Bounds on CFTs with algebras and AdS higher spin theories
The scaling dimension of the first excited state in two-dimensional conformal field theories (CFTs) satisfies a universal upper bound. Using the modular bootstrap, we extend this result to CFTs with algebras which are generically dual to higher spin theories in AdS. Assuming unitarity and modular invariance, we show that the conformal weights , of the lightest charged state satisfy and in the limit where the central charges , are large. Furthermore, we show that in this limit any consistent CFT with currents must contain at least one state whose charge obeys . We discuss hints on the existence of stronger bounds and comment on the interpretation of our results in the dual higher spin theory.
The modular bootstrap is a powerful tool used to constrain the spectrum of two-dimensional conformal field theories (CFTs). Like the conformal bootstrap, which relies on conformal and crossing symmetry of correlation functions Belavin et al. (1984), the modular bootstrap also relies on symmetry, namely invariance under large diffeomorphisms of CFTs defined on the torus.111A generalization of the modular bootstrap to higher dimensional CFTs is not known (for generalizations of modular invariance see refs. Cardy (1991); Belin et al. (2017); Shaghoulian (2016).) In contrast, the conformal bootstrap has been applied to CFTs in three and four dimensions, see e.g. Rattazzi et al. (2008); Poland et al. (2012); El-Showk et al. (2012).
A remarkable consequence of modular invariance is Cardy’s formula Cardy (1986). There, modular S-transformations, where is the complex structure of the torus, are used to relate the low () to the high () temperature limits of the partition function. As a result, the density of states at high temperature and large central charge is universal for any unitary, modular-invariant CFT.
Modular S-transformations yield further universal constraints on CFTs at large central charge Hellerman (2011); Hellerman and Schmidt-Colinet (2011); Keller and Ooguri (2013); Friedan and Keller (2013); Qualls and Shapere (2014); Qualls (2015a, b); Kim et al. (2016); Benjamin et al. (2016); Collier et al. (2016). These follow from the observation that, at the fixed point , there is an infinite number of constraints on the partition function of modular invariant CFTs Hellerman (2011),
In the presence of a chemical potential , these constraints have been generalized to Benjamin et al. (2016)
where denotes the S-transformed value of and stands for the arguments of the partition function.
In this paper we constrain the conformal weights of the lightest charged state in CFTs with algebras and prove the existence of a state whose mass-to-charge ratio obeys a universal upper bound. The algebra is a nonlinear extension of the Virasoro algebra which, in the semiclassical, large central charge limit, reproduces the asymptotic symmetries of higher spin theories in AdS Campoleoni et al. (2010).222In the context of the AdS/CFT correspondence Maldacena (1999); Gubser et al. (1998); Witten (1998), CFTs with algebras are generically dual to higher spin theories in AdS, which feature fields of spin Henneaux and Rey (2010); Campoleoni et al. (2010, 2011). Concrete examples of the duality are known for Gaberdiel and Gopakumar (2011) (see Gaberdiel and Gopakumar (2013) for a review). Assuming unitarity and modular invariance, we find that the conformal weights , of the first excited state carrying charge satisfy
in the limit where the central charges , are large. The bound (3) holds for CFTs with either one or two copies of the algebra. In particular, for a left-right symmetric theory where , the scaling dimension obeys
The bound (4) is shared by other (nonextremal) CFTs without supersymmetry.333Known exceptions satisfying stronger bounds are extremal CFTs with a holomorphically factorized partition function Witten (2007), and CFTs with or supersymmetry where the object of interest is the elliptic genus Gaberdiel et al. (2008); Benjamin et al. (2016). Indeed, it is also satisfied by the lightest uncharged state in CFTs with or without extended chiral symmetries Hellerman (2011); Friedan and Keller (2013); Qualls and Shapere (2014); Qualls (2015b), and by charged states in CFTs with abelian chiral symmetries Benjamin et al. (2016). In all of these cases, it is not possible to rule out the existence of stronger bounds inaccessible to analyses based on eqs. (1) and (2) with a finite number of derivatives. Indeed, ref. Collier et al. (2016) foregoes this assumption by considering a large number of derivatives (extrapolated to infinity) before taking the large-c limit; these limits do not seem to commute, and numerical evidence is presented for a stronger bound, where . One of the motivations for the present paper was to confirm whether the non-abelian and nonlinear structure of the algebra is constraining enough to reproduce similar bounds, even when the number of derivatives in eq. (2) is small. While our analysis shows that this is not the case, we discuss hints that suggest the existence of a stronger bound where .
