JHEP10(2017)089 3 − c . Thus J Springer June 27, 2017 : August 14, 2017 . By using the = 2 supercon- October 12, 2017 : : Supersymmetry, N expansion is more , values of XYZ J a,b all Received = . While the lowest pri- ultiplet with dimension eral lemma, that scalar J Accepted W Published on, such as the fixed point lar primary is in a semi-short e, we calculate the anomalous nal signal propagation in the ence of scalar semi-short states term in the effective theory on ng in a natural way, by ordinary -charge rse (WPI), University of Tokyo, kyo, R Published for SISSA by https://doi.org/10.1007/JHEP10(2017)089 follow from supersymmetry, but rather from [email protected] , and a superpotential and Masataka Watanabe not +1, a fact that cannot be seen directly from the a,b J is an unknown positive coefficient. The coefficient does X,Y,Z 3 3 − − c c ), where . = 2 superconformal symmetry with the large- 4 3 Shunsuke Maeda − J N a ( 1706.05743 O The Authors. + 3 c Conformal Field Theory, Effective Field Theories, Extended

We consider the operator spectrum of a three-dimensional − , J [email protected] , this proves the existence of scalar semi-short states at 3 J − c Lagrangian. The third-lowest scalar primary lies in a long m − +2 Kavli Institute for theKashiwa, Physics Chiba and 277-8582, Mathematics Japan ofDepartment the of Unive Physics, Faculty ofBunkyo-ku, Science, Tokyo 133-0022, University Japan of To E-mail: [email protected] b a the combination of J multiplication of local operators.at Combined large with the exist unitarity of moduli scatteringeffective and dynamics the of absence the ofsemi-short complex representations superlumi form modulus. a module We over also the chiral prove ri a gen XYZ is proportional to themoduli leading space. superconformal The interaction positivity of mary operator is a BPSrepresentation, scalar with primary, dimension the exactly second-lowest sca formal field theory with atheory moduli with three space chiral of superfields one complex dimensi Supersymmetric Effective Theories ArXiv ePrint: Keywords: powerful than the sum of its parts. Open Access Article funded by SCOAP Abstract: Simeon Hellerman, Operator dimensions from moduli existence of an effective theory ondimensions each of branch certain of low-lying moduli operators spac carrying large JHEP10(2017)089 9 1 4 7 9 24 25 28 29 30 32 32 11 14 15 16 23 23 24 26 28 18 21 ons eories that are strongly n families — discrete or cting free fields or solved – 1 – branch theory -wave one-particle state X s branch X R × 2 S E.4 A lastE.5 bit of closure Examples of the algebra E.1 Commutation relations E.2 Oscillator realization E.3 Perturbation theory C.1 One-boson one-fermionC.2 excitation Two-fermion excitation 4.3 BPS property4.4 and vanishing of Semi-short the property4.5 vacuum of correction the Correction to the two-particle energies 4.1 Dynamics on 4.2 Meaning of the “vev” 1 Introduction Generic CFT need notcontinuous be — weakly with coupled, limits inexactly that the in can sense some be of reduced more lying to i general weakly way. intera However even individual th E Semishort superalgebra C Energy correction to one-boson one-fermion and two-fermion excitati D Superconformal index and scalar semi-short multiplets 6 Conclusions A Notation B Uniqueness of the super-FTPR operator on flat space 5 Operator algebras and the semishort spectrum 3 Quantization of the effective 4 Corrections to operator dimensions Contents 1 Introduction 2 Effective theory of the JHEP10(2017)089 , . ] d 1 in J J on r √ 0 may ρ , and J 1 2 J << -charge ≡ , whose , c 3 and R ρ Λ 3 2 R c H << 1 − r y large at high scaling encodes the . In each case, the 3 = 2 supersymmetric 3 2 + the curvature vanishes, J N The common form of the = Φ 1 ng states of large int forces all Λ dependence ing, and only the use of the perators using radial quanti- W ppressed by powers of , when one considers the di- ], which suffered from a misuse of l charge. ) under the global symmetry. ctive theory describes a single nvariant vacua on -charge. That SUSY must be he Λ-independent terms to be 1 . the only BPS scalar primaries ationship between R nergy states on the sphere, and t vacua on flat spatial slices. As ing — the chiral ring modulo its = 2 SCFT with infinite (finitely is a scalar operator with dimen- . The theories considered in [ and the charge density J J urther determines the Λ-dependent N theory, the 00 T algebraically independent generators, 3 + (const k χ models is a consequence of the fact that -complex-dimensional moduli space, then = Φ . H ≡ k 3 3 2 → W ) where the unknown coefficients + with limits that are weakly coupled in this χ | 1 2 ρ − = Φ – 2 – J . Crucially, conformal invariance is a symmetry of ( W J O H ∼ | ] it was noted that three dimensional CFT with global -independent term is universal. 5 , taking the Wilsonian cutoff Λ to satisfy 0093 + J – (2) and . r 2 0 O transforming as − follows from the fact that the two theories are both describe relations. ] from the one originally appearing in [ π 1 2 6 = 2 superconformal J ℓ + J + 2 N 1 2 ]. Conversely, if there is a families of observables χ c 7 (2) model and the infrared fixed point of the + ≃ 3 2 O χ + scaling in the J itself. However the fact that the breaking is parametricall 3 2 3 2 ground state can only be c + φ J ] and related work [ J generators with = 1 expansions for ∆ ℓ J J + The The moduli space of vacua implies a degenerate spectrum when In [ In the case of the The case of an infinite chiral ring is different. For The value was corrected [ k 1 -function regularization. operator dimension of the lowest operator with charge and consequently the curvature is always relevant in the rel theory with a single chiral superfield Φ and superpotential sense. The simplest such limit to study is that of large globa the sphere. For a theory with a moduli space of vacua, low-lyi the effective theory at theclassically quantum scale level, invariant (indeed, which Weyl constrains invariant)terms t and f uniquely in terms of the Λ-independent terms. This effective theory can bezation used on to the calculate sphere dimensions of radius of o the large- cancel in observables, order by order in the renormalization group equation at the conformal fixed po or sion ∆ it is described by a nilpotent-free chiral ring with by an effective theory incompact the same scalar universality class. The effe Then both quantum effects and higher-derivative terms are su ζ large- symmetries simplify in thismensions sense of in low-lying some operatorsare familiar of the examples large critical global charge fact that SUSY is strongly spontaneously broken at large neither theory has a continuousa family result, of the Lorentz-invarian curvatureso of the the space relationship is between the irrelevant for energy high-e density differ between theories and the broken follows from the factare that 1 the and chiral ring truncates; coupled, may have holomorphic coordinate ring is thenilpotent “radical” elements of [ the chiral r generated) chiral ring, there is a moduli space of Lorentz-i cannot be understood ineffective terms theory of can the uncover structure it. of the chiral r JHEP10(2017)089 , ]. 1 ]), | 8 ] = 10 J g (1.1) X,Y,Z 1, we , ·| 9 (1) +1 << are all of expansion perturbed is a linear R model, the ∼ J R J /J descendant is he elements, J ) symmetry and 2 R = 2. The three 3 J ¯ X,Y,Z XYZ Q ℓ X,Y,Z uthors are [ = 4 SUSY in four − 2 its rator with the same -charge N R alous dimensions should fields = 0 bosonic components. , which is always BPS if e is the ¯ θ J s the U(1) = 1 and branch of moduli space, to branch”, that is, operators state on the sphere, via the etry. The chiral ring of this = uperfields is the Ricci scalar curvature Strassler type argument [ ing if they satisfy the BPS ons of k θ rge X respectively. X rwardly in a large- = 0, so that the moduli space . In this paper we will show that nman diagrams with loops have Z Ric . N ) es into a relationship ∆ Z YZ ] to study the anomalous dimensions J and their , and 1 = + , where of order 1. We will quantize the theory | Y ρ | J X,Y R XZ J X,Y,Z 2 1 + 1. + is a coupling constant with dimension [ + , the moduli space effective theory becomes a − = -charge on the “ J
– 3 – X g J R J 2 ( Ric 3 2 . XY J ≡ and ∆ where R R H ∼ J J 2 3 this theory flows to a superconformal theory that is strongly satisfies an appropriate quantization condition. − 2 g XY Z carry charge +1, respectively, and the J X perturbation theory. J = Z J . This theory has three independent U(1) global symmetries U W E <> ∀ X,Y, R model, which lacks these additional simplifications. J , X,Y,Z X p . At scales ,J ,Z 1 2 p R XYZ spatial slices and use the effective field theory of the + After the lowest operator,charge is which also is a BPS, scalar the and next-lowest lies ope in a “semi-short” multiplet — absent and its dimension is precisely J 2 In particular, we will extend the approach of [ We find some interesting results: The first examples of such a perturbation theory known to the a Possibly the simplest interacting theory with a moduli spac That is, the chiral ring consists of linear combinations of t ,Y We will use the same notation for the superfields • S p = 2 superconformal infrared fixed point of three free chiral s 2 (1). The theory has exactly one marginal operator that break X lowers the dimension of the moduli space (as shown by a Leigh- in the dimensions, as well as the planar approximation at large by a superpotential [mass] near-BPS operator dimensions can be calculated straightfo bound ∆ = Scalar superconformal primary operators are in the chiral r controlled tool: both higher-derivativetheir operators effects and suppressed Fey by powers of O coupled: the anomalous dimensions and OPE coefficients of the ought to be able tobe think computable of in them large- as “near-BPS”, and their anom compute the operator dimensionstate-operator realized as correspondence. the energy At of large the on of near-BPS scalar operators with large In those papers the authors used the additional simplificati theory is one-dimensional, corresponding to the case where but no marginal operators neutral under the full global symm a moduli space exists and N satisfy a relationship of the form under which generators a relationship that is exact for the lowest operator with cha of the spatial slice. In radial quantization, this translat { with combination of the three, consists of three branches, freely generated by JHEP10(2017)089 , X (2.1) (2.2) transforming as ]. As a result, the vative super-Weyl- pes: classical and X 11 modulus has a rela- operator is negative X lculable, but its sign is Λ and invariant under g multiplet and receives space. ator and scale as positive en scale invariant as terms for one of these fields alone. two fields are massive, with 0 rule follows from the fact cular directions where one of ussed in [ uation becomes the condition he 1PI effective action for rically smaller than the action e loop contributions to the RG -dependence. These terms are od as full-superspace integrals, e out a shell of modes, between q > ns that simply determine the Λ- must be exactly 1. 0. , I , ¯ 0 satisfied by the quantum terms is not θ q > 2 q − classical θd 1 . q > I 2 I and 3 d q − q , with the superpartner of J – 4 – Z − X = Λ and naively it would seem that Λ can appear to The effective theory of the
