On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles

Journal of High Energy Physics, Apr 2017

Recent work has demonstrated that black hole thermodynamics and information loss/restoration in AdS3/CFT2 can be derived almost entirely from the behavior of the Virasoro conformal blocks at large central charge, with relatively little dependence on the precise details of the CFT spectrum or OPE coefficients. Here, we elaborate on the non-perturbative behavior of Virasoro blocks by classifying all ‘saddles’ that can contribute for arbitrary values of external and internal operator dimensions in the semiclassical large central charge limit. The leading saddles, which determine the naive semiclassical behavior of the Virasoro blocks, all decay exponentially at late times, and at a rate that is independent of internal operator dimensions. Consequently, the semiclassical contribution of a finite number of high-energy states cannot resolve a well-known version of the information loss problem in AdS3. However, we identify two infinite classes of sub-leading saddles, and one of these classes does not decay at late times.

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On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles

Received: February On the late-time behavior of Virasoro blocks and a classi cation of semiclassical saddles A. Liam Fitzpatrick 0 1 3 Jared Kaplan 0 1 2 Open Access 0 1 c The Authors. 0 1 0 Charles Street, Baltimore, MD 21218 , U.S.A 1 Commonwealth Avenue , Boston, MA 02215 , U.S.A 2 Department of Physics and Astronomy, Johns Hopkins University 3 Department of Physics, Boston University Recent work has demonstrated that black hole thermodynamics and information loss/restoration in AdS3/CFT2 can be derived almost entirely from the behavior of the Virasoro conformal blocks at large central charge, with relatively little dependence on the precise details of the CFT spectrum or OPE coe cients. Here, we elaborate on the non-perturbative behavior of Virasoro blocks by classifying all `saddles' that can contribute for arbitrary values of external and internal operator dimensions in the semiclassical large central charge limit. The leading saddles, which determine the naive semiclassical behavior of the Virasoro blocks, all decay exponentially at late times, and at a rate that is independent of internal operator dimensions. Consequently, the semiclassical contribution of a nite number of high-energy states cannot resolve a well-known version of the information loss problem in AdS3. However, we identify two in nite classes of sub-leading saddles, and one of these classes does not decay at late times. AdS-CFT Correspondence; Black Holes; Conformal and W Symmetry; Con- 1 Introduction and summary Information loss and the late-time behavior of AdS correlators The semiclassical saddles of the Virasoro conformal blocks Summary of results The kinematic limit associated with late time behavior Solutions with A + B Solutions with A + B C 6= 0 at late times C ! 0 at late times A comment on degenerate operators Saddles at intermediate times Connecting the z Connecting early and late times in the z Semiclassical virasoro blocks for degenerate states An algebraic description of semiclassical degenerate correlators The monodromy method from the algebraic method A Comments on the generality of our results A.1 Forbidden singularities are forbidden from all Virasoro blocks A.2 Can forbidden singularities cancel between semiclassical blocks? A.3 Theories with larger symmetry algebras Saddles and their late-time behavior from the monodromy method Introduction and summary Black hole thermodynamics can be derived [1{10] as a universal consequence of the Virasoro symmetry algebra of AdS3/CFT2. Unitarity violation or `information loss' occurs in the semiclassical limit where Newton's constant1 GN ! 0 with GN E xed for bulk masses and 23c , where c is the central charge of the CFT2, but it is explicitly resolved [11, 12] by nonperturbative `e c' e ects. These phenomena occur independently within each irreducible representation of the Virasoro algebra. dashed lines drawn in to indicate past and future lightcones emanating from the operator OL(0). The lightcones appear as branch cuts in CFT correlators. These observations suggest a simple and unorthodox viewpoint: that in 2 + 1 dimensions, black hole thermodynamics, information loss, and its restoration essentially depend on only the Virasoro algebra (i.e. on the gravitational sector of AdS3), and more specifically on the behavior of the Virasoro conformal blocks. This means that some of the most fascinating features of quantum gravity in AdS3 appear to be largely independent of the spectrum and OPE coe cients of the CFT2 dual. This prospect opens up a powerful line of attack for understanding non-perturbative physics in quantum gravity through the determination of the non-perturbative behavior of the Virasoro conformal blocks. The present work will bolster these claims in two ways. First, we will explicitly identify all of the semiclassical `instantons' or `saddles' associated with the non-perturbative or `e c' e ects that can appear in the large c expansion of the Virasoro conformal blocks. Second, we will compute the late-time behavior of all semiclassical Virasoro blocks, with arbitrary internal and external states. We nd that in the heavy-light limit, which corresponds to a sub-Planck-mass object probing a BTZ black hole, the leading semiclassical Virasoro blocks all decay exponentially at late times, at a rate independent of the dimension of the exchanged state. Roughly speaking, this means that semiclassical high energy states do not play a privileged role in resolving information loss. However, in addition to the leading saddle, we will also identify two in nite classes of sub-leading semiclassical saddles, and one of these two classes of saddles does not decay at very late times. Before explaining these results we will review some background and motivation. Although any violation of unitarity in quantum gravity might be construed as `information loss', we would like to be more speci c and zero in on information loss associated with black holes. In this work we will focus on physical processes where initial data disappears because a high-energy microstate behaves too much like the thermal ensemble, i.e. as if the heavy state is a perfect thermal bath with an in nite number of degrees of freedom. This seems to be a good description of why encyclopedias are destroyed when they fall into semiclassical black holes. The information loss problem(s) we will discuss can be studied by observers who stay far away from a black hole, meaning that in the AdS/CFT context, the problem can be diagnosed using only CFT correlators [13], without reconstructing the AdS spacetime. Thus we are studying the `easier' information loss problem [11], rather than the `hard' problem of how to simultaneously maintain unitarity and e ective eld theory across the horizon of a black hole. From the point of view of CFT the problem has two parts: rst we must understand why the CFT appears non-unitary at large c, and then, from this vantage point, we need to identify the e ects that restore the information. We will be focusing on 4-point correlators in CFT2, which we will write as hOH (1)OH (1)OL(z)OL(0)i to emphasize that we are most interested in the physics of the heavy-light limit where a relatively light probe interacts with a black hole in AdS3. The dual AdS3 setup is pictured GN Mi are xed in the semiclassical large c limit. In this paper, unlike in many of our previous works, we will obtain results that hold for arbitrary values of both hcL and hcH . Our methods will pertain to the entire semiclassical regime. We recently discussed two sharp signatures of information loss [11] encoded in the heavy-light CFT correlator, `forbidden singularities' and the late time behavior [13] of the correlator. Here we will be focusing on the latter. If we consider a schematic decomposition of the heavy-light correlator in the OH OL ! OH OL OPE channel, which corresponds to conventional time slices of the cylinder pictured in gure 1, we nd hOH (1)OL(t)OL(0)OH ( 1)i = for some coe cients i and energy levels Ei. In the presence of a large number of terms, well-known statistical considerations apply (see [14] for a relevant, detailed discussion). In holographic CFTs, these suggest that the correlator will decay exponentially until it is of order e 21 SBH , where SBH is the heavy-state black hole entropy. After this point it enters a chaotic phase where it is typically very small, but occassionally returns to an order one value due to Poincare recurrences on timescales of order eeSBH . This channel seems to describe healthy unitary evolution as long as the i are discrete and nite, and in particular it is hard to see how information loss can occur. Individual unitary conformal blocks in this channel bear little resemblance to the physics of semiclassical gravity, which produces a continuous spectrum in the limit c ! 