Published for SISSA by Springer
Received: February 7, 2017
Accepted: March 23, 2017 Published: March 30, 2017
Hongbin Chen,a A. Liam Fitzpatrick,b Jared Kaplan,a Daliang Lia and Junpu Wanga
aDepartment of Physics and Astronomy, Johns Hopkins University, Charles Street, Baltimore, MD 21218, U.S.A.
bDepartment of Physics, Boston University,
Commonwealth Avenue, Boston, MA 02215, U.S.A.
E-mail: mailto:[email protected]
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Web End [email protected] , mailto:[email protected]
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Abstract: One can obtain exact information about Virasoro conformal blocks by analytically continuing the correlators of degenerate operators. We argued in recent work that this technique can be used to explicitly resolve information loss problems in AdS3/CFT2. In this paper we use the technique to perform calculations in the small 1=c / GN expansion:
(1) we prove the all-orders resummation of logarithmic factors / 1c log z in the Lorentzian
regime, demonstrating that 1=c corrections directly shift Lyapunov exponents associated with chaos, as claimed in prior work, (2) we perform another all-orders resummation in the limit of large c with xed cz, interpolating between the early onset of chaos and late time behavior, (3) we explicitly compute the Virasoro vacuum block to order 1=c2 and 1=c3 with external dimensions xed, corresponding to 2 and 3 loop calculations in AdS3, and (4) we derive the heavy-light vacuum blocks in theories with N = 1; 2 superconformal symmetry.
Keywords: 1/N Expansion, AdS-CFT Correspondence, Conformal and W Symmetry, Conformal Field Theory
ArXiv ePrint: 1606.02659
Open Access, c
[circlecopyrt] The Authors.
Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP03(2017)167
Web End =10.1007/JHEP03(2017)167
Degenerate operators and the 1/c expansion: Lorentzian resummations, high order computations, and super-Virasoro blocks
JHEP03(2017)167
Contents
1 Introduction and summary 1
2 Degenerate operators and heavy-light Virasoro blocks 4
3 Computing the 1/c expansion of the vacuum block 73.1 The heavy-light Virasoro vacuum block at order 1=c 103.2 The all-light Virasoro vacuum block at order 1=c2 and 1=c3 113.3 Integral formulas from the Coulomb gas 123.3.1 Leading order at large c 133.3.2 Expansion at order 1=c 13
4 All-orders resummations in the Lorentzian regime 144.1 Resummation of 1c log z e ects 154.2 Resumming leading singularities in 1
cz 17
5 Heavy-light super-Virasoro vacuum blocks at large c 205.1 The N = 1 super-Virasoro vacuum block 20
5.1.1 Brief review of 2d N = 1 SCFTs 20
5.1.2 N = 1 super-Virasoro vacuum blocks at leading oder 21
5.2 The N = 2 super-Virasoro vacuum block 24
5.2.1 Brief review of 2d N = 2 SCFTs 24
5.2.2 Super null-state equations 265.2.3 Solutions for general hL 28
A Summary of corrections to the vacuum block 30
B Direct derivation of leading logs in the Lorentzian regime 33
C Leading contribution to the vacuum blocks in N = 1 SCFTs 36
D Details of the N = 2 SCFT calculations 38
D.1 Superconformal Ward identities 38D.2 Leading contributions to the vacuum blocks 38D.3 Correlation functions with descendant component elds 39D.4 Decomposition of qh+1 41
1 Introduction and summary
The in nite dimensional Virasoro algebra profoundly contrains the dynamics of Conformal Field Theories (CFTs) in two dimensions. Certain \rational" theories have operator algebras that truncate, allowing them to be solved exactly. But despite their phenomenological relevance and beauty, rational theories are small islands in a largely uncharted sea of 2d CFTs. Furthermore, we can only study quantum gravity in AdS3 by analyzing CFTs with large central charge c, and relatively little is known about these irrational 2d CFTs.
{ 1 {
JHEP03(2017)167
Although it appears that large c CFTs cannot be solved exactly, it is still possible to take some of the methods [1, 2] that make rational CFTs tractable and apply them [3] to irrational theories. The reason is that correlation functions in any CFT2 can be decomposed into Virasoro conformal blocks Vhi;h;c(z) as1
[angbracketleft]O1(1)O2(1)O3(z)O4(0)[angbracketright] =Xh;h
Ph;hVhi;h;c(z)Vhi;h;c( z): (1.1)
The Virasoro blocks are the contributions to the Operator Product Expansion (OPE) of
O3(z)O4(0) from irreducible representations of the Virasoro algebra[Ln; Lm] = (n m)Ln+m +
c12n(n2 1) n+m;0 (1.2)
h are intermediate operator weights. When O1 = O2 and O3 = O4, a universal contribution in equation (1.1) is the
Virasoro vacuum block, which encapsulates the exchange of any number of pure AdS3 graviton states between the external operators.
The Virasoro blocks have turned out to be extremely useful as a source of information about gravity in AdS3, and in fact BTZ black hole [4] thermodynamics [5] emerges in a universal, theory-independent way from the heavy-light, large central charge limit of the Virasoro blocks [6{15]. Information loss from black hole physics appears to be due to the behavior of the blocks in this limit [3, 7, 11]. The blocks are also the basic components of the conformal bootstrap program [16{20]. Knowing their explicit forms would greatly assist the study of 2d CFTs and 3d gravity using the bootstrap [5, 21{26].
Each conformal block depends only on the quantum numbers (hi; h; c) of the representations involved and not on the speci c theory. A strategy for computing the blocks is to work with a theory where operator truncation occurs, and then use the fact that the result is theory-independent. This technique becomes even more useful when augmented by the fact that the conformal blocks are analytic functions of their de ning quantum numbers, so that one can compute the blocks for special values of the external dimensions hi and then analytically continue. In this paper, we will use this technique to develop an e cient method to compute and study the blocks order-by-order in a 1=c expansion, and to perform certain all-orders Lorentzian resummations.2
We will organize the series expansion in terms of the ansatz3
VhH;hL;0;c(z) = exp
hi are weights of the external operators Oi, while h;
1 c
~V for the normalized vacuum block, i.e. the vacuum block com-
~V begins with
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The hi;
m
hL c
"hL1
Xn;m=0
nfmn ( H; z)
#; (1.3)
1We have explicitly indicated the decomposition into a product of holomorphic and anti-holomorphic parts.
2Currently, closed form expressions for Virasoro blocks have been obtained in an expansion about various limits, such as h ! 1 [27], as well as at c ! 1 in the all light and the heavy-light limit [6{13]. In addition,
as a function of c and of intermediate operator dimensions, the Virasoro blocks are meromorphic functions with only simple poles. These properties imply recursion relations [27, 28] that e ciently compute the series expansion [29] of the vacuum blocks near z = 0 with generic hi, h, c.
3In this paper, we denote by V the vacuum Virasoro block component of the correlator
[angbracketleft]OL(0)OL(z)OH (1)OH (1)[angbracketright]
[angbracketleft]OH (1)OH (1)[angbracketright] , while we use
ponent of [angbracketleft]OL(0)OL(z)OH (1)OH (1)[angbracketright] [angbracketleft]OL(0)OL(z)[angbracketright][angbracketleft]OH (1)OH (1)[angbracketright]
~V = 1 + [notdef] [notdef] [notdef] in the small z expansion.
{ 2 {
where H = hHc is xed at large c, and we will compute the function fmn. This is a natural expansion to use in the semiclassical limit [6, 13, 27, 30, 31], which keeps only the terms with m = 0, but direct calculations [8] indicate that it is also justi ed to all orders in the 1=c expansion (see section 2 for a more detailed discussion). We explicitly compute up to order 1=c3, i.e. m + n 3, relegating many of the detailed forms of the functions fmn to
appendix A.
We have veri ed that our results match with a direct computation of the blocks to high orders in a series expansion in z, providing further direct evidence for the validity of our methods and for their application to information loss [3]. We also apply this method to compute the heavy-light super-Virasoro vacuum block in the case of N = 1 and N = 2
superconformal symmetry. With some straightforward but tedious work, the method could certainly be extended to theories with more supersymmetry, which have been studied recently using the bootstrap [24].
One physically interesting regime where our techniques prove to be particularly e ective is in the limit where z ! 0 in the Lorentzian region. This limit would be trivial in
the Euclidean regime,
z = z , where z ! 0 is the OPE limit of a conformal block and
is therefore dominated by the primary state contribution. However, the regime of small z becomes highly non-trivial after analytically continuing through a branch cut to the Lorentzian sheet. In particular, correlators in the Lorentzian regime depend on the order of the operators, and continuing along di erent paths before taking z ! 0 can produce
di erent results, which generally include singularities at small z.
This behavior is related to a variety of fascinating physical phenomena, including bulk singularities [32], black hole scrambling [33], and universal CFT causality constraints [34], to name a few. This regime was studied at subleading order in the large c, heavy-light limit of the vacuum block [35], where it was found that certain 1
c log(z) terms appear.
These were argued to be 1=c corrections to the power-law behavior of singular terms. In particular, at leading non-trivial order the growth of the singular terms is z1, which
after mapping to the thermal cylinder z = e2i(t+x)= corresponds to exponential growth with \Lyapunov" exponent 2
. In [35] a logarithmic correction at the next order in 1=c was argued to be the leading term in a correction to this exponent, shifting it to 2 (1+ 12c).
In section 4, we will prove that there are indeed an in nite series of terms of the form
1 cz
with exactly the correct coe cients to resum into a correction to the Lyapunov exponent. This might also be viewed as a quantum correction to the Regge trajectory. We will also provide a simple way to understand subleading logarithms, ie terms of the form 1
n
As noted in [35], there are also power-law corrections that are larger than the logarithmic corrections. In the limit c ! 1 with cz xed, there is an in nite sequence of terms
of the form (cz)n that survive. The \Lyapunov" regime, where the onset of scrambling rst takes place, is the regime of large cz and is well-described by the rst few terms in a 1=c expansion. However, eventually the behavior transitions to the \Ruelle" regime [36],
{ 3 {
JHEP03(2017)167
log(z) c
n
(1.4)
cm
log(z) c
with m > 1.
related to the decay of quasi-normal mode excitations around a BTZ black hole, and to describe this regime of small cz one must sum all the leading terms. As we will see, this series is asymptotic, so one must Borel resum it. We will show how to do this in subsection 4.2, with a remarkably simple result that interpolates between the \Lyapunov" regime and the \Ruelle" regime:
lim
c!1
cz xed
G(h1; h2; x) (x)2h1(2h2)2h1 1F1(2h1; 1 + 2h1 2h2; x): (1.5) It should be remembered, however, that non-vacuum blocks may also signi cantly a ect the behavior of the correlator at intermediate and late times.
The idea that makes these Lorentzian resummations possible is that analytic continuation from the Euclidean to the Lorentzian region simply transforms a degenerate vacuum block into a nite sum of degenerate blocks. In other words, when evaluated on the second (Lorentzian) sheet, the degenerate vacuum block V(1;s)(z) becomes a linear combination
of s degenerate blocks, which are to be evaluated on the rst (Euclidean) sheet. Once we understand the behavior of V(1;s)(z) in the Lorentzian region for all s, we can use this to
obtain the physical vacuum blocks with general hL. We justify and implement these ideas in section 4.
The outline of this paper is as follows. In section 2 we review degenerate operators and outline our method of computation. The in section 3 we use the method to compute the heavy-light vacuum Virasoro block at order 1=c, and the all-light Virasoro block up to order 1=c3, which would correspond to a 3-loop gravitational calculation in AdS3. We use two methods, one based on solving di erential equations, and another based on a 1=c expansion of the Coulomb gas formalism. In section 4 we state and prove various results on the resummation of logarithms and singularities in the Lorentzian regime, and discuss the application of these results to the study of quantum chaos. Finally, in section 5 we derive the super-Virasoro vacuum block for N = 1; 2 superconformal symmetry. Various
technical details have been relegated to the appendices.
2 Degenerate operators and heavy-light Virasoro blocks
In this section, we will review the properties of degenerate operators4 and explain how to use them to extract information concerning the Virasoro vacuum block in the large central charge or c 1 limit. A degenerate operator is a Virasoro primary operator with
null descendants, which means that some of its Virasoro descendants have vanishing norm. When discussing degenerate states it is useful to introduce a parameter b so that
c 1 + 6
In this work, we take the c ! 1 limit via b ! 1. In this notation, the simplest example
of a null state is the second level descendant
L21 + b2L2
4See [37] or [38] for more systematic reviews.
z2hLVhH;hL;0(z) = G
hH; hL; icz 12
+ G
hL; hH; icz 12
;
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b + 1 b
2: (2.1)
|h1;2[angbracketright] = 0: (2.2)
{ 4 {
One can check using the Virasoro algebra of equation (1.2) that the level 2 Gram matrix
hh[notdef]L21L21[notdef]h[angbracketright] [angbracketleft]h[notdef]L21L2[notdef]h[angbracketright]
hh[notdef]L2L21[notdef]h[angbracketright] [angbracketleft]h[notdef]L2L2[notdef]h[angbracketright] !
3
4b2 ;
the level two descendant in equation (2.2) is the corresponding null vector. In general, degenerate states can only occur for holomorphic dimensions satisfying the Kac formula
hr;s = b24 (1 r2) +
21 + b2z +b2 1 z
= 0: (2.5)
At b ! 1, O1;2 has dimension h1;2 ! 12 and is a light \probe" operator. The other
operator, OH, has arbitrary weight hH. Equation (2.5) is a version of the hypergeometric
di erential equation; it is an exact relation for this correlator and its conformal blocks. One of its solutions, corresponding to the vacuum conformal block, is given by
[angbracketleft]OH(1)OH(1)O1;2(z)O1;2(0)[angbracketright] [angbracketleft]OH(1)OH(1)[angbracketright][angbracketleft]O1;2(z)O1;2(0)[angbracketright]
; (2.6)
where H is a parameterization of the operator dimension hH and is related to its Coulomb gas charge:
H = 1
2b
Q
pQ2 4hH
; hH = b H(Q b H); Q b + b1: (2.7)
We will be interested both in the \light-light" limit, where hH is O(1), as well as in the
\heavy-light" limit where b2hH is xed in the large b limit. In the heavy-light limit, OH
represents a heavy operator generating a background probed by O1;2. More speci cally,
in a putative AdS3 dual description, OH will create either a de cit angle or BTZ black
hole [4]. At c ! 1 in the heavy-light limit, (2.6) simpli es to
e
1
2 tE sin(THtE)
TH ; (2.8)
where tE = log(1 z) is the Euclidean time and TH =
q24hHc 1 is the Hawking temperature of a BTZ black hole created by acting with OH on the vacuum.
