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Abstract We show that the entropy of strings that wind around the Euclidean time circle is proportional to the Noether charge associated with translations along the T-dual time direction. We consider an effective target-space field theory which includes a large class of terms in the action with various modes, interactions and α′ corrections. The entropy and the Noether charge are shown to depend only on the values of fields at the boundary of space. The classical entropy, which is proportional to the inverse of Newton’s constant, is then calculated by evaluating the appropriate boundary term for various geometries with and without a horizon. We verify, in our framework, that for higher-curvature pure gravity theories, the Wald entropy of static neutral black hole solutions is equal to the entropy derived from the Gibbons-Hawking boundary term. We then proceed to discuss horizonless geometries which contain, due to the back-reaction of the strings and branes, a second boundary in addition to the asymptotic boundary. Near this “punctured” boundary, the time-time component of the metric and the derivatives of its logarithm approach zero. Assuming that there are such non-singular solutions, we identify the entropy of the strings and branes in this geometry with the entropy of the solution to all orders in α′. If the asymptotic region of an α′-corrected neutral black hole is connected through the bulk to a puncture, then the black hole entropy is equal to the entropy of the strings and branes. Later, we discuss configurations similar to the charged black p-brane solutions of Horowitz and Strominger, with the second boundary, and show that, to leading order in the α′ expansion, the classical entropy of the strings and branes is equal exactly to the Bekenstein-Hawking entropy. This result is extended to a configuration that asymptotes to AdS.

Abstract In string theory, a thermal state is described by compactifying Euclidean time on a thermal circle S β 1 $$ {S}_{\beta}^1 $$ , of fixed circumference. However, this circumference is a dynamical field which could vary in space, therefore thermal equilibrium is not guaranteed. We discuss a thermal state of type II string theory near and above the Hagedorn temperature and show that the circumference of the thermal circle can indeed be fixed and stabilized in the presence of a uniform isotropic flux. We solve the equations of motion derived from an action that reproduces the tree-level string S-matrix. We find solutions with the topologies of S β 1 $$ {S}_{\beta}^1 $$ × S 2 × M $$ \mathcal{M} $$ d−2 at a fixed temperature, which include a space-filling winding-mode condensate and a uniform Neveu-Schwarz Neveu-Schwarz flux supported on S β 1 $$ {S}_{\beta}^1 $$ × S 2. The solutions that we find have either a linear dilaton or a constant dilaton, in which case, we find solutions with either a cosmological constant or a Ramond-Ramond flux. We then compare our solutions to the cigar and cylinder backgrounds associated with the SL(2, ℝ)/U(1) coset theory, which include a winding-mode condensate but without flux. We also compare and contrast our solutions with the non-uniform Horowitz-Polchinski solution, which also possesses a winding-mode condensate and is characterized by an approximate thermal equilibrium near the Hagedorn temperature.

Abstract We consider the backreaction of the winding condensate on the cigar background. We focus on the case of the SL(2, ℝ) k /U(1) cigar associated with, e.g., the near-horizon limit of k NS5 black-branes. We solve the equations of motion numerically in the large k limit as a function of the amplitude, A, of the winding mode at infinity. We find that there is a critical amplitude, A c = exp(−γ/2), that admits a critical solution. In string theory, the exact SL(2, ℝ) k /U(1) cigar CFT fixes completely the winding amplitude, A s , at infinity. We find that in the large k limit there is an exact agreement, A c = A s . The critical solution is a cigar with a puncture at its tip; consequently, the black-hole entropy is carried entirely by the winding condensate. We argue that, in the Lorentzian case, the information escapes the black hole through this puncture.

Abstract We calculate the entropy of an asymptotically Schwarzschild black hole, using an effective field theory of winding modes in type II string theory. In Euclidean signature, the geometry of the black hole contains a thermal cycle which shrinks towards the horizon. The light excitations thus include, in addition to the metric and the dilaton, also the winding modes around this cycle. The winding modes condense in the near-horizon region and source the geometry of the thermal cycle. Using the effective field theory action and standard thermodynamic relations, we show that the entropy, which is also sourced by the winding modes condensate, is exactly equal to the Bekenstein-Hawking entropy of the black hole. We then discuss some properties of the winding mode condensate and end with an application of our method to an asymptotically linear-dilaton black hole.

Abstract We discuss interacting, closed, bosonic and superstrings in thermal equilibrium at temperatures close to the Hagedorn temperature in flat space. We calculate S-matrix elements of the strings at the Hagedorn temperature and use them to construct a low-energy effective action for interacting strings near the Hagedorn temperature. We show, in particular, that the four-point amplitude of massless winding modes leads to a positive quartic interaction. Furthermore, the effective field theory has a generalized conformal structure, namely, it is conformally invariant when the temperature is assigned an appropriate scaling dimension. Then, we show that the equations of motion resulting from the effective action possess a winding-mode-condensate background solution above the Hagedorn temperature and present a worldsheet conformal field theory, similar to a Sine-Gordon theory, that corresponds to this solution. We find that the Hagedorn phase transition in our setup is second order, in contrast to a first-order transition that was found previously in different setups.

Abstract We show that the state of the Hawking radiation emitted from a large Schwarzschild black hole (BH) deviates significantly from a classical state, in spite of its apparent thermal nature. For this state, the occupation numbers of single modes of massless asymptotic fields, such as photons, gravitons and possibly neutrinos, are small and, as a result, their relative fluctuations are large. The occupation numbers of massive fields are much smaller and suppressed beyond even the expected Boltzmann suppression. It follows that this type of thermal state cannot be viewed as classical or even semiclassical. We substantiate this claim by showing that, in a state with low occupation numbers, physical observables have large quantum fluctuations and, as such, cannot be faithfully described by a mean-field or by a WKB-like semiclassical state. Since the evolution of the BH is unitary, our results imply that the state of the BH interior must also be non-classical when described in terms of the asymptotic fields. We show that such a non-classical interior cannot be described in terms of a semiclassical geometry, even though the average curvature is sub-Planckian.