Universal Bounds on Charged States in 2D CFT and 3D Gravity

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JHEP08(2016)041 Springer June 3, 2016 July 18, 2016 , a : August 4, 2016 : : Received Accepted Published 10.1007/JHEP08(2016)041 doi: [email protected] , and Shamit Kachru b Published for SISSA by [email protected] A. Liam Fitzpatrick , a and provide examples that parametrically saturate this bound. c . 3 Ethan Dyer, a 1603.09745 The Authors. c Conformal and W Symmetry, Global Symmetries

We derive an explicit bound on the dimension of the lightest charged state , [email protected] [email protected] Stanford Institute for Theoretical Physics, Via Pueblo, Stanford, CA, 94305Boston U.S.A. University Physics Department, Commonwealth Avenue, Boston, MA, 02215E-mail: U.S.A. b a Open Access Article funded by SCOAP ArXiv ePrint: We also prove that anya such minimal theory must lower containdimensional bound. a theories state of with We gravity. charge-to-mass comment ratio above on the implicationsKeywords: for charged states in three Abstract: in two dimensional conformal fieldthe theories bound with scales a global with abelian symmetry. We find that Nathan Benjamin, gravity Universal bounds on charged states in 2d CFT and 3d JHEP08(2016)041 6 18 4 5 – 1 – 14 16 8 10 4 20 19 ]. Turning on a small negative cosmological constant, the 1 13 13 11 17 1 Holographic approaches allow one to make this intuition precise and to extract quanti- B.1 Perturbative argument 3.4 Supersymmetry 4.1 Extremal lattices 4.2 Gravity theories with a large gap 3.1 Hellerman-type bound3.2 on charged spectrum Bounds mass on3.3 gap charge-to-mass ratio Bound from asymptotic growth 2.1 Derivation of transformation tative predictions by turning aof poorly-defined UV-complete question, that theories of of howary gravity, to into of formulate sharp the space-time. questions space implications about In of observables analyticity flat at and the other space,the axioms bound- the of dynamics the corresponding of S-matrix observable provide the a is theory path the to [ constraining S-matrix, and the about the spectrum or dynamicson of gravity some at high knowledge energies of wouldfor necessarily the the be low-energy contingent behavior spectrum. ofthe the Conversely, behavior insisting ultraviolet of on description the various ought theory criteria at to low-energies. impose non-trivial constraints on At low energies, theories of quantum gravityas are effective straightforward to field formulate quantitatively theories.gauge However, theories, the at high universal energiesfrom coupling the gravity of matter has content gravity of no the implies decoupling theory. that, limit This might unlike where lead it one to can expect be that precise separated statements C Transformation of characters 1 Introduction 5 Discussion and future directions A Modular forms B Current algebra and flavored partition function 4 Large gap examples 3 Bounds on the charged-spectrum gap Contents 1 Introduction 2 Modular transformations with currents JHEP08(2016)041 , ], m N 1 G 19 4 . L m for the mass pl CFTs, even those m Qm all ] and systematically improved ]. Such constraints clearly can 10 13 – 5 . Roughly, this bound is 3 ]. In fact, the structure of CFTs is sufficiently 4 – 2 – – 2 ], classical gravity predicts that it should collapse ] can therefore very roughly be summarized as the 18 [ 3 in the boundary CFT. Now, one can ask not only about 14 , J 10 (1) in units of the Planck scale. O ], where a rigorous and universal upper bound was found on the may not be well-described by GR at high energies, or where the . All graviton degrees of freedom are purely boundary excitations 3 17 2 – 14 is order /CFT 3 3 for black holes in AdS ], but so far a rigorous proof is lacking. A key result of this paper will be to is Newton’s constant. When the mass of a state is greater than the threshold N 21 1 of the lightest bulk degree of freedom in AdS G – N 8 L G 19 m ≥ This result has a clear connection to the Weak Gravity Conjecture (WGC) of [ Our main goal in this paper will be to extend this technique to the case where the A remarkable result along these lines was derived in [ The most powerful such methods are found in the case of the correspondence between BH arguments have been made thatmass light charged [ states must beprove present such below an some upper bound bound. in which exists in multiple forms but in any case is an upper bound on the spectrum of energies ofquestion is states, whether but there also must aboutthey be their can charged be states charges. made in A arbitrarily the heavy veryat theory and simple fixed at therefore but low finite effectively decoupled energies, elusive energy. from or Charged anywith if description black instead a holes gauge would field, seem toinconsistent, but exist since it as is in classical not such solutions obvious in a that GR theory a charged theory black with holes only can neutral states never is be actually produced. Various whose bulk duals in AdS curvature of AdS bulk theory has athere gauge is field conserved vector in current addition to gravity. This is dual to the assumption that in all theories ofa quantum CFT gravity; is in fact, usuallyis most discrete continuous. are whereas The not, thestatement results since spectrum that the of the of spectrum [ spectrum solutionscontain of black to of states holes general in states near relativity in threshold. (GR) theories Moreover, of the gravity result must, applies at to a bare minimum, mass where m and form a horizon, andno we guarantee can that suggestively any call specific it solution a to “black Einstein’s hole” equations state. is A actually priori, a there physical is state conclusions about the high-energylow-energy spectrum. dynamics as One can a eventheories, function derive i.e. properties of that that the make are only assumptions common very about to basic all the assumptions gravitational suchupon as numerically unitarity. in [ three-dimensional gravity in AntiTheories, i.e. de AdS Sitter spacewhose dynamics and are completely two-dimensional fixed Conformal byMoreover, the Field infinite modular conformal invariance symmetry of in two thehigh dimensions. theory energies at and finite the temperature relates spectrum the at spectrum low at energies. This makes it possible to draw sharp time structure is radicallydynamics different, of and a the Conformal boundary Fieldrigid Theory observables that [ now it comprise isall the possible theories full to of gravity derive inalso universal AdS be constraints at applied on low to and the the high dynamics full energies space and [ of spectrum CFTs, of which is a central area of study in its own right. theory apparently remains nearly unchanged locally; yet the global Anti de Sitter space- JHEP08(2016)041 , 3 At 3 (1.3) (1.1) (1.2) shows appears k k is 1. . J 3 ] for discussions 22 ] for discussions of how 23 , as one might expect , for non-chiral (which c due to peculiarities of three c 12 . 