We consider the
thermodynamics of the near-extremal NS5-brane in type IIA string
theory. The central tool we use is to map phases of
six-dimensional Kaluza-Klein black holes to phases of
near-extremal M5-branes with a transverse circle in
eleven-dimensional supergravity. By S-duality these phases
correspond to phases of the near-extremal type IIA NS5-brane. One
of our main results is that in the canonical ensemble the usual
near-extremal NS5-brane background, dual to a uniformly smeared
near-extremal M5-brane, is subdominant to a new background of
near-extremal M5-branes localized on the transverse circle. This
new stable phase has a limiting temperature, which lies above the
Hagedorn temperature of the usual NS5-brane phase. We discuss the
limiting temperature and compare the different behavior of the
NS5-brane in the canonical and microcanonical ensembles. We also
briefly comment on the thermodynamics of near-extremal Dp-branes
on a transverse circle.
1 Introduction
The physics of the NS5-brane is interesting for many reasons: It
is conjectured to have a novel non-gravitational string theory
living on its world-volume
[1, 2, 3]. The type
IIA NS5-brane world-volume theory has the (2,0) super conformal
field theory as its low energy limit [4].
Moreover, the NS5-brane wrapped on certain cycles describes pure
super Yang-Mills theory [5].
In this paper we consider the thermodynamics of the near-extremal
NS5-brane in type IIA string theory. We use as a starting point
that the type IIA NS5-brane is given non-perturbatively as an
M5-brane with a transverse circle. More specifically, we employ
the idea of [6, 7] that in order to
understand the thermodynamic phases of the type IIA NS5-brane it
is necessary to know all possible phases of M5-branes with a
transverse circle.^{1}^{1}1In this paper we consider only the near-extremal type
IIA NS5-brane, but our results can easily be extended to the full
non-extremal case. However, we restrict ourselves to the
near-extremal case since this has the most interesting physics and
since it is dual to a decoupled non-gravitational theory.
An important aspect of the NS5-brane is that the thermodynamics of
the usual near-extremal NS5-brane background (dual to an M5-brane
uniformly smeared on a transverse circle) has entropy as function
of energy S(E)=T−1hgE, where Thg is a
fixed temperature [8, 9]. Thus,
the thermodynamics of the NS5-brane is singular and has
Hagedorn-like behavior, supporting the idea that its
non-gravitational dual is a string theory. In
[10, 11] the idea was put forward that
the singular thermodynamics of the NS5-brane could be resolved by
computing a string one-loop correction to the background. This was
done in [12], with the result that the corrected
background has T>Thg and negative specific heat. In a
subsequent analysis the results of [12] have been
interpreted to indicate that one does not have a Hagedorn phase
transition and that the Hagedorn temperature is a limiting
temperature [13].
The main result of this paper is that in the canonical ensemble
the usual near-extremal NS5-brane background is thermodynamically
subdominant to a new eleven-dimensional background of
near-extremal M5-branes localized on the transverse circle. This
means that if one starts with the usual near-extremal NS5-brane
background this will decay and end up in the new phase which is
thermodynamically stable in that it has a positive specific heat.
That the new phase is dominant means in particular that the string
one-loop correction, or any other correction to the usual
near-extremal NS5-brane background, is subdominant to the effect
of the decay to the new eleven-dimensional phase. The new phase is
also dominant in the microcanonical ensemble for sufficiently
small energies.
However, the presence of the new stable phase also raise a puzzle.
The puzzle origins in the fact that the new phases we discuss in
this paper have a maximally possible temperature. Therefore, it is
not clear what happens if one tries to heat up the type IIA
NS5-brane to a temperature higher than this maximal temperature.
As we discuss in the paper, this is also connected to the fact
that the canonical and microcanical ensembles seem to be
inequivalent at large energies.
The new results in this paper on the thermodynamics of the
NS5-brane are obtained using the transformation of
[6, 7] that can take a phase of a
six-dimensional Kaluza-Klein black holes and transform it into a phase
for near-extremal M5-branes with a transverse circle. Our focus in
this paper is on phases without Kaluza-Klein bubbles since these
are the dominant phases in the phase diagram. Phases obtained from
the solutions with Kaluza-Klein bubbles in [14] are
considered in [15].
