Quantum states, thermodynamic limits, and entropy in M theory

Quantum states, thermodynamic limits, and entropy in M theory

M. C. B. Abdalla*

Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 - Sa˜o Paulo, SP, Brazil

A. A. Bytsenko†

Departamento de Fı´sica, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina-Parana´, Brazil and Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 - Sa˜o Paulo, SP, Brazil

M. E. X. Guimara˜es‡

Departamento de Matema´tica, Universidade de Brası´lia, Brası´lia, DF, Brazil

and Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 - Sa˜o Paulo, SP, Brazil 共Received 25 September 2003; published 1 March 2004兲

We discuss the matching of the BPS part of the spectrum for a共super兲membrane, which gives the possibility of getting the membrane’s results via string calculations. In the small coupling limit of M theory the entropy of the system coincides with the standard entropy of type IIB string theory共including the logarithmic correction term兲. The thermodynamic behavior at a large coupling constant is computed by considering M theory on a manifold with a topologyT2_⫻R9_{. We argue that the finite temperature partition functions共brane Laurent series} for p⫽1) associated with the BPS p-brane spectrum can be analytically continued to well-defined functionals. It means that a finite temperature can be introduced in brane theory, which behaves like finite temperature field theory. In the limit p→0 共point particle limit兲 it gives rise to the standard behavior of thermodynamic quan-tities.

DOI: 10.1103/PhysRevD.69.064002 PACS number共s兲: 04.70.Dy, 11.25.Mj, 11.25.Yb

I. INTRODUCTION

There are deep connections between a fundamental 共su-per兲membrane and 共super兲string theory. In particular, it has been shown that the Bogomol’nyi-Prasad-Sommerfield 共BPS兲 spectrum of states for type IIB string on a circle is in correspondence with the BPS spectrum of fundamental com-pactified supermembrane 关1,2兴. Brane thermodynamics can indicate nontrivial information about microscopic degrees of freedom and the behavior of quantum systems at high tem-perature. Finite temperature M theory defined on a manifold with topology T2⫻R9, at small and large string coupling constant regimes, has been considered recently in 关3,4兴. In the small radius of compactification limit M theory recovers the type IIB superstring thermodynamics. In that case the critical temperature coincides with the Hagedorn temperature 关3兴. There is a first order phase transition at a temperature less than the Hagedorn temperature with a large latent heat leading to a gravitational instability关5兴.

The purpose of the present paper is to consider the above mentioned problems, comparing small and large coupling re-gimes by considering M theory on a manifold with topology T2_⫻R9_{, where one of the sides of the}

T2 _{torus is the} Euclid-ean time direction 共fermions obey antiperiodic boundary conditions兲. We turned to the problem of asymptotic density of quantum states for fundamental p-branes already initiated in关6–9兴.

This paper is organized as follows: In Sec. II the

light-cone Hamiltonian formalism for membranes wrapped on a torus is summarized. The small coupling limit of M theory is considered in Sec. III, while the limit of large coupling con-stant is analyzed in Sec. IV. We calculate the entropy asso-ciated with a string and argue that there is an interesting possibility allowing for a finite temperature being introduced into the brane theory. Section V summarizes our findings and discusses the relevant results.

II. TOROIDAL MEMBRANES

Let us consider the light-cone Hamiltonian formalism for membranes wrapped on a torus in Minkowski space. A com-pactification of M theory with (⫺,⫹) spin structure, having the topology T2⫻R9, assumes that the dimensions X11,X10 are compactified on a torus with radii R10,R11 and two spa-tial membrane directions wind around this torus. The single-valued functions on the torus X10(␴,␳),X11(␴,␳), where

␴,␳苸关0,2␲), are the membrane world-volume coordinates: X10共␴,␳兲⫽m0R10␴⫹X˜10共␴,␳兲,

X11共␴,␳兲⫽R11␳⫹X˜11共␴,␳兲. 共1兲

The eleven bosonic coordinates are兵X0,Xi,X10,X11其 and the transverse coordinates Xi(␴,␳), i⫽1,2, . . . ,8 are all single-valued. The transverse coordinates can be expanded in a complete basis of functions on the torus, namely