3 of the current ∼ k . So the actual value of , ], and for a bulk scalar the weight k 24 √ [ . From the CFT point of view, where . c 1 k Q/ N and not √ 1) = 1 without loss of generality. We thank and is completely removed by canonically AdS G = ` c 2 6 The bound is numerically determined k 2 O 3 0 Q/ k x . An analogous mismatch arises in the 2 Q + ∼ c 3 = ∼ ( c , which is fairly small, but our main point π → 3 with 2 π (0) Q 1 Q √ J 4 + ) . , x ) is at At large central charge c ( 6 – 3 – π J 1 2 AdS 1 1.1 √ + 1 (supersymmetric) < ` 2 4 of modular transformations is actually used) or if c m 12 1 τ > and charges ≤ N k vacuum 2) = √ Q ∆ → − − is normalized so that the level mG J/ − (1) and is a rigorous proof that there must exist some state τ , the relation between mass and charge of extremal BHs is qualitatively . explicit and replace 3 Q = vacuum O 0 ∆ k knowledge of the charges that arise in the theory, the central charge ∆ J vacuum 1) supersymmetric theories with a U(1) current. Here, we can ], and it is unclear if this is a short-coming of the methods (for < Q is the weight of the vacuum. In terms of AdS quantities, this ∆ − , → for the lightest charged state. when − N π c J ∆ 14 12 , √ N N 4 1 mG − G a priori )(∆ = (1 Q 10 4 mG = ) CFTs our upper bound on the weight ∆ of the lightest charged state in . . Furthermore, boundary currents in 2d are dual to Chern-Simons gauge fields in AdS N c 2 of some state in the theory. vacuum Q m = ¯ ∆ Q ∼ c − vacuum M , it implies that there exists a state in the theory with charge to mass ratio satisfying, c It is notable that the upper bound ( Equivalently, one can keep Recall that in GR, the CFT central charge satisfies One might expect that the WGC should be qualitatively modified in AdS 3 2 1 satisfies (∆ we do not assume any up only in thenormalizing two-point function the of currents theto current, be invisible toDan the Harlow CFT for data encouraging we us are to emphasize using, this. and we can set which have no bulk degrees ofification freedom. of On extra the dimensions, other hand,of one if which might the versions expect WGC of is these the sufficientlyWGC peculiarities robust WGC might under are to compact- be robust be to modified irrelevant.modular compactification See in invariance of and [ dimensions, AdS/CFT thus and contexts. provide [ an In independent approach any to case, studying our the bounds WGC in are AdS a rigorous consequence of dimensions. In particular, indifferent, AdS there are physical theoriesalso that consider saturate stronger the bounds weaker onin bound. the gap the Partly to case the motivated lightest byconsider of charged this, a state we holomorphic that quantity can called bebound the obtained on elliptic the genus, and weight indeed of we the find lightest the charged improved state, in the theory with from the classical threshold forbounds black found holes in in [ AdS instance, only the subgroup This bound could alsofor likely a be better bound improved closer withis to further that a effort. ratio it of is One 1 parametrically than might expect or hope where ∆ translates to from the modular bootstrap andfind can a likely bound be improved on withlarge better numerics. We will also have, e.g., the theory asymptotes to and charge JHEP08(2016)041 , 5 ; in , we R (2.2) (2.3) (2.1) 4 . While pl is from the c M we examine a 4 ] argues for a stronger bound 21 , It is also possible to derive , . d ) 6 + 0 z J , we discuss the transformation τ, z y ). As we explain in detail in ap- cτ ( 2 24 Z 2.3 = ¯ c/ 5 0 d − 2 z + 0 ¯ z ¯ τ c ¯ L c ¯ q → − d 24 2 + c/ cz cτ − 0 – 4 – L , z q πik b d , one can consider the partition function graded by e J + + tr (1) at the Planck scale, whereas [ . For conciseness, however, we will simply refer to the abelian without making additional assumptions. In section ) = O ≡ cτ aτ R 0 c ) , z = 0 0 . The starting point of our analysis is that under modular τ τ, z τ ( ( πiz Z Z 2 → e is the level of the current algebra. τ k , we derive our bounds on the charged spectrum. In section ) are related to the central charge of the theory. The variable = 3 y 2.3 ) and in ( and 2.2 k ] for recent interesting arguments along these lines. In particular, all examples we present 25 πiτ , nor 2 e 21 c , this transformation property is independent of the particular theory, and only = B q The outline of the paper is as follows. In section One might also wonder how much more the gap to charged states can be lowered in Neither The explicit form of the partition function is of course theory dependent. It is only the transformation See e.g. [ 6 4 5 transformation in ( property which is universal.fact, since The our transformation analysis makes appliesdistinguish no as between assumption well the about to cases thegroup representations the U(1) as that non-compact “U(1)”. and arise abelian in the group theory, it does not pendix depends on the universal structure of U(1) currenthave algebra. a coupling for the gaugeonly field when that this is gauge coupling is small. tive example. 2.1 Derivation ofMost transformation of our analysis inpartition this function paper under is a based modular on transformation, the ( transformation property of the flavored the partition function transforms in a universal way: We will discuss the argument for this transformation below, and work through an illustra- where transformations, 2 Modular transformations withIn currents any CFT with athe conserved charge current of the current: of the partition functioncharacters. in In the section presencepresent of specific models a to chemical demonstratebounds potential, that that our and bounds are the are possible close corresponding we at to discuss large saturating potential the future optimal directions. principle by any methods, not necessarilyprove on those general used grounds here. that charged Inthis states particular, should may one enter might indeed parametrically hope below prove to few true counter-examples after that restricting demonstrate to it cannot certain be classes true of in theories, complete generality. JHEP08(2016)041 (2.6) (2.7) (2.4) (2.5) ) of the cur- z i ( = 1. There is k dzJ m, n + 1. Under the S ] for free fermions, | . γ } i τ H 29 nR , Whichever method one → R − m, n {z p 28 . 7 | τ ]. R 2 R R 2 p m , , 2 p | ¯ y 2 ) 27 ) L , p = = b πi y 26 2 τ, z i i ( 2 R 2 + p k ¯ q ( bos a π 2 L m, n m, n 2 Z | | p e 0 0 q 2 j ¯ ¯ τ ¯ L ¯ z k Z X ∈ − 2 τ a X z 1 m,n √ 2 – 5 – πi | , e ) = i τ 1 ( b` η + ) = | 0 2 m, n | , , z ). We have 0 } πa` i τ ) = τ − ( nR ( e of the torus, as shown in [ L τ, z + {z p m, n bos ` γ iτη ( | , the transformation of the partition function can be easily X Z 2 L R √ 2 m bos 2 p | /τ Z 1 = 2 theory. ) = = = sums. Combining this with the modular transformation property of N i i /τ n 1 → − − ( τ m, n m, n ). Here we have normalized our currents to have level | | η and 0 0 j 2.3 L m , the primary states under the U(1) current algebra are labeled by two integers, R ) on different cycles z ( , for momentum and winding. These states satisfy, function, ] for any chiral i J The flavored partition function is then given by, Let us review this explicitly in the case of the free boson. For a free boson on a circle η We thank Herman Verlinde for bringing this argument to our attention. Their argument is in several 30 7 m, n with a U(1) globalupper symmetry. bounds We on present the constraints weight of on the what lightest charged charges state, must and appear, theways and ratio more of satisfying weight and tothe charge. elegant, partition and function. more suggestive of how an argument might be generalized beyond or [ 3 Bounds on the charged-spectrumIn this gap section we derive bounds constraining what charged states must appear in a theory establishing ( nothing special about ourbeen choice worked of out free explicitly bosons, in and other indeed examples, this for transformation instance has see also [ to both the the This partition function istransformation, invariant under thecomputed transformation by applying the Poisson resummation formula, | rent prefers, once one knows that thesufficient transformation to property derive is it theory-independent in it a becomes particularly simple theory. of radius this transformation directly using algebraic properties of the modes JHEP08(2016)041 τ . , it U(1) (3.2) (3.3) (3.4) (3.1) gap and = 0 × ∆ z < + 1 4 i out of of the modular , in addition to Q , to the flavored 2 π i . z α 4 π Q } / 8 y , 1 ) i 09. Thus the theory ¯ τ ¯ h − . i ¯ q . In order for the sum 1 > ¯ h 6 i i = 0 π h − 2 i Q 4 τ q . i 3 / Q derivatives: combining the h i 15 π 1 ), to bring down factors of π + 1 . Q z ), that there must be charged z 4 32 i < . This gives 3.1 Q . i gap 2 i 2.2 − ( , i 2 i ∆ Q πi = ∆ 2 , with the following properties: Q π e i τ 344 > = 0 2 . 2 − τ i = Q e ) 0 are charged states with ∆ iπz ∆ e π = 0, i , τ ¯ ), such an operator gives a positive con- τ,z < 2 if if ∆ X ) z τ, {z − − ) ( e i 3.1 ¯ τ = 0 = to zero to obtain constraints on the charged ,Q F / τ, z i i z ∆ ( , , , ¯ h z – 6 – X i 0 0 F Z πi ( = 2 = > > α ,τ τ +2 i ) = 1 ) ) iπz /τ = =0 e ) ,Q ,Q z . However we will get surprisingly strong results using i ,τ