For comparison, we also briefly discuss the thermodynamics of
near-extremal Dp-branes on a circle. These cases have some
properties in common with the M5-brane case, but also differ
significantly in some other aspects. In particular, they do
neither exhibit a maximal temperature, nor a discrepancy between
the microcanonical and canonical ensembles.
The outline of this paper is as follows. In Section 2
we review some of the basic facts on near-extremal NS5-branes that
we build on in the rest of the paper. In Section 3 we
review how one maps phases of six-dimensional Kaluza-Klein black
holes to phases of near-extremal type IIA NS5-branes, and we use
recently obtained data to find the entire localized phase. In
Section 4 we discuss the new thermodynamics of the
near-extremal type IIA NS5-brane in the canonical and
microcanonical ensembles. In Section 5 we comment on
similarities in the thermodynamics of near-extremal Dp-branes on
a transverse circle. Finally, we present our conclusions in
Section 6.
2 Basic facts on near-extremal NS5-branes
In this section we briefly review basic facts about near-extremal
NS5-branes. Note that the results of
[6, 7] for near-extremal NS5-brane
on which we build in this paper, are not reviewed in
this section but instead in Sections 3 and
4.
One of the reasons why the thermodynamics of near-extremal
NS5-branes is interesting is the fact that this background is
believed to be dual to a non-gravitational theory. In the
decoupling limit of N coincident NS5-branes in type IIA string
theory the string length ls is kept fixed while the string
coupling gs goes to zero
[9, 16]. In this limit the dynamics
of the theory is believed to reduce to a string theory without
gravity, called Little String Theory (LST), or more precisely
(2,0) LST
of type AN−1 [1, 2, 3].^{2}^{2}2See
[17, 18, 19, 20, 21, 22, 13, 23]
for other interesting work on thermodynamics and correlators of
Little String Theory.
Consider now instead N non-extremal type IIA NS5-branes. Then
the decoupling limit, or near-extremal limit, is defined as
keeping ls fixed and taking gs to zero, while at the same
time keeping the energy above extremality fixed. This gives the
following background (in the string frame)
describing N coincident near-extremal NS5-branes^{3}^{3}3Note that the radial coordinate r is defined to be
dimensionless in (2.1).
Here ϕ is the dilaton and B2 is the Kalb-Ramond two-form
potential for the field strength H3 under which NS5-branes are
magnetically charged. The thermodynamics of this background is
T=Thg≡12π√Nls,S(E)=T−1hgE,EV5=r20(2π)5l6s.
(2.2)
This thermodynamics corresponds to the thermodynamic behavior of a
string theory at the Hagedorn temperature, Thg being
the Hagedorn temperature [8, 9].
The NS5-brane description of (2,0) LST is valid at high energies
E/V5≫Nl−6s.
The above background corresponds to the exact worldsheet CFT given by
H+3/U(1)×SU(2)N×R5, with H+3≡SL(2,C)N/SU(2)N,
which can be used to perform string computations in this background.
In particular, the string coupling expansion in the background becomes
an expansion around infinite energy in powers of 1/E
[24]. Thus, one expects the stringy features of
the world-volume theory of NS5-branes to appear in the high energy
regime E/V5≫Nl−6s.
As mentioned in the Introduction it was suggested in
[10, 11] that one could resolve the
singular thermodynamics of the near-extremal NS5-brane by
computing string corrections to the NS5-brane background. In
[12] this was done using the fact that the
near-extremal NS5-brane background is described by an exact
worldsheet CFT. The result of this computation is that the
corrected background has T>Thg, F>0 and negative specific heat. Subsequently,
the instability of this background was argued in
[13] to be a tad-pole instability since the
unstable mode has a potential proportional to −logE, driving
the mode to larger values of the energy. See the conclusions in
Section 6 for further comments on the high energy
regime and comparison to the results of this paper.
