The black hole interior and a curious sum rule

Amit Giveon 2 Nissan Itzhaki 0 Jan Troost 1 0 Physics Department, Tel-Aviv University , Ramat-Aviv, 69978, Israel 1 Laboratoire de Physique Theorique, Unite Mixte du CRNS et de l'E 2 Racah Institute of Physics, The Hebrew University , Jerusalem , 91904, Israel We analyze the Euclidean geometry near non-extremal NS5-branes in string theory, including regions beyond the horizon and beyond the singularity of the black brane. The various regions have an exact description in string theory, in terms of cigar, trumpet and negative level minimal model conformal field theories. We study the worldsheet elliptic genera of these three superconformal theories, and show that their sum vanishes. We speculate on the significance of this curious sum rule for black hole physics. Contents 1 Introduction The NS5-brane geometry The sum rule 4 Interpretations Introduction The Wick rotation of the Lorentzian black hole geometry to a Euclidean solution of gravity gives valuable information about quantum black holes. For the Schwarzschild solution, the Euclidean geometry reads ds2 = (1 2M/r)dt2 + 1 2M/r The region that corresponds to the exterior of the black hole r > 2M is known as the cigar geometry. From the cigar geometry, one reads off the relationship between the black hole temperature and mass and the value of the Euclidean action that corresponds to the entropy of the black hole [1], and one can fix the Hartle-Hawking wave function [2] associated with the Lorentzian extended geometry [3]. The region that corresponds to the interior of the black hole, r < 2M , does not appear to be physical. In the region between the horizon and the singularity, 0 < r < 2M , the space-time signature is (, , +, +). It includes two time directions (r and t). The big advantage of Wick rotation is thus lost; instead of getting a Euclidean space in which the path integral is well defined, we get a divergent path integral. The region beyond the singularity r < 0 is again more standard in that its signature is Euclidean. In this short note, we study an analogue of the Schwarzschild geometry which is the geometry near non-extremal NS5-branes. We consider the three regions identified above, after analytic continuation, and study those regions from the perspective of a string worldsheet theory. We establish a curious sum rule for the worldsheet elliptic genera associated to the three Euclidean regions. This could be viewed as an indication that in string theory the interior of the black hole plays an important role even after Wick rotation. The near horizon Euclidean geometry associated with k near extremal NS5-branes in the type II superstring is described (at 0 = 2) by the metric and dilaton [4]: where M is the energy density above extremality. There is an H-field flux on the threesphere corresponding to k NS5-branes. The geometry has a similar causal structure to the Schwarzschild geometry (1.1). In a first region I, outside the horizon r > 2M , we have a two-dimensional cigar geometry (times an SU(2)k Wess-Zumino-Witten model, a five-dimensional flat space, as well as a non-trivial dilaton profile). It has an exact conformal field theory description in terms of an SL(2, R)k/U(1) coset conformal field theory [57]. In a second region II, between the horizon and singularity, 0 < r < 2M , we obtain a bell geometry with all negative signature, corresponding to an SU(2)k/U(1) coset conformal field theory at a negative level k. The fact that the level is negative, and that we therefore have two time directions is also dictated by the vanishing of the total central charge of the string theory. The seeming singularity in the string coupling and metric are absent in the exact conformal field theory description (see e.g. [8, 9] for detailed discussions). Finally, region III, the region beyond the singularity, is obtained by setting r < 0. The resulting geometry is that of the trumpet, the vectorially gauged SL(2, R)k/U(1) coset conformal field theory (see e.g. [10] for a review). See figure 1 for a map of the regions in the Lorentzian black hole in Kruskal coordinates to the conformal field theory target spaces. For the regions outside the horizon and beyond the singularity, there are exact conformal field theory descriptions. The elliptic genera of these theories are known [1113]. It should be noted that the trumpet theory is T-dual to the Zk orbifold of the cigar (see e.g. [10]). The region between the horizon and the singularity corresponds to an N = 2 minimal model at a negative level. The elliptic genus of the conformal field theory with positive level was calculated longer ago [14]. We analytically continue the result into negative level k and we demonstrate below the curious fact that the sum of the three elliptic genera vanishes. The sum rule The identity we wish to prove reads: I + II + III cos(k) + MM(k) + orb(k) = 0 , Figure 1. The relationship between the Lorentzian black hole in Kruskal coordinates and the different regions obtained after Wick rotation. Regions I and III are Euclidean. Region II includes two time directions. n,w i mZ 1 z k1 qm n,w i q sk2 + (n4kkw)2 z nkkw qsk2 + (n+4kkw)2 . (3.3) The proof of the identity proceeds as follows. In a first step, we notice that the first and third term in equation (3.1) combine into a holomorphic expression. This is because of the identity (see the appendix of [15]): mZ rceoms = i131 X qkm2 z2m orb,rem . The second step in the proof uses the theory of elliptic functions, or the theory of Jacobi forms. In particular, we follow a proof of the fact that the sum of N = 2 minimal model characters and a ratio of theta-functions are equal [16]. Thus, we attempt to prove that the ratio of the two elliptic functions is one: We observe that both numerator and denominator have identical modular and elliptic transformations properties. The ratio is therefore an elliptic function of zk, and a modular form of weight zero. Moreover both numerator and denominator are holomorphic, due to the reasoning in step one of the proof. By inspection, it is clear that there are a finite number of poles and zeroes in the fundamental domain. It is therefore a ratio of a finite number of theta functions, which equals to one on the condition that the expression equals one in the q 0 limit [16]. This limit is easily taken, and the condition is verified. Therefore, we have proven the identity. Side remark. Note that in the special case of level k = 1 our identity encompasses a relation between the conifold and an analytically continued minimal model elliptic genus [17, 18]. It explains the relative factor of 12 between the two, which arises from the fact that at level k = 1, the cigar and its Zk orbifold are identical. Our identity also correctly captures the overall sign. The latter can be fixed from the Witten indices of the individual models. Interpretations A prosaic interpretation of the sum rule is that it is merely a curious mathematical fact. Indeed, the (...truncated)