h ∆ ∆ i − C − =0 F F F ) = 0 for simplicity, we can write the constraint of modular z z ( ( 2 z vacuum i

z i ∂ α α F Q ( ( F π ), and evaluate at 3 2 z α 2 πi ∂ 2 , except for the interval, 0 /τ, z/τ e 3.1 − 2 i 1 i | i Q X − = ( i 2 ,τ Z X derivatives of the modular relation, ( 1 π . This expression is negative for all 8 =0 i z = z ¯ h

i 0 = F + of each state, and then set 6 z i i ∂ h 3 Q π = 1 i 60 To prove a bound on the gap to the lightest charged state, our strategy (similarly We can take i X Acting on the modulartribution invariance from equation the ( vacuum whichstates. must Since be the canceled only by states a that negative have contribution from some to most bootstrapderivatives approaches) evaluated will at the be self-dual to point construct a linear operator This is positive formust all have some states with charge in this range. where ∆ to give zero, thethe theory neutral must states. therefore haveconstraints One some from can states six do and with even two better derivatives, one by ﬁnds, taking more spectrum. As atransformation simple equation ( example, take two derivatives with respect to Here the sum ispartition taken function. over individual Stronger states,characters, constraints each discussed contributing could in be appendix the derived simpler using single the state full expressions. Virasoro the charge It is immediately clear fromstates the in transformation property, the ( theory. Ascharges a must show warm-up, up. it is Settinginvariance worth ¯ as writing down some simple bounds on what 3.1 Hellerman-type bound on charged spectrum mass gap JHEP08(2016)041 i α F and is an (3.6) (3.8) (3.5) c . Fur- 12 κ gap = κ ≥ − , . π =2 ,β + 64 =0 are contours of the z κ κ

1 ) + 5 , 2 β, z 3 ( i πκ − F + 32) + 3 π An optimal analysis would seek κ 2 z + 24 8 ∂ (3.7) 3 3 128 β + 7 . Evaluated on the contribution κ ∂ , c 2 2 − 2 (∆) , 2 +¯ π 3 24 c O a πκ πκ πκ 16 4 . ∆)) + + ≡ + 64 22 2 π 3 β − κ κ + 12 . Also shown in figure 1 (∆) ∂ − κ ∆ 1 κ 1 0 3 16 3 , 64 ( 2 1 p means that any upper bound on the weight of a state is 3 κ – 7 – κ π 2 2 a − 2 2 + 13 κπ O 2 π Q Q π 2 , and κ + ≈ + π (8 > 2 ∆ z 64 ) is manifestly positive for all charged states as well iβ 2 πκ 2 κ π ∂ (3 + 4 3 c ,Q 24 0. Thus, an upper bound on the gap is given by the − − 2 π β ) in figure (∆) , p ∆ ∂ 3 2 3 κ (∆) + 1 ) + 2 2 + 0 < = F κ , κ + 24 p (∆) = 0: h ( 1 p 3 − √ 1 κ 3 πκ ¯ τ a α = 2, where one can see that the polynomial is negative only π p κ κ , ∆ 2 2 ≈ κ π + (∆) 32 κ, π iβ 2 π 1 2 2 ∆ ) = β − p (3 + 3 − κ π ∂ 16 2 ( = e π 0 (∆) which is a manifestly positive polynomial for ∆ ) at , π 1 0 1 24 produces the following polynomial: τ 1 p ,Q a gap 1 1 128 64 24 − ∆ α ) = ∆ F over the space of linear functionals subject to the above constraints on = = = ≡ = ( ,Q (∆) = 1 + (∆ + (∆) = α ∆ ) 0 2 0 2 0 1 , , , , i and for sufficiently small ∆. For a left-right symmetric theory, p p F ∆ 1 1 3 3 gap F ( π a a a a ( 2 Q α e α = 0, this gives Incidentally, the unitarity bound Q 8 . However, even with a small number of derivatives it is possible to obtain a functional This is plotted aspolynomial a function of for non-zero also an upper bound on its charge. is also manifestly positive,when so ∆ is veryin large. the The range only possible oflarger negative ∆ of contributions the where come two from solutions charged to states At thermore, at sufficiently large ∆, from a single state, where we have taken upper bound on the weightto of minimize the ∆ lightest chargedα state. satisfying them. Already quite non-trivial bounds are provided by the following example: immediately implies that such states must be present in the theory. This means ∆ JHEP08(2016)041 . i F Q (3.9) �� = 2; the κ defined in for large α �� /κ ) κ ( gap �� , show in red, dashed. κ Δ/κ κ=� = ∆ ) is negative for /κ �� . 3.6 at large c 1 κ ) in ( -�� O between the vacuum and the lightest ,Q ∆ κ + F ( + π 3 α 2 -�� ∆

gap

π �� + � 2 e c 6 ≈ – 8 – �� vacuum . The slope asymptotes to 2 ∆ c +¯ 24 − c �� ) κ ≡ ( κ , so the bound on the gap between the lightest charge state and gap c 12 ∆ κ 0. We will be most interested, however, in obtaining a bound for �� − > ) ,Q ∆ : an upper bound on the total gap ∆ F �� ( α Left , . Q 9 . c : the shaded region is where the polynomial

��

� � � � � � � �

We often think about theories with a given level and a quantized, order one U(1) charge. In these �� κ + Δ 9 large cases, our bound onratio, the and gives weight a of boundthe the that previous lightest scales subsection, charged with we state theadditional will central assumptions immediately be charge. on translates able quantization. By into to applying a derive the a bound linear similar operator on bound, techniques the of that holds more generally without any So far we have investigatedthe the lightest bounds charged on state. thegap gap It in in is charge, the interesting and ratio to thethe of also previous gap weight section ask in already to the what provides weight charge. weenough a of can bound For say on ﬁxed about this central ratio, a charge, as maximal for the large operator enough ∆ or small than we have usedcan here, be and obtained it this would way. be interesting to explore3.2 the optimal bounds Bounds on that charge-to-mass ratio the vacuum is More restrictive bounds can certainly be obtained by considering more derivatives of Right right edge asymptotes toIn a unitary vertical theories, line there (shown must in be blue, at dashed) least at one ∆ state inthe the vacuum shaded is region. at ∆ = Figure 1 charged state, as a function of JHEP08(2016)041 (3.13) (3.15) (3.16) (3.10) (3.12) (3.14) (3.11) . 2 κ 3 . − 3 2 √ ∆ . . In this case, πκ . κ , π κ κ, κ 3 √ . , π 4 − =2 ∼ 2 2 ,β ≤ Q 2 ∆ (∆) ∈ 2 1 ) =0 Q ) has a minimum in the allowed + ˜ 3 z p

