PHYSICAL REVIEW D 69, 064002 ~2004! Quantum states, thermodynamic limits, and entropy inM 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. Guimarães‡ Departamento de Matema´tica, Universidade de Brası´lia, Brası́lia, DF, Brazil and Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 - Sa˜o Paulo, SP, Brazil ~Received 25 September 2003; published 1 March 2004! We discuss the matching of the BPS part of the spectrum for a~super!membrane, which gives the possibility of getting the membrane’s results via string calculations. In the small coupling limit ofM theory the entropy of the system coincides with the standard entropy of type IIB string theory~including the logarithmic correction term!. The thermodynamic behavior at a large coupling constant is computed by consideringM theory on a manifold with a topologyT23R9. We argue that the finite temperature partition functions~brane Laurent series for pÞ1) associated with the BPSp-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 limitp→0 ~point particle limit! it gives rise to the standard behavior of thermodynamic quan- tities. DOI: 10.1103/PhysRevD.69.064002 PACS number~s!: 04.70.Dy, 11.25.Mj, 11.25.Yb el in om n o em g th tu tu he o r y ity t- a n- so- ting ed and or m- le- s: in a I. INTRODUCTION There are deep connections between a fundamental~su- per!membrane and~super!string theory. In particular, it has been shown that the Bogomol’nyi-Prasad-Sommerfi ~BPS! spectrum of states for type IIB string on a circle is correspondence with the BPS spectrum of fundamental c pactified supermembrane@1,2#. Brane thermodynamics ca indicate nontrivial information about microscopic degrees freedom and the behavior of quantum systems at high t perature. Finite temperatureM theory defined on a manifold with topology T23R9, at small and large string couplin constant regimes, has been considered recently in@3,4#. In the small radius of compactification limitM theory recovers the type IIB superstring thermodynamics. In that case critical temperature coincides with the Hagedorn tempera @3#. There is a first order phase transition at a tempera less than the Hagedorn temperature with a large latent leading to a gravitational instability@5#. The purpose of the present paper is to consider the ab mentioned problems, comparing small and large coupling gimes by consideringM theory on a manifold with topology T23R9, where one of the sides of theT2 torus is the Euclid- ean time direction~fermions obey antiperiodic boundar conditions!. We turned to the problem of asymptotic dens of quantum states for fundamentalp-branes already initiated in @6–9#. This paper is organized as follows: In Sec. II the ligh *Email address: mabdalla@ift.unesp.br †Email address: abyts@uel.br; abyts@ift.unesp.br ‡Email address: marg@unb.br; emilia@ift.unesp.br 0556-2821/2004/69~6!/064002~6!/$22.50 69 0640 d - f - e re re at ve e- cone Hamiltonian formalism for membranes wrapped on torus is summarized. The small coupling limit ofM theory is considered in Sec. III, while the limit of large coupling co stant is analyzed in Sec. IV. We calculate the entropy as ciated with a string and argue that there is an interes possibility allowing for a finite temperature being introduc into the brane theory. Section V summarizes our findings discusses the relevant results. II. TOROIDAL MEMBRANES Let us consider the light-cone Hamiltonian formalism f membranes wrapped on a torus in Minkowski space. A co pactification ofM theory with (2,1) spin structure, having the topologyT23R9, assumes that the dimensionsX11,X10 