Geometries arxiv: v1 [hep-th] 10 Jul 2007 Chang-Yong Liu Institute of Theoretical Physics, Academia Sinica, Beijing , China Abstract We use matrix model to study thermal phase in bubbling half-bps type IIB geometries with SO(4) SO(4) symmetry. In near horizon limit, we find that thermal vacuum of bubbling geometries have disjoint parts, and each part is one phase of thermal system. We connect the thermal dynamics of bubbling geometries with one dimension fermions thermal system. Finally, we try to give a new possible way to resolve information loss puzzle. lcy@itp.ac.cn 1

2 1 Introduction AdS/CFT [1, 2, 3] correspondence relates N =4 SU(N) Supersymmetric Yang-Mills(SYM) theory to type IIB string theory on the AdS 5 S 5 background. In N = 4 Super Yang-Mills theory, the important sector of half-bps operators with conformal dimension equal to the U(1) R charge have concrete realization of this map. These operators form a decoupled sector of N = 4 Super Yang-Mills which can be efficiently described by a gauged quantum mechanics matrix model with harmonic oscillator potential. The matrix model is completely integrable, see [4]. In [5] Lin, Lunin and Maldacena(LLM) explicitly constructed smooth (and without horizon) type IIB supergravity backgrounds holographically dual to chiral primary operators of N = 4 SYM. In[10], Hawking and Page studied the thermodynamics of four dimensional asymptotically AdS spaces with compact boundary S 2 S 1. Later Witten [9] used the AdS/CFT correspondence to relate the thermodynamics of gauge theories to the thermodynamics of asymptotically anti-de Sitter (AdS) spaces. In a quantum gravity theory the path integral should involve contributions from all spaces with a fixed asymptotic boundary. Multiple classical configurations are possible because of the general feature of boundary value problems in differential equations: there can be multiple solutions to the classical equations satisfying the same asymptotic boundary conditions. Using this approach, Hawking and Page found two different solutions of the Einstein equations: the AdS and the black hole AdS spaces. They considered a phase transition at some critical temperature. Above this temperature the black hole solution is thermodynamically preferred. For lower temperatures the (thermal) AdS solution is dominant. In LLM, the fermions discussed are characterized by having a step-function distribution in the two-dimensional phase space. In this picture, the data defining the geometry is captured by a distribution of incompressible droplets of fermions on a 1+1 dimensional phase space. Since the effective of the fermions is related to the Planck length of type IIB supergravity via =2πl 4 p, the semiclassical limit of the Fermi system in which it makes sense to talk about the geometry of the Fermi surface in phase space corresponds to the semiclassical limit of the type IIB geometry. They can be seen therefore as fermions at zero temperature. It is then natural to investigate how turning on the temperature affects the supergravity solution. The fermion at non zero temperature are described by a Fermi- Dirac distribution. The corresponding AdS bubbling solution is hyperstar[7, 8]. This supergravity background can be thought of as resulting from a coarse graining process of smooth half-bps geometries. 1

