Universal bounds on charged states in 2d CFT and 3d gravity

Abstract

Abstract

We derive an explicit bound on the dimension of the lightest charged state in two dimensional conformal field theories with a global abelian symmetry. We find that the bound scales with c and provide examples that parametrically saturate this bound. We also prove that any such theory must contain a state with charge-to-mass ratio above a minimal lower bound. We comment on the implications for charged states in three dimensional theories of gravity.

Published for SISSA by Springer Received: June 3, 2016 Accepted: July 18, 2016 Published: August 4, 2016

JHEP08(2016)041

Universal bounds on charged states in 2d CFT and 3d gravity

Nathan Benjamin,a Ethan Dyer,a A. Liam Fitzpatrickb and Shamit Kachrua

aStanford Institute for Theoretical Physics, Via Pueblo, Stanford, CA, 94305 U.S.A.

bBoston University Physics Department,

Commonwealth Avenue, Boston, MA, 02215 U.S.A.

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , [email protected]

Abstract: We derive an explicit bound on the dimension of the lightest charged state in two dimensional conformal eld theories with a global abelian symmetry. We nd that the bound scales with c and provide examples that parametrically saturate this bound. We also prove that any such theory must contain a state with charge-to-mass ratio above a minimal lower bound. We comment on the implications for charged states in three dimensional theories of gravity.

Keywords: Conformal and W Symmetry, Global Symmetries

ArXiv ePrint: 1603.09745

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP08(2016)041

Web End =10.1007/JHEP08(2016)041

Contents

1 Introduction 1

2 Modular transformations with currents 42.1 Derivation of transformation 4

3 Bounds on the charged-spectrum gap 53.1 Hellerman-type bound on charged spectrum mass gap 63.2 Bounds on charge-to-mass ratio 83.3 Bound from asymptotic growth 103.4 Supersymmetry 11

4 Large gap examples 134.1 Extremal lattices 134.2 Gravity theories with a large gap 14

5 Discussion and future directions 16

A Modular forms 17

B Current algebra and avored partition function 18B.1 Perturbative argument 19

C Transformation of characters 20

1 Introduction

At low energies, theories of quantum gravity are straightforward to formulate quantitatively as e ective eld theories. However, the universal coupling of gravity implies that, unlike gauge theories, at high energies gravity has no decoupling limit where it can be separated from the matter content of the theory. This might lead one to expect that precise statements about the spectrum or dynamics of gravity at high energies would necessarily be contingent on some knowledge of the low-energy spectrum. Conversely, insisting on various criteria for the behavior of the ultraviolet description ought to impose non-trivial constraints on the behavior of the theory at low-energies.

Holographic approaches allow one to make this intuition precise and to extract quantitative predictions by turning a poorly-de ned question, that of how to formulate the space of UV-complete theories of gravity, into sharp questions about observables at the boundary of space-time. In at space, the corresponding observable is the S-matrix, and the implications of analyticity and other axioms of the S-matrix provide a path to constraining the dynamics of the theory [1]. Turning on a small negative cosmological constant, the

{ 1 {

JHEP08(2016)041

theory apparently remains nearly unchanged locally; yet the global Anti de Sitter space-time structure is radically di erent, and the boundary observables now comprise the full dynamics of a Conformal Field Theory [2{4]. In fact, the structure of CFTs is su ciently rigid that it is possible to derive universal constraints on the dynamics and spectrum of all theories of gravity in AdS at low and high energies [5{13]. Such constraints clearly can also be applied to the full space of CFTs, which is a central area of study in its own right.

The most powerful such methods are found in the case of the correspondence between three-dimensional gravity in Anti de Sitter space and two-dimensional Conformal Field Theories, i.e. AdS3/CFT2. All graviton degrees of freedom are purely boundary excitations whose dynamics are completely xed by the in nite conformal symmetry in two dimensions. Moreover, modular invariance of the theory at nite temperature relates the spectrum at high energies and the spectrum at low energies. This makes it possible to draw sharp conclusions about the high-energy dynamics as a function of the assumptions about the low-energy spectrum. One can even derive properties that are common to all gravitational theories, i.e. that make only very basic assumptions such as unitarity.

A remarkable result along these lines was derived in [10] and systematically improved upon numerically in [14{17], where a rigorous and universal upper bound was found on the mass mL of the lightest bulk degree of freedom in AdS3. Roughly, this bound is mL [lessorsimilar]

1 4GN ,

where GN is Newton's constant. When the mass of a state is greater than the threshold mBH

18GN for black holes in AdS3 [18], classical gravity predicts that it should collapse and form a horizon, and we can suggestively call it a \black hole" state. A priori, there is no guarantee that any speci c solution to Einstein's equations is actually a physical state in all theories of quantum gravity; in fact, most are not, since the spectrum of states in a CFT is usually discrete whereas the spectrum of solutions to general relativity (GR) is continuous. The results of [10, 14] can therefore very roughly be summarized as the statement that the spectrum of states in theories of gravity must, at a bare minimum, contain black holes near threshold. Moreover, the result applies to all CFTs, even those whose bulk duals in AdS3 may not be well-described by GR at high energies, or where the curvature of AdS3 is order O(1) in units of the Planck scale.

Our main goal in this paper will be to extend this technique to the case where the bulk theory has a gauge eld in addition to gravity. This is dual to the assumption that there is conserved vector current J in the boundary CFT. Now, one can ask not only about the spectrum of energies of states, but also about their charges. A very simple but elusive question is whether there must be charged states in the theory at low energies, or if instead they can be made arbitrarily heavy and therefore e ectively decoupled from any description at xed nite energy. Charged black holes would seem to exist as classical solutions in GR with a gauge eld, but it is not obvious that a theory with only neutral states is actually inconsistent, since in such a theory charged black holes can never be produced. Various arguments have been made that light charged states must be present below some bound in mass [19{21], but so far a rigorous proof is lacking. A key result of this paper will be to prove such an upper bound.

This result has a clear connection to the Weak Gravity Conjecture (WGC) of [19], which exists in multiple forms but in any case is an upper bound on m

Qmpl for the mass m

{ 2 {

JHEP08(2016)041

and charge Q of some state in the theory.1 At large central charge c, for non-chiral (which have, e.g., c =

c) CFTs our upper bound on the weight of the lightest charged state in the theory asymptotes to

vacuum <

14GN for the lightest charged state.2 The bound is numerically determined from the modular bootstrap and can likely be improved with better numerics. We will also nd a bound on mGN

Q when Q is normalized so that the level k of the current J is 1.3 At large c, it implies that there exists a state in the theory with charge to mass ratio satisfying,

Q mGN >

c12 , as one might expect from the classical threshold for black holes in AdS3. An analogous mismatch arises in the bounds found in [10, 14], and it is unclear if this is a short-coming of the methods (for instance, only the subgroup ! 1 of modular transformations is actually used) or if

there are physical theories that saturate the weaker bound. Partly motivated by this, we also consider stronger bounds on the gap to the lightest charged state that can be obtained in the case of N = (1; 1) supersymmetric theories with a U(1) current. Here, we can

consider a holomorphic quantity called the elliptic genus, and indeed we nd the improved bound on the weight of the lightest charged state,

vacuum

vacuum

k

x2 and is completely removed by canonically normalizing the currents J ! J[prime] = J/pk and charges Q ! Q[prime] = Q/pk. So the actual value of k appears

to be invisible to the CFT data we are using, and we can set k = 1 without loss of generality. We thank Dan Harlow for encouraging us to emphasize this.

{ 3 {

c6 +

32 + O

1 c

; (1.1)

where vacuum = c12 is the weight of the vacuum. In terms of AdS quantities, this translates to m [lessorsimilar]

14p ; (c 1) : (1.2)

This bound could also likely be improved with further e ort. One might expect or hope for a better bound closer to a ratio of 1 than 1

4p , which is fairly small, but our main point is that it is parametrically O(1) and is a rigorous proof that there must exist some state

in the theory with mGN4p < Q.It is notable that the upper bound (1.1) is at c6 and not

JHEP08(2016)041

c12 + 1 (supersymmetric) : (1.3)

1One might expect that the WGC should be qualitatively modi ed in AdS3 due to peculiarities of three dimensions. In particular, in AdS3, the relation between mass and charge of extremal BHs is qualitatively di erent, M Q2. Furthermore, boundary currents in 2d are dual to Chern-Simons gauge elds in AdS3,

which have no bulk degrees of freedom. On the other hand, if the WGC is su ciently robust under compact-i cation of extra dimensions, one might expect these peculiarities to be irrelevant. See [22] for discussions of which versions of the WGC are robust to compacti cation of dimensions, and [23] for discussions of how WGC might be modi ed in AdS/CFT contexts. In any case, our bounds are a rigorous consequence of modular invariance and thus provide an independent approach to studying the WGC in AdS3.