*Email address: [email protected]

†_{Email address: [email protected]; [email protected]} ‡_{Email address: [email protected]; [email protected]}

Xi_{共␴,␳兲⫽}

冑

_␣

_⬘

兺

k,ᐉ X_共k,ᐉ兲i eik␴⫹iᐉ␳_, Pi共␴,␳兲⫽ 1 共2␲兲2

冑

_␣

_⬘

兺

_k,ᐉ P共k,ᐉ兲 i eik␴⫹iᐉ␳. 共2兲

In these equations␣

⬘

⫽(4␲2R11T2)⫺1, while T2is the mem-brane tension. The memmem-brane Hamiltonian in light-cone for-malism关3,10–12兴 is H⫽H0⫹Hint, where for bosonic modes of membrane the Hamiltonian takes the form:

␣

⬘

H0⫽8␲4␣

⬘

T2 2 R₁₀2 R₁₁2 m2 ⫹1 2

兺

n 关Pn i_P ⫺n i _⫹␻ km 2 _X n i_X ⫺n i _{兴, 共3兲} ␣

⬘

H_int⫽ 1 4g_A2

兺

共n1⫻n2兲共n3⫻n4兲Xn1 i _X n₂ j _X n₃ i _X n₄ j _{. 共4兲} In Eqs. 共3兲 and 共4兲 n⬅(k,ᐉ), n⫻n

⬘

⫽kᐉ

⬘

⫺ᐉk

⬘

, g_A2 ⬅R11 2 _(␣

_⬘

₎⫺1_⫽4␲2_R 11 3 _T 2, ␻k_ᐉ⫽(k2⫹m2ᐉ2R10 2_R 11 ⫺2₎1/2_{, and} (m,k,ᐉ,k

⬘

,ᐉ

⬘

)苸Z. The interaction term 共4兲 depends on the type IIA string coupling gA. Mode operators, related to basic

functions Xi(␴,␳), Pi(␴,␳), are X_共k,ᐉ兲i ⫽ 1 i

冑

2␻(k,ᐉ) 关␣_共k,ᐉ兲i _⫹␣˜ 共⫺k,⫺ᐉ兲 i _兴, P_共k,ᐉ兲i ⫽ 1

冑

2关␣共k,ᐉ兲 i _⫺␣˜ 共⫺k,⫺ᐉ兲 i _{兴, 共5兲} 共X(k,ᐉ) i _兲†_⫽X (⫺k,⫺ᐉ) i , 共P_(k,i _ᐉ)兲†⫽P₍i_⫺k,⫺ᐉ), 共6兲 and ␻(k,_ᐉ)⬅sgn(k)␻k_ᐉ. The canonical commutation

rela-tions read 关X共k,ᐉ兲i , P共k_⬘,ᐉ⬘兲 j 兴⫽i␦k⫹k⬘␦ᐉ⫹ᐉ⬘␦i j, 关␣(k,ᐉ) i ,␣_共kj _{⬘,ᐉ⬘兲}兴⫽␻(k,ᐉ)␦k⫹k⬘␦ᐉ⫹ᐉ⬘␦i j. 共7兲 The similar relations hold for the␣˜_(k,i _ᐉ). The mass operator becomes

M2⫽2p⫹p⫺⫺共pi兲2⫺p₁₀2⫽2共H0⫹Hint兲⫺共pi兲2⫺p10 2

. 共8兲 The Hamiltonian of the membrane is nonlinear, but there are two situations where one can simplify this Hamiltonian 共we shall consider these cases in the next sections兲:

共i兲 The limit gA→0.

共ii兲 The other limit of large gA.

III. ZERO TORUS AREA LIMIT OF M THEORY

The zero torus area limit of M theory onT2 leads to the asymptotic g_A→0 at fixed (R₁₀/R₁₁). In M theory it gives a

ten-dimensional type IIB string. More precisely, it has been shown 关1,12兴 that quantum states of M theory describe the ( p,q) strings bound states of type IIB superstring.

