Published for SISSA by

Springer

Received: March 31, 2011
Accepted: June 25, 2011
Published: August 29, 2011

Simeon Hellerman and Cornelius Schmidt-Colinet
Institute for the Physics and Mathematics of the Universe,
Todai Institutes for Advanced Study (TODIAS), The University of Tokyo,
Kashiwa, Chiba 277-8582, Japan

E-mail: simeon.hellerman.1@gmail.com,
cornelius.schmidt-colinet@ipmu.jp
Abstract: In this note we explore the application of modular invariance in 2-dimensional
CFT to derive universal bounds for quantities describing certain state degeneracies, suchas the thermodynamic entropy, or the number of marginal operators. We show that theentropy at inverse temperature 2π satisfies a universal lower bound, and we enumerate
the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for completely general CFT. We then restrict our attention to infrared-stable
CFT with moderately low central charge, in addition to the usual assumptions of modular
invariance, unitarity and discrete operator spectrum. For CFT in the range cL + cR < 48
with no relevant operators, we are able to prove an upper bound on the thermodynamic
entropy at inverse temperature
2π. Under the same conditions we also prove that a CFT
cL +cR
can have no more than 48−c
· exp{+4π} − 2 marginal deformations.
L −cR
Keywords: AdS-CFT Correspondence, Models of Quantum Gravity, Conformal and W
Symmetry
ArXiv ePrint: 1007.0756

c SISSA 2011

doi:10.1007/JHEP08(2011)127

JHEP08(2011)127

Bounds for state degeneracies in 2D conformal field
theoryContents
1

2 Energy and entropy bounds from modular invariance
2.1 Review of modular invariance
2.2 Review of previous work
2.3 A lower bound for the entropy at medium temperature

3
3
4
9

3 Meta-problem: why are upper bounds for state degeneracies hard?

9

4 Upper bounds for state degeneracies under certain conditions
4.1 A bound on the number of marginal operators
4.2 An upper bound on the entropy at medium temperature
4.3 Discussion

13
13
14
16

1

Introduction

Conformal field theory plays several distinct roles in our understanding of nature, including
quantum gravity. First, 2D CFT defines the classical solutions of the mysterious system
whose perturbative amplitudes are given by the dynamics of relativistic strings. Second ofall, conformal field theory defines the Hamiltonian of quantum gravity in spaces with negative cosmological constant and asymptotically anti-de Sitter boundary conditions. Thirdof all, conformal field theory describes fixed points of the renormalization group flow thatorganizes the behavior of all local quantum systems, including string theory in the lowenergy approximation. Beyond these important roles, CFT appears in many other guises
in various aspects of fundamental physics.
It is therefore of interest to understand as clearly as possible the gross features of the“landscape” of conformal field theories [3, 20]. Even the case that is by far best explored,that of two-dimensional CFT, is far from being understood in a systematic way.
Recently multiple independent lines of development have converged to probe the consistency conditions on the space of conformal field theories in extreme limits. In particular,
various papers over recent years have investigated the question of how big a gap can everbe opened in the spectrum of operator dimensions, above a universal sector defined byproducts of stress tensors.
This question is explored under various simplifying assumptions in different articles:in [4, 8, 9], the two-dimensional case is explored under the assumption of holomorphic
factorization of the Hilbert space; in [18] the two-dimensional case is explored again andextended superconformal symmetry is assumed; in [5, 6] the authors examine the maximum
possible dimension of the lowest-dimension operator appearing in the operator product–1–

JHEP08(2011)127

1 Introduction

1

In [7] the converse is shown: By making the central charge sufficiently large, it is possible to push
the gap to infinity and obtain a consistent limit where the theory contains only a sector of low-dimension
operators.

–2–

JHEP08(2011)127

expansion of two scalar operators whose dimensions are taken as given; in [2] the twodimensional case is probed once again and the dimension of the lowest primary operator
is bounded in a completely general CFT satisfying the minimal properties of unitarity anddiscrete operator spectrum.
In all cases the result is qualitatively the same: if one fixes the input parameters — in
two dimensions, the central charge, and in the case of [5, 6] the dimensions of the external
scalar operators in the OPE — the lowest-dimension operator that appears can never havea dimension higher than some universal bound.1 Strengthening the assumptions — adding
special conditions such as supersymmetry or holomorphic factorization — improves thebound numerically but does not qualitatively change the result.
In each case discussed above, the constraining principle is either modular invariance,or else associativity of the OPE, which is closely related. Both are consistency conditions
expressing the constraint that the theory must make sense when quantized in two different,
inequivalent channels — that is, foliating the space with two different time-slicings whose
leaves may be orthogonal to one another. It would appear that the condition of covarianceamong channels, or democracy among foliations, is a fundamental principle that makes it
impossible to pick and choose the spectrum of a CFT at will. In particular, the conditionof modular invariance, or channel covariance of the OPE, is incompatible with an attempt
to deform the spectrum in an extreme way.
In all cases discussed above, the gap in operator dimensions is the quantity under
examination, that is bounded by the principle of modular invariance, or channel covariancemore generally. There are other “extreme directions” in the space of possible spectra, inwhich we suspect it may be impossible to engineer the spectrum of a consistent CFT. In
particular, one expects that a CFT of fixed central charge ought not to have a numberof marginal deformations greater than some universal number depending on the central
charge. That is, for a CFT of central charge ctot and discrete spectrum, it would appear
likely that there may be a fundamental limit on the dimension of moduli space.
This idea is correlated with commonly held beliefs in mathematics and physics. In
physics, the holographic principle of ’t Hooft and Susskind [26, 27, 29] could be vitiatedby limits in which the number of massless species is pushed infinitely large [28, 29]. In
mathematics, it is thought that there is likely an upper bound on the Euler number ofcompact Calabi-Yau threefolds [30–32], which would follow immediately from a fully generalbound on the number of marginal operators of a superconformal field theory of central
charge cL = cR = c = 32 ĉ = 9.
In this note we derive rigorous bounds on certain state degeneracies: First, a lower
bound on the thermodynamic entropy at the inverse temperature β = 2π that maps toitself under the modular S-transformation. Second, we prove an upper bound on thenumber of marginal operators of a two-dimensional CFT of a given central charge, undercertain conditions. Under the same conditions we also derive an upper bound on thethermodynamic entropy at inverse temperature β = 2π.

