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QUANTUM STATE DENSITY AND CRITICAL TEMPERATURE IN M-THEORY M.C.B. ABDALLA, A.A. BYTSENKO, AND B.M. PIMENTEL 3 0 0 2 Abstract. Wediscusstheasymptoticpropertiesofquantumsta- n tes density for fundamental p branes which can yield a micro- − a scopicinterpretationofthethermodynamicquantitiesinM-theory. J The matchingofthe BPSpartofspectrumforsuperstringandsu- 9 permembrane gives the possibility of getting membrane’s results 2 via stringcalculations. Inthe weakcoupling limitofM-theorythe 1 critical behavior coincides with the first order phase transition in v standard string theory at temperature less then the Hagedorn’s 9 temperature TH. The critical temperature at large coupling con- 3 stantiscomputedbyconsideringM-theoryonmanifoldwithtopol- 2 ogy R9 T2. Alternatively we argue that any finite temperature 1 ⊗ 0 canbeintroducedintheframeworkofmembranethermodynamics. 3 0 1. Introduction / h t Recently deep connections between fundamental (super) membrane - p and (super) string theory have been found. In particular, it has been e h shown that the BPS spectrum of states for type IIB string on a circle : is in correspondence with the BPS spectrum of fundamental compact- v i ified supermembrane [1, 2]. Membrane thermodynamics can indicate X non trivial information about microscopic degrees of freedom and the r a behavior of quantum systems at high temperature. Finite temperature M-theory defined on a manifold with topology R9 T2, at weak and strong string coupling constant regime, has been ⊗ consideredrecently inRef. [3]. Inthesmallcircleradiuslimit(radiusof compactification) M-theory must recover the string thermodynamics. In type IIB superstring theory, which is associated with M-theory in the weak coupling limit, the critical temperature coincides with the Hagedorn temperature [3]. In fact, in string theory there is a first order phase transition at temperature less than T with a large latent H Date: February, 2001. We would like to thank Prof. J.G. Russo for very constructive discussion. A.A. BytsenkogratefullyacknowledgesFAPESP’sgrant(Brazil),andtheRussianFoun- dationfor Basic Research,grantNo. 01-02-17157. B.M. Pimentel thanks to CNPq for partial support. 1 2 M.C.B. ABDALLA,A.A. BYTSENKO,ANDB.M. PIMENTEL heat leading to a gravitational instability [4]. Note that the critical behavior in type IIA superstring theory, at large coupling limit, is not well understood. The purpose of the present paper is to consider again the above mentioned problems, comparing strong and weak coupling regims by considering M-theory on R9 T2, where one of the sides of the torus T2 is the Euclidean time dire⊗ction (fermions obey antiperiodic bound- ary conditions). We turned into the problem of asymptotic density of quantum states for fundamental p-branes already initiated in Refs. [5, 6, 7, 8]. In Section 2 we consider asymptotic expansions of generating func- tionsandthermodynamic quantities relatedto fundamental p branes. − The light-cone Hamiltonian formalism for membranes wrapped on a torus is summarized in Section 3. The