Hagedorn Temperature Research Articles

Strings in horizons, dissipation and a possible interpretation of the Hagedorn temperature

We consider the entanglement of closed bosonic strings intersecting the event horizon of a Rindler spacetime, and, by using some simplified (rather semiclassical) arguments and some elements of the string field theory, we show the existence of a critical temperature beyond which closed strings cannot be in thermal equilibrium. The order of magnitude of this critical value coincides with the Hagedorn temperature, which suggests an interpretation consistent with the fact of having a partition function that is ill defined for temperatures higher than it. Possible implications of the present approach for the microscopical structure of stretched horizons are also pointed out.

SEMICLASSICAL AND QUANTUM BLACK HOLES AND THEIR EVAPORATION, DE SITTER AND ANTI-DE SITTER REGIMES, GRAVITATIONAL AND STRING PHASE TRANSITIONS

An effective string theory in physically relevant cosmological and black hole space–times is reviewed. Explicit computations of the quantum string entropy, partition function and quantum string emission by black holes (Schwarzschild, rotating, charged, asymptotically flat, de Sitter dS and anti-de Sitter AdS space–times) in the framework of effective string theory in curved backgrounds provide an amount of new quantum gravity results as: (i) gravitational phase transitions appear with a distinctive universal feature: a square-root branch point singularity in any space–time dimensions. This is of the type of the de Vega–Sánchez transition for the thermal self-gravitating gas of point particles. (ii) There are no phase transitions in AdS alone. (iii) For dS background, upper bounds of the Hubble constant H are found, dictated by the quantum string phase transition. (iv) The Hawking temperature and the Hagedorn temperature are the same concept but in different (semiclassical and quantum) gravity regimes respectively. (v) The last stage of black hole evaporation is a microscopic string state with a finite string critical temperature which decays as usual quantum strings do in nonthermal pure quantum radiation (no information loss). (vi) New lower string bounds are given for the Kerr–Newman black hole angular momentum and charge, which are entirely different from the upper classical bounds. (vii) Semiclassical gravity states undergo a phase transition into quantum string states of the same system, these states are duals of each other in the precise sense of the usual classical–quantum (wave–particle) duality, which is universal irrespective of any symmetry or isommetry of the space–time and of the number or the kind of space–time dimensions.

Thermal AdS3, BTZ and competing winding modes condensation

We study the thermal physics of AdS(3) and the BTZ black hole when embedded in String theory. The exact calculation of the Hagedorn temperature in TAdS(3) is reinterpreted as the appearance of a winding tachyon both in AdS(3) and BTZ. We construct a dual framework for analyzing the phases of the system. In this dual framework, tachyon condensation and geometric capping appear on the same footing, bridging the usual gap of connecting tachyon condensation to modifications of geometry. This allows us to construct in a natural way a candidate for the unstable phase, analogous to a small black hole in higher dimensions. Additional peculiar effects associated with the Hagedorn temperature and the Hawking-Page transition, some to do with the asymptotic structure of AdS(3) and some with strong curvature effects, are analyzed and explained.

Thermodynamics of theories with sixteen supercharges in non-trivial vacua

We study the thermodynamics of maximally supersymmetric U(N) Yang-Mills theory on $\mathds{R}\times S^2$ at large $N$. The model arises as a consistent truncation of ${\cal N}=4$ super Yang-Mills on $\mathds{R}\times S^3$ and as the continuum limit of the plane-wave matrix model expanded around the $N$ spherical membrane vacuum. The theory has an infinite number of classical BPS vacua, labeled by a set of monopole numbers, described by dual supergravity solutions. We first derive the Lagrangian and its supersymmetry transformations as a deformation of the usual dimensional reduction of ${\cal N}=1$ gauge theory in ten dimensions. Then we compute the partition function in the zero 't Hooft coupling limit in different monopole backgrounds and with chemical potentials for the $R$-charges. In the trivial vacuum we observe a first-order Hagedorn transition separating a phase in which the Polyakov loop has vanishing expectation value from a regime in which this order parameter is non-zero, in analogy with the four-dimensional case. The picture changes in the monopole vacua due to the structure of the fermionic effective action. Depending on the regularization procedure used in the path integral, we obtain two completely different behaviors, triggered by the absence or the appearance of a Chern-Simons term. In the first case we still observe a first-order phase transition, with Hagedorn temperature depending on the monopole charges. In the latter the large $N$ behavior is obtained by solving a unitary multi-matrix model with a peculiar logarithmic potential, the system does not present a phase transition and it always appears in a ``deconfined'' phase.