{ 5 {
(2.3)
has a vanishing determinant when the holomorphic dimension satis es h1;2 = 12
14b2 (1 s2) +
1
2(1 rs) (2.4)
for positive integers r; s. This formula determines the values of dimension h when the Kac determinant, of which equation (2.3) is an elementary example, vanishes. Notice that r $ s
simply corresponds with b $ 1=b. For rational models, b2( pp[prime] ) is a rational number,
and consequently so are hr;s and c. In this work we will mainly be interested in general
(irrational) values of b and hr;s.
Null conditions such as (2.2) translate into di erential equations for the correlation functions involving a degenerate state. This follows because within a correlator with operators of dimension hi, the Virasoro generators Lm act as di erential operators due to the
stress tensor Ward identities. In the simplest case of O1;2, we have:
@2z +
JHEP03(2017)167
@z + b2hH(1 z)2 [angbracketleft]OH(1)OH(1)O1;2(z)O1;2(0)[angbracketright] [angbracketleft]OH(1)OH(1)[angbracketright][angbracketleft]O1;2(z)O1;2(0)[angbracketright]
Hb 2F1 1 + b2; 2 H; 2(1 + b2); z
= (1 z)
1
2
More generally, the vacuum block for the correlator [angbracketleft]OH(1)OH(1)Or;s(z)Or;s(0)[angbracketright] sat
is es a nite order di erential equation for all of the degenerate operators Or;s. Since the conformal blocks depend only on the parameters hi; hp; b, and not on the particular theory, this suggests that one can compute them in general by solving the resulting di erential equations. Of course, there is an obvious obstacle: the light weights hr;s are not quite
independent free parameters. We can dial their value by changing their indices r and s, but within some limitations. First, r and s must be positive integers, and at large c > 0 this means hr;s are always in the non-unitary regime. This is not as signi cant a limitation as it may seem, because the conformal blocks are meromorphic functions of hr;s (for a
detailed discussion see [3]). Thus, one can hope to analytically continue the blocks as a function of integer (r; s) to non-integer values.5 And in fact this method was used in [3] to study contributions to the vacuum block that are non-perturbatively small in the large c limit, which are associated with the resolution of information loss problems.
A second, more serious obstacle is that increasing r and s produces new di erential equations of increasingly high orders. Thus, solving for more values of hr;s requires solving increasingly complicated di erential equations of increasingly high order. We will see that this translates into increasing complexity in using the method to solve for the vacuum block at increasingly high orders in 1=c. Nevertheless, comparison with other methods [6{ 8] suggests that this may be the most e cient available procedure for determing the large c vacuum blocks, especially if one wishes to go beyond the semi-classical limit.
To be more precise about the method we use, we write a generic vacuum block (which will not involve any degenerate operators) in a double expansion in 1
c and hLc :6
VhH;hL;0;c(z) = exp
"hL1
Xn;m=0
1 c
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nfmn ( H; z)
m
hL c
#; (2.9)
where H = hHc . This ansatz can be justi ed as follows. In the semiclassical limit of c ! 1
with all hi=c xed, we have a great deal of evidence [6, 13, 27, 30, 31] that the vacuum block can be written as exp
c g(hLc ; hHc ; z) for some function g that is analytic in hL=c and hH=c in a neighborhood around the origin. This explains why equation (2.9) does not
5Continuing a function on the integers to the entire complex plane generally requires some additional knowledge of its behavior at 1; we will see that order-by-order in the large c expansions we employ in this
paper, the required information is provided by the OPE.
6A limitation of this method is that it is much more complicated to get results for general internal dimension hI of the conformal block. The reason for this is that once the external dimensions hL, hH are xed and OL is chosen to be a degenerate operator, then the dimensions of the allowed internal operators
are also xed to lie in a nite set. In principle, one could hope to get around this by using the fact that degenerate operators O1,s contain more and more operators in the OPE as s is increased, and in the limit
that s becomes large one would have access to a tower of operators with a discretum of dimensions. However, each order in 1/c has a complicated dependence on hI, in contrast to the simple polynomial dependence on the external dimensions hL, hH (for instance, the c ! 1 piece is the global block, which is independent
of hH and hL but a hypergeometric function 2F1(hI, hI, 2hI, z) of hI), so extracting this dependence from the discretum of exchanged operators really requires the entire in nite tower, which in turn requires solving the large c degenerate blocks in the s ! 1 limit. Thus we are focusing entirely on the vacuum Virasoro
block in this paper.
{ 6 {
contain terms such as e.g. h4L=c2, which would behave very di erently in the semi-classical limit. Corrections to the semi-classical limit can then be expanded in powers of 1=c, leading to equation (2.9).
The exponential form of the ansatz will be convenient for our purposes, but beyond the semiclassical limit it is not obligatory, and it would be just as natural to write the 1=c corrections in a power-series multiplying the exponential semiclassical part. One can also justify the ansatz to any order in z via a direct, brute force computation [8] of the vacuum block using the Virasoro algebra. Finally, note that although we have expanded the ansatz in a heavy-light limit, the conformal blocks will be symmetric under hL $ hH.
As mentioned previously, the vacuum block in equation (2.9) is analytic in hL and hH [3, 27]. Therefore, when taking hL = hr;s, we must recover the null state vacuum block such as solution (2.6). Matching order by order in 1
c and
hLc to these solutions, we can determine fmn( H; z). Note that our knowledge of the block in the heavy-light semiclassical limit strongly constrains its behavior at large values of the external dimensions, so it seems very unlikely that there are any ambiguities in the analytic continuation from hL = hr;s.
The method can be generalized to study theories with supersymmetry. In particular, we work out heavy-light large c limit of the holomorphic part of super-Virasoro vacuum blocks with N = 1; 2 supersymmetries in section 5. It turns out that the super-Virasoro
vacuum block of the lowest component elds in these theories do not get contributions from the fermionic supersymmetry generators at leading order of the large c limit, so they largely match with results extrapolated from [7], but it is interesting to understand the supermultiplet structure and the correlators of superconformal descendant elds.
Although the method is straightforward, it becomes quite tedious beyond the rst few orders in (2.9). When r is large, it becomes a non-trivial task to construct the null state di erential equation for 1;r, which is a complicated r-th order di erential equation whose exact solutions can be di cult to compute. But in speci c limits of physical interest, these equations simplify greatly and become extremely useful in determining key properties of the higher order quantum corrections. One example of these are the leading log terms in the 1
c corrections when all four operators are light, which we discuss in section 4. Such terms plays an important role in the growth of quantum chaos [33, 35, 39] and can be computed e ciently with the 1;r null state di erential equations.
Another very useful way to get higher order corrections in the large c limit is to use the Coulomb gas formalism, which provides a straight-forward construction of integral representations for the Virasoro blocks involving degenerate operators. We have used it in [3] in order to study the non-perturbative part of the vacuum Virasoro block in the large c asymptotic expansion. In this work, we will show that directly expanding the integrand in the Coulomb gas formalism provides an e cient way to obtain higher order terms in (2.9). This method is discussed in section 3.3.
3 Computing the 1/c expansion of the vacuum block
In this section, we will use the computational method explained in last section to calculate the higher order corrections to the Virasoro block. The idea is to assume that the general
{ 7 {
JHEP03(2017)167
heavy-light vacuum block V can be written as the ansataz (2.9). When OL is a degenerate
operator, V satis es a null-state di erential equation. At order
1cp , there are p + 1 fmn functions [notdef]f0;p; f1;p1; : : : ; fp;0[notdef]. Each one appears with a di erent power of hL in its
coe cient, i.e. log V hn+1Lfm;n. By (2.4), the degenerate operators h1;s with r = 1
have weights
h1;s = 12(1 s) +
1 cp
where J[notdef] are matrix generators of the spin (s 1)=2 representation of SU(2):
(J0)ij = 12(s 2i + 1) ij;
(J)ij =
(i(s i) i+1;j (i = 1; 2; : : : ; s 1);0 else : (3.5)
7More precisely, h1,s is not just a xed number but rather a xed function of c. However, because of the relation
h1,s = h(0)1,s + 1
b2 (h(0)1,s (h(0)1,s)2), (3.3)
where h(0)1,s = limc!1 h1,s, we are free to perform an expansion in powers of h1,s or in powers of h(0)1,s, since
the di erence between the two just corresponds to a rede nition of the fm,n functions.
{ 8 {
14b2 (1 s2)
1
2(1 s) +
32c(1 s2) + O
1 c2
; (3.1)
that are O(1) at c ! 1. For any choice of s, the operator O1;s produces a di erential
equation that we can solve for V and expand at large c to obtain the O(cp) term as
log V
h1;sfp;0 + h21;sfp1;1 + [notdef] [notdef] [notdef] + hp+11;sf0;p ; (3.2)
Unfortunately, for a single xed s, h1;s is just a number and therefore knowledge of (3.2) does not allow one to separate out the di erent contributions fmn.7 To accomplish this, one needs to take p+1 di erent degenerate operators, which give p+1 di erential equations to be solved for these p + 1 fmn functions.
This is the procedure that we will implement in sections 3.1 and 3.2 in order to obtain 1=cp corrections up to p = 3, corresponding to 3-loop gravitational e ects in AdS3. In section 3.3 we will study a di erent method that uses the Coulomb gas formalism to replace di erential equations with integrals.
A convenient and e cient formalism for keeping track of the null state di erential equations at large c was developed in [37, 40]. Let D1;s be the following matrix:
D1;s = J +
1
Xm=0
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J+ b2
m
Lm1; (3.4)
( i;j+1 (j = 1; 2; : : : ; s 1)0 else ;
[J+; J] = 2J0; [J0; J[notdef]] = [notdef]J[notdef]:
(J+)ij =
Then, the null state equation of motion is given by the equation f0 = 0 after eliminating f1; : : : ; fs1 from the equation
D1;s
0
B
B
B
B
@
f1 f2
... fs
1
C
C
C
C
A
=
0
B
B
B
B
@
f0
1
C
C
C
C
A
0 : (3.6)
At in nite c with hH held xed and O(1), one can manifestly drop all terms in the
sum in D1;s except for m = 0, so the null state manifestly becomes
Ls1O1;s = 0; (3.7)
and the in nite c di erential equation for the conformal block becomes
@szV(z) = 0; (3.8)
where the factor of zs1 arises because our convention for V(z) factors out the [angbracketleft]O1;s(z)O1;s(0)[angbracketright] two-point function. More generally, allowing hH to be O(c) with hH=c
xed, at in nite c the di erential equation for V(z) becomes [3, 37, 40]
2
64
Yk=(s1)+2j j=0,...,s1
@t k 2
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p1 24 H
375
e
s1
2 t
V(t) = 0; (3.9)
where t = log(1 z).
Let us see how this works in the simplest case, namely at lowest order in 1=c in (2.9). At order c0, logV hLf00 ( H; z), there is only one unknown function f00, which means
that we only need the di erential equation (3.9) with s = 2:8
d2 dt2
et2 eh1,2f00(z(t)); (3.11)
with h1;2 [similarequal] 12. In terms of z = 1 et, we obtain
f[prime][prime]00 = 12 H (1 z)2
+ 12 f[prime]00
1 24 H
4
2 : (3.12)
And the solution that corresponds to the vacuum block is
f00 ( H; z) = (1 2iTH) log (1 z) 2 log
1 (1 z)2iTH
2iTH
!: (3.13)
p24 H 1: This reproduces the result for the heavy-light limit of the vacuum
block rst found in [6].
8For later reference, the exact equation for the vacuum block for O1,2 is
@2z +
2 1 + b2z +b2 1 z
with TH = 1
2
@z + b2hH(1 z)2 z2h1,2V(h1,2, H, z) = 0. (3.10)
with b2 =
32(2h1,2+1) in this equation.
{ 9 {
3.1 The heavy-light Virasoro vacuum block at order 1/c
At order 1=c , there are two functions f01 and f10:
logV
hLc (f10 + hLf01)
which means that we need to use both the h1;2 null-state equation (3.10) and the h1;3 null-state equation:
0 = 1
z2
d3dz3 z2 +
4 1b2 2z1z z
!
d2 dz2 +
4( 1b2 +
2b4 )hH(2 z)
(1 z)3z
JHEP03(2017)167
+ 2 1b2 (9 13z + (5 + 2hH)z2) +
1b4 (3 + (z 3)z)
!
d dz
!
z2(1 z)2
z2h1,3V (h1;3; ; z)
(3.14)
with b2 =
2h1,3+1 in this equation and h1;3 [similarequal] 1 12c 156c2 + O 1=c3
in the large c
limit. The c0 order of equation (3.14) only involves f00 and the solution for it is exactly equation (3.12).
At order 1=c, equation (3.10) and (3.14) give the following two equations for f00; f01
and f10 :
0 = F [prime][prime]1 f[prime]00F [prime]1 8f[prime][prime]00 5f[prime]002
12(2z 1)
z(z 1)
f[prime]00
12(z + 1) (z 1)z2
0 = F [prime][prime][prime]2 3f[prime]00F [prime][prime]2 + 3
f[prime]200 f[prime][prime]00 +
8 H (z 1)2
F [prime]2 + 12f[prime][prime][prime]00 +
24 48z(z 1)z 72f[prime]00
f[prime][prime]00
+ 36f[prime]003 + 24(2z 1)
z(z 1)
f[prime]002 + 12((50 H + 3)z2 z 1)
(z 1)2z2
f[prime]00
+ 24 (25 H + 1)z3 2z + 1
(z 1)3z3
(3.15)
where we de ne
F1 = 2f10 f01; F2 = f10 f01:
Note that the di erential equations (3.15) involve the zeroth order term f00, which also appears at higher orders, since log V h1;sf00 = (1s2+1s24b2 )f00. There are a few signi cant simpli cations that occurred in the above equations. First, f10 and f01 show up only as a certain combination (F1 and F2) in each equation. The reason is that these equations come from the leading term in h1,sf10
c + h
21,sf01c . Since h1;s = 1s2 + O (1=c), the leading term in
h1,sf10c + h
21,sf01c is
1s
2 f10c +
1s
2
2 f01 c :
A similar phenomenon continues to be true for higher order calculations. This means that these di erential equations can be solved independently for F1 and F2. Second, only the derivatives of F1 and F2 show up in these equations. This allows one to solve for the
{ 10 {
derivatives rst, and then integrate. We have found this allows one to obtain a closed form expression for F1(z) directly using Mathematica; on the other hand, the di erential equation for F2 is too complicated to be solved this way. Since the solutions are known from previous work [11] (see also [12, 13] for semi-classical results), one can substitute them into equations (3.15) and verify them. For completeness, these solutions are included in appendix A.