Turning to lower energies, we have that using the IIA/M S-duality
the near-extremal NS5-brane can be seen as an M5-brane in M-theory
which is smeared uniformly in a transverse direction. The
background (2.1) is therefore equivalent to the following
M-theory background
(2.3)
where f is the same as given in (2.1) and C3 is the
three-form potential for the four-form field strength under which
the M5-brane is magnetically charged. This background describes
N coincident near-extremal M5-branes smeared on a transverse
circle. The transverse circle is parameterized by z which we
take to have period 2π. The background (2.3) is weakly
curved when N≫1. The near-extremal limit of non-extremal
M5-branes with a transverse circle is given by taking
lp→0 while rescaling the transverse directions.^{4}^{4}4The near-extremal limit of the NS5-brane and the one of
the smeared M5-brane are easily related using the S-duality
relations l3p=gsl3s and R11=gsls where
R11 is the radius of the eleventh direction which in this
case is the transverse circle that the M5-branes are smeared on.
The thermodynamics is identical to that of the near-extremal
NS5-brane given above, and this description of (2,0) LST is valid
for energies l−6s≪E/V5≪Nl−6s.
At low energy the background of near-extremal M5-branes smeared on
a transverse direction is unstable to decay to a background of
near-extremal M5-branes localized on the transverse circle
[7]. For very low energies the N near-extremal
NS5-branes thus enter a phase which is well-described as N
near-extremal M5-branes in eleven dimensional flat space. The dual
theory of this background is the (2,0) super conformal field
theory (SCFT). We shall return to this in the following.
3 NS5-brane phases from Kaluza-Klein black holes
In this section we briefly review the map of
[6, 7]^{5}^{5}5See [25, 26] for reviews.
by which one can obtain solutions for near-extremal M5-branes with
a transverse circle from phases of Kaluza-Klein black holes in six
dimensions. By six-dimensional Kaluza-Klein black holes we mean
pure gravity solutions of six-dimensional General Relativity that
at asymptotic infinity have the geometry of Kaluza-Klein
space-time M5×S1, M5 being five-dimensional
Minkowski space (see the reviews [27, 28]
for more on Kaluza-Klein black holes). As we discuss below, the
near-extremal M5-brane phases can be seen as phases of
near-extremal type IIA NS5-branes.
As described in [14, 27] there are two
classes of Kaluza-Klein black holes in six dimensions: Solutions
with a local S0(4) symmetry, and solutions without a local
SO(4) symmetry. The latter we shall briefly discuss in the
Conclusions, and these are ones that contain Kaluza-Klein bubbles.
For the former class of solutions, which do have a local SO(4)
symmetry and which are the ones of interest in this paper, it has
been proven [29, 30] that the metric
can be put in the ansatz [6]
ds2=−fdt2+AfdR2+AK3dv2+KR2dΩ23,f=1−R20R2.
(3.1)
Here A(R,v) and K(r,v) are two functions determining the
metric. R→∞ is the asymptotic region where A
and K go to one, while R=R0 is the event horizon
[6]. We have v≡v+2π, i.e. we take
the asymptotic circle to have unit radius for simplicity.
Using the map found in [6, 7] we can
now take any six-dimensional Kaluza-Klein black hole solution of
the form (3.1) and transform it to a solution of eleven
dimensional supergravity for N near-extremal M5-branes with a
transverse circle given by [6, 7]
(3.2)
Here f is as in (3.1). The supergravity solution
(3.2) is valid as long as N≫1, since this guaranties
the geometry to be weakly curved. Note that in the near-extremal
limit lp→0 while ls is finite. The relation
between the Planck length lp and the string length ls is
explained in the previous section. See
[6, 7] for more on the near-extremal
limit of M5-branes with a transverse circle.
The important point is now that the type IIA NS5-brane is
non-perturbatively defined as an M5-brane with a transverse
circle. Therefore, the phases of N coincident near-extremal
type IIA NS5-branes are in fact the phases of N coincident
M5-branes with a transverse circle. We can thus get insight into
the thermodynamics of near-extremal NS5-branes by examining
near-extremal M5-branes with a transverse circle as obtained from
phases of six-dimensional Kaluza-Klein black hole solutions.