= ], we begin by observing that the Wilsonian action I 2 σ = 0. 3 1 1 2 L Λ branch R has an expansion in powers of the cutoff over the UV ∆ J X = exp X → J X,Y,Z X (2) model in [ O . The scaling dimension of X ψ correction comes from a single insertion of the lowest-deri
3 σ Λ, the propagators are 6 7 is an operator of dimension 1 − δ correction to the energy of the third-lowest scalar primary J q − is an operator with − 3 1 − exp I I J definite. corrections to its dimension of order The third-lowest operator with the same charges lies in a lon invariant interaction term in the effective theory on moduli positive definite by virtue of the causality constraint disc The coefficient of this effective term is not perturbatively ca The It is important to note that the condition Terms in the effective Lagrangian can be classified into two ty Quantum terms are entirely dependent on the form of the regul → • • • X a universal rule in effectiveΛ field and theories: Λ when we integrat tively simple structure. In flat space, terms can be understo Structure of the effective action. for classical scaledependent invariance, terms with from the calculable Λ-independentthe terms. correctio three For fields the is parti nonzeromasses and above the the cutoff, other and two one vanish, can the obtain other an effective action scale defined by the scalar vevsequation, themselves. both the In finite this and regime Λ-dependentof th parts, a are classical paramet scale transformation, and the RG fixed point eq that the effective theory on moduli space is infrared-free. T where where quantum terms.the Classical Weyl terms transformation are independent of the cutoff negative powers in the Wilsonian action. Rather, the powers of the cutoff scalein Λ. the They action; are scaleof not invariance the Weyl-invariant is or form explicitly ev broken by the Λ at large values of the fields 2 Effective theory of the As in the case of the ψ JHEP10(2017)089 - ζ . c (2.4) (2.3) , with UV 0 limit. branch. This 0 limit of the < . The quantum → ect of integrating all for most prac- i 3 2 q | Λ << 3 X eating our theory as | IR with he Λ-dependent terms is an observable in the = E i , s can be formulated on an X )] e form e a renormalization group -dependent quantum terms le-free regulator such as UV Λ input, the cutoff-dependent erlying theory is conformal, nite in the Λ +1) lator has been fixed and the than the Λ c UV E ce a riance constrains them more ( ∆ not consider the Λ-dependent taking erators. Λ must be exactly cancelled by /E L , δ c i , and add local counterterms with (Λ − ∆ s to cancel the Λ-dependence from O by ln − [ i 3 ) q hem. i q a Weyl invariance constrains terms more Λ ( )Λ L . Then the fixed point equation for RG c i i . In practice, we can simply quantize the q X ∆ , with a redefinition of the fields to restore 2 3 | 1 − − = − X ) 3 | (3 Λ / Λ – 5 – δ ≡ IR 6=3 i c − E X in terms of ∆ (classical) = L +1) ) a a ( ( [RG] L L δ Λ δ (RG) δ Λ the same must be true of the effective theory of the 4 is the set of terms in the Lagrangian with canonical scaling d . The condition of conformal invariance dictates that the eff 3 2 + | is an operator of dimension ∆ +1) c a i ( ∆ X | O L = We do not need to know the concrete form of the quantum terms at For purposes of tree-level amplitudes we need not consider t Concretely, the RG evolution of the classical action is of th There is less to the quantum terms than meets the eye. We are tr There can also in principle be dependences of the form Λ At least, a smooth one of nonnegative scalar curvature. 3 4 UV rescaling the momenta by a factor of (1 E evolution allows us to solve for where a Wilsonian theory in a perturbatively controlled regime by the normalization of theequation which kinetic can term. be solvedfrom This to the derive allows the Λ-independent us coefficients classical of to the action, writ Λ order by order in turns out to be rather constraining for possible effective op classical action with a (sufficiently supersymmetric)positive cutoff powers of Λthere to is cancel no any danger divergences. of Since getting the the und wrong answer by doing this. out a shell of modes and lowering the cutoff from Λ to Λ where terms, then, comprise a sortconformal of invariance epiphenomenon: of once the theterms underlying regu contain theory no is independent taken information. as an Wilsonian effective action. So the Wilsonian action must be fi infrared theory. But the 1PI effective action is nothing more expanded around a nonzero vev, is therefore convergent, sin incorporated into the RG equation but for simplicity we omit t tical purposes. In correlationquantum functions, amplitudes their order only by role order i in strongly still.arbitrary Since geometry, the original CFT is Weyl invariant and function or dimensional regularization.terms further We in therefore this need paper. Weyl invariance and super-Weyl invariance. strongly than simple scale invariance, and super-Weyl inva at all, and for one-loop amplitudes it is simplest to use a sca JHEP10(2017)089 . 9 2 | (2.7) (2.8) (2.6) (2.5) X | and the , = 1 φ ∞ , which is 6  | R 2 φ | R × ], and take the 2 64 23 S 12 − n ] which in flat space µν 16 R ther. However it has a – rivatives. This is imme- d out [ cale invariant cannot be µν ) In principle, there may 12 uplings for this particular r the transformation o not have a super-Weyl- ensity (a conformal tensor R allowed by supersymmetry. SUSY. ions would be At the four-derivative level, 2.7 ear as terms in the effective + l coupling to the Ricci scalar, sytlin-Paneitz-Riegert (FTPR)
). This term is scale-invariant tives [ ¯ φ ompletion of the FTPR operator R µ 2 is the Ricci scalar. , ∇ 3 ∇ s, but potentially subtle when computing , 3 )( 1 φ 2 R 1 8 φ Ric 2 assical solution can attain 0 and µ ds ∂ − , is the radius of the sphere. The curvature 2 = ∇ 2 ν | ∂ 3 4 2 r R r ¯ 1 w ∇ φ | X  ≡ and ≡ – 6 – = µν 3 φ R R R × 4 2 FTPR 2 S − O is truly Weyl-invariant. ds R in the Lagrangian is scale-invariant or Weyl-invariant, we O Now let us discuss the interaction term and its effects on µν g x O 3 4 5 d |  g µ | ∇ p branch. is the Ricci tensor and + 5 R X 2 µν ) ∇ 3 2 ∇ ]: Ric  19 ¯ 1 φ . – = ( 2 | are the linear coordinates on 17 ≡ φ Weyl-invariant and therefore not conformally invariant ei µν µ | , 3 R w 12 not Defining the field The next term would be the kinetic term ( For purposes of this paper, we need only know the Lagrangian o There are no Weyl-invariant bosonic operators with three de First, let us see why certain low-derivative terms that are s FTPR This transformation is innocuous when calculating energie Ric 5 1 8 O Weyl-invariant completion obtained by adding+ the conforma but where couplings derived this way agree with the general form in ( conformally flat. So wegeometry could by have Weyl-transforming worked the out flat-space the FTPR curvature term co unde This term is alsoNote that Weyl-invariant when although we say it a is term of course dis where be terms in theinvariant completion. effective However theory the super-Weyl-invariant that c are Weyl-invariant but d form takes the form of a supersymmetrized operatortype of [ Fradkin-Te given a Weyl-invariant completionaction and of therefore a cannot CFT, app even without considering constraints due to mean this as a shorthandof that weight the 3), term so transforms that as a scalar d two-point functions of large-charge operators, when the cl The curvature couplings of the FTPR operator have been worke Leading interaction term. the spectrum of the physics may be sensitive to the singular branch points there. we see that the leading scale-invariant term in three dimens diately clear on thethere basis is a of unique Lorentz Weyl-invariant operator invariance with and four-deriva parity. JHEP10(2017)089 , ). X 21 (2.9) A.4 (2.13) (2.10) (2.11) (2.12) ( ) shall branch. α correction X . , J , J assical action β ¯ Φ 2 ψ µ
 ∂ ¯ Φ 2 1 ] that such a term Φ Φ φ , we obtain: µ -charge r 11 , ∂  R 1 φ ective theory of the t ≡ αβ on the spatial directions conserved as exact sym- let (see for instance [  , γ of the super-FTPR term ation of an effective field 4 i r r dental symmetries. These 9 Since we only interested for + 16 well as unitarity violation in r a massless field. A negative ture couplings of the fermions f large t (which we shall call lify the classification of states the leading large- of radius µ tive theory. When we calculate − ∇ 2 t super-FTPR ∂ µ separately αβ 2 (super-FTPR) 4-fermion γ I 4 r L + +   2 . , 2 1 ]. In flat space, it can be written as the ¯ ∇ α φ 2 , it is more efficient to Weyl-transform the 3 fields of the moduli space of the ψ 20 r 3  φ α R 2 branch theory 2 ψ 2 – 7 – r × − ). It has been pointed out [ φ, ψ  X 2 + 2 2
S t (super-FTPR) 2-fermion 2.6 r 1 super-FTPR 2 ∂ 4 L I i ∇ r ¯ 3 θ + − , we present the action of the global symmetries on the 2  1 2 − ¯ θd 1 φ ∇ 2 2 -number symmetries are d  ∇ effective theory to a considerable extent. ψ = β J ¯ Z  3 ψ is a chiral superfield, whose complete expression is given in ¯ φ β α In table ≡ ¯ ¯ ψ FTPR ψ (super-FTPR) 4-boson ··· L O − − + The four-derivative, zero-fermion term in the flat-space cl = = = = θψ 2 -number and √ φ super-FTPR + O φ is the full three-dimensional Laplacian, not the Laplacian ≡ super-FTPR 2 (super-FTPR) 2-fermion (super-FTPR) 4-boson (super-FTPR) 4-fermion O L L L ∇ By coupling this action to a background supergravity multip ]), we should in principle be able to derive the general curva Note that the 3 Quantization of the effective and operators in the large- metries in theseparate moduli boson- space and effective fermion-number conservation theory, laws not simp merely acci We now derive thebranch. Feynman rules Our for approach is the the quantization standard of approach the to eff the quantiz where Φ action directly from the flat-space expression. For a sphere to the dimension of the lowest unprotected scalar operator o fields of the UV description and on the moduli scattering, within the regimethe of spectrum, validity we of will the see effec that the positivity of the coefficien can only appear with asign positive would sign give in rise the effective to action superluminal fo signal propagation, as as well. However infrom practice, the curved working superspace outpurposes expression the of is component quite this form cumbersome. paper in the case of superspace invariant only. Sign constraint. comes entirely from the FTPR term ( Here Global symmetries. translate directly into a negative sign for the coefficient of 22 exists and has recently been written down [ JHEP10(2017)089 , , = X φ 2 φ +2) r 1 J (3.2) (3.1) F 4 n + = B will only n φ ( 3 ry are − into a “vev” | 0 3 4 Ric φ | 8 1 X uations, retaining ≡ super-FTPR = + O φ sufficient to refer to it ψ icit computation, there 2 gets an expectation value, 2 1 1 1 ∇ 0 0 φ − +1 − − − +1 into sically and mathematically fluctuation; however these iding Φ U(1) 0 flat space; by Weyl rescaling do not list them. In general, . We will be working in finite however we shall refer to it as φ 3 4 2 | roportional to the leading-order φ , 3 3 3 φ X ) | 1 / / / 0 f +2 − +1 +1 fermions, scales as into +4 +1 +1 U(1) + , where Λ is the cutoff (of unspecified F Φ = 0 in the usual way, imposing Feynman n =0 ¯ ϕ UV θ ( f R 2 3 3 3 2 = 2 1 / / / / / θ