3.2 The all-light Virasoro vacuum block at order 1/c2 and 1/c3
At order 1=c2, there are three functions f20; f11 and f02:
log V
JHEP03(2017)167
hLc2 (f20 + hLf11 + h2Lf02):
To fully determine them, one needs to solve the h1;2 and h1;3 null-state equations and also the h1;4 null state equation at order 1=c2. These equations are complicated, but at least one can expand them in terms of H hHc < 1 and obtain the result as an expansion in
H. De ne the expansion of fmn as
fmn =
1
Xk=0 k+1Hfmnk for m or n > 0
f00 = 2 log(z) +
1
(3.16)
where the 2 log(z) in f00 is because we include the prefactor z2hL in the de nition of
the vacuum block. Since the vacuum block V (hL; hH; z) is symmetric under the exchange
hL $ hH, in our convention, this means that fijk = fikj.
The liner H and 2H terms at order 1=c2 are
logV
Xk=0 k+1Hf00k
hLc2 H(f200 + hLf110 + h2Lf020) + 2H(f201 + hLf111 + h2Lf021)
:
At order 1H, using the symmetry under the exchange of hL and hH, we have f110 = f101; f020 = f002, which can be calculated by expanding f10 (A.1) and f00 (3.13) in terms of H. So the only unknown at this order is f200, which means that we only need to solve the h1;2 null-state equation at this order to get this term. The result is
f200 =1728(z2 1) ( (3) Li3(1 z))
z2 +
288Li2(z)(7(z 2)z 12(z 1) log(1 z))
z2
1728(z 2)Li3(z)z
144(z 1) log2(1 z)(6(z + 1) log(z) 7z + 7)
z2 + 1128
+ 12 242 z2 1
+ (z 2)z
log(1 z)
z2 +
288(z 2)(z 1)2 log3(1 z)
z3 :
(3.17)
We have also checked that these results do satisfy the h1;3 and h1;4 null-state equations.
At order 2H, only f021 = f012 can be determined by expanding the result we already have (that is, f01), and we need to solve the h1;2 and h1;3 null-state equations at this order to get f201 and f111. These results are complicated and given in appendix A.
{ 11 {
Using the symmetry fijk = fikj, we can also determine the liner H terms at order 1=c3 by just using the the h1;2 null-state equation. At this order,
logV
hL H
c3 (f300 + hLf210 + h2Lf120 + h3Lf030):
Since f210 = f201, f120 = f102 and f030 = f003, only f300 cannot be obtained by expanding
the results we already have, thats why we only need the h1;2 null-state equation. These results are also given in appendix A.
3.3 Integral formulas from the Coulomb gas
As we mentioned in the previous sections, computation of fmn at higher orders becomes extremely technically challenging, because upon the substitution hL ! hr;s one needs to
solve a di erential constraint equation of order rs. However, an integral representation of the solutions to constraint equations such as (2.5) are known, thanks to the Coulomb gas formalism [37, 41, 42]. This method makes it possible to write down explicit expressions for all fmn in terms of multiple elementary integrals.
Explicitly, the vacuum block component of [angbracketleft]O1,s(0)O1,s(z)OH(1)OH(1)[angbracketright] [angbracketleft]O1,s(0)O1,s(z)[angbracketright][angbracketleft]OH(1)OH(1)[angbracketright] , where O1;s is a
light degenerate operator, is given by the following integral representation:
~V1;s(z) = N1;s s1Yi=1 Z
1
0 dwi!
(1 z)(s1) HeI1,s ; (3.18)
where the action I1;s is
I1;s =
s1
Xi=1
JHEP03(2017)167
s 1b2 log
wi(1 wi)
2 H log(1 zwi)
2 b2
X1
i<j s1
log(wi wj) :
(3.19)
with H given by (2.7). We have also introduced a normalization factor N1;s such that ~V1;s(0) = 1. Notice that N1;s is independent of hH. Perturbatively in b, it is given by
N1;s(b) = 1 + 4(s 1)2 3(s 1)(s 2)
2b2 + O(b4) : (3.20)
In the limit b ! 1 with xed H, we can expand the integrand of (3.18) in 1=b:
~V1;s(z) = N1;s(b)(1 z)(s1) H Z
1
s1
0 Yi=1 dwi
!(1 zwi)2 H
[notdef]
1
Xk=01 k!b2k
s1
Xi=1(s 1)Ki
X1
i<j s1
2Uij
!k: (3.21)
To lighten the notation, we denote
Ki = log(wi(1 wi)) ; Uij = log [notdef]wi wj[notdef] : (3.22)
{ 12 {
In the rest of this section, we will show how to extract various fmn from the integral (3.21). The general strategy is very simple. Recall that we postulated the ansatz of the vacuum block to be
~VhH;hL;0;c(z) = z2hL exp "
hL
1
Xn;m=0
1 c
nfmn ( H; z)
m
hL c
#: (3.23)
When we set hL = h1;s = 1s2 + 1s24b2 in the above ansatz and compare it with (3.21), we can read o the fmn functions.
3.3.1 Leading order at large c
Let us begin by computing the well-known c = 1 heavy-light vacuum block as a warm-up.
Upon substitution hL ! h1;s, eq. (3.23) in leading order in b is simply z2hL exp(1s2f00).
Denoting X1;s the (s 1) dimensional integral in (3.21), the comparison implies
z2hLe
1s
2 f00 = N(0)1;sX(0)1;s =
s1
Yi=1
(1 z) H Z10 dwi(1 zwi)2 H ; (3.24)
where the superscript denotes the powers in 1
b2 , e.g. X1;s = X(0)1;s +
1b2 X(1)1;s + : : : . From the
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above equation one immediately obtains that
f00 = 2 log
(1 z) H (1 z)1 H(1 2 H) : (3.25)
Noting that in large b limit H ! 1
p124 H
2 + O(b2), we recognize the above equation
in agreement with (3.13).
3.3.2 Expansion at order 1/c
Now we arrive at the sub-leading order in c. They are two functions, f10 and f01, to be determined at this order. The comparison of (2.9) with (3.21) yields
1 s2
4 (f00 + 2 log z) +
1 s
2
f106 +f0161 s2 = N(1)1;sX(0)1;s + N(0)1;sX(1)1;sN(0)1;sX(0)1;s: (3.26)
In the above equation, N(0)1;s and N(1)1;s on the r.h.s. are obtained from (3.20), while X(0)1;2 and X(1)1;2 are represented by elementary integrals. Staring at (3.21), one nds that
X(0)1;s =
Zw
1 s1 ; X(1)1;s = X(0)1;s (s 1)2 Rw1 K1
Rw1 1 (s1)(s2)
Rw1
Rw2 U12 [
Rw1 1]2
!: (3.27)
Here we have used the abbreviation
Zwif(w1; : : : ; wn) (1 z) H Z10 dwi(1 zwi)2 Hf(w1; : : : ; wn) : (3.28)
{ 13 {
Combining all these pieces of information, one can easily solve for f10 and f01:
f10 = 18 6(f00 + 2 log z) 12
Rw1
Rw2 U12 [
Rw1 1]2;
Rw2 U12 [
Rw1 1]2; (3.29)
where f00 is given by (3.25). Now what remains to be computed are the two integrals in the expressions above. After some cumbersome but straightforward algebra, one has
Rw1 K1
Rw1 1=
10 dw(1 zw)2 H log
f01 = 12 + 6(f00 + 2 log z) + 24
Rw1 K1
Rw1 1 24
Rw1
w(1
w)
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10 dw(1 zw)2
H (3.30)
=
H + log 1z
2 + (1 z)
2 +
(0)( ) + log zz1 +
+
2F1
1; ; + 1; 11z z + 1 (1 z) 2F1 1; 1; 2 ; 1z
z
!(1 (1 z) )1 ;
Rw1
Rw2 U12 [
Rw1 1]2=
10 dw1
10 dw2
(1 zw1)(1 zw2)
2 H log [notdef]w1 w2[notdef]
R
10 dw(1 zw)2
H
2 (3.31)
= "i + 8(1 z) 2 log(z) + (1 z)2 i 2 log(z1z ) 1 1
2
+ (1 z)2
B(1 z; ; 0) 3B
11z ; ; 0
+ B
11z ; ; 0
3B(1 z; ; 0)
2
+ cot( ) 2H + (1 z)2 ( cot( ) 2H )
2
#
(1 (1 z) )2 ;
where B(x; ; 0) = x 2F1(1; ;1+ ;x)
is the incomplete Beta function, Hn is the harmonic function, is the Euler gamma constant, (x) = [prime](x)
(x) is the digamma function and the parameter is related to the Hawking temperature by p1 24 H = 2iTH. Hav
ing (3.30) and (3.31) plugged into the expression of f10 and f01 (3.29), it is straightforward to show that they match the results obtained in [11], which are also given in appendix A.
One can easily continue this procedure to higher orders in the 1=c expansion, but for brevity we spare the reader the details, since the lengthy 1=c2 and 1=c3 results have already been given in section 3.2 and in appendix A.
4 All-orders resummations in the Lorentzian regime
Our main focus in this section is to understand how the large c vacuum Virasoro block behaves in the Lorentzian regime. More spec cally, we are interested in the behavior of the block after the argument z is analytically continued across the branch cut emanating from z = 1 and then taken to small values of [notdef]z[notdef] on the second sheet. The behavior of
{ 14 {
Ruelle
)
(
-
Lyapunov
-
-
-
-
Figure 1. Plot of the behavior of 1 F (t) as a function of time t in the limit c ! 1 with cz
xed, with hL = hH = 12. F (t) is absolute value of the out-of-order correlator [angbracketleft]OLOHOLOH[angbracketright]
[angbracketleft]OLOL[angbracketright][angbracketleft]OHOH[angbracketright] , and
t log(cz=6). The initial \Lyapunov" growth and the later \Ruelle" decay are labeled as in [36].
We have plotted only the contribution of an approximation to the vacuum Virasoro block, but the result has the qualitative features expected of the full correlator.
CFT correlators in this regime has interesting implications for causality [34, 43{45] and a fascinating interpretation in terms of chaos [33, 35, 36, 39, 46{48].
In this section we will show that quantum corrections to the Lyapunov exponent resum to all orders, and that one can also resum the full 1
cz expansion in order to obtain an interpolation between the early onset of chaos and late time e ects associated with thermalization. These are the Lyapunov and Ruelle regions of gure 1. We refer the reader to [35] for a pertinent review of chaotic correlators and Lyapunov exponent bounds in the context of CFT2 at large central charge.
4.1 Resummation of 1
c log z e ects
Consider the Virasoro vacuum block in a large c expansion with external dimensions xed.
In a 1=c expansion, the leading correction near z 0 after analytically continuing around
the branch cut emanating from z = 1 is of the form
F (z) 1
n; (4.2)
JHEP03(2017)167
48ihLhH
cz + : : : (4.1) The rst term comes from the vacuum itself, while the second term is due entirely to the exchange of a single quasi-primary stress tensor or graviton state along with its global conformal descendants. The quantity F is the contribution of the vacuum block to the out of time order correlator [angbracketleft]OHOLOHOL[angbracketright] in a thermal background, normalized by the [angbracketleft]OHOH[angbracketright][angbracketleft]OLOL[angbracketright] ; it is plotted in gure 1.
As z decreases towards 0, the 1=c correction grows like z1 and becomes increasingly important. Similarly, higher order terms in 1=c can become important at su ciently small z as well. In this subsection we will show how to resum one set of contributions that grow large at small z, namely the terms that are leading logs in the 1=c expansion. That is, we will see that terms of the form (1=zc)(log(z)=c)n appear exactly in the combination
A cz1+ =c =
A zc
1
Xn=01 n!
{ 15 {
log(z) c
with constants A = 48ihLhH; = 12. We provide another derivation of this resum
mation in appendix B. We also checked the coe cients of these terms by analytically continuing the fm00 functions given in appendix A directly to the second sheet up to and including 1=c3 corrections. These e ects provide a quantum correction [35] to the Lyapunov exponents that characterize the early onset of chaos.
For degenerate external operators, there is a particularly transparent way of understanding this logarithmic resummation, because only a nite number of Virasoro blocks appear in any channel. The crucial point is that passing through the branch cut in z simply reshu es one linear combination of blocks into a di erent linear combination.9 In other words, on the second (Lorentzian) sheet, the vacuum block is equal to a sum of degenerate blocks evaluated on the rst (or Euclidean) sheet.
For the degenerate operator O1;s, the operators in the O1;s [notdef] O1;s OPE are degenerate
operators O1;p with p = 1; 3; : : : ; 2s 1. These have dimension
h1;p =
1
2(p 1)
Xq=0cq(h1;s; hH) b2q
where q (p1)2 and the fq(z) 1+O(z) parts of the blocks have a regular series expansion
around z 0. In the above, cq and fq are functions of b as well but we have factored out
explicit powers of b2 so that they have a nite limit at b ! 1. The reason this prefactor
of b2q must be present is that cq vanishes up to O(b2q+2), by the following argument.