1 [1]. We will be focusing instead on the OLOL ! OH OH OPE channel hOH (1)OL(t)OL(0)OH ( 1)i = where Vh are Virasoro conformal blocks and Ph;h are products of OPE coe cients. We emphasize that this expansion always converges [15] away from OPE limits. In this channel the familiar features of semiclassical quantum gravity, including the physics of BTZ black holes, emerge in the large c limit [1{11, 16{20]. In particular, information loss appears [11, 13] as the exponential decay of correlators at arbitrarily late times. Previously, this behavior has been demonstrated [8, 11] in the limit of small hcL and hI , where hI is the dimension or AdS3 energy of the intermediate state OI c OL(z)OL(0) appearing in the light-light and heavy-heavy OPEs. These analyses left open the question of whether the information loss problem might be ameliorated by a nite collection of heavy semiclassical states with large nite hcI , or perhaps by e ects suppressed by hcL . We will demonstrate that this is not the case. All semiclassical Virasoro blocks decay exponentially at su ciently late times, and the asymptotic rate is independent of the intermediate operator dimension hI . The semiclassical saddles of the Virasoro conformal blocks We will be investigating the Virasoro conformal blocks in the limit of large central charge. We can de ne these blocks very explicitly by inserting a sum over all the intermediate states that are related to each other by the Virasoro algebra. The sum over an irreducible representation of Virasoro can be written using an intermediate state projection operator VhI (z) = OH (1)OH (1) @ A OL(z)OL(0) where L m are Virasoro generators, N is a Gram matrix of normalizations, and hI is the dimension of the primary operator/state labeling the irreducible representation. This sum has a natural interpretation as an OPE expansion in small z, with intermediate states of dimension hI + n contribution as zhI +n. At nite values of c and external and internal operator dimensions hi and hI , the OPE has radius of convergence 1, and this formula provides a non-perturbative de nition of the blocks. Although one can perform some interesting calculations [3] using this de nition, the present work will be based on a very di erent approach to the Virasoro blocks. Remarkably, Virasoro blocks have a large c expansion [21, 22] reminiscent of the semiclassical expansion of a path integral. This means that at large c with xed ratios the perturbative large c expansion has the structure where f is nite as c ! 1. We have also separated out the internal primary dimension I from the external operator dimensions. One justi cation for this expansion is that Virasoro blocks should be computable using Chern-Simons theory [23{28], a fact we hope to return to in the future. More practically, (1.5) has been checked against the nonperturbative de nition (1.4) to high orders in a small z expansion, and in other ways in various speci c limits. Our main goal in this paper will be to compute and classify the f ( i; z) and all other semiclassical `instantons' or `saddles' in certain kinematic limits. In section 1.3 we will discuss the meaning of both the perturbation series in 1c and the semiclassical saddles; then in section 2.1 we will review how these saddles can be identi ed and classi ed near Equation (1.5) is an asymptotic series with zero radius of convergence in the small parameter 1c . This leads us to expect that there must be sub-leading semiclassical `instantons' or `saddles' that contribute to V at a non-perturbative level. The sharpest way to see that (1.5) is missing subleading saddles is to study cases where some of the external operators are degenerate, i.e. have null Virasoro descendants. In this case, the blocks can be computed explicitly and one can see that additional saddles appear [11] after analytically continuing z along certain paths in the complex plane. The sub-leading saddles eventually dominate over the leading saddle in certain regimes [12]. However, to place the subleading saddles in a more general context where the exact form of the blocks may not be known, it is useful to motivate them in terms of Borel resummation of asymptotic series and resurgence phenomena [29{39]. To summarize the logic, we can de ne a Borel series B(s) by sending gn ! n1! gn in equation (1.5). Then we can try to de ne a function V = c as the Borel transform, which reproduces the series expansion if we expand B order-byorder in s. If the Borel integral converges and has no singularities on the real axis, then it can be viewed as another de nition of the Virasoro block V, and could be explicitly veri ed by comparison with equation (1.4). Singularities of B(s) in the s-plane lead to branch cuts when V is analytically continued [31] in its various parameters, which include i, c, and the kinematic variable z. We expect that we can identify these singularities in the Borel plane with corresponding semiclassical saddles [11, 40]. If we were able to obtain V from a path-integral computation, then we could go further (see [27] for a relevant review), identifying the semiclassical saddles as solutions to speci c equations of motion, and interpreting singularities in the Borel plane as due to path-integral Stokes phenomena [27]. Either way, there is a picture where equation (1.5), supplemented with a prescription for the proper Borel or path integration contour, may be su cient to de ne V [29{42]. As we analytically continue away from that limit, we will eventually cross Stokes lines (or equivalently, encircle singularities in the Borel plane), and so we will be forced to add contributions from sub-leading saddles. Bringing the non-perturbative e ects into view, one can write an improved expansion for V as a double-sum, or \transseries" [29], V(hi; I ; c; z) = nd that V is dominated by the We can think of the subleading terms p > 0 as instanton contributions that ll in the interior of AdS3 with particular geometries. As we continue further in z, the coe cients of the various saddles will change as we cross Stokes lines. At any nite value of z, we may saddle with minimum Re[fp] that appears2 with a non-zero coe cient in equation (1.7). gure 5), then we cross more and more branch cuts. In this case we do not expect that any particular saddle will dominate V, particularly because the coe cients of these saddles will depend directly on t. This means that knowing the large time behavior of the individual saddles fp will not immediately tell us the large time behavior of the Virasoro blocks.3 We emphasize that these statements are not just hypothetical | they have already been demonstrated [11] in the case of simple degenerate operators. In those special cases, the Coulomb gas formalism [43{45] plays a role analogous to that conjectured for ChernSimons theory in the general case. This strongly suggests that the Chern-Simons description of Virasoro blocks must reduce to the Coulomb gas in the case of degenerate operators [28]. We have also shown that the semiclassical saddles can be used [12] to obtain a useful and explicit result for a CFT2 correlator. In that example we found that there were an in nite number of saddles, and they themselves needed to be Borel resummed in order to obtain a nite result.4 Classifying the semiclassical saddles provides important information about the behavior of the blocks away from Stokes lines. In particular, it tells us the behavior of each individual instanton, including the leading saddle which governs the large time behavior of the semiclassical Virasoro blocks. This makes it possible to prove that the leading semiclassical Virasoro blocks all decay at the same exponential rate in the heavy-light limit. A crucial consequence of this result is that the information loss problem must persist after including semiclassical conformal blocks for heavy states with hI Summary of results Writing semiclassical contributions to the Virasoro blocks as e 6c f( i; I ;z) with z nd two discrete in nite classes of = z(1 1. All of the sub-leading saddles may be interpreted as `additional angles' in AdS3, as depicted in 2This innocuous-seeming statement may be subtle in practice [12] if there are an in nite number of Stokes lines as t increases. 3At least not in the OH OH ! OLOL OPE channel, or any channel where we must cross more and more 4Somewhat mysteriously, the result matches an AdS2 computation (compare 6.57 of [46] with 4.14 dotted, respectively) and I = 1, which is the vacuum Virasoro block. For ease of comparison we have made an overall constant shift in each f to emphasize that the late-time exponential decay is completely independent of the intermediate operator dimension. See gure 13 for more details. We de ne L to be real and H = 2 iTH to be purely imaginary, as this is the case of interest for correlators probing BTZ black holes. The rst in nite class are the decaying saddles, with asymptotic of the form dec(n) = n(1 where n must be an integer, as discussed near equation (2.43). The leading semiclassical always dynamically chosen (by following the solutions from early to late times) so that decyas as jtj ! 1 for real L and real TH . The other in nite class are the oscillating saddles which approach osc(m) = at late times, where the function (t) is given in terms of an arbitrary integer m, and is speci ed more precisely in and around equation (2.51). For physical values of the external operator dimensions, osc approaches a real number at late times; this is what indicates that these saddles oscillate rather than decay. Near the OPE limit z 0 classi cation of saddles becomes very easy, as it depends solely on the power-law behavior of V(z) as z ! 