Turning to the dual theory, we note that higher spin theories admit black hole solutions carrying charge Gutperle and Kraus (2011) (see Ammon et al. (2013) for a review). These higher spin black holes obey an extremality bound which, in terms of CFT quantities, reads Gutperle and Kraus (2011); Bunster et al. (2014); Bañados et al. (2016)
where the charge and the central charge are assumed to be equal in the left and right-handed sectors. Thus, in the semiclassical, large central charge limit, the CFT bound (4) is larger than the lightest charged higher spin black holes.
Using the modular bootstrap, we also prove the existence of a state whose charge is bounded, in the large- limit, by
Thus, if , the CFT must feature a state with for some nonvanishing number . In particular, assuming a left-right symmetric CFT, eq. (6) implies the existence of a state in the dual higher spin theory whose mass-to-charge ratio satisfies444The case must be handled with care. If , then is still bounded by terms of , as in eq. (6). Otherwise if , then is bounded by , cf. eq. (61).
In eq. (7) we have defined where and denote the charges of the dual higher spin theory. The existence of a state satisfying eq. (7) is reminiscent of the weak gravity conjecture, which features a similar bound Arkani-Hamed et al. (2007).555For generalizations of the weak gravity conjecture to AdS spacetimes see e.g. Nakayama and Nomura (2015). In particular, for related studies in AdS that focus on the structure of the dual CFT see refs. Benjamin et al. (2016); Montero et al. (2016). However, in our case the bound cannot be interpreted as a measure of the relative strength between two forces since the dual higher spin theory contains only one coupling constant, namely . It would be interesting to understand the necessity of this bound from the point of view of the dual higher spin theory.
The bounds (4) and (6) constrain states in the CFT and, correspondingly, in the dual higher spin theory. However, the latter is deemed inconsistent if it contains light matter fields Perlmutter (2016). There are two exceptions: (i) pure higher spin gravity, an analog of pure three-dimensional gravity, which admits only boundary gravitons, boundary higher spin fields, and black holes; and (ii) higher spin theories with an infinite tower of higher spin fields dual to CFTs with algebras. If pure higher spin gravity exists, the bound (4) constrains the scaling dimension of the lightest black hole microstate, while (6) constraints the charge of at least one such microstate. On the other hand, we comment on the obstacles of extending our analysis to CFTs with algebras.
Ii Partition function and modular transformations
Let us begin by defining the partition function and deriving its modular transformation in CFTs with a chiral algebra. The symmetries of the theory are described by Zamolodchikov (1985)
where and represent the modes of the chiral components of the stress-energy tensor and , denotes the modes of the dimension-3 current , and is given by
The modes in eq. (11) are those of a composite operator which is responsible for the nonlinearity of the algebra,
We are interested in constraining the conformal weights , of states carrying charge, i.e. states such that
Therefore, we focus on the canonical partition function with an additional chemical potential for the dimension-3 current
Besides the algebra, the additional requirements made on the CFT are unitarity and modular invariance. The former restricts the coefficients in eq. (16) to be positive, while the latter leads to a modular invariant partition function when , i.e.
where is given by
with and . Once the chemical potential is turned on, the partition function is no longer modular invariant. While a closed-form expression for the transformed partition function is not known, it is possible to determine this transformation perturbatively around Gaberdiel et al. (2012); Iles and Watts (2016). In particular, note that the constraints given in eq. (2) are also expanded around this point and that, to recover the bound in eq. (3), we will only need terms at most quadratic in . The modular-transformed partition function thus reads
where denotes the torus one-point function of , and is given by Li et al. (2013)666The transformation (20) is also featured in ref. Kaneko and Zagier (1995) in the context of generalized Jacobi Theta functions.