The bounds proven herein, though requiring certain conditions to hold that are lessthan fully general, provide an illustration of a principle bounding state degeneracies fromabove, that the authors hope may apply in a broader set of circumstances.

2
2.1

Energy and entropy bounds from modular invariance
Review of modular invariance

where L0 and L̃0 are the zeroth right- and left-moving Virasoro generators, cR and cL are
the right- and left-moving central charges, and the complex structure τ lies in the upper halfplane. The torus can be thought of as a quotient of the complex plane C = {σ ≡ σ 1 + iσ 2 }by the identifications σ ∼ σ + 2π ∼ σ + 2πτ . The generators L0 and L̃0 are related toenergy H and momentum P1 by
L0 = 12 (H + P1 ) + c24R ,L̃0 = 12 (H − P1 ) + c24L .
We can represent this partition function as a path integral on a torus with metric!
!
g11 g12
1 Re τ
1= Im τ
g12 g22Re τ |τ |2
Modular invariance of the partition function is the statement that a local conformal field
theory has nothing to distinguish the various cycles of the two-torus other than the background metric itself. Therefore, partition functions on two distinct background metrics
differing by a large coordinate transformation should have the same partition function. A
large coordinate transformation acts on the cycles of the torus as
 
a b
,
SL(2, ZZ) ∋ Γ =
c d

aτ +b
which induces an action τ 7→ cτ+d on the complex structure of the torus.


 
0 −1
11
The modular group is generated by two elements, S ≡and T ≡
, which1 0
01
satisfy the relation (ST )3 = −1. For modular invariance to be a good symmetry of the
CFT at the quantum level, it suffices to check that the partition function transforms underboth S and T without anomalous phases.
For purposes of this note, as in [2], we shall not require invariance under the T transformation, but only under the S-transformation. Failure of the partition function to–3–

JHEP08(2011)127

The principle underlying our bounds is modular invariance, in particular invariance underthe modular S-transformation τ → − τ1 of the partition function of the CFT on a two-torus
with complex structure τ . The partition function of a two-dimensional CFT on such atorus can be written as



 
Z[τ ] = tr exp 2πiτ L0 − c24R − 2πiτ̄ L̃0 − c24L
,
(2.1)

2.2

Review of previous work
The medium-temperature expansion.

Going forward let us assume that the CFT is described by a unitary quantum mechanics(i.e., a Hermitean Hamiltonian with a positive definite norm on the Hilbert space), and thatthe spectrum of the Hamiltonian in finite volume is discrete. Thus the partition function
can be written asP(2.2)
Z[β] = tr ( exp{−βH}) = n exp{−βEn } ,where En are the discrete, real eigenvalues of the Hamiltonian H of the theory on a circleof length 2π.
By virtue of Cardy’s formula, the partition function and all of its derivatives areconvergent for any positive β. The pth derivative is equal to a sum of derivatives of
exponentials, that is
 p
P
∂
(2.3)
Z[β] = (−1)p tr (H p exp{−βH}) = (−1)p n Enp exp{−βEn } .
∂βiβexpression (2.1) reduces to the
For purely imaginary complex structure τ = 2π
usual thermodynamic partition function (2.2) at inverse temperature β. Then the Stransformation acts on the partition function as2Z[β] → Z[ 4πβ ] .