2Recall that in GR, the CFT central charge satis es c = 3[lscript]AdS2GN [24], and for a bulk scalar the weight satis es (

vacuum)(

2) = m2[lscript]2AdS.

3Equivalently, one can keep k explicit and replace Q with Q/pk. From the CFT point of view, where we do not assume any a priori knowledge of the charges that arise in the theory, the central charge k shows up only in the two-point function of the current, J(x)J(0)

One might also wonder how much more the gap to charged states can be lowered in principle by any methods, not necessarily those used here. In particular, one might hope to prove on general grounds that charged states should enter parametrically below Mpl. While

this may indeed prove true after restricting to certain classes of theories,4 we examine a few counter-examples that demonstrate it cannot be true in complete generality.

The outline of the paper is as follows. In section 2, we discuss the transformation of the partition function in the presence of a chemical potential, and the corresponding characters. In section 3, we derive our bounds on the charged spectrum. In section 4, we present speci c models to demonstrate that our bounds are close to saturating the optimal bounds that are possible at large c without making additional assumptions. In section 5, we discuss potential future directions.

2 Modular transformations with currents

In any CFT with a conserved current J, one can consider the partition function graded by the charge of the current:

Z(; z) tr

qL0c/24 q L0 c/24yJ0 ; (2.1)

where q = e2i and y = e2iz. The starting point of our analysis is that under modular transformations,

! [prime] =

a + bc + d; z ! z[prime] =

JHEP08(2016)041

zc + d; (2.2)

the partition function transforms in a universal way:5

Z([prime]; z[prime]) = eik

cz2 c+d

c z

2 c

+d

Z(; z): (2.3)

We will discuss the argument for this transformation below, and work through an illustrative example.

2.1 Derivation of transformation

Most of our analysis in this paper is based on the transformation property of the avored partition function under a modular transformation, (2.3). As we explain in detail in appendix B, this transformation property is independent of the particular theory, and only depends on the universal structure of U(1) current algebra.6 It is also possible to derive

4See e.g. [21, 25] for recent interesting arguments along these lines. In particular, all examples we present have a coupling for the gauge eld that is O(1) at the Planck scale, whereas [21] argues for a stronger bound

only when this gauge coupling is small.

5Neither c nor k in (2.3) are related to the central charge of the theory. The variable c is from the transformation in (2.2) and k is the level of the current algebra.

6The explicit form of the partition function is of course theory dependent. It is only the transformation property which is universal. The transformation applies as well to the non-compact abelian group R; in fact, since our analysis makes no assumption about the representations that arise in the theory, it does not distinguish between the cases U(1) and R. For conciseness, however, we will simply refer to the abelian group as \U(1)".

{ 4 {

this transformation directly using algebraic properties of the modes

H dzJ(z) of the current J(z) on di erent cycles of the torus, as shown in [26, 27].7 Whichever method one prefers, once one knows that the transformation property is theory-independent it becomes su cient to derive it in a particularly simple theory.

Let us review this explicitly in the case of the free boson. For a free boson on a circle of radius R, the primary states under the U(1) current algebra are labeled by two integers,

|m; n[angbracketright], for momentum and winding. These states satisfy,

j0[notdef]m; n[angbracketright] =

m2R + nR

| {z }

|m; n[angbracketright] ;

j0[notdef]m; n[angbracketright] =

m2R nR

| {z }

|m; n[angbracketright]

JHEP08(2016)041

pL

pR

(2.4)

L0[notdef]m; n[angbracketright] =

p2L

2 [notdef]m; n[angbracketright] ;

L0[notdef]m; n[angbracketright] =

p2R

2 [notdef]m; n[angbracketright] :

The avored partition function is then given by,

Zbos(; z) = 1

| ()[notdef]2

Xm,n2Zqp2L2 qp2R2 ypL ypR : (2.5)

This partition function is invariant under the transformation ! + 1. Under the S

transformation, ! 1=, the transformation of the partition function can be easily

computed by applying the Poisson resummation formula,

Xke

a (k+ b2i )2 ; (2.6)

to both the m and n sums. Combining this with the modular transformation property of the function, (1=) = pi (). We have

Zbos([prime]; z[prime]) = ei

z2

X[lscript]ea[lscript]2+b[lscript] = 1 pa

z

2

Zbos(; z) ; (2.7)

establishing (2.3). Here we have normalized our currents to have level k = 1. There is nothing special about our choice of free bosons, and indeed this transformation has also been worked out explicitly in other examples, for instance see [28, 29] for free fermions, or [30] for any chiral N = 2 theory.

3 Bounds on the charged-spectrum gap

In this section we derive bounds constraining what charged states must appear in a theory with a U(1) global symmetry. We present constraints on what charges must appear, and upper bounds on the weight of the lightest charged state, and the ratio of weight to charge.

7We thank Herman Verlinde for bringing this argument to our attention. Their argument is in several ways more satisfying and elegant, and more suggestive of how an argument might be generalized beyond the partition function.

{ 5 {

3.1 Hellerman-type bound on charged spectrum mass gap

It is immediately clear from the transformation property, (2.2), that there must be charged states in the theory. As a warm-up, it is worth writing down some simple bounds on what charges must show up. Setting

z = 0 for simplicity, we can write the constraint of modular

invariance as

0 = Z(1=; z=) e

iz2

Z(; z)

e2i(Qizhi)/+2i hi/ eiz2 e2i(Qiz+hi hi )

i | {z }

Fi(,

,z)

=

: (3.1)

Here the sum is taken over individual states, each contributing qhi

q hiyQi, to the avored

partition function. Stronger constraints could be derived using the full Virasoro [notdef] U(1)

characters, discussed in appendix C. However we will get surprisingly strong results using the simpler single state expressions.

We can take z derivatives of the modular relation, (3.1), to bring down factors of the charge Qi of each state, and then set z to zero to obtain constraints on the charged spectrum. As a simple example, take two derivatives with respect to z of the modular transformation equation (3.1), and evaluate at z = 0, = i. This gives

1 82

Xi@2zFi

z=0,=i

=

JHEP08(2016)041

Xie2 i

Q2i 14 = 0 ; (3.2)

where i = hi +

hi. This expression is negative for all Q2i < 1=4. In order for the sum to give zero, the theory must therefore have some states with Q2i > 1=4, in addition to the neutral states. One can do even better by taking more z derivatives: combining the constraints from six and two derivatives, one nds,

Xi

1603 @6zFi z=0,=i

32 @2zFi z=0,=i

=

Xie2 i

323Q6i15 82Q4i + 1 = 0 :

(3.3)

This is positive for all Q2i, except for the interval, 0:344 [lessorsimilar] Qi [lessorsimilar] 1:09. Thus the theory must have some states with charge in this range.

To prove a bound on the gap to the lightest charged state, our strategy (similarly to most bootstrap approaches) will be to construct a linear operator out of z and derivatives evaluated at the self-dual point z = 0; = i, with the following properties:

(Fvacuum) = 1;

(F ,Q) > 0; if Q = 0;

(F ,Q) > 0; if > gap: (3.4)

Acting on the modular invariance equation (3.1), such an operator gives a positive contribution from the vacuum which must be canceled by a negative contribution from some states. Since the only states that have (F ,Q) < 0 are charged states with < gap, it

{ 6 {

immediately implies that such states must be present in the theory. This means gap is an

upper bound on the weight of the lightest charged state.8 An optimal analysis would seek to minimize gap over the space of linear functionals subject to the above constraints on . However, even with a small number of derivatives it is possible to obtain a functional satisfying them. Already quite non-trivial bounds are provided by the following example:

(Fi)

a1,0@ + a1,2@ @2z + a3,0@3 + a3,2@3 @2z

z=0, =2

Fi ( ; z)

;

a1,0 = 1

128 323 3 642 2 22 128 3

;

JHEP08(2016)041

a1,2 = 1

64 162 3 + 24 2 + 13 + 64

;

a3,0 = 1

242(3 + );

a3,2 =

1

242 ; (3.5)

where we have taken = i

2 ;

= i 2 , and

c+ c

24 . Evaluated on the contribution Fi from a single state, produces the following polynomial:

(F ,Q) = e2

p0( ) + Q2p1( )

p0( ) = 1 + ( + )(3 + 4( ))2

64

p1( ) = 3 2

3 22

116 162 3 + 24 2 + 5 + 64

: (3.6)

At Q = 0, this gives p0( ) which is a manifestly positive polynomial for . Fur

thermore, at su ciently large ,

p1( ) 3 2 + O( ) (3.7) is also manifestly positive, so (F ,Q) is manifestly positive for all charged states as well when is very large. The only possible negative contributions come from charged states in the range of where p1( ) < 0. Thus, an upper bound on the gap is given by the larger of the two solutions to p1( ) = 0:

gap( ) = p2

p (82 3 + 12 2 + 7 + 32) + 3 4

+

32 + O

1

: (3.8)

This is plotted as a function of in gure 1. Also shown in gure 1 are contours of the polynomial e2 (F ,Q) at = 2, where one can see that the polynomial is negative only for non-zero Q and for su ciently small . For a left-right symmetric theory, = c

12 and

8Incidentally, the unitarity bound h + c

24 > Q

2

2 means that any upper bound on the weight of a state is

also an upper bound on its charge.