weak coupling limit of M-theory isconsidered inSection 4. We calculate the one-loopfree energy associated to strings (and p branes). The limit of strong coupling constant is analized in Section−5. In thermodynamics of M-theory on R9 T2 ⊗ we find the critical temperature, which coincides with Hagedorn temperature also obtained in Ref. [3]. However we argue that a more interesting possibility allows for a finite temperature to be introduced into the membrane theory. Finally we end up with some remarks. 2. Asymptotics of quantum states in p-brane thermodynamics Let us consider multi-component versions of the classical generating functions for partition functions, namely G (z) = [1 exp( zω (a,g))]±q, (2.1) ± n ± − n∈Zp/{0} Y where z = y +2πix, z > 0, q > 0 and ω (a,g) is given by n ℜ 1/2 ω (a,g) = a (n +g )2 , (2.2) n j j j ! j X g , and a are some real numbers. j j In the context of thermodynamics of fundamental p branes, classi- − cal generating functions G (z) can be regarded as a partition function ± where z β is the inverse temperature. ≡ In this paper we shall be working only with the temperature dependent part of the free energy F(β). In the following we write down the QUANTUM STATE DENSITY IN M-THEORY 3 statistical free energy F(β) (both for p branes and supersymmetric − p branes), the internal energy E and the entropy S − 1 F (β) = log[G (β)], (2.3) p − −β 1 F (β) = log[G (β)G (β)], (2.4) sp + − −β ∂ ∂ E = [βF(β)], S = β2 F(β). (2.5) ∂β ∂β However, in order to calculate the above quantities we need first to know the total number of quantum states which can be described by the quantities Ω (n) defined by ± ∞ K (t) = Ω (n)tn G ( logt), (2.6) ± ± ± ≡ − n=0 X where t < 1, and n is a total quantum number. The Laurent inversion formula associated with the above definition has the form 1 Ω (n) = K (t)t−n−1dt, (2.7) ± ± 2π√ 1 − I where the contour integral is taken on a small circle about the origin. We shall use the results of Meinardus [9, 10, 11] that can be easily generalizedtothevector-valuedfunctionsofthe(2.1)type(formorede- g tail see Ref. [8]). The p-dimensional Epstein zeta function Z (z,ϕ) p|h| associatedwiththequadraticformϕ[a(n+g)] = (ω (a,g))2 for z > p n ℜ is given by the formula g ...g Z 1 p (z,ϕ) = ′(ϕ[a(n+g)])−z/2 p h ...h 1 p (cid:12) (cid:12) n∈Zp (cid:12) (cid:12) X (cid:12) (cid:12)exp 2π√ 1(n,h) , (2.8) (cid:12) ×(cid:12) − where (n,h) = p n h , h are real numbers and the prime on ′ i=1 i i i (cid:2) (cid:3) means to omit the term n = g if all the g are integers. For z < p, j g P − ℜ P Z (z,ϕ) is understood to be the analytic continuation of the right p|h| g hand side of the Eq. (2.6). The functional equation for Z (z,ϕ) p|h| reads g Γ(p−z) Zp h (z,ϕ) = (deta)−1/2π21(2z−p) Γ(z2) exp[−2π√−1(g,h)] (cid:12) (cid:12) 2 (cid:12) (cid:12) (cid:12) (cid:12) h (cid:12) (cid:12) Z (p z,ϕ∗), (2.9) × p g − (cid:12) − (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 M.C.B. ABDALLA,A.A. BYTSENKO,ANDB.M. PIMENTEL and ϕ∗[a(n+g)] = a−1(n +g )2. Formula (2.9) gives the analytic j j j j g continuation of the zeta function. Note that Z (z,ϕ) is an