κ ∆ 2 − ) κQ )) vacuum 3 3 Q 2 3.15 κ π )+2 . times a relatively simple polynomial: ∆ π 1 ∆ 4 κ , β, z 3 − 3 ( − ∆ + as , 2 κ i , and thus such states have a very small κ π 3 κ vacuum 3 3 F 2 (∆+ ) + 2 2 π α π 3 (3 2 2 2 (∆) + z κ ∆ − κ 4 ∆ π 2 κ 0 3 κ ∂ e / ( 2 ˜ p ∆) 3 4 − π π 1 π − . )) − 4 κ 2 , we have, – 9 – 2 κ κ ∆ ∆ 4 π > κ ≈ (( (∆ + ) + 2 ≈ π ∆ 2 2 2 ) (∆) + (∆) κ ) 0 1 − q − Q ) + ,Q e p p ,Q ∆) i 2 ∆ ∆ ≤ /Q F κ ) − ( F F (∆ + ( ) = ( (3 α κ 2 large, and divide up our analysis into the three regimes ˜ α ˜ ( α κ vacuum ,Q ( (∆) = (∆) = ∆ ∆ κ ∆) ∆ (1). ∆ 0 1 π | 2 π ) = π vacuum ˜ ˜ 2 vacuum 4 p p F i 2 π 4 − e ( Q − e F | ∆ ∆ ˜ α ( κ ∼ O ≈ ( ˜ α − − , we have, ) κ /κ ∆ ,Q ≥ ∆ 2 F Q ( ˜ α ∆ , and ∆ π κ 2 e independent term is again positive, while the second term can be negative for ∆ κ, Q The most interesting states are those with ∆ For small ∆, that is ∆ For large ∆, ∆ To this end, deﬁne a new linear functional ˜ Indeed we see that thecharge. largest gap in weight per unit charge scales linearly with the central Though not uniform inrange of ∆, ∆. the The right quantity hand of interest side is of thus ( bounded by, The suﬃciently small ∆. For the total expression to be negative we must have, This is negative onlymass-to-charge for ratio states (∆ with which is positive for all states. cusing on howcan the therefore mass-to-charge look bounds mainly∆ scale at in the large central charge limit. We As before, we want to investigate for what states these polynomials can be negative, fo- Acting on a single state, this again produces JHEP08(2016)041 . If gap 10 (3.19) (3.20) (3.21) (3.18) (3.17) ˜ ∆. transforms as 4 W . Then the sum in . gap . . ) j =0 j z ˜ ∆

β 2 +∆ growing with large − i + ∆ )) e | 1) theories. This method is i τ (∆ , ˜ ˜ ∆ ∆ ( . β . ) Z D D − ¯ | τ 2 z e 12 ∂ = (1 ), it follows that τ, ˜ j 4 c/ ( ∆ = ∆ 3( ) = 0 4 has − Q N 2.1 β − 0 ( W 0 gap + 4 , X W ) 0 j 2 =0 W d ∆ z Q 0

≥ i 2 ) + → ˜ ¯ ∆ τ Q β . 6 cτ τ, ( – 10 – ˜ ∆ ) = − , we have, ) = lim β Z = ( β i 4 π 4 z β − ( 2 ( e ∂ 0 Q 0 0 ) ˜ ¯ τ ∆ ¯ τ W , ). In fact, the situation is even better: even if this particular W iβ/ 0 C i,j τ, τ = X ( = ˜ ∆ 3.17 1 2 4 →∞ X Z lim τ β 2 W = π β 12, then, 4 ) = ) = 12. We will show that this is inconsistent for large enough ∆ above, we will assume transformation, ¯ τ β 4 ( c/ 0 S 4 τ, ( > c/ W − 4 W W W gap gap , 0 for imaginary 4 ˜ ∆ = ∆ W In order to see this, it is instructive to consider an abstract function which is invariant Assume for contradiction that the first charged state has weight ∆ starts at This assumption is tantamount to an asymptotic non-cancellation assumption between the combination 4 10 of partition functionscombination appearing had in a cancellation, ( weusing could these construct higher higher order ordercancel modular to modular objects, invalidate forms. the and argument.a rerun Obviously, In the combination we argument that this could did repeat case,cancellations. not this a cancel. as different many Thus times particular to as combination invalidate needed would this until have reaching argument would to require an infinite number of To make contact with we further take ∆ under the real modular In the last line we have written the sumW over weights, Considering Note that this function vanishes ifmation there properties are of no the charged states. flavored partition Using function the ( modular transfor- in a 2d CFT.than More using accurately, the it presence of is the∆, analogous vacuum we to to will ascertain the show the inverse asymptotic thatlight of growth a charged of Cardy’s state. non-vanishing states argument; asymptotic at We rather large charge will density argue implies for the this presence by of considering a the following object, We next present an alternate methodsubject for to deriving a a bound non-cancellation on hypothesis.assumption, the Although gap the this in advantage the argument is charged will sector, argument require both a we that mild will extra it usemore is to analogous very derive to simple, the stronger original and bounds argument it for due is to Cardy similar for in the style asymptotic density to of an states 3.3 Bound from asymptotic growth JHEP08(2016)041 − . 6 gap c/ , (3.24) (3.23) (3.22) 0 1) ) trans- − ] tells us 31 τ, z ), and thus ( β ( − NS 0 Z W . and , , ) ¯ c 24 , ¯ c 24 , − , invalidating the applicability 0 4 − τ, z ¯ ¯ c 0 ( 24 L ˆ ¯ ¯ c W ¯ 24 L q − . Since the modular discrimi- behavior of ¯ 0 q + NS 0 − ) is optimal. One motivation J ¯ 0 . 0 L J ¯ y β ¯ L q ,Z 3 y ¯ c 0 q ) / 24 3 J c 0 → ∞ 3.23 24 / J − y ) 1 0 − . y β τ, z c β 0 24 L , the final value theorem [ )) ( ) ( c 24 L q 6 + 1 ˜ 1) supersymmetry and a U(1) current. β 4 − − ∆ β R q ( 0 R , ( − c/ 0 R D 4 F L W ) under (some congruence subgroup of) ,Z ∆ F L q + ˆ ) ( W q = + R L = (1 R L F F 2.3 = F F τ, z 1) 1) ( ) = , not to be confused with the weight ∆, only has zeroes N 1) 1) – 11 – + β − − R 2 ∆ ( ( ( − − The advantage of considering the elliptic genus is vacuum Z 4 ( ( π β , that we are interested in, we divide by the mod- ) to the appropriate power to create an invariant 4 has to grow as ˆ ∆ 4 W π 12 . 4 2 − W is given by the small , we must have a maximal gap to charged states of ¯ c 4 ˆ R,R R,R NS,R NS,R 24 W 3 ˆ ˜ W iβ/ ∆ gap β/ ( = D − ∆ 0 24 e ¯ η L ) transform as ( ) = Tr ) = Tr ) = Tr ) = Tr ∼ 3 / τ, z τ, z τ, z τ, z 3.24 1 ) = ( ( ( ( β + − )) R R ( + − NS NS β Z Z , as otherwise we would introduce extra poles in ( Z Z ∆ ) as the Laplace transform of ∆ β ∼ ∞ ( ˜ ∆ behavior of 0 1) supersymmetry we can define a holomorphic quantity called the elliptic 0 , 6. = 0, contradicting our assumed growth. This tells us we must have ∆ W → c/ ˜ ∆ β = (1 11 D ≤ ). In particular, the functions N Z , →∞ The functions in ( To apply this to the function, In running this argument, it is crucial that Note, here the left-moving fermion number for the NS vacuum is conventionally defined as ( ˜ ∆ vacuum 11 12 SL(2 at the cusp of the final value theorem. In all of thesymmetric functions ground above, states the at right-movingthat sector it gets is contributions a onlythe holomorphic from gap modular super- to form, the so lightest we charge can state. use the power of holomorphy to bound genus. This theoryspin has structures fermions, we so canfollowing when consider elliptic we genera, depending put on it boundary on conditions. a torus, We there thus are define four the different 3.4 Supersymmetry As we have mentioned,for we this don’t conjecture believe comesconsider the the from case bound considering of ( theoriesWith a with 2d CFT additional with symmetry. both We can The above argument tells usnant that behaves as ( function: that the large lim ∆ ular discriminant, Thinking of JHEP08(2016)041 , ) ), τ τ ∞ ( ( i = 0. R 4 4 (3.28) (3.29) (3.25) (3.26) (3.30) (3.27) = E τ W = 0 can τ as . τ 4 ) ) and − τ τ ) has no poles ( τ 0 2 ( about . E . NS w 4 above the (NS-NS) =0 − =0 z W τ c