Let us consider string theory in Euclidean space 共time coordinate X0 is compactified on a circle of circumference

␤). The presence of coordinates compactified on circles gives rise to winding string states. The string single-valued function X0(␴,␶) admits an expansion:

X0共␴,␶兲⫽x0⫹2␣

⬘

p0␶⫹2R0w0␴⫹X˜共␴,␶兲, 共9兲 where p0⫽ᐉ₀(R₀)⫺1, ᐉ₀,m₀苸Z. The Hamiltonian and the level matching constraints become

H⫽␣

⬘

pi 2_⫹m0 2 R0 2 ␣

⬘

⫹ ␣

⬘

ᐉ0 2 R₀2 ⫹2共NL⫹NR⫺aL⫺aR兲⫽0, NL⫺NR⫽ᐉ0m0, 共10兲

where aL,aR are the normal ordering constants, which

rep-resent the vacuum energy of the (1⫹1)-dimensional field theory. In the case of type II superstring the number opera-tors in the m0⫽⫾1 sector read

NL⫽

兺

n⫽1 ⬁

冋

␣_⫺ni _␣ n i_⫹

冉

n⫺1 2

冊

S⫺n a S_na

册

, NR⫽

兺

n⫽1 ⬁

冋

␣˜ ⫺n i _␣˜ n i_⫹

冉

n⫺1 2

冊

˜S⫺n a S ˜ n a

册

, 共11兲

where a⫽1, . . . ,8. The normal-ordering constants are the same as in the NS sector of the NSR formulation, i.e. aL

⫽aR⫽1/2.

A. The entropy in type II string theory

To begin our discussion of the entropy in string theory we recall that the semiclassical quantization of p-branes, com-pactified on a manifold with topologyM⫽Tp_⫻RD⫺p_{, leads}

to the ‘‘number operators’’ N_n with n⫽(n₁, . . . ,n_p)苸Zp_.

Therefore, let us consider multi-component versions of the classical generating functions for partition functions, namely

G_⫾共z兲⫽

兿

n_苸Zp/_兵0_其

关1⫾exp„⫺z␻n共a,g兲…兴⫾⌳, 共12兲

where Rz⬎0, ⌳⬎0, ␻n(a,g) is given by ␻n(a,g)

⫽(兺ᐉaᐉ(nᐉ⫹gᐉ)2)1/2, gᐉ, and aᐉ are some real numbers.

In the context of thermodynamics of fundamental p-branes, classical generating functions G_⫾(z) can be re-garded as a partition function associated to Fermi共or Bose兲 modes, where z⬅␤ is the inverse temperature. In order to calculate the thermodynamic quantities we need first to know the total number of quantum states which can be described by the functions⍀_⫾(N) defined by

K⫾共t兲⫽

兺

N⫽0

⬁

where t⬍1, and N is a total quantum number. The Laurent inversion formula associated with the above definition has the form ⍀⫾共N兲⫽ 1 2␲i

冖

dtt ⫺N⫺1_K ⫾共t兲, 共14兲

where the contour integral is taken on a small circle about the origin. The p-dimensional Epstein zeta function Zp兩h

g_兩(z,

␸) associated with the quadratic form ␸关a(n⫹g)兴 ⫽„␻n(a,g)…2 for Rz⬎p is given by the formula

Zp

冏

g1. . . gp h1. . . hp

冏

共z,␸兲⫽

兺

⬘

n_苸Zp „␸关a共n⫹g兲兴…⫺z/2_e2␲i(n,h)_, 共15兲 where (n,h)⫽兺_ip_⫽1nihi, hi are real numbers and the prime

on 兺

⬘

means to omit the term n⫽⫺g if all the gi are

inte-gers. ForRz⬍p, Zp兩h g_兩(z,

␸) is understood to be the analytic continuation of the right-hand side of the Eq.共15兲. The func-tional equation for Zp兩h