entire P p|h| function in the complex z plane except for the case when all the h i − g are integers. In this case Z (z,ϕ) has a simple pole at z = p with p|h| residue 2πp/2 A(p) = , (2.10) (deta)1/2Γ(p/2) which does not depend on the winding numbers g . Furthermore one i g has Z (0,ϕ) = 1. p|h| − Proposition 2.1. In the half-plane z > 0 there exists an as- ℜ ymptotic expansion for G (z) uniformly in x as y 0, provided ± → argz π/4 and x 1/2 and given by | | ≤ | | ≤ G (z) = exp q A(p)Γ(p)ζ (1+p)z−p Z g (0,ϕ)log2+ (yc+) , + R − p|0| O (2.11) (cid:8) (cid:2) (cid:3)(cid:9) G (z) = exp q A(p)Γ(p)ζ (1+p)z−p Z g (0,ϕ)logz − + p 0 − | | +(d(cid:8)/d(cid:2)z)Z g (z,ϕ) + (yc−) , (2.12) p|0| |(z=0) O where 0 < c ,c < 1 and ζ (s) ζ (s) is the Riemann zeta function, + − − ≡ R (cid:3)(cid:9) ζ (s) = (1 21−s)ζ (s). + R − By means of the asymptotic expansion of K (t) for t 1, which is ± → equivalent to the expansion of G (z) for small z, and using formulae ± (2.11) and (2.12) one arrives at a complete asymptotic limit of Ω (n): ± Theorem 2.1 (Meinardus; also see Ref. [8]). For n one has → ∞ Ω±(n) = C±(p)n(2qZp|g0|(0,ϕ)−p−2)/(2(1+p)) 1+p exp [qA(p)Γ(1+p)ζ (1+p)]1/(1+p)np/(1+p) [1+ (n−κ±)], ± × p O (cid:26) (cid:27) (2.13) C±(p) = [qA(p)Γ(1+p)ζ±(1+p)](1−2qZp|g0|(0,ϕ))/(2p+2) g exp q(d/dz)Z (z,ϕ) p|0| |(z=0) , (2.14) × [2π(1+p)]1/2 (cid:2) (cid:3) p C δ 1 κ = min ± , δ , (2.15) ± 1+p p − 4 2 − (cid:18) (cid:19) and 0 < δ < 2. 3 QUANTUM STATE DENSITY IN M-THEORY 5 Using Eqs. (2.13) of the Theorem 2.1 and assuming 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 p brane states as well as for super p branes and they read − − ̺(M)dM 2C±(p)M(1+2p−2qZp|g0|(0,ϕ))/(1+p)exp b±(p)M12+pp , (2.16) ≃ h i 1 1 b (p) 1+ [qA(p)Γ(1+p)ζ (1+p)]1+p . (2.17) ± R ≡ p (cid:18) (cid:19) 1 1 b (p) 1+ [qA(p)Γ(1+p)(ζ (1+p)+ζ (1+p))]1+p . (2.18) sp + − ≡ p (cid:18) (cid:19) In the supersymmetric case we had to deal with product of generating functions G (β) G (β). + − × This result has a universal character for all p branes. With the − help of Theorem 2.1. we can compute the complete p brane state − density (2.16), including the prefactors C (p) and the factors b (p), ± ± depending on the dimension of the embedding space. There are branes which do no wind around the torus. Those cannot be treated in the semiclassical approximation. The free energy of the zero winding sector, for infinitely small Planck length, can be associated with the energy of supergravity theory. Our goal, however, is the calculation of the temperature dependent state density of branes. An attempt to compare the asymptotic states density of branes and the density of states of neutral black holes has been made in Ref. [5, 6]. The prefactor for the degeneracy of black hole states at mass level represents general quantum field corrections to the state density. The asymptoticbehaviourofclassicalentropyofnear-extremalblackbranes coincides with the asymptotic degeneracy of some weakly interacting fundamental p-brane excitation modes. The asymptotic density of states is consistent with the entropy of 2p near-extremal p-branes (indeed