Let us now determine the transformation of the partition function up to quadratic order in . To zeroth order, the partition function is modular invariant, cf. eq. (17). This is also true at linear order where we have
This equation follows from the fact that (i) only the zero mode contributes to the one-point function of primary fields on the torus; and (ii) for any primary field with weights and , the torus one-point function transforms as
Next, we consider the transformation of the partition function at quadratic order in . Following refs. Iles and Watts (2014, 2016), we first construct a composite primary field such that (see Appendix A for details)
where is the identity operator. We then note that the modular transformations of and are easily found: the former is given by eq. (22) while the latter follows directly from eq. (17). In this way, the modular transformation of is determined entirely from the symmetry algebra and the modular invariance of the CFT. Deferring details of the derivation to Appendix A we find
where the function is given by
and , denote the Eisenstein series,
We conclude that, up to quadratic order in , the modular transformation of the partition function is given by
Equation (28) generalizes the corresponding equation in CFTs with a chiral algebra. There, the transformation of the partition function is known to all orders Kraus and Larsen (2007)777The partition function is given by eq. (15) with the zero mode of a dimension-1 current and a chemical potential whose modular transformation is given by .
and its expansion around reproduces eq. (28) with . In contrast, the fact that in eq. (25) depends not only on , but also on and , is a consequence of the non-abelian and nonlinear structure of the algebra.
Iii Bounds on charged states
We now have the necessary ingredients to derive bounds on the charged states of CFTs with algebras. The latter follow from the constraints imposed by invariance of the CFT under modular S-transformations, cf. eq. (2).888Recall that modular S-transformations correspond to and in eqs. (18), (20), and (28). These constraints may be written as
where is a differential operator given by
For convenience, in the remainder of the text is evaluated at and .
Unitarity restricts the coefficients in eq. (30) to be positive. Thus, the modular constraints are satisfied by any nontrivial operator if the corresponding function admits both positive and negative values. In particular, assume that and exist such that is positive for all and , and is negative otherwise. Then, eq. (30) is satisfied only if states with and are included in the spectrum. As a consequence, the conformal weights of the first excited state cannot be larger than and Hellerman (2011).
The condition on considered above must be modified slightly in order to constrain the conformal weights of charged states. Following ref. Benjamin et al. (2016), our goal is to find operators such that,
for the smallest possible and . The first and second conditions tell us that there is at least one positive contribution to eq. (30), that of the vacuum, which can only be cancelled by the contribution of charged states. On the other hand, condition (iii) tells us that the charged states necessary to satisfy eq. (30) obey an upper bound on their conformal weights, namely and . The behavior of the functions considered in Sections III.1 and III.2 is illustrated in fig. 1.
iii.1 Bound on
Let us first consider the bound on the conformal weight of the lightest charged state. Note that it is not difficult to find an operator satisfying condition (iii). The simplest example is
For this choice of the function is not positive definite, however. The goal then is to find additional contributions to such that conditions (i) and (ii) hold. This inevitably increases the value of given in eq. (34) and weakens the bound to the value reported below.