z ) must have a term of 12

2 τ 2 1 (mod 2) ( )) )) R 4 ]. ≡ τ, z W 32 d τ, z ( [ (2), note that about ( . 0 + 4 ) − + NS R , E Z Z ) , z , 2 , 2 z z τ ) , c ( ∂ ∂ 4 ) diverges at most as + 1 , , and 3( 3( 1 τ τ E τ ( 0 2 ) ) 2 ( respectively. These are deﬁned as − − − c R 4 E θ /τ, z/τ + NS + W =0 =0 1 Z z z 2 NS

4 − ) ) + 1. ) ( πic τ 24 above the vacuum, then ; this then means that there must be at least and diverges at most as 1 (mod 2) W 2 ( c − 12 0 2 NS 4 c − τ, z ∞ (2). Moreover, the only contributions to τ, z 1 – 12 – 12 ≡ τ i ∞ ( Z e ( 0 E i (2) is generated by the functions 0 (mod 2) 0 (mod 2) 1 b + R (2), and Γ 0 c − = NS 0 = Z + ≡ ≡ ) = Z ) = ) = 4 z τ Γ 4 z τ τ , ∂ ) = ( ∂ ) ) ) τ, z τ, z τ R , b , a , c 4 ), we see that Z ( ( ( ) ) ) , R + ), one can show τ, z R 4 W Z Z Z − τ, z NS ( , , , Z ( . 3.28 W Z + R 2 − SL(2 NS 3.27 Z c , Z SL(2 SL(2 SL(2 ≡ (2) ≡ ) ∈ ∈ ∈ and 0 τ ) = 0. . In particular, any meromorphic function that transforms with weight ( is a weight four modular form under Γ 1 ! ! ! τ ) and ( c τ R ( 4 A R 4 ) also only gets contributions from charged states. In particular, as we’ve W NS c d c d c d a b a b a b 4 W 3.26 τ ( W . Thus using ( ) under Γ ( ( ( NS 4 ) when expanded about ∞ (2) that has no poles at i q 0 W ≡ ≡ ≡ ( 2.3 = θ O Γ τ (2) (2) 0 0 is finite, as the lightest Ramond sector states have weight zero. The only question is Suppose we have a theory with the first charged state at least The ring of modular forms under Γ To see that Now let us consider the following function: Γ Γ under Γ R 4 assumed the first charged stateabout shows up This means it can be written as for some constants, where we define Note that the behavior about vacuum. From ( defined in appendix w be written as a linear combination of products of W This is a weight 4come modular from form charged under states. Γ at Our least basic strategy is toone show charged that state ofvacuum, dimension we one must have above a the charged RR state vacuum. by Thus, relative to the NS-NS These functions transform into each other via forms as ( JHEP08(2016)041 c 12 , a (1) R 4 (4.4) (4.1) (4.2) (4.3) O ∼ , and (3.31) c W vac ∆ − ]. ). Thus, in q 39 ( . O , 2 chiral bosons compactified 2 ~v c = . ~v may hold in general, even in the h + 1 c 12 . . 6802. c ) 24 . Such lattices are known to exist for ∼ z c Z ( + 2 Λ 1 = ) can start at is k > ∈ ∈ c , 12 k i∂φ k 3.30 − they have not been constructed. As mentioned = vacuum currents, under which the only charged operators – 13 – ∗ for k h c + 1 (supersymmetric) ~v ) = , ~v k · z − ) c ∗ ( ,. . . . 12 z 2 ( ~v J even self dual lattice with the smallest norm non-zero ~ φ ~ gap φ ≤ · = 24 , c ∂ c i~v c , Λ e -expansion ( in the gravity picture. It seems natural to conjecture that ~ φ h q N i∂ 1 . Consider any one of these currents, G ) = vacuum 8 − z ~v , form a set of ]. Appealing to more standard string theory examples, we also ∆ ( ~ V φ ~v ∼ 33 ]. This CFT has a spectrum consisting of the vertex operators, − V , is a rank i∂ + 1, and so these examples are tight for chiral CFTs [ c m − ∆ ], however for larger c 38 24 , 36 ≤ 37 – 163264 [ 34 above the NS-NS vacuum, we thus get a bound to the first charged state of ≥ vacuum c c h 12 having length squared, − ∗ ~v gap h 13 The differentials, A consistent chiral CFT can be constructed by considering The improvement by a factor of 2 compared to our non-supersymmetric bounds brings The highest order in the It can actually be shown that chiral CFTs satisfy a stricter bound on the weight of the lightest charged = 1,2, and 3 [ 13 The gap to the firstthus, charged operator is given by the gap in the norm of vectors in Λ state, as well as the differentials, are the vertex operators, above, they do not exist for sufficiently large on such a lattice [ vector, We will be focusedk on the case consider a gravitational theory in flatAdS space, space. and discuss the D1-D5 system in highly4.1 curved Extremal lattices An extremal lattice, Λ 4 Large gap examples In this section wefactors. provide some One examples class ofSuch of theories examples lattices which is realize can given our beto by bound exist explicitly free up for bosons to constructed compactified for on small extremal central lattices. charge and are known not this into line withcorrespond the to threshold masses for BTZa black bound holes, upper since bound dimensionsnon-supersymmetric of on case. ∆ charged states of order charged state must appear byvacuum dimension is at least one above the RR vacuum. Since the RR JHEP08(2016)041 0. in c 6 (4.6) → (4.5) (1). ∼ . Still, O R . As 2d M . Planck 11 , Planck M M ∼ Planck . in 11d Planck units. M , the only mass scale M 11 R ∼ ` = R small, and the only charged states in 11 components of the 11d metric. . This easily generalizes to lower µ 11 , string g , . Planck 3 (1) at a mass scale M G AdS / 2 string 2 O L g ∼ 3 – 14 – = (1). 10 = , O R c , and a BPS bound relates the mass of the lightest Planck R M , and the lightest charge has mass 11 , Planck 14 . We provide two examples below. It is important to stress that in M Planck , the long distance theory is a weakly curved gravity theory in ten dimen- M R Consider M-theory compactified on a circle of radius ∼ At long distances, one then has gravity coupled to an abelian gauge field in 10d flat There is a Kaluza-Klein gauge field arising from the Half-BPS states carrying this charge do exist. They are the Kaluza-Klein gravitons on We think the answer is no. One can easily provide examples of gravity theories which In light of the Brown-Henneaux formula In fact, in 10d we can shrink the circle, thereby taking 14 which is stronger thancurved the gravity Planckian theories. bound, at least not one whichthe applies theory to are all D0coupling; branes, weakly however, which the remain gauge above coupling the remains Planck scale. This provides an example at small string the circle, or D0-brane boundin states the in BPS the formula IIA is string. When space, with a lightestdimensions, charge by at compactifying onshows a that Planck one radius cannot torus, derive rather a than stronger a bound single on circle. the mass This of the lightest charged state This gauge field becomes theBut Ramond-Ramond photon it of is type IIA presentcharged string KK for theory modes as all of values a of given charge to the radius of the circle. For any finite sions. each, our ability to makeexact controlled BPS statements mass depends formulae, as on we extended work supersymmetry inExample and regimes 1. where some sizeAt or very coupling large radius, is the of reinterpret theory the reduces to radius 11d in supergravity. terms At of very the small type radius, IIA one can string coupling, via one might wonder — istheories? a stronger absolute bound possible in weakly curvedare gravitational thought toappearing be at fully consistent, yet have abelian gauge fields with the first charges our bound is sufficient toAdS guarantee units this (the expectation. highest Charged value consistent states with at masses the bound), are at a mass It is expected for awill variety of exhibit reasons charged that quantum matterCFTs gravity are with theories (sometimes) with charge dual U(1) to of gaugecompare weakly fields to curved this 3d gravity, expectation? one can ask: how does our bound 4.2 Gravity theories with a large gap JHEP08(2016)041 (4.8) (4.9) (4.7) (4.10) (4.11) volume 4 T , and the 6 g . The problem of finding c , whereas what we want to . c 1 (4.12) gravity theories, via the rela- ∼ is a coset space 3 duality, coming from the D1-D5 1 5 4 SO(5) 2 . Q Q T . 4 × 5 T . Unfortunately, this example comes . 