1 +2 − +1 it/ Φ e +2 +2 +2 +1 − U(1) <equations of motion. By Lorentzit invariance this is must automatically be the so in case on are no corrections quadratic in fluctuations, modulo terms p JHEP10(2017)089 . 3 m − β , k field J (3.5) (3.3) (3.4) ψ  F ∝ ··· 3 0 , 6 F φ 2 − |  , k 0 , the . 1 φ 0 3 | t . αβ k i φ − γ by dotted lines. k . i J r 3 ¯ ψ P − ∝ + 1 2 J µ 2 − 3 and ∝ 0 F m ∇ e “vev” . We will focus on the hich means computing φ 2 − ψ α r 1 µ αβ J 3 0  ψ γ 4 φ α is f corrections by writing the r ψ 9 ynman propagator. -charge), as well as low-lying tate is described by a classical 16    ns around the classical solution R 2 3 0 ¯ r − 1 ¯ F φ 4 FTPR vertices with 2 t ∂  − m 2 2 , in units of 2 4 2 r r r Having decomposed the field into vev ∇ + + t 2 + ∂ 2 t , and therefore as ∇ i i r ∂ 2 3 k and the quartic vertices scale as r 3 − 2 P − – 9 – 2  / by solid lines and that of − 2 5 α − m with radius ¯ − ∇ 3 0 F ψ 2 2 ¯ ) J φ α 2 − 2 + | ¯ ψ S 0 ∝ 2 t ∇ φ ( ∂ 5 | =  − − | 2 0 fields. A diagram with 3 0 ¯ F ¯ φ  φ | ψ 2 R α . ¯ ψ 3 = × -scaling of diagrams. 4 − and 2 J − J f S = ∝ As we will explain later when we give a precise definition of th The 4-point bosonic vertex is Now we are goingoperator dimensions to as study energies the on theory in radial quantization, w lowest states with given global charges (in particular the excited states above the lowest.solution We with will a see particular that symmetry, the and lowest that s the fluctuatio 4 Corrections to operator4.1 dimensions Dynamics on The vertex with two fermions and one bosonic fluctuation is and the vertex with four fermions and no bosonic fluctuations lines on each vertex, will scale as so the cubic vertices scale as satisfies Feynman boundary conditions, thus has theFeynman usual rules Fe and Hereafter we denote the propagation of and fluctuations, we canFeynman most rules for easily the understand the scaling o JHEP10(2017)089 | | - . 2 3 J 0 0 in } by R for r X X ure) | | ¯ 3 J X ation /r it/ 2 { and ng a given There is a X , because of the | charge equal to X assical solutions of | -charge, the lowest Z ather, we are simply , which is simply the R e effective action for X . e conserved charge rties of the moduli space and west, are calculable in a J e lowest classical solution ower of e quantum properties of the e global symmetry action is n derivatives of h a particular combination of rge on for r, controls the leading large- as an input in constraining the ution(s) carrying a given value X,Y, using dimensional analysis. All in the denominator to render the frared. In particular, each branch vature) in the numerator costs an R | J depending on time as exp X -symmetry rotation is exactly 1 | R on symmetric group permuting the three that saturates the lower bound on the | 3 0 is the radius of the sphere. , which are universal, the amplitude S R . X R X,Y,Z r | 1 J J × 2 , always preserves a “helical” symmetry, i.e., J S depends on the unknown form of the Wilsonian – 10 – , and then two other branches of solutions, with } R r , where J in the helical solution. Thus we conclude that superpotential, there is no such solution with more branches, and focus exclusively on the properties of 3 | /r 0 Z it/ R is invariant under a combined time translation and for a given 2 X J | { R ω XYZ ≥ and J exp E Y 0 and X E (or two) in the denominator. Thus the derivative (and curvat = at leading order, with a coefficient depending on the normaliz 2 . These branches of solutions have | 3 4 Z X ) X -charge, turned on at a time, so the lowest classical solutions carryi | r J R . In the case where the global symmetry is an or dJ dE Y -charge, respectively. Due to the ≡ X,Y,Z R ω -charge have a helical symmetry, with branch. R Unlike the values of Due to the presence of the This solution exists regardless of the form of the terms in th We emphasize that we are not assuming any relation between cl X replaced by . This follows from a general fact in classical mechanics: th -charge are simply the numerator, dressed with the appropriate power of expansion of the Lagrangian is also an expansion in inverse p term scale invariant. So eachadditional additional derivative power (or of cur free kinetic term withasymptotics conformal of coupling the to magnitude the of Ricci scala is proportional to ( energy for a given underlying conformal invariance of the theory,action. which we It use follows that the leading term in the effective acti effective action to describe the largeof charge moduli states in space the has in exactlyglobal one charges. light complex scalar field wit the UV free XYZusing model, known and structure solutions of the of moduli the space moduli and space the EFT. known R prope action. However we can estimate the value of scale invariant bosonic terms take the form of polynomials i of the helical solution as a function of times their branches, we can ignore the the Classical solutions withparticular lowest family of energy classical for solutions a on given global charge. of the kinetic term and corrections that are subleading at la lowest state andperturbation states series of with low expansion parameter excitation number above the lo are weakly coupled when the global charges are large. Thus th classical solution with a given any value of the amplitude.of That the is, the lowest classical sol given by a combined symmetry underPoisson a brackets. time translation Furthermore, and the action angular of frequency th of th than one of with a given value of a conserved charge X R symmetry rotation, and the angular frequency of the X JHEP10(2017)089 : ¯ ≡ φ In ¯ f (4.3) (4.2) (4.5) (4.1) (4.4) (4.6) exact and φ and i φ − h φ . } rix elements. ≡ 2 | f v | on functions of + { ed two-point function iden- eigenstate with expectation ue of a charged operator in , is a creation operator for an elation between the vacuum are exactly Gaussian. There- i lassical solution for the scalar J rrelation function for charged φ ds only if the state is an breaking of global symmetries. . n a “background” given by the 0 ) = exp