3 So far the matrix model used to study LLM geometries is harmonic oscillator potential. We consider more general matrix potential to study thermal phase. In near horizon limit, we find that thermal vacuum of bubbling geometries have disjoint parts, and each part is one of thermal phase. We connect the thermal dynamics of bubbling geometries with one dimension fermions thermal system. The paper is organized as follows. In section 2, we review the LLM geometries. In section 3, we study giant gravitons in AdS 5 S 5 in order to understand the bubbling geometries. In section 4, complex matrix model and vacuum structure have be given. In section 5, we carefully study the thermal dynamics aspects of bubbling geometries. 2 Review the LLM Geometries In this section, we review the LLM description of the AdS 5 S 5 geometry and the haf-bps fluctuation modes. These modes is giant gravitons in AdS 5 S 5. All half-bps geometries of type IIB supergravity which preserve the SO(4) SO(4) R symmetry are obtained by LLM [5]. These half-bps geometries are given by ds 2 = h 2 (dt + V i dx i ) 2 + h 2 (dy 2 + dx i dx i ) + ye G dω ye G d Ω 2 3 (2.1) h 2 = 2ycosh(G), y y V i = ε ij j z (2.2) y( i V j j V i ) = ε ij y z (2.3) z = 1 tanhg 2 (2.4) F = db t (dt + V ) + B t dv + d ˆB (2.5) F = d B t (dt + V ) + B t dv + dˆ B (2.6) B t = 1 4 y2 e 2G, B t = 1 4 y2 e 2G (2.7) d ˆB = 1 4 y3 3 d( z y 2 ), dˆ B = 1 4 y3 3 d( z 1 2 y 2 ) (2.8) where i = 1, 2 and 3 is the flat space epsilon symbol in the three dimensions parameterized by y, x 1, x 2. We see that the full solution is determined in terms of a single function z. This function obeys the linear equation. i i z + y y ( yz y ) = 0 (2.9) Since the product of the radii of the two 3-spheres is y, we would have singularities at y = 0 unless z has a special behavior. It turns out that the solution is non-singular as 2

4 long as z = ± 1 2 on the y = 0 plane spanned by x 1, x 2. These two signs corresponds to the fermions and the holes, and the x 1, x 2 plane corresponds to the phase space. In [11], the author give a precise map between the complex matrix modle in the SYM side and the droplet configurations in the bubbling AdS geometries in the literature. The quantization condition on the total area A of the droplets is related to the fiveform flux N as follows: with: A 2π = N (2.10) = 2πl 4 p (2.11) The flux N coincides with the number of fermions. For better understood LLM geometries, we consider the giant gravitons in AdS 5 S 5. 3 Giant Gravitons in AdS 5 S 5 Giant gravitons(gg) in AdS 5 S 5 have been identified with a particular class of half-bps operators of N = 4 SYM theory[4, 6]. N = 4 SYM theory in four dimensions includes a vector field A µ, four complex Weyl fermions λ αa, and six real scalars φ I (where I is an index in the 6 of SU(4) R ). For convention, the scalars are written as three complex scalars Φ I = 1 2 (φ I + iφ I+3 ), with I = 1, 2, 3, and all the fields transform in the adjoint representation of U(N). Giant gravitons are identified with Schur polynomial in Φ I, and Schur polynomial are defined as χ (n,r) (Φ) = 1 n! σ S n χ R (σ)φ i1 σ(i1 ) Φin σ(i n) (3.1) Where i is the U(N) indices, taking values from 1 to N. The totally symmetric representation U of the symmetric group S n correspond to a giant gravitons in AdS 5 and the totally antisymmetric representation U of the symmetric group S n correspond to a giant gravitons in S 5. U representations of S n have the associated Young diagrams: one row of boxes. U have the associated Young diagrams which is a column with m boxes, and m N but of order N. In CFT, the half-bps giant gravitons operators with = J are described by the gauged matrix model with action S 0 = 1 2 dttr{(d t Φ) 2 Φ 2 } (3.2) 3

5 with Φ a hermitian matrix. The natural ground state of the fermions in this matrix model is the circular droplet that corresponds to AdS 5 S 5 on the Type IIB side. At finite temperature, the only relevant part of the gauge field A is the time-component, and we consider only U(N)-invariant states. Gauge invariance requires that the effective action must be expressed in terms of products of trφ n with n an integer. Φ is the zero mode of Polyakov loop on S 3 [15]. N= 4 SYM partition function on S 3 at a temperature T can be reduced to an integral over a unitary U(N) matrix Φ, Z(λ, T) = dφe S eff(φ) (3.3) and β Φ = Pexp(i Adτ) (3.4) 0 Because global gauge transformations which are periodic in the Euclidean time direction up to Z N factors, S eff (Φ) has a Z N symmetry: A generic term in S eff (Φ) will have the form Φ e 2πi N Φ (3.5) trφ n 1 trφ n2 trφ n k, n 1 + n n k = 0(mod N) (3.6) These terms is the deformations of N= 4 SYM matrix model. These terms can make the gauge symmetry breaking. U(N) n U(N i ) (3.7) Young diagrams associated giant gravitons also break. This is a kind of melting Young diagrams. Schur polynomial have factorization form. i=1 χ (N,R) (Φ) = χ (N1,R 1 )(Φ)χ (N2,R 2 )(Φ) χ (Nn,R n)(φ) (3.8) Where R represent the symmetry group is n i=1 U(N i), and R i represent the symmetry group is U(N i ). So the Hilbert space of giant gravitons must be: of H = n H i (3.9) i=1 4