Decoupling limits of 𝒩 = 4 super Yang-Mills on ℝ ×S3

We find new decoupling limits of = 4 super Yang-Mills (SYM) on × S3 with gauge group SU(N). These decoupling limits lead to decoupled theories that are much simpler than the full = 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 = 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 = 4 SYM on × S3 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 = 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.

Hagedorn strings and correspondence principle in AdS3

Motivated by the possibility of formulating a strings/black hole correspondence in AdS space, we extract the Hagedorn behavior of thermal AdS3 bosonic string from 1-loop partition function of SL(2, R) WZW model. We find that the Hagedorn temperature is monotonically increasing as the AdS radius shrinks, reaches a maximum of order of string scale set by the unitarity bound of the CFT for internal space. The resulting density of states near the Hagedorn temperature resembles the form as for strings in flat space and is dominated by the space-like long string configurations. We then argue a conjectured strings/black hole correspondence in AdS space by applying the Hagedorn thermodynamics. We find the size of the corresponding black hole is a function of the AdS radius. For large AdS radius a black hole far bigger than the string scale will form. On the contrary, when the AdS and string scales are comparable a string size black hole will form. We also examine strings on BTZ background obtained through SL(2, Z) transformation. We find a tachyonic divergence for a BTZ black hole of string scale size.

Spectrum and thermodynamic properties of two-dimensionalN=(1,1)super Yang-Mills theory with fundamental matter and a Chern-Simons term

We consider $\mathcal{N}=(1,1)$ super Yang-Mills theory in $1+1$ dimensions with fundamentals at large ${N}_{c}$. A Chern-Simons term is included to give mass to the adjoint partons. Using the spectrum of the theory, we calculate thermodynamic properties of the system as a function of the temperature and the Yang-Mills coupling. In the large-${N}_{c}$ limit there are two noncommunicating sectors, the glueball sector, which we presented previously, and the mesonlike sector that we present here. We find that the mesonlike sector dominates the thermodynamics. Like the glueball sector, the meson sector has a Hagedorn temperature ${T}_{H}$, and we show that the Hagedorn temperature grows with the coupling. We calculate the temperature and coupling dependence of the free energy for temperatures below ${T}_{H}$. As expected, the free energy for weak coupling and low temperature grows quadratically with the temperature. Also the ratio of the free energies at strong coupling compared to weak coupling, ${r}_{s\ensuremath{-}w}$, for low temperatures grows quadratically with $T$. In addition, our data suggest that ${r}_{s\ensuremath{-}w}$ tends to zero in the continuum limit at low temperatures.

High temperature properties of the MIT bag model

It is shown that there are thermodynamic inaccuracies in the MIT ‘bag’ model of hadrons at high excitation where the one parameter, the pressure, was used to fix the scale of the baryon mass. For high excitation there is a limiting (Hagedorn) temperature, and these two constants imply, via the adiabatic conditions, that the volume is also constant.

Hagedorn divergences and tachyon potential

We consider the critical behavior for a string theory near the Hagedorn temperature. We use the factorization of the worldsheet to isolate the Hagedorn divergences at all genera. We show that the Hagedorn divergences can be resummed by introducing double scaling limits, which smooth the divergences. The double scaling limits also allow one to extract the effective potential for the thermal scalar. For a string theory in an asymptotic anti-de Sitter (AdS) spacetime, the AdS/CFT correspondence implies that the critical Hagedorn behavior and the relation with the effective potential should also arise from the boundary Yang-Mills theory. We show that this is indeed the case. In particular we find that the free energy of a Yang-Mills theory contains ``vortex'' contributions at finite temperature. Yang-Mills Feynman diagrams with vortices can be identified with contributions from boundaries of moduli space on the string theory side.