† he one-point functions, by a free h Ian Swanson, and used to estimate

} | ] he usual Feynman diagrammatic a

¯ ( ] v f tation value, to calculate arbitrary i [ ] v i 0 [

v ≡ , f, [ J

† |

} |

] a † ]. ! v ] +1 a [ | O{ J J v J dv · 23 ) can be written 0 [ v v h

√ v s [ { v 4.1 = ≡ I J

= X v ] 1 exp – 11 – v − N [

= ) -oscillator). ≡

] φ }

v πi [ ¯

φ ]

] v , a [ v φ, [

= (2

i ] O{ -wave, then the coherent state

v J s [ ] | , the definition (

v [ i and to show that the latter approximates the former in the 1 2

0 | 6 − v J ) † a ( ! i ≡ N , the charge of the built from the oscillators have the property that ] field in the J φ 1 v ¯ [ J √ φ φ | φ, = i J | denotes limit, with calculable corrections. have the same correlation functions as the vacuum correlati J

J ¯ φ This state has exactly Gaussian correlators, with a connect Using It remains to compare the expectation values in a large-

This calculation was developed by one of the authors (SH) wit 6 − ¯ free fields can be constructed as coherent states. If charge eigenstate. However states with exactly Gaussian co classical solution represented by the coherentfree-field state correlation expec functions inperturbation the theory coherent with state. a So “vev” t given by a nontrivial free c tical to that ofand the coherent-state (uncharged) two-point vacuum. function is Itclassical simply follows solution. a that shift We the of can r therefore t use Feynman diagrams i where we have defined fore free fields and correlation functions of the oscillators in the state excitation of the φ has the property that values in a coherent state (Here, large- corrections to the energies of rotating relativistic string Free-field matrix elements with a “vev”finite are volume, there coherent is state of mat courseThis no such can thing be as spontaneous seena easily state from of the definite fact charge, that is always the zero. expectation This val statement hol 4.2 Meaning of the “vev” and inverted to give JHEP10(2017)089 J 6= 0, (4.8) (4.7) (4.9) (4.10) O . J oherent- J corrections. as a double d as a series . If in a coherent J that commute J O O O O J matrix elements, be-

] = d can be evaluated by J ] luated by saddle point; the charge, is O , v erators he coherent-state matrix [ ] ˆ

J, A consequence of this rep- J in the state [ n are suppressed by powers ) O ory in the coherent state in erent state. i

. O ] ( J x h. The first of the two contour , Fock state expectation value to ) w F [ ) subleading large- v y · pproximated at leading order in n w | O |