Restricting to contain up to 5 derivatives with respect to we find
where we recall that , cf. eq. (12). The sign of in eq. (35) guarantees that is positive for large . On the other hand, the coefficients are unwieldy functions of the central charge that render positive for all . Their values are given explicitly in Appendix B. The corresponding function can be written as
where possible choices for and are given by
and the and coefficients read
The coefficients in eq. (33) are determined by demanding the strongest possible bound on the conformal weight of charged states at leading order in .999The coefficients, and correspondingly the functions and , are not unique and other alternatives exist that lead to the same large- bound when the number of derivatives in eq. (33) is small, i.e. of the same order considered in eq. (35). While the function is manifestly positive, admits two real roots for positive values of the central charge. In particular, the function shares the behavior illustrated in fig. 1 and condition (iii) is satisfied with given by101010The largest root of eq. (38) yields a bound valid for all . The expression is cumbersome so we only report its value at large .
where the Eisenstein series at evaluates to
Thus, unitarity and modular invariance imply that the lightest charged state in CFTs with algebras satisfies
in the limit where the central charge is large.
iii.2 Bound on
We now find an operator that yields an upper bound on the conformal weight of the lightest charged state. As before, we note that it is not difficult to satisfy condition (iii). Consider for example,
for which the would-be bound is given by
Not surprisingly, the operator (45) violates conditions (i) and (ii). The latter demand the addition of several terms to which, once again, weaken the would-be bound given in eq. (46) to that derived below.
Up to normalization, the coefficients of the first three terms in eq. (47) guarantee that the contribution of the charged and uncharged states to is positive for large and . The remaining coefficients, whose values are given in Appendix B, depend on both central charges and render positive for large c (see below). The corresponding function reads
where and are given by
The term in eq. (49) is -dependent and given by
The fact that in eq. (47) involves twice as many terms as in eq. (35) follows from the nontrivial dependence of , for any integer , on both and . This can be ultimately traced back to the non-abelian and nonlinear nature of the algebra, i.e. to the -dependence of the transformed partition function (28). Clearly, the function is positive and satisfies conditions (i) and (ii) for , i.e. for all . Furthermore, is proportional to for large and admits two real zeroes. Hence the function features the dip shown in fig. 1, and condition (iii) is satisfied with given by
Thus the conformal weight of the lightest charged state in CFTs with algebras is bounded by
in the limit where the central charges and are large.
We conclude this section by noting that in the presence of an additional algebra, denoted here by , the modular transformation of the partition function may be modified by terms proportional to , the chemical potential for the charge. In this case the function in eq. (32) receives corrections proportional to . The operators , in eqs. (33) and (47) are blind to these modifications, however, since they are evaluated at the self-dual point , . This implies that the bounds (44) and (54) hold also for theories with an additional algebra, the latter of which is necessary in the duality to higher spin theories discussed in Section I.
iii.3 Bound on
Let us now use the modular bootstrap to prove the existence of a state whose mass-to-charge ratio is bounded from above by . In order to achieve this, the prescription given by conditions (i) – (iii) must be modified in the obvious way. We now look for operators such that
for the largest possible . Conditions (i*) – (iii*) tell us that the constraint imposed by modular invariance on the partition function, namely eq. (30), cannot be satisfied unless the spectrum contains at least one charged state with .
In principle, one can construct operators satisfying conditions (i*) – (iii*) so that is independent of the conformal weight. This would result in bounds constraining only the charges of states in the theory. However, we find that this is not possible in CFTs with algebras if the number of derivatives in is small. The obstruction originates from the and dependence of the modular-transformed partition function, which we recall is a direct consequence of the nonlinear and non-abelian structure of the algebra. In particular, when operators with up to two derivatives with respect to and up to five derivatives with respect to are considered, it is not possible to derive bounds that constraint only the charges. This pattern, which seems to extend to higher derivatives in the modular parameter, may be broken by higher derivatives in the chemical potential. In this case one would need to consider the transformation of the partition function to higher orders.