S Mp+1). In the limit β 0 ∼ → (T ) the entropy of fundamental objects may be identified with → ∞ log(Ω (n)), while the internal energy is related to n. Therefore, from ± Eqs. (2.13), (2.14) and (2.3) - (2.5) one has F (T) qA(p)Γ(p)ζ (1+p)Tp+1, (2.19) p R ≃ − F (T) qA(p)Γ(p)[ζ (1+p)+ζ (1+p)]Tp+1, (2.20) sp + − ≃ − E pqA(p)Γ(p)ζ (1+p)Tp+1, (2.21) p R ≃ E pqA(p)Γ(p)[ζ (1+p)+ζ (1+p)]Tp+1, (2.22) sp + − ≃ 6 M.C.B. ABDALLA,A.A. BYTSENKO,ANDB.M. PIMENTEL S (1+p)qA(p)Γ(p)ζ (1+p)Tp, (2.23) p R ≃ S (1+p)qA(p)Γ(p)[ζ (1+p)+ζ (1+p)]Tp. (2.24) sp + − ≃ Eliminating the quantity T among equations above one gets 1+p 1 p Sp [qpA(p)Γ(p)ζR(1+p)]1+p Ep1+p, (2.25) ≃ p 1+p 1 p Ssp [qpA(p)Γ(p)(ζ+(1+p)+ζ−(1+p))]1+p Es1p+p. (2.26) ≃ p Thus the behavior of the entropy can be understood in terms of the degeneracy of some interacting fundamental p brane excitation modes. − Generally speaking the p brane approach can yield a microscopic in- − terpretation of the entropy. 3. Toroidal membranes In this section we will consider the light-cone Hamiltonian formalism for membranes wrapped on a torus in Minkowski space. Such a com- pactificationofM-theorywith(-,+)spinstructure, havingthetopology R9 T2, assumes that the dimensions X11, X10 are compactified on a ⊗ torus with radii R , R and two spatial membrane directions wind 10 11 around this torus. The single-valued functions on the torus X10(σ,ρ),X11(σ.ρ), where σ,ρ [0,2π) are the membrane world-volume coordinates, have the ∈ form X10(σ,ρ) = m R σ+X10(σ,ρ), X11(σ,ρ) = R ρ+X11(σ,ρ). (3.1) 0 10 11 The eleven bosonic coordinates are X0,Xi,X10,X11 and in addition e { } e the transverse coordinates Xi(σ,ρ), i = 1,2,...,8 are all single-valued. Transverse coordinates can be expanded in a complete basis of functions on the torus, namely Xi(σ,ρ) = √α′ Xi eikσ+iℓρ, (3.2) (k,ℓ) k,ℓ X 1 Pi(σ,ρ) = Pi eikσ+iℓρ. (3.3) (2π)2√α′ (k,ℓ) k,ℓ X Here α′ = 4π2R T −1, while T is the membrane tension. 11 2 2 ThemembraneHamiltonianinlight-coneformalism[12,13,14,15,3] (cid:0) (cid:1) can be written as follows: H = H + H . In particular, for bosonic 0 int modes of membrane the explicit Hamiltonian is QUANTUM STATE DENSITY IN M-THEORY 7 1 α′H = 8π4α′T2R2 R2 m2 + PiPi +ω2 XiXi , (3.4) 0 2 10 11 2 n −n km n −n n X(cid:2) (cid:3) 1 α′H = (n n )(n n )Xi Xj Xi Xj . (3.5) int 4g2 1 × 2 3 × 4 n1 n2 n3 n4 A X In Eqs. (3.4) and (3.5) n (k,ℓ), n n′ = kℓ′ ℓk′, g2 R2 (α′)−1 = 4π2R3 T , ω = (k2 + m≡2ℓ2R2 R−2×)1/2, and (−m,k,ℓ,Ak′,≡ℓ′) 11 Z. The 11 2 kℓ 10 11 ∈ interaction term (3.5) depends on the type IIA string coupling g . A Mode operators, related to basic functions Xi(σ,ρ), Pi(σ,ρ), are 1 1 Xi = αi +αi , Pi = αi αi , (k,ℓ) √ 2ω (k,ℓ) (−k,−ℓ) (k,ℓ) √2 (k,ℓ) − (−k,−ℓ) (k,ℓ) − (cid:2) (cid:3) (cid:2) (3(cid:3).6) e e Xi † = Xi , Pi † = Pi , (3.7) (k,ℓ) (−k,−ℓ) (k,ℓ) (−k,−ℓ) and ω sign(k)ω . The canonical commutation relations read (k,ℓ) (cid:0) (cid:1)kℓ (cid:0) (cid:1) ≡ X(ik,ℓ),P(jk′,ℓ′) = √−1δk+k′δℓ+ℓ′δij, (3.8) (cid:2) (cid:3) [α(ik,ℓ),α(jk′,ℓ′)] = ω(k,ℓ)δk+k′δℓ+ℓ′δij. (3.9) We have similar relations for the αi . (k,ℓ) Finally the mass operator takes the form e M2 = 2p+p− (pi)2 p2 = 2(H +H ) (pi)2 p2 . (3.10) − − 10 0 int − − 10 The Hamiltonian of membrane is non linear. But there are two situations where one can simplify this Hamiltonian: (i) The limit g 0. A → (ii) The other limit of large g . A We shall consider these two cases in the next two sections. 