0. However, the connection between the classi cation of saddles near z 1 appears to be rather complicated, and we have not fully mapped it out. All semiclassical saddles have a leading large time behavior that is independent of the intermediate operator dimension hI . This somewhat surprising fact accords with all prior calculations. The dependence on hI appears only at sub-leading order in the late time limit, controlling the rate at which (t) approaches its asymptotic value. In we show the time-dependence of f for a variety of leading and sub-leading saddles to illustrate the range of possible behaviors. Some saddles can grow before they ultimately decay; it would be interesting to understand the parametric details of this phenomenon. The outline of this paper is as follows. Section 2 is devoted to an analytic classi cation of the saddles near z 1. We review the monodromy method and the classi cation near z saddles near z 0 in section 2.1. Then in section 2.2 we explain why the behavior of the 1 is su cient to understand their late time behavior. Finally in section 2.3 we solve the monodromy problem analytically near z 1 and classify the possible late time behaviors of the saddles with section 2.4. In section 3 we connect the saddles near z 1, and study their time-dependence and the way they approach their asymptotic behavior at late times. We match our analytic solutions to numerics, and in the process obtain many consistency checks. In section 4 we use a very di erent `algebraic' method to compute the semiclassical saddles associated with correlators of degenerate external operators. We also provide a partial derivation of the monodromy method, based on analytic continuation from the algebraic method, in section 4.2. We conclude with a discussion in section 5. In appendix A we make some more detailed comments about the generality of our results. Saddles and their late-time behavior from the monodromy method In this section we will classify all semiclassical saddles contributing to the Virasoro conformal blocks, and we will calculate their late-time behavior. We show that the leading semiclassical Virasoro blocks all decay exponentially, and at a rate that is independent of intermediate operator dimensions. We also identify two much larger classes of semiclassical solutions; one class decays even more rapidly at late times, while the other approaches a constant magnitude. These computations are tractable because the late-time behavior of the semiclassical Virasoro blocks can be determined by focusing on the region of small jz 1j, as illustrated in gure 5. So our strategy will be to solve the monodromy method directly in this kinematic region, keeping the phase of z First, in section 2.1, we will review the monodromy method and classify all saddles near the OPE limit z ! 0. In section 2.2 we explain the kinematics of the late time limit, and then in section 2.3 we perform the relevant monodromy method computations in this limit. In section 2.4 we classify the solutions analytically and determine their late time behavior, while in section 3 we analyze the behavior of the solutions at intermediate times. A remarkably e ective method for investigating the semiclassical functions fp( i; I ; z) is the \monodromy method" developed5 in [21, 22]. It involves solving the following math 5For detailed reviews see e.g. appendix C of [1] or appendix D of [47]; sometimes the method is known as \the method of auxiliary parameters". problem, which is formulated in terms of the auxiliary parameter (z) de ned by z)@zfp( i; I ; z): for some 2 2 matrix M . The auxiliary parameter (x) is xed by the condition that M where T (y; z) is the value the classical stress tensor takes at position y, while z is the holomorphic cross-ratio of equation (1.1). In this paper, we will be focused on the case 1 = 2 L; 3 = 4 H , mainly for simplicity but also because this is the case where the identity block can contribute. Then, T (y; z) = There will be two solutions, 1 and 2, and if we track their behavior as y follows a closed path encircling 0 and z, as pictured in gure 3, they must transform into a new linear combination according to By inserting into the conformal correlator an additional operator 2;1 that has a degenerate level 2 Virasoro descendant, (L 2 di erential equation 00(y) + T (y; z) (y) = 0; where hI is the intermediate operator dimension. Since the product of the eigenvalues is 1, we can simplify this to a constraint on the trace of M . The main challenge then is to determine M as a function of . This problem is di cult because solutions to equation (2.2) do not have a simple integral representation for general values of the parameters. It was solved by Zamolodchikov in the limit ! 1, and more recently by us [1] and others [6, 8, 48] in a perturbative expansion in L and I . The general problem is equivalent to the question of the monodromy of Heun's functions [49], and to the connection problem of Painleve IV [50], and the solution is not known in analytic form. Now suppose that one has obtained M as a function of . At this point, the reader may be wondering which of the distinct saddles fp (with xed values of all s) is related to . The answer is all of them! For generic values of parameters, solving the monodromy condition will produce an in nite number of solutions for . One of these solutions will be the corresponding to the leading saddle, but in fact the other solutions correspond to subleading saddles. In this way, the function M [ ] contains an enormous amount of information about the Virasoro (y), the solutions to equation (2.2), in order to de ne a 2 , holding the key not only to the leading semiclassical behavior but also to other non-perturbative e ects as well, as we discussed in section 1.3. One can argue that the sub-leading saddles should all have the same monodromy as the leading saddle by applying the original logic of the monodromy method to the full non-perturbative series (1.7). Adding 2;1(y) in the correlator produces the sequence the main point being that because 2;1 is a light (i.e. h2;1 O(1)) operator, its presence can shift the perturbative parts gp;n but not the non-perturbative pieces fp. Acting with the degenerate combination L 2 + 2(2h +1) L 1 produces the di erential equation (2.2) for p, but with given by the derivative of the corresponding fp. To leading order at large central charge, the monodromy of the total combination tot must still be given by the matrix M as y encircles 0 and z. However, it is manifest that the saddles do not mix under this monodromy, since they are completely independent of y. So each individually have a monodromy matrix M , and thus M [ = z(1 z)@zfp] is the same 6This argument is not completely rigorous since in fact it has never been proven directly from the de nition of the blocks even that the leading exponential in the large c limit (with i; I xed) grows like O(c), though there are by now a large number of highly non-trivial consistency checks of this behavior; we are assuming this O(c) scaling holds for all subleading saddles as well. The second unproven assumption is that the addition of the 2;1 light degenerate operator does not a ect the O(c) part of the exponentials. Part of the motivation for this assumption comes from Liouville theory where the action is O(c) and one can see explicitly that adding light operators creates an O(1) shift rather than an O(c) shift; however, this last statement concerns the full correlator and it is not clear how to turn it into a proof for the individual blocks. One might also be more skeptical of this assumption for the subleading saddles than for the leading ones, since for any given value of parameters, and to leading order at large c, only one saddle will dominate, while the others will be negligible; it is not clear if one can exhibit a (perhaps unphysical) region in kinematic or parameter space where each saddle dominates. 12 L = hinst = This small z behavior looks naively like an OPE singularity for an exchanged operator of weight is just the weight of the block itself, p = 0 : hinst = hI : hinst = (2p + 1)2): This is exactly the large c weight of the O2p+1;1 degenerate operators [11, 12]. The monodromy equation at small y immediately implies values of M [ ] at z = 0 are = 1 p1+4 48 L , i.e. the eigenz = 0 : eigenval(M )[ ] = Comparing with (2.5), we see that to have the correct monodromy, the square root term in (2.8) must be I + 2p for an integer p. So, This relation in turn gives us the small z behavior of the \saddles", Although we cannot solve for M [ ] in complete generality, we can immediately solve for it in the limit z 0. This limit includes the OPE limit, though it is more general since it also includes small z on every sheet after analytic continuation. To obtain M [ ] in monodromy as y encircles 0 and z (now equal), it is su cient to keep just the leading power, The appearance of negative weights may be surprising, in particular because this means that the saddles will produce stronger singularities at z 0 than the identity block OPE singularity. On further re ection, however, these negative weights and their corresponding singularities are in fact a necessary consequence of the structure of the large c expansion of the blocks. The point is that the subleading saddles are not present on the rst sheet in the complex z plane (we will de ne this region more precisely in later sections), but rather are generated upon analytic continuation. Passing to higher sheets, one does indeed stronger singularities in a large c expansion than the OPE singularity [12, 51{53]. In fact, we have already used the behavior of these saddles on the second sheet [12] to successfully compute the behavior Lorentzian correlators associated with chaos [51] in 2d CFTs [12]. Heavy states in AdS back-react on the geometry, and static eigenstates create geometries that at long distances look like de cit angles or BTZ black holes [1]. In the case of saddles contributing to the vacuum Virasoro block, the conformal weight hinst are negative, so the expression for the de cit