2 Q × 1 /CFT 1 , 3 5 Q , SO(5) Q 1 1 5 / p 5 /CFT Q 6 Q Q 3 → ∞ 5) g Q 0 , 3 5 branes leaves a worldvolume unbroken (4,4) 1 5 = α S – 15 – Q D v × = p SO(5 3 6 \ g ) 2 AdS ,Q CFTs and is likely challenging. However, at present it Z AdS R c 5; , → ∞ (wrapped) 1 5 Q SO(5 Q 2d CFTs with sparse spectrum and weakly curved gravity. So c examples with a large gap to charged states is similar to the problem 1 and 3 D 1 Our bound is more directly related to AdS Q . Before inserting the branes and taking the near-horizon limit, the moduli 4 T is in a hypermultiplet and we are free to choose its value. Validity of the 6d 6 g Consider, then, the scaling limit The two moduli of significance for us are the 6d string coupling The resulting near-horizon solution is Via the attractor mechanism, the tensor multiplet scalars take fixed values in the near- So, let us discuss the original example of AdS We will try to give some sense of whether a counter-example may or may not exist . In string units, the volume is given by while supergravity description requires weak AdS curvature, i.e. The radius of thecompactification), AdS given by space and the sphere are equal (as is standard inv Freund-Rubin horizon geometry, independent ofat our will. choices. The hypermultiplet scalars can be tuned Inserting the supersymmetric theory on thedivided black into background string tensor in multiplet and six hypermultiplet5 scalars dimensions. come of this from The supersymmetry; tensor 25 multiplets real and moduli 20 can from hypermultiplets. be system on space of compactifications of type IIB string theory on close to our bound onlycompare at to small is AdS the length parametric andweakly dependence curved thus at on AdS small the boundof at constructing large very sparseis large unclear whether thiscompactifications is methods. a fundamental limit, or just a limitation of available controlled tionship between large one could ask — instronger that bound more available? limited context, could it be thatby there discussing is one a canonical (parametrically) example of AdS Example 2. JHEP08(2016)041 . 6 , (4.13) Planck ]. Adding M 45 ∼ (1). So the 6d string 15 ∼ O v . Unfortunately, as the AdS 3 . (1) O – 16 – . (1) in Planck units. It is clear that the problem g O = 3 S R = AdS R of states, one could start to constrain the representations of states Q mass, we need to further reduce on the S 3 ], to name just a few of the many such analyses in this rapidly growing area. 44 – 40 , ), by taking 14 4.11 Another potentially powerful extension would be to correlation functions in higher We also expect that these methods could be generalized to bound other quantities We can produce theories where the lightest charged states are at the Planck mass in Now, consider the KK U(1) gauge fields on the torus. The story is similar to that of See e.g. [ 15 dimensions. This paperCFTs this has is equivalent focused toin a on four-point a the correlation chemical function partition potential of is function, twist equivalent operators but to [ inserting in Wilson two-dimensional lines in this correlation function. make richer statements aboutbounding the the spectrum charge of charges.in In the theory. particular, It rather wouldgroups, than be certain simply very representations interesting must for appearthe instance representations in to that the show appear spectrum, that in or for the to certain low-energy find symmetry spectrum relations with between those at high energies. be possible, and itnumerically sophisticated would machinery be of interesting recent bootstrap to approaches. return tobesides these those bounds considered with here.but the when For much one, its more we symmetry have is focused part only of a a single larger conserved non-abelian current, group, then one should be able to to put concrete boundsholographic, 2d on CFT. the Interpreted in spectrumstates terms of of must gravitational be charged duals, present these statesthere imply in in must that the charged a exist theory general, statesconcrete at not with lower the bound. charge-to-mass necessarily Planck ratio scale Foranalytically (in the transparent or at units most lower, the cost part, of and of we leaving the that the have furthermore Planck constraints attempted weaker scale) than to should above ultimately make a our analysis more AdS radius. 5 Discussion and futureWe directions have demonstrated that the partition function with a chemical potential can be used is that the Freund-Rubinan construction external by sphere definition ties indimensional the (AdS) the AdS Planck geometry. radius scalecharges to due So under the to a at KK radius the large gauge of externalissue, AdS field. but sphere, radius, More we will elaborate are the always constructions not dilution lower can aware the partially of of surmount any gap the this where to lower- we would calculably saturate our bound at large To read off the AdS and sphere radius are tied, the three dimensional massthis is example, well but below only thelimit, Planck by ( mass. considering highly curved theories outside of the supergravity In this limit, the 6dand supergravity theory Planck is scales weakly are curved, comparable. while Example 1; the lightest charges will be KK modes with six-dimensional masses while simultaneously selecting JHEP08(2016)041 (A.1) (A.2) = 2. d . ) ) . τ τ ) are defined as ( ( n n 4 6 τ n n q q ( q q ]. E E 6 Such constraints are therefore inter- 3 5 4 6 − − ) ) E n n 47 1 1 d d 17 + + = 2 supersymmetry are of additional in- =1 =1 ∞ ∞ X X n n cτ cτ ) and N τ ( 504 4 = ( = ( – 17 – − E b b d d + + + + ) = 1 + 240 ) = 1 τ τ cτ cτ aτ aτ ( ( 4 6 E E 4 6 E E ], section 3.1.4 for a very rough sketch of such an argument in If this is correct, then it would provide a practical way of including non- 46 16 ] for various approaches to this question. 49 , 48 Finally, bounds on the number of BPS operators at a given weight and charge in a See for instance [ See [ 16 17 They transform as used in this paper. The Eisenstein series support of the National Science Foundation via grant PHY-1316699. A Modular forms For convenience, we reproduce the definitions and relevant properties of several functions Herman Verlinde, Roberto Volpato, andHarlow, Xi Ami Yin Katz, for useful Christophsupported discussions. Keller, by We and also a Alex thank Maloney Stanfordsupported Dan for by Graduate the comments Fellowship NSF on under and grant aof an PHY-0756174. draft. Energy NSF ALF is Office NB Graduate supported of by is Fellowship. the Science US under ED Department Award is Number DE-SC-0010025. SK acknowledges the Acknowledgments We thank Andy Cohen,Jared Thomas Dumitrescu, Kaplan, Guy Ami Gur-Ari, Katz, Dan Harlow, Alex Daniel Maloney, Jafferis, Greg Moore, Eric Perlmutter, Cumrun Vafa, 2d superconformal field theoryterest, with as at they least wouldof have the a corresponding topological target-space interpretation Kählermanifold. esting as geometrically, bounds and on modularcomplementary the bootstrap to Hodge approaches other numbers may approaches. provide information that is property of the correlator under crossingpurely in through the knowledge presence of of the such currentof Wilson two-point its lines function, anomalies. can or be in derived evenlocal dimensions line in operators terms in the conformalabout bootstrap, the potentially theory accessing important that information would be invisible otherwise [ Optimistically, one may hope that even in this more general case, the transformation JHEP08(2016)041 ∈ ), τ ( 0 2 (B.1) (A.7) (A.8) (A.4) (A.5) (A.6) c d a b E are polynomials ). We also define Z ), defined as , ∞ τ i ( 2 E = . ) τ d .... + . 2 + (2) defined as matrices cτ ) q ( 0 , τ c . ( ) 6 n 6 iπ n (A.3) q ) n q n E n τ, z ( q − nq ) + − nq 1 + Z τ 1728 1 3 − ( d ) + 196884 2 2 ) called Γ + τ (2) are generated by the functions =1 ∞ =1 (1 cz ( ∞ q 1 Z E 0 X n cτ X n 4 , 2 =1 ) ∞ πi E Y n d 24 e 1 – 18 – = 24 + − = q 744 = 24 0 cτ ) − τ , z ) = function, which is a modular function of weight 0 with ( 3 0 ) = 1 + 24 ) = 1 ) = ( ) τ η τ τ τ τ J ( τ ( ( ( ( 0 η 2 2 4 b Z d ∆ E ) = E E τ + + ( ∆ cτ aτ ). ) = τ ( 2 A.1 J E . even. Modular forms under Γ ∞ c i = τ ), defined in ( τ ) with ( ). 4 Z τ , E Finally, we consider the subgroup of SL(2 ( J relies on the universaldetails. structure Here we of demonstrate the this current in gory algebra, detail. rather than any theory specific and B Current algebra and flavoredWe partition have argued function that the transformation property, SL(2 defined as Holomorphic modular invariantin functions with poles only at We also define the Klein-invariant a pole at This is not quite a modular form, as it transforms as We are also occasionally interested in the second Eisenstein series the Dedekind eta function as and the modular discriminant as Together, they generate the ring of modular forms invariant under SL(2 JHEP08(2016)041 . 0 L (B.4) (B.5) (B.6) (B.7) (B.2) (B.3) 24 c/ − 0 L . q as part of its 2 0 ] for instance). J Tr =0 z 2 50