6 From here, we say that the thermal phase have many parts. All this is obtained just from SY M side. In gravity side, when we put N D 3 together, the supergravity have the form ds 2 = f 1/2 ( dt 2 + dx dx dx 2 3) + f 1/2 (dr 2 + r 2 dω 2 5), (3.10) In the near horizon region, the geometry becomes: F 5 = (1 + )dtdx 1 dx 2 dx 3 df 1, (3.11) f = 1 + R4 r, 4 (3.12) R 4 4πg s α 2 N (3.13) ds 2 = r2 R 2( dt2 + dx dx dx 2 3) + R 2dr2 r 2 + R2 dω 2 5 (3.14) This is the geometry of AdS 5 S 5. When we put N 1 D 3 branes together, N 2 D 3 branes together, N n D 3 branes together, it s have multi-center solutions. In the near horizon limit, this geometry gives a disjoint sum of n copy of AdS 5 S 5 geometries. AdS 1 5 S 5 AdS 2 5 S 5 AdS n 5 S 5 (3.15) Where AdS i 5 S5 is the AdS 5 S 5 space that be produced by N i D 3 branes. So in the gravity side, we also have the factorization form. We take the Hilbert space H i associate with AdS i 5 S5. In next section, we study this carefully. 4 Complex matrix Model and Vacuum structure(thermal Phase) In this section, we take the thermal matrix model (3.3)-(3.6) just the deformations of N= 4 SYM and don t consider its thermal dynamics. We can receive a general property of thermal phase. We take the a general matrix tree potential to be: W tree (Φ) = n k=0 g k k + 1 Φk+1 (4.1) Where the gauge group is U(N) and the matrix integral is : Z = dφexp( 1 TrW(Φ)) (4.2) g s 5

7 The classical vacuum structure is determined by the extrema of W tree (z), seen as a holomorphic function of a complex variable. If we assume that the n extrema are all distinct: W tree(z) = g n n (z a i ), a i a j for i j, (4.3) i=1 then in the classical vacuum the eigenvalues of Φ will partition into groups corresponding to every extremum and the fluctuations around these vacua will all be massive. If we have N i eigenvalues equal to a i, the classical gauge symmetry is spontaneously broken in the following way: where U(N) N = n U(N i ) (4.4) i=1 n N i (4.5) i=1 The N i is also fiveform flux in bubbling geometries. This is viewed as N i D 3 branes at a i point. At each point, we have AdS/CFT correspondence. In [13] also see [14], the matrix model is described by a complex curve : y 2 m = W (x) 2 + f n 1 (x) (4.6) that defines a hyperelliptic Riemann surface, where f depends on the gauge coupling and the N i. If we assume that k = n, is a double cover of the x-plane. With W being of degree n, the right hand side of (4.6) has 2n zeroes; the projection of to the complex x-plane is branched at these 2n points. We need N i and N j D 3 branes separated a large distance. The vacuum structure is that the polynomial W has n roots a i and small f. A pair of roots a ± i are near each a i in the vacuum polynomial W (x) 2 + f(x), we connect the a ± i with branch cuts, so that y m is a single-valued function away from these cuts. The one-form ydx is directly related to the eigenvalue density. The solution of the matrix model then takes the form of a set of period integrals of the meromorphic one-form ydx S i = 1 F 0 ydx, = ydx (4.7) 2πi A i S i B i Where a cycle A i surrounding the i th cut and A i, B i are canonically conjugated cycles on the Riemann surface(4.6). In LLM, the quantization condition on the total area A of the droplets is related to the fiveform flux N i, the gauge symmetry spontaneously broken is related to the droplets broken to several parts. 6