angle is also negative [1, 54]: = 2 Therefore we expect that saddles to look like surplus angles (depicted in gure 4) on intermediate slices of AdS3 . Going beyond the leading behavior of each saddle at small z is straightforward and can be determined directly from the small z expansion of (1.4). This is because all dependence on the index p for the saddle enters in the combination ( I + 2p)2, so the corresponding In all examples where we can compute (z), it has a nite radius of convergence around 0. However, branch cuts develop at larger z, and we now turn to methods that will allow us to determine its behavior far from the OPE sheet. The kinematic limit associated with late time behavior We want to study the con guration of CFT operators depicted in gure 5, which can be interpreted as the 2-pt function of OL in the background of an energy (dilatation) hOH (1)jOL(t1)OL(t2)jOH ( Here the CFT lives on a cylinder, and t = t1 t2 is a Lorentzian time separation. The OH operators act in the in nite past and future and create a primary state, and so the correlator will be independent of the average time t1 + t2. Thus t and a relative angular coordinate on the cylinder will be the only physical variables. z = 1 gure depicts a generic con guration of the Lorentzian heavy-light correlator, with dashed lines drawn in to indicate past and future lightcones emanating from the operator OL(0). Due to the cylindrical geometry, as t increases the operator OL(t) must pass through the future lightcone of OL(0) at regular intervals. From the point of view of the conventional z-plane, depicted on the right, the multi-sheeted CFT correlator transitions to a di erent sheet for each 2 increment of t. obtained by taking j1 zj small. The semiclassical Virasoro blocks depend on jtj 1 through the by expanding at small jz 1j while keeping the phase of this quantity arbitrary. See gure 5 for the interpretation of this time-dependence in AdS. z = 1 This Lorentzian correlator can be obtained from the usual Euclidean 4-pt function, hOH (1)OH (1)OL(z; z)OL(0)i; by the substitution z ! 1 e it i and a simple overall rescaling, which is necessary to pass from the plane to the cylinder. Perhaps it is surprising that OH (1) and OH (0) in the Euclidean plane immediately produce the desired eigenstates in the in nite Lorentzian future and past. This follows from the standard i prescription: to pick out the lowest-energy state created by OH , we must give time a small imaginary part as we push the operator insertion into the in nite future or past, and we are left with limT ! 1 OH (e T ) which becomes OH (0) or OH (1). The analytic continuations of OL(0) and OL(z) are conventional; for a nice review in this context see [53]. Thus as t ! t + 2 , the z coordinate encircles 1. Because both correlators and conformal blocks typically have branch cuts from 1 to 1, we will pass onto a new sheet in the complex z-plane, as depicted in gure 5. The late time behavior will therefore be governed by the change in the value of the correlator or Virasoro block between sheets. In section 2.1, we saw that it is useful to compute the semiclassical Virasoro blocks The dependence on I has dropped out entirely. Continuing to late-time decay of the semiclassical Virasoro block in agreement with previous results. However, this analysis neglected two important ques7Strictly speaking, we will see that (z) for z 1 has non-trivial dependence on the phase of 1 z, so it is a bit of an abuse of notation at this point to write (1). However, we will see that in the limit t ! e it) approaches a constant, which is all that is needed in the present discussion. Astute readers might also worry about encountering forbidden singularities, but these are only present on the Euclidean sheet, and do not interfere with the large time analysis. At late times, z will encircle 1 again and again, so that the leading late-time behavior will be given by the residue 2 i (1). This dramatically simpli es our task to the computation of a single number!7 As an example of how this works, let us re-interpret the results of [1] in this light; we will also note two potential pitfalls. In [1] we found the leading saddle has I H (1 to leading order in L and I , where can take the limit z ! 1 unambiguously to nd 24 H . When H is positive and real, we f (z) = z) H (1 + 1 (1 z) H p 1 (1) generic values of y and (B) the region where jy monodromy matrix by matching the two solutions in an intermediate regime. 1, and then we construct the full we know that the semiclassical blocks are exponentially decaying, rather than exponentially We will deal with the rst question by explicitly computing the leading non-analytic pieces of (z) as a function of z 1. In our example here, it is easy to see that when z = 1 H is imaginary, at late times the non-analytic pieces such as (1 vanish at late times.8 We will obtain similar results for the general semiclassical blocks. The resolution to the second issue relates closely to that of the rst. Just as the perturbative (in ipped sign at t ! 1 as compared to t ! +1, the general solutions will also be dominated by di erent terms in these two regimes, so that we nd decay rather than growth whenever jtj ! 1. We also demonstrate the transition between these two behaviors numerically in section 3. z)@zf (z), where V = e 6c f(z), to leading order In this section we will compute (z) in an expansion in small j1 zj. As we have discussed in section 2.2, this is su cient to determine the late time behavior of the Virasoro conformal blocks in the semiclassical limit is only the absolute value of this quantity which is presumed small. z can be large, as it Our strategy will be to divide the monodromy path pictured in gure 7 into two regions where (A) y is far from z and 1 and (B) where y and z both approach 1.9 Remarkably, it is possible to solve equation (2.2) exactly in both limits. Then we will compute the full monodromy matrix M as the product of four matrices: the monodromy from circling 0 using solutions in region (A), a matching matrix between the two regions, the monodromy from region (B) and encircling z, and a nal matching matrix. We begin by studying equation (2.2) in region A, and so we take the limit z ! 1 immediately. From gure 7, this should be a good approximation when y is far from both 8Or for the opposite choice of sign of H , these terms grow in such a way that there is a cancellation between numerator and denominator in equation (2.19), so we obtain identical late-time behavior. 9More precisely, region A is de ned as the limit z ! 1 with y xed, and region B as the limit z ! 1 z and 1, but it must break down as y passes between these two points because T (y; z) has di erential equation after a re-de nition of parameters10 as y encircles 0, which is labeled as region A in gure 3. To solve in region B, we will change coordinates to so that region B is the limit jZj 1 with W xed, and the path B shown in just W ! e2 iW . Expanding equation (2.2) to leading order at small Z leads to another hypergeometric di erential equation, with solutions 12 = W C=2(1 22 = W 1 C2 (1 which are completely independent of Z. Once again, it is very easy to determine these solutions' relevant monodromy because we need only expand at small W , where the solutions behave as simple power-laws. In fact, as a 2 2 matrix the monodromy from encircling z in region B is identical to that from circling 0 in region A. The only remaining challenge is to match the solutions between region A and B in the regime where they are both valid, namely jW j accomplished with some standard hypergeometric identities,11 and can be summarized by 1. This can be 10Here and throughout this section, C is a hypergeometric parameter unrelated to the central charge. 11For reference, it is useful to note that 2F1(A; B; C; y) = B + C + 1; 1 y) B + C) 2F1(A; B; A + B 2F1(A; B; C; y) = (1 y) A B+C 2F1(C can be useful when matching. We will write the solutions as 11 = yC=2(1 21 = y 1 C2 (1 y) 21 (A+B C+1) 2F1(A; B; C; y); These solutions have the simple monodromy matrix H = L = by its trace. To leading order at small Z, the trace can be written Tr(M ) = 2 where it is convenient to separate out the coe cients of the di erent powers of Z that appear at this order: = Mmatch in the regime of overlapping validity of the two solutions. The result Mmatch is fairly complicated algebraically, but the end result is given in terms of it by the product. M = 2 matrix Mmatch de ned by The product m+m terms of sines: m+ = C) (A + B (A) (B) (A C + 1) (B that appears in the Z-independent term can be written purely in 4 sin( A) sin( B) sin( (A C)) sin( (B sin2( (A + B Equation (2.33) is a key result in this paper, and it contains the information that we will need to determine the late time behavior of all Virasoro conformal blocks. We expect that deviations from equation (2.33) at higher order in Z will take the form of a power series in Z multiplying each of the terms of order Z0; ZA+B C , and ZC A B. This means that the the terms in the trace (2.33). As depicted in gure 6, the limit jZj ! 0 should not be z has a phase that depends on Lorentzian time. Furthermore, as time evolves Z repeatedly crosses branch cuts, which typically extend from 1 to 1. These observation are crucial for understanding the relative size of the terms equation (2.33), since even for xed jZj their relative size will be time-dependent. 