) ) , τ πi i ( 0 | (2 Z )) c 2 d τ . 2 . ( − , + c − 0 Z ) and then present the general L n 2 L 0 τ cτ 24 ( J c z 2 c J 2 n c/ Z 18 πi 24 q . − − − ( 2 c J c/ 1 24 q n − O 2 − + 2 24 ], for example. J 0 q∂ J n L c/ 51 ) + =0 − q − ) is well known (see [ z τ 0 J

24 ( = 0, but we refrain from writing this be- L ) 1 ↔ c/ q Z τ z ≥ B.1 Tr ) ( τ − X n 2 z ∂ q Z ) ( Tr 2 z 1 T = = πi ∂ + 2 ≥ c 2 – 19 – 2 0 X n 0 2 2 J − ) ) L ) d = ) = (2 πi z 24 ( τ 0 + c/ 2 ( ) about 0. The transformation rule can be veriﬁed at each 2 − J Z 0 0 cτ z 2 z O +2(2 ) L ( } 2 ∂ q 2 ) = 0 = ( O z ( J ( 2 Tr =0 24 24 z O

c/ c/ ) − − 0 ) 0 0 τ τ ( L L At quadratic order we have, ( Z {z q q 2 Z z 2 ∂ z ∂ Tr Tr 2 2 ) ) ) order by order in derivatives are always evaluated at πi πi ). Such an operator is given by, τ z (2 | ( B.1 Z = (2 ≡ 2 z ) The second and thirdthird terms term on we the have, second line can be simplified. Starting with the We can compute the torus one point function of which has a zero mode, In order to dozero this, mode, we as would well as likeprimary other to one known find point contributions. functions a This transform,∂ primary is and that convenient as thus contains we can know solve how for the transformation of The low, to avoid clutter. We wantcomputing, to check this second order transformation by explicitly τ ( This style of computation is similar to that presented in [ 2 18 O F for instance the free boson, for whichQuadratic the rule Order. ( Our strategy willfunction be ( to calculateorder the using transformation the property structuretheory. of of We demonstrate the the this current explicitly flavored algebra atargument. partition quadratic without As order the any in rule knowledge is of theory independent the we can particular thus read it off from any theory we like, B.1 Perturbative argument JHEP08(2016)041 ], ), ). 52 ). B.1 A.6 (B.8) (B.9) τ (B.10) (B.11) ( Z 2 z ∂ ) are modular τ ( ) written in ( Z . , τ τ ) ) ( ∂ . Thus the modular τ τ 2 . 0 ( ( n ) E J − Z Z τ ) J ( d n Z τ ) and J ) + ∂ τ τ 24 ( c 2 ( 2 , c/ cτ 2 ) O ( + − E τ 24 0 F ) ( of an individual character. In [ L πic − τ q Z ( 1 τ 1 12 2 n q n E ). = Tr ) + 2 − τ nq τ → − n ( 2 ( 1 q n ) Z τ Z J 2 z : = = 2 z n πi ∂ ∂ − S 2 J n ) , we look for a primary operator which contains – 20 – d J 24 n ) + (2 m c/ 0 + − τ − J ( J 0 2 only depend on the universal current algebra, and so cτ 24 L 24 O q c/ c/ F = ( − m 0 − 0 0 J L 0 Tr L ) = q 24 τ q 1 τ ≥ c/ ( X n − Z Z modes. The traces over these terms can be evaluated, as they 0 2 Tr z 2 z L ∂ ∂ q m J To compute at arbitrary order we can replicate the argument style used and m L In deriving this, we used the fact that both as part of its zero mode. In addition it will contain terms of weight zero built m 0 properties of Tr at each order, thecan transformation compute rule the transformation is ruleand identical in so in the it any case must theory. also of be the In free correct particular, boson. for we any This theory gives with ( a U(1) symmetry. above. To compute Tr J out of were in the quadratic case,differential using only operators the acting current algebra on to traces reduce them with to fewer modular powers of as desired. forms of weight 2, as well as the anomalous transformation of We are now in a position to write down the transformation properties of Putting this together, we can solve for and the definition of the Eisenstein series to write, while for the second term we use, that further development ofgroup this could approach allow to onewhich include could to then the construct be representations image usedfunction. as of under modules Virasoro In the to be plus the full added case modularuseful to modular of method. add invariance, additional chiral In states theories, the the Rademacher general full partition sums non-holomorphic case, indeed the make major this obstacle a is viable that and the image C Transformation of characters Modular invariance can bespectrum thought of the of theory. asgeneral theory One a is way sharp to to look relationthe at build the between transformation some image the of additional under UV characters intuition on of and the the the relation Virasoro IR in algebra a were derived. One might hope General Order. JHEP08(2016)041 . πiz 2 (C.6) (C.2) (C.4) (C.5) (C.1) e 1 and current = ≥ , one can × y to get the n , n 4 n