If we expand at large b, we know that the b2q+2 term is a (q 1)-th order polynomial
in h1;s, and therefore given by q coe cients.10 These coe cient can be xed by looking at the OPE of the q degenerate operators [notdef]O1;s[notdef]1 s q. From the above description of the
O1;s[notdef] O1;s OPE, we know that none of the operators [notdef]O1;s[notdef]1 s q contains the O1;2q+1
operator, therefore this operator does not appear at O(b2q+2) or lower. But, cq is just
the OPE coe cient for the O1;2q+1 operator; therefore the lowest order where it appears
is b2q.
Now, to see explicitly the behavior of leading logs, sub-leading logs, sub-sub-leading logs, etc, we can simply expand in large b and look for terms of order (b2 log(z))n, b2(b2 log(z))n, etc. Logarithms manifestly arise only from expanding an exponent of z in the above expression, so any term of the form
(b2)m(b2 log(z))n (4.5)
9One way of understanding this is that crossing symmetry z ! 1 z acts as a linear operator that
changes blocks in one channel into blocks in the other channel. In the other channel, taking z around 1 acts on each block by simply introducing a phase (1 z)hI ! e2ihI (1 z)hI given by the weight hI of the
corresponding primary operator. Transforming back to the original channel again acts with the inverse of the rst linear operator, producing a linear combination of blocks in the original channel.
10These coe cients are functions of z and hH.
{ 16 {
1 + 12b2(p + 1) : (4.3)
where we recall c 6b2 1. So for a given value of s, analytic continuation of z around 1
transforms the vacuum block into a linear combination of terms of the form
~V(1;s)(z)
s1
1zq(1+b2(q+1)) fq(h1;s; hH; z); (4.4)
JHEP03(2017)167
must come from expanding an exponent n times after expanding the prefactor up to m-th order. There are manifestly no terms with m = 0. Terms with m = 1 must clearly come from the rst term, q = 1, and are the leading logs. Consequently, we immediately see that all these leading logs arise from the expansion of the term
c1(h1;s; hH)f1(h1;s; hH; z)
b2z1+2b2 (4.6)
and thus manifestly just resum back to this form. This result holds for all values of s. Since we expect that the vacuum block V is analytic in hL, and because this result obtains
for hL = h1;s for all s, we expect that it also holds if we analytically continue to general hL. The correction to the power-law in the denominator is
1 ! 1 + 2b2 = 1 +
12c + O(c2); (4.7)
proving equation (4.2) with = 12. This also provides a quick alternative check of the magnitude and sign of the correction to this (Lyapunov) exponent.
The above considerations also make it easy to understand the e ect of sub-leading logs, sub-sub-leading logs, etc. For instance terms with m = 2 must come from either the rst term or the second term in (4.4), and therefore are of the form
z2h1,s[angbracketleft]OH(1)OH(1)O1;s(z)O1;s(0)[angbracketright] (4.8)
1
Xn=0 (b2)2
[c1f1(z)]O(b2) (2b2 log(z))n
zn! +
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[c2f2(z)]O(b0) (6b2 log(z))n
z2n!
!:
It is easy to expand in large b to obtain similar higher order results.
4.2 Resumming leading singularities in 1
cz
Resumming the leading logarithms tells us something about the functional form of the large c expansion, but because of the power-law singularities (cz)n, the leading logs
never dominate the behavior of the blocks. In this subsection, we will derive and resum the leading (cz)n singularities, which do give the dominant behavior at small z in the limit c ! 1 with cz xed.
The arguments in the previous subsection already provide a signi cant amount of information on the coe cients of these singularities in equation (4.4): they are polynomials in hL and hH of order n, they have to vanish when hL is a degenerate operator h1;s with s n, and they have to be symmetric in hL $ hH. This in fact completely determines the
coe cients cq(hL; hH) of equation (4.4) up to an hH; hL-independent prefactor:
cq(hL; hH) = aq(2hL)q(2hH)q; (4.9)
where aq depends only on q and not on hH or hL. To obtain its value, we just need to calculate it for some chosen hH, in the limit c ! 1. A convenient choice is hH = Hc
{ 17 {
= =
=
)
)
(
(
-
-
-
-
-
-
-
-
-
-
JHEP03(2017)167
Figure 2. Plots comparing the exact behavior from eq. (4.14) (black, dashed) for 1 F (t) in the
limit c ! 1 with cz xed, to the heavy-light approximation (4.10) (red, solid). Left: hL = hH = 12, Right: hL = 32 ; hH =
310 . F (t) and t are as in gure 1. Note that both curves only include contributions from the vacuum block, neglecting double-trace operators which could a ect an AdS3 calculation.
xed, followed by H small, since in that case we know from the form of the heavy-light blocks in the c ! 1 limit that, on the second sheet [33, 35], the vacuum block is [6]
z2hLV(z)
11 24ihHcz !2hL
: (4.10)
Series expanding in 1=c, we can read o the cq coe cients in this limit and determine the prefactor aq, with the result11
cq(hL; hH) = (2i)q(2hH)q(2hL)q
q! (4.11)
Substituting these coe cients back into the sum over singular terms
1
Xq=0cq(hL; hH)b2qzq ; (4.12)
we see that the sum on q is an asymptotic series, ie it has zero radius of convergence. One can nevertheless Borel resum it:
B(t) =
1
Xq=0cqtqq! = 2F1(2hL; 2hH; 1; 2it): (4.13)
Performing the Borel integral
tb2z )dt, we obtain a relatively compact expression for the resummation of the leading singular terms:
lim
R
1
0 etB(
(z2hL)V(z) = G
hH; hL; icz
12
+ G
hL; hH; icz 12
(4.14)
c!1
cz xed
11Note that since the approximation (4.10) retains some of the hH-dependence and all of the hL-dependence of the coe cients cq in its 1/c series expansion, this also provides a non-trivial consistency check of equation (4.9).
{ 18 {
where
G(h1; h2; x) (x)2h1(2h2)2h1 1F1(2h1; 1 + 2h1 2h2; x): (4.15) This might be compared with the integral formulas from [47] derived from AdS physics.
As one might expect, we see that the singular terms all resum into something that shuts down at z 0. The two terms above decay like z2hH and z2hL, respectively. Suggestively,
these exponents would naively correspond to the contributions from a OHOH double-trace
operator and a OLOL double-trace operator. This is closely related to the fact that if one
takes the expression for the vacuum block in the heavy-light limit
V /
and promotes it to a periodic function of (which the full correlator must be) by adding all its images under ! + 2n, then this generates additional contributions in the
conformal block decomposition that behave like double-trace operators in the OLOL OPE. It is interesting that, unlike the global conformal blocks, the Virasoro conformal blocks thereby \know" about double-trace operator contributions in the same channel OLOL !
OH OH as the vacuum.
Adopting the nomenclature of [36], the above expression interpolates between the \Lyapunov" regime, where c is large with cz xed and large, and the \Ruelle" regime, where c is large with cz xed and small. For hH = hL, the expression simpli es somewhat:
lim
hH!hL
JHEP03(2017)167
2hL
TH sin2(TH(t + i)
2hL
TH sin2(
TH(t i)
(4.16)
G(hL; hH; x) + G(hH; hL; x) = x2hLU(2hL; 1; x): (4.17)
where U(a; b; x) is a con uent hypergeometric function.12 It is particularly simple at hL = 1=2, since U(1; 1; x) = ex (0; x). In gure 1, we have plotted the resulting behavior for the correlator (only including the vacuum block contributions) interpolating between the Lyapunov and Ruelle regime for hL = hH = 12 . In gure 2, we compare the behavior of the vacuum block with that of the approximate formula (4.10) from the heavy-light limit.
Although all of these plots only include vacuum block contributions, they seem to agree with qualitative expectations for the behavior of the full correlator.
We make one nal comment on the relation of this result to the heavy-light limit. One open question has been whether or not taking the heavy-light limit, then analytically continuing around z 1, and nally taking c large with hL; hH; and cz xed is the same as
simply analytically continuing the exact Virasoro block and then taking the limit c large with hL; hH; and cz xed. So far, all indications are that these di erent orders of limits do commute for the O(1=c) singular term (4.1), which was the main interest of [33], but in
the above we see explicitly that they do not commute for most other terms. In particular, taking the heavy-light limit followed by small hH completely discards the contribution
12For b /
2 Z,
U(a, b, x) = (b 1)
(a) z1b1F1(a b + 1, 2 b, x) +
(1 b)
(a b + 1) 1
F1(a, b, x) (4.18)
{ 19 {
in (4.15) that decays like z2hH, since by inspection we see that (4.10) contains only the
(cz)2hL piece at small cz. This is perhaps not so surprising, since the full result has
to be symmetric under hL $ hH, but taking the heavy-light limit breaks this symmetry
and makes the O(z2hH) contributions become formally non-perturbative e2 Hclog(z). By
contrast, by working out the exact coe cient of the leading singularities, we have kept the hH $ hL symmetry at all stages of the computation.
5 Heavy-light super-Virasoro vacuum blocks at large c
Similar to the case of non-supersymmetric CFTs that we have being discussing so far, in two-dimensional superconformal theories (SCFTs) there are degenerate operators whose correlators satisfy super null-state di erential equations. In this section, we will use these super null-state equations to calculate the large c heavy-light super-Virasoro vacuum block for these degenerate operators, and then analytically continue the result to obtain the super-Virasoro vacuum block for operators with general conformal dimensions. Speci cally, we will focus on the holomorphic part of the Neveu-Schwarz (NS) sector of 2d N = 1 [49{54]
and N = 2 [55{60] SCFTs (see e.g. [61] for a review of these theories). Previously, the N = 1 super-Virasoro blocks in NS sector have been studied using recursion relations [62{
64], while those of N = 2 are less investigated [24, 65].5.1 The N = 1 super-Virasoro vacuum block5.1.1 Brief review of 2d N = 1 SCFTsIn the N = 1 super-space, a point is denoted by Z (z; ), where is a Grassmann
variable. A primary super eld h(Z) of conformal dimension h can be expanded in terms of as h(Z) = h(z)+ h+12 (z), where h(z) and h+
JHEP03(2017)167
12 (z) are two component elds with conformal dimension h and h + 12, respectively . In the NS sector, the energy-momentum super eld T (Z), which has conformal dimension 3=2, can be expanded around the origin as
T (Z) =
Xn2Z1zn+2 Ln; (5.1)
where the fermionic generators Gr are the supersymmetry generators and the bosonic generators Ln are Virasoro generators. The (anti-)commutation relations between these generators are:
[Ln; Lm] = (n m) Ln+m +
c12 n3 n
Xr2Z+ 1212zr+3=2 Gr +
n+m;0;
{Gr; Gs[notdef] = 2Lr+s +
c 3
r2 1 4
r+s;0;
[Ln; Gr] =
n2 r
Gn+r; m; n 2 Z; r; s 2 Z +1 2:
(5.2)
The singular terms in the OPE of T (Z1) and (Z2) are
T (Z1) (Z2)
h 12
Z212
(Z2) + 1
2Z12 D2 (Z2) +
12
Z12 @2 (Z2);
{ 20 {
where Zij = zij i j; zij = zi zj and Di = @ i + i@zi: Descendant super elds are obtained by acting on a primary with Ln and Gr for n; r > 0. From the above OPE,
one can derive that correlation functions with one descendant super eld can be written in terms of a di erential operator acting on the correlation functions with only primary super elds [54] via
h(Ln 1)(Z1)X[angbracketright] = Ln [angbracketleft]( 1)(Z1)X[angbracketright] ; [angbracketleft](Gr 1)(Z1)X[angbracketright] = Gr [angbracketleft]( 1)(Z1)X[angbracketright] ; (5.3) where X = 2(Z2) [notdef] [notdef] [notdef] N(ZN) is an assembly of primary super elds, and i has conformal
dimension hi. These two super-di erential operators are
Ln =
N
Xi=2Zni1[(1 n)(hi +12 i1Di) + Zi1@zi] [angbracketleft] 1(Z1)X[angbracketright] ;
Gr =
N
Xi=2Z(r+12 )i1 [(2r 1)hi i1 + Zi1(Di 2 i1@zi)] [angbracketleft] 1(Z1)X[angbracketright] : (5.4)
N-point functions of the super elds FN [angbracketleft] 1(Z1) 2(Z2) [notdef] [notdef] [notdef] N(ZN)[angbracketright] should be in
variant under the global superconformal transformations generated by L[notdef]1; L0; G[notdef]
1
2 , which
leads to the superconformal Ward identities [53]:
L1 :
N
Xi=1@ziFN = 0;
G
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1
2 ; G
1
2 :
N
Xi=1(@ i i@zi)FN =
N
Xi=1(2hi i + zi( i@zi @ i))FN = 0;
L0 :
N
Xi=1(2zi@zi + 2hi + i@ i)FN = 0;
L1 :
N
Xi=1(z2i@zi + zi(2hi + i@ i))FN = 0:
(5.5)
Due to these constraints, the two-point function is xed (up to normalization) to be
h 1 (Z1) 2 (Z2)[angbracketright] =
1 Z2h121
h1;h2 = 1 z2h121
+ 1 2 2h1 z2h1+121
! h1;h2: (5.6)
where each term on the r.h.s. corresponds to a two-point function of the component elds.
5.1.2 N = 1 super-Virasoro vacuum blocks at leading oderThe heavy-light super-Virasoro vacuum block V L L H H is the contribution to the heavy-
light four-point function [angbracketleft] L (Z1) L [angbracketleft]Z2[angbracketright] H [angbracketleft]Z3[angbracketright] H (Z4)[angbracketright] from an irreducible representation of the superconformal algebra whose highest weight state is the vacuum [notdef]0[angbracketright]. In the
following calculation, we will take the heavy-light limit, meaning that
H
hHc ; hL xed as c ! 1:
{ 21 {
Our result of this part is V L L H H given in (5.8) with fhL and ghL given in (5.17)
and (5.19).