SEMICLASSICAL (QUANTUM FIELD THEORY) AND QUANTUM (STRING) DE SITTER REGIMES: NEW RESULTS

We compute the quantum string entropy S s (m, H) from the microscopic string density of states ρ s (m, H) of mass m in de Sitter space–time. We find for high m (high Hm → c/α') a new phase transition at the critical string temperature T s = (1/2πk B )L cl c2/α', higher than the flat space (Hagedorn) temperature t s (L cl = c/H, the Hubble constant H acts at the transition, producing a smaller string constant α' and thus, a higher tension). T s is the precise quantum dual of the semiclassical (QFT Hawking–Gibbons) de Sitter temperature T sem = ħ c/(2πk B L cl ). By precisely identifying the semiclassical and quantum (string) de Sitter regimes, we find a new formula for the full de Sitter entropy S sem (H), as a function of the usual Bekenstein–Hawking entropy [Formula: see text]. For L cl ≫ ℓ Planck , i.e. for low [Formula: see text] is the leading term, but for high H near c/ℓ Planck , a new phase transition operates and the whole entropy S sem (H) is drastically different from the Bekenstein–Hawking entropy [Formula: see text]. We compute the string quantum emission cross-section σ string by a black hole in de Sitter (or asymptotically de Sitter) space–time (bhdS). For T sem bhdS ℓ T s (early evaporation stage), it shows the QFT Hawking emission with temperature T sem bhdS (semiclassical regime). For T sem bhdS → T s , σ string exhibits a phase transition into a string de Sitter state of size [Formula: see text], [Formula: see text], and string de Sitter temperature T s . Instead of featuring a single pole singularity in the temperature (Carlitz transition), it features a square root branch point (de Vega–Sanchez transition). New bounds on the black hole radius r g emerge in the bhdS string regime: it can become r g = L s /2, or it can reach a more quantum value, r g = 0.365 ℓ s .

The Hagedorn temperature in a decoupled sector of AdS/CFT

AbstractWe match the Hagedorn/deconfinement temperature of planar 𝒩 = 4 super Yang‐Mills (SYM) on ℝ × S3 to the Hagedorn temperature of string theory on Ads5 × S5. The match is done in a near‐critical region where both gauge theory and string theory are weakly coupled. On the gauge theory side we are taking a decoupling limit found in [7] in which the physics of planar 𝒩 = 4 SYM is given exactly by the ferromagnetic XXX1/2 Heisenberg spin chain. We find moreover a general relation between the Hagedorn/deconfinement temperature and the thermodynamics of the Heisenberg spin chain. On the string theory side, we identify the dual limit which is taken of string theory on a maximally symmetric pp‐wave background with a flat direction, obtained from a Penrose limit of Ads5 × S5. We compute the Hagedorn temperature of the string theory and find agreement with the Hagedorn/deconfinement temperature computed on the gauge theory side. Finally, we discuss a modified decoupling limit in which planar 𝒩 = 4 SYM reduces to the XXX1/2 Heisenberg spin chain with an external magnetic field.