{ 7 {

=

+

JHEP08(2016)041

-

-

/

Figure 1. Left: an upper bound on the total gap gap + between the vacuum and the lightest

charged state, as a function of c+c24. The slope asymptotes to 2 at large , show in red, dashed.

Right: the shaded region is where the polynomial e2 (F ,Q) in (3.6) is negative for = 2; the

right edge asymptotes to a vertical line (shown in blue, dashed) at = = gap( )= for large Q. In unitary theories, there must be at least one state in the shaded region.

the vacuum is at = c12, so the bound on the gap between the lightest charge state and

the vacuum is

gap( ) vacuum

c6 +

32 + O

1 c

: (3.9)

More restrictive bounds can certainly be obtained by considering more derivatives of Fi than we have used here, and it would be interesting to explore the optimal bounds that can be obtained this way.

3.2 Bounds on charge-to-mass ratio

So far we have investigated the bounds on the gap in charge, and the gap in the weight of the lightest charged state. It is interesting to also ask what we can say about a maximal gap in the ratio of weight to charge. For xed central charge, the operator de ned in the previous section already provides a bound on this ratio, as for large enough or small enough Q, (F ,Q) > 0. We will be most interested, however, in obtaining a bound for large c.9

9We often think about theories with a given level and a quantized, order one U(1) charge. In these cases, our bound on the weight of the lightest charged state immediately translates into a bound on the ratio, and gives a bound that scales with the central charge. By applying the linear operator techniques of the previous subsection, we will be able to derive a similar bound, that holds more generally without any additional assumptions on quantization.

{ 8 {

To this end, de ne a new linear functional ~

as

(Fi) = (Fi) +

: (3.10)

Acting on a single state, this again produces e2 times a relatively simple polynomial:

~

(F ,Q) = e2

~p0( ) + Q2~p1( ) ;

~p0( ) = p0( ) + 2

2 3;

~p1( ) = p1( ) 23 3 :

4 3@2zFi( ; z) z=0, =2

~

(3.11)

As before, we want to investigate for what states these polynomials can be negative, focusing on how the mass-to-charge bounds scale in the large central charge limit. We can therefore look mainly at large, and divide up our analysis into the three regimes ; , and = O(1).

For large , , we have,

e2 ~

(F ,Q)

2

4 3 + 3 Q2 2 ; (3.12)

which is positive for all states.For small , that is vacuum , we have,

e2 ~

(F ,Q) 23 3

14 Q2

JHEP08(2016)041

: (3.13)

This is negative only for states with Q2 > 1=4, and thus such states have a very small mass-to-charge ratio ( vacuum)=Q .

The most interesting states are those with vacuum . In this case,

e2 ~

(F ,Q)

2

4 ( )2( + ) + 2 3

+ Q23 2 3 2

: (3.14)

The Q independent term is again positive, while the second term can be negative for su ciently small . For the total expression to be negative we must have,

Q2 (

)2( + ) + 2 3

4 ( (3 2 2)); 2

; p3 : (3.15)

Though not uniform in , the right hand side of (3.15) has a minimum in the allowed range of . The quantity of interest is thus bounded by,

vacuum

|Q[notdef]

vacuum

q(( )2( + )+2 3)4( (3 2 2))

4p : (3.16)

Indeed we see that the largest gap in weight per unit charge scales linearly with the central charge.

{ 9 {

3.3 Bound from asymptotic growth

We next present an alternate method for deriving a bound on the gap in the charged sector, subject to a non-cancellation hypothesis. Although this argument will require a mild extra assumption, the advantage is both that it is very simple, and it is similar in style to an argument we will use to derive stronger bounds for N = (1; 1) theories. This method is

more analogous to the original argument due to Cardy for the asymptotic density of states in a 2d CFT. More accurately, it is analogous to the inverse of Cardy's argument; rather than using the presence of the vacuum to ascertain the asymptotic growth of states at large , we will show that a non-vanishing asymptotic charge density implies the presence of a light charged state. We will argue for this by considering the following object,

W4(;

) = Z(;

JHEP08(2016)041

: (3.17)

Note that this function vanishes if there are no charged states. Using the modular transformation properties of the avored partition function (2.1), it follows that W4 transforms as

W4 [prime];

[prime]

z=0

3(@2zZ())2 z=0

)@4zZ(;

)

= (c + d)4W4(;

) : (3.18)

Considering W4 for imaginary = i =2, we have,

W4( ) = 12Xi,j

Q4i 6Q2iQ2j + Q4j

e ( i+ j)

(3.19)

=

X~

C~ e ~ :

In the last line we have written the sum over weights, ~

= i + j.

Assume for contradiction that the rst charged state has weight gap. Then the sum in

W4 starts at ~

= gap c=12. We will show that this is inconsistent for large enough gap.

In order to see this, it is instructive to consider an abstract function which is invariant under the real modular S transformation,

W0

42 = W0( ) =

X~ 0,gapc/12D~ e ~ : (3.20)

To make contact with W4 above, we will assume W0 has [notdef]D~

[notdef] growing with large

~

.10 If

we further take 0,gap > c=12, then,

lim

!1

W0( ) = lim

!0

W0( ) = 0 : (3.21)

10This assumption is tantamount to an asymptotic non-cancellation assumption between the combination of partition functions appearing in (3.17). In fact, the situation is even better: even if this particular combination had a cancellation, we could construct higher order modular objects, and rerun the argument using these higher order modular forms. In this case, a di erent particular combination would have to cancel to invalidate the argument. Obviously, we could repeat this as many times as needed until reaching a combination that did not cancel. Thus to invalidate this argument would require an in nite number of cancellations.

{ 10 {

Thinking of W0( ) as the Laplace transform of D~

that the large ~

~

vacuum c=6.

To apply this to the function, W4, that we are interested in, we divide by the modular discriminant, ( ) = 24(i =2) to the appropriate power to create an invariant function:11

^

W4( ) = W4( )

( ( ))1/3

^

W4

The above argument tells us that ^

W4 has to grow as ! 1. Since the modular discrimi

nant behaves as ( ( ))1/3 e /3, we must have a maximal gap to charged states of

gap vacuum = c=6 + 1=3 : (3.23)

3.4 Supersymmetry

As we have mentioned, we don't believe the bound (3.23) is optimal. One motivation for this conjecture comes from considering theories with additional symmetry. We can consider the case of a 2d CFT with both N = (1; 1) supersymmetry and a U(1) current.

With N = (1; 1) supersymmetry we can de ne a holomorphic quantity called the elliptic

genus. This theory has fermions, so when we put it on a torus, there are four di erent spin structures we can consider depending on boundary conditions. We thus de ne the following elliptic genera,

Z+R(; z) = Tr R,R

(1)FRqL0

c24 yJ0 q L0

c24 ;

ZR(; z) = Tr R,R

(1)FL+FRqL0

c24 yJ0 q L0

c24 ;

Z+NS(; z) = Tr NS,R

(1)FRqL0

c24 yJ0 q L0

c24 ;

ZNS(; z) = Tr NS,R

(1)FL+FRqL0

c24 yJ0 q L0

c24 : (3.24)

In all of the functions above, the right-moving sector gets contributions only from super-symmetric ground states at

L0 =

c24 .12 The advantage of considering the elliptic genus is that it is a holomorphic modular form, so we can use the power of holomorphy to bound the gap to the lightest charge state.