As the four-point function, the super-Virasoro vacuum block V L L H H also satis es the superconformal Ward identities. There are eight coordinate variables (four Grassmann even and four Grassmann odd) in V L L H H and it satis es ve global superconformal
Ward identities, which means that there are only three independent superconformal invariants, two Grassmann even and one Grassmann odd. The two Grassmann even invariants that we choose are [50]
x0
Z12Z34
Z13Z24 ; x1
Z14Z23
Z13Z24 (1 x0) : (5.7)
It is easy to verify that x21 = 0 and superformal Ward identities x V L L H H (which is
Grassmann even) to be of the following general form:
V L L H H = 1Z2hL21Z2hH34
[fhL (x0) + x1ghL (x0)] : (5.8)
The conformal dimensions of the degenerate elds in the NS sector of an N = 1 SCFTs
can be parameterized by
hr;s = [(m + 2) r ms]2 48m (m + 2) ; c =
32
12m (m + 2) r; s 2 Z+; r s 2 2Z: (5.9)
and the corresponding null-state is at level rs
2 . The rst non-trivial null state
(r = 1; s = 3) is
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22h1;3 + 1L1G1=2 G3=2 [notdef]
1;3[angbracketright] = 0; (5.10)
with h1;3 = 12 3c +O 1=c2
in the large c limit. If L = 1;3 in the heavy-light four-point function [angbracketleft] L L H H[angbracketright], then
22h13 + 1L1G1=2 G3=2
1;3 (Z1) 1;3 (Z2) H (Z3) H (Z4)
= 0: (5.11)
Using (5.4), we get a null-state equation satis ed by the four-point function, which also satis ed by the super-Virasoro vacuum block. Simplifying this equation using the super-conformal Ward identities (L1 becomes just @z1 and G1=2 becomes just D1 = @ 1 + 1@z1),
we nd
(2@z1 (@ 1 + 1@z1) 2h1;3 + 1 +4
Xi=2
Z1i1 (@ i i@zi + 2 1@zi) + 2hi i1Z2i1
)
V 1,3 1,3 H H = 0:
This is a super-di erential equation with two unknown function f(x1) and g(x1). To solve it, we can expand it in terms of is and require that all the coe cients of is equal to zero. First, we can send the zis to (0; z; 1; 1), in which case, x0 and x1 becomex0 ! z + 1 2 z 1 3;
x1 ! 1 2 1 3 + 2 3:
{ 22 {
Expanding the super-di erential equation in terms of is, we get two di erential equations from the coe cients of 1 and 2 (di erential equations from coe cients of other is are dependent with these two). In the large c limit, with h1;3 = 12 3c +O 1=c2
and H = hHc
xed, the leading order (c0) of these two equations are13
(z 1)2
zf[prime][prime]h1,3(z) + 2f[prime]h1,3(z)
+ z Hfh1,3(z) = 0;
z
f[prime][prime]h1,3(z) + g[prime]h1,3(z) + 2f[prime]h1,3(z) + gh1,3(z) = 0:
Solving these equations and xing the constants of integration to match the expansion of the vacuum block in terms of small z, we nd
fh1,3 (z) = z1e
1
2 f00(z); (5.12)
gh1,3 (z) = 1
z
fh1,3 (z)
z f[prime]h1,3 (z) : (5.13)
where f00 (z) is de ned in equation (3.13). These solutions only apply to hL = h1;3 in the large c limit. But the appearance of f00(z) in fh1,3(z) gives us some hints for how to analytically continue to nd fhL(z) for general hL, which is what we are going to do in the following. After getting fhL(z), we can use it to obtain ghL(z) without using the null-state equations.
Expanding both sides of the vacuum block of the super elds (5.8) in terms of is and
matching the coe cients of is, we can obtain relations between vacuum blocks of the component elds14 and the functions fhL(z) and ghL(z):
VLLHH =z2hLfhL (z) ; (5.14)
V L LHH = z2hL f[prime]hL(z)
2hLfhL(z)z + ghL(z)
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: (5.15)
where VLLHH is from the term without i in it and V L LHH is from the coe cient
of 1 2. In (5.15), the minus sign in front is due to the fact that anti-commutes with
. Using equation (5.14) for h1;3 = 12 + O (1=c) in the leading large c limit, we have V1,31,3HH = zfh1,3 = e
1
2 f00(z), which suggests that for general hL, we should have
VLLHH = ehLf00(z): (5.16)
Using equation (5.14) again, we have
fhL (z) = z2hLehLf00(z): (5.17)
13For later reference, the exact di erential equations are
f[prime][prime]h1,3 + 2(3 z)h1,3 + 3z 1
2(z 1)z
f[prime]h1,3
(2h1,3 + 1) hH
(z 1)2
fh1,3 + 2h1,3 + 1
2(z 1)z
gh1,3 = 0
f[prime][prime]h1,3 + (6 4z)h1,3 + 2z 1
2(z 1)z
f[prime]h1,3 + g[prime]h1,3 + z 2(z 2)h1,3
2(z 1)z
gh1,3 = 0
14These vacuum blocks are normalized such that the rst term of the small z expansion of a vacuum block VOL(0)OL(z)OH(1)OH(1) is [angbracketleft]OL(0)OL(z)[angbracketright].
{ 23 {
From equation (5.16), one can see that the super-Virasoro vacuum block VLLHH in
N = 1 SCFTs is the same as the vacuum block in non-susy CFTs at leading order of the
large c limit. We explain in detail why this is true in appendix C, but the basic point is that in this limit, only the pure Virasoro generators contribute to the sum over intermediate states in this block.
To get ghL(z), we need to know V L LHH. At leading order of the large c limit,
the only di erence between V L LHH and VLLHH (up to normalization) is that the
conformal dimensions of the light operators are di erent (hL = hL; h L = hL + 12).15 Since we know VLLHH = ehLf00(z), we can immediately see that
V L LHH = 2hLe(
hL+ 12 )f00(z); (5.18)
where the prefactor 2hL is due to our convention of the vacuum block and can be read o from the two-point function of super elds (5.6). Equating the above vacuum block to (5.15), we nd
ghL (z) = 2hLz2hLe
(hL+ 12 )f00(z) + 2hLfhL (z)z f[prime]hL (z) : (5.19)
One can check that setting hL = 12 gives us back gh1,3 (5.13).
Having the expressions for fhL and ghL, we can restore their argument to x0, then other super-Virasoro vacuum blocks of the component elds can be read o from the expansion of V L L H H (5.8) in terms of the i variables.
5.2 The N = 2 super-Virasoro vacuum block5.2.1 Brief review of 2d N = 2 SCFTsIn the N = 2 superspace, a point is denoted by Z z; ;
, where z is the usual complex coordinate, while and are two Grassmann coordinates. The energy-momentum super eld can be expanded as
J (Z) = J (z) + G (z) G (z) + 2T (z) : (5.20)
where J(z) is the U(1) R-current. The mode expansions are de ned in the usual way
J(z) =
Xn2ZJnzn+1 ; G(z) = Xr2Z+ 12Grzr+32; G = Xr2Z+ 12Grzr+32; T (z) =
JHEP03(2017)167
Xn2ZLn zn+2 :
15This point can be seen from the commutation relations of the Virasoro generators with these component elds (C.1), and at leading order of large c limit, only Virasoro generators contribute to these two vacuum blocks.
{ 24 {
The full N = 2 superconformal algebra of these generators takes the following form:
[Lm; Ln] = (m n) Lm+n +
c12 m3 m
m+n;0;
[Lm; Gr] =
m2 r Gm+r;
Lm; Gr =
m2 r Gm+r;
[Jm; Jn] = c
3m m+n;0; [Lm; Jn] = nJm+n;
[Jm; Gr] = Gm+r;
Jm; Gr = Gm+r;
Gr; Gs = 2Lr+s + (r s) Jr+s +c 3
r2 1 4
r+s;0
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{Gr; Gs[notdef] =
Gr; Gs = 0; m; n 2 Z; r; s 2 Z +1 2:
(5.21)
A super eld (Z) can be expanded in terms of and as
qh (Z) = qh (z) + q1h+12 (z) + q+1h+12 (z) + qh+1 (z) ; (5.22)
where the superscripts and subscripts are the conformal dimensions and U (1) charges of the component elds. The OPE of J (Z1) and (Z2) is
J (Z1) (Z2)
2h 12 12
Z212
(Z2) + 12D2 12D2
Z12 (Z2) +
2 12 12
Z12 @z2 (Z2) +
qZ12 (Z2):
where the super derivatives and super-translationally invariant distance are
Di = @ i + i@zi; Di = @ i + i@zi; Zij zij i j i j; (5.23)
with zij = zi zj, ij = i j and ij = i j.
The highest weight states in the NS sector are characterized by their eigenvalues under L0 and J0 :
L0 [notdef] [angbracketright] = h [notdef] [angbracketright] ; J0 [notdef] [angbracketright] = q [notdef] [angbracketright] ; (5.24)
and they satisfy
Ln [notdef] [angbracketright] = Jn [notdef] [angbracketright] = Gr [notdef] [angbracketright] = Gr [notdef] [angbracketright] = 0; for n; r > 0: (5.25)
Acting on primary super eld with Ln; Gr; Gr; Jn(n; r > 0), we get the descendent super elds. Using the OPE of J and , one can show that the correlation function with
one descendant super eld can be written in terms of a super-di erential operator acting on a correlation function with only primary elds:
h(Ln 1)(Z1)X[angbracketright] = Ln [angbracketleft] 1(Z1)X[angbracketright] ; [angbracketleft](Jn 1)(Z1)X[angbracketright] = Jn [angbracketleft] 1(Z1)X[angbracketright] ; h(Gr 1)(Z1)X[angbracketright] = Gr [angbracketleft] 1(Z1)X[angbracketright] ;
(Gr 1)(Z1)X = Gr [angbracketleft] 1(Z1)X[angbracketright] :
(5.26)
{ 25 {
where X = 2(Z2) [notdef] [notdef] [notdef] N(ZN) is an assembly of primary elds with conformal dimension
hi and U(1) charge qi. These super-di erential operators are [60]
Ln =
N
Xi=2 Zni1
(1 n)
hi + 12 i1Di +12 i1Di + Zi1@zi qi2 i1 i1Z1i1n(1 n) ;
Jn =
N
Xi=2Zni1 i1Di i1Di + 2 i1 i1@zi + qi 2hi i1 i1nZ1i1
;
Gr =
N
Xi=2Zr1 2 i1
r
1
2
i1(2hi + qi + i1Di) + Zi1(Di 2 i1@zi)
;
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Gr =
N
Xi=2Zr1 2 i1
r
1
2
i1(2hi qi + i1Di) + Zi1(Di 2 i1@zi)
:
(5.27)
N-point correlation functions of the primary super elds should be invariant under the global super-conformal transformations generated by L[notdef]1; L0; J0; G[notdef]
1
2 ; G[notdef]
1
2 , which leads
to the superconformal Ward identities (D.1). These Ward identities completely x the two-point functions to be
h 1(Z1) 2(Z2)[angbracketright] =
1 Z2h121
eq2
12 12
Z12 q1+q2;0 h1;h2
= 1
z2h121
+ q2
z2h1+121
1 1 + 2h1 + q2 z2h+121
1 2 + 2h1 q2 z2h1+121
1 2
+ q2
z2h1+121
2 2 + 2h1 (2h1 + 1) z2h1+221
! q1+q2;0 h1;h2; (5.28)
up to a normalization constant. Each term in the above equation corresponds to a two-point function of the component elds. Notice that only the two-point function of the lowest component eld is normalized as usual.
5.2.2 Super null-state equations
The heavy-light super-Virasoro vacuum block V
1 1 2 2
qLL qLL qHH qHH is the contribution to the heavy-light four-point function [angbracketleft] qLL(Z1) qLL(Z2) qHH(Z3) qHH(Z4)[angbracketright] from an irreducible
representation of the superconformal algebra whose highest weight state is the vacuum [notdef]0[angbracketright].
In this paper, we will take the following heavy-light limit:
hL; qL; H
hHc ; q
qHc xed as c ! 1:
Our main result of this part is V
qLL qLL qHH qHH given in (5.29), with F (x0; x1; x2; x3; x4)
given in (5.31) and the gi;hL functions given in next subsection 5.2.3.
Superconformal Ward identities x the vacuum block (and the four-point function) to take the following form [58]
V qLL qLL qHH qHH =
1 Z2hL21Z2hH34
exp
qL 12 12Z12 + qH 34 34 Z34
F (x0; x1; x2; x3; x4) ; (5.29)
{ 26 {
where F (x0; x1; x2; x3; x4) is a function of ve superconformal invariants
x0 = Z12Z34Z13Z24 ; x1 =
Z14Z23
Z13Z24 + x0 1;
x2 = 23 23
Z23 +
34 34
Z34
24 24
Z24 ;
x3 = 12 12
Z12 +
24 24
Z24
14 14
Z14 ; (5.30)
x4 = 13 13
Z13 +
34 34
Z34
14 14
Z14 :
It is easy to verify that these super-conformal invariants satisfy the relations
x31 = 0; x22 = 0; x23 = 0; x24 = 0; x1x2 = x1x3 = x1x4 = 0;
x2x3x0 = x2x4; x2x3 = x2x4 + x3x4; x21 = 2x2x3x0 (1 x0) : which means that the most general form of F (x0; x1; x2; x3; x4) can be written as
F = g0;hL(x0)+x1g1;hL(x0)+x2g2;hL(x0)+x3g3;hL(x0)+x4g4;hL(x0)+x2x3g5;hL(x0): (5.31)
The conformal dimensions of the degenerate elds in the NS sector of N = 2 SCFTs
can be parameterized by16 [60]
hr;s = r2 18 t
rs
4 +
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8t ; c = 3 3t r 2 Z+; s 2 2Z+: (5.32)
For each degenerate eld with dimension hr;s, there is a null- eld at level rs
2 . The rst
non-trivial null-state (r = 1; s = 2) is:
h(q 1) L1 (2h1;2 + 1) J1 + G12 G1 2
i
q1;2
s2 1
8t
4q2 1
E
= 0; (5.33)
with h1;2 = c3q2
62c = 12 + 3
(q21)
2c + O 1=c2
: Notice that the U(1) charge q is a free parameter here. If hL = h1;2 in the heavy-light four-point function, then
D
(qL 1)L1 (2h1;2 + 1)J1 + G
1
2 G
1
2
qL1;2(Z1) qL1;2(Z2) qHH(Z3) qHH(Z4)
E
= 0:
Using equations (5.26), we get a super-di erential equation satis ed by the four-point function, which is also satis ed by the vacuum block V
qLL qLL qHH qHH . Simplifying this super-di erential equation using the superconformal Ward identities (D.1) (L1 ! @z1,
G
1
2
! D1), we nd
(qL 1) @z1 (2hL + 1) J1 + D1D1
V qL1,2 qL1,2 qHH qHH = 0; (5.34)
with J1 given in (5.27) and D1; D1 given in (5.23).