12The series coe cients of the higher order terms in Z can have singularities at some values of A; B; C, and for such values the higher order terms would not be negligible. The fact that the non-analyticity is captured by eq. (2.33) means that the size of such correction terms can be diagnosed on the rst sheet of the complex z plane, where a numeric analysis of the monodromy di erential equation is straightforward. and t are real. The approximations leading to the formula (2.33) for the trace 1, but t can take any value. So the magnitude We can parameterize the real and imaginary parts of Z as in gure 6, Z = e it; quantity depends on depends on time t and generically transitions from small values to large values around unless the imaginary part of A + B C vanishes. Of course, this C = so it is not quite correct to treat A + B C as a constant for all t. We will see, however, that all solutions have the property that A; B; C approach xed values at large t. We will nd two classes of solutions: those with A + B C 6= 0 at large times, and those where A + B C ! 0. For the former, from (2.37) we expect that either jZA+B C or its inverse will diverge at late times. So our strategy will be to nd all solutions for assuming that one of jZ (A+B C)j is large, and then to check for self-consistency of this assumption once we have the solutions in hand. Then we will study the case A+B nding a separate class of solutions to the monodromy condition. is quite complicated; we will solve it numerically in section 3, demonstrating the validity of our analytic solutions, and showing how they interpolate between small and large times. Solutions with A + B Let us assume that A + B C 6= 0 at late times I drastically simpli es because it is dominated by the singular term from (2.33). Therefore the leading order equation for (z) is either m+ = 0; Z(C A B) = 0; Z (C A B) and both are completely independent of I ! In fact, the value of I only becomes relevant when we study the rate at which either m coe cients vanish only when one or more of the functions in their denominators become in nite. It is then straightforward to write down the solutions to m = 0 in terms of A; B; C. To translate these solutions into solutions for in terms of H ; L, note that the inverse of the set of equations (2.22) has eight solutions. This is because (2.22) is invariant under the following symmetries: U1 : (A; B; C) ! (B; A; C); U2 : (A; B; C) ! (C U3 : (A; B; C) ! (1 as a function of H ; L, but that the value of A + B = 0 have the same C for each solution depends on whether it is a zero of m+ or m . Without loss of generality, we can therefore study all solutions to m+m = 0, which means that we must have sin( A) sin( B) sin( (A C)) sin( (B C)) = 0; using equation (2.22), we obtain the following two in nite towers of solutions, each parameterized by an integer n: and the full eight transformations generated by them. The rst of these leaves m C alone, whereas the second and third interchange the signs of m and (A+B = +1; late times; t ! 1; early times; t ! The two towers are related to each other by to simultaneously ipping the signs of both H . Note that n ! 1 We are not quite done because we must now check that these solutions are selfconsistent with the assumption on the magnitude of ZA+B C . For this, we have to keep = 0, since this determines the value of A + B C. After some straightforward book-keeping, one nds that n is equivalent B; m+ = 0 = 0 = sgn n where n and the sign on H correspond to those in (2.43). Consistent solutions are those satisfying (2.39), i.e. jZA+B C = 0 or jZC A B j ! 0 if m+ = 0. There are a number of di erent cases to treat depending on the relative sizes of the imaginary and real parts of H and L. For the sake of brevity, here we will assume that hH > 2c4 > hL, which includes the physically interesting case corresponding to a probe correlator in a black hole background. We can then also take Im( H ) 2 TH > 0 without loss of generality. requirement (2.39) as Now we have Im(A + B determined by the sign of Im(A + B C). Using (2.44), we can summarize the consistency block between z 1. Legend for the curves is as in gure 9: (black, solid ) is the large /small z approximation, (blue, dotted ) is the small j1 zj approximation, and (red, dashed ) is the complex conjugate of the entire equation simply sends t ! t and A; B; C ! A ; B ; C . There is no e ect on the r.h.s. since I is always either pure real or pure imaginary.14 By inspection of (2.22), conjugating A; B; C just conjugates , and has no e ect on real z = 0 is this case. For the choice of parameters L = 0:99, the initial value of the leading saddle at = 1 2 L . As discussed in the previous subsection, one can identify which saddles and (2.33), with the exact solution transitioning from one to the other approximation. This the saddles to late times using (2.33). In gure 12, we show the trajectory of several saddles that end on one of the late-time solutions with non-vanishing imaginary part. These are the solutions from equation (2.43), point of each curves always lies on the real axis. By comparison with gure 11, one can 0, i.e. it is the leading OPE saddle. A similar analysis can be applied to non-vacuum blocks. In gure 13, we show the leading saddle for the non-vacuum block with I = i=2 and I = 5i=4. In these cases, the early and late times, so it has negative (positive) imaginary part at t ! 1 ( 14We have used the fact that the m are just products of functions whose arguments are linear 15In fact, this conclusion is more general than the small j1 zj regime. Starting from the monodromy di erential equation (2.2), (2.3), the complex conjugate of any solution (y; z) is a solution to the same simply changes the sign of the exponent of the eigenvalues of the monodromy matrix, which does not a ect its overall trace. Thus if is a solution at t is a solution at t, regardless of the magnitude of j1 n = 2 n = 1 n = n = 3 n = 2 n = 3 n = 2 n = 2 n = 0 n = as we interpolate between early and late time behaviors in the decaying (2.43) class. Recall that Im( ) (the vertical axes) determines the exponential decay or growth rate of the Virasoro blocks. The upper left plot shows the full region covering all solutions, and the remaining three plots show scaled up regions of the rst plot for better visibility. Parameters are chosen to be H = i; L = 0:99; j1 = n(1 L H for n = zj = 0:01. Dots indicate the analytic solutions 2; 1; : : : 3; each path interpolates between two points with the same n and opposite choice of sign for H . All of these saddles decay exponentially its contribution to the Virasoro block decays asymptotically to 0 at t ! 1. However, it has a \ gure 8" pattern where the imaginary part of changes sign at intermediate t, and therefore the saddle actually grows for a period of time before decaying. This is visible in z)) to obtain the and subsequent decay are extremely rapid. that have non-zero imaginary parts asymptotically, and some are from the oscillating class (2.51) that have zero imaginary parts asymptotically. In gure 14, we show sev 1. While there may exist a simple rule for which values of p map to saddles in the decaying vs oscillating classes, we have not found such a rule and the most we can say is that for generic values of I ; L; and H , the two classes of solutions are interspersed with each other as one looks at greater and greater jpj. - respectively). Left: interpolation between z 1. Legend for the curves is as in gure 9: (black, solid ) is the large /small z approximation, (blue, dotted ) is the small j1 zj approximation ( becomes complex where the curve ends), and (red, dashed ) is the exact numeric result. Middle: Trajectory of the p = 0 saddle at zj = 0:011 from t = 1 (where Im( ) > 0) to t ! 1 matches the value at grow before their ultimate late-time decay. 1 in the left plot. Right : plot of the term f6 = 1c log(V) in the exponent of the block at xed f (0) = 0. Parameters zj = 0:022, shown at various levels of magni cation. Black dots indicate the asymptotic positions of decaying saddles, however most saddles are seen to be oscillating saddles in these plots. Parameters are L = 0:99; H = i; I = 0:97; j1 zj = 0:022. Semiclassical virasoro blocks for degenerate states In this section we will discuss a di erent method of calculation using degenerate states and operators. By de nition, these operators are annihilated by a polynomial in the Virasoro generators L n, which means that their correlators satisfy di erential equations. Expandxed degenerate operator Or;1, the resulting algebraic equations for e cient to study both numerically and analytically, and so give us the solutions for the semiclassical saddles for all values of z. In particular, at any z, the problem of nding the behavior of the saddles is reduced to nding the eigenvalues of a nite-dimensional matrix, which is numerically very e cient. In the limit that z ! 1, we can solve these algebraic equations for every hr;1 degenerate state. For degenerate operators this limit is much simpler than for general operators, because the z ! 1 limit no longer depends on the phase of 1 z. Rather, the z ! 