The relevant physical quantities for the near-extremal M5-brane
solutions with a transverse circle of the form (3.2) are
given as
[6, 7]^{6}^{6}6Note that these relations imply the Smarr formula 5TS=2(3−r)E [7].
where E is the energy, r is the relative tension (see
[31, 7] for definition), T is the
temperature and S is the entropy. We define here the number
χ for a given solution by K(R,v)=1−χR20/R2+⋯ in the asymptotic region R→∞, and Ah
is the value of A on the horizon: Ah=A(R,v)|R=R0. We
can then relate the physical quantities (3.3) for
near-extremal M5-branes with a transverse circle to the physical
quantities of the corresponding six-dimensional Kaluza-Klein black
hole by the map [7]
Here μ is the rescaled dimensionless mass, n is the relative
tension, t is the rescaled dimensionless temperature and s is
the rescaled entropy of the six-dimensional Kaluza-Klein black hole,
as defined in [32, 27].^{7}^{7}7For six-dimensional Kaluza-Klein black hole solutions
with metric of the form (3.1) we have
[6, 27]
μ=R20[32−χ],n=1−6χ3−2χ,t=1√AhR0,s=√AhR30.
We now consider the solutions for N near-extremal M5-branes with
a transverse circle obtained from transforming six-dimensional
Kaluza-Klein black hole solutions with local SO(4) symmetry, via
the map from (3.1) to (3.2). The Kaluza-Klein black
holes with local SO(4) symmetry come in three branches: The
uniform phase, the non-uniform phase and the localized phase. We
therefore naturally obtain the following three phases for N
near-extremal type IIA NS5-brane, i.e. for N near-extremal
M5-branes with a transverse circle [7]:
Uniform phase. This phase corresponds to N
near-extremal M5-branes uniformly smeared on a transverse circle.
This is the background given by (2.3). As discussed in
Section 2 this background corresponds to the usual
near-extremal NS5-brane background given by (2.1). The
relative tension is r=1/2 for the uniform phase. Note that from
the six-dimensional Kaluza-Klein black hole point of view the uniform
phase corresponds to the uniform black string phase.
Non-uniform phase. The non-uniform phase corresponds
to the new phase discussed in Ref. [7] of
near-extremal M5-branes non-uniformly distributed on the
transverse circle. The non-uniform phase emanates from the uniform
phase at the critical energy^{8}^{8}8Note that the critical energy
is obtained from the neutral Gregory-Laflamme mass using the map
(3.4).Ec/V5=1.54⋅(2π)−5l−6s. The slope in the (E,r) diagram (Figure
1) at this point is given by r(E)≃1/2−0.39⋅(2π)5l6s(E−Ec)/V5 [7].
Note that the non-uniform phase is mapped from the neutral
non-uniform black string phase in six dimensions. The solutions
for the neutral non-uniform black string were obtained numerically
in [33] and the behavior near the
Gregory-Laflamme mass was found in [34].
Localized phase. The localized phase corresponds to a
background of near-extremal M5-branes localized on the transverse
circle. For E→0 we have that r→0
corresponding to the fact that the background is well-described by
near-extremal M5-branes in eleven dimensional flat space (this
background is dual to (2,0) SCFT). The first correction to the
background due to the presence of the transverse circle was
obtained in Ref. [7] using the analytical
results on neutral small black holes on cylinders of
[35] (see also
[36, 37]). The slope of the curve
for the localized phase at the origin is given by r(E)≃[9ζ(3)/(25π2)](2π)5l6sE/V5 [7].
Beyond the analytical results for small energies, the localized
phase is known numerically by mapping the numerical solutions
for localized black holes in six-dimensional
Kaluza-Klein space obtained in Ref. [38]. These
newly obtained data is in fact what enables us in this paper to
gain new insights into the thermodynamics of the NS5-brane, as we
shall see in Section 4.
We have depicted the three phases^{9}^{9}9Note that for the non-uniform and localized phase there
are extra phases in the form of copies since the solutions can be
copied k times on the circle
[39, 30, 7].
At the level of solutions this means that given the solution (3.2)
we obtain for any k=2,3,...
a new solution with A′(R,v)=A(kR,kv), K′(R,v)=K(kR,kv)
and R′0=R0/k. This solution will have the physical quantities
E′=E/k2, r′=r.
in an (E,r) diagram in Figure
1. Note that part of the phase diagram for the
near-extremal type IIA NS5-brane was obtained in
[7], but here we add the extra numerical data of
[38].
Figure 1: Relative tension versus energy for the uniform (red),
non-uniform (blue) and localized (magenta) phase of near-extremal
NS5-branes.