The functions in (3.24) transform as (2.3) under (some congruence subgroup of) SL(2; Z). In particular, the functions Z+R(; z); ZR(; z); Z+NS(; z); and ZNS(; z) trans-

11In running this argument, it is crucial that , not to be confused with the weight , only has zeroes at the cusp ! 0 1, as otherwise we would introduce extra poles in

^

W4, invalidating the applicability

12Note, here the left-moving fermion number for the NS vacuum is conventionally de ned as (1)c/6.

{ 11 {

, the nal value theorem [31] tells us

behavior of D~

is given by the small behavior of W0( ), and thus

lim~

!1 D

= 0, contradicting our assumed growth. This tells us we must have 0,gap

JHEP08(2016)041

42 = ^W4( ) :

(3.22)

of the nal value theorem.

forms as (2.3) under 0(2); SL(2; Z); 0(2), and respectively. These are de ned as

0(2)

(

! 2

! 2

! 2

These functions transform into each other via

Z+R(; z) = ZNS(1=; z=);

ZNS(; z) = e

2ic

24 Z+NS( + 1; z): (3.26)

Now let us consider the following function:

W R4() Z+R(; z)@4zZ+R(; z) z=0

3(@2zZ+R(; z))2 z=0

: (3.27)

This is a weight 4 modular form under 0(2). Moreover, the only contributions to W R4() come from charged states. Our basic strategy is to show that W R4() must have a term of at least O(q) when expanded about = i1; this then means that there must be at least

one charged state of dimension one above the RR vacuum. Thus, relative to the NS-NS vacuum, we must have a charged state by c

12 + 1.

The ring of modular forms under 0(2) is generated by the functions E[prime]2() and E4(), de ned in appendix A. In particular, any meromorphic function that transforms with weight w under 0(2) that has no poles at = i1 and diverges at most as w about = 0 can

be written as a linear combination of products of E[prime]2 and E4 [32].

To see that W R4 is a weight four modular form under 0(2), note that about = i1,

W R4 is nite, as the lightest Ramond sector states have weight zero. The only question is the behavior about = 0.

Suppose we have a theory with the rst charged state at least c

12 above the (NS-NS)

1

: (3.29)

Note that W NS4() also only gets contributions from charged states. In particular, as we've assumed the rst charged state shows up c

12 above the vacuum, then W NS4() has no poles about = i1. Thus using (3.28), we see that W R4() diverges at most as 4 as = 0.

This means it can be written as

W R4() = c1E[prime]2()2 + c2E4() ; (3.30)

for some constants, c1 and c2.

{ 12 {

a b c d

SL(2; Z); c 0 (mod 2)

);

0(2) (

a b c d

SL(2; Z); b 0 (mod 2)

);

(

a b c d

SL(2; Z); a + b 1 (mod 2); c + d 1 (mod 2)

): (3.25)

JHEP08(2016)041

vacuum. From (3.26) and (3.27), one can show

W R4() = 1

4 W NS4

; (3.28)

where we de ne

W NS4() ZNS(; z)@4zZNS(; z) z=0

3(@2zZNS(; z))2 z=0

The highest order in the q-expansion (3.30) can start at is O(q). Thus, in W R4, a charged state must appear by dimension at least one above the RR vacuum. Since the RR vacuum is c

12 above the NS-NS vacuum, we thus get a bound to the rst charged state of

vacuum

c12 + 1 (supersymmetric) : (3.31)

The improvement by a factor of 2 compared to our non-supersymmetric bounds brings this into line with the threshold for BTZ black holes, since dimensions of vac

c 12

correspond to masses m

18GN in the gravity picture. It seems natural to conjecture that a bound upper bound on charged states of order

c12 may hold in general, even in the

non-supersymmetric case.

4 Large gap examples

In this section we provide some examples of theories which realize our bound up to O(1) factors. One class of examples is given by free bosons compacti ed on extremal lattices. Such lattices can be explicitly constructed for small central charge and are known not to exist for c 163264 [33]. Appealing to more standard string theory examples, we also

consider a gravitational theory in at space, and discuss the D1-D5 system in highly curved AdS space.

4.1 Extremal lattices

An extremal lattice, c, is a rank c even self dual lattice with the smallest norm non-zero vector, ~v having length squared,

~v [notdef] ~v =

c12 + 2 : (4.1)

We will be focused on the case c = 24k for k 2 Z. Such lattices are known to exist for

k = 1,2, and 3 [34{36], however for larger k they have not been constructed. As mentioned above, they do not exist for su ciently large k, k > 6802.

A consistent chiral CFT can be constructed by considering c chiral bosons compacti ed on such a lattice [37, 38]. This CFT has a spectrum consisting of the vertex operators,

V[vector]v(z) = ei[vector]v[notdef][vector](z) ; ~v 2 c h[vector]v =

~v 2

2 ; (4.2)

JHEP08(2016)041

as well as the di erentials, i@

, @~

~ 2,. . . .

, form a set of c currents, under which the only charged operators are the vertex operators, V[vector]v . Consider any one of these currents,

J(z) = i@1(z) : (4.3)

The gap to the rst charged operator is given by the gap in the norm of vectors in c, and

thus,13

h c,gap hvacuum =

The di erentials, i@

~

c24 + 1 : (4.4)

13It can actually be shown that chiral CFTs satisfy a stricter bound on the weight of the lightest charged state, hgap h

vacuum

c24 + 1, and so these examples are tight for chiral CFTs [39].

{ 13 {

4.2 Gravity theories with a large gap

It is expected for a variety of reasons that quantum gravity theories with U(1) gauge elds will exhibit charged matter with charge of O(1) at a mass scale M [lessorsimilar] MPlanck. As 2d

CFTs are (sometimes) dual to weakly curved 3d gravity, one can ask: how does our bound compare to this expectation?

In light of the Brown-Henneaux formula

c = 3LAdS2G ; (4.5)

our bound is su cient to guarantee this expectation. Charged states at masses M c6 in AdS units (the highest value consistent with the bound), are at a mass MPlanck. Still, one might wonder | is a stronger absolute bound possible in weakly curved gravitational theories?

We think the answer is no. One can easily provide examples of gravity theories which are thought to be fully consistent, yet have abelian gauge elds with the rst charges appearing at MPlanck. We provide two examples below. It is important to stress that in each, our ability to make controlled statements depends on extended supersymmetry and exact BPS mass formulae, as we work in regimes where some size or coupling is of O(1).

Example 1. Consider M-theory compacti ed on a circle of radius R in 11d Planck units. At very large radius, the theory reduces to 11d supergravity. At very small radius, one can reinterpret the radius in terms of the type IIA string coupling, via

R = g2/3string : (4.6)

For any nite R, the long distance theory is a weakly curved gravity theory in ten dimensions.

There is a Kaluza-Klein gauge eld arising from the 11 components of the 11d metric. This gauge eld becomes the Ramond-Ramond photon of type IIA string theory as R ! 0.

But it is present for all values of R, and a BPS bound relates the mass of the lightest charged KK modes of a given charge to the radius of the circle.

Half-BPS states carrying this charge do exist. They are the Kaluza-Klein gravitons on the circle, or D0-brane bound states in the IIA string. When R = `11, the only mass scale

in the BPS formula is MPlanck,11, and the lightest charge has mass MPlanck,11.

At long distances, one then has gravity coupled to an abelian gauge eld in 10d at space, with a lightest charge at MPlanck,10 MPlanck,11. This easily generalizes to lower dimensions, by compactifying on a Planck radius torus, rather than a single circle. This shows that one cannot derive a stronger bound on the mass of the lightest charged state which is stronger than the Planckian bound, at least not one which applies to all weakly curved gravity theories.14

14In fact, in 10d we can shrink the circle, thereby taking gstring small, and the only charged states in

the theory are D0 branes, which remain above the Planck scale. This provides an example at small string coupling; however, the gauge coupling remains O(1).

{ 14 {

JHEP08(2016)041

Example 2. Our bound is more directly related to AdS3 gravity theories, via the relationship between large c 2d CFTs with sparse spectrum and weakly curved gravity. So one could ask | in that more limited context, could it be that there is a (parametrically) stronger bound available?

We will try to give some sense of whether a counter-example may or may not exist by discussing one canonical example of AdS3/CFT2. Unfortunately, this example comes close to our bound only at small AdS length and thus at small c, whereas what we want to compare to is the parametric dependence on the bound at large c. The problem of nding weakly curved AdS3 examples with a large gap to charged states is similar to the problem of constructing very sparse large c CFTs and is likely challenging. However, at present it is unclear whether this is a fundamental limit, or just a limitation of available controlled compacti cations methods.