1 limit corresponds to a crossed channel OPE limit, and the semiclassical solutions simply approach values corresponding to the allowed dimensions of operators in this crossed channel. We will see that we reproduce the results of the large c limit of such dimensions derived in [11, 55]. By analytic continuation of these results, we obtain an alternate derivation for the in nite class of solutions derived in section 2.4.1. The algebraic method can also be used it to provide a partial derivation of the monodromy method itself, as we explain in section 4.2. An algebraic description of semiclassical degenerate correlators Bauer, Di Francesco, Itzykson, and Zuber have developed a systematic method [56, 57] (for a review see [45] section 8.2 and exercise 8.8) for obtaining the combination of Virasoro generators that annihilate degenerate states. When studying degenerate states, it is convenient to the write the central charge in terms of a paramter b via c = 1 + 6 b + We will take the limit of large c via the limit of large b. Using this notation, the degenerate states have dimensions hr;s = parameterized by the positive integers r; s; we will be focusing on states with dimension hr;1. Now let us de ne the null state equations. Let Dr;1 be the following matrix:16 Dr;1 = where J are matrix generators of the spin (r 1)=2 representation of SU(2): (J )ij = (J+)ij = (J0)ij = ( i;j+1 (j = 1; 2; : : : ; r i) i+1;j (i = 1; 2; : : : ; r [J+; J ] = 2J0; [J0; J ] = The factor of b12 compensates for the single power of b2 or h that will be obtained when L m 1 acts on a semiclassical Virasoro block. The simplest example of this formalism is the case r = 2, where we obtain 2;1(b) = det 12 are annihilated by this 16This formula di ers from the analogous one in [11, 45, 56, 57] by some factors of b; the di erence is equivalent to rescaling the representations (4.4) for L by factors of b so that the algebra is unchanged, and rescaling Dr;1 itself by an overall power of b. We have also taken b ! b 1, which is a choice of convention Note that J and J+ are nilpotent. The degenerate state equation is obtained by eliminating f1; : : : ; fs 1 from the equations Formally, this can be re-written as 0 = We can obtain a di erential operator that annihilates the four-point function and commuting all Lns to the right. The Lm act on an operator O(z) via 0 = hhr;1j ( r;1)y Or;1(0)O(x)O(y)i [Lm; O(z)] = zm (h(m + 1) + z@z) O(z): The Ln in yr;1 all annihilate Or;1(0) because it is a primary and all n > 0. So within the correlator, we need only act the Virasoro generators on O(x) and O(y), which we take to have dimension h. We would like to obtain the resulting di erential equation in the semiclassical limit. We will approximate the correlator as itself as hhr;1jOr;1(0)O(x)O(y)i = identifying z = 1 . We wish to keep only the leading results at large b2 / c / hL. Raising hhr;1j with an Lm acts on the correlator (4.10) as the di erential operators Lm = xm+1@x + ym+1@y + hL(m + 1)(xm + ym): We are taking hL / b2, so we can ignore actions of the derivatives in Lm that do not produce factors of b2 or hL. This means that we can ignore actions of the derivatives in Lm on kinematic factors such as xn and yn within Ln, and so e ectively when acting on the correlator in equation (4.8). Using these simpli cations, we nd that when the Lm act on the correlator, ym+1 + (m + 1) (xym Using equation (4.6) and taking y ! 1 and x ! 1 z leads to the determinant formula 0 = det J where we have replaced (z) = z(z 1)f 0(z) and performed the sum over m in equation (4.14) in closed form to simplify the result. The factors of b12 have canceled against factors of b2 and h to produce a result that is xed in the semiclassical limit. Crucially for the following, the elements of the above matrix are all now just numbers, and so the determinant is no longer de ned formally but is instead just has its standard meaning. This fact allows us to multiply the matrix inside the determinant by any invertible matrix, since doing so does not change the condition that the determinant vanishes. In particular, we can multiply by (1 z)J+), yielding 0 = det hH(z) + (z)i; H(z) = (1 Equation (4.15) is exactly the condition that (z) is the set of eigenvalues of of H(z)! This equation also provides an rth order algebraic equation for (z) that can be solved in closed form for the rst several values of r. As a check, let us consider what happens in the limit z ! 1. In this case, we know all solutions for should reduce to operators allowed by the fusion rules of the degenerate operator. The dimensions of allowed operators are most easily seen in the Coulomb gas formalism for the shift of the weights of operators when they fuse with a degenerate operator at large central charge.17 That is, a degenerate operator Or;1 can fuse with a general operator OL to make a new operator O0 with weight h0 satisfying h0 = hL where Q = (b + 1=b) and the integers n = 0; : : : ; Coulomb gas charge of OL by related to h0 by Equation (4.16) follows immediately from the fact that fusing with Or;1 can shift the 1) 2b for any n in (4.17). The parameter (1) = h0 since the OPE singularity at z z)h0 hL hr;1 . Taking the limit b ! 1, one reduces to the following tower of pairs of solutions: For comparison, take (4.15) in the limit z ! 1, in which case it greatly simpli es to (1) = 0 = det (1 L = 4bh2L , analytically continuing r = 17See e.g. [11, 55] for similar observations in the large c limit. This can be solved in closed form for all values of r, yielding the pairs of solutions in (4.19). H , and letting n range over all integers. The ambiguity of sign just corresponds to the choice of sign of square roots in can also see that these results accord with the derivation of BTZ quasi-normal modes given in [11, 55]. So in the semiclassical limit our algebraic method matches our results from the monodromy method. Note however that we have not obtained the other in nite class of semiclassical solutions discussed in section 2.4.2. The formula (4.15) is quite useful since it allows us to nd the instantons at general z for OH degenerate by computing the eigenvalues of a matrix. Such computations are fairly e cient, and so we can quickly get a sense of how the instantons behave for a large range of r and hL values. For instance, in 0, and the pairwise merging of eigenvalues at z the solutions merge in pairs, they move o into the complex plane. Once we turn on a non-zero hL, the behavior becomes much more interesting. For develop complex parts. This is shown in gure 16 and 17. Perhaps surprisingly, the degenerate operators have only \decaying" saddles, i.e. saddles in the class (2.43) that pick up non-zero imaginary parts at z indication that the generation of non-decaying saddles under analytic continuation in z is more subtle than that of the decaying solutions, or it could be an indication that instantons for conformal blocks of degenerate operators simply do not contain key information about the generic case. The monodromy method from the algebraic method We would like to `analytically continue' results from the methods of the previous section section in order to re-derive the monodromy method for the vacuum conformal block. However, the methods of section 4.1 involved deriving di erential equations of rth order from an r r matrix appropriate for hr;1 degenerate states. Throughout it was crucial that r be an integer | so how can we analytically continue the dimensions of a matrix? This is easy if we interpret the matrix as a product of lie algebra generators, and then generalize to an in nite dimensional representation of the group. So our rst step will be to reformulate the method using a representation where J ; J0 act as di erential operators in the case where r is an integer. Our starting point will be equation (4.15), which states that a certain matrix, written in terms of the su(2) matrices J+ and J , must have a zero eigenvalue. For integer r we used the spin r 1 representation. In order to treat general values of r, we will use the following su(2) representation = iy@y2 + i(1 J+ = iy J0 = y@y + where we chose a particularly simple J+ because it appears most frequently in equation (4.15). The space of vectors of the theory is by de nition spanned by the eigenvectors of J0 with eigenvalues that this exactly corresponds to the case where the vector space that J ; J0 act on is polynomials of degree r. Naively, in this representation, the statement that the determinant of Dr;1 vanishes is just the condition that `matrix' H(z) + (z) of equation (4.15) has a zero eigenvector. However, because the elements of H(z) are no longer pure numbers, we must be more careful and return to the \formal" de nition of the determinant and consider how to process it correctly. So let us analyze eq. (4.14) in this context. The formal de nition of the determinant in this expression is that Dr;1 acting on the our new representation, this just means that the condition for is that Dr;1 acting on the zero eigenvalue, it does change whether or not the functions that it acts on are degree-r polynomials or some other space of functions. The point of this discussion is that it tells us only a certain space of functions are acceptable as zero eigenfunctions of the matrix H(z) + (z). Uplifting from the space is not obviously straightforward. When L = 0, one of the solutions to the monodromy matrix is in fact the nite order polynomial in (4.22) (after factoring out some simple prefactors, see below), so the uplift is trivial. In any case, though, once we move away from integer r it is necessary to justify what in nite dimensional space of functions one should allow and we do not have a sharp argument using the present method that it should be the space of functions with the proper monodromy. A somewhat ad hoc generalized criterion would be that the series expansion in y should converge in a particular region; this is clearly satis ed by nite order polynomials, and any cycles in the region of convergence would have trivial monodromy as a necessary consequence. Demanding that the operator H(z) + (z) has a zero eigenvector is equivalent to the di erential equation y (y + 1) (yz + 1) (y) = 0 1 r (y; z), where we recall that H = hcH = 6hbH2 , and we sent z to simplify the result. To analytically continue, we need only for the function identify r = H = 6 H + y (y + 1) (yz + 1) (y) = 0: Finally, let us send z back via z ! 