–4–

(2.4)

JHEP08(2011)127

be invariant under the T -transformation is easily understood in the Hamiltonian framework
as a failure of the momentum P1 to be quantized in integer units; the expression (2.1) isinvariant under τ → τ + 1 if and only if every state in the spectrum has P1 ∈ ZZ.
The S-transformation is both more obscure and more robust.
It is obscure in the sense that it is completely non-manifest in the Hamiltonian formalism. That is to say, a Hamiltonian formulation of a quantum theory depends on afoliation of spacetime, with Hilbert spaces associated to the individual leaves and Hamiltonian flow implementing linear transformations between the Hilbert spaces on different
leaves. But the S-transformation is not a canonical transformation, and does not preserve
even a single leaf of any smooth foliation of the torus, even up to Hamiltonian flow. The
S-transformation, even when an exact symmetry of the system, is not realized as an action
on the Hilbert space of quantum states.
On the other hand, the S-transformation is a good symmetry of the quantum theory under very broad conditions: if the theory has a Poincaré-invariant path integral
formulation in terms of local variables, then the path integral automatically respects the
S-transformation at the quantum level. There are many known theories that satisfy all
axioms of CFT except symmetry under the S-transformation; however all such examples
involve imposing projections directly on a Hilbert space and do not have partition functions
defined by a path integral over local variables. In this paper we shall restrict our attention
to modular-invariant CFT, though only invariance under the S-transformation is necessary
for our results to hold.

A modular invariant partition function is invariant under this transformation:2Z[β] = Z[ 4πβ ] .

(2.5)(β∂β )p Z[β]|β=2π = 0 .

(2.6)

It follows immediately [2] thatfor any positive odd p. Written in terms of the energies En , this can be written
n exp{−2πEn } fp (En ) = 0 ,where fp (E) is a pth order polynomial defined byp∂
fp (E) ≡ exp{+2πE} β ∂βexp{−βE}

(2.7)

β=2π

.

(2.8)

Explicit expressions for low p are:f1 (E) = −2πE

f3 (E) = −(2πE)3 + 3(2πE)2 − (2πE)(2.9)

More generally, if F (x) is any odd function of x, then we define a derived function∂fF (E) ≡ exp{+2πE} F β ∂β.
· exp{−βE}(2.10)
β=2π

The polynomials fp are just the derived functions corresponding to F (x) = xp .
For any odd F the derived function fF (E) satisfies
Pn exp{−2πEn }fF (En ) = 0 ,(2.11)

where {En } is the spectrum of a modular invariant CFT. For some purposes it is convenientto think of this condition in terms of a density
ρ(E) ≡ exp{−2πE}Pn δ(E − En ) ,(2.12)

which is exp{−2πE} times the usual spectral density. Then we can write the condition (2.11) asR

dE ρ(E) fF (E) = 0(2.13)

for an f derived from any odd F . By virtue of Cardy’s formula, the partition function isreal analytic as a function of β for any β. Therefore the conditions (2.7) for all odd p are
not only a consequence of invariance under the S-transformation, but when taken together
are sufficient to imply it as well.
The condition (2.7) follows directly from writingβ ≡ 2π exp{s} ,–5–

JHEP08(2011)127Pnoting that the S-transformation acts as s 7→ −s, and expanding the equation (2.5) to pth
order in s. Since we are expanding the partition function in the neighborhood of β = 2π,
which is intermediate between the high-temperature β → 0 and low-temperature β → ∞régimes, we refer to the expansion in s as the medium-temperature expansion [2]. The
medium-temperature expansion has proven to be useful in deriving general constraints on
the spectrum of a modular invariant conformal field theory [2, 10, 11].
Review of the upper bound on ∆1 .

tot
∆1 ≤ c12+ 0.48 ,(2.14)

ctot ≡ cL + cR

(2.15)

whereand ∆1 is the dimension of the primary operator of lowest dimension other than the identity
itself. For ctot < 24 − 18
π ≃ 18.270, the proof is completely elementary and does not dependon any use of representation theory of the Virasoro algebra. We recall the proof here.
Fix a value of the central charge ctot less than 24 − 18
π . Now consider the cubic
polynomial
F (x) ≡ x (x2 − 4π 2 E02 + 6πE0 − 1) ,(2.16)

where E0 is the ground state energy
tot.
E0 = − c24(2.17)

Using expressions (2.9), the derived polynomial isfF (E) = f3 (E) − (4π 2 E02 − 6πE0 + 1)f1 (E)= −8π 3 E (E − E0 ) (E − E+ ) ,(2.18)

3
where E+ ≡ 2π
− E0 . By equation (2.13), the quantity
R∞
E0 fF (E)ρ(E)dE(2.19)

3
tot∆1 < ∆+ ≡ 2π
+ c12.

(2.20)

must vanish. The measure ρ(E) is positive and the derived polynomial fF (E) vanishes
at E0 and is negative for E > E+ . If all excited energy levels were to be E+ or higher,
then the quantity (2.19) could not vanish, as it would receive only negative contributions.
3− E0 . Translating into
The lowest excited level E1 must therefore be lower than E+ = 2πoperator dimensions via ∆ = E −E0 , we find that the lowest-dimension operator other than3
the identity can have dimension no higher than ∆+ ≡ E+ − E0 = 2π
− 2E0 . Using (2.17)
we write the bound as–6–

JHEP08(2011)127

In [2] we used the medium-temperature expansion to prove that any conformal field
theory satisfying unitarity and modular invariance, with a discrete spectrum, satisfies theboundFor ctot < 24 − 18
π the value of ∆+ is less than 2, and it follows that the operator of
dimension ∆1 cannot be a descendant of the identity; it must therefore be primary or elsethe descendant of a primary operator of even lower dimension. So in this range of central
charge, we learn that equation (2.20) can be taken to apply specifically to the dimensionof the lowest primary operator above the identity.
Medium-temperature equations with characters.