So, let us discuss the original example of AdS3/CFT2 duality, coming from the D1-D5 system on T 4. Before inserting the branes and taking the near-horizon limit, the moduli space of compacti cations of type IIB string theory on T 4 is a coset space

SO(5; 5; Z)\ SO(5; 5)= SO(5) [notdef] SO(5) : (4.7) Inserting the Q1 D1 and Q5 (wrapped) D5 branes leaves a worldvolume unbroken (4,4)

supersymmetric theory on the black string in six dimensions. The 25 real moduli can be divided into background tensor multiplet and hypermultiplet scalars of this supersymmetry; 5 come from tensor multiplets and 20 from hypermultiplets.

Via the attractor mechanism, the tensor multiplet scalars take xed values in the near-horizon geometry, independent of our choices. The hypermultiplet scalars can be tuned at will.

The resulting near-horizon solution is

AdS3 [notdef] S3

The radius of the AdS space and the sphere are equal (as is standard in Freund-Rubin compacti cation), given by

R2AdS = [prime]g6

Q1

Q5 1 (4.12)

JHEP08(2016)041

Q1Q5[notdef] T4 : (4.8)

pQ1Q5 : (4.9)

The two moduli of signi cance for us are the 6d string coupling g6, and the T 4 volume v. In string units, the volume is given by

v = Q1

Q5 ; (4.10)

while g6 is in a hypermultiplet and we are free to choose its value. Validity of the 6d supergravity description requires weak AdS curvature, i.e.

g6

pQ1Q5 1 : (4.11)

Consider, then, the scaling limit

Q1 ! 1; Q5 ! 1;

{ 15 {

while simultaneously selecting

g [lessorsimilar] O(1) : (4.13) In this limit, the 6d supergravity theory is weakly curved, while v O(1). So the 6d string

and Planck scales are comparable.

Now, consider the KK U(1) gauge elds on the torus. The story is similar to that of Example 1; the lightest charges will be KK modes with six-dimensional masses MPlanck,6.

To read o the AdS3 mass, we need to further reduce on the S3. Unfortunately, as the AdS and sphere radius are tied, the three dimensional mass is well below the Planck mass.

We can produce theories where the lightest charged states are at the Planck mass in this example, but only by considering highly curved theories outside of the supergravity limit, (4.11), by taking RAdS = RS3 = O(1) in Planck units. It is clear that the problem is that the Freund-Rubin construction by de nition ties the AdS radius to the radius of an external sphere in the geometry. So at large AdS radius, the dilution of the lower-dimensional (AdS) Planck scale due to the external sphere, will always lower the gap to charges under a KK gauge eld. More elaborate constructions can partially surmount this issue, but we are not aware of any where we would calculably saturate our bound at large AdS radius.

5 Discussion and future directions

We have demonstrated that the partition function with a chemical potential can be used to put concrete bounds on the spectrum of charged states in a general, not necessarily holographic, 2d CFT. Interpreted in terms of gravitational duals, these imply that charged states must be present in the theory at the Planck scale or lower, and that furthermore there must exist states with charge-to-mass ratio (in units of the Planck scale) above a concrete lower bound. For the most part, we have attempted to make our analysis more analytically transparent at the cost of leaving the constraints weaker than should ultimately be possible, and it would be interesting to return to these bounds with the much more numerically sophisticated machinery of recent bootstrap approaches.15

We also expect that these methods could be generalized to bound other quantities besides those considered here. For one, we have focused only a single conserved current, but when its symmetry is part of a larger non-abelian group, then one should be able to make richer statements about the spectrum of charges. In particular, rather than simply bounding the charge Q of states, one could start to constrain the representations of states in the theory. It would be very interesting for instance to show that for certain symmetry groups, certain representations must appear in the spectrum, or to nd relations between the representations that appear in the low-energy spectrum with those at high energies.

Another potentially powerful extension would be to correlation functions in higher dimensions. This paper has focused on the partition function, but in two-dimensional CFTs this is equivalent to a four-point correlation function of twist operators [45]. Adding in a chemical potential is equivalent to inserting Wilson lines in this correlation function.

15See e.g. [14, 40{44], to name just a few of the many such analyses in this rapidly growing area.

{ 16 {

JHEP08(2016)041

Optimistically, one may hope that even in this more general case, the transformation property of the correlator under crossing in the presence of such Wilson lines can be derived purely through knowledge of the current two-point function, or in even dimensions in terms of its anomalies.16 If this is correct, then it would provide a practical way of including nonlocal line operators in the conformal bootstrap, potentially accessing important information about the theory that would be invisible otherwise [47].

Finally, bounds on the number of BPS operators at a given weight and charge in a 2d superconformal eld theory with at least N = 2 supersymmetry are of additional in

terest, as they would have a topological interpretation as bounds on the Hodge numbers of the corresponding target-space Kahler manifold.17 Such constraints are therefore interesting geometrically, and modular bootstrap approaches may provide information that is complementary to other approaches.

Acknowledgments

We thank Andy Cohen, Thomas Dumitrescu, Guy Gur-Ari, Dan Harlow, Daniel Ja eris, Jared Kaplan, Ami Katz, Alex Maloney, Greg Moore, Eric Perlmutter, Cumrun Vafa, Herman Verlinde, Roberto Volpato, and Xi Yin for useful discussions. We also thank Dan Harlow, Ami Katz, Christoph Keller, and Alex Maloney for comments on a draft. NB is supported by a Stanford Graduate Fellowship and an NSF Graduate Fellowship. ED is supported by the NSF under grant PHY-0756174. ALF is supported by the US Department of Energy O ce of Science under Award Number DE-SC-0010025. SK acknowledges the support of the National Science Foundation via grant PHY-1316699.

A Modular forms

For convenience, we reproduce the de nitions and relevant properties of several functions used in this paper. The Eisenstein series E4() and E6() are de ned as

E4() = 1 + 240

1

a + bc + d = (c + d)4E4()

a + bc + d = (c + d)6E6(): (A.2)

16See for instance [46], section 3.1.4 for a very rough sketch of such an argument in d = 2.

17See [48, 49] for various approaches to this question.

{ 17 {

JHEP08(2016)041

Xn=1n3qn 1 qn

E6() = 1 504

1

Xn=1n5qn1 qn: (A.1)

They transform as

E4

E6

Together, they generate the ring of modular forms invariant under SL(2; Z). We also de ne the Dedekind eta function as

() = q

1

24

1

Yn=1(1 qn) (A.3)

and the modular discriminant as

() = ()24 = E4()3 E6()21728 : (A.4)

We are also occasionally interested in the second Eisenstein series E2(), de ned as

E2() = 1 24

1

JHEP08(2016)041

Xn=1nqn1 qn: (A.5)

This is not quite a modular form, as it transforms as

E2

a + bc + d = (c + d)2E2() + 6ci (c + d): (A.6)

We also de ne the Klein-invariant J function, which is a modular function of weight 0 with a pole at = i1.

J() = E4()3

() 744 =

1q + 196884q + : : : : (A.7)

Holomorphic modular invariant functions with poles only at = i1 are polynomials

in J().

Finally, we consider the subgroup of SL(2; Z) called 0(2) de ned as matrices a b

c d

2

SL(2; Z) with c even. Modular forms under 0(2) are generated by the functions E[prime]2(), de ned as

E[prime]2() = 1 + 24

1

Xn=1nqn1 + qn (A.8)

and E4(), de ned in (A.1).

B Current algebra and avored partition function

We have argued that the transformation property,

Z [prime]; z[prime]

= ei

cz2c+d Z(; z) ; (B.1)

relies on the universal structure of the current algebra, rather than any theory speci c details. Here we demonstrate this in gory detail.

{ 18 {

B.1 Perturbative argument

Our strategy will be to calculate the transformation property of the avored partition function (B.1) order by order in z about 0. The transformation rule can be veri ed at each order using the structure of the current algebra without any knowledge of the particular theory. We demonstrate this explicitly at quadratic order in z and then present the general argument. As the rule is theory independent we can thus read it o from any theory we like, for instance the free boson, for which the rule (B.1) is well known (see [50] for instance).