1 (y) = 0: this factor becomes / (1 cycles where y circles 0 and z but not 1. This coincides exactly with the di erential equation associated with the monodromy method, equations (2.2) and (2.3), once we shift our de nition of by whereas in the monodromy method of section 2 we absorb the z 2hL factor into the definition of f (z). The factor of y 2 (y; z) introduces a non-trivial around certain cycles, but tracking our changes of variables one sees that y) 2 , so that no monodromy is introduced by this factor on We have now derived the di erential equation of the monodromy method by analytically continuing results relevant to degenerate external states. This derivation started from the case of degenerate external operators, i.e. H = r was an integer, so H was restricted to speci c values. However, in the nal formulation, the parameter H appears only as an argument in the di erential equation (4.25), which is trivial to analytically continue by H take arbitrary complex values. More signi cantly, the complete formulation of the monodromy method also requires the statement that the space of allowed functions generalizes to the space of functions with trivial monodromy around certain cycles. We have made some heuristic comments to try to motivate this last step, but we do not have a proof along these lines that this is the correct analytic continuation. Perhaps the best that can be said is that such a generalization seems natural, and in any case is the one that reproduces the standard monodromy method. Even with the AdS/CFT correspondence in hand, it has been di cult to resolve the most conceptually fascinating conundrums of black hole physics | while AdS/CFT may be an exact description of quantum gravity in principle, it has yet to become one in practice. Some of the di culty arises from limitations in our understanding of the AdS/CFT dictionary, in particular how or even if one can generally reconstruct bulk observables in terms of CFT dynamics. However, another major obstacle is that we often do not know how to explicitly compute key observables of the boundary CFT. Such quantities include boundary CFT correlators related to \easier" versions of the information loss problem [11, 13]. Recently, there has been remarkable progress towards an understanding of the robust features of AdS3/CFT2 using methods closely related to the conformal bootstrap [1{4, 8, 10{12, 16{20, 51, 52, 58{70]. An organizing principle underlies many of these results: black hole physics emerges directly from the structure of the Virasoro conformal blocks at large central charge, and is largely independent of the precise details of the CFT spectrum and OPE coe cients. This suggests that it may be possible to understand information loss and unitarity restoration without solving any speci c holographic CFT. To understand AdS3 quantum gravity we appear to need all of the foundational principles of conformal symmetry, modular invariance, locality, and quantum mechanics, but not so much more. As a starting point, it was important to understand how to reproduce semiclassical results on the geometry and thermodynamics of strongly-coupled gravitational backgrounds directly from CFT [1, 2, 8, 16, 51, 59, 69, 71]. In other words, it was crucial to see how unitary CFTs mimic the information-destroying e ects of black holes, pointing the way towards an understanding of what is missing from the semiclassical gravity description. We have [11, 12] explicitly computed some of the `e c' e ects responsible for the restoration of unitarity. However, there remain important kinematical regimes, such as late Lorentzian times, where tractable and su ciently accurate approximations to the Virasoro blocks are not yet known. In this paper we have focused on identifying and classifying non-perturbative e ects in AdS3 gravity through the study of the remarkable semiclassical `saddles' of Virasoro conformal blocks. The motivation for this investigation was twofold. First, black hole information loss manifests as unitarity violation in semiclassical correlation functions, and this violation is present at the level of the individual Virasoro blocks. Studying the large c saddles teaches us about the nature of the exponentially small corrections involved in the restoration of unitarity. The most important lesson from the present analysis is that the leading semiclassical contribution to Virasoro blocks with hH > 2c4 > hL decays exponentially18 at late times, and at a universal rate independent of internal operator dimensions, as we illustrated with gure 2. This feature was observed previously for conformal blocks of light external and intermediate states [1], but it was far from obvious that it would persist for all semiclassical blocks. We have also identi ed other saddles that either decay or approach a constant magnitude at late times. The latter may play an important role in resolving information loss, but as we explained in section 1.3, their mere existence is not su cient by itself. It would be very interesting to understand how the semiclassical saddles that we have uncovered here relate to the non-perturbative resolution of forbidden singularities, and to the `master equation' [11] that seems to be a rst step towards a determination of the true late-time behavior of the vacuum Virasoro block. Second, we hope that our analysis may provide key insights towards a complete path integral description of the Virasoro blocks [23{28]. Ideally, one could start directly from an action, possibly the Chern-Simons action for holomorphic `gravitons', together with a coherent, self-contained set of rules for when particular saddles contribute. Knowledge of the saddles likely gives us a very precise indication of what semiclassical e ects will look like in a gravitational context. The simple reason for this optimism is that the saddles directly indicate the corresponding background value for the boundary stress tensor, and we can attempt to extend the stress tensor to the bulk metric. We expect our metrics should be vacuum metrics since in the semiclassical limit sources are localized on 18Note the order of limits: a semiclassical block with nite hI will decay exponentially at su ciently late times. An in nite sum over blocks including hI ! 1 might not decay; this deserves further study. geodesics. A general vacuum metric can be written ds2 = L(y)L(y) dydy + where L(y) and the boundary stress tensor T (y; z) are related by T (y; z) = and T (y; z) is given in terms of by (2.3). Restoring the explicit dependence on the positions xi of the OH and OL operators, T (y; z) can be written19 T (y; xi) = x15x25x35x45 z) L)z= xx1123xx3244 # ; y. Interestingly, generic saddles therefore depend on the positions x3; x4 of the probe operators OL even in the limit L ! 0. Knowledge of the O(c) part, i.e. log V = su cient to determine their absolute contribution to the blocks, since there is an unknown O(1) prefactor. This prefactor vanishes for all but the leading saddle in the z limit, and to ascertain the contribution from the subleading saddles we need to identify Stokes lines/walls. This should be possible, since Stokes lines occur when leading and subleading saddles have equal imaginary parts. However, obtaining a sharp prediction for the prefactor of the subleading saddles is more challenging. The prefactor is in principle known quasi-analytically for blocks with degenerate external operators from the crossing matrices [43, 44, 72], but the existence of the non-decaying saddles which do not appear in degenerate conformal blocks suggests that perhaps there are limitations to analytic continuation from degenerate states. Alternatively, it may be possible to extract the prefactor from the results of [73, 74] for the braiding matrices of Virasoro conformal blocks. 6c f , of the saddle exponents is of course not Acknowledgments We would like to thank Hongbin Chen, Tom Hartman, Ami Katz, Zohar Komargodski, Daliang Li, Miguel Paulos, Jo~ao Penedones, Eric Perlmutter, Martin Schmaltz, Julian Sonner, Douglas Stanford, Mithat Unsal, Matt Walters, Huajia Wang, Junpu Wang, and Sasha Zhiboedov for useful discussions, and the GGI for hospitality while parts of this work were completed. ALF is supported by the US Department of Energy O ce of Science under Award Number DE-SC-0010025. JK has been supported in part by NSF grants PHY-1316665 and PHY-1454083, and by a Sloan Foundation fellowship. ALF and JK are supported by a Simons Collaboration Grant on the Non-Perturbative Bootstrap. Comments on the generality of our results In this paper and other recent work [1{3, 10{12] we have been developing an understanding of black hole thermodynamics, information loss, and its resolution in AdS3/CFT2. This 19More precisely, this is the contribution to hOH(x1)OH(x2)OL(x3)OL(x4)T (y)i from the saddle (z). hOH(x1)OH(x2)OL(x3)OL(x4)i means that we are interested in `irrational' or non-integrable CFT2 at large central charge, and speci cally theories that may have a dual interpretation involving nite temperature black holes in AdS3. Our results are extremely general because they follow entirely from the structure of the Virasoro algebra. Roughly speaking, our results pertain to any theory where the Virasoro algebra is not embedded in a much larger algebra. This means that our results are not relevant for integrable theories or free theories, but will apply to virtually all other large c theories. In this appendix we will