16Besides hr,s, there are other degenerate elds whose conformal dimensions can be parameterized by hk = kq + 12 t(k2 14 ), k 2 Z + 12 and having a null eld at level [notdef]k[notdef], but these will not be used in this paper.
! D1 and G
1
2
{ 27 {
To solve this super-di erential equation, we can expand it in terms of is and is to get six independent di erential equations to solve for the six unknown functions17 g0;h1,2(z); [notdef] [notdef] [notdef] , g5;h1,2(z). These solutions gi;h1,2(z) only apply to those vacuum blocks whose light operators are degenerate operators with hL = h1;2. To get gi;hL(z) for general hL we need to analytically continue these solutions, as what we did for the non-susy Virasoro blocks. But in the non-susy case, there was only one unknown function and we already knew its anzatz for general hL (2.9), so things were easier there. Here, we have six gi;h1,2(z) functions and some of them are complicated and hard to know how to analytically continue them. But it turns out that once we solve the di erential equation for g0;h1,2(z), then analytically continue the solution to get g0;hL(z), we can derive the other gi;hL(z) functions from it, which will be shown in next subsection 5.2.3. The equation that only involves g0;h1,2(z) is
g[prime][prime]0;h1,2(z) +
6qL q z1+ 2 z
g[prime]0;h1,2(z) +6z H + 3 q 3z q2L1
q + (z2)qL
(z 1)2z
g0;h1,2(z) = 0:
The solution is
g0;h1,2(z) = z1e
1
2 ~f(z) (1 z)3 qqL ; (5.35)
where
~f(z) = (1 ~
) log (1 z) 2 log
1 (1 z)~ ~
; (5.36)
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with =
q1 24 H + 36 2q. In the above solution, the constants of integration have been xed such that the rst term in the expansion of g0;h1,2(z) in small z is 1, which corresponds to the vacuum block.
5.2.3 Solutions for general hLIn this subsection, we are going to analytically continue g0;h1,2 to get g0;hL, then use it to derive the other gi;hL functions. Expanding the ansatz (5.29) in terms of is and is, we can express the vacuum blocks of the component elds in terms of gi;hL(z):18
VqLLqLLqHH qHH = z2hLg0;hL; (5.37)
V qL1L qL+1LqHH qHH = z(2hL+1)
(qL 2hL) g0;hL + zg1;hL + g3;hL + zg[prime]0;hL ; (5.38)
V qL+1L qL1LqHH qHH = z(2hL+1)
(qL + 2hL) g0;hL zg1;hL + g3;hL zg[prime]0;hL ; (5.39)
VqLL qLLqHH qHH = z(2hL+1)
qLg0;hL g3;hL +zz 1 g2;hL
; (5.40)
V qLLqLLqHH qHH = z(2hL+1) (qLg0;hL + g3;hL + zg4;hL) ; (5.41)
V qLL qLLqHH qHH = z2hL
g[prime][prime]0;hL 4hLz g[prime]0;hL + g1;hL
+ 2hL (2hL + 1)
z2 g0;hL
+2g[prime]1;hL +
1(1 z)z
(qLg2;hL + g5;hL) + qL
z g4;hL
: (5.42)
17Again, we send the coordinates zi to (0, z, 1, 1), in which case, x0 ! z + [notdef] [notdef] [notdef] , where [notdef] [notdef] [notdef] represents
terms proportional to i, i or their products.
18These vacuum blocks are normalized such that the rst term of the small z expansion of a vacuum block VOL(0)OL(z)OH(1)OH(1) is [angbracketleft]OL(0)OL(z)[angbracketright].
{ 28 {
The basic idea of these derivations is to derive the vacuum blocks on the l.h.s. , then solve the above equations to get the functions gi;hL(z) on the r.h.s. .
First, the most important function is g0;hL(z), which is associated with
VqLLqLLqHH qHH . From equation (5.37), for hL = h1;2 = 12 + O(1=c), we have
VqL1,2qL1,2qHH qHH = zg0;h1,2 = e
1
2 ~f(z)(1 z)3 qqL, which suggests that for general hL,
we should have
VqLLqLLqHH qHH = ehL ~f(z)(1 z)3 qqL: (5.43)
Indeed, this matches the Virasoro vacuum block for CFT2s with a global U (1) symmetry, which have been computed in [7]. The fact that this super-Virasoro vacuum block only gets contributions from the Virasoro generators and U(1) generators at leading order of the large c limit can be seen from the commutation relations of these generators with the component eld , as we explain in the appendix D.2. Using equation (5.37) again, we have
g0;hL (z) = z2hLehL ~f(z) (1 z)3 qqL : (5.44)
Next, to get g1;hL(z) and g3;hL(z), we need to know the blocks V
qL1
L qL+1LqHH qHH
and V
qL+1L qL1LqHH qHH . At leading order of the large c limit, the only di erences between these two blocks and V
JHEP03(2017)167
qLL qLLqHH qHH are that the conformal dimensions and U(1) charges of the light elds are di erent (note that the conformal dimensions of L and L are hL+ 12, while that for L is hL), which means that we can change the parameters accordingly in the expression of V
qLL qLLqHH qHH to get these two blocks:
V qL1L qL+1LqHH qHH = (2hL qL)e
(hL+ 12 ) ~f(z) (1 z)3 q(qL+1) z2hL1gqL+1(z);
V qL+1L qL1LqHH qHH = (2hL + qL)e
(hL+ 12 ) ~f(z) (1 z)3 q(qL1) z2hL1gqL1(z): where the prefactor 2hL qL is due to our convention of the de nition of the vacuum
blocks and can be read o from the two-point function (5.28). Equating these two blocks to equations (5.38) and (5.39) respectively, we can solve for g1;hL(z) and g3;hL(z)
g1;hL = 1
2z 4hLg0;hL 2zg[prime]0;hL gqL+1 gqL1
; (5.45)
g3;hL =12 (2qLg0;hL gqL+1 + gqL1) : (5.46)
The remaining functions g2;hL(z), g4;hL(z) and g5;hL(z) are related to the vacuum blocks V
qLL qLLqHH qHH , V
qLL qLLqHH qHH and V
qLL qLLqHH qHH (5.40){(5.42), respectively.
As is shown in appendix D.4, qLL can be written as descendant elds plus a Virasoro and U(1) primary
qLL(z) =
12h2 3q2
2ch 3q2
(J1qLL)(z) +
q(c 6h)
2ch 3q2
(L1qLL)(z) + ~
qLL(z) (5.47)
The Virasoro and U(1) primary part ~
qLL has conformal dimension hL + 1 and U(1) charge qL, which are the same as qLL. Using this decomposition, we can calculate these three
{ 29 {
vacuum blocks from V
qLL qLLqHH qHH . Some details for performing these calculations are given in appendix D.4. Equating these three vacuum blocks to equations (5.40), (5.41)
and (5.42), we can solve for g2;hL(z), g4;hL(z) and g5;hL(z). At leading order of the large c limit, these functions are
g2;hL =
3z q q2L 4h2L
g0;hL + 2(z 1)hLg3;hL + (z 1)zqLg[prime]0;hL
2hLz ; (5.48)
g4;hL = g2;hL g3;hL; (5.49)
g5;hL = (z 1)z
4
(z 1)q2Lh2L+ 4
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g[prime][prime]0;hL + (z 1) 2zg[prime]1;hL 4hLg1;hL + qLg4;hL
qLg2;hL
(z 1) 6z qqL(4h2L q2L) + q2L (2(z 2)hL z) + 16h3L 4 h2L
g[prime]0;hL
+ 4h2L q2L
2 2hLz
(z2e ~f 1)(1 z) (2hL + 1) + 3(z 2)z qqL
4h2L q2L
+ 9 2qz2 2hL
!g0;hL:
(5.50)
with ~f in g5;hL given in (5.36). One can easily check that setting hL = h1;2 [similarequal] 12, these
gi;hL(z) functions will become gi;h1,2(z) and they are indeed the solutions to the null-state equation (5.34) at leading order of the large c limit. Restoring the argument of gi;hL to x0, other super-Virasoro vacuum blocks of the component elds can be read o from the expansion of V
qLL qLL qHH qHH in terms of is and is. Weve checked the rst few terms of the expansion of these other blocks in terms of small z, and they match the results from the direct calculation of these blocks.19
Acknowledgments
We would like to thank David Gross, Tom Hartman, Gary Horowitz, Ami Katz, Don Marolf, Dan Roberts, Edgar Shaghoulian, Douglas Stanford, and Matt Walters for valuable discussions. ALF is supported by the US Department of Energy O ce of Science under Award Number DE-SC-0010025. JK and HC are supported in part by NSF grants PHY-1316665 and PHY-1454083, and JK is supported by a Sloan Foundation fellowship. We would also like to thank the GGI in Florence for hospitality as this work was completed.
A Summary of corrections to the vacuum block
In this appendix, we will list results concerning the large c expansion of the vacuum block.
At order c0 and 1=c, f00 (3.13) and f10; f01 are known in closed form. The functions f10 and f01 are known from the work of [11] to be
f10 = csch2
t 2
2
[notdef]
3 e tB et; ; 0
+ e tB et; ; 0
+ e tB et; ; 0
+ e tB et; ; 0
19By direct calculation, we mean to calculate the vacuum block by inserting the vacuum state and its descendants into the four-point function, and then summing over all these contributions.
{ 30 {
+ 1
2 + cosh( t)
1 2 + 6H + 6H + 6i 5 + 12 log
2 sinh
t2 + 5
t
13 2 1
coth t
2
2 + 12 log
2 sinh t
2
!;
t2
B(et; ; 0) + B(et; ; 0) + B(et; ; 0) + B(et; ; 0) 2
+H + H + 2 log
2 sinh
f01 = 6
csch2
t2 + i
+ 2
log
JHEP03(2017)167
sinh
t 2
csch
t2 + 1 : (A.1)
where B(x; ; 0) = x 2F1(1; ;1+ ;x)
is the incomplete Beta function, z 1 et, =
p1 24 H, Hn is the harmonic function.
At order 1=c2, we calculated the order H and 2H terms in the expansion of the vacuum block in the parameter H = hHc , and at order 1=c3, we calculated the linear H terms.
At order 1=c2,
log V
hL c2
1
Xk=0 k+1H f20k + hLf11k + h2Lf02k
:
The linear H terms are f200, f110 = f101 and f020 = f002. This rst term is given in equation (3.17), while the last two terms can be obtained from the expansion of f10 and
f00. The 2H terms are
f201 =432 (z 1)z(15z 46) + 42(z((z 2)z 10) + 12)
log2(1 z)
z3
10368(z 2)2Li2(z)2
z2
+ 864 (9z 46)(z 1)2 + (4z(15 2(z 2)z) 72) log(z)
log3(1 z)
+ 864(z(z(z(5z 44) + 103) 96) + 33) log4(1 z)
z4 +
z3
+ 5184Li2(z) log(1 z)(z(z(7z 32) + 32) + 2(z 9)(z 2)(z 1) log(1 z))
z3
+ 5184Li3(1 z)(z(z(5z 14) + 16) 4(z 3)(z 1)(z + 2) log(1 z))
z3
+ 10368Li3(z)((z 2)z + 2((z 8)z + 8) log(1 z))
z2 +
20736(z 2)Li4(1 z)
z
Li4 zz1 + Li4(z)
z2 +
12960(z 2)Li2(z) z
+ 216 (z 2)z2 + 96(6 5z) (3) 42(z(5z 14) + 16)z
log(1 z)
+
20736((z 6)z + 6)
z3
144 525z2 + 180(z(5z 14) + 16) (3) + 84(z 2)z
5z2
+ 2592(z(5z 14) + 16) log(z) log2(1 z)
z2 ; (A.2)
{ 31 {
f111 =864 3z z2 8z + 7
+ 82 2z2 9z + 8
log2(1 z)
z3 +
5184(z 2)Li2(z) z
41472(z 1)Li2(z)2 z2
+ 3456((z 1)((z 17)z + 21) 6(z(2z 9) + 8) log(z)) log3(1 z)
z3
+ 10368((z 7)z + 7) log(z) log2(1 z)
z2
+ 1728(z(z(z(3z 44) + 127) 136) + 51) log4(1 z)
z4
41472((z 6)z + 6)Li3(z) log(1 z)
z2
+ 20736Li2(z) log(1 z)(z((z 9)z + 9) + (z((z 14)z + 34) 22) log(1 z))
z3
+ 20736Li3(1 z)(z((z 7)z + 7) 2(z(2z 9) + 8) log(1 z))
z3
+
JHEP03(2017)167
2 ((z 7)z + 7) (3) z2
+ 432 3(z 2)z2 + 96(z(2z 9) + 8) (3) 82((z 7)z + 7)z
41472((z 6)z + 6)
Li4(z) + Li4 zz1
z2 + 20736
log(1 z)
z3 ; (A.3) and f021 = f012 can be obtained from the expansion of f01.
At order 1=c3,
log V
hL c3
1
Xk=0 k+1H f30k + hLf21k + h2Lf12k + h3Lf03k
;
The linear H terms are
f300 =864(2z(z(z2 8z + 17) 14) + 9) log4(1 z)
z4
20736(z 1)Li2(z)2 z2
+ 216 log2(1 z) 82(z 2) z2 2
108z z2 1
log(z) + 73z(z 1)2
z3
864(z 2) log3(1 z) 4 2z2 3
log(z) 9(z 1)2
z3
+ 432Li2(z) 73(z 2)z2 + 24(z 1) log(1 z)((6 4z) log(1 z) 9z) z3
+ 5184 z2 1
Li3(1 z)(9z + 4(z 2) log(1 z))
z3
20736(2z 3)Li4
z z1
z2
5184(z 2)Li3(z)(9z + 4(z 2) log(1 z))
z2 +
20736(z 3)(z 1)Li4(z) z2
+ 20736(z 2)Li4(1 z)
z +
192 1215 z2 1
(3) + 320z2 64(z 2)z
5 z2
+ 12 (z 2) z2 1728 (3)
+ 6482z z2 1
log(1 z)
z3 ; (A.4) f210 = f201, and f120 = f102, f030 = f003 can be obtained from the expansions of f10 and
f00, respectively.