4. Zero torus area limit of M-theory (g 0) A → The zero torus area limit of M-theory on T2 is related to the asymptotic g 0 at fixed (R /R ). Such a limit in M-theory leads A 10 11 → to a ten-dimensional type IIB string theory. More precisely, it has been shown [1, 15] that quantum states of M-theory describe the (p,q) strings bound states of type IIB superstring theory. We will consider in this section string theory at finite temperature associated with the g 0 limit of M-theory. A → 8 M.C.B. ABDALLA,A.A. BYTSENKO,ANDB.M. PIMENTEL 4.1. Critical temperature in type II string theory. Let us consider string theory in Euclidean space (time coordinate X0 is compactified on a circle of circumference β). The presence of coordinates compactified on circles gives rise to winding string states. The string single-valued function X0(σ,τ) admits an expansion: X0(σ,τ) = x0 +2α′p0τ +2R w σ +X(σ,τ), (4.1) 0 0 where p0 = ℓ (R )−1, ℓ , m Z. The Hamiltonian and the level 0 0 0 0 ∈ e matching constraints becomes m2R2 ℓ2 H = α′p2 + 0 0 +α′ 0 +2(N +N a a ) = 0, (4.2) i α′ R2 L R − L − R 0 N N = ℓ m , (4.3) L R 0 0 − where a ,a are the normal ordering constants, which represent the L R vacuum energy of the 1+1 dimensional field theory. In the case of type II superstring the number operators in the m = 1 sector read 0 ± ∞ ∞ 1 1 N = αi αi+(n )Sa Sa , N = αi αi+(n )Sa Sa , L −n n −2 −n n R −n n −2 −n n n=1 n=1 X(cid:2) (cid:3) X(cid:2) (4.(cid:3)4) e e e e where i = 1,...,8, a = 1,...,8. The normal-ordering constants are the same as in the NS sector of the NSR formulation, i.e. a = a = 1/2. L R The critical temperature of string can be obtained as usual by de- termining the radius R at which appear infrared instabilities. In the 0 following we reproduce the critical Hagedorn temperature using the general state density formulation (2.16) - (2.18), which is more suit- able for generalizing to membrane theory. Indeed, in the string case p = 1 the inverse critical temperature becomes: 1 β = b (1) = [12ζ (2)A(1)]1/2 = 2π2A(1) 1/2, (4.5) cr sp R 2 where the the factor 1/2 is related to the right-(cid:2)and left-(cid:3)moving modes (N = N ) of closed string. Taking into account that A(1) = 2 and L R restoring the α′ dependence one finds the critical Hagedorn’s temperature 1 T = . (4.6) H 2π√α′ QUANTUM STATE DENSITY IN M-THEORY 9 This result is well known. In general the Hagedorn’s temperature depends on the normal-ordering constants a , a and has the form L R T−1 = 2π 2α′(a +a ). H L R 4.2. Thepone-loop free energy of strings. Let usconsider thesemi- classical free energy associated with fundamental compactified (super) p branes (which is known to be divergent) embedded in flat D- d−imensional manifolds with topologies M = S1 Tp RD−p−1. For the ⊗ ⊗ simplest quantum field model the free energy associated