g_兩(z, ␸) reads Zp

冏

g h

冏

共z,␸兲⫽ ␲(1/2)(2z⫺p) 共det a兲1/2 ⌫

冉

p⫺z 2

冊

⌫

冉

₂z

冊

⫻e⫺2␲i(g,h)_Z p

冏

h ⫺g

冏

共p⫺z,␸*兲, 共16兲 and ␸*关a(n⫹g)兴⫽兺_ᐉa_ᐉ⫺1(n_ᐉ⫹g_ᐉ)2. Equation 共16兲 gives the analytic continuation of the zeta function. Note that Zp兩h

g_{兩(z,␸) is an entire function in the complex z plane}

ex-cept for the case when all the hi are integers. In this case

Zp兩h

g_{兩(z,␸) has a simple pole at z⫽p with residue A(p)}

⫽2␲p/2_{关(det a)}1/2_{⌫(p/2)兴⫺1, which does not depend on the} winding numbers g_ᐉ. Furthermore one has Z_p兩_hg兩(0,␸) ⫽⫺1.

By means of the asymptotic expansion of K_⫾(t) for t

→1, which is equivalent to the G_⫾(z) expansion for small z, one arrives at a complete asymptotic limit of⍀_⫾(N) 关6–9兴:

⍀⫾共N兲N→⬁⫽C⫾共p兲N[2⌳Zp兩0 g_{兩(0,␸)⫺p⫺2]/2(1⫹p)} exp

再

1⫹p p 关⌳A共p兲⌫共1⫹p兲␨⫾共1⫹p兲兴 1/(1⫹p)_Np/(1⫹p)

冎

_关1⫹O共N⫺␬_⫾兲兴, 共17兲 C⫾共p兲⫽关⌳A共p兲⌫共1⫹p兲␨⫾共1⫹p兲兴[1⫺2⌳Zp兩0 g_{兩(0,␸)]/共2p⫹2兲}exp关⌳共d/dz兲Zp兩0 g_兩共z, ␸兲兩(z⫽0)兴 关2␲共1⫹p兲兴1/2 , 共18兲

where ␨_⫺(z)⬅␨R(z) is the Riemann zeta function, ␨⫹(z)

⫽(1⫺21_⫺z)␨

R(z), ␬_⫾⫽p/(1⫹p)min„C_⫾( p)/ p⫺␦/4,1/2

⫺␦…, and 0⬍␦⬍2/3. Using Eqs. 共17兲 and 共18兲 and assum-ing linear Regge trajectories, i.e. the mass formula M2⫽N for the number of brane states of mass M to M⫹dM, one can obtain the asymptotic density for共super兲p-brane states.

In fact, for linear Regge-like trajectories the partition function always diverges. This IR divergence in the partition function might be regularized by some effects of brane theory, for example, like imposing U-duality 共see, for ex-ample,关13兴兲 or choosing nonlinear behavior of Regge trajec-tory共say, M(1⫹p)/p or something similar兲.

These results can be used in the context of the brane method’s calculation of the ground state degeneracy of sys-tems with quantum numbers of certain BPS extreme black holes关14–17兴. The brane picture gives the entropy in terms of partition functionsG_⫾(z) for a gas of species of massless quanta. In fact for unitary conformal theories of fixed central charge c Eq. 共17兲 represents the degeneracy of the state ⍀(N) with momentum N and for N→⬁ one has 关4,18兴

S共N兲⫽log ⍀共N兲⯝S0⫹A共p,c兲log共S0兲, 共19兲

where S0⫽A0

冑

cN and A0 is a real number. It gives the growth of the degeneracy of BPS solitons for NⰇ1. Note

that in the case of zero modes the dependence of the loga-rithmic correction A(p,c) on an embedding spacetime can be eliminate关19兴.