Quadratic Order. At quadratic order we have,

@2zZ([prime])

z=0

= (c + d)2

JHEP08(2016)041

z=0

z=0

@2zZ()

+ 2i c

c + dZ()

: (B.2)

The z derivatives are always evaluated at z = 0, but we refrain from writing this below, to avoid clutter. We want to check this second order transformation by explicitly computing,

@2zZ() = (2i)2 Tr

qL0c/24J20 : (B.3)

In order to do this, we would like to nd a primary that contains J20 as part of its zero mode, as well as other known contributions. This is convenient as we know how primary one point functions transform, and thus can solve for the transformation of @2zZ(). Such an operator is given by,

O2(z) = J2(z)

2c T (z) $

J21 2c L2

[notdef]0[angbracketright] ; (B.4)

which has a zero mode,

Xn 1JnJn 2c L0 : (B.5)

We can compute the torus one point function of O2.18

FO2() (2i)2Tr

qL0c/24(O2)0 = (2i)2Tr

qL0c/24J20

| {z }

@2

z Z()

(O2)0 = J20 + 2

+2(2i)2

Xn 1 Tr

qL0c/24JnJn

2c (2i)2Tr

qL0c/24L0

:

(B.6)

The second and third terms on the second line can be simpli ed. Starting with the third term we have,

Tr

qL0c/24L0 = qc/24 q@q(qc/24Z())

= @Z() + c

24Z() ;

(B.7)

18This style of computation is similar to that presented in [51], for example.

{ 19 {

while for the second term we use,

Tr qL0c/24JnJn = qn Tr

qL0c/24JnJn

= nqn

(B.8)

1 qn

Z() ;

and the de nition of the Eisenstein series to write,

Xn 1 Tr

qL0c/24JnJn = 1 E2()24 Z() : (B.9)

Putting this together, we can solve for @2zZ().

@2zZ() = FO2() + (2i)2

E2()

JHEP08(2016)041

Z() : (B.10)

We are now in a position to write down the transformation properties of @2zZ() .

@2zZ [prime]

= (c + d)2@2zZ() + 2ic(c + d)Z(); (B.11)

as desired.

In deriving this, we used the fact that both FO2() and @Z() are modular

forms of weight 2, as well as the anomalous transformation of E2() written in (A.6).

General Order. To compute at arbitrary order we can replicate the argument style used above. To compute Tr qL0c/24Jm0

12 +

2c @

, we look for a primary operator which contains Jm0 as part of its zero mode. In addition it will contain terms of weight zero built out of Lm and Jm modes. The traces over these terms can be evaluated, as they were in the quadratic case, using only the current algebra to reduce them to modular di erential operators acting on traces with fewer powers of J0. Thus the modular properties of Tr qL0c/24Jm0

only depend on the universal current algebra, and so at each order, the transformation rule is identical in any theory. In particular, we can compute the transformation rule in the case of the free boson. This gives (B.1), and so it must also be correct for any theory with a U(1) symmetry.

C Transformation of characters

Modular invariance can be thought of as a sharp relation between the UV and the IR spectrum of the theory. One way to build some additional intuition on the relation in a general theory is to look at the image under S : ! 1 of an individual character. In [52],

the transformation of characters of the Virasoro algebra were derived. One might hope that further development of this approach to include the image under the full modular group could allow one to construct representations of Virasoro plus modular invariance, which could then be used as modules to be added to add additional states the full partition function. In the case of chiral theories, Rademacher sums indeed make this a viable and useful method. In the general non-holomorphic case, the major obstacle is that the image

{ 20 {

under S produces a continuous, rather than a discrete, spectrum, and it is not clear how to systematically correct this. Moreover, since the image of a single character is an integral over a continuum of characters up to arbitrarily high weight, for the analysis to be \closed" in a sense one must also characterize the modular image of in nite sums over characters as well. Despite these caveats, we nd the results of [52] to provide some useful guidance in thinking about modular transformations of non-holomorphic theories. In this appendix, we will therefore consider the modular transformation of an individual Virasoro [notdef] current

algebra character, which we describe below. We assume the existence of both left and right U(1) currents for ease of exposition.

The holomorphic Virasoro [notdef] U(1) A ne Kac-Moody algebra is given by,[Lm; Ln] = (m n)Lm+n +

c12m(m2 1) m+n,0; [Lm; Jn] = nJn+m;

[Jm; Jn] = mk n+m,0 ; (C.1)

and similarly for the anti-holomorphic algebra. If c > 2, the full irreducible representations of the Virasoro and current algebra are generated by all combinations of Jn; n 1 and

Ln; n 2, acting on the primary states, as well as the anti-holomorphic modes. Since

these do not change the total U(1) charge, and they raise the L0 eigenvalue by n, one can immediately write the characters as products of the characters ~J and ~T under the two sectors separately:

~J(q) =

1

JHEP08(2016)041

Yn=111 qn;

~T (q) = qh

1

Yn=111 qn

! (

1 q vacuum

1 h > c24 )

: (C.2)

The full character is

~h,Q,c(q; y) = yQ~T (q)~J(q)

q) (C.3)

where we have graded over the U(1) left- and right-moving charges Q;

Q with y = e2iz.

It is convenient to multiply by the modular invariant function

(i

)1/4 ()

y Q

~T (

q)

~J(

4

to get the

\reduced" characters:

^

~(q; y) = [notdef][notdef](q

q)

1 12

(

yQ

y Qqh

q h h > c24;

h > c24;

): (C.4)

We want to consider what happens if we add an extra non-vacuum state to a theory. We can focus on the left-moving part of the reduced character

^

~(; z) = (i)1/2e2i(EL+zQ): (C.5)

Under S, this character gets mapped to

e2i

c 6

(1 q)(1

q) vacuum

z2

(i)1/2e2iEL/e

2izQ

: (C.6)

{ 21 {

Our goal is to decompose this into an integral over the untransformed characters times a density of states (E; Q):

Z

dE[prime]LdQ[prime](E[prime]L; Q[prime])(i)1/2e2i(E[prime]L

+zQ[prime]) : (C.7)

Integrating both sides against

R

dze2iQ[prime][prime], we obtain

Z

dE[prime]Le2iE[prime]L (E[prime]; Q[prime][prime]) = 1

(i)e2iEL/ Z

dze2i

c 6

+Q[prime][prime]z zQ

z2

2

=

p3 pic

e2iEL/e

3i(QQ[prime][prime])

c : (C.8)

This is just the left-moving piece of the full character; multiplying by the corresponding right-moving piece, we nd

Z

dE[prime]LdE[prime]R(E[prime]L; E[prime]R; Q[prime][prime];

EL

+ ER

Q[prime][prime])e2i(E[prime]L+ E[prime]R) = 3 c[notdef][notdef]

e2i

e3i(QQ[prime][prime])2c + 3i(

Q Q[prime][prime] )2 c

:

(C.9)

If we assume that the theory satis es charge conjugation symmetry, then for each state with charge (Q;

Q) and energy (EL; ER), there is another state with charge (Q;

Q) and

energy (EL; ER). Adding these two contributions together, their image under S has a spectrum given by

Z

dE[prime]LdE[prime]R(E[prime]L;E[prime]R;Q[prime][prime];

JHEP08(2016)041

EL

+ ER

Q[prime][prime])e2i(E[prime]L+ E[prime]R)= 3 c[notdef][notdef]

e2i

e3i(Q2+Q[prime][prime]22)2c + 3i(

Q2+ Q[prime][prime]2 )2 c

: (C.10)

To bring this into a more natural form, we can massage it a little to be

Z

dE[prime]LdE[prime]R(E[prime]L; E[prime]R; Q[prime];

Q[prime])e

2i

2cos

6c (QQ[prime][prime]+

Q Q[prime][prime])

E[prime]L 3Q

2 +

E[prime]R 3 Q

02

2c

= 3

c[notdef][notdef]

e

2i

1

EL 3Q 2 2c

+ 1

ER 3 Q22c [notdef] 2 cos 6

c (QQ[prime] +

Q[prime])

: (C.11)

Clearly, it is natural to de ne the variables

L EL

3Q2

2c ;

Q22c : (C.12)

In terms of these variables, the above relation takes the simple form

Z

d[prime]Ld[prime]R([prime]L;[prime]R; Q[prime];

R ER

3

2i

Q[prime])e2i([notdef][prime]L+

[notdef]L

+

[notdef]R

[notdef][prime]R) = 6 c[notdef][notdef]

e

cos

6(QQ[prime] +

Q Q[prime]) c

:

(C.13)

This has reduced to the transformation for the case Q = 0, up to an extra cos factor, and with the E's are replaced by's. But that is exactly the transformation that was

{ 22 {

derived in [52]19 Adopting their result (and keeping track of our slightly di erent integration measure), we nally arrive at

([prime]L;[prime]R; Q[prime];

Q[prime]) = 12c (

cos

[prime]L) ([prime]R) 1

6c (QQ[prime] +

Q Q[prime])

qL[prime]L) cosh(4i

qR[prime]R)

q[prime]L[prime]R cosh(4i

: (C.14)

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.