8 At above vacuum, the gauge symmetry is beaking. Like section 3 analysis, we say that Hilbert space and gravity have factorization form. In each a i, we have AdS/CFT correspondence. So at each a i, we have gauged matrix harmonic oscillator potential. Near each critical points the tree potential can be taken to be harmonic potential or inverted harmonic potential. That is tree level potential. When we take quantum effects into consider, the potential can be deformed. At the other hand, the f n 1 (x) coefficients are determined by the S i. In other words, in order to decide the coefficients of the function f n 1 (x), we must give each critical point S i. So inverted harmonic potential can be deformed to become harmonic potential locally. In the the saddle-point approximation, we also have disconnected vacuum with harmonic potential. Each part of vacuum is one of thermal phase. In the eigenvalue basis, we choose the matrix Φ to be diagonal. This is an open string description of the system[4]. Let us label the eigenvalue of Φ as λ i, then the eigenvaluedependent part of the measure becomes: [dφ] = i dλ i 2 (λ) (4.8) where (λ) = i<j (λ i λ j ) is the Vandermonde determinant. When we consider the kinetic term 1 2 Φ 2, the kinetic term of the Hamiltonian is 1 (λ) i d 2 dλ 2 i (λ i ) (4.9) acting on wave function χ(λ i ). By redefining the wave function as follows: χ(λ i ) Ψ(λ i ) = (λ i )χ(λ i ) (4.10) For harmonic potential V (x) = x2, the Hamiltonian is 2 H = 1 { λ 2 2 i + λ 2 i } (4.11) i we have a simpler Hamiltoniain but the new wave function Ψ is fermionic, since interchange of any two eigenvalues gives a minus sign. This is N free fermions in a given potential. An orthogonal basis for the N-particle wave functions is given by Slater determinants of one particle wave functions for the Harmonic oscillator. We can describe a state by drawing a Young diagram. In this subsection, we connect above idea with bubbling geometries. Because the matrix Φ is Hermitian, the eigenvalue is real. So the fermions fill a domain on the real 7

9 axis. We call this result that fermions are confined in a domain on the real axis. The domain might consist of several disconnected components known as cuts. In the case of more than one components one speaks of a multi-cut solution. In the present case they are at most n of these cuts. We will denote the corresponding intervals in the complex x plane as A i, i = 1,, n. For example, when W(x) = x 2 /2, the only parameter in the large N limit will be the t Hooft coupling S = g s N. In this limit the eigenvalues will spread out from their classical locus λ I =0 at the minimum of the potential well into a single cut( a, a) along the real case. This is that fermions move in one-dimensional harmonic potential. This is a map between the complex matrix model in the SYM side and the droplet configurations in the bubbling AdS geometries. Because fermion fill a single cut( a, a) along the real case and potential is W(x) = x 2 /2, the classical one-body hamiltonian is given by h cl = 1 2 p q2 and classical phase density U cl take the values 0 or 1. Regions in which U cl =1 are called droplets. The 1/2 BPS bubbling geometries is fixed by the topology of the droplets. So we have the map between the complex matrix model in the SYM side and the droplet configurations in the bubbling AdS geometries. If the potential have the form(3.1), we will consider the saddle-point approximation. At each critical point, fermions spread out and make the each critical point became a cut. So locally fermions live in one-dimensional harmonic potential and each have a bubbling geometries correspondence. 5 Thermal Dynamics aspects of Bubbling Geometries We have obtained the thermal phase structure. We also connect the bubbling geometries in each phase space with fermions in each correspondence cut. This fermions move in a domain on the real axis and can be taken to be a one dimension dynamics system. In spite of all the fermions is one dimension dynamics system, the gravity side remain have disjoint parts. In general, fermions in one dimension fill the cuts of Riemann surface of (4.6) and its potential be different in different cuts. In a given temperature T, when we adopt units in which = k B = 1, the probability distribution as a function of the energy H(p, q) = ǫ is given by the Fermi-Dirac distribution: n FD (ǫ) = 1 e (ǫ µ)/t (5.1)

