Asymptotic Density of Open p-brane States with Zero-modes included

We obtain the asymptotic density of open p-brane states with zero-modes included. The resulting logarithmic correction to the p-brane entropy has a coefficient - \frac{p + 2}{2 p}, and is independent of the dimension of the embedding spacetime. Such logarithmic corrections to the entropy, with precisely this coefficient, appear in two other contexts also: a gas of massless particles in p-dimensional space, and a Schwarzschild black hole in (p + 2)-dimensional anti de Sitter spacetime.

1. The asymptotic density of states ρ(N) at level N, N ≫ 1, for p-branes compactified on (S 1 ) p ×R D−p has been be calculated within the semiclassical quantisation scheme [1,2,3,4]. (For various applications of this result see the review article in [4] and [2,5].) The corresponding p-brane entropy, given by lnρ(N), then has a logarithmic correction with a particular coefficient X, which depends on the dimension D of the embedding spacetime.
The correct counting of the asymptotic density of states must also include the zero-mode states. They have been included for the open string case (p = 1) in [6,7]. As a consequence, the logarithmic correction coefficient X becomes independent of D and is given by X = − 3 2 . Such logarithmic corrections to the entropy, with precisely this coefficient, − 3 2 , have appeared in other contexts also: in (1 + 1)-dimensional conformal field theories [6,7], and in the entropies for (2 + 1) and (3 + 1) dimensional black holes calculated using the spin network formalism [8,9].
In this paper, using the results of [4], we obtain the asymptotic density of states for open p-branes. We then include the zero-modes following the methods of [6,7]. We find that the logarithmic correction coefficient X becomes independent of the dimension D of the embedding spacetime, and is given by X = − p+2 2p . Logarithmic corrections to entropy also arise for statistical mechanical systems due to statistical fluctuations [10]. Using the results of [10], we find that logarithmic corrections to the entropy, with precisely the same coefficient as that obtained in the open p-brane case, namely − p+2 2p , appear in two other contexts also: a gas of massless particles in p-dimensional space, and a Schwarzschild black hole in (p+2)-dimensional anti de Sitter spacetime [10]. This paper is organised as follows. In section 2, we briefly present the results of [4] and, using them, obtain the asymptotic density of states for open p-branes. In section 3, we include the zero-modes. In section 4, we show that such logarithmic corrections with precisely the same coeffiecient appear in other contexts also. In section 5, we conclude by mentioning a few issues for further study.
2. The asymptotic density of states ρ(N) at level N, N ≫ 1, for p-branes compactified on (S 1 ) p × R D−p can be calculated within the semiclassical quantisation scheme, and is of the form where δ, A, B, and C are constants. δ was obtained in [1,2] and the correct expressions for the remaining constants A, B, and C in [4]. The corresponding p-brane entropy S(N) is given by where the leading term S 0 = AN δ and X = B δ is the coefficient of the logarithmic correction to the entropy.
We now present briefly the results of [4]. See [4] for details. In the semiclassical quantisation, the total number operator N can be written in the proper time formalism as where d = (D − p − 1), n = (n 1 , n 2 , · · · , n p ) ∈ Z p , 0 = (0, 0, · · · , 0), and N i n are number operators [11]. For the sake of simplicity, we have set the p-brane tension to unity and taken all the circles in (S 1 ) p to be of unit radius. Let ρ(N) be the number of independent eigenstates of the total number operator N with eigenvalue N. Its generating function F (z) is given by Inverting the above relation then gives ρ(N) in terms of F (z): where the integration contour is a small circle around the origin. Using Meinardus theorem [12] and the properties of Epstein zeta function, an asymptotic expression for F (z) can be obtained in the limit Re(z) → 0 which is sufficient to obtain ρ(N) in the limit N ≫ 1. The asymptotic expression where a, b, and c are constants. The contour integral in (5), and thus ρ(N), can then be evaluated by saddle point method. For F (z) of the form given in equation (6), the saddle point is located at z = z 0 = N ap − 1 p+1 and ρ(N), in the limit N ≫ 1, is given by In our case, a = 2d Γ(p)π and ζ is the Riemann zeta function; the constant c is given explicitly in [4] and is not required for our purposes. See [4] for further details.
We now obtain the asymptotic density of states for an open p-brane using the above results. In this case modes n and −n together contribute to one standing wave (SW) mode of the open p-brane and, hence, should be counted only once. Thus, the corresponding total number operator is given by The corresponding generating function F o (z) is given by where the last equality follows since ω n = ω −n and the product in the last expression includes both n and −n. Note that the generating function F o (z) is identical to that in equation (4) with d there replaced by d 2 . It therefore follows that the density of states ρ o (N) for an open p-brane in the limit N ≫ 1 is given by (7), but now with d replaced by d 2 . Explicitly, in the limit N ≫ 1, , c is as given in [4]  in obtaining (10). Note that ρ(N) is of the form given in equation (1) with δ = p p+1 and B = − D+1 2(p+1) , and that string theory result [3] is obtained upon setting p = 1.