The terms f200,f201,f111 and f300 were derived for the rst time in this work. Weve checked these expressions against a direct small z expansion up to O(z9) using the methods
{ 32 {
of [2]. Weve also analytically continue these results to the second sheet and checked that the they do contain the rst few terms of (4.2) and (4.12). Under this analytic continuation, the various logarithms and polylogarithms have monodromies
log(1 z) ! log(1 z) 2i;Lin(z) ! Lin(z) +
2i (n 1)!
logn1(z);
(A.5)
Lin(1 z) ! Lin(1 z); Lin
zz 1
!
Lin
zz 1
2i (n 1)!
logn1
zz 1
;
JHEP03(2017)167
which can be derived from Lin(z) =
R
t dt and Li1(z) = log(1 z).
B Direct derivation of leading logs in the Lorentzian regime
In subsection 4.1, we presented a proof that the \leading logs" in the Lorentzian regime resum to form a correction to the leading singularity (cz)1 that appears at O(1=c) in a
large c expansion. The proof given was somewhat indirect, however, and in this appendix we will give another proof that is more cumbersome, but more directly connected to the structure of the di erential equations for the degenerate operators that are used order-by-order in 1=c in the rest of the paper. In this appendix, for convenience we de ne
~V(z) = z2hLV(z); (B.1)
so that for the vacuum block, ~V(z) z!0! 1 on the rst sheet.
From equation (3.7), at large c the null equation of motion for the degenerate operator
O1;s takes the form
(Ls1 + O(1=c))O1;s = 0: (B.2)
In terms of ~V, (B.2) translates into the di erential equation
(@sz + O(1=c))
zs1 ~V(z)
z 0
Lin1(t)
= 0: (B.3)
We will organize the solution to (3.8) in a series expansion of ~V:
~V(z)
~V0(z) +
1 c
~V2(z) + : : : : (B.4)
The lowest-order term then obey the following di erential equation
@sz(zs1 ~V0(z)) = 0; (B.5)
whose general solution takes the form
~V0(z) =
s1
~V1(z) +
1 c2
Xi=0cizi (B.6)
{ 33 {
with s free coe cients ci. Of course, the relevant solution for the vacuum at c ! 1 is
c0 = 1; ci[negationslash]=0 = 0. But equally importantly, when we work to higher orders, all the solutions
above will continue to be homogeneous solutions, and there will also be one particular solution at each order that arises because of the \source" from the lower order terms.
There is a drastic simpli cation that occurs if we are interested only in the leading log terms. First, notice that none of the homogeneous solutions (B.6) have logarithms in them. As a result, logarithms can be produced only by the \particular" solutions, which are integrals of the lower-order solutions. More precisely, the leading logarithms arise from integrating the lowest order solution and never introducing any \homogeneous" terms, since doing so would reduce the power of the logarithm. Therefore, we can simply perform our analysis directly on the second sheet (where the di erential equation must still be satis ed), and the unknown integration constants that enter at each step will not contaminate the leading logs.
Using the expression (3.9), it is straightforward to extend this argument to leading logs in the heavy-light limit as well. At in nite c, the general solution to (3.9) is
~V(t) = e
1s
2 t
s1
2j s 12 p1 24 H : (B.7)
These are all exponentials in t, i.e. powers in (1 z). Therefore, logarithms of z can arise
only from integrating source terms that are generated from the solution at lower orders.
Let us see how this works in practice, and along the way we will illustrate some points. For simplicity, we will begin by solving for the leading logarithms in the conformal block
[angbracketleft]OHOHO1;2O1;2[angbracketright]. Once we have gone through this case, it will be easy to see how to
generalize to arbitrary degenerate operators. The exact equation of motion for ~V(z) is 0 = (z 1)
(4(z 2)h1;2 + 4z 2)~V[prime](z) + 3(z 1)z~V[prime][prime](z) 2z (2h1;2 + 1) ~V(z)hH(B.8)
This can be solved in closed form by a hypergeometric function, but to illustrate our points we will solve it in a 1=c expansion. At leading order it is just (B.5) with s = 2. At next order, it is
@2z(z ~V1(z)) =
6hHz
(1 z)2
JHEP03(2017)167
Xj=0cj exp
t
; (B.9)
which is easily solved:
~V1(z) = c0 +
6(z 2)hH log(1 z)
z (B.10)
We x c0 and c1 on the rst sheet by demanding that ~V1 have the correct behavior (i.e.,
have leading term / z in a small z expansion), and then analytically continuing to the
second sheet and taking small z to nd the small z behavior on the second sheet. Doing this, we nd c0 = 12hH; c1 = 0 on the rst sheet. Analytically continuing, this means that on the second sheet,
c1 = 24ihH; c0 = 12ihH: (B.11)
{ 34 {
c1z
Note that at this order, there are no logarithms log(z) in a small z expansion, even on the second sheet:
~V1(z) =
c1z + (c0 12hH) + O(z): (B.12)
To see the emergence of logarithms, we have to work to the next order in 1=c. The equation of motion for ~V2 is
@2z(z ~V2(z)) =
6hHz
(1 z)2
6 z ~V1(z)hH + (z 2)(z 1) ~V[prime]1(z)
(z 1)2
(B.13)
This can also be solved in closed form. It again has two free parameters corresponding to the two homogeneous solutions, which we can x the same way we xed them for the free parameters in ~V1. However, we can instead apply an argument that will easily generalize
to all higher orders, which is to expand the above equation of motion at small z directly on the second sheet:
@2z(z ~V2(z)) = 6c1
(1 hH) +
The solution to the above equation of motion is again easily determined:
~V2(z) = 6c1
2log(z)
z + log(z) + O(z)
JHEP03(2017)167
1z +
2z2 + O(z)
: (B.14)
z + d0: (B.15)
We do not need to determine the integration constants d0; d1, because they do not contaminate the leading logs! Since the integration constants are always coe cients of the homogeneous solutions, this feature manifestly continues to all higher orders as well.
The above explicit demonstration was speci c to the O1;2 block, but it is straightfor
ward to generalize to general degenerate operators. For all degenerate operators O1;s, the 1=c piece ~V1 is the same universal function (B.10) (in fact, it is just the global conformal
block for the stress tensor), with a coe cient that is linear in hs;1:
~V1;s =
2hHh1;s
c z22F1(2; 2; 4; z): (B.16)
We do not have to appeal to our knowledge that this is the stress tensor conformal block; (B.10) is a derivation of ~Vs;1 since we know
~V1;s =
limc!1 h1,sh1,2
+ d1
~V1;2. This means
that generally, on the second sheet we have ~V1 is given by
c(s)1 =
lim
c!1
h1;s
h1;2
24ihH = 24ihH(s 1);
c(s)0 =
(12ihH) = 12ihH(s 1): (B.17)
Thus, ~V2 is generally given on the second sheet at small z by
@sz
zs1 ~V2 = 12As z~V1(z)z2 + O(1=z)
= 12c(s)1Asz2 + O(1=z); (B.18)
{ 35 {
lim
c!1
h1;s
h1;2
(where As depends on s but will be determined momentarily). We have taken advantage of the fact that z ~V1(z) is regular at z ! 0 since c1=z was the most singular term generated
at this order, and so by scaling z ~V1(z)=z2 is the most singular term generated in the null
equation of motion above. The solution to (B.18) is clearly
~V2(z) = 12c(s)1
log(z) z
As(s 2)!
+ O(log(z)): (B.19)
We can easily x As since ~V2 is completely determined for any h1;s by just the two function
~V1;2 and
~V2;2; therefore once we calculate
~V2 for two values of s, we know it for all s.
A simple computation shows that A2 = A3 = 1. Demanding consistency of the above equation with all r immediately xes
As(s 2)!
= 1: (B.20)
Finally, to get the leading logs, we can just iterate at higher orders, since the only way to get double logs is to integrate single logs (which rst appear in ~V2), and the only way
to get triple logs is to integrate double logs (which rst appear in ~V3), etc. So for instance,
in the equation of motion for ~V3, we can just look at
~V2 in the source terms, since this is
the only contribution that has a single log. But the relation between ~V3 and
~V2 at leading
order in 1=c is the same as the relation between ~V2 and
~V1 at leading order in 1=c:
@sz
zs1 ~V3 = 12(s 2)!z ~V2(z)z2 + O(1=z); (B.21)
and so on. Keeping track of just the most singular leading log terms, we see that at each order
~Vn(z) 12c1
logn1(z)
z(n 1)!
JHEP03(2017)167
logn(z)
zn! +O(z0; log(z)); (B.22) which proves that the leading singularity in the leading logs exactly exponentiates to all orders.
C Leading contribution to the vacuum blocks in N = 1 SCFTsIn this section, we are going to prove that the heavy-light vacuum block VLLHH in
N = 1 SCFTs is the same as the vacuum block in non-supersymmetric CFTs at leading
order of the large c limit, meaning that it only gets contributions from the pure Virasoro generators at this order.
The commutators of the symmetry generators with the component elds of a super eld (Z) = h (z) + h+12 (z) are
[Ln; (z)] = zn [h (n + 1) + z@z] ;
[Ln; (z)] = zn
h + 12
(n + 1) + z@z
+O(z0; log(z)) !
~Vn+1(z) 12c1
;
(C.1)
[Gr; (z)] = zr+
1
2 ;
{Gr; (z)[notdef] = zr
1
2 [h (2r + 1) + z@z] ; n 2 Z; r 2 Z +
1
2:
{ 36 {
The vacuum block VLLHH is the contribution to [angbracketleft]H(1)H(1)L(z)L(0)[angbracketright] from
an irreducible representation of the super-Virasoro algebra whose highest weight state is the vacuum [notdef]0[angbracketright]. The vacuum state is annihilated by Ln and Gr for n 1 and r 12.
Besides the vacuum state, other states in this representation are the descendants of the vacuum, which can be obtained by acting on the vacuum with Ln and Gr for n 2
and r
32 . To get the vacuum block, we can insert a projection operator into the four-point function:
VLLHH = [angbracketleft]H(1)H(1)P0L(z)L(0)[angbracketright] hH(1)H(1)[angbracketright]
: (C.2)
At leading order of the large c limit, we can use the approximate projection operator
n1
Pjl=1 rl. Notice that in the above equation (C.3), at each level N + R, we should only sum over independent states. For example, at level 3, we only have
L3, because G
Pil=1 nl and R =
32 G
Consider a state Grj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1 [notdef]0[angbracketright], its contribution to VLLHH is
H (1) H (1) Grj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1[notdef]0 0[notdef]Ln1[notdef] [notdef] [notdef] LniGr1 [notdef] [notdef] [notdef] GrjL (z) L (0)
hH(1)H(1)[angbracketright]
Ln1 [notdef] [notdef] [notdef] LniGr1 [notdef] [notdef] [notdef] GrjGrj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1
(C.4)
In the large c limit, the normalization factor in the denominator scales as
Ln1 [notdef] [notdef] [notdef] LniGr1 [notdef] [notdef] [notdef] GrjGrj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1
cN+R
because the commutation of each pair of generators Gr with Gr or Ln with Ln will give
us one power of c (5.2). In the numerator,
0[notdef]Ln1 [notdef] [notdef] [notdef] LniGr1 [notdef] [notdef] [notdef] GrjL (z) L (0) is orderO(1), because the commutation of these generators with L will not give us c or hH. And the remaining part in (C.4) scales as
H (1) H (1) Grj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1[notdef]0
hH(1)H(1)[angbracketright]
{ 37 {
JHEP03(2017)167
P0
0[notdef]L
[notdef] [notdef] [notdef] LniGr1 [notdef] [notdef] [notdef] Grj
Ln1 [notdef] [notdef] [notdef] LniGr1 [notdef] [notdef] [notdef] GrjGrj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1 : (C.3)
with ni 2 Z and rj 2 Z+ 12, because the states Grj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1 [notdef]0[angbracketright] are orthogonal
with each other at this order.20 We can arrange the order of the generators such that ni [notdef] [notdef] [notdef] n1 2 and rj [notdef] [notdef] [notdef] r1 32. Denote the level of each state as N + R, where
N =
X{ni;rj[notdef]
Grj [notdef] [notdef] [notdef] Gr1Lni [notdef] [notdef] [notdef] Ln1[notdef]0
32 = L3 and shouldnt be included.
hN+R=2H: (C.5)
The reason is that when we commute one Ln with H well get one power of hH, but
we need to commute two Grs with H to get one power of hH as can be seen from
the commutation relations (C.1). So in the heavy-light limit, with H = hH
c xed, the
contribution of (C.4) will be order O(cR=2). This means that at order c0 (that is, R = 0), the heavy-light vacuum block VLLHH in N = 1 SCFTs will only get contributions from
the pure Virasoro generators, which make it the same as that in non-susy CFTs at leading order. This is also true for the vacuum blocks V L LHH.
20The proof is similar to that for non-susy CFTs with only Virasoro generators, see appendix B of [6].
D Details of the N = 2 SCFT calculationsD.1 Superconformal Ward identities
N-point functions FN [angbracketleft] 1(Z1) 2(Z2) [notdef] [notdef] [notdef] N(ZN)[angbracketright] in N = 2 SCFTs satisfy the following
eight superconformal Ward identities
L1 :
N
Xi=1@ziFN = 0;
L0 :
N
Xi=1(2zi@zi + 2hi + i@ i + i@ i)FN = 0;
L1 :
N
JHEP03(2017)167
Xi=1(z2i@zi + zi(2hi + i@ i + i@ i) + qi i i)FN = 0;
J0 :
N
Xi=1( i@ i i@ i + qi)FN = 0 (D.1)
G
1
2 ; G
1
2 :
N
Xi=1(@ i i@zi)FN =
N
Xi=1(@ i i@zi)FN = 0;
G1
2 :
N
Xi=1[zi(@ i i@zi) i(2hi + qi + i@ i)]FN = 0;
G1
Xi=1[zi(@ i i@zi) i(2hi qi + i@ i)]FN = 0:
Speci cally, the three identities corresponding to L1, G
1
2 and G
2 :
N
1
2 were used in the
simpli cation of the super null-state equation (5.34).