are compactified on a torus with radiiR10,R11 and two spa- tial membrane directions wind around this torus. The sing valued functions on the torusX10(s,r),X11(s,r), where s,rP@0,2p), are the membrane world-volume coordinate X10~s,r!5m0R10s1X̃10~s,r!, X11~s,r!5R11r1X̃11~s,r!. ~1! The eleven bosonic coordinates are$X0,Xi ,X10,X11% and the transverse coordinatesXi(s,r), i 51,2, . . . ,8 are allsingle- valued. The transverse coordinates can be expanded complete basis of functions on the torus, namely ©2004 The American Physical Society02-1 fo s ic - r er an en e les ed p- ra- e we the ely tal to ow ibed ABDALLA, BYTSENKO, AND GUIMARÃ ES PHYSICAL REVIEW D69, 064002 ~2004! Xi~s,r!5Aa8 ( k,, X~k,, ! i eiks1 i ,r, Pi~s,r!5 1 ~2p!2Aa8 ( k,, P~k,, ! i eiks1 i ,r. ~2! In these equationsa85(4p2R11T2)21, while T2 is the mem- brane tension. The membrane Hamiltonian in light-cone malism@3,10–12# is H5H01H int , where for bosonic mode of membrane the Hamiltonian takes the form: a8H058p4a8T2 2R10 2 R11 2 m2 1 1 2 ( n @Pn i P2n i 1vkm 2 Xn i X2n i #, ~3! a8H int5 1 4gA 2 ( ~n13n2!~n33n4!Xn1 i Xn2 j Xn3 i Xn4 j . ~4! In Eqs. ~3! and ~4! n[(k,,), n3n85k,82,k8, gA 2 [R11 2 (a8)2154p2R11 3 T2 , vk,5(k21m2,2R10 2 R11 22)1/2, and (m,k,,,k8,,8)PZ. The interaction term~4! depends on the type IIA string coupling gA . Mode operators, related to bas functionsXi(s,r),Pi(s,r), are X~k,, ! i 5 1 iA2v (k,,) @a~k,, ! i 1ã~2k,2, ! i #, P~k,, ! i 5 1 A2 @a~k,, ! i 2ã~2k,2, ! i #, ~5! ~X(k,,) i !†5X(2k,2,) i , ~P(k,,) i !†5P(2k,2,) i , ~6! and v (k,,)[sgn(k)vk, . The canonical commutation rela tions read @X~k,, ! i ,P ~k8,,8! j #5 idk1k8d,1,8d i j , @a (k,,) i ,a ~k8,,8! j #5v (k,,)dk1k8d,1,8d i j . ~7! The similar relations hold for theã (k,,) i . The mass operato becomes M252p1p22~pi !22p10 2 52~H01H int!2~pi !22p10 2 . ~8! The Hamiltonian of the membrane is nonlinear, but th are two situations where one can simplify this Hamiltoni ~we shall consider these cases in the next sections!: ~i! The limit gA→0. ~ii ! The other limit of largegA . III. ZERO TORUS AREA LIMIT OF M THEORY The zero torus area limit ofM theory onT2 leads to the asymptotic gA→0 at fixed (R10/R11). In M theory it gives a 06400 r- e ten-dimensional type IIB string. More precisely, it has be shown @1,12# that quantum states ofM theory describe the (p,q) strings bound states of type IIB superstring. Let us consider string theory in Euclidean space~time coordinateX0 is compactified on a circle of circumferenc b). The presence of coordinates compactified on circ gives rise to winding string states. The string single-valu function X0(s,t) admits an expansion: X0~s,t!5x012a8p0t12R0w0s1X̃~s,t!, ~9! wherep05,0(R0)21, ,0 ,m0PZ. The Hamiltonian and the level matching constraints become H5a8pi 21 m0 2R0 2 a8 1 a8,0 2 R0 2 12~NL1NR2aL2aR!50, NL2NR5,0m0 , ~10! whereaL ,aR are the normal ordering constants, which re resent the vacuum energy of the (111)-dimensional field theory. In the case of type II superstring the number ope tors in them0561 sector read NL5 ( n51 ` Fa2n i an i 1S n2 1 2DS2n a Sn aG , NR5 ( n51 ` F ã2n i ãn i 1S n2 1 2D S̃2n a S̃n aG , ~11! where a51, . . . ,8. Thenormal-ordering constants are th same as in the NS sector of the NSR formulation, i.e.aL 5aR51/2. A. The entropy in type II string theory To begin our discussion of the entropy in string theory recall that the semiclassical quantization ofp-branes, com- pactified on a manifold with topologyM5Tp3RD2p, leads to the ‘‘number operators’’Nn with n5(n1 , . . . ,np)PZp. Therefore, let us consider multi-component versions of classical generating functions for partition functions, nam G6~z!5 ) nPZp/$0% @16exp„2zvn~a,g!