On the latetime behavior of Virasoro blocks and a classification of semiclassical saddles
Received: February
On the latetime 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 nonperturbative 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 highenergy states cannot resolve a wellknown version of the information loss problem in AdS3. However, we identify two in nite classes of subleading saddles, and one of these classes does not decay at late times.
AdSCFT Correspondence; Black Holes; Conformal and W Symmetry; Con
1 Introduction and summary
Information loss and the latetime 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 latetime 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 nonperturbative physics in quantum gravity through the
determination of the nonperturbative 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 nonperturbative or `e c'
e ects that can appear in the large c expansion of the Virasoro conformal blocks. Second,
we will compute the latetime behavior of all semiclassical Virasoro blocks, with arbitrary
internal and external states.
We nd that in the heavylight limit, which corresponds to a subPlanckmass 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 subleading 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 highenergy 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 nonunitary 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 4point 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 heavylight 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
heavylight 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 heavylight 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,
wellknown 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 heavystate 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 lightlight and heavyheavy 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 nonperturbative 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 ChernSimons 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 subleading semiclassical `instantons'
or `saddles' that contribute to V at a nonperturbative 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 subleading 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
orderbyorder 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 splane 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 pathintegral
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 pathintegral 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 subleading saddles. Bringing the nonperturbative e ects into view,
one can write an improved expansion for V as a doublesum, 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 nonzero 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 ChernSimons
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 heavylight 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 subleading saddles may be interpreted as `additional angles' in AdS3, as depicted in
2This innocuousseeming 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 latetime 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 powerlaw 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 subleading order in the late
time limit, controlling the rate at which (t) approaches its asymptotic value. In
we show the timedependence of
f for a variety of leading and subleading 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 timedependence 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 latetime 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 latetime 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 latetime 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 crossratio 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
nonperturbative e ects as well, as we discussed in section 1.3.
One can argue that the subleading saddles should all have the same monodromy as
the leading saddle by applying the original logic of the monodromy method to the full
nonperturbative 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 nonperturbative 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 nontrivial 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 backreact 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 2pt 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 heavylight 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 zplane,
depicted on the right, the multisheeted 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 timedependence in AdS.
z = 1
This Lorentzian correlator can be obtained from the usual Euclidean 4pt 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 lowestenergy 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 zplane, 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
latetime 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 nontrivial 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 latetime 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 reinterpret 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 nonanalytic
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 nonanalytic 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 latetime 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 rede 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 powerlaws. 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 Zindependent 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 timedependent.
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 nonanalyticity 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 selfconsistency 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 bookkeeping, 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 latetime
solutions with nonvanishing 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 nonvacuum blocks. In gure 13, we show the
leading saddle for the nonvacuum 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 nonzero 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 latetime 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 nitedimensional 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 rewritten as
0 =
We can obtain a di erential operator that annihilates the fourpoint 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 quasinormal 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 nonzero 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 nonzero imaginary parts at z
indication that the generation of nondecaying 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 rederive 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 degreer
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 nontrivial
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 stronglycoupled 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 informationdestroying 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 nonperturbative 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 nonperturbative resolution of forbidden singularities, and to the `master
equation' [11] that seems to be a rst step towards a determination of the true latetime
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 ChernSimons action for holomorphic `gravitons', together with a
coherent, selfcontained 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 quasianalytically for blocks with degenerate external operators from the crossing
matrices [43, 44, 72], but the existence of the nondecaying 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 DESC0010025. JK has been supported in part by NSF grants
PHY1316665 and PHY1454083, and by a Sloan Foundation fellowship. ALF and JK are
supported by a Simons Collaboration Grant on the NonPerturbative 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 nonintegrable 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 heavylight 4pt
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 4pt function approximates the 2pt
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 2pt function
(ie in the canonical ensemble), but are forbidden from 4pt functions (which can be viewed
as 2pt 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, SimmonsDu 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 nonOPE
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 heavylight conformal
blocks can never have unitarityviolating 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 blockbyblock.
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 heavylight 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 heavylight 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 heavylight 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 Walgebra
symmetries, but free CFTs with a large number of fundamental
elds provide a more
straightforward testcase. The authors of [68] gave interesting examples involving orbifolds
of free theories relevant to the free limit of the D1D5 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 heavylight 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 heavylight 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) ChernSimons 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 orderbyorder 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 rewrite 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. It would be interesting to see
this cancellation more explicitly.
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