References

[1] A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity

and an IR obstruction to UV completion, http://dx.doi.org/10.1088/1126-6708/2006/10/014

Web End =JHEP 10 (2006) 014 [http://arxiv.org/abs/hep-th/0602178

Web End =hep-th/0602178 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0602178

Web End =INSPIRE ].[2] S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical

string theory, http://dx.doi.org/10.1016/S0370-2693(98)00377-3

Web End =Phys. Lett. B 428 (1998) 105 [http://arxiv.org/abs/hep-th/9802109

Web End =hep-th/9802109 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9802109

Web End =INSPIRE ].[3] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253

[http://arxiv.org/abs/hep-th/9802150

Web End =hep-th/9802150 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9802150

Web End =INSPIRE ].[4] J.M. Maldacena, The Large-N limit of superconformal eld theories and supergravity, http://dx.doi.org/10.1023/A:1026654312961

Web End =Int. J.

http://dx.doi.org/10.1023/A:1026654312961

Web End =Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [http://arxiv.org/abs/hep-th/9711200

Web End =hep-th/9711200 ]

[http://inspirehep.net/search?p=find+EPRINT+hep-th/9711200

Web End =INSPIRE ].

[5] A.L. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of Long-Distance AdS Physics

from the CFT Bootstrap, http://dx.doi.org/10.1007/JHEP08(2014)145

Web End =JHEP 08 (2014) 145 [http://arxiv.org/abs/1403.6829

Web End =arXiv:1403.6829 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1403.6829

Web End =INSPIRE ].[6] A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Du n, E ective Conformal Theory

and the Flat-Space Limit of AdS, http://dx.doi.org/10.1007/JHEP07(2011)023

Web End =JHEP 07 (2011) 023 [http://arxiv.org/abs/1007.2412

Web End =arXiv:1007.2412 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.2412

Web End =INSPIRE ].[7] Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, http://dx.doi.org/10.1007/JHEP11(2013)140

Web End =JHEP 11 (2013)

http://dx.doi.org/10.1007/JHEP11(2013)140

Web End =140 [http://arxiv.org/abs/1212.4103

Web End =arXiv:1212.4103 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1212.4103

Web End =INSPIRE ].[8] A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Du n, The Analytic Bootstrap and

AdS Superhorizon Locality, http://dx.doi.org/10.1007/JHEP12(2013)004

Web End =JHEP 12 (2013) 004 [http://arxiv.org/abs/1212.3616

Web End =arXiv:1212.3616 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1212.3616

Web End =INSPIRE ].[9] L.F. Alday and J.M. Maldacena, Comments on operators with large spin, http://dx.doi.org/10.1088/1126-6708/2007/11/019

Web End =JHEP 11 (2007)

http://dx.doi.org/10.1088/1126-6708/2007/11/019

Web End =019 [http://arxiv.org/abs/0708.0672

Web End =arXiv:0708.0672 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0708.0672

Web End =INSPIRE ].[10] S. Hellerman, A Universal Inequality for CFT and Quantum Gravity, http://dx.doi.org/10.1007/JHEP08(2011)130

Web End =JHEP 08 (2011) 130

[http://arxiv.org/abs/0902.2790

Web End =arXiv:0902.2790 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0902.2790

Web End =INSPIRE ].[11] T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in

the Large c Limit, http://dx.doi.org/10.1007/JHEP09(2014)118

Web End =JHEP 09 (2014) 118 [http://arxiv.org/abs/1405.5137

Web End =arXiv:1405.5137 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1405.5137

Web End =INSPIRE ].[12] N. Benjamin, S. Kachru, C.A. Keller and N.M. Paquette, Emergent space-time and the

supersymmetric index, http://dx.doi.org/10.1007/JHEP05(2016)158

Web End =JHEP 05 (2016) 158 [http://arxiv.org/abs/1512.00010

Web End =arXiv:1512.00010 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1512.00010

Web End =INSPIRE ].[13] N. Benjamin, M.C.N. Cheng, S. Kachru, G.W. Moore and N.M. Paquette, Elliptic Genera

and 3d Gravity, http://arxiv.org/abs/1503.04800

Web End =arXiv:1503.04800 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1503.04800

Web End =INSPIRE ].

19See their equation (23).

{ 23 {

JHEP08(2016)041

[14] D. Friedan and C.A. Keller, Constraints on 2d CFT partition functions, http://dx.doi.org/10.1007/JHEP10(2013)180

Web End =JHEP 10 (2013) 180

[http://arxiv.org/abs/1307.6562

Web End =arXiv:1307.6562 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.6562

Web End =INSPIRE ].[15] J.D. Qualls, Universal Bounds on Operator Dimensions in General 2D Conformal Field

Theories, http://arxiv.org/abs/1508.00548

Web End =arXiv:1508.00548 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1508.00548

Web End =INSPIRE ].[16] J.D. Qualls, Universal Bounds in Even-Spin CFTs, http://dx.doi.org/10.1007/JHEP12(2015)001

Web End =JHEP 12 (2015) 001 [http://arxiv.org/abs/1412.0383

Web End =arXiv:1412.0383 ]

[http://inspirehep.net/search?p=find+EPRINT+arXiv:1412.0383

Web End =INSPIRE ].

[17] J.D. Qualls and A.D. Shapere, Bounds on Operator Dimensions in 2D Conformal Field

Theories, http://dx.doi.org/10.1007/JHEP05(2014)091

Web End =JHEP 05 (2014) 091 [http://arxiv.org/abs/1312.0038

Web End =arXiv:1312.0038 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1312.0038

Web End =INSPIRE ].[18] M. Baados, C. Teitelboim and J. Zanelli, The Black hole in three-dimensional space-time,

http://dx.doi.org/10.1103/PhysRevLett.69.1849

Web End =Phys. Rev. Lett. 69 (1992) 1849 [http://arxiv.org/abs/hep-th/9204099

Web End =hep-th/9204099 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9204099

Web End =INSPIRE ].[19] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes and

gravity as the weakest force, http://dx.doi.org/10.1088/1126-6708/2007/06/060

Web End =JHEP 06 (2007) 060 [http://arxiv.org/abs/hep-th/0601001

Web End =hep-th/0601001 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0601001

Web End =INSPIRE ].[20] T. Banks and N. Seiberg, Symmetries and Strings in Field Theory and Gravity, http://dx.doi.org/10.1103/PhysRevD.83.084019

Web End =Phys. Rev.

http://dx.doi.org/10.1103/PhysRevD.83.084019

Web End =D 83 (2011) 084019 [http://arxiv.org/abs/1011.5120

Web End =arXiv:1011.5120 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1011.5120

Web End =INSPIRE ].[21] D. Harlow, Wormholes, Emergent Gauge Fields and the Weak Gravity Conjecture, http://dx.doi.org/10.1007/JHEP01(2016)122

Web End =JHEP 01

http://dx.doi.org/10.1007/JHEP01(2016)122

Web End =(2016) 122 [http://arxiv.org/abs/1510.07911

Web End =arXiv:1510.07911 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1510.07911

Web End =INSPIRE ].[22] B. Heidenreich, M. Reece and T. Rudelius, Sharpening the Weak Gravity Conjecture with

Dimensional Reduction, http://dx.doi.org/10.1007/JHEP02(2016)140

Web End =JHEP 02 (2016) 140 [http://arxiv.org/abs/1509.06374

Web End =arXiv:1509.06374 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1509.06374

Web End =INSPIRE ].[23] Y. Nakayama and Y. Nomura, Weak gravity conjecture in the AdS/CFT correspondence,

http://dx.doi.org/10.1103/PhysRevD.92.126006

Web End =Phys. Rev. D 92 (2015) 126006 [http://arxiv.org/abs/1509.01647

Web End =arXiv:1509.01647 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1509.01647

Web End =INSPIRE ].[24] J.D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic

Symmetries: An Example from Three-Dimensional Gravity, http://dx.doi.org/10.1007/BF01211590

Web End =Commun. Math. Phys. 104

http://dx.doi.org/10.1007/BF01211590

Web End =(1986) 207 [http://inspirehep.net/search?p=find+J+%22Comm.Math.Phys.,104,207%22

Web End =INSPIRE ].[25] C. Cheung and G.N. Remmen, Infrared Consistency and the Weak Gravity Conjecture,

http://dx.doi.org/10.1007/JHEP12(2014)087

Web End =JHEP 12 (2014) 087 [http://arxiv.org/abs/1407.7865

Web End =arXiv:1407.7865 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1407.7865

Web End =INSPIRE ].[26] R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 Conformal Field Theories on Riemann

Surfaces, http://dx.doi.org/10.1007/BF01224132

Web End =Commun. Math. Phys. 115 (1988) 649 [http://inspirehep.net/search?p=find+J+%22Comm.Math.Phys.,115,649%22

Web End =INSPIRE ].[27] E.P. Verlinde and H.L. Verlinde, Chiral Bosonization, Determinants and the String Partition

Function, http://dx.doi.org/10.1016/0550-3213(87)90219-7

Web End =Nucl. Phys. B 288 (1987) 357 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B288,357%22

Web End =INSPIRE ].[28] E. Kiritsis, String theory in a nutshell, Princeton University Press (2007).[29] L.