 ≡ J J d

(1 + +1 dJ i J dv ] ≡ h  xy v v ] [ − m e J [ +1 J ) J = dw expansion for definite- O | O | ] w ( mn n J w y A – 12 – R [ m ] is given by 1 2 x J ≡ h [ 0 II ) ] ≥ mn O 2 J ( [ X R : − ) m,n A ) is 1, and all the other coefficients are given by the gener- carries a definite charge, i.e., [ 0 J matrix elements in charged Fock states are approximated O π ( ≥ J 00 O F ] = X R m,n J [ ) ] = (2 O J ( [ ) A O ( ] as the expectation value of an uncharged operator is large, with fluctuation corrections that can be calculate Fock matrix element is given by a double contour integral of c cl A J J J [ ) O ( F ; and is uncharged, the remaining contour integral is nonzero, an J . Assume WLOG that O ˆ Fluctuation corrections to the saddle-pointof approximatio state matrix elements; element in the coherent state where the expectation value of The leading saddle-point approximation is simply given by t The double contour integral for a neutral operator can be eva The definite- J . Define If One combination of the two integrals simply projects onto op First we write the expectation value • • • • 1 J saddle point when vanish. then clearly its expectation valueintegrals in simply implements Fock states the projection must vanis that causes the Then the Fock expectation value in ating function with where the leading coefficient cause as we will nowat see, leading large- order by matrix elements inRelationship the between corresponding Fock states coh andresentation coherent is states. that expectation values for Fock states are a contour integral, state of classical charge equal to finite volume. This is relevant to the large- field, is simply a way of doing time-ordered perturbation the To see this concretely, we need the following facts: by expectation values in coherent states, up to (calculable JHEP10(2017)089 at . If k 2 k | ) (4.13) (4.12) φ | is well- -charge / φ 3 4 2 k expansion. M/J X M try vanish to ≡ e /J φ gin is “irrelevant” -field space of size X is visible asymptotically, , e sense that the details of ] perturbation theory. ntization rule for he moduli space geometry J [ J . t encodes the conical deficit . So one would expect the ) corrections go as ( ] J O . ( J J [ ity, must necessarily correspond rge- f field space, nor do fixed-energy F ′′′ o all orders in the 1 n ] (4.11) conical outside of a region of field  JF J [ ) d 3 1 asymptotic expansion in the effective dJ O The change of variables ( k  J A ] + , m exactly 0 J ] [ J ≥ ] is simply equal to the coherent-state expec- J k X [ ′′′′ J ′′ [ mn F 0 – 13 – 2 R , ] = contribution to the Fock-state expectation value, A ] J k JF J [ k J = to be a multiple of 3 , in which case the number of 1 2 1 8 ) [ (equivalently, a neighborhood of − m O − F X ( X J 1 2 − J 2, so there are only a finite number of nonzero terms at n A + ] = ] = ] = + n/ that modify the moduli space effective action in a neigh- M J J J ] = [ [ [ (compared to the infrared scale) but singular at the origin. , the field doesn’t live at the origin or anywhere near it. 2 ≤ 1 0 2 J , [ J 1 ) A A A X m H O ( k -charge quantization. A φ . Concretely, if we expand is some ultraviolet scale. If the correction terms scale lik J = 0 unless , then the corrections to observables from the modified geome M 1 2 . This is the precise sense in which the singularity at the ori ] is the relative-order -field space of size mn J J φ [ physics: at large ) R < M O J ( k | A φ | ), where On the one hand, this expectation is entirely accurate, in th However the conical deficit is a property of the geometry that 4 3 + -excitations is an integer, and in particular a multiple of 4 then tation value. Conical deficit and space As expected, the leading approximation and the first few contributions are the two resolutions of the geometry are both the “resolution” of the singularity are indeed irrelevant t all orders in where long distances in field space, then the corresponding large- borhood of singularity of the change of variables to be irrelevant in la Note that a given order in The classical helical solution neverperturbations comes near of the origin the o helical solution in the limit of large behaved at large values of Any two physically well-defined resolutionsto of different the singular Hamiltonians is precisely the propertyat of large the vev. quantum effective theory For tha purposes of deriving a large- where the vev is large compared to the infrared scale. The qua and the effective theory should know about all properties of t field theory, one may simply take for large- M φ JHEP10(2017)089 . . 0 IR φ } ) (4.14) r >>E (2 2 | it/ φ | { , the super-FTPR , ution to quadratic = exp 3 is not immediately el, the bosonic and φ in principle have cor- = 0 , ion to all orders in the r 1 1 φ 2 ates, as well as semiclas- the quantization rule, one . A nontrivial test of the e is therefore uncorrected  li space effective action to J 4 lly uncorrected at absolute sm adapted to quantization particular coefficients, none s their contributions to the BPS states unaffected at the r n the energy spectrum to any 9 s more transparent. o pieces quadratic in fermions classical solution. Nonetheless istency check, to build further 16 uency l and one-loop level. By general First, as a consistency check, we − ion uniformly satisfies 2 t ∂ We now check the one-loop energy of 2 4 r ) does indeed combine to give zero when + 2 2.11 ∇ 2 r 3 effective theory, in terms of a logarithmic super- 2 – 14 – over bosonic and fermionic fluctuations with fre- φ , the only effect of the conical deficit is to alter the ground state. ) even without any further nontrivial Bose-Fermi − L 3 ω ], these energies must remain uncorrected, and equal 2 J −
; otherwise the Lagrangian is completely unaffected by ± 2 27 ground state. J but it does not: as noted in section L – ∇ 2 J 24 −  J , as it must be to all orders in ¯ 1 2 φ − = J (Φ). In terms of ln (bosonic) FTPR ≡ O L ground state, by expanding the action around the helical sol (which is relative order J , with the sign appropriate to the statistics. At the free lev 2 -charge of the state. However even at the classical level, it ω expansion would be to verify the cancellation of the correct − R J J Next, we shall compute the one-loop energies of the ground st To verify that the only effect of the conical deficit is to alter quency order in fluctuations, and summing fermionic fluctuations are pairedvacuum energy at cancel each mode frequency, byrected mode. and the frequencies thu The at super-FTPR order term could the large- loop expansion. It mayabout be that the some helical type ground of state superfield would formali make such cancellation One-loop energy of the large- sical and one-loop energiesconfidence of in first-excited our methods. states as a cons large- cancellation. So we seeup that to the and energy including of order the BPS ground stat This gives us somecompute confidence energies in consistently. the applicability of the modu 4.3 BPS property andThe vanishing classical of energy the of vacuumshall the correction examine large- the energies ofmultiplet-shortening the arguments BPS [ states, at the classica field defined as apparent that the super-FTPR termclassical leaves the level. energies The of the termof is which a individually sum vanishes ofthe for many sum the contributions of with helical the terms ground in state the FTPR expression ( term, when expanded around theor helical in solution, contains bosonic n fluctuations,order and thus the energy is automatica evaluated on the helical solution: for any spherically homogeneous helical solution with freq to the can simply repeat any calculation in the periodicity of the imaginary part of the deficit. We conclude thatorder the in conical perturbation deficit theory, has so no long effect as o the classical solut JHEP10(2017)089 , φ 2 − J with an -multiplet ¯ φ -wave. At the s ]. This operator . field is not a well- ds, because it the rested to calculate 29 . The lowest state  , X J -branch persist down r it 2 28 X excitation. hort representations at + ¯ x, can be seen to cancel φ btained by shifting the arge  ges is visible at the level of may see the agreement for f normal-ordered operators S state breaks time-reversal le ave at least two ith the appropriate angular ained by acting on the BPS r the index in the appendix, formal index. rove in many cases that the ical symmetry of the ground or the effective theory of the expression is well-defined and exp is the operator ordering of the s of the current [ ∗ perturbing Hamiltonian does not g simply described in terms of time- X = 0 mode, i.e., the . So up to and including order = ℓ 2 ¯ φ − J + 1. field, can be shown to be protected by such J ¯ states. φ – 15 – J -wave one-particle state expansion. However the one-loop correction actually s , and we have seen that the frequency is unaffected r , J 1 2 + 1, and then cancelling it by adding a single quantum  = J r quantum numbers to fill out a full long representation at . This representation in terms of the it 2 ω 4 3 excitation, both in the → X ) − that has a positive-energy mode excited. The one-loop cor- ¯ φ ¯ J  X φ = 1 modes of the X ( exp ℓ ∝ g = K , and U(1) ground state with the same U(1) quantum numbers, is the state R φ J with the rather than excitation and -wave: ¯ φ s +1 φ J φ = 0: the “moment map” operator is semi-short on general groun in the The existence of semishort states with the appropriate char The one-particle states with nonzero spin are also in semi-s Heuristically, the first-excited state can be thought of as o Since this state is not a BPS chiral primary, one might be inte J ¯ branch, namely φ to superconformal primary whose descendant is the U(1) the superconformal index; weas have well included as an expression itsherself. fo expansion It to is several interesting orders, to so note that that the the reader semi-short state annihilation oscillators ordered to theaffect right, the and energy thus the of the semishort state, which has only a sing free level, each has frequency an argument. This prediction is also verified by the supercon weak but finite coupling.vacuum For instance, the vector states obt ordered terms. Thein more the convenient description Hilbert is space in on terms the sphere: o all operators appearing h rection to the energyexplicitly. of The the one statequartic nontrivial term: coming aspect since from of the a thisinvariance spontaneously, vev the quartic cancellation corresponding ordering of verte to operators the is coherent not BP additional above the large- can be thought ofX as the conformal potential K¨ahler itself f semi-short property persists,momentum, because U(1) there are no states w the free-field level. At the interacting level, it is easy to p by the interaction term up to and including order defined, controlled operator forprecise general in matrix purposes, elements but between this large- Next we turn to the computation of first-excited energies at l charge of the vacuum from state; it is Note that this linearized solution does not preserve the hel 4.4 Semi-short property of the the energy of the first excited state is simply of vanishes. corrections to its energy in the large- JHEP10(2017)089 - ) is ¯ Q 3 − 4.17 (4.17) (4.18) (4.19) (4.15) (4.16) J , the ¯ ψ to the state Say we calculate the en- me dependence, i.e., how y correction at order on that this diagram should se one scheme that is compat- s are changed into ¯ φ . By expanding in VEV and fluctua- J φ are as follows: 3 − – 16 – J s on top of ¯ φ , the expression is meaningful, and the diagram ( R × 2 S this is ).

. The only diagram that contributes to the energy correction J ¯ φ φ 4.15

¯ ψ ¯ ψ is

J The above argument holds even when some of the φ

¯ φ ¯ the diagram ( ψ while for you regularize and renormalize loop integralsible — with once supersymmetry we choo on gives exactly zero. Hence the only contribution to the energ Incidentally, we know from thevanish: argument in the last subsecti Note that these Feynman diagrams with loops in them have sche and descendant of Semi-short property means noergy correction disconnected to diagrams. the state with two 4.5 Correction to the two-particle energies tions, two diagrams to consider at order JHEP10(2017)089 : ), is

t J ( ¯ f φ f

(4.20) (4.22) (4.21) (4.23) (4.24) ¯ φ ¯ φ and . In ap- J = 2 super- f X ctuation N . and the radius π t 6 α | 0 e interacting theory 12 ϕ | n terms of − , , dered rather than time-   = ¯ f f )

˙ i i istency of a smaller algebra ment in the last subsection. he BPS states 2 2 ementing the f 0 can be evaluated and gives ¯

h respect to f − −  † , and it can be derived directly aa. times the Hamiltonian density) , which corresponds to

f − ) a i ¯ J Π Π † b i 0 π 2 | φ a   ˙ ¯

† ff −  ( π π a ¯ † ψ † i 4 4 2 i ¯ a aa ¯ Π + ψ a . 0 ) √ √ 2 )( | b + ˙ b f † ˙ = =   f 2 − a ˙ ¯ ¯ ¯ − of the sphere to 1, as we shall continue to † f f Now let us calculate the energy correction f and † † i 2 6 a int r | , a =

, a , b 0 2 ( )( aaa ˙ J 24 2 b f ϕ | – 17 – αH   | φ int 2 × F

¯ 0 Π + . ¯ f f − f h L 1 4 + ¯ 6 π i i φ ×  2 2 † C 6 α | ¯ 0 , as it should be due to supersymmetry. Detailed a ψ π π 0 − (

× 24 H +  ϕ ˙ J | 6 F 0 ¯ | Π + Π + φ ˙ = 4 = = 4 ¯

0 aa 1 − L F   0

¯ ϕ φ | H int π π ¯ = = = 0 φ H , respectively. 4 4

H 0 − f L √ √ int i 2 π L 6 L | α = = + 0 6 ˙ † † ϕ f b | a by expanding in VEV and fluctuations. Here note that we only −