10 where µ is the fermi energy. When fermions fill only one cut, the fermion energy is smooth, and µis determined by the normalization condition which gives In very small temperature T, fermi energy becomes 0 dǫ e (ǫ µ)/t + 1 = N (5.2) µ = Tln(e N/T 1) (5.3) µ = N + O(Te N/T ) (5.4) When fermion fill many cuts, the thermal system have energy gap. In different cuts, the thermal property is different. In gravity side, each part have different entropy. For simplify, we consider an effective action of the form: S(g) = 1 dttr{(d t Φ) 2 + Φ 2 g 2 2 Φ4 } (5.5) and the coefficient g in the action is functions of t Hooft s coupling λ and T. When T is small, the g is also small. Its potential is : The potential have three critical points x = 0, ± W(x) = 1 2 x2 + g 4 x4 (5.6) 1 g. Each point can be deformed to became a cut. Unfortunately, when we compute the leading contribution to the free energy, the calculations are in general quite involved. So we take approximation. In our case, we get just three cut in real axis. Because the potential have Z 2 symmetry, cuts at 1 x = ± have this symmetry and each cut be a fractional part. we call this cuts is I g ±, and cut at x=0 is I 0. So the potential have two distinct part that can t be related by Z 2 symmetry. In AdS/CFT point there can be more than one saddle point in the range of integration, and when there is we should sum e ISUGRA over the classical configurations to obtain the saddle-point approximation to the gauge theory partition function. Multiple classical configurations are possible because of the general feature of boundary value problems in differential equations: there can be multiple solutions to the classical equations satisfying the same asymptotic boundary conditions. The solution which globally minimizes I SUGRA is the one that dominates the path integral. In our case the matrix have three saddle-point, two are related by Z 2 symmetry, so we have two disjoint bubbling geometries. 9

11 In classic limit, the Hamiltonian of a gas of N non-interacting fermions is : H(p, q) = 1 2 p2 1 2 q2 + g 4 q4 (5.7) At T = 0, fermions fill all the cuts and n(ε) = 1 and the energy level have a gap in semi-classical limit. let us denote E 00, E 01 as minimum and maximum energy value of fermions in cut I 0, and E 11, E 10 as maximum and minimum energy value of fermions in cut I ± separately. E = E 00 E 11 (5.8) In T = 0, E 01 = µ is the Fermi energy. The probability distribution is the Fermi-Dirac distribution, but the energy level isn t smooth. In cut I ±, because of E, we have n FD (ǫ) 1. This just like T = 0 distribution. The topology of the solution is AdS 5 S 5 fragmentation, see [12] and it s thermal AdS space. In I 0, the distribution is (5.2), we use method in [7, 16] to study. Given arbitrary smooth function f(ǫ) such that defined I as I = 0 f(ǫ)dǫ e (ǫ µ)/t + 1 converges for small T. In here, we let ǫ minimum = 0 in cut I 0. we have (5.9) I = µ The total energy of the Fermi gas is given by: E = 0 0 f(ǫ)dǫ + π2 6 T 2 f (µ) + O(T 4 ) (5.10) ǫdǫ e (ǫ µ)/t + 1 N π2 6 T 2 (5.11) N 0 is the total fermion number in I 0. The first term is clearly the ground state energy of the N 0 fermions. We now evaluate the entropy of the fermion gas. Partition function Z in the continuous limit is: Z = exp[ N 0µ T + Using the definition F = TlnZ. One obtains the free energy 0 dǫln(1 + e (ǫ µ)/t )] (5.12) F = N 0 µ E (5.13) The entropy is then given by the relation F = E TS from which we get S = 2E N 0µ T (5.14) 10