3.
We now include the zero-modes and obtain the resulting asymptotic density of open p-brane states. The complete Hamiltonian for a p-brane in the proper time formalism is given by where p 2 is the transverse momentum square opeartor and N is the total number operator given in (8), and they both commute with each other [4]. The correct counting of the total number of states ρ(N) must include the zero-mode states also, namely those corresponding to the transverse momentum. For open strings (p = 1), ρ(N) with zero-modes included has been calculated in [6,7] in the limit N ≫ 1. The resulting ρ(N) is still given by equation (1) with δ = 1 2 as before, but now with B = − 3 4 independent of the dimension D of the embedding spacetime. See [6,7] for further discussions.
The total number of states ρ(N) for open p-branes, with zero-modes included, can be calculated for other values of p also in the limit N ≫ 1. ρ(N) is still given by equation (5), but now the relevent generating function F (z), with zero-modes included, is given by where the second equality follows since the operators p 2 and N commute with each other and F o (z) is the generating function given in equation (9). The momentum integral can be evaluated easily and results in an extra zdependent factor given by  (13) where the constants a and c are as in equation (10). Clearly, ρ(N) is of the form given in equation (1) with δ = p p+1 as before, but now with B = − p+2 independent of the dimension D of the embedding spacetime. The open p-brane entropy S(N) with zero-modes included is thus given by where S 0 = AN p p+1 . The coefficient of the logarithmic correction to the open p-brane entropy with zero-modes included is now given by − p+2 2p and is independent of the dimension D of the embedding spacetime. Note that without zero-modes included it is given by − D+1 2p , as can be seen from equations (2) and (10), and depends on D.
Similarly, it turns out that logarithmic corrections, with precisely the coefficient given in equation (14), namely − p+2 2p , also appear in two other contexts. Recently, logarithmic corrections to entropies of statistical mechanical systems, arising due to statistical fluctuations, have been obtained in [10]. One calculates the density of states ρ(E) as an inverse Laplace transformation of the partition function in the canonical ensemble. Statistical fluctuations can then be incorporated naturally. Then, ρ(E)∆ is the number of states with energy in the range E ± ∆ 2 where ∆ depends on the precision with which the system is prepared and, in particular, is independent of E. The entropy S(E) is therefore given by S(E) = lnρ(E) + (const).
The result of [10] is that for a system at temperature T , with specific heat C (which must be positive for this formalism to be applicable), one obtains for its entropy where S 0 is the leading term. See [10] for details. Now, consider a gas of massless particles in p-dimensional space. Then Equation (15), therefore, gives For a Schwarzschild black hole, the specific heat is negative and, hence, the above formalism is inapplicable [10]. However, a Schwarzschild black hole of sufficiently large mass in a d-dimensional anti de Sitter spacetime (AdS d ) has positive specific heat. Consider an AdS p+2 Schwarzschild black hole of mass M. For sufficiently large M, one has [13] S 0 ∝ r p where r + is the horizon. It then follows that C ∝ r p + . Equation (15), therefore, gives See [10]) for details. For AdS 3 , see also [8,9]. From equations (16) and (17), we see that logarithmic corrections to the entropy of a gas of massless particles in p-dimensional space, and to that of an AdS p+2 Schwarzschild black hole, both have a coefficient − p+2 2p which is precisely the same as that obtained in the open p-brane case with zero-modes included.
5. To summarise, we have obtained the asymptotic density of open pbrane states with zero-modes included. The corresponding open p-brane entropy has a logarithmic correction, with a coefficient − p+2 2p . Such logarithmic corrections, with precisely the same coefficient, also appear for a p-dimensional gas and for an AdS p+2 Schwarzschild black hole where the corrections arise due to statistical fluctuations.
The relation of a p-dimensional gas to AdS p+2 Schwarzschild black hole, for p = 1, 2, 3, and 5, can be understood in the context of AdS/CFT duality [14] as that of a boundary conformal field theory at high temperature [13]. In light of the present results, one may explore the relations between quantum/semi classical p-branes, p-dimensional gas, and AdS p+2 spacetime in more detail. In particular, it will be interesting to know if the values of p are restricted for quantum/semi classical p-branes, or if a duality exists between p-dimensional gas and AdS p+2 spacetime for any value of p.
As mentioned earlier, the coefficient X = − 3 2 , corresponding to p = 1, also appears for the entropy of a (3 + 1) dimensional black hole calculated using the spin network formalism [8]. In this formalism, one considers punctures, each carrying a spin J puncture , and counts the number of spin singlet states, namely those states with J(total) = 0. The leading term (and a part of the logarithmic correction) in the entropy corresponds to the number of states with J z (total) = 0. However, such states include states with J(total) = 0 also. A correct counting, that counts states with J(total) = 0 only, then leads to the coefficient − 3 2 for the logarithmic correction to the entropy [8]. It will be interesting to find if a similar interpretation exists for the logarithmic correction coefficient X = − p+2 2p for other values of p also.