J = ¯ Π := φ -waves accordingly, we get the Lagrangian density for the flu

s E ¯ φ ∆ ¯ to φ and f ¯ f i 2 becomes , we demonstrate how the closure of the operator algebra in th E − int ˙ ¯ is set equal to unity. We derive the Hamiltonian (4 f H Note that the form of the operator perturbation is normal-or 2 S Π := conformal algebtra; in fact,generated it by can half be the seen just generators, from namely the those cons preserving t of the system from this Lagrangian. The conjugate momentum i pendix have to care aboutTruncating spatially uniform field because of the argu ordered. This is afrom consistency condition the for necessity supersymmetry of the existence of a set of operators impl Above we have for simplicity set the radius and As always we define creation and annihilation operators as do in the rest of this section. Dots represent derivative wit of to the state Two-particle states energy correction. We then evaluate the energy correction to the state the same number as that of Likewise, the energy correction to calculations are given in the appendix JHEP10(2017)089 , 2 in α (5.1) (5.2) conjugate hort joins = 2 SCFT † ], ) ↓ N 27 ¯ Q – ( : oted in section ↓ 24 ≡ ¯ nge the axis of the Q ] and thus the order ↑ S 11 een the parameter . In this section we will J semishort one-particle en- tor dimension. A positive shing of these contributions rgy is unrenormalized to all mishort energy cancels, and In any unitary . rgy (non)correction. We have ates with the correct quantum rcharge hilated by hat our formalism implements g, and as a result associativity arge must satisfy [ he Hamiltonian in a toy model, , so in general we have the BPS = 0 R -descendants of the primary state, , to fill out a long multiplet. This o the anomalous dimensions are the O 2 due to the structure of the operator J ↓ ¯ Q ¯ J Q → − . 3 . R s and J ⇔ + ¯ Q R , the energy shifts of the one-boson, one-fermion values of 3 O – 18 – ≥ C s + 1 at large sector, and also the two-fermion state with both ∆ + R low J 2 1 O be fully superconformal primary; we need only assume we can send R O = ¯ Q = ), note that disconnected contributions coming from vac- to those at low ℓ O multiplet down to its zero mode on the sphere. J ∆ 4.24 ¯ with φ Q mode. These are the 1 2 = perturbation theory: the energy can only change if the semis ℓ J must be positive due to a superluminality constraint [ implies a negative anomalous dimension, and vice versa. As n α α .) contribution to the anomalous dimension is negative. ↓ By acting with an SO(3) rotation on the supercharge we can cha We also draw the attention to the relative negative sign betw We have also calculated, in appendix In the matrix element ( ¯ 3 Q − algebra: scalar semishorts formwill a relate module semishorts over the at chiral high rin Nonsingularity of certain OPE structure functions. spin, and by exchanging in three dimensions, all superconformal primary operators to the Lagrangian, and thevalue first for correction to the non-BPS opera see that this fact has consequences at with the inequality saturated if and only if the state is anni establishes that there is a scalar semishort at sufficiently l (In fact, we need not evenit assume is annihilated by the energy- and R-charge-lowering supe with other states to form anumbers, long that multiplet, and are there near are no energy st ∆ = state with thefermions fermion in in the the the sign of J uum bubbles and propagatorfollow corrections from are the absent; nonrenormalization theergy, of respectively. vani the vacuum energy and there is an algebraicorders argument in that large- the scalar semishort ene 5 Operator algebras andIn the this semishort section we spectrum returnseen to the explicitly question that of the the semishort one ene loop correction to the scalar se respectively. The first-order interaction contributionssame t as for the two-bosonsuperconformal state, symmetry which consistently. nontrivially checks t obtained by truncating the directly dictates the form of the operator perturbation of t JHEP10(2017)089 s = = = 2 2 , (5.9) (5.7) (5.6) (5.3) (5.5) (5.4) 1 ∆ BPS super- vanish with a − i ↑ ite limit, f 1 ¯ Q BPS ance, scale ∆ R ) in the OPE − i 5.9 ∆ , and in particular . . There is also the i ≡ arges satisfy ∆ s erator is annihilated ), but its i γ SSS + 1 term ( 5.2 R of R-charge aries at coincident points eir OPE is automatically e nonsingular terms in a + , spin SSS i perconformal primary oper- R ··· BPS -scaling BPS = , O σ R he scalar semi-shorts. We cannot cative structure of the chiral ring. (0) SSS = (0) + i has ∆ . O α i ′ SSS ) f ¯ ′ BPS Q σ O (0)+(lesssingular) (5.8) ( , R , i s , , dimension ∆ α , and therefore does saturate it: + f . 2 ↑ + Q = ′ BPS R ¯ S i | = 0 O X , and so all the structure functions + R i 1 = ∆ . The dimensions of the operators are ∆ ∋ · · · O 1 s ′ BPS − – 19 – α | R R ≥ | ¯ σ SSS ≥ Q (0) = (0) = ∆ i R , 2 i , i.e. a scalar superconformal primary satisfying γ ). ∼ | and also SSS 3 2 . The function O R ↓ 3 i ) O 5.2 SSS s ¯ ) + σ (0) Q ( O σ 1 ( SSS -charge SSS = 0, does not saturate the bound ( O -charge O R R BPS ) R ) we have σ O SSS = ( O +1. In this case, there can be singular terms in the OPE, such a 5.3 with ¯ Q BPS SSS O SSSSP is a scalar. In the latter case the structure function has a fin SSS ]. The argument follows immediately from rotational invari R i ∆ O 31 O = is annihilated by , . So by ( 30 2 is a BPS scalar primary of ∆ SSS ) R are two chiral ring elements, then their dimensions and R-ch ↑ ( SSSSP are operators of R-charge 2 , − = 0 but not ′ BPS i is the total spin. This bound is saturated if and only if the op 1 O satisfying 1 O s O O and ∆ R = 0 unless O SSS Now we consider the operator product of a BPS scalar We would like to establish that this smooth and nonvanishing If The scalar chiral primaries have a ring structure because th For instance, an element of the chiral ring, i.e., a scalar su . Their OPE is of the form − 2 σ O , i 2 BPS 1 ¯ nonsingular term where R scalar semi-short at the OPE is nonsingular, anddefines the an product associative multiplication of which two is BPS the scalar multipli prim where R which will be of principal interest to us in this section. partner third component of spin equal to defines an associative multiplication ofdraw the chiral this ring on conclusion t directly from the OPE above, because th invariance, and the BPS formula ( ∆ nonsingular [ bound where by one of the four energy raising supercharges ator A scalar semi-short operator has dimension equal to its Q JHEP10(2017)089 , γ | σ | (5.13) (5.12) (5.11) (5.10) (5.15) , then ) vanish of scalar operator RHS 1 2 s 5.14 SSS . . , so ·O descendant of a -descendant, we 2 1 ↑ ¯ α Q (0) 3 2 ), so by the same ¯ = Q ¯ = 0) β is a spin- B Q − σ 5.2 O α . ≡ A | µ 0 -charge of the operator RHS O the r.h.s. is σ σ | s SSS and side of ( form a module over the R ≥ ) R descendant, we can define l terms, singular and not, ↑ n never appear in a parity- αβ µ ( SSSSP rily the further justification, as one ↑ − γ ned by the OPEs of primary . O ¯ ) Q r is of the form , a fact synonymous with the (0) (5.14) er, so s B α A BPS ctures, RHS ver by taking a O s R ) A − + 1 + RHS (0) + (vanishing at ]. (const appearing in their OPE must scale as (0) + (const , γ 32 SSS i α A ∋ ′ = ∆ 7 3 2 f R α SSSSP O ) satisfy the BPS formula ( + O + (0) – 20 – ), A SSSSP SSS +1 are the dimension and 5.3 SSS BPS model. The result is that the OPE of a chiral ring ∆ R α SSSSP R -descendants of scalar semishorts form a module over ·O SSS (const − O ¯ + Q ) = α R ) indirectly: by taking the ∋ σ ¯ ( Q + BPS XYZ BPS 5.9 RHS (0) ∆ ≡ R (0) = (const) BPS ∆ BPS − ≥ O R α SSSSP α SSSSP = α SSSSP RHS O RHS O ) O defined as ) ∆ ∆ σ σ ( γ RHS ( ≡ 0. Therefore the R and γ BPS BPS → is the scalar sem-short superpartner and the operator O O and σ BPS O RHS α SSSSP = 4 this module structure was argued for in [ O D From this, it follows that the scalar semishorts themselves Both By virtue of the BPS bound, all other operators on the right-h In 7 the dimension satisfies the inequality ( can establish associativity of ( semi-shorts. an associative action of the chiral ring on the superpartner scalar semi-short, as shown by the following argument. in the limit arguments as above, any structure functions on the right hand side of the OPE. If the spin of the operator on It can be shown thatsymmetric the theory second such of the aselement two the with tensor the structures superpartner ca of a scalar semishort operato with the exponent where saturating the BPS bound. Any such operator is again necessa the commutative ring of the chiral primaries: This allows for two possible Lorentz-invariant tensor stru generally satisfy associativity when taken together. Howe generic OPE are not in general associative; only the sum of al In the case where the r.h.s. is a scalar semishort superpartn where ∆ chiral ring. Naively thisexpects would that the appear OPEs to ofoperators. follow descendants without are For any completely conformal determi invariance this is indeed the case JHEP10(2017)089 , is ¯ φ φ +1 = (5.16) J branch. 4 3 ¯ ) φφ ansion in vial, fully ¯ X X = 2 super- X N -quanta in the . We have seen φ 2 1 f superconformal ]. Multiplying a picture yields the + J 33 J + 2) module over the chiral R SSS x to several orders and alar semishort J t that is proportional to ric version of the FTPR hiral ring on scalar semi- tors in the ·O fficient of the super-FTPR uli space of the ↑ eory: it implies the existence conformally covariant OPEs, the large- ¯ scales as Q perspace [ does indeed formally commute to vanish, for any nonnegative to make logical sense of equa- tive interaction terms are sup- · the identity is the only function | the case where one of the three hree copies of superspace, yields φ ring on the module of semishorts | +1 . Algebraically, this state could in BPS k BPS +1 O acting on any one of the three points. ¯ φφ J ·O α . To do this, we treated the theory in ¯ 1 ¯ D φφ R −
can of course in principle be seen via the J ↑ 2 3 k ¯ Q -wave is protected because its superconformal ∼ is large, and s ≡ – 21 – | X φ , the operator | J ↑ ′ SSS ¯ -wave, on top of a sea of (2 Q descendant. The third-lowest scalar primary can be O s , using the effective description! By associativity, then, ] and has to candidate state with the correct quantum 2 = J when ¯ Q 27 3 4 -charge – | SSS X 24 X as well. Starting with the moment map operator ( | J × O excitation in the = generalize; there exist multiparameter families of nontri ¯ | φ of the intermediate products φ low to obtain a scalar semishort BPS | quanta in the O φ not ¯ φ and large any model in three dimensions, to first nontrivial order in an exp R J , justifying the above definition. However, the uniqueness o XYZ ↑ . ). ¯ times with Q k -charge J R 5.16 The property of scalar semishort operators, that they form a The presence of scalar semishorts for all Heuristically, we can talk directly about the action of the c -wave. This state is in a long multiplet and has an energy shif nonvanishing for sufficiently large of scalar semishorts at ring, has a remarkable consequence for the spectrum of the th term. By arguments based on unitarity and causality, the coe the coefficient of the first interaction term — the supersymmet principle vanish: a priori theneed representation not of be the chiral faithful. However we have seen already that the sc that the state with one representation is semi-short [ same conclusion with a far less laborious method. 6 Conclusions In this note we have computed the dimensions of certain opera s radial quantization, and used the effective theory on the mod understood as two numbers to play the role of its on three copies of superspace that is annihilated by supercovariant three-point function by such aanother function supercovariant of three-point t function.operators However, is in a BPS scalar primary, this issue does not arise: superconformal index; andverify in the the prediction. appendix However it we should expand be emphasized the that inde In this theory, both quantumpressed corrections by and powers higher-deriva of we act this argument does superconformally invariant functions of three points in su existence and uniqueness of the conformal blocks. For super value of conformal it is impossible for three-point functions with onetion chiral ( primary is necessary through shorts through the combined operation Since the chiral ring is annihilated by large JHEP10(2017)089 - ral negative ] and references expansion for 2” conference, 37 hoped that this , /J ch Fellowship for 36 -charge expansion d > R -charge as well. This orld Premier Interna- BPS states with large X ntaneously broken and t Numbers JP22740153, e Fundamental Laws of o the dynamics of other y shift must be mpute a 1 onformal field theory that numbers. l Jafferis, and Markus Luty ion of holomorphy with the li space EFT is the ratio of he low-energy effective the- mensions the anomalous dimension of ese two expansions within a alculations of this work were ediate, and it is slightly more ed in the language of effective the large- lei Institute during the “Con- Japan; by the JSPS Program imply an inverse power of the ate the Circulation of Talented um extracted with some effort ts at low p, see for instance [ once we can verify the existence of rely on any weak coupling in the underlying — i.e., anomalous dimensions and operator 8 – 22 – not expansion, in the case where a moduli space exists, J ], despite the two expansions resting on rather different 35 , -charge, the module structure of the semishorts over the chi 34 X expansion is more powerful than the sum of its parts. It may be J As a consistency check, we have verified that the appropriate In moduli space dynamics, superconformal invariance is spo One major virtue of the large- For a review of modern developments in the conformal bootstra 8 -charge do in fact exist. For chiral ring elements this is imm ring, immediately implies the existence of scalar semishor nontrivial for semi-shorts scalarscalar states. semishorts at Interestingly, high X Nature, the Burke Institute atformal Caltech, Field and Theories and the Renormalization Galileo Group Gali Flows in Di tional Research Center Initiativefor (WPI Advancing Initiative), Strategic MEXT, International NetworksResearchers; and to also Acceler supported inJP26400242. part by JSPS SM KAKENHI andYoung Gran Scientists. MW acknowledge SH the is support grateful by to JSPS the Resear Harvard Center for th interesting superconformal theories as well. Acknowledgments The authors are deeply gratefulfor to valuable Thomas discussions. Dumitrescu, Danie The work of SH is supported by the W prediction agrees withfrom explicit the calculations superconformal of index.large- the Thus it spectr appears thecombination combinat of points of view may be used to gain insights int properties of near-BPS states in a controlled fashion. properties of the theoryory. can Such be perturbative computed computationsdynamics; perturbatively do the in perturbative t parameterthe in infrared the to context the oftotal modu ultraviolet charge energy of the scale, state. which Thus here our is framework can s be used to co therein. algebra structure constants — withfield those that theory can on be the express moduli space of vacua. logical arguments. Itunified would framework be of interesting operator to dimensions with understand large th quantum is that it givesare us expressed the in tools “bootstrap”-like to language connect properties of a superc definite. There is an interesting formal similarity between term in the action must be positive, and as a result, the energ for hospitality while this work was in progress. Some of the c of the anomalous dimension,operators and the with large-spin large expansion spin of [ JHEP10(2017)089 2 g , ). 1 , so and R
2.9 (B.1) (A.6) (A.1) (A.2) (A.3) (A.4) (A.5) 3 α D ariables , σ 1 , σ 2 . iσ ) . x µ developed by M. ( = ∂ φ . satisfy the Clifford ∂x µ β β α ∂ β α β δ θ µ ) Φ = 0, where α β ∂ µ α ) µν The metric on flat 2 γ α µ ¯ η mmetric dimension-3 op- ¯ D θ ) γ 2 µ . θ γ ¯ µ = 2 diffgeo.m Φ, modulo the leading-order 1 4 ( ∂ . β δ − + any odd number of superderiva- ) αβ the super-FTPR operator ( =0
) µ α ¯ θ ) ¯ µ θ γ ∂ = x ( γ ∂ θ 0. ( , δ ,

package ψ α Φ I ≃ ) µ is defined by := 2 α ν = 1 ∂ ¯ ¯ = 2 ( Φ
γ D α TM D µ ¯ 2 θ ¯ 2 θ ¯ β