IV. LARGE STRING COUPLING LIMIT

We now focus on the case 共ii兲 mentioned at the end of Sec. II by letting the constant gA being large. In this limit

R10,R11 are large with fixed (R10/R11) and the nonlinear interacting Hamiltonian is multiplied by the small constant g_A⫺2so that it can be considered perturbatively. In the leading order of perturbative theory in g_A⫺2 the interaction term can be dropped and the solution of the membrane equations of motion takes the form关3兴

Xi共␴,␳,␶兲⫽xi⫹␣

⬘

pi␶⫹

冑

⫺␣

⬘

2n_⫽(0,0)

兺

eiwn␶ ␻n ⫻关␣n i_eik␴⫹iᐉ␳_⫹␣˜ n i_e⫺ik␴⫺iᐉ␳_{兴. 共20兲}

The momentum components in the X10and X11directions are p₁₀⫽(ᐉ₁₀/R₁₀), and p₁₁⫽(ᐉ₁₁/R₁₁), where ᐉ₁₀,ᐉ₁₁苸Z. The nine-dimensional mass operator reads

M2⫽ᐉ10 2 R₁₀2 ⫹ ᐉ11 2 R₁₁2 ⫹ m₀2R₁₀2 ␣⬘2 ⫹ 1 ␣

⬘

兺

k,ᐉ 共␣共⫺k,⫺ᐉ兲 i ␣共k,ᐉ兲i ⫹␣˜ 共⫺k,⫺ᐉ兲 i _␣˜ 共k,ᐉ兲 i _{兲. 共21兲}

The level-matching conditions are 关3,20兴 N_␴⫹⫺N_␴⫺ ⫽m0ᐉ10, N␳⫹⫺N␳⫺⫽ᐉ11, and N_␴⫹⫽

兺

ᐉ⫽⫺⬁ ⬁

兺

k⫽1 ⬁ _k ␻kᐉ␣共⫺k,⫺ᐉ兲 i _␣ 共k,ᐉ兲i , N_␴⫺⫽

兺

ᐉ⫽⫺⬁ ⬁

兺

k⫽1 ⬁ _k ␻kᐉ␣ ˜ 共⫺k,⫺ᐉ兲 i ␣ ˜ 共k,ᐉ兲 i , 共22兲 N_␳⫹⫽

兺

ᐉ⫽1 ⬁

兺

k_⫽0 ⬁ _ᐉ ␻kᐉ关 ␣_{共⫺k,⫺ᐉ兲}i _␣ 共k,ᐉ兲 i _⫹␣˜ 共⫺k,⫺ᐉ兲i ␣˜共k,ᐉ兲i 兴, N_␳⫺⫽

兺

ᐉ⫽1 ⬁

兺

k⫽0 ⬁ _ᐉ ␻kᐉ关␣共⫺k,⫺ᐉ兲 i _␣ 共k,⫺ᐉ兲 i _⫹␣˜ 共⫺k,⫺ᐉ兲 i _␣˜ 共k,ᐉ兲 i _兴. 共23兲 Let us define the quantum oscillator operator Hˆ as

Hˆ⫽

兺

k,ᐉ 共:␣共⫺k,⫺ᐉ兲 i

␣共k,ᐉ兲i :⫹:␣˜共⫺k,⫺ᐉ兲i ␣˜共k,ᐉ兲i :兲, 共24兲

where the annihilation operators␣_(k,i _ᐉ),␣˜_(k,i _ᐉ)are determined for k⬎0 and ᐉ苸Z, and k⫽0, ᐉ⬎0. In Eq. 共24兲 the normal ordering means taking the annihilation operators to the right. The relation is 共see 关3兴兲 H⫽Hˆ⫹2(D⫺3)E, E ⫽(1/2)兺k,ᐉ␻kᐉ, where the constant energy shift 2(D

⫺3)E (E is the Casimir energy兲 represents the purely bosonic contribution to the vacuum energy of the (2⫹1)-dimensional field theory. In the case of supersymme-try preserving boundary conditions for fermions the contri-butions to the vacuum energy coming from bosonic and fer-mionic fields cancel out 关20,21兴.