• the conformal family of the vacuum, generated by the independent statesQ

Nm (L̃
Ñn |0i ,−n )m,n≥2 (L−m )(2.21)

and• the conformal family of a generic primary with dimension ∆, generated by the independent statesQÑn |∆i ,Nm (L̃
−n )m,n≥1 (L−m )(2.22)

with the Nm , Ñn running over all possible Nm , Ñn ≥ 0 in each case.
Each type of conformal family is then spanned by different monomials in Virasoro
raising operators acting on the primary state, with each monomial raising the energy ofP
the vacuum by an amount n n Nn + nÑn . The contributions of the two families to thepartition function are then given byχv (β) exp{−E0 β}(2.23)χg (β) exp{−(∆ + E0 )β}(2.24)

for the vacuum family, andfor the family of the generic primary, with the vacuum and generic characters given byQχv (β) ≡ n≥2 (1 − exp{−nβ})−2 ,(2.25)Qχg (β) ≡ n≥1 (1 − exp{−nβ})−2 ,–7–

JHEP08(2011)127

For higher central charge a similar bound can be proven using roughly the same argument, with the contributions to the partition function organized according to full representations of the Virasoro algebra rather than individual energy levels. The proof for
ctot > 24 − 18
π uses some elementary facts about the representation theory of the Virasoro algebra, in particular formulae for the characters of the Virasoro algebra for various
representations [12–15].
It is natural to generalize the derived polynomials fp (E) to derived polynomials fp | χwith respect to characters χ of the Virasoro algebra. In this context, we will only consideriβ.
characters restricted to the imaginary axis of their argument, χ(β) with τ = 2π
Our simplifying assumptions, that cL , cR > 1 and that there are no holomorphic orantiholomorphic operators, guarantee that there are only two types of representations of
the Virasoro algebra for a unitary CFT with discrete spectrum [12, 13, 15]:

respectively. Then the full partition function can be written asP
Z[β] = χv (β) exp{−βE0 } + χg (β)( ∞n=1 exp{−βEn }) ,(2.26)

fF | χ (E) ≡ exp{+2πE}
F (β∂β ) · [χ(β) exp{−βE}]|β=2π .
χ(2π)(2.27)

where the sum in the second term runs over energies En ≡ ∆n + E0 of primary statesalone. Then invariance under the modular S-transformation β → 4π 2 /β can be expressedin terms of energies of primary states. For a given function F (x) and any character χ,
define the derived functions

χv (2π) exp{−2πE0 }fF | χv (E0 ) +P∞

n=1 χg (2π) exp{−2πEn } fF | χg (En ) = 0 ,(2.28)

for any odd F . To express this condition in terms of measures, defineρv (E) ≡ χv (2π) exp{−2πE}δ(E − E0 )(2.29)andρg (E) ≡ χg (2π) exp{−2πE}P∞

n=1 δ(E − En ) ,(2.30)

where n runs over non-vacuum primary states. Then the condition for modular invarianceisR

dE ρv (E)fF | χv (E) +R

dE ρg (E)fF | χg (E) = 0(2.31)

for any odd F . For an arbitrary character χ(β), the derived polynomials fp | χ corresponding
to low-order monomials F (x) = xp areχ′ (2π)χ(2π)
χ′ (2π)32+3(2.32)
f3 | χ (E) = −(2πE) + (2πE) 6π
χ(2π)




′′
′′′
′′
χ′ (2π)χ′ (2π)2 χ (2π)
3 χ (2π)
2 χ (2π)−(2πE) 12π+12π+1 + 8π
+12π
+2π
χ(2π)
χ(2π)
χ(2π)
χ(2π)
χ(2π)

f1 | χ (E) = −(2πE) + 2π

In [2] we use this structure to derive a bound on the weight of the lowest non-vacuum
primary dimension ∆1 that applies for arbitrarily high values of the total central charge.
We refer the reader to [2] for details. We want to emphasize, however, that the generalized
proof is not much more complicated than the elementary proof for low central charge that
we have reviewed in the previous subsection.

–8–

JHEP08(2011)127

Then the invariance under the modular S-transformation can be expressed by the condition
that

2.3

A lower bound for the entropy at medium temperature

The most straightforward consequence of modular invariance is to give a lower, rather thanan upper bound for the thermodynamic entropy of the canonical ensemble, particularly atinverse temperature β = 2π.
From the first order of the medium-temperature expansion, it follows that a universal
1. The
lower bound holds for the thermodynamic entropy at medium temperature β −1 = 2π
entropy σ is related to the partition function via(2.33)

Using the derived function f1 (E) = −2πE in equation (2.7), we havehEi|β=2π = 0 .

(2.34)

By unitarity, every contribution to the partition function is positive, so the value of Z isbounded below by its vacuum contribution exp{−2πE0 }, so we haveσ|β=2π ≥ −2πE0 = πc12tot .