D 2 ¯ ¯ ¯ D θγ α D . One may choose ( D + ( ) ¯ θd β D 1 β 2 2 x 16 , x, θ, δ ( θ d ) β – 23 – ↔ 2 , 2 Φ = ν F ¯ = θ D 1 µ 2 0 and β γ α 2 √  ¯ ( θ θ 2. The Dirac matrices ( ∂ θ D δ 2 , ≃ ∂x α − α = 1 β ) ¯ Z , θd ) Φ ) + D ¯ θ

µ 2 2 x x β β d γ ( ( D ¯ α I φ D = 0 ) , µ θφ µ = ( α ∂ µ 2 Z γ ¯ β Φ, because such operators are always equal to ones containin
( D √ ¯ θ α  µ − } ν ) + α is symmetric in θγ x ∂ , γ ( ∂θ δβ µ +) with φ − ǫ γ , δ . A chiral superfield Φ { := = 2 α + ) ¯ , ψ
α µ . We define the complex conjugation on products of Grassmann v 1 ¯ θ − µ ), we have γ ¯ D ψ γ ]. = A.3 = x, θ, := ( is expanded as 38 ∗ ∗ )
) µ 2 ¯ αβ Φ θ µ = diag ( γ ψ γ 1 From ( ψ are the superderivatives, µν x, θ, η α ¯ as ( Then, B Uniqueness of theWe super-FTPR would like operator to on showerator flat here constructed that space on with flat four space superderivatives, there except is for no supersy is They satisfy the anticommutation relation, Φ only even number of superderivatives acting on a single Φ or that ( D Headrick [ algebra, tives acting on a single Φ or First of all, we do not have to consider operators containing superspace equations of motion, A Notation Here we summarize the notation used in the bulk of this paper. done with the help of the excellent Mathematica The normalization for the Berezinian integral is and it is convenient to note that, up to total derivatives, JHEP10(2017)089 is (4) 1 (B.2) (B.3) (C.1) index O . µ ∂ on excita- µ αβ γ equations of mo- . = 2 ¯ Φ 2 µ

∂ ¯ Φ β ) ing operators possibly Φ ¯ − Φ , s D ( (+) µ , iv v u ∂  α order superspace equations − − − D erivatives, and it is nothing ∼  ntaining four superderivatives = = =  ymmetric dimension-3 operator + c.c. s v Φ (+) (+) 2 ¯ = 0 (C.2) µ Φ Φ . ¯ v u S β ∂ = 0 (C.3) β 0 ¯ Φ / ∇ D  ψ β α r ), since α ,i ¯ v µ D ¯ 2 Φ ¯ s α 2 ∂ ) D α : S 3 2.9
¯ u 2 x Φ ψ 1 D Φ ( ¯ ∇ Φ Φ β α s α β Φ Φ , ) v D β i † ∼ s α sr γ D γ ¯ δ D α Φ – 24 – π ¯ µ 1  + ( D ¯ Φ 4 ) + ∂ ¯ ¯ 3 θ θ β x 2 2 Φ Φ ( = ψ µ t α s θd θd state must be done as follows: ∂ α r ∂ u 2 2 u α β s α d d ) s α ∼ , , γ β 0 ψ ) ¯ , u ± Z Z s γ − Φ ( (+) ( = β X iu s v v to be unity throughout this section. := := D 2 = = = α S ¯ ) ) Φ (4) (4) ¯ ) = s 1 2 D 3 − − x u O O ( ( Φ ( 2 Φ ¯ u u β α S Φ = 0. These identities are useful in decreasing the number of 0 ψ / D α ∇ γ α D ¯ D α ¯ D ” we mean modulo total superderivatives, the leading-order ∼ is equivalent to the super-FTPR operator ( tions By the above consideration, we conclude that only the follow (4) 2 survive, Quantization of the lowest spin C.1 One-boson one-fermion excitation C Energy correction to one-boson one-fermion and two-fermi We assume the radius of tion, and numerical coefficients. So,with there is four only superderivatives one supers onbut flat the space unique super-Weyl modulo completion total of superd the FTPR operator. also equivalent to the super-FTPR operator, because We also make use of the equation of motion for O At the end of the quantization process we set structures. For instance, one canacting show on that a any candidate single co Φof motion. in any order vanishes modulo the leading- Here, by “ and especially JHEP10(2017)089 0 ϕ 2 (C.6) (C.4) (C.8) (C.7) (C.9) (C.5) it/ (C.11) (C.10) e , = β 0 ψ φ  3 0 F φ  t αβ , , iγ α  s 3 ψ + γ φ α 2 and spin-0 contribution † s µ / ψ γ f ) ∇ and that of the fermion field ˙ ¯  + f e, as it should be from super- , i µ on convention we get the free αβ 4 1 s ) , γ is β ¨ 4.5 − L  †  s − i ¯ which includes 2-fermion 2-boson s f β 2 − 0 . γ ( |    2 S 3 † which includes 4-fermion interaction s ˙ L 3 0 † − + γ × ± ¯ i ∇ . 3 ¯ F φ = 1 β J = 6 − † ψ X + | s +
} 0 a 0 J α ). Making use of the equation of motion + 3 † s 2 t × ϕ ). Making use of the fact that β ∂ | ¯ 12 L † s + 2 ψγ a π − ) t β γ, γ 4.20 (2 b 24 {  −  4.20 ¯ L i∂ 3 ± − 6 × β – 25 – = | = ¯ = † π 3 0 ψ 6 X s } − | a ¯ 4 φ β † ϕ E π 0 2 | ¯ ψ 4 = (2 2 S ϕ ∆ | 6 = | ∇ β, β × 0 − { 6 4) ϕ + , Dirac | 0 = πα 2 t (0 and then taking only the spin-1 π int H ∂ 2) 0 , H − − = 4 ϕ (2 2 contribution for the fermion field, we get int 2 = /  4) H , it/ α 2) , (0 e int ¯ ψ (2 int 4 = αH H × 0 φ πα 4 − = is a proportionality constant as in ( is a proportionality constant as in ( 2) , α α (2 int αH where interaction is given by where for the fermion and the boson field, respectively, we get with the commutation relation being which leads to the energy correction to the state given above, we get and the fact that Using the quantization of the boson field given in section and taking only the spin-1 C.2 Two-fermion excitation The interaction term in the Hamiltonian of order This agrees with the energysymmetry. correction to the two-boson stat is given by Dirac Hamiltonian, as a normalisation condition. According to this quantizati The interaction term in the Hamiltonian of order JHEP10(2017)089 ts, (D.2) (D.1) (D.3) (C.12) (C.14) ). In a D.1 , in (  are the fugacities z 1 − YZ  is t 2 YZ i ) (C.13) t YZ / model to confirm that ], 0 1 J YZ | − t X − ) β , 40 † − X te, as it should be from − + , J X ) ) and β variable plet cancel between them- ! t x, t β β ) − − † + 3 iplets can contribute to the 39  + s X XYZ β β β descendant of the semi-short t + β f + + n YZ e ↑ e superconformal index for the † ∆+ − β β + , t ¯ + γ x Q 3 + + n X † † + −  s γ γ , and , t † † − − − -derivatives contribute to the index, 2 † + n )( γ γ YZ R , ↑↑ S γ t x † † † + + + − 6 2 ∂ ( | )( γ γ β / ∆ 0 ) is the so-called letter index, α 1 F † † + z − )( )( ϕ X − 1 β n | β F 12 † † + + YZ . † π − β β 1) + 3 , t x, t β =1 † † ∞ − − / − X n −  4 X + β β ( + – 26 – x γ

= f  1 2 + + − + x, t − x γ γ E ( t + + ( ) + − γ γ F ∆ 6 − γ − | X − − ) and the normalization condition, we get ( 0 1 = exp 3 γ γ 2 / ϕ ( ( 12 ) := Tr | 2 1 x, t π 2 2 C.1 ( π 8 i 1 π π tx YZ f 4 2 − − , t -derivatives that contribute to the index. When the index is , contributions from BPS multiplets have positive coefficien = X = = = ↑↑ x ∂ ) := ) := ˙ ¨ L L 4) , x, t ¯ ¯ ¯ LL L L ( (0 YZ int , respectively. x, t ( , t H f YZ X XYZ I x, t ( . Then by using ( F is the third component of the spin on 3 ψψ becomes and U(1) s = 4) X , L (0 model is given by the following plethystic exponential [ int H Here, and whereas contributions from theselves. other In states a in given the scalar BPS semi-short multi multiplet, it is the expanded with respect to BPS multiplet, the BPS primary operator and its primary operator and its for U(1) Because of the superconformalsuperconformal algebra, index only and protected therefore mult it is independent of the supersymmetry. D Superconformal index andIn scalar this semi-short appendix multiplets we check the superconformal index for the which agrees with the energy correction to the two-boson sta where and the resulting energy correction to the two-fermion stat scalar semi-short multiplets reallyXYZ exist in the theory. Th JHEP10(2017)089  and  (D.4) ), be- ↑↑ 2 YZ t ∂ 2 YZ Z 1 2 5.2 -axis. 1 t model in / ↑↑ z 1 X as follows: X ∂ t t symmetries. x + + XYZ 2 2 YZ effective theory. / Y X 1 X YZ 2 YZ t t t ↑↑ J t ∂ along the + + 2 1 umbersome a and we X 2 ) with (descendants of) t ex into characters of the l grounds. For instance, X 3 and U(1) 2 X  / ltiplets, and therefore we t d + ↑↑  3 ∂ expand the superconformal / X 10 er, by brute force. We also the large- 8 x perators satisfying ( 2 YZ 2 + x ( ation about operators in short o establish its agreement with 1 t Z s just by expanding the index. O fficients. The operators ). However, some contributions − bute with a positive sign, while X s to the index separately, noting 3 t 5 YZ s in a full and explicit calculation / ries, including the 5 t  ute with a negative sign. In prac- 1 2 Z 10 + / x 5 X descendants of the moment map op- ) are nontrivial, however. These can- YZ Z ( t 2 t 3 X 1 ↑ / 2 ↑↑ 2 YZ Y t O t ¯ / ∂ + Q 10 1 X t 2 x + 5 ) is expanded with respect to / ( 5 X 5 YZ 2 Y + t t 2 X O X t 2 Y D.1 . /  + 1 X
YZ ↑↑ 3 – 27 – 5 t t 4 ∂ / 5 X 4 x X t + x + 2 X O ) and at X + YZ X x 3 X D ↑↑ 2 t / D 2 ) is due to the +  ↑ / ∂ 8 ↑↑ ↑ 2 1 X YZ ¯ Q − ¯ x t t Q  x + ( Z∂ ( YZ  t up to and including O 1 Z O 2 + 2 YZ / 1 t 4 YZ 4 x 1 X 3 YZ 1 t Z 3 Z t t X 1 t 2 X 2 ↑↑ Z X 1 t / t + 3 X Y ∂ + t 2 + / + Y , which trivially exist because of the U(1) 1 X YZ + X 4 t t 2 YZ t 2 X t 2 4 YZ Y t Y 3 YZ t / YZ ) + 3 X 3 YZ Y t ↑↑ + D t t 2 + X / X YZ t 1 X 2 X X∂ 4 + t t 4 X , t  X t 3 and 3   3 X X /  X t 3 3 ) with respect to 2 / / 3 X  / x 10 10 8 2 x, t D ( D.1 x x x x + 1 + − + + We do not do this, since the organization into characters is c To see the existence of the scalar semi-short multiplets, we In principle, the superconformal index contains all inform are the partial derivatives and supercharges with spin +1 an XYZ = 3. These cancellations can be removed by organizing the ind = 1 ↑ I not be descendants of the moment map operators on dimensiona BPS operators. The superconformal index ( erators the spectrum of semishort representations as computed with are just calculating the some particular terms in the index t The negative contributions at index ( D particular short representations that appear. from semi-short multiplets arecannot canceled see by all those theSo, from contributions we BPS from separate mu semi-short theseidentify multiplet two all kinds the of positive contributions contributions order up by to ord and including or semi-short representations. In principle,of even the the index term do not necessarily correspond one-to-one with o superpartners of scalar semishorts,tice, for cancellations instance, occur contrib frequently in many familiar theo cause cancellations can occur. Chiral ring elements contri Rather, we list both thethe positive cancellations and negative as contribution they occur. The negative contribution at whereas those from semi-short multiplets have negative coe Q JHEP10(2017)089 , . J i 2 se 0 de- ise, X has | and X · ↑ X J (D.5) 3 i J 4 0 ¯ Q D X φ X | D 0 . ¯ φ 1 0 mal algebra . − φ | and J σ | X · 3, respectively. In must automatically / X ul mainly as a conve- D ··· onzero scalar semishort out semishort operators -short state with given , , emishorts at large 2 as many supercharges as um directly at sufficiently f as ing block of the algebra, ∆, nded above. So we see that X ··· able at large 3 and 7 · can be thought of as / X a to a finite degrees of freedom,