In the presence of membrane excitation states with non-trivial winding numbers around the target space torus the spectrum of the light-cone Hamiltonian is discrete关3,20,21兴. Let the Euclidean time coordinate X0 play the role of X10. Then fermions will obey antiperiodic boundary conditions around X0 but periodic boundary conditions around X11. In the m₀⫽⫾1 sector fermions are antiperiodic under the re-placement ␴→␴⫹2␲ 共while periodic under ␳→␳⫹2␲). The Hamiltonian operator becomes

H⫽ᐉ0 2 R0 2⫹ ᐉ11 2 R11 2 ⫹ R₀2 ␣⬘2 ⫹ 1 ␣

⬘

„Hˆ⫹2共D⫺3兲E…, 共25兲 where Hˆ⫽

兺

n 关:␣⫺n i _␣ n i_:⫹:␣˜ ⫺n i _␣˜ n i_:⫹␻ k⫹1/2,ᐉ共:S_⫺n a _S n a_: ⫹:S˜_⫺na S ˜ n a :兲兴, 共26兲 and E⫽E_B⫹E_F⫽1 2

兺

k,ᐉ 共␻kᐉ⫺␻k⫹1/2,ᐉ兲, ␻kᐉ⫽

冉

k2⫹ ᐉ2 g_{e f f}2

冊

1/2 . 共27兲

A. Brane thermodynamics presented in M theory

Let us consider semiclassically the partition function as-sociated with fundamental p-branes 共which is known to be divergent兲 embedded in flat D-dimensional manifolds. For the standard quantum field model the free energy associated with bosonic 共b兲 and fermionic 共f兲 degrees of freedom has the form 共see, for example, 关8,9兴兲

F(b, f )共␤兲⫽⫺␲p共det A兲1/2

冕

0 ⬁ ds⌶(b, f )共s,␤兲 共2s兲(D⫺p⫹2)/2⌰

冋

g 0

册

⫻共0兩⍀兲e⫺sM0 2_/2␲ , 共28兲 where ⌶(b)_{共s,␤兲⫽␪} 3

冉

0

冏

i␤2 2s

冊

⫺1, ⌶( f )_{共s,␤兲⫽1⫺␪} 4

冉

0

冏

i␤2 2s

冊

, 共29兲

and ␪3(␯兩␶) and␪4(␯兩␶)⫽␪3(␯⫹12兩␶) are the Jacobi theta

functions. Here A⫽diag(R₁⫺2, . . . ,R⫺2_p ) is a p⫻p matrix. The global parameters R_ᐉ characterizing the non-trivial to-pology appear in the theory due to the fact that the coordi-nates x_ᐉ(ᐉ⫽1, . . . ,p) obey the conditions 0⭐x_ᐉ⬍2␲R_ᐉ. The number of topological configurations of quantum fields is equal to the number of elements in group H1(M;Z2), that is, the first cohomology group with coefficients in Z2. The multiplet g⫽(g1, . . . ,gp) defines the topological type of

field共i.e., the corresponding twist兲, and depends on the field type chosen in M, g_ᐉ⫽0 or 1/2. In our case H1₍

M;Z2) ⫽Z2

p_{and so the number of topological configurations of real}

scalars共spinors兲 is 2p_.

We follow the notations and treatment of 关22兴 and intro-duce the theta function with characteristics a,b for a,b 苸Zp_,

⌰

冋

a

b

册

共z兩⍀兲⫽_n

兺

_苸Zp

e␲i[(n⫹a)⍀(n⫹a)⫹2(n⫹a)(z⫹b)]. 共30兲

In this connection⍀⫽(si/2␲2)diag(R₁2, . . . ,R_p2).