…#6L, ~12! where Rz.0, L.0, vn(a,g) is given by vn(a,g) 5((,a,(n,1g,)2)1/2, g, , anda, are some real numbers. In the context of thermodynamics of fundamen p-branes, classical generating functionsG6(z) can be re- garded as a partition function associated to Fermi~or Bose! modes, wherez[b is the inverse temperature. In order calculate the thermodynamic quantities we need first to kn the total number of quantum states which can be descr by the functionsV6(N) defined by K6~ t !5 ( N50 ` V6~N!tN[G6~2 log t !, ~13! 2-2 nt a ou n e ic at e e QUANTUM STATES, THERMODYNAMIC LIMITS, AND . . . PHYSICAL REVIEW D69, 064002 ~2004! wheret,1, andN is a total quantum number. The Laure inversion formula associated with the above definition h the form V6~N!5 1 2p i R dtt2N21K6~ t !, ~14! where the contour integral is taken on a small circle ab the origin. The p-dimensional Epstein zeta functio Zpuh gu(z,w) associated with the quadratic formw@a(n1g)# 5„vn(a,g)…2 for Rz.p is given by the formula ZpUg1 . . . gp h1 . . . hpU~z,w!5 ( 8 nPZp „w@a~n1g!#…2z/2e2p i (n,h), ~15! where (n,h)5( i 51 p nihi , hi are real numbers and the prim on (8 means to omit the termn52g if all the gi are inte- gers. ForRz,p, Zpuh gu(z,w) is understood to be the analyt continuation of the right-hand side of the Eq.~15!. The func- tional equation forZpuh gu(z,w) reads n on n c an sy ac s ss ra at 06400 s t ZpUg hU~z,w!5 p (1/2)(2z2p) ~deta!1/2 GS p2z 2 D GS z 2D 3e22p i (g,h)ZpU h 2gU~p2z,w* !, ~16! and w* @a(n1g)#5(,a, 21(n,1g,)2. Equation ~16! gives the analytic continuation of the zeta function. Note th Zpuh gu(z,w) is an entire function in the complexz plane ex- cept for the case when all thehi are integers. In this cas Zpuh gu(z,w) has a simple pole atz5p with residueA(p) 52pp/2@(deta)1/2G(p/2)#21, which does not depend on th winding numbers g, . Furthermore one hasZpuh gu(0,w) 521. By means of the asymptotic expansion ofK6(t) for t →1, which is equivalent to theG6(z) expansion for smallz, one arrives at a complete asymptotic limit ofV6(N) @6–9#: V6~N!N→`5C6~p!N[2LZpu0 gu(0,w)2p22]/2(11p)expH 11p p @LA~p!G~11p!z6~11p!#1/(11p)Np/(11p)J @11O~N2k6!#, ~17! C6~p!5@LA~p!G~11p!z6~11p!# [122LZpu0 gu(0,w)]/~2p12! exp@L~d/dz!Zpu0 gu~z,w!u(z50)# @2p~11p!#1/2 , ~18! oga- n f nt ing s of where z2(z)[zR(z) is the Riemann zeta function,z1(z) 5(12212z)zR(z), k65p/(11p)min„C6(p)/p2d/4,1/2 2d…, and 0,d,2/3. Using Eqs.~17! and ~18! and assum- ing linear Regge trajectories, i.e. the mass formulaM25N for the number of brane states of massM to M1dM, one can obtain the asymptotic density for~super!p-brane states. In fact, for linear Regge-like trajectories the partitio function always diverges. This IR divergence in the partiti function might be regularized by some effects of bra theory, for example, like imposing U-duality~see, for ex- ample,@13#! or choosing nonlinear behavior of Regge traje tory ~say,M (11p)/p or something similar!. These results can be used in the context of the br method’s calculation of the ground state degeneracy of tems with quantum numbers of certain BPS extreme bl holes@14–17#. The brane picture gives the entropy in term of partition functionsG6(z) for a gas of species of massle quanta. In fact for unitary conformal theories of fixed cent chargec Eq. ~17! represents the degeneracy of the st V(N) with momentumN and forN→` one has@4,18# S~N!5 logV~N!.S01A~p,c!log~S0!, ~19! where S05A0AcN and A0 is a real number. It gives the growth of the degeneracy