discuss the generality of our results more precisely, and explain why they do not apply to integrable theories. Forbidden singularities are forbidden from all Virasoro blocks In recent work [11] we discussed two signatures of information loss in heavy-light 4-pt correlators: late Lorentzian time behavior, which has been the major focus in this paper, and forbidden singularities in the Euclidean region. Let us now focus on the latter; for a more extensive discussion see [11]. Forbidden singularities arise because the CFT 4-pt function approximates the 2-pt function of a light probe eld in a BTZ black hole background. These correlators are thermal, which means that they are periodic in Euclidean time. In particular, this means that the OPE singularity has an in nite sequence of image singularities under tE ! tE +n for any integer n, where tE is the Euclidean time and = T1H . However, from the point of view of the CFT correlator, these are not OPE singularities, rather they are extraneous, It should be noted that these singularities are allowed in a true thermal 2-pt function (ie in the canonical ensemble), but are forbidden from 4-pt functions (which can be viewed as 2-pt functions in a pure state background) and also from the microcanonical ensemble. The singularities can only emerge after summing over an in nite number of external states. We outlined an argument [11] that forbidden singularities are forbidden not just from full CFT correlators, but also from individual Virasoro blocks. To clarify the situation, here we provide a more complete proof. As we previously noted [11], all of the hard work was done by Maldacena, Simmons-Du n, and Zhiboedov (MSZ) in section 7 and appendix D of [15], so we recommend that interested readers consult that reference for details. We will simply summarize their results. hO1(1)O2(1)O1(z)O2(0)i in the O1(z)O2(0) OPE channel can only have OPE sinIn particular, the Virasoro blocks have an expansion where (z) is a universal prefactor with only OPE singularities, the an > 0 for all n, and 12;12(z) = q(z) = e where K(z) is an elliptic integral of the rst kind. for some bn that are not necessarily positive. We would like to bound these coe cients to prove convergence for jqj < 1. For this purpose let us consider the 2 2 matrix of Virasoro conformal blocks O2(1)O2(1) Ph (O1(z)O1(0); O2(z)O2(0)) where Ph is a formal projector onto the primary state h and all of its Virasoro descendants. This matrix must be positive de nite. Equivalently, the linear combination of correlators h(O1(1)O1(1) + O2(1)O2(1)) (O1(z)O1(0) + O2(z)O2(0))i The full CFT correlator must be nite for jqj < 1, and this condition obtains for all z away from OPE singularities, including after analytically continuing z arbitrarily far from the original Euclidean sheet of the complex plane. Because the coe cients of the individual Virasoro blocks as well as the coe cients an > 0, this necessarily implies that the Virasoro blocks themelves must converge for jqj < 1. Thus these blocks cannot have any non-OPE singularities. Note that all of these results assume nite values of hi; h; c, so they do not However, this is not quite what we wanted, because we are studying the Virasoro blocks appropriate for the OPE limit O1(z)O1(1) rather than the O1(z)O2(0) channel. The relevant block will still have a q expansion (transforming now to z ! 1 z for convenience) 11;22(z) = can be interpreted as a (somewhat unusual) inner product of normalizable states in MSZ's pillow metric [15]. This fact bounds the absolute value of jVh proving convergence of equation (A.3) for all jqj < 1. Thus the heavy-light conformal blocks can never have unitarity-violating forbidden singularities at nite values of operator dimensions h and the central charge c. Can forbidden singularities cancel between semiclassical blocks? In section A.1 we proved that the forbidden singularities that arise in the semiclassical approximation to the Virasoro blocks cannot be present in the exact Virasoro blocks. This is su cient to demonstrate that a major problem associated with black hole physics must be resolved block-by-block. But one can ask if it is nevertheless possible for forbidden singularities to cancel between the distinct semiclassical Virasoro blocks that make up a full CFT correlator in semiclassical level, but these singularities cancel in the linear combination P speci c CFTs. In other words, perhaps blocks Vi all have forbidden singularities at the appears in correlators. We will see below that in integrable theories this can occur [68, 75], but we do not believe it occurs in the irrational CFT2 of interest to the study of quan In any case, one can immediately argue that in large c CFT2 where the stress tensor is the only conserved current, there are forbidden singularities that can never cancel in this way. If we study semiclassical heavy-light correlators in the limit z ! 0, which is the lightcone OPE limit [76, 77], the vacuum Virasoro block dominates over all other are therefore conserved currents. But in the heavy-light semiclassical limit the vacuum Virasoro block has forbidden singularities [11], so no other semiclassical Virasoro block can Note that this argument does not require any assumption about the sparseness of the light operator spectrum, which is relevant to other situations where the vacuum Virasoro block takes center stage [8, 16, 71]. The argument depends only on the behavior of the Virasoro blocks in large c heavy-light limit, and the assumption that T (z) is the only conserved current in the CFT. Theories with larger symmetry algebras It is interesting to understand how the result from appendix A.2 can break down in theories where the Virasoro algebra is contained within a larger symmetry algebra. In such theories, it may be possible for the forbidden singularities to cancel among distinct Virasoro blocks [68, 75]. Examples include WZW theories and rational CFTs with W-algebra symmetries, but free CFTs with a large number of fundamental elds provide a more straightforward test-case. The authors of [68] gave interesting examples involving orbifolds of free theories relevant to the free limit of the D1-D5 system. Here we will study an extremely simple example: a CFT2 with c 1 and a U(1) symmetry,20 for which the conformal blocks are already known (see section 3.5 of [2] for the derivation and relevant discussion). In such a theory, the stress tensor T can be we can also write the scaling dimensions of operators and states as T = T sug + T (0) h = h(0) + where k is the level and q is the U(1) charge of an operator. 2 The theory may include states with large 2qk / c but small or vanishing values of h(0). Very naively, we might expect that heavy-light Virasoro blocks involving such states appear thermal when 2qk2 > 2c4 . But this is impossible if h(0) is small. In fact, the explicit form of the heavy-light U(1) + Virasoro blocks is [2] q Thus if h(L0) = h(0) = 0, then the blocks are simply z kL (1 H z) k , which clearly cannot be thermal. We only see a Hawking temperature when h(0) > 2c4 , and this condition is H 20We thank Simeon Hellerman for discussions of this example at the 2015 Bootstrap workshop. VT +J (c; hi; qi; z) = VT independent of qH . This is consistent with expectations from AdS3 theories with a bulk U(1) Chern-Simons gauge eld [2]. Since equation (A.8) represents a correlator in a theory with a symmetry containing Virasoro, it can be expanded in terms of pure Virasoro blocks (ignoring the U(1) current algebra), just as correlators in theories with Virasoro symmetry can be expanded in the global or sl(2) conformal blocks. This suggests that in theories with a symmetry larger than Virasoro, it is possible to see cancellations between Virasoro blocks that eliminate forbidden considering do not need to be positive, since this quantity is not the square of a real number. The Virasoro blocks with hI ; hL Vh(z) = 2F1 (hI ; hI ; 2hI ; 1 z) ) (A.9) 24hH as usual, and we have normalized the block to begin with zh 2hL in We can explicitly decompose the U(1) + Virasoro vacuum block into pure Virasoro qH qL for notational a series expanion in z. hI =0 = 1 2F1 (hI ; hI ; 2hI ; 1 where in this case 12cqH2 . This can be matched order-by-order to the sum where hI only take integer values, and PhI are the Virasoro block coe cients, which are products of Virasoro primary OPE coe cients. This matching is tedious but straightforward to any xed order; the rst few terms are P0 = 1 P1 = P2 = P3 = The P1 term is simply the coe cient of the exchange of the Virasoro block of the U(1) current J (z), while the higher terms involve a combination of J n generators (which include the Sugawara stress tensor within their algebra). Thus these coe cients could also be computed directly using matrix elements of the current algebra. We can also re-write them entirely in terms of qL and qH , in which case we nd that at large c P0 = 1; P1 = Q; P2 = ; P3 = ; P4 = 2 ; P5 = Thus at large c, the PhI depend only on Q, but incorporating the other terms will be important for an expansion in the Virasoro blocks with xed qH2 =c at large c. In any case, the point of this exercise is that we can express a full block of the form of equation (A.8) in terms of the relevant pure Virasoro blocks, meaning that at least at a formal level, the forbidden singularities cancel in this sum. 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A. Liam Fitzpatrick, Jared Kaplan. On the late-time behavior of Virasoro blocks and a classification of semiclassical saddles, Journal of High Energy Physics, 2017, 72, DOI: 10.1007/JHEP04(2017)072