rt space. At any rate, however, D n is useful in understanding the J , (0) + X J

X · , upper the indices of the operators X is not the semishort operator X A 0 | J : D σ J | X J effective theory simply agree with those of X J J with , the state X , and is nonzero. This gives information about and can be calculated semiclassically from an X X – 28 –

defines an associative multiplication, and since 4.2 D J J · D C X X

J ∋ X X , and the coefficient function is singular, (0) D J J C X X , the Hermitian conjugation. We are here doomed to dismiss ) † terms are naturally identified with the which are the , for sufficiently large σ -number current. However this description is not fully prec variables, these semishorts can be represented as ( J 2 X in the state X X φ t X is the weight-1 scalar semishort “moment map” operator, who 3 D X / X 10 D D x with − X at the very least, because of the fact that the bosonic confor D and K X , where as explained in section t or as well. 3

. For the consistency with the notation in 6= 0, then all the lower-dimension products / 2 methods combined with associativity, yield information ab J , the power of supersymmetric representation theory is usef 8 J P J x X J J

X Heuristically, these semi-short operators can be thought o We emphasize that, for purposes of understanding the spectr The most important thing we learn directly is that there is a n Combined with the power of associativity, the existence of s − 0 · ¯ φ X 0 φ either can only be unitary representedwe in are infinite-dimension trying Hilbe topossible, find under a such a truncation constraint. of the algebra that contains the operator dimension, and with dagger assigned. E.1 Commutation relations We work in radial quantization — then we have, as a basic build terms of the almost-free scendants of semishort operators of spin 0 and dimension 5 because the leading term in the OPE of the descendant is the spin-1 respectively, where but rather the chiral primary charge E Semishort superalgebra Let us try to set upwhich a formalism are of creation truncating the and superalgebr annihilationvanishing operators. of This the sectio 1-loop energy correction to the BPS and semi at low be nonzero as well. Thislarge- is in agreement with the index as expa more consequences: since the product D nience: the explicit computations ofthe the index, large- with the spectrum computation becoming more reli the index only asymptotically. large expectation value of where the structure constant in the OPE of JHEP10(2017)089 ). E.8 (E.3) (E.2) (E.4) (E.9) (E.7) (E.1) (E.8) (E.5) (E.6) (E.10) . and also the R J is given in ( ] = 0 a ≡ , s , J † β . Now these two have a that has the above ) t are preserved by the . Q ) , −− α ( † α a αβ ] = [∆ , −− . We shall call the bosonic β ( a x convention as σ † α with the transformation law Q Q 2 . -charge Q 1 1 2 2 Q S α R tion the following way, J, s the generator that changes the b − − − . αβ and † ) n by ǫ ) . a R σ ] = ] = ] = = ±± units. For instance, the we denote = β ( ∆ † α † α † α ∓∓ β † ∆] = [ on the σ (++) α ( α † α Q  Q Q Q Q R ¯ Q J, φ , , Q σ [ Q a J, a βα αβ [ s [∆ ǫ σ [ (++) α 2 1 ± Q . Note that here we will take the convention . . a ,  a † α † α = = s b , a b = – 29 – † i β 2 1 † α ) ,  b b † α ) a † α charge by − s b Q ±± + α ( α ±± R a α ( b a βα † a αβ † α , a , , ,Q σ Q a b σ a † α  s 1 1 2 2 2 1 b h = 0 , ) + = 2 superconformal algebra, we have the following commu- (++) α β = = = J o Q a α Q † β J N ∆ = − , s , ≡ Q Q α , c a βα , s α σ α (∆ † α Q Q 2 2 1 1 Q abc Q αβ δ n iǫ 2 units, and the = = = / of the angular momentum ] = + ] = + ] = + i ∆ } o α α α the same transformation law under spatial rotations, which b and the fermionic oscillator 3 β σ † β s † † α Q Q Q , s Q Q b , , a , , a J, s α [ the generator that raises both dimension ∆ and α s a, a h [∆ [ Q Q -wave mode of a free antichiral superfield { s n (++) ↑ Now, according to the In order to specify such a subalgebra of the full superalgebr We would now like to restrict our attention to generators tha Q oscillator tation relations for these generators chosen to preserve: Hereafter, by using these conventions, we simplify our nota that assigns of the E.2 Oscillator realization Let us deal with the case where we have a single free multiplet Then the oscillator realization of the superalgebra is give Then we have 3-component by to be related by conjugation, but as we choose the spinor inde property, let us fix some conventions. Denote by dimension by BPS states, that is, we are only going to consider JHEP10(2017)089 : ǫ (E.19) (E.18) (E.17) (E.15) (E.21) (E.22) (E.25) (E.24) (E.23) (E.12) (E.13) (E.16) (E.11) (E.14) (E.20) , ) ǫ ( † β . reserved at order , ) . . , , Q , independent of ǫ β β ) ( b b ǫ (0) † α a αβ ′ (0) ( , σ † α † α † α Q a a αβ αβ α 2 2 1 1 σ σ Q Q Q a , b hange to the generators, , 1 1 1 1 2 2 2 2 − − − ≡ ≡ constant − − − , † α † α , l, and correspond merely to
] = 0 q
2 . Q )] = )] = )] = . a 2 mutation and anticommutation ǫ ] = + ] = ] = ] = ǫ ǫ ǫ ǫ ormations on Hilbert space; we
( ( ( to be α α α , s 2 αβ † α † α † α ) ǫ δ O J ǫ , b , b ) J, a O [ Q Q Q a a J, b = [ , , s s O [ [ ) ). On the oscillators they act as a J, [∆( E.9 . ǫ [ o s and [ † β (0) + a (0) + ′ E.9 )–( s [∆( , b = 0 ′ α † α (0) + α ′ )–( a Q Q b E.4 ǫ ǫ ∆ dǫ n ds ǫ E.4 , , – 30 – = , , (0) (0) + (0) + a ′ = 0 α † α dǫ s dJ ˜ = 1 , } ∆ Q Q ) ) i a ǫ βα , , ǫ † ( ( σ † α † β † β β , b b , , ) = ) = ) = ∆(0) + , ) † ǫ ǫ ǫ Q , a, a † α (0) = c Q ) to be ǫ ( ( ′ a b a a , ) + βα βα h s ) ǫ ( , , ) ǫ α † α ǫ 1 1 2 2 σ σ ∆( ∆ α ǫ a βα ( ( Q Q 2 2 1 1 abc σ α ˜ − Q ∆( (0) (0) † α ≡ 1 1 2 2 ′ α α ′ Q i ǫ ), ∆( Q αβ ǫ ˜ { δ ] = + ] = ] = ] = ( ∆ Q Q † † α † α † α † α ) satisfy the algebra ( = = = ≡ ≡ , b , b Q ] = + ′ -derivative of the algebra ( } } a a J, a α α J, b b )] = + )] = + )] = + ǫ [ ) ) [ s s ǫ ǫ ǫ ), ∆ q ǫ ǫ [ [ E.10 Q ( ( ( ǫ , s ( ( ( α α α a β β † α s [ Q Q Q Q Q , , Q , , ) a J, ) ) ǫ [ s ǫ ǫ , and demand that the structure of the superalgebra still be p [ ( ( ǫ α α [∆( Q Q { { . So define and take the first ǫ first-order redefinitions of thefix variables much of induced this by ambiguity transf by the condition that Many first-order perturbations of the algebra are unphysica Then use the notation E.3 Perturbation theory Now let us set usat perturbation order theory. We wish to make a small c The generators ( relations where we have taken the oscillators to satisfy canonical com JHEP10(2017)089 s ). n by E.29 (E.26) (E.27) (E.31) (E.32) (E.34) (E.29) (E.33) (E.28) (E.30) involve -neutral. ). Solving J -charge. E.29 ly solve ( R , † β , q A , a ] αβ , † β , ˙ σ † [2] † α Q A 1 2 † α q O e in a sufficient condition + ) = 0 , − − Q . † ′ ith equation ( β are the same as those of the . ∆ αβ † Q , this equation tells us that [2] nt perturbations: [ which is scalar and . For real ǫ a , ] = ] = ↔ ′ † [2] s ′ αβ − † α † α → ·O ≡ A δ α , q q O ∆ ′ † 1 2 = , 2 J † α † a ∆ J, αβ Q + ( [ † α s Q δ [ Q q − } α † β = , . Q = q ′ α † α 1 2 , , † α ˜ q q ∆ † α a q 1 2 2 1 − s Q αβ to be a scalar with vanishing − { δ [2] O ) just express that the transformation laws of O – 31 – = and , α ] = ] = + ) = } ′ ′ J β α Q [2] , ∆ ∆ E.26 † α q , , . , , , , A ↔ α † α ·O Q ] † β i β β 1 2 α α Q Q [2] q Q [ [ ). Contracting it with Q will become clear shortly. So then we have + { Q α ] = 0 O = ′ − − + ( a βα , [2] ′ , [2] ↓ Q + ] ] = ˜ σ α ∆ } α E.30 α † α ∆ , } β α αβ 1 2 Q q a q q † β [ ǫ q q for acting by commutation or anticommutation. We denote thi just corresponds to changing the Hamiltonian by conjugatio , , s q → O , ) just constrains · contributes a total derivative to ∆ , ≡ α ≡ O [∆ [∆ α [2] 2 α Q O ] = + ] = + ] = [ Q ′ { O q E.26 Q α α { ∆ q q = 0. This gives a set of “easy” perturbation equations, which , , a a J, ǫ [ s s [ [ ↓ = means it’s just an ansatz. But this ansatz does automatical , and also define Let us solve the hard perturbation equations. First, start w Now consider equation ( Q an infinitesimal unitary transformation parametrized by so an imaginary part of commutators with the fixed generators The “easy” perturbation equations ( corresponding unperturbed generators. An imaginary part of The easy equations ( The meaning of the subscript the perturbed generators under the “fixed” generators this in full generalityway, i.e. may this be equation difficult, is but solved by can be at least don We can use the notation The symbol and evaluate at and “hard” perturbation equations which involve two differe by JHEP10(2017)089 ) ) ) is E.21 E.40 of the ) (and (E.40) (E.35) (E.36) (E.38) (E.39) (E.37) ′ E.37 in terms ) without ′ E.31 ) and ( E.30 . Thus ( ǫ -term type. The E.20 D ]. ) and ( 22 , which correspond to basis, we can fix that , der in perturbations ∆ formula for ∆ [0] E.29 21 O tions on the perturbation of ilation operator on the right. It comes from eq. ( an be verified directly on the e of [ . y, to see concretely how inter- bations come out automatically [0] α . uations ( q ′ . . 1 2 ∆ − O [2] = 0 αβ O [0] . δ ] α [2] ] = + , the only remaining content of ( ′ [2] ′ = ·O O ), is Q ] O ∆ ] ∆ , γ , [2] α † α α = E.3 Q ; this equation is forced by ( Q Q , ·O – 32 – Q [ ǫ , [ † [2] † γ ] 1 2 † β α O Q − define [ Q ] [ = ) reads Q 2 1 , α ′ a αβ † β q ↓ , is just an operator realization of superspace perturba- ∆ = ) is not present in flat-space SUSY, and corresponds to , σ ) to Q E.31 . To see this it is simplest to note that this equation is [ [0] [2] [∆ [0] O O E.39 E.36 O ) we get playing the role of the superspace integrand of ) at first order in ) reads: [0] )). Equation ( E.35 E.23 E.30 O E.32 The ansatz we’re going to make, to solve ( Hamiltonian. In particular, we willnormal-ordered, see with that at all least such one pertur semishort zero mode annih second term on the r.h.s. of ( Now we would like to apply our formula to some examples of via the gradedfollows Jacobi automatically from identity, the other and first-orderimposing closure this eq further must conditions on hold the order perturbation. byE.5 or Examples This equation does notthe impose any generators. further independent Ingenerators condi principle constructed it from follows automatically and c a nontrivial background curvature of superspace inE.4 the sens A lastThere bit is of one closure last of nontrivialits equation the conjugate that algebra ( must be satisfied. equivalent to the statement that of four supercharges acting on With convention ( So now equation ( ambiguity by simply taking the convention Since ultimately we only care about the system up to change of the superspace integrand ofaction superspace terms perturbation made theor from the semishort zero mode correspond to the commutator ( The first term would be present in flat-space SUSY. Indeed, the tion theory, with Since we can take equation ( JHEP10(2017)089 . ), we (E.43) (E.44) (E.47) (E.45) (E.48) (E.41) (E.46) (E.42) ing the E.10 So the sim-  γ . b , , † γ The most general γ γ b b b 1 + p α − b a a p p 2 † a † a − 2 a . Then using ( . p  − a γ a p † b E 1 †  a α a − 1 p b p = rator simply vanish: † α , † , † α . b − γ a b [0] b ≡ E γ † γ  1 † γ . 1 † γ a, b b b O a γ b  − α † [0] γ κ b p b E 1) a . b 1) a α † O γ † α + 1 b . + . a b − b − + 2 − 2 † γ p 2 2 † γ † γ = 0 p [0] p p b α b a − b a a ( † ′ ( b p p 2 O a 2 † † + a 1 † γ − − a † γ a 2 a 2 − E − − E b p 2 Q p α = ∆ a − E 4 2 E a E p a † γ b 2 a p + a † † α † † α 1 α 1 Q + † – 33 – a p b a = ≡ b − − Q a a 3 4 1 p a p † α 2 = p † † b [0] a † α − [0] † α =  a p p a b † γ b a † † O O [0] α b † α E 2 2 a a E E † α b vanish, as we found earlier. q 2 O  b ′ = κ 2 †   = = 1 1) p 1) a 1 1 p p of the operator Hamiltonian is + α † α [2] E − ′ − − q q − − O p [0] p 2 p 1 1 and ∆ O Now let us work out the formulae for the quartic perturbation = 2    + ( ′ † α ∆ a ∆ = = 2 ( = = , − ′ † α α α [2] q q q ∆ [0] 1 1 O 1 O you can write down made from the bosonic oscillator, preserv − − 1 − γ E E − E Q [0] E † γ O Q = [2] = 1, note that O p Then we have and find that So then R-symmetry, is The first-order modification ∆ For and the perturbation of the supercharges and dilatation ope plest sort of deformation to add would of course be Example: perturbation corresponding to quadratic deformations perturbation More general perturbations with a single semishort multiplet. Quartic perturbation. We define JHEP10(2017)089 as So † Q (E.49) (E.52) (E.53) (E.57) (E.56) (E.51) (E.59) (E.58) (E.55) (E.54) (E.50) and tiplet. individually. Q J -number. How- c . The composite -neutral operators † is actually a rather R a . z † ˆ , have the same SUSY B, ˆ A α ˆ . 3 2 † A, or njugate to make it Her- too. This means it com- ˆ γ B ˆ , , A αβ − − ˆ 1 2 B δ − 1 ations such as is the energy-raising, rather − = 0 = = 0 = ] = ] = p z and ˆ † is really just a ˆ † ˆ ˆ β β A B . A A ˆ ˆ ˆ , , A z ˆ , let us define it so that B B † α p 1 A † α z ˆ † α † α p − A , we have Q [∆ [∆ q ˆ q Q A † α Q Q † † , and not with ∆ and ˆ . 1 ˆ -charge. We can therefore make new A A zb J p − ) † R γ q . ˆ p † A − ˆ ≡ B , up to a constant. q ˆ A 0 † † α − ˆ + † φ γ ˆ A q B ˆ p B ˆ 1 A ] = +1 pq 1 p = ( q – 34 – † , z i = have been introduced, is that † = ˆ p A , but the same frequency as z † ˆ [ a A [0] 2 p [0] Now we introduce the BPS zero mode. Let us call it q † O and vanishing O z, z ˆ , A γ . † , , = + φ /r ˆ Q ˆ † B A, ∆ , ˆ 2 3 h [0] A α , , αβ and at the free level, it has ∆ = + O ˆ δ B γ 1 2 -charge as only commutes with ∆ Q = 0 = = 0 = R ] = + ] = + † γ z † † † † ˆ β β respectively. Defining A ˆ ˆ ˆ ˆ ˆ Q = + A A B B B α † , , α α α J Q Q Q Q [∆ [∆ a, a . Since it is a BPS primary field, it also commutes with the has frequency 1 J commute with the whole superalgebra, † − † za has the same z, z ≡ † z ˆ A So, Since From the point of view of our small superalgebra, this operat One major difference, now that , but we shall think of it as corresponding to than the energy-lowering part of object representations as interesting perturbations out of this operator. Note that this is only possible in a unitary theory because So, since we have not yet specified the normalization of well. So, from the point of view of the small superalgebra, ever, on the other hand, General bosonic perturbations involving a semishort and a BPS mul funny object. It has mutes with ∆ and so now we can make all sorts of fascinating R-symmetric perturb and This is non-Hermitean, butmitean. we So can we always have add the Hermitean co z The BPS zero mode multiplet. no longer necessarily commute with ∆. In particular we have JHEP10(2017)089 φ (E.60) (E.61) (E.62) ]. ommons , γ SPIRE ˆ B tation value in IN 1 [ p − ˆ p A ˆ , , A ]. q γ γ † 1 p ˆ ˆ ˆ B B A − ˆ A 1 q ]. α †
q ˆ − B chanics, these are higher- ˆ q † p ]. SPIRE A 2 ˆ ˆ A † A γ IN − − redited. p ˆ 2 f the semishort multiplet not B 2 + 1-dimensional superfields ][ SPIRE p  ˆ regardless of the form of the − A q q IN and the free chiral superfield + SPIRE Semiclassics, Goldstone bosons 1 − Compensating strong coupling with † ˆ ][ p − ¯ 2 IN + φ On the CFT operator spectrum at A pq q ˆ p arXiv:1612.08985 † A p † ][ α . , ˆ q ˆ A γ + † B − ˆ ˆ 2 † † B γ . γ A , q q p γ ˆ α ˆ B B 1 p ˆ ˆ B B a + 1) 1) 2 α + q − the entire semi-short multiplet. This much † ˆ − q p B q α − vanishes; we normalized the perturbation, for − p a pq 2 ˆ ˆ  p q q B A − + ( arXiv:1505.01537 ( ≡ So now we see, p 1 [ 2 – 35 – + p ˆ − − A − − [0] p α q nor arXiv:1611.02912 2 † p ˆ = α ˆ [ A O p B ˆ ˆ − ˆ A A B arXiv:1610.04495 q 1 1 [ [0] † 1 † α 1 − − ˆ q annihilates ˆ p − A O − B † q ˆ p κ † α † A ˆ ˆ † A γ (2015) 071 ), which permits any use, distribution and reproduction in ˆ A ˆ 1 A B at the beginning. If instead we hadn’t, and we had defined ˆ + † α B q − † † α ˆ † γ q B 12 ˆ † 1) (2017) 011 ˆ [0] ˆ pq A B B ˆ A O (2017) 059 − 1 2) 1 1) 06 † α ∆ and the perturbed generators are: q q p ˆ − − − B − 04 JHEP − q q p [2] q 1)( , CC-BY 4.0 2) O + [0] = = − JHEP 1)( − p O This article is distributed under the terms of the Creative C p , ( γ [2] [2] q JHEP − , Abelian scalar theory at large global charge O O Q p pq + ( + α † α † γ p = , the Hamiltonian perturbation not only has vanishing expec Q Q Q + ( ′ = ( = = = ∆ ′ α † α ´ lae-ame O.Alvarez-Gaum´e, Loukas, D. Orlando and S. Reffert, [2] ∆ q q O large charge large global charge and CFT data The nonzero modes of the free antichiral superfield [1] S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, [2] L. [3] A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold,[4] O. Loukas, References any medium, provided the original author(s) and source are c The expressions for This formula of course assumes neither spin multiplets, obtained byon Kaluza-Klein the reduction sphere. of the Open Access. should also be included; from the point of view of quantum meh perturbation convenience, by dividing by Attribution License ( the semishort multiplet, itstronger simply condition would seemjust to to guarantee first the order, protection but o to all orders in perturbation theory. then we would have had Protection of the semishort state. JHEP10(2017)089 ]. , ]. , f ies SPIRE ]. ] IN SPIRE ]. Yang-Mills -curvature IN ][ waves from , , , Q ][ , = 4 pp rs in SPIRE , ]. N ]. IN ence (1984) 56 ][ rbitrary Causality, analyticity larization SPIRE SPIRE IN [ IN B 134 arXiv:0803.4331 , charge in ]. [ ][ arXiv:1506.09063 hep-th/0602178 ]. . [ [ R ]. math-ph/0201030 . 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