Decoupling limits of N=4 super Yang-Mills on R x S^3

We find new decoupling limits of N=4 super Yang-Mills (SYM) on R x S^3 with gauge group SU(N). These decoupling limits lead to decoupled theories that are much simpler than the full N=4 SYM but still contain many of its interesting features. The decoupling limits correspond to being in a near-critical region, near a point with zero temperature and critical chemical potentials. The new decoupling limits are found by generalizing the limits of hep-th/0605234 to include not only the chemical potentials for the SU(4) R-symmetry of N=4 SYM but also the chemical potentials corresponding to the SO(4) symmetry. In the decoupled theories it is possible to take a strong coupling limit in a controllable manner since the full effective Hamiltonian is known. For planar N=4 SYM on R x S^3 all the decoupled theories correspond to fully integrable spin chains. We study the thermodynamics of the decoupled theories and find the Hagedorn temperature for small and large values of the effective coupling. We find an alternative formulation of the decoupling limits in the microcanonical ensemble. This leads to a characterization of certain regimes of weakly coupled N=4 SYM in which there are string-like states. Finally, we find a similar decoupling limit for pure Yang-Mills theory, which for the planar limit leads to a fully integrable decoupled theory.Comment: 48 pages, 1 figure; added references, published versio

Thermodynamics of Squashed Kaluza-Klein Black Holes and Black Strings -- A Comparison of Reference Backgrounds --

We investigate thermodynamics constructed on different background reference spacetimes for squashed Kaluza-Klein (SqKK) black hole and electrically charged black string in five-dimensional Einstein-Maxwell system. Two spacetimes are possible to be reference spacetimes giving finite gravitational classical actions: one is four-dimensional Minkowski times a circle and the other is the KK monopole. The boundary of the SqKK black hole can not be matched perfectly to that of the former reference spacetime because of the difference in topology. However, the resultant classical action coincides with that calculated by the counterterm subtraction scheme. The boundary of the KK monopole has the same topology with that of the SqKK black hole and can be matched to the boundary of the black hole perfectly. The resultant action takes different value from the result given by using the former reference spacetime. After a brief review of thermodynamic quantities of the black hole solutions, we calculate thermodynamic potentials relevant for several thermodynamic environments. The most stable state is different for each environment: For example, the KK monopole is the most stable state in isothermal environment with fixed gravitational tension. On the other hand, when the size of the extra-dimension is fixed, the Minkowski times a circle is the most stable. It is shown that these two spacetimes can be reference spacetimes of the five-dimensional black string.Comment: 28 pages; references added, typo corrected;version accepted for publication in Class. Quantum Gra

On non-uniform smeared black branes

We investigate charged dilatonic black $p$-branes smeared on a transverse circle. The system can be reduced to neutral vacuum black branes, and we perform static perturbations for the reduced system to construct non-uniform solutions. At each order a single master equation is derived, and the Gregory-Laflamme critical wavelength is determined. Based on the non-uniform solutions, we discuss thermodynamic properties of this system and argue that in a microcanonical ensemble the non-uniform smeared branes are entropically disfavored even near the extremality, if the spacetime dimension is $D \le 13 +p$, which is the critical dimension for the vacuum case. However, the critical dimension is not universal. In a canonical ensemble the vacuum non-uniform black branes are thermodynamically favorable at $D > 12+p$, whereas the non-uniform smeared branes are favorable at $D > 14+p$ near the extremality.Comment: 24 pages, 2 figures; v2: typos corrected, submitted to Class.Quant.Gra

Formation of Five-Dimensional String Solutions from the Gravitational Collapse

We study the formation of five-dimensional string solutions including the Gregory-Laflamme (GL) black string, the Kaluza-Klein (KK) bubble, and the geometry with a naked singularity from the gravitational collapse. The interior solutions of five-dimensional Einstein equations describe collapsing non-isotropic matter clouds. It is shown that the matter cloud always forms the GL black string solution while the KK bubble solution cannot be formed. The numerical study seems to suggest that the collapsing matter forms the geometries with timelike naked curvature singularities, which should be taken cautiously as the general relativity is not reliable in the strong curvature regime.Comment: 17 pages, 10 figures, LaTeX, to appear in Class. Quant. Grav., a appendix and some discussions added, title change

New Phase Diagram for Black Holes and Strings on Cylinders

We introduce a novel type of phase diagram for black holes and black strings on cylinders. The phase diagram involves a new asymptotic quantity called the relative binding energy. We plot the uniform string and the non-uniform string solutions in this new phase diagram using data of Wiseman. Intersection rules for branches of solutions in the phase diagram are deduced from a new Smarr formula that we derive.Comment: 19 pages, 6 figures, v2: typos corrected, v3: refs. added, comment on bounds on the relative binding energy n added in end of section