(2.35)We have given an elementary proof of a universal lower bound for the thermodynamic
entropy in a modular invariant 2D CFT at a particular temperature. On the other hand
we shall see in the next section that there can be no fully general upper bound on thethermodynamic entropy or on the microstate degeneracies that would hold without imposing additional assumptions on the CFT. Understanding the issues involved will help us toformulate a useful set of additional assumptions on the CFT as we go forward.

3

Meta-problem: why are upper bounds for state degeneracies hard?

In the previous section we reviewed the derivation of an upper bound on the energy ofthe first excited energy level. Should it not be possible, then, to prove an upper boundon entropy — thermodynamic entropy or quantum mechanical degeneracies — using
similar techniques? Let us examine, briefly, a few reasons why the type of argument in theprevious section cannot be generalized very easily to give a fully general upper bound for
entropy or quantum mechanical degeneracies, without imposing additional assumptions
on the CFT as inputs.
(a) The homogeneity problem.
The homogeneity problem is a meta-problem with any candidate for a method tobound the entropy above, using invariance under the S-transformation. Suppose we hadsome equation of type (2.13) that would always be violated if the entropy — the thermodynamic entropy or the quantum mechanical degeneracy, in some energy range, according
to whatever definition — were to be sufficiently high. We can argue by contradiction thatno such equation can ever exist.
Suppose such an equation did exist, that ruled out the possibility of a modular invariant spectrum with effective state degeneracy greater than nmax , by whatever definition.

–9–

JHEP08(2011)127

σ = ln(Z) + β hEi .

But we could always take k copies of a modular invariant spectrum with effective statedegeneracy n < nmax , such that k n > nmax . Taking k copies of a modular invariant spectrum automatically yields a modular invariant spectrum, so it is clear that there can be no
direct constraint from modular invariance bounding the effective state degeneracy above,
without using additional inputs.
(b) The continuum problem at the vacuum.

(c) The hyperfine structure problem of character corrections.
To evade the continuum problem without assuming a minimum gap in the spectrum
as an input, we could try using other consistency conditions of the CFT; in particular,
we could try using the organization of the spectrum into representations of the Virasoro
algebra. However this approach immediately runs into the problem that the differences
between different characters of the Virasoro algebra are numerically very tiny.

– 10 –JHEP08(2011)127

One way around the homogeneity problem is to use the fact that the spectrum underconsideration is not arbitrary, but corresponds to the spectrum of a good CFT, satisfying
all the usual CFT axioms including cluster decomposition. Together with unitarity and thestate-operator correspondence, cluster decomposition implies that there is a unique lowest
totstate, with energy E0 = − c24.
Then it may be possible in principle to find an upper bound on the entropy (or thedegeneracy of some states in some range of energy) that evades the homogeneity problem
by using the fact that the degeneracy of the vacuum is never greater than 1.
To exploit this fact, it would be necessary to find an odd function F such that thederived function fF (E) gets a positive contribution from the vacuum, a negative contribution from the states of interest, and a sum of contributions from all other states thatis bounded above independently. (In particular, it would not do to pick an F such thatfF (E0 ) = 0, as we did to prove the universal upper bound on E1 .)This type of approach to bounding the entropy suffers from a separate meta-problem
that we shall call the continuum problem. By the continuum problem, we mean that such a
proof could never apply in cases where the spectrum develops an approximate continuumof states with energies an arbitrarily small amount above the vacuum. In such a case thecontinuum would contribute with the same sign as the vacuum (by continuity of f (E))
with an unboundedly large coefficient, due to the presence of an arbitrarily numerous setof levels in the range between E0 and E0 + ǫ.
Of course, no CFT under our consideration ever has a strict continuum; we are always
assuming that our CFT have a discrete operator spectrum, or equivalently discrete spectrum of the Hamiltonian in finite volume. But many CFT are known to come in families
with singular limits where the limiting spectrum has a continuum of some kind. In particular, many familiar moduli spaces of CFT have limits in which the CFT can be thoughtof as a sigma model with volume approaching infinity. So the continuum problem is a
general no-go principle for upper bounds on CFT degeneracies of any sort that do not use
additional consistency conditions of the CFT, or assume a minimum gap in the spectrum
above the vacuum.

f1 | χv (E) = −2πE + 4π
f1 | χg (E) = −2πE + 4π
The two differ only byn exp{−2πn}n=2 1− exp{−2πn}P∞n exp{−2πn}n=1 1− exp{−2πn}

P∞

−2 .
f1 | χg (E) − f1 | χv (E) = 4π 1−exp{−2π}exp{−2π} ≃ 2.35 × 10

(3.1)

(3.2)

Furthermore, a minimal criterion for overcoming the continuum problem is that the valuelimof E→Ef(E) should have a different sign from fF | χv (E0 ). In the case F (x) = x, it
0 F | χgis impossible for the two to differ in sign unless E0 > 0, which would be inconsistent withpositivity of the central charge, and thus with unitarity.
(d) The fine structure problem at large central charge
There is a separate problem in attempting to bound the entropy from above by using
invariance of the partition function under the modular S-transformation, that is particularly acute when the central charge becomes large. To see the nature of the problem,
consider in particular an attempt to bound the degeneracy of marginal operators in thelimit of large central charge. When the central charge is large, it becomes increasingly
difficult to find derived functions fF (E) that are positive for E = E0 , and negative for all
E ≥ E0 + 2. In the limit E0 → −∞ with E/E0 held fixed, the derived function fF (E) isto good approximation equal to F (−2πE), which is an odd function. For F (x) = xp , wehavefp (E) = (−2πE)p + O(E p−1 ).