On the other hand, the nonlinear structure of the algebra allows us to construct operators satisfying conditions (i*) – (iii*) with linear in the conformal weight. An operator with at most two derivatives in satisfying these requirements is given by
The corresponding function reads
where is manifestly positive and given by
Since eq. (56) is positive for all , the constraint (30) cannot be satisfied — and a contradiction is reached — unless one or more states exist with . Thus, the spectrum of CFTs with currents must feature at least one charged state for which
Let us now note that the states of CFTs with algebras are constrained by unitarity to satisfy Mizoguchi (1989)
This implies that, with the exception of uncharged states with , all the other states in the theory must scale linearly with the central charge. In particular, if then, in the large central charge limit, the bound (58) reads
Thus the CFT must contain at least one state whose ratio between conformal weight and charge, , is bounded from above by . On the other hand, if the bound (58) becomes
We may expect operators featuring a larger (but still finite) number of derivatives to exist that yield stronger bounds at finite , but preserve the leading order behavior observed in eq. (60). In this context we note that, considering up to five derivatives in and two derivatives in , it is possible to find several operators satisfying conditions (i*) – (iii*) for large enough . These operators contain terms like and but yield weaker bounds in the large- limit.
In a left-right symmetric theory where , the bound on the lightest charged state may be written as
This bound is shared by the lightest charged and uncharged states in other CFTs without supersymmetry Hellerman (2011); Friedan and Keller (2013); Qualls and Shapere (2014); Qualls (2015b); Benjamin et al. (2016). The analyses performed in these references are also based on eqs. (1) and (2) with either a small or large — but always bounded — number of derivatives. Therefore, we expect a more systematic approach, one where the operators , contain a greater but still finite number of derivatives, to leave the bound unchanged at leading order in . On the other hand, we do expect such an analysis to yield a stronger bound at subleading order in the central charge.
except that conditions (i) and (ii) do not hold. Hence, the operators and must contain additional terms that render the and functions positive. If the number of , derivatives in these operators is bounded from above, then some of the additional terms must be proportional to and they are responsible for weakening the would-be bound (63). The reason for this is that vanishes when is even and this reduces the number of polynomials in and we can use to make and positive. A priori, this is not a problem if the number of , derivatives is unbounded, in which case terms proportional to may not be necessary. Thus, relaxing the assumption on a finite number of derivatives before taking the large-c limit, as done in ref. Collier et al. (2016), may lead to the stronger bound suggested by eq. (63).
We conclude with comments on the generalization of our results to CFTs with algebras, the latter of which feature currents of dimension (see Appendix A). As before, we would like to constrain the conformal weights of the first excited state charged under the dimension-3 current. Therefore, we use the partition function defined in eq. (15).111111A similar setup was considered in ref. Gaberdiel et al. (2012) where the high temperature limit of the partition function was shown to match the free energy of higher spin black holes Kraus and Perlmutter (2011). The modular-transformed partition function is no longer given by eq. (28), however. This is to be expected, as the commutator features contributions from the modes of the dimension-4 current , cf. eq. (68). Following the steps described in Section II and Appendix A, the partition function now transforms as
The term in eq. (64) gives an additional contribution to in eq. (32) which is proportional to the charge corresponding to the dimension-4 current. This term leaves the contribution of the charged states to and in eqs. (36) and (48) unchanged, so condition (iii) still holds. In contrast, the contribution of the uncharged states is no longer positive due to terms linear in . Thus, the and operators given in Section III satisfy conditions (i) – (iii) only in the sector of zero -charge and the bounds derived in the previous section cannot be extended to CFTs with algebras. To derive similar bounds, it may be necessary to extend the transformation of the partition function to terms quartic, or higher, in .121212Note that turning on a chemical potential for the dimension-4 current, and considering only terms linear in in the transformation of the partition function (64), does not help in removing the -dependent terms from and . In this case, additional currents beyond the dimension-4 current featured in eq. (64) will contribute to .
It is a pleasure to thank Nico Wintergerst for helpful discussions and comments on the manuscript. This work was supported by a grant from the Swedish Research Council.
Appendix A Modular transformation of
In this Appendix we derive the modular transformation of for CFTs with algebras. For generality, we consider the algebra which features an infinite number of currents of dimensions in addition to the stress-energy tensor. These algebras are characterized by a free parameter such that, when is an integer greater than two, it is possible to truncate the currents with and reduce the algebra to .