) − under the two τ with J ( T . η ¯ χ Q 4 ) / 1 , Q, ) , ¯ c 24 and 0 . iτ ( n, − J eigenvalue by

) (C.3) + ) χ 0 c m q 24 (¯ L δ ¯ h > J − , 1) . χ . c )¯ 24 ) − q τ vacuum (¯ − h > ] to provide some useful guidance 2 zQ πizQ T 2 + q m ¯ 52 e χ 2, the full irreducible representations L ( ¯ 1 Q − h > /τ m τE y L ( 1 ) vacuum )¯ c > c ¯ q q πi 12 ( ¯ h 2 πiE ¯ q J − 2 e + !( h 2 χ − / q ) n n e 1 ¯ q q )(1 2 Q ) + ( / q , ¯ y 1 – 21 – 1 m , T − iτ 0 , Q − n L − χ 1 y ) ) q m m, Q n 1 + (1 + iτ y − =1 n ( n ∞ ) = ( Y n − ( 2 1 τ z U(1) Aﬃne Kac-Moody algebra is given by, nJ

1 ) = c 6 m 12 τ, z h =1 − mkδ ∞ ) ( × Y πi q n ¯ q ˆ 2 χ q, y q ( − ( | ] = ( ] = ] = e τ n n n ) = ) = | q q ( ( h,Q,c ,J ,J ,L J T χ m m m χ χ ) = J L L [ [ [ q, y ( ˆ χ 2, acting on the primary states, as well as the anti-holomorphic modes. Since produces a continuous, rather than a discrete, spectrum, and it is not clear how to , this character gets mapped to ≥ S S We want to consider what happens if we add an extra non-vacuum state to a theory. The holomorphic Virasoro , n n − Under “reduced” characters: We can focus on the left-moving part of the reduced character The full character is where we have gradedIt over the is U(1) convenient left- to and multiply right-moving by charges the modular invariant function L these do not change theimmediately total write U(1) the charge, characters andsectors as they separately: products raise of the the characters and similarly for the anti-holomorphicof algebra. the If Virasoro and current algebra are generated by all combinations of we will therefore consider thealgebra modular character, transformation which we of describe an below.U(1) individual currents We Virasoro assume for the ease existence of of exposition. both left and right systematically correct this. Moreover,over a since continuum the of characters image up ofin to a a arbitrarily high single sense weight, character one foras is the must well. analysis an also to integral Despite characterize be these “closed” thein caveats, modular thinking we about image find modular of the transformations infinite results of sums of non-holomorphic theories. over [ characters In this appendix, under JHEP08(2016)041 . 2 ) . 2 ¯ τ ) ¯ 2 τ 00 (C.7) (C.8) (C.9) ) and 00 has a ) ¯ ¯ 0 Q (C.10) (C.11) (C.12) (C.13) Q ¯ ¯ τ ¯ τ Q + c ¯ c − S Q 2 ¯ − Q ¯ ¯ ( Q Q ( πi Q, πi 3 + 3 . c 0 − + + 2 τ ) 2 zQ ) ) τ 0 2 QQ 00 − τ ( ¯ z Q Q 2 π 00 00 cτ − ¯ Q 6 . Q Q Q ( cτ . + 2 + + ) 2 πi 2 τ 0 τ 3 Q z ) ( 00 e c 6 00 Q cos πi . ¯ 3 cτ . − Q QQ ) R ¯ τ 0 e ( 2 Q πi ¯ ( E 2 Q c R 2 0 ¯ τ c π ˜ Q c ¯ 2 E πi 2 zQ + Q − R + 6 ¯ 3 3 τ 3 + L 00 + E τ e = 0, up to an extra cos factor, 0 L − E L + − τ dze ˜ 0 R /τ E L τE Q τ QQ E L R ( πi E ( Z 2 E πi πi τ 2 cos π − c 2 πiE 2 /τ πi 6 +¯ e 2 e ≡ − 2 × | L 2 − e − / τ 2 R e 3 | 0 | 1 c e ˜ c 2 τ πiE ) Q | E 6 | 3 2 2 τ c c 3 3 icτ | iτ 2cos ¯ − − Q = 2 c 3 e 0 L − √ )( × ) = , we obtain 0 ) – 22 – − E = 0 R 00 √ ) ) , 1 R iτ 0 R 0 R 2 τ ,Q ( E τE − ), there is another state with charge ( ˜ c E 0 πiQ L +¯ Q R 2 τ τE 2 ¯ ’s. But that is exactly the transformation that was 1 τ 0 L 3 πi E = +¯ +¯ − 2 ˜ ( + E 0 L ,E 0 L ) = e ρ τE − ˜ 0 ( ) 00 E L 0 2 τ τE dze L c πi ( E ( ¯ Q 2 Q 2 ,Q dQ 3 R E πi , 0 πi e 0 0 L 2 2 − ) E e e ≡ 00 L ( ) ) 0 ,Q ¯ E dE ρ Q 00 L ¯ 0 R , 0 L Q ˜ ¯ E Q 1 τ 00 , Z , ): 0 00 ,E 0 L ,Q πiτE πi ,Q ,Q 2 2 0 R E 0 R e E,Q 0 R ( − ( ˜ 0 L ) and energy ( e ρ E ,E ρ | ,E ¯ , 0 R 0 ’s are replaced by Q L ). Adding these two contributions together, their image under τ 0 0 L L 3 dE | E R E ˜ c E E ( dE Q, ( ( Z ρ 0 L ,E ρ ρ = 0 R L 0 0 R R dE ˜ E E dE dE d 0 L Z 0 0 L L ˜ E dE dE d Z Z Z This has reduced toand the with the transformation for the case In terms of these variables, the above relation takes the simple form Clearly, it is natural to define the variables To bring this into a more natural form, we can massage it a little to be If we assume thatwith the charge ( theory satisfiesenergy charge ( conjugation symmetry, thenspectrum for given by each state This is just theright-moving left-moving piece, piece we find of the full character; multiplying by the corresponding density of states Integrating both sides against Our goal is to decompose this into an integral over the untransformed characters times a JHEP08(2016)041 ]. ) ] (2013) 0 R (C.14) Int. J. SPIRE ˜ , (2007) E ]. 11 IN R ˜ ][ 11 E (2011) 130 q ]. SPIRE 08 JHEP ]. IN πi (1998) 253 Elliptic Genera , ]. ][ JHEP 2 , ]. 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