(3.3)

Thus if we take an odd polynomial F (x) such that fF (E) is positive at E = E0 but negativeat E0 + 2, then it will tend to be positive again by the time it reaches E = −(E0 + 2), if|E0 | is large.

– 11 –JHEP08(2011)127

As in [2], simplify the discussion by assuming that cL and cR are both greater than1, and that there is no chiral algebra of the theory other than the Virasoro algebra. The
condition (2.31) then treats the vacuum conformal family, which contributes to ρv , differently from the conformal families of the other primaries, which contribute to ρg . Therefore
it is possible in principle to find odd F such that the derived polynomials fF | χv (E0 ) andfF | χg (E) contribute with different sign, even when E is arbitrarily close to E0 . In practice,
however, it seems difficult to generate a useful bound this way.
An approach to overcoming the continuum problem based on the difference in Virasoro
representations between that of the vacuum and that of the generic primaries lying in
the continuum just above the vacuum, would need to exploit the differences between thederived functions fF | χv and fF | χg , evaluated at E0 . But the absolute difference between
the derived functions for the two different types of conformal family is numerically small,1.
being proportional to exp{−2π} ≃ 535
For instance, suppose we want to take the lowest-order polynomial possible, F (x) = x1 .
Then the two derived polynomials are

The even part of the function fF (E) is a kind of “fine structure” of subleading orderat large central charge. But it is only by tuning the form of the function F (x) to enhance
the contribution of the even part of fF relative to the odd part, that one has any hope ofobtaining a derived function with properties that could imply a bound.
(e) The resolutional problem.

22

and 2R α(2+ǫ). The region
between two spheres centered at the origin, with radii 2R α(2−ǫ)′′
RR
of interest is an annular region of radius 2 √ ′ , circumference 4π √ ′ and thickness Rǫ
α′ . Theααlattice points are distributed with unit density in the plane, so the annular region contains
2ǫ
of order 4πRlattice points. Thus, no matter how small ǫ is chosen to be, the radius R canα′always be made sufficiently large that the number of almost-marginal operators — scalar
operators with dimensions between 2 − ǫ and 2 + ǫ can be made as large as desired.
The five problems described above are not entirely logically independent from one
another. The strict version of problem (a) can be solved trivially by assuming cluster
decomposition, but even then this solution is “unstable” against turning into problem (b)under a small perturbation of the spectrum. Problem (b) can in principle be solved byorganizing the partition function using characters of the Virasoro algebra to separate thevacuum from the continuum. But in practice one runs into problem (c), that the effect ofthe distinct characters at β = 2π is so small that it is not easy to exploit the separatenessof these contributions to derive a bound.
Problems (a)-(c) all concern the difficulty of controlling unwanted contributions
to (2.7)–(2.13) from energy levels near the vacuum E0 . Problem (d) concerns the difficultyof controlling unwanted contributions from energy levels much higher than the marginal
operators at E0 + 2, and problem (e) concerns the difficulty of controlling unwanted contributions from levels arbitrarily close to E0 + 2.
The purpose of discussing these meta-problems is not only to warn ourselves away from
too-naive attempts to prove a bound using modular invariance, but also to guide ourselvesin choosing a favorable set of additional assumptions on the CFT that will allow us to
avoid these persistent difficulties.

– 12 –JHEP08(2011)127

The resolutional problem is the difficulty in formulating a suitable definition of theentropy of marginal operators that is robust against small perturbations of the spectrum.
Certainly it seems quite unlikely that there should be any unitary conformal field theory,100with discrete spectrum and central charges cL = cR = 2, say, with 1010 marginal operators, for instance. And yet, if we alter the question slightly to ask whether there might bea unitary conformal field theory with discrete spectrum and central charges cL = cR = 2,
100
and 1010 operators between dimensions 2 − ǫ and 2 + ǫ, for arbitrarily small values of ǫ,
the answer is that, yes, there certainly do exist such CFT. Simply consider a sigma modelwith target space T 2 , with cycles of length 2πR, and the kinetic term for the target space
1ijaacoordinates X a normalized as L = 4πα′ g (∂iX )(∂jX ).
The theory contains scalar primary operators of the form On1 ,n2 ≡: exp{ina Xa /R} :,√α′ 2
α′ , the set of operators withwhich have dimension ∆ = 2R
2 na . For R much larger than2−ǫ ≤ ∆ ≤ 2+ǫ corresponds to the set of integer pairs
in the Euclidean planeq(n1 , n2 ) lying q