SEMICLASSICAL (QFT) AND QUANTUM (STRING) ANTI-DE SITTER REGIMES: NEW RESULTS

We compute the quantum string entropy Ss(m, H) from the microscopic string density of states ρs(m, H) of mass m in Anti-de Sitter space–time. For high m, (high Hm → c/α′), no phase transition occurs at the Anti-de Sitter string temperature Ts = (1/2πkB)L cl c2/α′, which is higher than the flat space (Hagedorn) temperature ts. (L cl = c/H, the Hubble constant H acts as producing a smaller string constant α′ and thus, a higher tension). Ts is the precise quantum dual of the semiclassical (QFT) Anti-de Sitter temperature scale T sem = ℏc/(2πkBL cl ). We compute the quantum string emission σ string by a black hole in Anti-de Sitter (or asymptotically Anti-de Sitter) space–time (bhAdS). For T sem bhAdS ≪ Ts (early evaporation stage), it shows the QFT Hawking emission with temperature T sem bhAdS (semiclassical regime). For T sem bhAdS → Ts, it exhibits a phase transition into a Anti-de Sitter string state of size [Formula: see text], and Anti-de Sitter string temperature Ts. New string bounds on the black hole emerge in the bhAdS string regime. The bhAdS string regime determines a maximal value for H : H max = 0.841c/ls. The minimal black hole radius in Anti-de Sitter space–time turns out to be rg min = 0.841ls, and is larger than the minimal black hole radius in de Sitter space–time by a numerical factor equal to 2.304. We find a new formula for the full AdS entropy S sem (H), as a function of the usual Bekenstein–Hawking entropy [Formula: see text]. For L cl ≫ ℓ Planck , i.e. for low H ≪ c/ℓ Planck , or classical regime, [Formula: see text] is the leading term with its logarithmic correction, but for high H ≥ c/ℓ Planck or quantum regime, no phase transition operates, in contrast to de Sitter space, and the entropy S sem (H) is very different from the Bekenstein–Hawking term [Formula: see text].

Magnetic Heisenberg-chain/pp-wave correspondence

We find a decoupling limit of planar N=4 super Yang-Mills (SYM) on R x S^3 in which it becomes equivalent to the ferromagnetic XXX_{1/2} Heisenberg spin chain in an external magnetic field. The decoupling limit generalizes the one found in hep-th/0605234 corresponding to the case with zero magnetic field. The presence of the magnetic field is seen to break the degeneracy of the vacuum sector and it has a non-trivial effect on the low energy spectrum. We find a general connection between the Hagedorn temperature of planar N=4 SYM on R x S^3 in the decoupling limit and the thermodynamics of the Heisenberg chain. This is used to study the Hagedorn temperature for small and large value of the effective coupling. We consider the dual decoupling limit of type IIB strings on AdS_5 x S^5. We find a Penrose limit compatible with the decoupling limit that gives a magnetic pp-wave background. The breaking of the symmetry by the magnetic field on the gauge theory side is seen to have a geometric counterpart in the derivation of the Penrose limit. We take the decoupling limit of the pp-wave spectrum and succesfully match the resulting spectrum to the low energy spectrum on the gauge theory side. This enables us to match the Hagedorn temperature of the pp-wave to the Hagedorn temperature of the gauge theory for large effective coupling. This generalizes the results of hep-th/0608115 to the case of non-zero magnetic field.

Non-BPS D9-Branes in the Early Universe

We have investigated the finite temperature systems of non-BPS D-branes and D-brane-anti-D-brane pairs in the previous papers. It has been shown that non-BPS D9-branes and D9-anti-D9 pairs become stable near the Hagedorn temperature on the basis of boundary string field theory. This implies that there is a possibility that these spacetime-filling branes exist in the early universe. We study the time evolution of the universe in the presence of non-BPS D9-branes on the basis of boundary string field theory in this paper. We try to construct the following scenario for the early universe: The universe expands at high temperature and the open string gas on the non-BPS D9-branes dominates the total energy of the system at first. The temperature decreases as the universe expands. Then the non-BPS D9-branes become unstable at low temperature and decay through tachyon condensation. We obtain some classical solutions for Einstein gravity and dilaton gravity in the very simple cases.