We assume that the free energy is equivalent to a sum of the free energies of quantum fields which are present in the

modes of a p-brane. The factor exp(⫺sM₀2/2␲) in Eq. 共28兲 should be understood as Tr exp(⫺sM2/2␲), where M is the mass operator of the brane and the trace is taken over an infinite set of Bose-Fermi oscillators Nn. The one-loop-like

contribution for the共super兲p-brane can be evaluated making use of the Mellin-Barnes representation for the partition function共energy integral兲 and in this formalism the generat-ing function reads

G_⫾共z兲⫽Tr关e⫺zM2 兴⫾⫽ 1 2␲i

冕

R s⫽s0 ds⌫共s兲Tr关zM2兴_⫾⫺s, Tr关zM2兴_⫾⫺s⫽ 1 ⌫共s兲

冕

0 ⬁ dtts⫺1G_⫾共tz兲. 共31兲 One can use some substraction procedure for the divergent terms in G_⫾(z) in order to procede with the regularization scheme共see for detail Ref. 关23兴兲. To simplify our calculation we set a_ᐉ⫽R_ᐉ⫽1, and the final result for the free energy is 关24,25兴 F_⫾共␤兲⯝⫺Q共D,p兲

兺

k⫽1 ⬁ ⌫

冉

pk⫹共1⫺p兲 2

冊

⌫共k兲 ⫻␨⫾共2pk⫹1⫺p兲x1⫹p(2k⫺1), 共32兲

where Q(D, p)⫽⌳A(p)⌫(p)关y(p)兴⫺1⫺p, with ⌳⫽D⫺p ⫺1, and x␤⫽y共p兲⬅

冋

⌳23 p⫺2␲( p⫺1)/2⌫

冉

p⫹1 2

冊

␨R共p⫹1兲

册

1/2p . 共33兲 The asymptotic expansion of⌫(s) for large value of 兩s兩 has the form

⌫共s兲⫽共2␲兲1/2_ss⫺1/2_e⫺s_„1⫹O共s⫺1_{兲…, 兩arg s兩⬍␲,}

共34兲 and for p⬎1 the power series 共32兲 is divergent for any x⬎0.

B. The analytic continuation of a brane Laurent series and the thermodynamic limit

In principle, the power series共32兲 is divergent, neverthe-less one can construct its analytic continuation. Let us define for 兩z兩⬍⬁ two series

W_⫾共z兲⫽

兺

k⫽0 ⬁

_冑

_␲␯ ⫾共k;p兲 ⌫共k⫹1兲⌫

冉

pk⫹p⫹2 2

冊冉

z 2

冊

p(2k⫹1)⫹1 , 共35兲 where the factors␯_⫾(k; p) have the form

␯⫺共k;p兲⫽共⫺1兲pk⫹1,

␯⫹共k;p兲⫽␯⫺共k;p兲关1⫺2⫺p(2k⫹1)⫺1兴.

共36兲 For finite variable z these series converge and the conver-gence improves rapidly with the increasing of the integer number p. Let z⫽ᐉ•2␲x, then we get the series

兺

ᐉ⫽1 ⬁ W_⫾共ᐉ•2␲x兲⫽

兺

ᐉ⫽1 ⬁

兺

k⫽0 ⬁

_冑

_␲␯ ⫾共k;p兲共ᐉ␲x兲p(2k⫹1)⫹1 ⌫共k⫹1兲⌫

冉

pk⫹p⫹2 2

冊

. 共37兲 Now, if we commute the共up to now divergent兲 sum ⌺_ᐉwith the sum ⌺k, new extra terms of the type x⫺1W⫾( p) will

appear on the right-hand side of Eq. 共37兲. Therefore, the result is

兺

ᐉ⫽1 ⬁ W_⫾共ᐉ2␲x兲⫹x⫺1W_⫾共p兲 ⫽

兺

k_⫽0 ⬁ _␲␯ ⫾共k;p兲 ⌫共k⫹1兲⌫

冉

pk⫹p⫹2 2

冊

␨R关⫺p共2k⫹1兲兴 x⫺1⫺p(2k⫹1) ⫽sin

冉

␲₂p

冊

兺

k⫽1 ⬁ ⌫

冉

pk⫹1⫺p 2

冊

⌫共k兲 ␨⫾共2pk⫹1⫺p兲 x⫺1⫺p(2k⫺1) , 共38兲 where W_⫾( p) is an integer function of p 共see, for example, 关24兴兲. In the second equality the functional equation for

␨R(s) has been used.