4Upper bounds for state degeneracies under certain conditions

In this section we will prove a bound on the number of marginal operators in an infraredstable CFT with cL + cR less than 48. We will also derive an upper bound on the thermodynamic entropy at inverse temperature 2π, under the same conditions. We conclude witha discussion of the possibility of bounding the number of marginal operators under othersets of assumptions, such as extended superconformal symmetry.
4.1

A bound on the number of marginal operators

0 = E0 exp{−2πE0 }+(E0 + 2)(N + 2) exp{−2π(E0 + 2)}
X+(E0 + ∆) exp{−2π(E0 + ∆)}(4.1)

where the sum in the third term runs over all operators with dimension ∆ > 2. For
tot
is positive, so the only negative contribution in
ctot < 48, the quantity E0 + 2 = 48−c
24equation (4.1) is the first. Multiplying (4.1) through by exp{+2π(E0 + 2)} we obtain
0 < (E0 + 2)(N + 2)
< (E0 + 2)(N + 2) += −E0 exp{+4π} ,X(E0 + ∆) exp{−2π(∆ − 2)}(4.2)

giving us a boundN<cL + cR
48 − cL − cR· exp{+4π} − 2 .

– 13 –(4.3)

JHEP08(2011)127

Keeping in mind the meta problems discussed in the section above, we will now choosesome simple assumptions that will allow us to avoid problems (a) through (e). We can
avoid problems (a)-(c) by assuming cluster decomposition, and the absence of a continuumof states just above E0 : in fact the bound is simplest if we restrict our considerations to
infrared stable fixed points of the renormalization group — CFT with no relevant operators
at all other than the identity. We will avoid problem (d) by restricting our considerations
to CFT with moderately low central charge — say, less than 24 on the left and on the right,
so ctot < 48. We need not make any additional assumption in order to evade problem (e)— in fact we will see that the assumptions of infrared stability and ctot < 48 are enoughto prove a bound; problem (e) is avoided automatically.
Under these assumptions it is possible to prove a bound using only the first order in themedium-temperature expansion. Let N be the number of primary operators of dimension
2. The degeneracy at energy E0 + 2 is then N + 2, with the 2 extra operators coming fromthe left- and right-moving stress tensor. The number N includes both scalar primaries ofweight (1, 1) as well as spin-1 operators of weights (3/2, 1/2) and (1/2, 3/2). The simple
method we are using is not sufficiently refined to distinguish these, but an upper boundon N necessarily bounds the number of scalar primary operators, so it will not matter too
much that we do not distinguish them.
Taking F (x) = −(x/(2π)) gives fF (E) = E, and we have the equation

N max187, 869
204, 820
223, 026
242, 633
263, 809
286, 749
311, 684
338, 885
368, 678
401, 449
437, 671
477, 916
522, 897
573, 500
630, 850

ctot
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48

N max
696, 394
772, 020
860, 251
964, 525
1, 089, 652
1, 242, 587
1, 433, 754
1, 679, 541
2, 007, 257
2, 466, 059
3, 154, 262
4, 301, 267
6, 595, 278
13, 477, 309∞

Table 1. Maximal number (4.3) of marginal operators in a stable unitary CFT for integer valuesof the central charge in the range (4.4).

This result depends on the assumption that there are no operators with dimensions ∆between 0 and 2. Due to the earlier result [2], reviewed in the second section, this can only
be the case if ctot > 24 − 18
π ≃ 18.27. So the interesting range of central charge, wherea useful bound may be proven from modular invariance at first order in the mediumtemperature expansion, is∼ 18.27 < ctot < 48 .

(4.4)

For integer values of ctot in this range, we give the maximum possible number of marginal
operators in a CFT with no relevant operators in table 1. As an immediate corollary of
our bound, note that the resolutional problem, problem (e) of the previous section, has
resolved itself automatically: through modular invariance as expressed at first order inthe medium-temperature expansion, the assumption of no relevant operators other than
the identity implies the absence of a continuum of states with dimensions near 2. If thelatter did exist, then equation (4.1) would receive an unboundedly large number of positive
contributions going as (E0 + 2 + ǫ) exp{−2π(E0 + 2 + ǫ)}, for ǫ made arbitrarily small,without offsetting negative contributions; if there were a (near)-continuum close to ∆ = 2
but no states between ∆ = 0 and ∆ = 2, then the first-order medium-temperature condition
for modular invariance could not be satisfied.
4.2

An upper bound on the entropy at medium temperature

To complement our discussion of a lower bound for thermodynamic entropy in the second
section, we would like to establish an upper bound on the thermodynamic entropy of thecanonical ensemble at medium temperature, β = 2π, under certain conditions. We impose– 14 –JHEP08(2011)127

ctot
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33

the same assumptions on our CFT as we did to derive the upper bound on the degeneracy of marginal operators in the previous subsection: That is, in addition to unitarity,
cluster decomposition, discrete operator spectrum and invariance under the modular Stransformation, we also assume that ctot < 48 and that there are no relevant operators
other than the identity.
As in the previous subsection, we need consider only the leading nontrivial order inthe medium-temperature expansion in order to derive an interesting bound. Using the factthat E0 < 0, we can write E0 = −|E0 |, and equation (4.1) can be written asEnn≥1 |E0 | exp{−2π(En + |E0 |)} .