Matching the Hagedorn temperature in AdS/CFT correspondence

We match the Hagedorn/deconfinement temperature of planar $\mathcal{N}=4$ super Yang-Mills (SYM) on $\mathbb{R}\ifmmode\times\else\texttimes\fi{}{S}^{3}$ to the Hagedorn temperature of string theory on ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. The match is done in a near-critical region where both gauge theory and string theory are weakly coupled. The near-critical region is near a point with zero temperature and critical chemical potential. On the gauge-theory side we are taking a decoupling limit found in Ref. [7] in which the physics of planar $\mathcal{N}=4$ SYM is given exactly by the ferromagnetic $XX{X}_{1/2}$ Heisenberg spin chain. We find moreover a general relation between the Hagedorn/deconfinement temperature and the thermodynamics of the Heisenberg spin chain and we use this to compute it in two distinct regimes. On the string-theory side, we identify the dual limit for which the string tension and string coupling go to zero. This limit is taken of string theory on a maximally supersymmetric pp-wave background with a flat direction, obtained from a Penrose limit of ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{S}^{5}$. We compute the Hagedorn temperature of the string theory and find agreement with the Hagedorn/deconfinement temperature computed on the gauge-theory side.

Strings, black holes and the extreme energy cosmic rays

In a large class of string and string inspired models the string excitation and black hole pictures invoked as an explanation of trans-GZK cosmic ray events are equivalent at high excitations. Single particle inclusive distributions are thermal, asymptotically at the Hagedorn temperature. Some observational consequences are briefly explored.

Matrix models for D-particle dynamics and the string/black hole transition

For a generic two-dimensional 0A string background, we map the Dirac–Born–Infeld action to a matrix model. This is achieved using a canonical transformation. The action describes D0-branes in this background, while the matrix model has a potential which encodes all the information of the background geometry. We apply this formalism to specific backgrounds: for Rindler space, we obtain a matrix model with an upside-down quadratic potential, while for AdS2 space, the potential is linear. Furthermore we analyse the black hole geometry with RR flux. In particular, we show that at the Hagedorn temperature, the resulting matrix model coincides with the one for the linear dilaton background. We interpret this result as a realization of the string/black hole transition.

ON THE SPECTRUM OF PRIMORDIAL DENSITY FLUCTUATIONS

The problem of the origin and spectrum of cosmic density fluctuations is discussed, especially with reference to the heterotic superstring theory of Gross et al. It is shown that primordial variation of the gravitational constant, due to its renormalization by higher-derivative terms [Formula: see text] dependent on the moduli, or the Harrison mechanism applied at the Hagedorn temperature T H = 1.14 × 1017 GeV (where we have argued that the four-dimensional theory decompactifies to higher dimensions), both naturally give rise to the scale-invariant spectrum of Zeldovich with a density contrast δ ~ 10-4-10-3, as required by the indeterminacy principle and for galaxy formation in the Universe.

Interferometry search for new forms of matter in A+A collisions

A method allowing studies of the hadronic matter at the early evolution stage in A+A collisions is developed. It is based on an interferometry analysis of approximately conserved values such as the averaged phase-space density (APSD) and the specific entropy of thermal pions. The plateau found in the APSD behavior vs collision energy at SPS is associated, apparently, with the deconfinement phase transition at low SPS energies; a saturation of this quantity at the RHIC energies indicates the limiting Hagedorn temperature for hadronic matter.

Random walks and the Hagedorn transition

We study details of the approach to the Hagedorn temperature in string theory in various static spacetime backgrounds. We show that the partition function for a {\it single} string at finite temperature is the torus amplitude restricted to unit winding around Euclidean time. We use the worldsheet path integral to derive the statement that the the sum over random walks of the thermal scalar near the Hagedorn transition is precisely the image under a modular transformation of the sum over spatial configurations of a single highly excited string. We compute the radius of gyration of thermally excited strings in $AdS_D\times S^n$. We show that the winding mode indicates an instability despite the AdS curvature at large radius, and that the negative mass squared decreases with decreasing AdS radius, much like the type 0 tachyon. We add further arguments to statements by Barbon and Rabinovici, and by Adams {\it et. al.}, that the Euclidean AdS black hole can thought of as a condensate of the thermal scalar. We use this to provide circumstantial evidence that the condensation of the thermal scalar decouples closed string modes.