of BPS solitons forN@1. Note e - e s- k l e that in the case of zero modes the dependence of the l rithmic correctionA(p,c) on an embedding spacetime ca be eliminate@19#. IV. LARGE STRING COUPLING LIMIT We now focus on the case~ii ! mentioned at the end o 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 consta gA 22 so that it can be considered perturbatively. In the lead order of perturbative theory in gA 22 the interaction term can be dropped and the solution of the membrane equation motion takes the form@3# Xi~s,r,t!5xi1a8pit1A2 a8 2 ( nÞ(0,0) eiwnt vn 3@an i eiks1 i ,r1ãn i e2 iks2 i ,r#. ~20! The momentum components in theX10 andX11 directions are p105(,10/R10), and p115(,11/R11), where ,10,,11PZ. The nine-dimensional mass operator reads 2-3 h y he e tri fe o th ns re s- ated s - rdi- lds f d al of the ABDALLA, BYTSENKO, AND GUIMARÃ ES PHYSICAL REVIEW D69, 064002 ~2004! M25 ,10 2 R10 2 1 ,11 2 R11 2 1 m0 2R10 2 a82 1 1 a8 ( k,, ~a~2k,2, ! i a~k,, ! i 1ã~2k,2, ! i ã~k,, ! i !. ~21! The level-matching conditions are@3,20# Ns 12Ns 2 5m0,10, Nr 12Nr 25,11, and Ns 15 ( ,52` ` ( k51 ` k vk, a~2k,2, ! i a~k,, ! i , Ns 25 ( ,52` ` ( k51 ` k vk, ã~2k,2, ! i ã~k,, ! i , ~22! Nr 15 ( ,51 ` ( k50 ` , vk, @a~2k,2, ! i a~k,, ! i 1ã ~2k,2, ! i ã~k,, ! i #, Nr 25 ( ,51 ` ( k50 ` , vk, @a~2k,2, ! i a~k,2, ! i 1ã~2k,2, ! i ã~k,, ! i #. ~23! Let us define the quantum oscillator operatorĤ as Ĥ5( k,, ~ :a~2k,2, ! i a~k,, ! i :1:ã~2k,2, ! i ã~k,, ! i : !, ~24! where the annihilation operatorsa (k,,) i ,ã (k,,) i are determined for k.0 and,PZ, andk50, ,.0. In Eq.~24! the normal ordering means taking the annihilation operators to the rig The relation is ~see @3#! H5Ĥ12(D23)E, E 5(1/2)(k,,vk, , where the constant energy shift 2(D 23)E (E is the Casimir energy! represents the purel bosonic contribution to the vacuum energy of t (211)-dimensional field theory. In the case of supersymm try preserving boundary conditions for fermions the con butions to the vacuum energy coming from bosonic and mionic fields cancel out@20,21#. In the presence of membrane excitation states with n trivial winding numbers around the target space torus spectrum of the light-cone Hamiltonian is discrete@3,20,21#. Let the Euclidean time coordinateX0 play the role ofX10. Then fermions will obey antiperiodic boundary conditio aroundX0 but periodic boundary conditions aroundX11. In the m0561 sector fermions are antiperiodic under the placements→s12p ~while periodic underr→r12p). The Hamiltonian operator becomes H5 ,0 2 R0 2 1 ,11 2 R11 2 1 R0 2 a82 1 1 a8 „Ĥ12~D23!E…, ~25! where 06400 t. - - r- n- e - Ĥ5( n @ :a2n i an i :1:ã2n i ãn i :1vk11/2,,~ :S2n a Sn a : 1:S̃2n a S̃n a : !#, ~26! and E5EB1EF5 1 2 ( k,, ~vk,2vk11/2,,!, vk,5S k21 ,2 ge f f 2 D 1/2 . ~27! A. Brane thermodynamics presented inM theory Let us consider semiclassically the partition function a sociated with fundamentalp-branes~which is known to be divergent! embedded in flatD-dimensional manifolds. For the standard quantum field model the free energy associ with bosonic~b! and fermionic~f! degrees of freedom ha the form ~see, for example,@8,9#! F (b, f )~b!52pp~detA!1/2E 0 ` dsJ (b, f )~s,b! ~2s!(D2p12)/2 QFg 0G 3~0uV!e2sM0 2/2p, ~28! where J (b)~s,b!5u3S 0U ib2 2s D21, J ( f )~s,b!512u4S 0U ib2 2s D , ~29! and u3(nut) and u4(nut)5u3(n1 1 2 ut) are the Jacobi theta functions. HereA5diag(R1 22 , . . . ,Rp 22) is a p3p matrix. The global parametersR, characterizing the non-trivial to pology appear in the theory due to the fact that the coo natesx,(,51, . . . ,p) obey the conditions 0