P

(4.5)

We are assuming 0 < ctot < 48, and that 2 ≤ ∆1 < ∆2 < · · · . Then using En = ∆n + E0 =tot
, we have∆n − c24
0 ≤ E1 ≤ E2 · · ·

(4.6)

E1
So the nth term on the right hand side of (4.5) is no smaller than |E
exp{−2π(En +|E0 |)}.
0|Summing terms, we find

P
E1 P
En
n≥1 exp{−2π(En + |E0 |)} ≤n≥1 |E0 | exp{−2π(En + |E0 |)} = 1 .
|E0 |

(4.7)

|E0 |
n≥1 exp{−2πEn } ≤ E1 exp{+2π|E0 |} .

(4.8)

0|
Multiply each side by |E
E1 exp{+2π|E0 |} to derive the inequality

P

We may have no information about the specific value of E1 , but under our assumptions we
do know that E1 ≥ 2 − |E0 | > 0, soP|E0 |n≥1 exp{−2πEn } ≤ 2−|E0 | exp{+2π|E0 |} .

(4.9)

Adding the vacuum contribution exp{−2πE0 } = exp{+2π|E0 |} yields an upper bound forthe full partition function at β = 2π:
Z[2π] =

X

n≥0

≤



exp{−2πEn }

|E0 |
1+
2 − |E0 |

exp{+2π|E0 |} =n πc o
48
tot
.
exp +
48 − ctot
12(4.10)The bound (4.10) on the partition function in turn gives us an upper bound for the thermodynamic entropy of the canonical ensemble at medium temperature. Using the fact that
< E >= 0 at β = 2π and combining with the lower bound (2.35), we have:
πctot
πctot
12 ≤ σ|β=2π ≤ 12 + ln

– 15 –

48
48−ctot.

(4.11)JHEP08(2011)127

1=

4.3

Discussion

Acknowledgments
We are very grateful to many people for valuable discussions and comments on the draft, including Michael Douglas, Matthias Gaberdiel, Zohar Komargodski, Hirosi Ooguri, Matthew
Sudano and Erik Tonni. We would also like to thank the Physics Department at Virginia– 16 –JHEP08(2011)127

We have proven a bound on the number of marginal operators for a general unitary CFTwith discrete spectrum, no relevant operators other than the identity, and central charge
in a certain range, cL + cR < 48. We have also proven upper and lower bounds for the thermodynamic entropy at inverse temperature 2π, under the same assumptions. Ultimately,
we would like to improve the bounds, both by extracting more refined information, and byproving similar bounds under weakened hypotheses.
In terms of more refined information, it would be good to be able to bound the numberof scalar marginal operators alone, without including spin-1 operators of weight 2 into thesame count. It seems possible that this could be done by using refined medium-temperatureequations that consider the partition function as a function of τ and τ̄ instead of Im(τ )alone, and using the stronger condition (τ̄ ∂τ̄ )p1 (τ ∂τ )p2 Z[τ ]|τ =i = 0 for p1 + p2 odd.
As far as weakening the assumptions is concerned, it would be good to be able tobound the thermodynamic entropy and microscopic state degeneracies for arbitrarily highcentral charge. Also, the condition that there be no relevant operators is a strong one, and
it is certainly desirable to derive limits on state degeneracies without this assumption. Attempts to weaken either of these two conditions will necessarily meet some of the difficulties
enumerated in the third section. New ideas may be required to circumvent those.
In one particular circumstance, the restriction to theories without relevant operators
seems particularly natural. Two-dimensional CFT with extended supersymmetry and integrally quantized U(1) charges play an important role as theories representing string propagation in spaces with unbroken spacetime SUSY, for instance on Calabi-Yau threefolds.
These CFT may be a particularly tractable special class in which the number ofmarginal operators may be bounded. First of all, these theories possess a large chiral algebra — at least N = 2 superconformal symmetry, together with spectral flow generators [25].
They have low central charge, circumventing the fine-structure problem discussed earlier.
As for the continuum problem, conformal sigma models on Calabi-Yau spaces admit nonthermal boundary conditions under which the partition function can be evaluated, which
project out the contribution from the near-continuum of operators that may be present
with dimension close to 0. Each non-thermal boundary condition for the partition function
nonetheless has simple modular transformation properties. Among the possible boundary
conditions for the partition function are those describing the elliptic genus [24, 25]. Interesting work has been done recently [18, 21–23] on the derivation of general consistency
conditions for elliptic genera. Perhaps the medium-temperature techniques discussed herecombined with results such as [18, 21] may yield useful information about the still mostlyuncharted landscape of Calabi-Yau manifolds. We hope the present note will provide clues
for further progress in that direction and others.

Tech for hospitality while this work was in progress. S. H. is supported by the World
Premier International Research Center Initiative (WPI Initiative), MEXT, Japan; and by
a Grant-in-Aid for Scientific Research (22740153) from the Japan Society for Promotionof Science (JSPS). C. S.-C. is supported by a fellowship from the Swiss National Science
Foundation (SNF).

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