D.2 Leading contributions to the vacuum blocks
Similar to the reasoning of N = 1 (appendix C), the vacuum block V
qLL qLLqHH qHH
in N = 2 SCFTs will not get contribution from the generators Gr and Gr at leading
order of large c limit, which makes it the same as the vacuum block of a theory with only Virasoro and U(1) symmetry at this order. This can be seen from the commutation relations of these symmetry generators with the lowest component eld (z) of a super eld qh (Z) = qh (z) + q1h+12 (z) + q+1h+12 (z) + qh+1 (z) : For simplicity, in the following subsections, we only keep the superscripts and subscripts when necessary.
Commutation relations of the generators with the component eld (z) are
[Ln; (z)] =zn [(n + 1) h + z@z] ; [Jn; (z)] =qzn;
[Gr; (z)] =zr+
1
2 ;
(D.2)
The last two commutators are exactly the same as that of the fermionic generator Gr with in N = 1 SCFTs (C.1), which upon the same reasoning means that when summing
over descendant states of the vacuum to get V
qLL qLLqHH qHH , those states having Gr
Gr; (z) =zr+12 :
{ 38 {
or Gr in them will not contribute at leading order of the large c limit. We can also easily see that some other vacuum blocks, such as V
qLL qLLqHH qHH , V
qLL qLLqHH qHH and
V qLL qLLqHH qHH , also only get contributions from Virasoro and U(1) generators. This point will be used in the calculation of subsection D.4. Note that to construct the projection operator for N = 2, the Hermiticity conditions among these generators are Ln = Ln,
Jn = Jn, Gr = Gr, Gr = Gr. And the vacuum [notdef]0[angbracketright] in N = 2 is annihilated by
Ln; Jm; Gr; Gr for n 1; m 0; r 12.
For completeness, the commutation relations of other component elds are
[Ln; (z)] = zn
h + 12
(n + 1) + z@z
;
JHEP03(2017)167
Ln; (z) = zn
h + 12
(n + 1) + z@z
;
[Ln; (z)] = zn [(h + 1) (n + 1) z + z@z] + 12n(n + 1)qzn1;
{Gr; (z)[notdef] =
Gr; (z) = 0
Gr; (z) = zr1 2
r + 12
(2h + q) + z@z
+ zr+12 ;
Gr; (z) = zr1 2
r + 12
(2h q) + z@z
zr+12 ;
[Gr; (z)] = zr
1
2
r + 12
(2h + q + 1) + z@z
; (D.3)
Gr; (z) = zr1 2
r + 12
(2h q + 1) + z@z
;
[Jn; (z)] = (q + 1) zn ;
Jn; (z) = (q 1) zn
[Jn; (z)] = qzn + 2hnzn1:
D.3 Correlation functions with descendant component elds
In this subsection, we are going to derive the relationships between correlation functions with descendant elds and correlation functions with only primary elds. These relationships are also true for the corresponding vacuum blocks. Speci cally, we only consider the lowest component primary eld qh and its descendants that are relevant to our calculation.
For correlation functions involving (L1) (z), since (L1) (z) = @z (z), we have
h(L1) (z) X[angbracketright] = @z [angbracketleft](z)X[angbracketright] (D.4)
where X is an assembly of primary or descendant component elds. If there are more than one (L1), we just need to take the derivatives in succession with respect to the
coordinate of each (L1).
{ 39 {
For correlation functions involving only one descendant (Jn), we have
h(Jn) (z1) Y [angbracketright] =
1
2i
Iz1 dz (z z1)n [angbracketleft]J (z) (z1) Y [angbracketright]
=
1
2i
N
Xi=2
Izidz (z z1)nqi [angbracketleft] (z1) Y [angbracketright] z zi
=
N
(D.5)
Xi=2qi [angbracketleft] (z1) Y [angbracketright] (zi z1)n
where Y = 2 (z2) [notdef] [notdef] [notdef] N (zN) is an assembly of primary elds with conformal dimensions
hi and U (1) charge qi, and we have used the OPE J(z)i(zi) qii(zi)zzi in the second line.
For correlation functions involving two (Jn)s, we need to know the OPE
J (z) (Jn) (w), which can be written as
J(z)(Jn)(w) =Xk>0(Jk;n)(w)(z w)k+1+ Xk 0 (Jk;n)(w) (z w)1k
(D.6)
In the rst sum, since [Jk; Jn] = c3k kn;0 and (Jk)(w) = 0 for k > 0, only the term with
k = n is non-zero. In the second sum, only the term with k = 0 is singular. So we have
J(z)(Jn)(w)
(Jn;n)(w) (z w)n+1
+ (J0;n)(w) z w
JHEP03(2017)167
(D.7)
where q is the U(1) charge of (and Jn will not change the U(1) charge) and means that
in the r.h.s. we omit terms that are regular. For n = 1, the OPE of J (z) with (J1) (w) is
J (z) (J1) (w)
c 3
nc 3
(w)(z w)n+1
+ q(Jn)(w) z w
(w) (z w)2
+ q (J1) (w)(z w)
(D.8)
In the calculation of this paper, we only need [angbracketleft](J11) (z1) (J12) (z2) Y [angbracketright] with Y = 3 (z3) [notdef] [notdef] [notdef] N (zN) an assembly of primary elds. Using the above OPE, we have
h(J11) (z1) (J12) (z2) Y [angbracketright] =
1
2i
Iz1 dz [angbracketleft]
J (z) 1 (z1) (J12) (z2) Y [angbracketright] z z1
=
1
2i
Iz2dzz z1
1 c32(z z2)2+ q2 (J12)(z z2)
Y
1
2i
N
Xi=3
Izidzz z1qi [angbracketleft]1 (J12) Y [angbracketright] z zi
= c
3 [angbracketleft]
12Y [angbracketright] (z2 z1)2
Xi=2qi [angbracketleft]1 (J12) Y [angbracketright] zi z1
= c
3 [angbracketleft]
12Y [angbracketright] (z2 z1)2
N
N
+
N
Xi=2
Xj=1;j[negationslash]=2qiqj [angbracketleft]12Y [angbracketright] (zi z1) (zj z2)
(D.9)
where in the second line, we used the OPE of J(z)(J12)(z2) and J(z)Y (or equa
tion (D.5)), and in the last line, we used equation equation (D.5).
{ 40 {
D.4 Decomposition of qh+1
In this subsection, well show that qh+1with conformal dimension h + 1 and U(1) charge q can be written as
qh+1(z) =
12h2 3q2
2ch 3q2
(J1qh)(z) +
q(c 6h)
2ch 3q2
(L1qh)(z) + ~
qh+1(z); (D.10)
where ~
qh+1 is a Virasoro and U(1) primary with conformal dimension h+1 and U(1) charge q, in the sense that L0~
qh+1 = (h + 1)~
qh+1, J0~
qh+1 = q~
qh+1 and Ln~
q = Jn~
q = 0; n 1.
In the following calculation, for simplicity, we only keep the superscripts and subscripts when necessary.
q can be obtained by acting on the lowest component eld q with G
1
2 and G
JHEP03(2017)167
1
2 :
q = 12
G12 G12 G12 G1 2
q =
L1 G12 G12 q: (D.11)
Suppose q can be written as
q = AJ1q + BL1q + ~
q; (D.12)
where A and B are two constants depending on h and q, and ~
q is a Virasoro and U(1) primary. Acting on (D.11) and (D.12) with L1 and J1, we get two equations
L1
L1 G12 G12 q = L1 AJ1q + BL1q + ~ q ;
J1
L1 G12 G12 q = J1 AJ1q + BL1q + ~ q :
Using the commutation relation of these generators (5.21), we have
qq = (Aq + 2hB) q;
2hq =
Ac3 + Bq
q:
Solving these equations, we get
A = 12h2 3q2
2ch 3q2
; B = q(c 6h)
2ch 3q2
: (D.13)
which give us the decomposition as equation (D.10). Note the A is invariant but B changes sign when q is changed to q, so the decomposition of q is
q = AJ1q BL1q +
~
q: (D.14)
The commutation of ~
q with the Virasoro and U(1) generators can be derive from those of q (D.3):
[Ln; ~
q(z)] = [Ln; q(z)] A[Ln; (J1q)(z)] B[Ln; (L1q)(z)];
[Jn; ~
q(z)] = [Jn; q(z)] A[Jn; (J1q)(z)] B[Jn; (L1q)(z)]:
(D.15)
{ 41 {
The commutation relations of J1q(z) on the r.h.s. can be derive from the OPE of T (w)
and J(w) with J1q(z):
T (w)(J1q)(z)
q(z) (w z)3
+ (h + 1)(J1)(z)(w z)2
+ @z(J1)(z) w z
;
(D.16)
J (w) (J1q) (z)
c 3
(z) (w z)2
+ q (J1) (z) (w z)
:
and the results are
[Ln; (J1q)(z)] = zn1
1
2(n + 1)nqq + (h + 1)(n + 1)z(J1q) + z2@z(J1q)
JHEP03(2017)167
[Jn; (J1q)(z)] = zn1
c3nq + zq(J1q)
:
(D.17)
The commutation relations of Ln and Jn with L1q(z) = @zq(z) are just the derivative
of the commutation relations of Ln and Jn with q(z) given in (D.2):
[Ln; (L1q) (z)] = zn1[n (n + 1) h + (n + 1) z@z + z2@2z]q; [Jn; (L1q) (z)] = qzn1(n + z@z)q:
(D.18)
Putting everything together, we nally get
[Ln; ~
q(z)] = zn[(h + 1) (n + 1) + z@z]~
q; [Jn; ~
q(z)] = qzn~
q: (D.19)
Comparing these two commutations with those for qh (D.2), we can see that under the action of Virasoro and U(1) generators, qh+1 acts like qh but with conformal dimension h + 1.
To derive the normalization of two-point function [angbracketleft]
~
q(z1)~
q(z2)[angbracketright], we need to use the
two-point function of q and q, which can be read o from the two-point function of two super elds (5.28):
D qh+1 (z1) qh+1 (z2)
E= 2h (2h + 1)z2h+221: (D.20)
Substituting the decompositions of qh+1 and qh+1 in the above two-point function, we can express [angbracketleft]
~
q(z1)~
q(z2)[angbracketright] as
[angbracketleft]
q q
A2
(J1q)(J1q) + B2
(L1q)(L1q)
~
q(z1)~
q(z2)[angbracketright] =
(L1q)(J1q) : (D.21)
The terms on the r.h.s. are easy to calculate using the equations derived in last subsection D.3 and the two-point function [angbracketleft]q(z1)q(z2))[angbracketright] =
1z2h21 . The results are as follow
(J1q)(J1q) =q2 + c3 z2h+221;
(L1q)(L1q) =@z1@z2 1z2h21= 2h (2h + 1) z2h+221;
(J1q)(L1q) =@z2 q [angbracketleft]qq[angbracketright]z21 =(2h + 1) q z2h+221;
(L1q)(J1q) =@z1 q [angbracketleft]qq[angbracketright]z12 = (2h + 1) q z2h+221:
AB
(J1q)(L1q) + AB
(D.22)
{ 42 {
Plugging these equations and equation (D.20) back in equation (D.21), we get
[angbracketleft]
~
qh+1 (z1) ~
qh+1 (z2)[angbracketright] = 4h2
q2
2ch
+ c 3 2h + q2
2ch 3q2
1 z2h+221
: (D.23)
Using the decomposition (D.10), we can calculate V
qLL qLLqHH qHH , V
qLL qLLqHH qHH
and V
qLL qLLqHH qHH from V
qLL qLLqHH qHH , and then equate these blocks to equations (5.40), (5.41) and (5.42), to solve for g2;hL(z), g4;hL(z) and g5;hL(z), respectively. As we said in subsection D.2, at leading order of the large c limit, these blocks only get contributions from Virasoro and U(1) generators. Some details for calculating these blocks are as follow:
1. In these calculations, we need to use the relationship between vacuum blocks with descendant elds and vacuum blocks with only primaries. These relationships are the same as those for the corresponding correlation functions, which are derived in subsection D.3.
2. The heavy-light vacuum blocks with one light operator being ~
and the other light
operator being vanish, V~
qLL qLLqHH qHH = V
qL L ~
JHEP03(2017)167
qLLqHH qHH = 0. The reason is
just because ~
L and L have di erent conformal dimensions (h~ L = hL + 1), and the two-point functions of them vanishes, [angbracketleft]
~
qLqL[angbracketright] = [angbracketleft]qL
qL[angbracketright] = 0.
3. At leading order of the large c limit, the only di erence (up to normalization) between the vacuum blocks V~
qL L ~
~
qLL qLLqHH qHH is that the conformal dimension of the light operators in the former is hL + 1 while that of the latter is hL.
So we can just change hL to hL + 1 in the expression of V
qLL qLLqHH qHH to get
qLLqHH qHH and V
(2hL + 1) 4h2L q2L
2hL e(hL+1) ~f(z)(1 z)3 qqL; (D.24)
where the prefactor here is just the prefactor in (D.23) in the large c limit.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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Abstract
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Abstract
One can obtain exact information about Virasoro conformal blocks by analytically continuing the correlators of degenerate operators. We argued in recent work that this technique can be used to explicitly resolve information loss problems in AdS3/CFT2. In this paper we use the technique to perform calculations in the small 1/c G N expansion: (1) we prove the all-orders resummation of logarithmic factors ...... in the Lorentzian regime, demonstrating that 1/c corrections directly shift Lyapunov exponents associated with chaos, as claimed in prior work, (2) we perform another all-orders resummation in the limit of large c with fixed cz, interpolating between the early onset of chaos and late time behavior, (3) we explicitly compute the Virasoro vacuum block to order 1/c 2 and 1/c 3 with external dimensions fixed, corresponding to 2 and 3 loop calculations in AdS3, and (4) we derive the heavy-light vacuum blocks in theories with ...... = 1, 2 superconformal symmetry.
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