Phase structure of matrix quantum mechanics at finite temperature

We study matrix quantum mechanics at finite temperature by Monte Carlo simulation. The model is obtained by dimensionally reducing 10d U(N) pure Yang-Mills theory to 1d. Following Aharony et al., one can view the same model as describing the high temperature regime of (1+1)d U(N) super Yang-Mills theory on a circle. In this interpretation an analog of the deconfinement transition was conjectured to be a continuation of the black-hole/black-string transition in the dual gravity theory. Our detailed analysis in the critical regime up to N=32 suggests the existence of the non-uniform phase, in which the eigenvalue distribution of the holonomy matrix is non-uniform but gapless. The transition to the gapped phase is of second order. The internal energy is constant (giving the ground state energy) in the uniform phase, and rises quadratically in the non-uniform phase, which implies that the transition between these two phases is of third order.Comment: 17 pages, 9 figures, (v2) refined arguments in section 3 ; reference adde

Thermodynamics of Large N Gauge Theories with Chemical Potentials in a 1/D Expansion

In order to understand thermodynamical properties of N D-branes with chemical potentials associated with R-symmetry charges, we study a one dimensional large N gauge theory (bosonic BFSS type model) as a first step. This model is obtained through a dimensional reduction of a 1+D dimensional SU(N) Yang-Mills theory and we use a 1/D expansion to investigate the phase structure. We find three phases in the \mu-T plane. We also show that all the adjoint scalars condense at large D and obtain a mass dynamically. This dynamical mass protects our model from the usual perturbative instability of massless scalars in a non-zero chemical potential. We find that the system is at least meta-stable for arbitrary large values of the chemical potentials in D \to \infty limit. We also explore the existence of similar condensation in higher dimensional gauge theories in a high temperature limit. In 2 and 3 dimensions, the condensation always happens as in one dimensional case. On the other hand, if the dimension is higher than 4, there is a critical chemical potential and the condensation happens only if the chemical potentials are below it.Comment: 37 pages, 4 figures; v2: minor corrections, references added; v3: minor corrections, to appear in JHE

Phases of Thermal N=2 Quiver Gauge Theories

We consider large N U(N)^M thermal N=2 quiver gauge theories on S^1 x S^3. We obtain a phase diagram of the theory with R-symmetry chemical potentials, separating a low-temperature/high-chemical potential region from a high-temperature/low-chemical potential region. In close analogy with the N=4 SYM case, the free energy is of order O(1) in the low-temperature region and of order O(N^2 M) in the high-temperature phase. We conclude that the N=2 theory undergoes a first order Hagedorn phase transition at the curve in the phase diagram separating these two regions. We observe that in the region of zero temperature and critical chemical potential the Hilbert space of gauge invariant operators truncates to smaller subsectors. We compute a l-loop effective potential with non-zero VEV's for the scalar fields in a sector where the VEV's are homogeneous and mutually commuting. At low temperatures the eigenvalues of these VEV's are distributed uniformly over an S^5/Z_M which we interpret as the emergence of the S^5/Z_M factor of the holographically dual geometry AdS_5 x S^5/Z_M. Above the Hagedorn transition the eigenvalue distribution of the Polyakov loop opens a gap, resulting in the collapse of the joint eigenvalue distribution from S^5/Z_M x S^1 into S^6/Z_M.Comment: 40 pages text + 15 pages appendix, 3 figures, latex; v2: one minor typo corrected and typeset in JHEP format; v3: computation of saddle points in Sec. 4.2 improved, discussion of stability of saddle points added in Sec. 6.2, minor changes, ref. adde

New Horizons for Black Holes and Branes

We initiate a systematic scan of the landscape of black holes in any spacetime dimension using the recently proposed blackfold effective worldvolume theory. We focus primarily on asymptotically flat stationary vacuum solutions, where we uncover large classes of new black holes. These include helical black strings and black rings, black odd-spheres, for which the horizon is a product of a large and a small sphere, and non-uniform black cylinders. More exotic possibilities are also outlined. The blackfold description recovers correctly the ultraspinning Myers-Perry black holes as ellipsoidal even-ball configurations where the velocity field approaches the speed of light at the boundary of the ball. Helical black ring solutions provide the first instance of asymptotically flat black holes in more than four dimensions with a single spatial U(1) isometry. They also imply infinite rational non-uniqueness in ultraspinning regimes, where they maximize the entropy among all stationary single-horizon solutions. Moreover, static blackfolds are possible with the geometry of minimal surfaces. The absence of compact embedded minimal surfaces in Euclidean space is consistent with the uniqueness theorem of static black holes.Comment: 54 pages, 7 figures; v2 added references, added comments in the subsection discussing the physical properties of helical black rings; v3 added references, fixed minor typo

Black Holes in Higher-Dimensional Gravity

These lectures review some of the recent progress in uncovering the phase structure of black hole solutions in higher-dimensional vacuum Einstein gravity. The two classes on which we focus are Kaluza-Klein black holes, i.e. static solutions with an event horizon in asymptotically flat spaces with compact directions, and stationary solutions with an event horizon in asymptotically flat space. Highlights include the recently constructed multi-black hole configurations on the cylinder and thin rotating black rings in dimensions higher than five. The phase diagram that is emerging for each of the two classes will be discussed, including an intriguing connection that relates the phase structure of Kaluza-Klein black holes with that of asymptotically flat rotating black holes.Comment: latex, 49 pages, 5 figures. Lectures to appear in the proceedings of the Fourth Aegean Summer School, Mytiline, Lesvos, Greece, September 17-22, 200