The new form of F(␤) is

F_⫺共␤兲⯝ Q共D,p兲 sin

冉

␲p 2

冊

兺

k⫽0 ⬁ _共⫺1兲pk_␲␨ R关⫺p共2k⫹1兲兴 ⌫共k⫹1兲⌫

冉

pk⫹p⫹2 2

冊

⫻

冉

␤ y共p兲

冊

⫺1⫺p(2k⫹1) . 共39兲

In the string case ( p⫽1) the corresponding series in Eq. 共32兲 can be resummed into the trigonometrical form using the identities

兺

k⫽1 ⬁ ␨⫺共2k兲x2k⫽ 1 2 „1⫺␲x cot共␲x兲…,

兺

k⫽1 ⬁ ␨⫹共2k兲x2k⫽␲ x 4 tan

冉

␲x 2

冊

. 共40兲

The finite radius of convergence, 兩x兩⬍1, of the Laurent se-ries corresponds to the Hagedorn temperature in string ther-modynamics 共see for detail Ref. 关23兴兲. Using trigonometric relations, formulas 共40兲 display a certain periodicity in the temperature. The physical meaning of that behavior is still obscure. The thermal dependence in Eqs. 共39兲, 共40兲,

corre-sponding to the quantum modes in two dimensions near the Hagedorn instability, can be interpreted as an indication of a vast reduction of the fundamental degrees of freedom in string theory 关5兴.

The divergent series in Eq. 共32兲 for the case p⬎1, when reexpressed on the left-hand side of Eq.共38兲, remains well-defined for finite temperature. Note that the series 共39兲 has smooth␤→0,⬁ limits. For example, when␤→⬁ we get

F_⫺共␤兲⯝A1共D,p兲␤⫺1⫺p⫹A2共D,p兲␤⫺1⫺3p⫹O共␤⫺1⫺5p兲, 共41兲 where A_ᐉ(D, p)(ᐉ⫽1,2) depends on the dimension of the embedding spacetime. The statistical internal energy E ⫽(⳵/⳵␤)„␤F_⫺(␤)… and the entropy S⫽␤2(⳵/⳵␤)F_⫺(␤) can be easily calculated using Eq.共39兲. The␤ behavior has a similar dependence which is similar to the one found for the string case in关5,26兴. Note that in the p→0 limit 共point par-ticle limit兲 Eq. 共39兲 leads to the standard thermodynamic behavior for both F,E⬃T.

V. CONCLUSIONS

The result 共17兲, 共18兲 has a universal character. We can compute state density, including the prefactors C_⫾( p), de-pending on the dimension of the embedding space. There are deep connections between strings and p-branes; at least they should be considered as different limits of a more general M theory. Indeed, string results may be obtained via

membrane-string correspondence and vice versa. Therefore, even being not a fundamental theory of p-branes it may provide new deep insights in the understanding of string theory and con-sistent formulation of M theory.

In this paper we had dealt with the same discrete mem-brane spectrum as it has been used in the memmem-brane-string correspondence. We analyzed the logarithmic correction to the entropy in the small string coupling limit of M theory. Note that in the large string coupling limit of M theory, com-pactified on manifold with topology T2丢R9_{, the analytic} continuation of a brane Laurent series has been given in its explicit form. Physically, it means that a finite temperature can be introduced in the theory and a membrane共if it can be quantized semiclassically兲 behaves like an ideal gas of quan-tum modes, which corresponds to a field theory at finite tem-perature 共zero critical temperature兲. Finally note that in the limit p→0 共point particle limit兲 the standard behavior of thermodynamic quantities has been obtained.

ACKNOWLEDGMENTS

A.A.B. and M.E.X.G. would like to thank Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo共FAPESP/Brazil兲 for partial support and the Instituto de Fı´sica Teo´rica 共IFT/ UNESP兲 for kind hospitality. M.C.B.A. and A.A.B. would like to thank the Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico 共CNPq/Brazil兲 for partial support.

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