with bosonic and fermionic degrees of freedom has the form (see for example Refs. [16, 7, 8]) ∞ F(b,f)(β) = πp(detA)1/2 ds(2s)−(D−p+2)/2Ξ(b,f)(s,β) − Z0 g sM2 Θ (0 Ω)exp 0 , (4.7) × 0 | − 2π (cid:20) (cid:21) (cid:18) (cid:19) where √ 1β2 √ 1β2 Ξ(b)(s,β) = θ 0 − 1, Ξ(f)(s,β) = 1 θ 0 − , 3 4 2s − − 2s (cid:18) (cid:19) (cid:18) (cid:19) (cid:12) (cid:12) (4.8) (cid:12) (cid:12) and θ (ν τ) and θ (cid:12)(ν τ) = θ (ν + 1 τ) are the Jacobi the(cid:12)ta functions. 3 | 4 | 3 2| Here A = diag(R−2,...,R−2) is a p p matrix. The global parameters 1 p × R characterizing the non-trivial topology of M appear in the theory j due to the fact that coordinates x (j = 1,...,p) obey the conditions j 0 x < 2πR . The number of topological configurations of quantum j j fie≤lds is equal to the number of elements in group H1(M;Z ), that is, 2 the first cohomology group with coefficients in Z . The multiplet g = 2 (g ,...,g ) defines the topological type of field (i.e., the corresponding 1 p twist), anddepends onthefield typechosen inM, g = 0or1/2. Inour j case H1(M;Z ) = Zp and so the number of topological configurations 2 2 of real scalars (spinors) is 2p. We follow the notations and treatment of Ref. [17] and introduce the theta function with characteristics a,b for a,b Zp, ∈ a Θ (z Ω) = exp π√ 1(n+a)Ω(n+a) b | − (cid:20) (cid:21) n∈Zp X (cid:2) +2π√ 1(n+a)(z+b) . (4.9) − In this connection Ω = (s√ 1/2π2)diag(R2,...,R2). − 1 (cid:3) p The above mentioned method of the free energy calculation admits a subsequent development for extended objects. We will assume that 10 M.C.B. ABDALLA,A.A. BYTSENKO,ANDB.M. PIMENTEL 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( sM2/2π)inEq. (4.7)shouldbeunderstoodasTrexp( sM2/2π), − 0 − where M is the mass operator of the brane and the trace is taken over (b) (f) an infinite set of Bose-Fermi oscillators N ,N . n n Theone-loopfreeenergyoffieldscontainedina(super)p branecan − be evaluated making use the Mellin-Barnes representation for the energy integral F(β) = dsF(s,D)Tr[M2](D−s)/2 (see Ref.[18]) and ℜs=c the mass operators R 1 Tr e−tM2 = dsΓ(s)Tr[tM2]−s, (4.10) 2π√ 1 h i − Zℜs=s0 1 ∞ Tr[M2]−s = dtts−1Tr e−tM2 , (4.11) Γ(s) Z0 h i Tr e−tMp2 = G (t), Tr e−tMs2p = G (t)G (t), (4.12) − + − h i h i G (t) = [1 exp( tω )]±(D−p−1). (4.13) ± n ± − n∈Zp/{0} Y One can use some substraction procedure for the divergent terms in G (t), G (t) in order to procede regularization (see for detail Ref. − + [18]). To simplify the calculation we put a = R = 1 and the final j j result for the free energy [19, 20] is: ∞ Γ pk + 1−p F (β) Q(D,p) 2 ζ (2pk +1 p)x1+p(2k−1), (4.14) p R ≃ − Γ(k) − (cid:0) (cid:1) k=1 X ∞ Γ pk + 1−p F (β) Q(D,p) 2 ζ (2pk+1 p)x1+p(2k−1), (4.15) sp + ≃ − Γ(k) − (cid:0) (cid:1) k=1 X where Q(D,p) is an integer function and 1 x = β−1y(p) β−1 (D p 1)23p−2πp−21Γ p+1 ζR(p+1) 2p . ≡ − − 2 (cid:20) (cid:18) (cid:19) (cid:21) (4.16) The first terms of the leading behavior of the series (4.14), (4.15) (k = 1) have to coincide with formulae (2.19) and (2.20) at q = D p 1. − − Therefore we have Q(D,p) = (D p 1)A(p)Γ(p)[y(p)]−1−p. (4.17) − −