Alvarez-Gaum e, G.W. Moore and C. Vafa, Theta Functions, Modular Invariance and

Strings, http://dx.doi.org/10.1007/BF01210925

Web End =Commun. Math. Phys. 106 (1986) 1 [http://inspirehep.net/search?p=find+J+%22Comm.Math.Phys.,106,1%22

Web End =INSPIRE ].[30] T. Kawai, Y. Yamada and S.-K. Yang, Elliptic genera and N = 2 superconformal eld theory,

http://dx.doi.org/10.1016/0550-3213(94)90428-6

Web End =Nucl. Phys. B 414 (1994) 191 [http://arxiv.org/abs/hep-th/9306096

Web End =hep-th/9306096 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9306096

Web End =INSPIRE ].[31] B. Rasof, The initial- and nal-value theorems in laplace transform theory, http://dx.doi.org/10.1016/0016-0032(62)90939-0

Web End =J. Franklin Inst.

http://dx.doi.org/10.1016/0016-0032(62)90939-0

Web End =274 (1962) 165 .[32] W.A. Stein, Modular Forms, A Computational Approach, Graduate Studies in Mathematics,

American Mathematical Society (2007) and online at

http://wstein.org/books/modform/modform/

Web End =http://wstein.org/books/modform/modform/ .[33] P. Jenkins and J. Rouse, Bounds for Coe cients of Cusp Forms and Extremal Lattices, http://dx.doi.org/10.1112/blms/bdr030

Web End =Bull.

http://dx.doi.org/10.1112/blms/bdr030

Web End =London Math. Soc. 43 (2011) 927 [http://arxiv.org/abs/1012.5991

Web End =arXiv:1012.5991 ].

{ 24 {

JHEP08(2016)041

[34] J. Leech, Notes on sphere packings, http://dx.doi.org/10.4153/CJM-1967-017-0

Web End =Can. J. Math. 19 (1967) 251 .[35] G. Nebe, Some cyclo-quaternionic lattices, http://dx.doi.org/10.1006/jabr.1997.7163

Web End =J. Algebra 199 (1998) 472 .[36] G. Nebe, An even unimodular 72-dimensional lattice of minimum 8, J. Reine Angew. Math.

673 (2012) 237 [http://arxiv.org/abs/1008.2862

Web End =arXiv:1008.2862 ].[37] M. Caselle and K.S. Narain, A New Approach to the Construction of Conformal Field

Theories, http://dx.doi.org/10.1016/0550-3213(89)90129-6

Web End =Nucl. Phys. B 323 (1989) 673 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B323,673%22

Web End =INSPIRE ].[38] L. Dolan, P. Goddard and P. Montague, Conformal Field Theory of Twisted Vertex

Operators, http://dx.doi.org/10.1016/0550-3213(90)90644-S

Web End =Nucl. Phys. B 338 (1990) 529 [http://inspirehep.net/search?p=find+J+%22Nucl.Phys.,B338,529%22

Web End =INSPIRE ].[39] N. Benjamin, E. Dyer, A.L. Fitzpatrick, A. Maloney and E. Perlmutter, Small Black Holes

and Near-Extremal CFTs, http://arxiv.org/abs/1603.08524

Web End =arXiv:1603.08524 [https://inspirehep.net/record/1436342

Web End =INSPIRE ].[40] R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in

4D CFT, http://dx.doi.org/10.1088/1126-6708/2008/12/031

Web End =JHEP 12 (2008) 031 [http://arxiv.org/abs/0807.0004

Web End =arXiv:0807.0004 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.0004

Web End =INSPIRE ].[41] D. Poland, D. Simmons-Du n and A. Vichi, Carving Out the Space of 4D CFTs, http://dx.doi.org/10.1007/JHEP05(2012)110

Web End =JHEP 05

http://dx.doi.org/10.1007/JHEP05(2012)110

Web End =(2012) 110 [http://arxiv.org/abs/1109.5176

Web End =arXiv:1109.5176 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1109.5176

Web End =INSPIRE ].[42] S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Du n and A. Vichi, Solving

the 3D Ising Model with the Conformal Bootstrap, http://dx.doi.org/10.1103/PhysRevD.86.025022

Web End =Phys. Rev. D 86 (2012) 025022

[http://arxiv.org/abs/1203.6064

Web End =arXiv:1203.6064 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1203.6064

Web End =INSPIRE ].[43] S. El-Showk and M.F. Paulos, Bootstrapping Conformal Field Theories with the Extremal

Functional Method, http://dx.doi.org/10.1103/PhysRevLett.111.241601

Web End =Phys. Rev. Lett. 111 (2013) 241601 [http://arxiv.org/abs/1211.2810

Web End =arXiv:1211.2810 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1211.2810

Web End =INSPIRE ].[44] F. Kos, D. Poland and D. Simmons-Du n, Bootstrapping Mixed Correlators in the 3D Ising

Model, http://dx.doi.org/10.1007/JHEP11(2014)109

Web End =JHEP 11 (2014) 109 [http://arxiv.org/abs/1406.4858

Web End =arXiv:1406.4858 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1406.4858

Web End =INSPIRE ].[45] E. Witten, Three-Dimensional Gravity Revisited, http://arxiv.org/abs/0706.3359

Web End =arXiv:0706.3359 [http://inspirehep.net/search?p=find+EPRINT+arXiv:0706.3359

Web End =INSPIRE ].[46] G. Moore, Trieste Lectures on Mathematical Aspects of Supersymmetric Black Holes,

http://www.physics.rutgers.edu/~gmoore/TriesteLectures_March28_2008.pdf

Web End =http://www.physics.rutgers.edu/ http://www.physics.rutgers.edu/~gmoore/TriesteLectures_March28_2008.pdf

Web End =gmoore/TriesteLectures March28 2008.pdf .

[47] O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional

gauge theories, http://dx.doi.org/10.1007/JHEP08(2013)115

Web End =JHEP 08 (2013) 115 [http://arxiv.org/abs/1305.0318

Web End =arXiv:1305.0318 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.0318

Web End =INSPIRE ].[48] C.A. Keller and H. Ooguri, Modular Constraints on Calabi-Yau Compacti cations, http://dx.doi.org/10.1007/s00220-013-1797-8

Web End =Commun.

http://dx.doi.org/10.1007/s00220-013-1797-8

Web End =Math. Phys. 324 (2013) 107 [http://arxiv.org/abs/1209.4649

Web End =arXiv:1209.4649 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.4649

Web End =INSPIRE ].[49] M.-A. Fiset and J. Walcher, Bounding the Heat Trace of a Calabi-Yau Manifold, http://dx.doi.org/10.1007/JHEP09(2015)124

Web End =JHEP 09

http://dx.doi.org/10.1007/JHEP09(2015)124

Web End =(2015) 124 [http://arxiv.org/abs/1506.08407

Web End =arXiv:1506.08407 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1506.08407

Web End =INSPIRE ].[50] P. Kraus, Lectures on black holes and the AdS3=CF T2 correspondence, Lect. Notes Phys.

755 (2008) 193 [http://arxiv.org/abs/hep-th/0609074

Web End =hep-th/0609074 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0609074

Web End =INSPIRE ].[51] N.J. Iles and G.M.T. Watts, Characters of the W3 algebra, http://dx.doi.org/10.1007/JHEP02(2014)009

Web End =JHEP 02 (2014) 009

[http://arxiv.org/abs/1307.3771

Web End =arXiv:1307.3771 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.3771

Web End =INSPIRE ].[52] C.A. Keller and A. Maloney, Poincar e Series, 3D Gravity and CFT Spectroscopy, http://dx.doi.org/10.1007/JHEP02(2015)080

Web End =JHEP 02

http://dx.doi.org/10.1007/JHEP02(2015)080

Web End =(2015) 080 [http://arxiv.org/abs/1407.6008

Web End =arXiv:1407.6008 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1407.6008

Web End =INSPIRE ].

{ 25 {

JHEP08(2016)041