12 For small T, we gets: S π2 3 T (5.15) At low temperature regime, the half-bps geometry dual to the Fermi-Dirac gas is hyperstar [8]. As we know, the entropy of a gravitational solution in d dimensions is given by the celebrated Bekenstein-Hawking formula: S = A d 4G d (5.16) where A d is the area of the horizon. The hyperstar seems to have a naked singularity, it is expected that α corrections to the equations of motion might generate a finite-area stretched horizon. The entropy is still given by (5.16) but now A d = A sh is the area of the stretched horizon. With these corrections hyperstar is black hole. So the half-bps geometry dual to the Fermi-Dirac gas in I 0 just be AdS Schwarzschild black hole. We get the same thermal phase as Hawking and Page. We now try to resolve information loss puzzle. Classically, black holes are stable and black, which means that nothing can ever escape from inside the horizon. The entropy of black holes always increases in classical processes like the collision of two black holes. The black-hole entropy should be taken into account in the second law of thermodynamics, ds/dt 0 (5.17) Hawking showed that quantum mechanics implies that black holes emit particles. This leads to the problem of information loss. Hawking argued that this lead to non-unitary evolution. When they are many disjoint AdS space, we take them as a whole. These are isolated system satisfying the second law of thermodynamics and a single black hole needn t satisfy. When an black hole of this has a finite temperature, it emits thermal radiation that is Hawking radiation. Eventually the isolated system are in thermodynamic equilibrium. Only small black holes that have horizons of zero area, and hence a naked singularity, in the supergravity approximation exist. The small black holes acquire horizons of finite area when stringy corrections to the supergravity approximation are taken into account, like hyperstar. The total entropy of these isolated system satisfy the second law of thermodynamics(5.17). These thermal isolated system can be described by one dimension fermions thermal system. On the other hand, these thermal isolated system are dual to a unitary gauge theory. So it s a possible way to resolve the information loss puzzle. 11

13 6 Conclusions We use matrix model to study thermal phase in bubbling half-bps type IIB geometries with SO(4) SO(4) symmetry. This is a development of LLM geometries beyond harmonic potential. In near horizon limit, we find that thermal vacuum of bubbling geometries have disjoint parts, and each part is one of thermal phase. We connect the thermal dynamics of bubbling geometries with one dimension fermions thermal system. It s maybe useful for our understood quantum gravity like information loss puzzle. In future, we may study how thermal phase transition in this method. Acknowledgements Thanks my advisor Yu-Qi Chen give me help. We also have benefitted from useful conversations with Fu-Qiang Xu and other teachers. References [1] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/ [2] S.S Gubser, I.R. Klebanov and A.M. Polyakov, Phys. Lett. B 428 (1998) 105, hepth/ [3] E. Witten, Adv. Theor. Math. Phys. 2 (1998) 253, hep-th/ [4] D. Berenstein, JHEP 0407, 018 (2004),hep-th/ [5] H. Lin, O. Lunin and J. Maldacena, Bubbling AdS space and 1/2 BPS geometries, JHEP 0410, 025 (2004) hep-th/ [6] Marco M. Caldarelli, Pedro J. Silva, Giant gravitons in AdS/CFT (I): Matrix model and back reaction. Published in JHEP 0408:029,2004. e-print: hep-th/ [7] A. Buchel, Coarse-graining 1/2 BPS geometries of type IIB supergravity, arxiv:hep-th/ [8] J. Simon, The library of Babel: On the origin of gravitational thermodynamics, arxiv:hep-th/

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